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address: |
John von Neumann-Institut für Computing NIC,\
Deutsches Elektronen-Synchrotron DESY,\
15738 Zeuthen, Germany\
and\
Deutsches Elektronen-Synchrotron DESY,\
22603 Hamburg, Germany\
E-mail: [email protected]
author:
- 'G. Schierholz'
title: 'Polarized Structure Functions and the GDH Integral from Lattice QCD[^1]'
---
DESY 04-198
Introduction
============
The Gerasimov-Drell-Hearn (GDH) integral, which is written as $$\begin{split}
I_{\rm GDH}(Q^2) &=\int_{\nu_0}^\infty \frac{d\nu}{\nu} \left[
\sigma^{\stackrel{\rightarrow}{\Leftarrow}}(\nu,Q^2) -
\sigma^{\stackrel{\rightarrow}{\Rightarrow}}(\nu,Q^2)\right]\\
&=\frac{8\pi^2 \alpha}{Q^2} \int_0^{x_0} dx\, \frac{1}{\sqrt{1+\gamma^2}}\,
\tilde{A}_1 F_1 \\ &=
\frac{16\pi^2 \alpha}{Q^2}\int_0^{x_0} dx\,
\frac{g_1(x,Q^2) -\gamma^2
g_2(x,Q^2)}{\sqrt{1+\gamma^2}} \,,
\end{split}
\label{igdh}$$ where $\nu_0=m_\pi+({m_\pi^2+Q^2})/{2 m_N}$, $x_0={Q^2}/{2 m_N \nu_0}$ and $\gamma^2={4 m_N^2 x^2}/{Q^2}$, connects the GDH sum rule at $Q^2=0$ to the Bjorken and Ellis-Jaffe sum rules at large values of $Q^2$. The spin asymmetry $\tilde{A}_1$ is known over a large kinematical region for proton, deuterium and helium targets, which allows to compute $I_{\rm GDH}(Q^2)$ down to $Q^2 \approx 1$ GeV$^2$, separately for the proton and the neutron.
The GDH integral is of phenomenological interest for several reasons. It involves both polarized structure functions of the nucleon, $g_1$ and $g_2$, and thus tests the spin structure of the proton and the neutron. Furthermore, the GDH integral provides a link between the nucleon state at high and at low resolution, allowing us to study the transition from an assembly of quasi-free partons to strongly coupled quarks and gluons. In particular, we hope to learn about the structure and magnitude of higher-twist contributions. This requires, however, that higher-twist contributions set in gradually and before $Q^2$ reaches $\approx$ 1 GeV$^2$, that is before the operator product expansion breaks down. In order to match the predictions for $I_{\rm GDH}(0)$, the GDH sum rule, with the Bjorken and Ellis-Jaffe sum rules, a strong variation of $I_{\rm GDH}(Q^2)$ with increasing $Q^2$ is anticipated.
Lattice QCD is in the position to address these questions. In this talk I shall confront measurements of $I_{\rm GDH}(Q^2)$ with recent lattice results.
Polarized Structure Functions
=============================
Let me recapitulate what we know about the nucleon’s polarized structure functions $g_1$ and $g_2$, which enter in (\[igdh\]), first.
A direct theoretical calculation of structure functions is not possible. Using the operator product expansion, we may relate moments of structure functions in a twist or Taylor expansion in $1/Q^2$, $$\begin{aligned}
2\int_0^1 dx\,x^n g_1(x,Q^2) &=& \frac{1}{2}\, e_{1,n}(Q^2/\mu^2,g(\mu^2))\,
a_n(\mu) + O(1/Q^2)\,, \label{ope1}\\[0.5em]
2\int_0^1 dx\,x^n g_2(x,Q^2) &=& \frac{n}{2(n+1)}
\left[e_{2,n}(Q^2/\mu^2,g(\mu^2))\,d_n(\mu)\right. \label{ope} \\[0.5em]
&-& \left.e_{1,n}(Q^2/\mu^2,g(\mu^2))\,a_n(\mu)\right] + O(1/Q^2)\,, \nonumber
$$ to certain matrix elements of local operators $$\begin{split}
\langle \vec{p},\vec{s}|
{\mathcal O}^{5}_{ \{ \sigma\mu_1\cdots\mu_n \} }| \vec{p},\vec{s} \rangle
&= \frac{1}{n+1}a_n^q\, [ s_\sigma p_{\mu_1} \cdots
p_{\mu_n}
+ \cdots -\mbox{traces}]\,, \\[0.5em]
\langle \vec{p},\vec{s}| {\mathcal O}^{5}_{ [ \sigma \{ \mu_1 ] \cdots
\mu_n \} } | \vec{p},\vec{s} \rangle
&= \frac{1}{n+1}d_n^{\,q}\, [ (s_\sigma p_{\mu_1} -
s_{\mu_1}p_\sigma) p_{\mu_2}\cdots p_{\mu_n} \\
&\hspace*{3.85cm} + \cdots -\mbox{traces}]\,,
\end{split}$$ where $${\mathcal O}^{5}_{\sigma\mu_1\cdots\mu_n}
= \left(\frac{i}{2}\right)^n\bar{q}\gamma_{\sigma} \gamma_5
\mbox{\parbox[b]{0cm}{$D$}\raisebox{1.7ex}{$\leftrightarrow$}}_{\mu_1}
\cdots \mbox{\parbox[b]{0cm}{$D$}\raisebox{1.7ex}{$\leftrightarrow$}}_{\mu_n}
q -\mbox{traces}\,.$$ In parton model language $$\vspace*{0.25cm}
a_n^q=2 \int_0^1 d x\, x^{n} \large[\raisebox{-0.35cm}{$\underbrace{
\raisebox{0.35cm}{$q_\uparrow(x,\mu^2)
-q_\downarrow(x,\mu^2)$}}$}\large]
=2\Delta^n q \,,$$
in particular $a_0^u=2\Delta u$, $a_0^d=2\Delta d$, while $d_n^q$ has twist three and no parton model interpretation.
In the following I will restrict myself to nonsinglet and valence quark distributions due to lack of space. These quantities show little difference between quenched and full QCD calculations, so that I can further restrict myself to quenched results.
Let us first look at the structure function $g_1$. In Table 1 I compare the lattice results for the lower moments of $\Delta q(x,Q^2)$ [@QCDSF] with the corresponding phenomenological (experimental) numbers [@BB]. The quoted result for $\Delta u -\Delta d \equiv g_A$ has been taken from a recent, ‘proper’ extrapolation to the chiral limit [@HPW], shown in Fig. 1. By and large we find good agreement.
Let us next look at the structure function $g_2$. This differs from $g_1$ by twist-three contributions. From (\[ope\]) we readily see that $g_2$ fulfills the Burkhardt-Cottingham sum rule $$\int_0^1 dx\, g_2(x,Q^2) = 0\,.$$
$$\begin{tabular}{c|l|l}
Moment & \multicolumn{1}{c|}{Lattice~\cite{QCDSF}} &
\multicolumn{1}{c}{Experiment~\cite{BB}} \\
\hline
$\Delta u_v$ & $\phantom{-}0.889(29)$ & $\phantom{-}0.926(71)$ \\
$\Delta d_v$ & $-0.236(27)$ & $-0.341(123)$ \\[1ex]
$\Delta^1 u_v$ & $\phantom{-}0.198(8)$ & $\phantom{-}0.163(14)$ \\
$\Delta^1 d_v$ & $-0.048(3)$ & $-0.047(21)$ \\[1ex]
$\Delta^2 u_v$ & $\phantom{-}0.041(9)$ & $\phantom{-}0.055(6)$ \\
$\Delta^2 d_v$ & $-0.028(3)$ & $-0.015(9)$ \\[1ex]
$\Delta u - \Delta d$ &
$\phantom{-}1.25(7)$ &
$\phantom{-}1.267(142)$
\\[1ex]
$\Delta^1 u - \Delta^1 d$ & $\phantom{-}0.246(9)$ & $\phantom{-}0.210(25)$
\\[1ex]
$\Delta^2 u - \Delta^2 d$ & $\phantom{-}0.069(9)$ & $\phantom{-}0.070(11)$\\
\hline
\end{tabular}
$$
\
[Table 1. Comparison of lattice and experimental values of the lower moments of $\Delta q(x,Q^2)$, defined in (6), in the $\overline{MS}$ scheme at $Q^2=4$ GeV$^2$. The subscript $v$ refers to valence quarks.]{}
The first nontrivial moment of the twist-three contribution, $d_1^q$, can be related to the tensor charge $\delta q$ of the nucleon [@Paul], $$\begin{split}
d_1^{\,q}\, (s_\mu p_\nu - s_\nu p_\mu) &= \langle \vec{p},\vec{s}|\bar{q}\!
\left(\gamma_\mu \gamma_5\,
\mbox{\parbox[b]{0cm}{$D$}\raisebox{1.7ex}{$\leftrightarrow$}}_{\!\nu}
- \mbox{\parbox[b]{0cm}{$D$}\raisebox{1.7ex}{$\leftrightarrow$}}_{\!\mu}
\gamma_\nu \gamma_5 \right)\!q | \vec{p},\vec{s} \rangle \\[0.75ex]
&= -\frac{i}{2}\, \langle \vec{p},\vec{s}|\bar{q}\!
\left(\sigma_{\mu\nu}\gamma_5 \,
\mbox{\parbox[b]{0cm}{$\slashed{D}$}\raisebox{1.7ex}{$\rightarrow$}}
+ \mbox{\parbox[b]{0cm}{$\slashed{D}$}\raisebox{1.7ex}{$\leftarrow$}}
\sigma_{\mu\nu} \gamma_5 \right)\!q | \vec{p},\vec{s} \rangle \\[0.75ex]
&= i\, m_q\, \langle \vec{p},\vec{s}|\bar{q}\sigma_{\mu\nu}\gamma_5 q|
\vec{p},\vec{s} \rangle \\[0.75ex]
&= \frac{2 m_q}{m_N}\, \delta q\, (s_\mu p_\nu - s_\nu p_\mu)\,,
\end{split}$$ where $m_q$ is the mass of the quark. Thus we have $$d_1^q(Q^2) = \frac{2 m_q}{m_N} \delta q(Q^2)\,,$$ which vanishes in the chiral limit ($m_q \rightarrow 0$). The second moment, $d_2^q$, has been computed on the lattice [@QCDSF2]. The result is shown in Fig. 2, separately for the proton and the neutron, and found to be in good agreement with experiment [@dex]. From (\[ope1\]) and (\[ope\]) we obtain $$\int_0^1 dx\, x^2 g_2(x,Q^2) + \frac{2}{3}\int_0^1 dx\, x^2 g_1(x,Q^2) =
\frac{1}{6} d_2\,.$$ Given the fact that $d_2$ is small, and $d_1$ even vanishes in the chiral limit, we derive that the Wandzura-Wilczek relation [@WW] $$\begin{aligned}
\int_0^1 dx\, x^n g_2(x,Q^2) &=& -\frac{n}{n+1}\int_0^1 dx\, x^n
g_1(x,Q^2)\,,\\[0.5ex]
g_2(x,Q^2) &=& \int_{x}^1 \frac{dy}{y}\, g_1(y,Q^2) - g_1(x,Q^2)
\label{wand}\end{aligned}$$ holds to better than $O(5\%)$, except perhaps at very large $x$, which we are not interested in here. It is needless to say that $g_2$ in (\[wand\]) satisfies the Burkhardt-Cottingham sum rule as well. The structure function $g_1(x,Q^2)$ is obtained from the parton distributions $\Delta q(x,Q^2)$ by $$g_1(x,Q^2) = \frac{1}{2} \sum_q e_q^2 \int_{x}^1 \frac{dy}{y}\,
e_1(y,Q^2)\,
\Delta q\left(\frac{x}{y},Q^2\right)$$ with $$\int_0^1 dy \, y^n e_1(y,Q^2) = e_{1,n}(1,g(Q^2))\,.$$
Higher-Twist Contributions
==========================
Contributions of twist four (and higher) have been studied on the lattice, either through calculations of appropriate nucleon matrix elements [@ht], or by evaluating the operator product expansion directly on the lattice [@htope]. Higher-twist contributions are generally found to be small. Four-fermion operators, for example, schematically drawn in Fig. 3, which account for diquark effects in the nucleon, contribute $$\int_0^1 dx\, F_2^{(4)}(x,Q^2)\big|_{{\bf 27}, I=1} = - 0.0005(5)\,
\alpha_s(Q^2)\,
\frac{m_N^2}{Q^2} \,.$$ The reason for quoting the flavor 27-plet structure function here, rather than the nucleon (octet) one, is that the underlying four-fermion operator does not mix with operators of lower dimension, which makes it a clean prediction.
The GDH Integral
================
We are ready now to address the GDH integral. Following our previous discussion, we may assume that higher-twist contributions are small, and that the nucleon’s second polarized structure function $g_2$ is well approximated by the Wandzura-Wilczek form (\[wand\]). Moreover, given the fact that the lattice predictions for $a_n^q$, compiled in Table 1, are in good agreement with the phenomenological numbers quoted, we may base our further discussion on the parameterization of $g_1(x,Q^2)$ given in Ref. 2.
As already mentioned, I will restrict myself to the nonsinglet GDH integral, which corresponds to proton minus neutron target. In Fig. 4 I show recent results from the Hermes collaboration [@hermes]. I compare this result with the theoretical predictions. The solid line represents the full integral, as given by the bottom line of (\[igdh\]), including $g_1$ and $g_2$, while the dashed line represents the integral $$\frac{16\pi^2 \alpha}{Q^2}\int_0^1 dx\,
g_1(x,Q^2) \equiv \frac{16\pi^2 \alpha}{Q^2} \Gamma_1\,.
\label{bj}$$
Equation (\[bj\]) corresponds to the limiting case $x_0 = 1$ and $\gamma =
0$. The dashed curve can hardly be distinguished from the solid curve over the whole kinematical range, $1 \lesssim Q^2 \leq \infty$, which tells us that the (nonsinglet) GDH integral is insensitive to $g_2$ and merely tests the Bjorken sum rule $$\begin{split}
\Gamma_1 = \int_0^1 dx\, g_1(x,Q^2) &= \frac{1}{6} g_A
\left[1-\frac{\alpha_s(Q^2)}{\pi}
-3.58\, \left(\frac{\alpha_s(Q^2)}{\pi}\right)^2\right. \\ &- 20.22\,
\left.\left(\frac{\alpha_s(Q^2)}{\pi}\right)^3 + \cdots \right] \,.
\end{split}$$
At larger values of $Q^2$ the data fall below the curve. This may be due to the fact that the $x$ range covered shrinks to $0.2 \leq x \leq 0.8$ in the highest-$Q^2$ bin. In order to draw quantitative conclusions, one certainly would need more precise data. So far we can only say that the experimental data are consistent with leading-twist parton distributions all the way down to $Q^2 \approx 1$ GeV$^2$, and with the Bjorken sum rule in particular.
In Fig. 5 I show the same figure on an enhanced scale, together with the predictions of chiral perturbation theory [@bhm] and the predicted value of the GDH sume rule. Both curves appear to meet in a sharp peak at $Q^2 \approx 0.4$ GeV$^2$. This value lies much below the range of validity of the operator product expansion. Likewise it lies beyond the applicability of chiral perturbation theory, so that no firm statement about the transition from large $Q^2$ to the resonance region can be made.
Conclusions
===========
To learn anything new, if possible at all, from the GDH integral, we need better experimental data. At present the GDH integral does not teach us anything quantitative about higher-twist contributions. Lattice calculations, on the other hand, indicate that the GDH integral is well represented by the Bjorken sum rule down to $Q^2 \approx 1$ GeV$^2$. More precise lattice data on moments of $g_1$ and $g_2$, including sea quark effects, will become available soon [@soon]. In order to match scaling and resonance region, it appears that one will have to resort to model building.
Acknowledgement {#acknowledgement .unnumbered}
===============
I like to thank Johannes Blümlein, Helmut Böttcher, Meinulf Göckeler, Roger Horsley, Dirk Pleiter and Paul Rakow for discussions and collaboration.
[99]{}
M. Göckeler, R. Horsley, E.-M. Ilgenfritz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, Phys. Rev. D53, 2317 (1996); M. Göckeler, R. Horsley, L. Mankiewicz, H. Perlt, P. Rakow, G. Schierholz and A. Schiller, Phys. Lett. B414, 340 (1997); S. Capitani, M. Göckeler, R. Horsley, H. Perlt, D. Petters, D. Pleiter, P.E.L. Rakow, G. Schierholz, A. Schiller and P. Stephenson, Nucl. Phys. Proc. Suppl. 79, 548 (1999); M. Göckeler, R. Horsley, W. Kürzinger, H. Oelrich, D. Pleiter, P.E.L. Rakow, A. Schäfer and G. Schierholz, Phys. Rev. D63, 074506 (2001); M. Göckeler et al., in preparation.
J. Blümlein and H. Böttcher, Nucl. Phys. B636, 225 (2002).
T.R. Hemmert, M. Procura and W. Weise, Phys. Rev. D68, 075009 (2003).
This derivation is due to Paul Rakow. Equation (9) is still to be verified numerically.
Last reference in Ref. 1.
P.L. Anthony et al. \[E155 Collaboration\], Phys. Lett. B458, 529 (1999).
S. Wandzura and F. Wilczek, Phys. Lett. B72, 195 (1977).
M. Göckeler, R. Horsley, B. Klaus, D. Pleiter, P.E.L. Rakow, S. Schaefer, A. Schäfer and G. Schierholz, Nucl. Phys. B623, 287 (2002).
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M. Göckeler et al., in preparation.
[^1]: nvited Talk given at 2004, une 2-5, 2004, orfolk,
|
---
author:
- 'Johan E. Lindberg'
- 'Jes K. J[ø]{}rgensen'
- 'Joel D. Green'
- 'Gregory J. Herczeg'
- Odysseas Dionatos
- 'Neal J. Evans II'
- Agata Karska
- 'Susanne F. Wampfler'
bibliography:
- 'herschel\_rcra\_arxiv.bib'
date: 'Received July 1, 2013; accepted November 27, 2013'
subtitle: '*Herschel*/PACS observations from the DIGIT key programme'
title: Warm gas towards young stellar objects in Corona Australis
---
[The effects of external irradiation on the chemistry and physics in the protostellar envelope around low-mass young stellar objects are poorly understood. The Corona Australis star-forming region contains the R CrA dark cloud, comprising several low-mass protostellar cores irradiated by an intermediate-mass young star.]{} [We study the effects on the warm gas and dust in a group of low-mass young stellar objects from the irradiation by the young luminous Herbig Be star R CrA.]{} [*Herschel*/PACS far-infrared datacubes of two low-mass star-forming regions in the R CrA dark cloud are presented. The distribution of CO, OH, H$_2$O, \[\], \[\], and continuum emission is investigated. We have developed a deconvolution algorithm which we use to deconvolve the maps, separating the point-source emission from the extended emission. We also construct rotational diagrams of the molecular species.]{} [By deconvolution of the *Herschel* data, we find large-scale (several thousand AU) dust continuum and spectral line emission not associated with the point sources. Similar rotational temperatures are found for the warm CO ($282\pm4$ K), hot CO ($890\pm84$ K), OH ($79\pm4$ K), and H$_2$O ($197\pm7$ K) emission, respectively, in the point sources and the extended emission. The rotational temperatures are also similar to what is found in other more isolated cores. The extended dust continuum emission is found in two ridges similar in extent and temperature to molecular millimetre emission, indicative of external heating from the Herbig Be star R CrA.]{} [Our results show that a nearby luminous star does not increase the molecular excitation temperatures in the warm gas around a young stellar object (YSO). However, the emission from photodissociation products of H$_2$O, such as OH and O, is enhanced in the warm gas associated with these protostars and their surroundings compared to similar objects not suffering from external irradiation.]{}
Introduction
============
One of the open questions in low-mass star formation is how the irradiation from intermediate-mass stars affects the chemistry, temperature, and excitation conditions in the warm gas around low-mass young stellar objects. With the resolution of the *Herschel Space Observatory*, superior to that of previous far-infrared telescopes, in combination with deconvolution algorithms, we can now address this question. Both the spectral line emission from the gas and the dust continuum emission from the warm regions peak in the far-infrared (FIR) part of the electromagnetic spectrum. CO, the second most abundant molecule in the interstellar medium (ISM) after H$_2$, has a large number of transitions in this band. In addition, water and its related species, OH, have their most important transitions in this band.
With the advent of the *Herschel Space Observatory* [@pilbratt10], FIR observations with unprecedented spatial and spectral resolution have been made available. The *Herschel* observations of low-mass YSOs reveal numerous lines of CO, H$_2$O, and OH, along with atomic lines like \[\] and \[\] [e.g. @herczeg12; @kristensen12; @green13]. In most studied sources, the CO rotational diagrams can be fitted with warm and hot components, with rotational temperatures of about 300 K and 900 K, respectively [@green13; @karska13; @manoj13]. The OH and H$_2$O emission is usually characterised by somewhat lower rotational temperatures around 100 K [@goicoechea12; @herczeg12; @wampfler13]. Most of these studies of low-mass YSOs have targeted isolated embedded objects. To better understand low-mass star formation in more dynamic environments, studies of small groups of resolvable embedded objects are warranted.
This paper presents PACS [Photodetector Array Camera and Spectrometer; @poglitsch10] maps of the R Coronae Australis (R CrA) dark cloud, which harbours one of the closest star-forming regions, located at a distance of 130 pc [@neuhauser08]. The cloud is named for the young star R CrA, which has spectral classifications ranging from F5 to B5 (e.g., F5: @hillenbrand92; A0: @manoj06; B8: @bibo92; B5: @gray06). The cloud was mapped in CO by @loren79, who found an elongated cloud extended over about 2 by 0.5 pc and peaking near R CrA. Higher resolution maps of C$^{18}$O with the SEST (Swedish-ESO Submillimetre Telescope) revealed several dense molecular clumps with masses between $2~M_{\odot}$ and $50~M_{\odot}$ near R CrA [@harju93]. Surveys of the clumps have revealed a number of embedded protostars, with IRS7 in the clump to the southeast of R CrA and IRS5 in the clump to the west. @taylor84 report that IRS7 is the most reddened source in the region, having a visual extinction of more than 25 mag. @brown87 split IRS7 into two sources separated by 14, using VLA 6 cm observations. For a more complete description and references, see @neuhauser08.
The R CrA region (including IRS7) has previously been studied at FIR wavelengths using the ISO telescope [@lorenzetti99; @giannini99], detecting lines from CO, OH, , and . However, these studies were limited by the ISO angular resolution of 80, which made it impossible to separate the sources in the region. @sicilia13 present 100[ m]{} and 160[ m]{} photometry maps of the CrA region observed with *Herschel*/PACS. The IRS7 clump is positioned with its centre between the two Herbig Ae/Be stars R CrA and T CrA, and harbours a handful of Class 0/I Young Stellar Objects (YSOs). @nutter05 report detections of four cores within the IRS7 and IRS5 clumps in SCUBA 450[ m]{} and 850[ m]{} data: SMM 1A, with no mid-IR counterpart, proposed to be a pre-stellar core; SMM 1B, coincident with the mid-IR source IRS7B; SMM 1C, a Class 0 protostar; and SMM 4, coincident with the IRS5 clump. @peterson11 report SMA 1.3 mm point-source continuum detections at the positions of SMM 1B (IRS7B), SMM 1C, and SMM 4 (IRS5N). The IRS5 clump contains two protostellar sources, IRS5A and IRS5N. IRS5 is situated at a slightly greater projected distance from R CrA than is IRS7. IRS5A is not detected in SMA 1.3 mm continuum observations, whereas IRS5N shows significantly fainter continuum emission than the IRS7 sources [95 mJy in IRS5N, 320 mJy in IRS7B; @peterson11].
Through single-dish APEX observations, @schoier06 found elevated H$_2$CO and CH$_3$OH abundances and rotational temperatures in IRS7A and IRS7B, which were suggested to be caused by increased internal heating or outflows. @vankempen09a used mid-$J$ CO observations to find an EW outflow centred at IRS7A, but also found that the CO line fluxes in the region are too high to originate from heating by the embedded protostars, and proposed that the heating originates from R CrA. This was confirmed by @lindberg12, who found large-scale ($\sim 10\,000$ AU) H$_2$CO emission heated to 40–60 K by external irradiation from R CrA. The IRS5 sources are found to be less affected by the irradiation from R CrA (the H$_2$CO rotational temperature in a 29 beam is 47 K for IRS7B and 28 K for IRS5A; Lindberg et al., in prep.).
In Sect. 2 we will describe the methods of observations and data reductions and show the first results. In Sect. 3 follows a discussion of our deconvolution algorithm and the results thereof. In Sect. 4 we investigate models of the excitation conditions of the gas. In Sect. 5 we discuss the interpretations of our results, and in Sect. 6 we list our conclusions.
Observations and data description
=================================
In this section we first describe the observations and methods of data reduction. A more detailed description can be found in @green13. We then discuss the first results of these reductions.
Observation setup and data reduction
------------------------------------
{width="1.0\linewidth"}\
[l l l l l l]{} Region & RA & Dec & P.A. & Observing IDs & Date of observation\
& (J2000.0) & (J2000.0) & \[\]\
IRS7A & 19:01:55.3 & $-$36:57:17.0 & $-0.22$ & 1342206990, 1342206989 & 2010-10-23\
IRS7B & 19:01:56.4 & $-$36:57:28.3 & 2.47 & 1342207807, 1342207808 & 2010-11-02\
IRS5A & 19:01:48.1 & $-$36:57:22.7 & 2.40 & 1342207806, 1342207805 & 2010-11-01\
Integral-field spectroscopy observations in the far infrared (FIR) regime that cover parts of the R CrA star-forming region were performed by the Photodetector Array Camera and Spectrometer [PACS; @poglitsch10] on board the ESA *Herschel Space Observatory* [@pilbratt10], with a spectral range from 55 m to 210 m. The observations were carried out in range-spectroscopy mode. The spectral resolution varies between 0.013 m and 0.13 m ($\Delta \varv
\approx 55$–318 km s$^{-1}$, $\lambda/\Delta\lambda \approx
950$–$5500$).
These observations are part of the “Dust, Ice and Gas in Time” (DIGIT; PI: N. Evans) Open-Time Key Programme, a survey of the change of FIR spectral features through the evolution of young stellar objects. For details of the DIGIT Key Programme, see @green13. The observed PACS fields are shown compared to the point-source positions and a Spitzer 4.5[ m]{} image in Fig. \[fig:overview\], and the telescope pointings are found in Table \[tab:obsparam\]. The configuration of the $9\farcs4\times9\farcs4$ spaxels shown in Fig. \[fig:overview\] is a simplified model: in reality, no spaxels overlap. This is taken into account in the deconvolution method described in Sect. \[sec:deconv\_meth\].
The spectral line fluxes in each spaxel were calculated using the HIPE 8.0.2489 reduction of the data, but corrected by the continuum value in the HIPE 6.1 reduction. The HIPE 8.0.2489 reduction was found to provide on average $\sim 40$% better signal-to-noise ratio (spanning from almost no improvement up to about a factor 2 at specific wavelengths), whereas the HIPE 6.1 reduction produced more accurate and reliable spectral energy distributions (SEDs) by $\sim20\%$. These SEDs are consistent with PACS photometry within $\sim10\%$ [@green13]. The method of combining these two reductions is thoroughly described in @green13. For the continuum flux densities, the HIPE 6.1 reduction was used.
For each spectral line in each spaxel, a first-degree polynomial baseline calculated from the surrounding line-free channels was subtracted from the spectrum. The total line flux was then calculated by summing the channel flux densities within a typical linewidth (mostly 0.1–0.5[ m]{}) and multiplying this sum by the channel width. Of the OH doublets, only four out of nine are resolved. In the unresolved cases, the combined total flux for both OH lines is calculated, and half of this value is used for each of the components in the rotational diagram analysis. The statistical errors are calculated from the rms noise around each spectral line (any lines weaker than 3$\sigma$ are ignored) to get the total standard deviation of the spectral line flux. If nothing else is stated, all errors given in this paper are at the 1$\sigma$ level. In addition to the statistical error, a systematic calibration error of 20% of the flux (see @green13) is used when calculating quantities such as the rotational temperatures and total number of molecules from the rotational diagrams.
The two bands below 103 m are considerably noisier than the bands above this wavelength, and between approximately 94 m and 103 m the noise makes line fluxes very difficult to estimate. The spectrometer suffers from leakage in the wavelength ranges 70–73[ m]{}, 98–105[ m]{}, and above 190[ m]{}, which produces ghost images of lines from the next higher grating order. Fluxes at these wavelengths are thus less reliable than at other wavelengths [@herczeg12].
In addition to the Herschel data, we also used the ISIS spectrograph on the *William Herschel Telescope* (WHT) on 6 August 2012 to obtain a low-resolution ($R\sim1\,000$) optical spectrum of R CrA covering 3000–10000 Å. The spectrum was flux calibrated against the spectrophotometric standard LTT 7987. The analysis of this spectrum will be discussed in Sect. \[sec:rcra\_spect\].
First look
----------
In this section, we provide a first look at the PACS continuum and spectral maps before they are treated by our deconvolution algorithm.
### Continuum maps
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{important_maps_7_1.png} &
\includegraphics[width=0.48\linewidth]{important_maps_7_2.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{important_maps_5_1.png} &
\includegraphics[width=0.48\linewidth]{important_maps_5_2.png} \\
\end{array}$
The total emission continuum maps of IRS7 and IRS5 at three representative wavelengths can be found in Figs. \[fig:important\_7\]–\[fig:important\_5\], and continuum maps at 20[ m]{} steps can be found in Figs. \[fig:contmaps7\]–\[fig:contmaps5\]. These continuum maps show extended emission in the order of 30–60 in size. The larger sizes are found in the longer wavelengths (which could be suggested to be attributed only to the larger beam size at these wavelengths, but the shapes of the emission suggest differently; see also Sect. \[sec:continuum\_pomac\]). The highest continuum flux densities in the strongest illuminated spaxels are found around 85[ m]{}in IRS7 and 105[ m]{} in IRS5, but the spatially integrated flux density is strongest around 120[ m]{} in IRS7 and 155[ m]{} in IRS5 (since the longer wavelength data have more extended PSFs, which spreads the signal over a larger solid angle). It is, however, difficult to draw conclusions about the stages of evolution of the individual sources from the total emission, since it is a combination of several compact sources and any extended emission. For such a study, deconvolution of the emission is necessary.
The continuum emission in the IRS7 region is very extended in comparison to the line emission (see Sect. \[sec:results\_line\_maps\]), and has at least some signal across most of the two PACS fields that cover the region (see Figs. \[fig:important\_7\]–\[fig:important\_5\]). Without any deconvolution, it is impossible to distinguish the point sources detected in mid-IR and mm data (IRS7A, IRS7B, SMM 1C, and R CrA).
IRS5 consists of two separate sources, of which the northern source IRS5N is detected in SMA continuum, whereas the southern IRS5A is detected in *Spitzer* continuum and line emission. It is difficult to distinguish the two sources in the PACS data.
### Spectral line maps {#sec:results_line_maps}
{width="1.0\linewidth"}\
The sum of the PACS spectra of the spaxels closest to IRS7A and SMM 1C is shown in Fig. \[fig:irs7a\_spectrum\]. The detected species are indicated by different colours. The total emission spectral line emission maps of some important lines are found in Figs. \[fig:important\_7\]–\[fig:important\_5\], and all spectral line emission maps can be found in Figs. \[fig:comaps7\]–\[fig:atomicmaps5\]. The CO emission in IRS7 is generally found to be more extended than the OH and H$_2$O emission. The emission from the atomic species and detected in IRS7 is even more extended. In particular, the \[\] line emission peaks off-source in a position east of IRS7A. Compared to the CO, OH, and H$_2$O emission in the IRS7 cloud, which is mainly centred on the three point sources, the \[\] emission in IRS7 is more extended in the EW-direction. The \[\] emission is similar to the CO and OH emission, but apparently much stronger in SMM 1C than in IRS7A. There is also a relatively strong band of extended emission in a band W and NE of IRS7A. For IRS5, the spectral line emission seems point-like.
The CO, OH, and H$_2$O emission detected by PACS in the IRS7 field is centrally peaked on the three protostellar sources in the IRS7 field (SMM 1C, IRS7A, and IRS7B), with the strongest emission from SMM 1C and IRS7A. This is consistent with the centrally peaked (but also extended) CO $J=6\rightarrow5$ and HCO$^+$ emission found by @vankempen09a [@vankempen09b], but different from the very extended H$_2$CO and CH$_3$OH emission detected in SMA (Submillimeter Array) and APEX (Atacama Pathfinder Experiment) observations [@lindberg12]. None or very little FIR line emission is seen around SMM 1A, despite the fact that this region shows very strong H$_2$CO and CH$_3$OH emission in the SMA mm data.
Image analysis {#sec:psfcorr}
==============
As is clearly seen in the total emission continuum and line maps (Figs. \[fig:important\_7\]–\[fig:important\_5\]), the emission originating from the different sources in the IRS7 and IRS5 fields cannot be easily separated, and a deconvolution method needs to be applied. To test the hypothesis that most of the emission can be accounted for by the previously known point sources, we need to deconvolve the data with the point-spread function (PSF) of the observations.
Deconvolution method {#sec:deconv_meth}
--------------------
The effective diameter of the *Herschel Space Observatory* is 3.28 m. The *Herschel* PSF is slightly triangular projected on the sky due to the three-point mount of the telescope dish (see Fig. \[fig:pacsbeam\]), but it can be roughly approximated by a Gaussian with a full width at half maximum (FWHM) between 4 and 12 depending on the wavelength (for our deconvolution algorithm we instead use PSFs constructed from the simulated telescope PSF; see Appendix \[app:pomac\]). The PACS spectrometer detector array consists of 25 spaxels, positioned in a square 5$\times$5 pattern. The separations between the spaxel centres are on average approximately 925. This has the effect that emission from a point source will spill over to adjacent spaxels, particularly for the longer wavelengths, since the telescope PSF will be larger than the spaxel size. This is normally easy to correct for when observing a single well-centred point source, because it is then in principle sufficient to use the central spaxel flux (or flux density) multiplied by a wavelength-dependent PSF correction factor[^1]. However, if observing off-centred point sources, several point sources in the same field, or extended emission (or combinations of these), the interpretation of the data becomes more difficult. We find several point sources in the IRS7 field, some of them not well-centred on spaxel centres, and also signs of extended emission in the PACS field, and therefore need another method to separate the emission from the different possible origins. The IRS7 field was observed in two partially overlapping PACS pointings, which creates an almost Nyquist-sampled map for all but the shortest wavelengths in the overlap region.
To be able to distinguish point-source emission from more extended emission, the signal must be deconvolved from the PSF. However, the spatial resolution of the signal is limited by the design of the PACS instrument, only providing 25 data points for this signal per pointing. We have developed a method that still can provide deconvolution of point sources from the PACS data, called POMAC[^2]. The method is based on the CLEAN algorithm [@hogbom74], often used to deconvolve undersampled maps in radio interferometry. Due to the low number of data points, our method relies on *a priori* knowledge about the point-source positions. In this case, we employ ALMA data from @lindberg13_alma, and SMA and *Spitzer* data from @peterson11 to establish the positions of the YSOs with an accuracy of $\lesssim1$. Based on the larger sample of more isolated sources [@green13], we expect that a major part of the emission seen with *Herschel*, both continuum and line, will originate from the point sources seen at mid-infrared and submillimetre wavelengths, and we use the deconvolution to test this hypothesis. The CLEAN algorithm is used with the modification that it is only allowed to identify these pre-defined point sources as sources of emission. The algorithm then iterates over the PACS data with customary break criteria (such as avoiding subtraction below the noise floor in any spaxel). After this, the residual map can be studied to identify previously unknown point sources as well as extended emission. Repeating the process after adding new point sources will eventually leave all extended emission in the residual map (however, still convolved with the PSF). The algorithm was tested on the PACS spectrometer continuum of the disc source HD 100546 [@sturm10], which is not expected to differ significantly from a point source in the continuum. The deconvolution produced results within errors of those obtained using the PACS PSF correction factor across the whole PACS band, and no significant residuals were noted. A more detailed explanation of the POMAC algorithm, including a description of how the telescope PSFs were generated, can be found in Appendix \[app:pomac\].
### Definition of point sources
The sources treated in this paper have been studied in several previous papers, giving them many different names. To facilitate comparison with other work, a list of the most common names for these point sources found in the literature can be found in Table \[tab:pointsource\]. We will maintain the name usage of @lindberg12.
[l l l l l l]{} Name & RA & Dec & Other names\
& (J2000.0) & (J2000.0) &\
IRS7A & 19:01:55.33 & $-$36:57:22.4 & IRS7W, IRS7\
SMM 1C & 19:01:55.31 & $-$36:57:17.0 & SMA 2, Brown 9\
IRS7B & 19:01:56.42 & $-$36:57:28.4 & IRS7E, SMM 1B,\
& & & SMA 1\
CXO 34 & 19:01:55.78 & $-$36:57:27.9 & FP-34\
R CrA & 19:01:53.67 & $-$36:57:08.0 & &\
IRS5A & 19:01:48.03 & $-$36:57:22.2 & CrA-19, IRS5ab\
IRS5N & 19:01:48.47 & $-$36:57:14.9 & CrA-20, SMM 4,\
& & & FP-25\
To match the line and continuum emission with compact objects we rely on *a priori* position data from other observations with better resolution, thus obtained in other wavebands. To cover all sources that could be visible in the FIR, we used both longer (submillimetre) and shorter (mid-infrared) wavelength observations to identify possible point sources. @peterson11 used SMA 226 GHz continuum observations to identify four continuum peaks in the IRS7 and IRS5 regions (IRS7B, SMA2 / SMM 1C, IRS5N, and R CrA); and Spitzer 4.5[ m]{} observations to find five continuum peaks in the same fields (R CrA, CrA-19 / IRS5A, IRS7B, IRS7A, and CXO 34). We identified four of these sources (IRS7B, SMM 1C, IRS7A, and CXO 34) in ALMA Cycle 0 observations of the 342 GHz continuum centred at IRS7B [@lindberg13_alma]. The ALMA coordinates are of superior accuracy, and will be used for these sources. The coordinates for these sources can be found in Table \[tab:pointsource\].
From these point-source positions, we selected the six most prominent sources (only excluding CXO 34, which is almost an order of magnitude weaker at 4.5 m than the second weakest infrared source) to use for the deconvolution of both the continuum and the line emission. The excluded source CXO 34 is not only very weak, but also situated between the much stronger sources IRS7B and IRS7A, and any far-infrared emission originating from CXO 34 would not be possible to disentangle from the emission of the surrounding sources.
IRS7A and SMM 1C have an angular separation of only 5, i.e. not spatially resolved by the PACS array. We attempted to separate the emission from IRS7A and SMM 1C, but we found that the results were unreliable, also for the shorter wavelengths. Therefore, we have decided to treat IRS7A and SMM 1C together, summing the fluxes given for the two sources. This is not ideal, however, since they are suggested to be of different types (Class 0 and Class I, respectively). Since the line and continuum emission in this region is extended in the north-south direction it is likely that both sources contribute to this emission.
Results of the continuum deconvolution {#sec:continuum_pomac}
--------------------------------------
Deconvolved and non-deconvolved spectra of the point sources are found in Fig. \[fig:spectra\_pomac\]. When comparing the non-deconvolved and deconvolved spectra, we find that the spectral lines are stronger compared to the continuum in the deconvolved data (especially in the IRS7 sources), indicating that the spectral line emission is less extended than the continuum emission, a pattern which is also seen in the larger DIGIT embedded objects sample [hereby referred to as “the DIGIT sample”; @green13], although data of more crowded regions [e.g. Serpens; @dionatos13] or strong outflow sources [e.g. L1448-MM; @lee13] show spatial extent. As in the other sources in the DIGIT sample, the emission from \[\] is more extended than that from the other species. We also find that IRS5N only shows a few spectral lines, but has a continuum as strong as that of IRS5A. The bumps in the spectra around 180[ m]{} are caused by an instrument leakage effect.
{width="0.48\linewidth"}\
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{irs7b_spect_pomac.png} &
\includegraphics[width=0.48\linewidth]{rcra_spect_pomac.png} \\
\includegraphics[width=0.48\linewidth]{irs5a_spect_pomac.png} &
\includegraphics[width=0.48\linewidth]{irs5n_spect_pomac.png} \\
\end{array}$
The continuum deconvolution is performed on averaged line-free spectral boxes every 5[ m]{}. As a sanity check, these results are compared to the continuum level found in the channel-by-channel deconvolution (see above and Fig. \[fig:spectra\_pomac\]), and the two methods are found to be in good agreement.
The total emission maps as well as the deconvolution residual maps at three different wavelengths can be found in the first six panels of Figs. \[fig:important\_7\]–\[fig:important\_5\]. Continuum total emission maps and residual maps in 20[ m]{} steps can be found in Figs. \[fig:contmaps7\]–\[fig:contmaps5\].
Running the POMAC deconvolution algorithm on the continuum data leaves strong residual emission in a shape that remains constant across most of the PACS spectral band, except for the shortest wavelengths. We find only a little continuum residual emission in the centre of the IRS7 field, but strong residual emission is found in two ridges north and south of the YSOs. This could be an effect of the stop criteria used by the POMAC algorithm, which assume that all emission on the point-source positions shall be attributed to the point sources, and should thus not leave considerable residuals on-source. We did several POMAC runs with different stop criteria. None of them could reproduce a smooth continuum residual, but either showed the ridges or left significant emission on the point source positions. The continuum residual at 110[ m]{} is shown in Fig. \[fig:cont\_extreg\], which is used to define the residual regions Res Nc and Res Sc (c for continuum). These regions are chosen to coincide with the spaxels with a residual continuum spectral flux density of at least 30 Jy at 110[ m]{}. Also shown are the ridges of molecular gas detected in H$_2$CO and CH$_3$OH [@lindberg12], which bear resemblance in shape and position to the PACS continuum residuals. Res Sc coincides partially with the pre-stellar core SMM 1A detected in SCUBA submm observations [@nutter05]. These similarities indicate that the deconvolution algorithm and the used stop criteria produce accurate results. Also in the IRS5 field there is some residual continuum emission, but only at wavelengths longer than 100[ m]{}.
![The 110[ m]{} continuum residual map (coloured contours) of IRS7 overplotted on the H$_2$CO $3_{03}\rightarrow2_{02}$ emission (greyscale) from SMA+APEX observations [@lindberg12]. The orange dots show the PACS spaxel centres. Contour levels are at 5 Jy intervals and go from light blue to dark blue in colour as the level of emission increases (i.e., the residual emission is at a minimum near the centre of the figure). The red dashed lines illustrate our definition of the two extended (residual) ridge regions (Res Nc and Res Sc) in IRS7 continuum. The crosses show the point sources used for the deconvolution.[]{data-label="fig:cont_extreg"}](ridge_regs_new.png){width="1.0\linewidth"}
Results of the spectral line deconvolution {#sec:line_pomac}
------------------------------------------
For the deconvolution of the spectral lines, the flux of each spectral line in each spaxel is first measured after subtraction of linear baselines, producing total line flux maps. These total line flux maps are then the input data to the POMAC algorithm. The line strengths are somewhat higher (approximately 20%) compared to those found if extracting line strengths from the continuum-subtracted channel-by-channel-deconvolved spectra (Fig. \[fig:spectra\_pomac\]). This is due to the stop criteria, making the algorithm reach the noise floor earlier in the channel-by-channel data than in the line flux data, since the S/N level is lower in the individual channels than for the total line fluxes.
The line flux maps as well as the deconvolution residual maps of seven important spectral lines can be found in the 14 last panels of Figs. \[fig:important\_7\]–\[fig:important\_5\]. All total line intensity maps and residual maps can be found in Figs. \[fig:comaps7\]–\[fig:atomicmaps5\] in Appendix \[app:linemaps\]. The extracted point source line fluxes are listed in Table \[tab:herschel\_lineparams\]. Note that, as in the case of the continuum deconvolution, the POMAC algorithm will attribute all on-source emission to the point sources, and not leave residuals at the point-source positions. One could also assume that the point sources are sitting on a plateau of extended emission, but attempts to model that situation with different stop criteria have been unsuccessful. In any case, since the residual emission is much fainter than the point-source emission, such a solution would not change the results of excitation analysis of the point sources dramatically. It is difficult to give an exact estimate on this contribution, since the residual emission is primarily found west of the IRS7 point sources, and a smooth distribution of the extended emission around these sources is thus not possible. For CO, the molecule with the most prominent extended emission, the point sources line fluxes would be $\lesssim25\%$ lower assuming a flat distribution of the extended emission as strong as the residual west of the point sources, but it shall be noted that this is a worst-case scenario. Our estimates show that the errors on the rotational temperatures (Sect. \[sec:rotdiag\_analysis\]) would increase by $\sim60\%$ assuming this scenario. The reported extended emission should, on the other hand, be seen as a lower limit on the amount of extended gas.
Some spectral lines suffer heavily from line blending, and will not be considered in the further analysis. These are listed in Table \[tab:blended\]. In addition, only spectral lines between 55[ m]{}and 100[ m]{}, and 103[ m]{} and 195[ m]{} are considered, due to leakage and/or a high noise level in the outer parts of the bands. The instrument suffers from leakage also in the ranges 70–73[ m]{} and 98–105[ m]{}, so line strengths in these bands are less reliable than those in other bands.
[l l l l l]{} Species & Transition & Wavelength & Blend & Transition\
& & \[m\] & &\
o-H$_2$O & $6_{25}\rightarrow5_{14}$ & 65.2 & OH & $^2\Pi_{3/2}(J=9/2-\rightarrow7/2+)$\
CO & $J=35\rightarrow34$ & 74.9 & o-H$_2$O & $7_{25}\rightarrow6_{34}$\
o-H$_2$O & $7_{25}\rightarrow6_{34}$ & 74.9 & CO & $J=35\rightarrow34$\
CO & $J=31\rightarrow30$ & 84.4 & OH & $^2\Pi_{3/2}(J=7/2+\rightarrow5/2-)$\
OH & $^2\Pi_{3/2}(J=7/2+\rightarrow5/2-)$ & 84.4 & CO & $J=31\rightarrow30$\
o-H$_2$O & $6_{25}\rightarrow6_{16}$ & 94.6 & o-H$_2$O & $4_{41}\rightarrow4_{32}$\
o-H$_2$O & $4_{41}\rightarrow4_{32}$ & 94.7 & o-H$_2$O & $6_{25}\rightarrow6_{16}$\
CO & $J=23\rightarrow22$ & 113.4 & o-H$_2$O & $4_{14}\rightarrow3_{03}$\
o-H$_2$O & $4_{14}\rightarrow3_{03}$ & 113.5 & CO & $J=23\rightarrow22$\
p-H$_2$O & $3_{22}\rightarrow3_{13}$ & 156.2 & o-H$_2$O & $5_{23}\rightarrow4_{32}$\
o-H$_2$O & $5_{23}\rightarrow4_{32}$ & 156.3 & p-H$_2$O & $3_{22}\rightarrow3_{13}$\
### CO emission patterns
We find CO emission associated with all the pre-defined point sources in the deconvolution, although only very faint emission is found to originate from IRS5N.
Studying the residual maps (Figs. \[fig:cont\_extreg\] and \[fig:comaps7\]–\[fig:comaps5\]), considerable extended CO emission is found southwest, east, north, and west of the point sources in the IRS7 field for the lower-*J* CO lines ($J \lesssim 25$). For the higher-*J* CO lines, the S/N is too low to find more than traces of such emission, but this will be further discussed in Sect. \[sec:co\_extended\_ex\]. In the IRS5 field, the CO emission is completely point-like, leaving no residuals after the deconvolution.
The relatively strong extended emission in the SW part could also be explained by a CO point source in a position not associated with any YSO. No corresponding point source has been identified in the SMA data, in the *Spitzer* data, or in any source catalogue, but we cannot rule out that this is a very faintly emitting YSO. However, we do not consider this to be a point source, and the excitation conditions of this emission will be treated together with that of the other extended CO emission in Sect. \[sec:co\_extended\_ex\]. This peak bears resemblance to an outflow front, but it does not align well with the EW outflow found by @vankempen09a. The origin is thus uncertain.
The CO line emission in IRS5 is well-centred on the spaxel corresponding to IRS5A (see CO emission maps and CO residual maps in Figs. \[fig:important\_5\] and \[fig:comaps5\]), and when running the POMAC code on this data assuming IRS5A as the only point source, only marginal residuals are found. It is thus reasonable to believe that IRS5N produces only a very low amount of CO emission.
### OH and H$_2$O emission patterns
The OH and H$_2$O (p-H$_2$O and o-H$_2$O) line emission will be treated in the same section due to their similar emission pattern. IRS7A+SMM 1C are strong emitters of lines from these molecules, whereas IRS7B is much weaker in the OH and H$_2$O transitions. The OH, p-H$_2$O, and o-H$_2$O line emission maps and residual maps are found in Figs. \[fig:ohmaps7\]–\[fig:oh2omaps5\]. There is less extended emission in the OH and H$_2$O line data compared to the CO data, in particular, there is no considerable extended emission southwest of IRS7. There is, however, some extended OH and H$_2$O emission west of IRS7.
As in the case of CO, IRS5A seems to be the dominant emitter of OH and H$_2$O in the IRS5 field, IRS5N not contributing any significant emission.
### Atomic line emission patterns
Compared to the CO, OH, and H$_2$O emission in the IRS7 cloud, which is mainly centred on the three point sources, the \[\] emission is more extended in the EW-direction and peaks in different positions from the other spectral lines in the IRS7 field. However, the \[\] PACS data of protostellar sources often suffer from emission in the off-positions. We have investigated the signal using the two different nod positions, and the general structure is similar but not identical in these two data sets, which suggests that the detected \[\] morphology may partly be an observational effect. The \[\] data could thus be unreliable, and will not be further discussed.
The \[\] emission peaks on the point sources, but also shows strong extended emission in the whole IRS7 field, with residual peaks similar to those of OH and H$_2$O, indicating that they trace the same extended gas. This is a good indicator of large-scale PDR (photo-dissociation region) activity or alternatively outflow-associated shocks, since H$_2$O can be photo-dissociated into OH and O [@hollenbach97]. The \[\] data do not suffer from off-position emission to a significant level. The line ratio between the 145[ m]{} and 63[ m]{} \[\] lines varies between 0.06 and 0.11 for the point sources, which is also indicative of a strong radiation field (i.e. PDR activity; @kaufman99).
Around IRS5, however, the \[\] and \[\] emission can be explained by two point sources centred at IRS5A and IRS5N.
Comparing line and continuum emission
-------------------------------------
Using our deconvolution algorithm POMAC, we find that the FIR line emission mostly originates from the (sub)mm/mid-IR continuum point sources (but there is also some CO and OH line emission from residual regions: Res SWl, Res El, Res Nl, and Res Wl), whereas the FIR continuum shows a much more extended shape (see Fig. \[fig:cont\_extreg\]). After using the deconvolution algorithm we find that most of the continuum emission not associated with point sources can be found in two ridges extending in the east-west direction, positioned north and south of the YSO point sources. These ridges coincide with molecular (H$_2$CO and CH$_3$OH) emission detected in millimetre data, proposed to be heated by external irradiation from the Herbig Be star R CrA [@lindberg12].
![The dashed lines illustrate our definition of the four extended (residual) regions Res SWl, Res El, Res Nl, and Res Wl in IRS7 CO and OH. The contour map shows the CO $J=19\rightarrow18$ residual map. The red dots show the PACS spaxel centres. Contours are $3\sigma = 1.55 \times 10^{-17}$ W m$^{-2}$.[]{data-label="fig:co_extreg"}](ext_regs.png){width="1.0\linewidth"}
The northern ridge (Res Nc and Res Nl) coincides not only with the northern H$_2$CO ridge observed by @lindberg12, but also to some extent with HCO$^+$ $J=3\rightarrow2$ emission [@groppi07] and X-ray emission [@forbrich07]. Another possible explanation for the physical conditions found in the region could thus be that the gas is dominated by X-ray irradiation.
Analysis
========
Analysis of the spectral energy distributions {#sec:sed_analysis}
---------------------------------------------
The spectral energy distributions (SEDs) of the low-mass point sources, can be found in Fig. \[fig:seds\], where continuum data points from *Spitzer* and SCUBA measurements as well as the deconvolved *Herschel*/PACS spectra have been included. Bolometric luminosities are calculated from integration of a first degree spline fit to the SED data points (including line-free points across the *Herschel* spectrum) and bolometric temperatures are calculated from the mean frequencies of these spline fits [see @myers98]. The results are shown in Table \[tab:sed\]. For the sources with strong submm emission, the SCUBA data have been linearly extrapolated to allow for integration of the SEDs up to 1.3 mm [cf. @jorgensen09]. The SED of R CrA is treated in Sect. \[sec:rcra\_spect\].
The luminosities of the observed low-mass sources (IRS7A+SMM 1C, IRS7B, IRS5A, and IRS5N) are all in the order $1~L_{\odot}$–$10~L_{\odot}$. All these sources fulfil the $L_{\mathrm{bol}}/L_{\mathrm{submm}} < 200$ criterion for Class 0 sources [@andre93]. On the other hand, the low-mass sources all have bolometric temperatures that fall in the Class I range, except IRS5N, which is a Class 0 source. However, as discussed previously, IRS7A+SMM 1C is a binary unresolved by *Herschel*, where the components have very different mid-IR and submm spectral energy distributions. It is thus likely that the bolometric temperature of SMM 1C is lower, and that of IRS7A is higher.
Differences between infrared and submillimetre continuum emission are also found between IRS5A and IRS5N in the IRS5 field. It has been suggested that IRS5A, being a binary ($\sim 100$ AU), has all the dust in the disc cleared away [@jensen96; @peterson11], explaining why it is not detected by SMA mm observations. According to @peterson11, both IRS5A and IRS5N are Class I sources or younger, with IRS5N having a steeper mid-IR spectral slope $\alpha$ than IRS5A. The observations of IRS5 can be used as a comparison for the IRS7 sources, since they should be less affected by the irradiation from R CrA. IRS5A has a mid-IR luminosity a few times higher than that of IRS7A and IRS7B; however, it shows only moderate line emission in the FIR (*Herschel*) data.
[l l l l]{} Source & $T_{\mathrm{bol}}$ & $L_{\mathrm{bol}}$ & $L_{\mathrm{bol}}/L_{\mathrm{submm}}$\
& \[K\] & $[L_{\odot}]$\
IRS7A+SMM 1C & 79 & 9.1 & 99\
IRS7B & 89 & 4.6 & 48\
R CrA & 889 & 53.4 & ...\
IRS5A & 209 & 1.7 & 135\
IRS5N & 63 & 0.7 & 55\
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{sed_irs7a_both.png} &
\includegraphics[width=0.48\linewidth]{sed_irs7b.png} \\
\includegraphics[width=0.48\linewidth]{sed_irs5a.png} &
\includegraphics[width=0.48\linewidth]{sed_irs5n.png} \\
\end{array}$
![Spectral energy distributions of the continuum residual regions Res Nc and Res Sc within the PACS band. The shading shows the 20% calibration uncertainty of the *Herschel*/PACS spectrometer.[]{data-label="fig:sed_res"}](sed_res.png){width="1.0\linewidth"}
The spectral energy distributions of the residual structures Res Nc and Res Sc within the PACS band (Fig. \[fig:sed\_res\]) are found to peak around the same wavelength as typical embedded YSOs do (such as the Class 0 source IRS5N). The SED peak around 100[ m]{} corresponds to a black-body dust temperature of $\sim40$ K.
### Spectral classification and SED of R CrA {#sec:rcra_spect}
The WHT optical spectrum of R CrA is consistent with a photosphere of a star with spectral type B3–A0, by comparison to the @pickles98 compilation of photospheric templates. The higher Balmer lines are seen in absorption and are used for the spectral comparison. The large range in acceptable spectral types is caused by the possibility of emission and absorption affecting the Balmer line equivalent widths. Strong emission is detected in H$\alpha$ and H$\beta$. P-Cygni and inverse P-Cygni absorption are detected in some lines by @brown13. Both emission and absorption may affect the equivalent widths in the photospheric lines. A high resolution spectrum would be required to improve the spectral type.
This spectral type is consistent with the most reliable literature spectral types of B5 [@gray06] and B8 [@bibo92], which were also obtained using blue spectra. Other spectral types range from A0–F5 [e.g. @joy45; @greenstein47; @hillenbrand92; @vieira03] but are typically based on red spectra, which are much less sensitive to the spectral type of hot stars. The spectrum cannot be well fit with the median interstellar extinction law, using a total-to-selective extinction ratio $R_{\mathrm{V}}=3.1$. For the @weingartner01 extinction law with $R_{\mathrm{V}}=5.5$, the $A_{\mathrm{V}}=4.5$ for an A0 spectral type and 5.3 for a B3 spectral type. A higher $A_{\mathrm{V}}$ may be obtained with a higher $R_{\mathrm{V}}$ [@manoj06], however such a high $R_{\mathrm{V}}$ is not necessary to explain the shape of the optical spectrum.
The V magnitude at the time of our observation was $\sim 13.2$ mag., as measured in our spectrum. The parameters B6 spectral type, $A_{\mathrm{V}}=5.0$, and $d=130$ pc lead to a luminosity of $22~L_{\odot}$ – much smaller than that inferred from the total SED and much smaller than that expected for a young B star. The V-band magnitude is variable by $\sim 3$ magnitudes [@bibo92]. At its brightest, the star could be $350~L_{\odot}$, assuming no change in the measured extinction. Alternatively, the measured luminosity may be much lower than the real luminosity if the star is seen edge-on, as found in the Robitaille models (see below).
The SED of R CrA including SAAO, 2MASS, ISO SWS, and *Herschel*/PACS data points, is shown in Fig. \[fig:robitaille\] (blue data points and spectra). R CrA is found to have a bolometric temperature of $\sim900$ K, making it a Class II YSO; and a luminosity of $53~L_{\odot}$, significantly lower than the previous value $99~L_{\odot}$–$166~L_{\odot}$ [@bibo92 the lower value is from integration of the SED and the higher value is from a model of the extinction-free SED]. This discrepancy largely owes to the fact that @bibo92 used KAO data for the FIR data points, and the large KAO beam included most of the R CrA cloud. As a result, the KAO 100[ m]{} flux is 7 times higher than our deconvolved value for R CrA measured with PACS. Another contribution to the large spread in the spectral classification data in the literature (F5 to B5) could be variability of the source [see e.g. @herbst99].
We here attempt to constrain the physical properties of R CrA by the use of another method: using a database of SED models of YSOs [@robitaille06] and an online[^3] fitting tool [@robitaille07], we find that the observed SED can best be explained by a source with stellar mass $M\approx6~M_{\odot}$, a stellar temperature corresponding to a B3 star, a total stellar luminosity $L\approx900~L_{\odot}$, a nearly edge-on disc (inclination $80\degr$), and an extinction of $A_{\mathrm{V}} \approx 3.3$. The fit is found in Fig. \[fig:robitaille\], together with the fifth and tenth best fits from this model. The ten best fits all correspond to sources with $L_{\mathrm{bol}}\gtrsim480~L_{\odot}$ and nearly edge-on discs. We also investigate the model SED of a star with the same properties as the best fit, but with a face-on disc. It is found to have flux densities more than an order of magnitude higher in the UV/optical and a few times higher in the infrared/submm than the edge-on counterpart. Thus, assuming that R CrA behaves like this model star, it will heat some parts of the surrounding regions much more efficiently than other parts, perhaps giving rise to the ridge-like structures of heated gas and dust. Differences in the density distribution could also contribute to the uneven temperatures in the region. Since the SED database is not exhaustive, and due to the large number of free parameters, the use of this model and the resulting interpretation could be unreliable. @lindberg12 estimate a minimum luminosity for R CrA of $100L_{\odot}$ in order to heat the molecular gas to the measured temperatures.
![The R CrA SED (blue) with optical (SAAO) and 2MASS data points, and ISO SWS and *Herschel*/PACS spectra. Note that this SED has a different $x$-axis than the SEDs in Fig. \[fig:seds\]. Overplotted are the best, fifth best, and tenth best fits from the Robitaille model (red solid, dashed, and dotted). The best fit corresponds to Robitaille model ID 3011150, with a disc observed at an inclination of $\sim81\degr$, an $A_{\mathrm{V}} = 3.27$, the distance 130 pc, and apertures similar to those of the instruments used for the actual observations. The sawtooth pattern at long wavelengths originates from the model being not very accurate at these wavelengths. The ten best fits all correspond to sources with luminosities of at least $480~L_{\odot}$ and high inclinations ($>80\degr$). For the PACS fit, the continuum flux densities at 70[ m]{}, 100[ m]{}, and 160[ m]{} have been used. []{data-label="fig:robitaille"}](robitaille.png){width="1.0\linewidth"}
Rotational diagram analysis {#sec:rotdiag_analysis}
---------------------------
If the line emission is optically thin, the flux of a spectral line can be converted into a population of molecules in the upper state of the rotational transition it represents. Rotational temperatures and the total number of emitting molecules can be estimated by fitting a line to a plot of upper-state population versus upper-state energy [@goldsmith99; @green13]. To evaluate the line flux of each spectral line for each YSO, one could either use the line flux in the spaxel closest to the YSO (with a wavelength-dependent correction factor[^4] applied), or the POMAC method described in Sect. \[sec:psfcorr\] and Appendix \[app:pomac\]. When studying an isolated point source, the difference in the results of these two methods should be small, at least if the amount of extended emission is reasonably low. However, in a field with several YSOs (like IRS7), the emission from the YSOs would spill over into each other’s spaxels, so that a strong emitter could influence the measured flux in a weaker nearby source. This spill-over contribution is minimised when the POMAC method is used. In this section, all rotational diagrams are calculated with fluxes estimated from the POMAC algorithm.
### CO – point-source emission {#sec:co_point_rot}
In the non-deconvolved YSO spaxels, up to 28 CO lines are detected, from the $J=13\rightarrow12$ line at 200[ m]{} to the $J=40\rightarrow39$ line at 66[ m]{}. However, some CO lines, including the $J=13\rightarrow12$ line, lie in the leakage spectral region [@green13], and others are blended with other spectral lines (see Table \[tab:blended\]). These lines will not be used in the rotational diagrams. We also detect five $^{13}$CO lines ($J=14\rightarrow13$ to $J=21\rightarrow20$; three of the eight lines in this range are blended with stronger spectral lines).
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{irs7a_both_co_ex.png} &
\includegraphics[width=0.48\linewidth]{irs7b_co_ex.png} \\
\includegraphics[width=0.48\linewidth]{rcra_co_ex.png} &
\includegraphics[width=0.48\linewidth]{irs5a_co_ex.png} \\
\includegraphics[width=0.48\linewidth]{irs5n_co_ex.png} \\
\end{array}$
The rotational temperature found by this method will correspond to the kinetic temperature if the cloud is homogeneous and in LTE, and the spectral lines are optically thin. However, in the case of CO, two rotational temperatures fit the data much better than does a single rotational temperature (see [@green13] for a justification of the use of a two-component fit; and also [@manoj13]; [@karska13]; [@dionatos13]). In the case of CrA, the warm component has a temperature of approximately 300 K for all point sources, whereas the hot component has a larger spread around 650–1400 K (mainly due to a lower S/N), which both are in the same order as in many other embedded YSOs [see e.g. @green13; @manoj13; @herczeg12].
In Table \[tab:rottemp\_co\], the calculated CO rotational temperatures and total number of molecules are given along with the same properties of other similar sources in the literature. The errors on the temperatures and numbers of molecules are calculated assuming a systematic error of 20% on the line fluxes – even though the line fluxes are from the same spectrum, the systematic errors are assumed to be independent. The CO rotational diagrams are found in Fig. \[fig:co\_ex\].
The CO rotational temperatures towards IRS5A established from the POMAC fluxes are found to be within errors of those derived from just using the central spaxel flux and the PSF correction factor, as expected for this relatively isolated point source. IRS5N has five times as faint CO emission as IRS5A, but has a similar CO rotational temperature in the warm component. No hot component is detected in IRS5N (if present at a ratio relative to the warm component seen in other sources, it would be below the detection limit).
[l l l l l]{} YSO & $T_{\mathrm{warm}}$ & $\mathcal{N}_{\mathrm{warm}}$ & $T_{\mathrm{hot}}$ & $\mathcal{N}_{\mathrm{hot}}$\
& \[K\] & \[$10^{48}$\] & \[K\] & \[$10^{48}$\]\
IRS7A+SMM 1C & $294\pm16$ & $33.0\pm3.0$ & $\phantom{0}682\pm\phantom{0}34$ & $2.7\phantom{0}\pm0.3$\
IRS7B & $273\pm14$ & $12.0\pm1.1$ & $\phantom{0}710\pm\phantom{0}54$ & $0.51\pm0.07$\
R CrA & $287\pm16$ & $\phantom{0}5.4\pm0.5$ & $\phantom{0}992\pm\phantom{0}91$ & $0.27\pm0.03$\
IRS5A & $293\pm17$ & $\phantom{0}3.1\pm0.3$ & $1417\pm780$ & $0.12\pm0.04$\
IRS5N & $283\pm24$ & $\phantom{0}0.7\pm0.1$ & ... & ...\
Res SWl & $285^{+15}_{-33}$ & $\phantom{0}5.0\pm0.5$ & $\phantom{0}653^{+68}_{-69}$ & $0.35\pm0.06$\
Res El & $287^{+16}_{-33}$ & $\phantom{0}1.7\pm0.2$ & $1015\pm198$ & $0.09\pm0.02$\
Res Nl & $281^{+15}_{-32}$ & $\phantom{0}3.3\pm0.3$ & $\phantom{0}898\pm334$ & $0.11\pm0.06$\
Res Wl & $253^{+12}_{-26}$ & $13.0\pm1.2$ & $\phantom{0}751^{+93}_{-94}$ & $0.40\pm0.07$\
CrA point-source average & $286\pm\phantom{0}3$ & $10.8\pm5.2$ & $\phantom{0}950\pm148$ & $0.90\pm0.52$\
CrA extended average & $277\pm\phantom{0}7$ & $\phantom{0}5.8\pm2.2$ & $\phantom{0}829\pm\phantom{0}69$ & $0.24\pm0.07$\
NGC 1333 IRAS 4B & 280 & 40 & 880 & 3\
Serpens SMM1 & $337\pm40$ & ... & $\phantom{0}622\pm\phantom{0}30$ & ...\
Serpens SMM3/4 average & $260\pm10$ & $49\phantom{.0}\pm6$ & $\phantom{0}800\pm\phantom{0}60$ & $2.2\phantom{0}\pm0.6$\
DIGIT average & $355\pm\phantom{0}3$ & $\phantom{0}5.2\pm0.4$ & $\phantom{0}814\pm\phantom{0}29$ & $1.63\pm0.20$\
HOPS average & $288\pm14$ & ... & $\phantom{0}735\pm\phantom{0}37$ & ...\
We also perform a rotational diagram fit for the $^{13}$CO data in IRS7A+SMM 1C (see Fig. \[fig:13co\_ex\]). From the five detected lines, we get a rotational temperature of $266\pm35$ K, which is consistent with the $^{12}$CO temperature $294\pm16$ K. The number of molecules is found to be $(8.90\pm1.58)\times10^{47}$, but if the temperature is constrained to the $^{12}$CO value the number of molecules becomes slightly lower, $7.47\times10^{47}$. With the $^{13}$CO rotational temperature constrained to the $^{12}$CO value, we find the $^{12}$CO/$^{13}$CO abundance ratio to be $44\pm9$, corresponding to an optical depth of $0.56\pm0.24$ assuming the local ISM $^{12}$C/$^{13}$C value of $77\pm7$ [@wilson94]. We adopt a CO line width of $\sim7.5$ km s$^{-1}$ from *Herschel* HIFI observations of the $^{12}$CO $J=16\rightarrow15$ line (Kristensen et al. in prep.). This value is comparable to the $^{12}$CO $J=7\rightarrow6$ quiescent component line width [8 km s$^{-1}$; @vankempen09a]. We use RADEX [@vandertak07], a non-LTE radiative transfer code for calculations of line strengths in isothermal homogeneous interstellar clouds, to find that this marginally optically thick result is consistent with a $^{12}$CO column density of $\sim10^{18}$ cm$^{-2}$, corresponding to a size of the emitting region in the order of a few arcseconds ($\sim500$ AU).
We have also investigated whether the data can be fitted with a single kinetic temperature component of much higher temperature and lower density, as suggested by @neufeld12. RADEX calculations assuming the same line width as above (7.5 km s$^{-1}$) and the column density derived above ($\sim10^{18}$ cm$^{-2}$) give a single kinetic temperature component with a best fit at $\sim5\,000$ K ($>2500$ K with a $1\sigma$ certainty) for an H$_2$ density $(2.5\pm0.5)\times10^4$ cm$^{-3}$ for IRS7A+SMM 1C. For the best solution, the lowest-$J$ ($J\lesssim17$) lines are marginally optically thick ($\tau \sim 1$), while the lines with $J\gtrsim 22$ have optical depths $\tau \ll 1$. The reduced $\chi^2$ value for the best fit is 1.1.
![$^{13}$CO rotational diagram of IRS7A+SMM 1C from the deconvolved PACS data. $\mathcal{N}$ is the total number of $^{13}$CO molecules in each source given the rotational fit.[]{data-label="fig:13co_ex"}](irs7a_both_13co_ex.png "fig:"){width="1.0\linewidth"}\
### CO – extended emission {#sec:co_extended_ex}
To investigate if the extended CO emission in the IRS7 region shows any variation in temperature, it was grouped into four areas: southwest, east, northwest, and west of the YSOs (see Fig. \[fig:co\_extreg\]). We call the four residual emission regions Res SWl, Res El, Res Nl, and Res Wl (l for spectral line). Res Nl overlaps with the central part of the continuum residual region Res Nc, and Res SWl overlaps with the western part of Res Sc (see Fig. \[fig:cont\_extreg\]). By computing the residual spectral line emission in these regions we can produce rotational diagrams for this extended emission.
The resulting rotational diagrams are found in Fig. \[fig:co\_ex\_res\]; as for the point-source emission, a warm and a hot component with rotational temperatures around 300 K and 900 K, respectively, are found. The exact properties of the fits are included in Table \[tab:rottemp\_co\]. The rotational temperatures of the residual emission are within the errors of the point-source rotational temperatures.
On one hand, the extended emission is not deconvolved, and one can thus argue that a PSF correction factor needs to be applied to this data. On the other hand, since these fluxes are sums of emission from several (2–6) spaxels (see Fig. \[fig:co\_extreg\]), they are clearly in less need of PSF correction than point-source data. However, to take this issue into account, we calculate the rotational temperatures of the extended emission also using the PACS standard PSF correction factor from the PACS manual, and use the result when establishing the lower boundary of the error estimate of these temperatures.
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{res1_co_ex.png} &
\includegraphics[width=0.48\linewidth]{res2_co_ex.png} \\
\includegraphics[width=0.48\linewidth]{res3_co_ex.png} &
\includegraphics[width=0.48\linewidth]{res4_co_ex.png} \\
\end{array}$
### OH and H$_2$O
The excitation conditions of the related species, OH and H$_2$O, are treated together in this section.
For the unresolved OH doublets, the sum of both lines is measured and divided by 2. In the rotational diagram fits we exclude the same lines as @wampfler13. These excluded lines are the 119[ m]{} doublet (which in similar sources is found to be an optically thick transition), the 84.4[ m]{} line (CO blend), and the 98[ m]{} and 55[ m]{} doublets (lines in leakage regions). They are plotted with open circles in the rotational diagrams.
The OH rotational diagrams of the point sources are found in Fig. \[fig:oh\_ex\], and those of the extended emission in Fig. \[fig:oh\_ex\_res\]. The derived parameters are listed in Table \[tab:rottemp\_oh\], where they are also compared to some other embedded sources in the literature. The CrA sources have fairly uniform excitation temperatures, and do not differ significantly from the other sources in the literature. We also produce OH rotational diagrams of the four extended regions (Fig. \[fig:oh\_ex\_res\]). The OH temperatures in the extended emission are similar to the point sources within $3\sigma$.
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{irs7a_both_oh_ex.png} &
\includegraphics[width=0.48\linewidth]{irs7b_oh_ex.png} \\
\includegraphics[width=0.48\linewidth]{rcra_oh_ex.png} &
\includegraphics[width=0.48\linewidth]{irs5a_oh_ex.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{res1_oh_ex.png} &
\includegraphics[width=0.48\linewidth]{res2_oh_ex.png} \\
\includegraphics[width=0.48\linewidth]{res3_oh_ex.png} &
\includegraphics[width=0.48\linewidth]{res4_oh_ex.png} \\
\end{array}$
[l l l]{} YSO & $T$ & $\mathcal{N}$\
& \[K\] & \[$10^{45}$\]\
IRS7A+SMM 1C & $83\pm3$ & $132\phantom{.0}\pm12$\
IRS7B & $89\pm4$ & $\phantom{0}16.7\pm\phantom{0}1.5$\
R CrA & $99\pm5$ & $\phantom{00}7.1\pm\phantom{0}0.7$\
IRS5A & $80\pm3$ & $\phantom{0}17.2\pm\phantom{0}1.5$\
Res SW-l & $76\pm4$ & $\phantom{00}9.5\pm\phantom{0}1.1$\
Res E-l & $68\pm4$ & $\phantom{00}7.2\pm\phantom{0}1.2$\
Res N-l & $66\pm2$ & $\phantom{0}19.1\pm\phantom{0}2.1$\
Res W-l & $72\pm2$ & $\phantom{0}26.1\pm\phantom{0}2.5$\
CrA point-source average & $88\pm4$ & $\phantom{0}43.3\pm25.7$\
CrA extended average & $71\pm2$ & $\phantom{0}15.5\pm\phantom{0}3.8$\
NGC 1333 IRAS 4B & $60$ & 130\
Serpens SMM1 & $72\pm8$ & ...\
Serpens SMM3/4 average & $88\pm2$ & $\phantom{0}26\phantom{.0}\pm\phantom{0}3$\
DIGIT & $83\pm3$ & $\phantom{0}24\phantom{.0}\pm\phantom{0}3$\
For H$_2$O, we assume an ortho-to-para ratio of 3 [@herczeg12]. This assumption is accounted for in the rotational diagrams (Figs. \[fig:h2o\_ex\]–\[fig:h2o\_ex\_res\]). The H$_2$O rotational diagrams show quite large spreads, which are mainly due to subthermal excitation effects and optical depth effects on some of the lines [@herczeg12]. The apparent shift between ortho and para lines might be caused by either of these effects, or by an ortho-to-para ratio lower than 3, but without more elaborate radiative transfer models it is impossible to distinguish between these scenarios. The derived rotational temperatures and total numbers of molecules are shown in Table \[tab:rottemp\_h2o\]. As in the case of OH, non-LTE radiative transfer models can be used to resolve the optical depth effects [@herczeg12]. Another method would be to exclude spectral lines suspected to be optically thick. The H$_2$O excitation temperatures are higher than the OH temperatures. The higher H$_2$O temperature in R CrA than in the other sources is likely caused by a smaller number of detected lines, which enhances the temperature since the stronger lines are optically thick. The observed H$_2$O temperatures do not vary significantly from the DIGIT average.
As in the CO and OH cases, the H$_2$O rotational temperatures are similar in the extended emission and in the point sources.
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{irs7a_both_h2o_ex.png} &
\includegraphics[width=0.48\linewidth]{irs7b_h2o_ex.png} \\
\includegraphics[width=0.48\linewidth]{rcra_h2o_ex.png} &
\includegraphics[width=0.48\linewidth]{irs5a_h2o_ex.png} \\
\includegraphics[width=0.48\linewidth]{irs5n_h2o_ex.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{res1_h2o_ex.png} &
\includegraphics[width=0.48\linewidth]{res2_h2o_ex.png} \\
\includegraphics[width=0.48\linewidth]{res3_h2o_ex.png} &
\includegraphics[width=0.48\linewidth]{res4_h2o_ex.png} \\
\end{array}$
[l l l]{} YSO & $T$ & $\mathcal{N}$\
& \[K\] & \[$10^{45}$\]\
IRS7A+SMM 1C & $197\pm\phantom{0}4$ & $\phantom{0}10.1\pm0.3$\
IRS7B & $174\pm\phantom{0}4$ & $\phantom{00}2.3\pm0.1$\
R CrA & $235\pm\phantom{0}9$ & $\phantom{00}1.3\pm0.1$\
IRS5A & $185\pm\phantom{0}5$ & $\phantom{00}1.2\pm0.1$\
IRS5N & $195\pm20$ & $\phantom{00}0.5\pm0.1$\
Res SWl & $181\pm\phantom{0}5$ & $\phantom{00}2.0\pm0.1$\
Res El & $201\pm\phantom{0}4$ & $\phantom{00}8.6\pm0.3$\
Res Nl & $174\pm\phantom{0}4$ & $\phantom{00}2.3\pm0.1$\
Res Wl & $235\pm\phantom{0}9$ & $\phantom{00}1.3\pm0.1$\
CrA point-source average & $197\pm\phantom{0}9$ & $\phantom{00}3.1\pm1.6$\
CrA extended average & $198\pm12$ & $\phantom{00}3.5\pm1.5$\
NGC 1333 IRAS 4B & 110/220 & 100\
Serpens SMM1 & $136\pm27$ & ...\
Serpens SMM3/4 average & $105\pm\phantom{0}6$ & $\phantom{0}25\phantom{.0}\pm3$\
DIGIT & $194\pm20$ & $\phantom{00}7.7\pm2.6$\
### Comparison of the rotational diagrams
The rotational temperatures of the three molecules studied in the FIR data (CO, OH, and H$_2$O) are all different (see Tables \[tab:rottemp\_co\]–\[tab:rottemp\_h2o\]). There are, however, relatively small spreads among the warm CO and OH temperatures, respectively. The warm CO temperature average is significantly lower in the CrA point-source sample than in the DIGIT sample [@green13], but in agreement with the HOPS (*Herschel* Orion Protostar Survey) sample [@manoj13]; whereas the hot CO is in agreement between the CrA and DIGIT samples, but higher than in the HOPS sample. The OH and H$_2$O temperatures are similar between the CrA and DIGIT samples, where the DIGIT sample average for OH has been recalculated for consistency using only the OH lines in our fits, and thus does not agree with the average value given by @green13.
Comparing the number of molecules per source between the CrA and DIGIT samples [@green13] shows that the average number of CO molecules is larger in the CrA sample, the average number of OH molecules is larger (but within errors) in the CrA sample, and the average number of H$_2$O molecules is lower in the CrA sample. Comparing these results could however be biased, since it was not possible to construct OH and H$_2$O rotational diagrams for all sources in the DIGIT sample where CO rotational diagrams could be made. Instead, calculations of line ratios in the whole DIGIT sample will be a better tracer of any difference in abundance ratios (see Sect. \[sec:lineratios\]).
Discussion {#sec:discussion}
==========
Survey of source properties
---------------------------
In Table \[tab:yso\_prop\], some important properties of the studied point sources and extended line emission regions are tabulated along with the DIGIT sample averages. The properties of the sources in the CrA sample are found to be fairly typical for low-mass embedded protostars. Since IRS7A and SMM 1C cannot be separated in the PACS data it is difficult to draw any conclusions about the classes of the separate sources. However, their different appearance in continuum data of other bands (IRS7A is detected in mid-IR but not mm; SMM 1C is detected in mm but not mid-IR) and their combined $T_{\mathrm{bol}} = 80$ K point towards SMM 1C being a Class 0 source. IRS5N is definitely a Class 0 object, but the other low-mass sources in the sample cannot consistently be assigned to Class 0 or Class I.
The SEDs of the extended continuum emission (in Res Nc and Res Sc; see Fig. \[fig:sed\_res\]) are similar to that of the Class 0 source IRS5N, which indicates that this gas has similar temperatures to those of very young protostellar cores. The dust black-body temperature of the two continuum ridges is found to be 40–50 K, which is consistent with the H$_2$CO temperature, estimated to be 40–60 K in @lindberg12. @lindberg12 showed that these temperatures cannot be caused by radiation from the low-mass protostars, but can instead be explained by external irradiation from R CrA.
[l l l l l l l l l c c l]{} YSO & & & & & & Mid-IR/FIR/mm & FIR/mm & Class\
& $T_{\mathrm{rot}}$ & $\mathcal{N}$ & $T_{\mathrm{rot}}$ & $\mathcal{N}$ & $T_{\mathrm{rot}}$ & $\mathcal{N}$ & $T_{\mathrm{bol}}$ & $L_{\mathrm{bol}}$ & continuum & lines\
& \[K\] & \[10$^{48}$\] & \[K\] & \[10$^{45}$\] & \[K\] & \[10$^{45}$\] & \[K\] & \[$L_{\sun}$\] & detected & detected\
IRS7A & 294/682 & 35.7 & 83 & 132 & 197 & 10.1 & 79 & 9.1 & yes/yes/no & yes/yes & 0/I\
SMM 1C & – & – & – & – & – & – & – & – & no/yes/yes & yes/yes & 0/I\
IRS7B & 273/710 & 12.5 & 89 & 17 & 174 & 2.3 & 89 & 4.6 & yes/yes/yes & yes/yes & 0/I\
R CrA & 287/992 & 5.7 & 99 & 7.1 & 235 & 1.3 & 889 & 53 & yes/yes/faint & yes/yes & II/III\
IRS5A & 293/1417 & 3.2 & 80 & 17 & 185 & 1.2 & 209 & 1.7 & yes/yes/yes & yes/... & 0/I\
IRS5N & 283/... & 0.7 & ... & ... & 195 & 0.5 & 63 & 0.7 & no/yes/yes & no/... & 0\
Res SW-l & 285/653 & 5.0 & 76 & 9.5 & 181 & 2.0 & ... & ... & ... & ... & ...\
Res E-l & 287/1015 & 1.7 & 68 & 7.2 & 201 & 8.6 & ... & ... & ... & ... & ...\
Res N-l & 281/898 & 3.3 & 66 & 19 & 174 & 2.3 & ... & ... & ... & ... & ...\
Res W-l & 253/751 & 13.0 & 72 & 26 & 235 & 1.3 & ... & ... & ... & ... & ...\
DIGIT & 355/814 & 7.0 & 83 & 24 & 194 & 7.7 & 167 & 6 & ... & ... & ...\
Excitation conditions
---------------------
As pointed out earlier, the CO excitation temperatures in the point sources are in good agreement with those found in larger samples of low-mass embedded objects [@green13; @karska13; @manoj13]. One possible explanation for the excitation conditions being similar towards the externally irradiated protostars and in sources not subject to external irradiation is that the irradiation from R CrA is not substantial enough to dramatically change the properties of the high-temperature gas. The excitation conditions of the extended molecular line emission found across the IRS7 field are consistent with those of the compact objects, and do not change significantly across the IRS7 field. The appearance of such extended emission is, however, unusual. The excitation diagrams could be explained by a single-temperature non-LTE fit [@neufeld12], assuming a low gas density ($n\sim10^4$ cm$^{-3}$), and that the gas is collisionally excited to high temperatures ($T\sim5\,000$ K). However, the observation of strong and extended H$_2$CO and CH$_3$OH emission in the field makes this low-density scenario unlikely. Furthermore, H$_2$ densities $\lesssim10^6$ cm$^{-3}$ do not fit with the H$_2$CO optical depth derived by @lindberg12. Another possibility is that the molecular gas towards the point sources and the extended gas are excited by different excitation mechanisms.
Comparing FIR and mm spectral line data {#sec:fir_mm_compare}
---------------------------------------
Strong CO $J=6\rightarrow5$ and $J=7\rightarrow6$ emission found on an east-west line centred at IRS7A @vankempen09a are consistent with the residual CO regions Res El and Res Wl, although the FIR data is more dominated by the point-source emission. The major difference between the morphology of the H$_2$CO and CH$_3$OH mm line data [@lindberg12] and the FIR (*Herschel*) line data is that most of the FIR line emission is well-aligned with the mm and mid-IR continuum point sources, while the mm lines appear in more extended structures, which are not centred on these point sources. However, the residual PACS continuum emission after deconvolution of the point-source emission (corresponding to extended dust continuum emission) shows shapes very similar to the high-temperature H$_2$CO ridges observed in the mm (see Fig. \[fig:cont\_extreg\]). The H$_2$CO rotational temperatures measured in the SMA/APEX mm data range from 30 to 100 K, but non-LTE modelling shows that the physical temperatures probably are in the order of 40–60 K [@lindberg12]. The PACS SEDs of these ridges show black-body temperatures ($\sim40$–$50$ K; see Fig. \[fig:sed\_res\]) consistent with the H$_2$CO temperatures measured. The H$_2$CO ridges are observed on relatively large scales ($\sim8\,000$ AU), and are not associated with the point sources.
The POMAC algorithm shows that most of the FIR molecular line emission originates from the mid-IR/(sub)mm point sources IRS7A, SMM 1C, IRS7B, R CrA, and IRS5A. This emission is similar in excitation to what is found towards sources that are not subject to external irradation. However, through the deconvolution we also find extended line emission in the IRS7 region. Interestingly, the CO, OH, and H$_2$O excitation conditions of the extended emission resemble those near the protostars (see Tables \[tab:rottemp\_co\]–\[tab:rottemp\_h2o\]), and suggest high temperatures also on these relatively large scales ($>1\,000$ AU from the protostars). Such extended line emission, in particular the hot CO and OH emission, is unusual around low-mass embedded objects [see e.g. @vankempen10; @green13; @karska13].
Still, the H$_2$CO emission detected by SMA/APEX is even more extended, and not associated with the point sources. Regardless of the lower spatial resolution of the PACS data, it is certain that the FIR and mm line emission have different origins.
Water and oxygen chemistry {#sec:lineratios}
--------------------------
Assuming that the region around R CrA exhibits PDR-like conditions, the OH abundance should be enhanced with respect to the H$_2$O abundance [see e.g. @walsh13]. We thus want to establish whether the OH/H$_2$O ratio is enhanced in these sources compared to other sources in similar stages of evolution. A first-order comparison can be made by analysing the ratios of certain OH and H$_2$O spectral lines in the DIGIT sample of embedded protostars [@green13]. So far, no systematic study of OH and H$_2$O excitation diagrams and abundances in low-mass embedded objects is available. To avoid biases introduced by different amounts of detected lines and different methods of extracting the abundances, we instead compare the ratios of individual spectral line luminosities.
We need to compare OH and H$_2$O lines suspected to be optically thin, that are detected in many sources, that have small PSFs (short wavelengths) or at least similar PSFs (similar wavelengths), and that have similar upper-level energies to remove any bias. Thus, in the first four panels of Fig. \[fig:lineratios\], we plot four different line ratios of three OH lines and three H$_2$O lines. The wavelengths differ by less than 20[ m]{} and the upper level energies by less than 80 K for each of the ratios. Details on the transitions are found in the figure. We find that all the CrA sources have an OH/H$_2$O line ratio higher than most other DIGIT embedded objects, which is indicative of PDR activity [@hollenbach97]. Also the extended regions have relatively high OH/H$_2$O ratios. We have also compared the OH (c.f. Fig. 9 in [@wampfler13]), H$_2$O, and CO ($J=16\rightarrow15$; cf. Fig. 22 in [@green13]) line luminosities to the bolometric luminosities of the DIGIT and WISH embedded sources, and found that the sources in CrA fall within the scatter around the linear correlation for all three lines, although on the higher end in the OH and CO case. We find that the enhanced OH/H$_2$O ratio mainly is due to an increase of the OH flux rather than a decrease of the H$_2$O flux.
From dissociation of OH, O should be a major destruction product, so we would also expect an enhancement of the \[\] line strengths compared to H$_2$O and OH in a PDR [@hollenbach97]. In the third row of diagrams in Fig. \[fig:lineratios\], we compare the line ratios of the \[\] 63.2[ m]{} line to one H$_2$O line and one OH line with similar wavelengths and upper-level energies. We see a very strong enhancement of \[\], in particular in the extended regions, supporting the hypothesis of a PDR induced by external irradiation.
In the last row of Fig. \[fig:lineratios\], the \[\] 63.2[ m]{} line flux is compared to two CO lines. In the comparison of \[\] and CO it is more difficult to accommodate our ambition of using similar wavelengths and upper-level energies – we could have used the \[\] 145[ m]{} line, but this was not desirable for S/N reasons. We find that the \[\] flux is enhanced in the CrA sources also with respect to the CO flux.
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{ratio_ohh2o_0.png} &
\includegraphics[width=0.48\linewidth]{ratio_ohh2o_1.png} \\
\includegraphics[width=0.48\linewidth]{ratio_ohh2o_2.png} &
\includegraphics[width=0.48\linewidth]{ratio_ohh2o_3.png} \\
\includegraphics[width=0.48\linewidth]{ratio_oih2o_0.png} &
\includegraphics[width=0.48\linewidth]{ratio_oioh_0.png} \\
\includegraphics[width=0.48\linewidth]{ratio_oico_0.png} &
\includegraphics[width=0.48\linewidth]{ratio_oico_1.png} \\
\end{array}$
The higher \[\]/OH, \[\]/H$_2$O, and OH/H$_2$O line ratios could also indicate a later stage of evolution [cf. Class II sources in @podio12], but the low bolometric temperatures and $L_{\mathrm{bol}}/L_{\mathrm{submm}}$ ratios of the sources indicate that they are Class 0/I sources, and the (in many cases) higher line ratios found in the extended emission compared to the point sources indicate that the heating is external in its origin.
Conclusions
===========
We study *Herschel*/PACS line and continuum maps of the low-mass star-forming region R CrA subject to strong irradiation from the nearby Herbig Be star R CrA. In addition, we deconvolve the maps to study the point-source and extended contributions to the emission. Our main results are the following:
1. FIR continuum emission is found not only at the (sub)mm and mid-IR continuum point sources, but also (somewhat fainter) in two ridges north and south of the IRS7 protostars. These correlate in position with H$_2$CO and CH$_3$OH mm emission, and the continuum emission peaks give temperatures (40–50 K) similar to the rotational temperature of the H$_2$CO emission [@lindberg12], both suggesting that the extended FIR continuum emission traces the dust associated with the externally irradiated material.
2. The rotational temperatures of the warm CO component ($286\pm3$ K), the hot CO component ($950\pm148$ K), OH ($88\pm4$ K), and H$_2$O ($197\pm9$ K) measured towards the continuum point sources are consistent with or lower than those found in larger samples of similar sources, suggesting that the excitation conditions of the dense gas close to the protostars are not affected by the external irradiation. A $^{13}$CO rotational diagram suggests that the mid-$J$ $^{12}$CO lines are marginally optically thick ($\tau\sim0.6$).
3. CO, OH, and H$_2$O emission not associated with any of the previously known continuum point sources is detected, and shows excitation conditions similar to the gas near the protostars ($277\pm7$ K for the warm CO component, $829\pm69$ K for the hot CO component, $71\pm2$ K for OH, and $198\pm12$ for H$_2$O). The warm gas thus exists on much larger scales than can be explained by heating from the low-mass YSOs. One possible explanation is that this emission traces radiatively excited low-density gas, but detections of high density tracers such as H$_2$CO and CH$_3$OH challenge this hypothesis. The extent of the FIR molecular emission is larger than previously seen in any low-mass protostellar sources.
4. When comparing the IRS7 and IRS5 fields – the former with a smaller angular separation from the irradiating Herbig Be star R CrA than the latter – we find that the two fields have similar average rotational temperatures of the warm CO component (285 K and 288 K, respectively), OH (90 K and 80 K, respectively), and H$_2$O (200 K and 190 K, respectively). However, more extended emission (both line and continuum) is seen in the IRS7 field than in the IRS5 field. The higher level of irradiation from R CrA in IRS7 than in IRS5 does thus not significantly affect the rotational temperatures, but the possible link between extended emission and external irradiation needs to be investigated further.
5. The OH/H$_2$O, \[\]/H$_2$O, \[\]/OH, and \[\]/CO line ratios are enhanced in the CrA point sources and extended gas compared to other embedded objects, which is similar to what has previously been seen in PDRs [@hollenbach97]. Typically, these line ratios are enhanced by a factor of 1.5–4.0 in the CrA sources.
To further study the origin of the excitation conditions in protostellar cores and their surroundings, we propose similar investigations of any extended emission in PACS observations of embedded objects, including other sources in regions of potential strong external irradiation. This could for instance include the Orion sources discussed by @manoj13, and the isolated sources in the DIGIT sample [@green13].
This research was supported by a grant from the Instrument Center for Danish Astrophysics (IDA) and a Lundbeck Foundation Group Leader Fellowship to JKJ. Research at Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen’s programme of excellence. Support for this work, part of the *Herschel* Open Time Key Project Program, was provided by NASA through an award issued by the Jet Propulsion Laboratory, California Institute of Technology. The *William Herschel Telescope* is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof[í]{}sica de Canarias. We thank Nienke van der Marel for helping to obtain the optical spectrum of R CrA. The authors would also like to thank the anonymous referee, Lars Kristensen, and Tim van Kempen for their useful suggestions and comments which improved the quality of the paper.
------------------------------ -------------------------------------------------------- ------------ ---------------------------------- --------------------------------- --------------------------------- --------------------------------- -------------------------------- --
Species Transition Wavelength IRS7A+SMM 1C IRS7B R CrA IRS5A IRS5N
\[m\] \[$10^{-18}$ W m$^{-2}$\] \[$10^{-18}$ W m$^{-2}$\] \[$10^{-18}$ W m$^{-2}$\] \[$10^{-18}$ W m$^{-2}$\] \[$10^{-18}$ W m$^{-2}$\]
OH $^2\Pi_{1/2}(J=9/2+\rightarrow7/2-)$ 55.891 $\phantom{00}352\pm24$ $\phantom{00}85\pm\phantom{0}9$ ... $\phantom{00}34\pm10$ ...
OH $^2\Pi_{1/2}(J=9/2-\rightarrow7/2+)$ 55.950 $\phantom{00}686\pm24$ $\phantom{00}83\pm\phantom{0}9$ $\phantom{0}164\pm17$ $\phantom{00}93\pm10$ $\phantom{0}36\pm\phantom{0}9$
p-H$_2$O $4_{31}\rightarrow3_{22}$ 56.325 $\phantom{00}347\pm29$ ... ... ... ...
o-H$_2$O $9_{09}\rightarrow8_{18}$ 56.816 $\phantom{00}460\pm30$ $\phantom{00}97\pm20$ $\phantom{0}372\pm39$ $\phantom{00}30\pm\phantom{0}9$ ...
p-H$_2$O $4_{22}\rightarrow3_{13}$ 57.637 $\phantom{00}431\pm39$ ... ... $\phantom{00}69\pm\phantom{0}9$ ...
o-H$_2$O $4_{32}\rightarrow3_{21}$ 58.699 $\phantom{00}547\pm27$ $\phantom{0}105\pm10$ ... $\phantom{00}66\pm\phantom{0}8$ ...
p-H$_2$O $7_{26}\rightarrow6_{15}$ 59.987 $\phantom{00}146\pm17$ ... $\phantom{00}67\pm12$ ... ...
$^3$P$_1\rightarrow$ $^3$P$_2$ 63.184 $14735\pm46$ $3914\pm26$ $2399\pm26$ $1961\pm19$ $809\pm16$
o-H$_2$O $8_{18}\rightarrow7_{07}$ 63.324 $\phantom{00}712\pm25$ $\phantom{00}97\pm11$ ... $\phantom{00}77\pm\phantom{0}9$ ...
p-H$_2$O $8_{08}\rightarrow7_{17}$ 63.458 $\phantom{00}354\pm31$ $\phantom{00}61\pm13$ ... ... ...
OH $^2\Pi_{3/2}(J=9/2-\rightarrow7/2+)$ 65.132 $\phantom{0}1341\pm30$ $\phantom{0}247\pm\phantom{0}9$ $\phantom{0}185\pm13$ $\phantom{0}163\pm\phantom{0}9$ ...
o-H$_2$O $6_{25}\rightarrow5_{14}$ 65.166
OH $^2\Pi_{3/2}(J=9/2+\rightarrow7/2-)$ 65.279 $\phantom{0}1197\pm28$ $\phantom{0}227\pm\phantom{0}8$ $\phantom{0}140\pm12$ $\phantom{0}118\pm\phantom{0}7$ $\phantom{0}16\pm\phantom{0}3$
CO $J=40\rightarrow39$ 65.686 $\phantom{000}88\pm21$ ... $\phantom{00}47\pm10$ ... ...
o-H$_2$O $7_{16}\rightarrow6_{25}$ 66.093 $\phantom{00}387\pm20$ $\phantom{00}71\pm\phantom{0}7$ ... $\phantom{00}34\pm\phantom{0}6$ ...
o-H$_2$O $3_{30}\rightarrow2_{21}$ 66.438 $\phantom{00}662\pm20$ $\phantom{0}110\pm14$ $\phantom{0}128\pm13$ $\phantom{00}74\pm10$ ...
p-H$_2$O $3_{31}\rightarrow2_{20}$ 67.089 $\phantom{00}401\pm24$ $\phantom{00}90\pm\phantom{0}8$ ... ... $\phantom{0}27\pm\phantom{0}9$
o-H$_2$O $3_{30}\rightarrow3_{03}$ 67.269 $\phantom{00}200\pm19$ $\phantom{00}35\pm\phantom{0}5$ ... ... ...
CO $J=39\rightarrow38$ 67.336 $\phantom{00}117\pm19$ $\phantom{00}29\pm\phantom{0}6$ $\phantom{00}34\pm\phantom{0}9$ ... ...
CO $J=38\rightarrow37$ 69.074 $\phantom{00}192\pm17$ ... $\phantom{00}86\pm\phantom{0}7$ ... ...
CO $J=37\rightarrow36$ 70.907 $\phantom{00}173\pm18$ $\phantom{00}47\pm\phantom{0}7$ ... ... ...
p-H$_2$O $5_{24}\rightarrow4_{13}$ 71.067 $\phantom{00}329\pm12$ $\phantom{00}64\pm\phantom{0}6$ $\phantom{00}40\pm\phantom{0}6$ $\phantom{00}22\pm\phantom{0}6$ ...
OH $^2\Pi_{1/2}(J=7/2-\rightarrow5/2+)$ 71.171 $\phantom{0}1268\pm23$ $\phantom{0}235\pm\phantom{0}7$ $\phantom{0}193\pm\phantom{0}9$ $\phantom{0}138\pm\phantom{0}7$ ...
OH $^2\Pi_{1/2}(J=7/2+\rightarrow5/2-)$ 71.215
p-H$_2$O $7_{17}\rightarrow6_{06}$ 71.540 $\phantom{00}271\pm13$ $\phantom{00}42\pm\phantom{0}6$ $\phantom{00}27\pm\phantom{0}7$ $\phantom{00}20\pm\phantom{0}5$ ...
o-H$_2$O $7_{07}\rightarrow6_{16}$ 71.947 $\phantom{00}624\pm13$ $\phantom{0}110\pm\phantom{0}6$ $\phantom{00}52\pm\phantom{0}7$ $\phantom{00}54\pm\phantom{0}5$ $\phantom{0}18\pm\phantom{0}5$
CO $J=36\rightarrow35$ 72.843 $\phantom{00}140\pm17$ ... ... ... ...
CO $J=35\rightarrow34$ 74.890 $\phantom{00}362\pm19$ $\phantom{00}74\pm\phantom{0}7$ ... ... ...
o-H$_2$O $7_{25}\rightarrow6_{34}$ 74.945
o-H$_2$O $3_{21}\rightarrow2_{12}$ 75.381 $\phantom{00}861\pm22$ $\phantom{0}133\pm\phantom{0}7$ $\phantom{00}71\pm\phantom{0}8$ $\phantom{00}90\pm\phantom{0}7$ ...
CO $J=34\rightarrow33$ 77.059 $\phantom{00}204\pm19$ ... ... ... ...
o-H$_2$O $4_{23}\rightarrow3_{12}$ 78.742 $\phantom{00}630\pm27$ $\phantom{0}102\pm12$ $\phantom{00}60\pm11$ $\phantom{00}95\pm\phantom{0}6$ ...
p-H$_2$O $6_{15}\rightarrow5_{24}$ 78.928 $\phantom{00}101\pm18$ $\phantom{00}53\pm\phantom{0}7$ ... $\phantom{00}30\pm\phantom{0}5$ ...
OH $^2\Pi_{1/2}(J=1/2-)\rightarrow$ $^2\Pi_{3/2}(J=3/2+)$ 79.116 $\phantom{0}1741\pm28$ $\phantom{0}250\pm14$ $\phantom{0}175\pm13$ $\phantom{0}264\pm\phantom{0}8$ ...
OH $^2\Pi_{1/2}(J=1/2+)\rightarrow$ $^2\Pi_{3/2}(J=3/2-)$ 79.179
CO $J=33\rightarrow32$ 79.360 $\phantom{00}162\pm15$ ... ... ... ...
CO $J=32\rightarrow31$ 81.806 $\phantom{00}290\pm15$ $\phantom{00}37\pm\phantom{0}6$ $\phantom{00}57\pm\phantom{0}7$ ... ...
o-H$_2$O $6_{16}\rightarrow5_{05}$ 82.031 $\phantom{00}740\pm22$ $\phantom{0}162\pm\phantom{0}9$ $\phantom{00}48\pm11$ $\phantom{00}83\pm\phantom{0}6$ ...
p-H$_2$O $6_{06}\rightarrow5_{15}$ 83.284 $\phantom{00}397\pm21$ $\phantom{00}68\pm10$ ... ... $\phantom{0}31\pm\phantom{0}5$
CO $J=31\rightarrow30$ 84.411
OH $^2\Pi_{3/2}(J=7/2+\rightarrow5/2-)$ 84.420 $\phantom{0}1474\pm18$ $\phantom{0}327\pm14$ $\phantom{0}172\pm11$ $\phantom{0}223\pm\phantom{0}7$ ...
OH $^2\Pi_{3/2}(J=7/2-\rightarrow5/2+)$ 84.597 $\phantom{0}1444\pm20$ $\phantom{0}306\pm14$ $\phantom{0}197\pm11$ $\phantom{0}250\pm\phantom{0}8$ ...
o-H$_2$O $7_{16}\rightarrow7_{07}$ 84.767 $\phantom{00}107\pm15$ ... ... $\phantom{00}53\pm\phantom{0}7$ ...
CO $J=30\rightarrow29$ 87.190 $\phantom{00}421\pm17$ $\phantom{00}73\pm10$ ... $\phantom{00}77\pm\phantom{0}6$ ...
p-H$_2$O $3_{22}\rightarrow2_{11}$ 89.988 $\phantom{00}597\pm18$ $\phantom{00}89\pm11$ ... $\phantom{00}85\pm\phantom{0}5$ ...
CO $J=29\rightarrow28$ 90.163 $\phantom{00}413\pm14$ $\phantom{00}46\pm\phantom{0}8$ $\phantom{00}34\pm\phantom{0}8$ $\phantom{00}37\pm\phantom{0}5$ ...
CO $J=28\rightarrow27$ 93.349 $\phantom{00}587\pm15$ $\phantom{0}106\pm\phantom{0}9$ $\phantom{00}69\pm10$ $\phantom{00}56\pm\phantom{0}6$ ...
p-H$_2$O $5_{42}\rightarrow5_{33}$ 94.210 $\phantom{000}87\pm13$ ... ... ... ...
o-H$_2$O $6_{25}\rightarrow6_{16}$ 94.644 $\phantom{00}351\pm18$ $\phantom{00}46\pm\phantom{0}7$ $\phantom{00}50\pm\phantom{0}9$ $\phantom{00}35\pm\phantom{0}7$ ...
o-H$_2$O $4_{41}\rightarrow4_{32}$ 94.705
p-H$_2$O $5_{15}\rightarrow4_{04}$ 95.627 $\phantom{00}501\pm14$ $\phantom{00}64\pm11$ ... $\phantom{00}58\pm\phantom{0}8$ ...
OH $^2\Pi_{3/2}(J=3/2+)\rightarrow$ $^2\Pi_{1/2}(J=5/2-)$ 96.271 $\phantom{00}405\pm17$ $\phantom{00}48\pm12$ $\phantom{00}57\pm14$ $\phantom{00}37\pm\phantom{0}8$ ...
OH $^2\Pi_{3/2}(J=3/2-)\rightarrow$ $^2\Pi_{1/2}(J=5/2+)$ 96.363
CO $J=27\rightarrow26$ 96.773 $\phantom{00}534\pm19$ $\phantom{0}136\pm\phantom{0}9$ ... $\phantom{00}44\pm\phantom{0}7$ ...
OH $^2\Pi_{1/2}(J=5/2+\rightarrow3/2-)$ 98.737 $\phantom{0}1282\pm30$ $\phantom{0}285\pm19$ ... $\phantom{00}83\pm21$ ...
OH $^2\Pi_{1/2}(J=5/2-\rightarrow3/2+)$ 98.764
o-H$_2$O $5_{05}\rightarrow4_{14}$ 99.493 $\phantom{00}783\pm34$ $\phantom{0}123\pm18$ ... $\phantom{00}87\pm20$ ...
CO $J=25\rightarrow24$ 104.445 $\phantom{00}665\pm21$ $\phantom{0}145\pm\phantom{0}7$ $\phantom{0}150\pm\phantom{0}4$ $\phantom{00}50\pm\phantom{0}6$ ...
o-H$_2$O $2_{21}\rightarrow1_{10}$ 108.073 $\phantom{00}604\pm10$ $\phantom{00}92\pm\phantom{0}8$ $\phantom{00}83\pm\phantom{0}4$ $\phantom{00}87\pm\phantom{0}3$ $\phantom{00}9\pm\phantom{0}3$
CO $J=24\rightarrow23$ 108.763 $\phantom{00}720\pm11$ $\phantom{0}157\pm\phantom{0}6$ $\phantom{0}126\pm\phantom{0}5$ $\phantom{00}65\pm\phantom{0}4$ ...
p-H$_2$O $5_{24}\rightarrow5_{15}$ 111.628 $\phantom{00}131\pm12$ ... ... ... ...
CO $J=23\rightarrow22$ 113.458 $\phantom{0}1636\pm13$ $\phantom{0}489\pm\phantom{0}8$ $\phantom{0}231\pm\phantom{0}3$ $\phantom{0}181\pm\phantom{0}3$ $\phantom{0}21\pm\phantom{0}3$
o-H$_2$O $4_{14}\rightarrow3_{03}$ 113.537
p-H$_2$O $5_{33}\rightarrow5_{24}$ 113.948 $\phantom{000}63\pm\phantom{0}9$ $\phantom{00}18\pm\phantom{0}5$ ... ... ...
CO $J=22\rightarrow21$ 118.581 $\phantom{0}1066\pm11$ $\phantom{0}337\pm\phantom{0}9$ $\phantom{0}193\pm\phantom{0}3$ $\phantom{0}104\pm\phantom{0}2$ $\phantom{0}28\pm\phantom{0}3$
OH $^2\Pi_{3/2}(J=5/2-\rightarrow3/2+)$ 119.234 $\phantom{0}1886\pm17$ $\phantom{0}227\pm10$ $\phantom{0}202\pm\phantom{0}6$ $\phantom{0}221\pm\phantom{0}4$ $\phantom{0}26\pm\phantom{0}4$
OH $^2\Pi_{3/2}(J=5/2+\rightarrow3/2-)$ 119.441
o-H$_2$O $4_{32}\rightarrow4_{23}$ 121.722 $\phantom{00}123\pm\phantom{0}7$ ... ... ... ...
CO $J=21\rightarrow20$ 124.193 $\phantom{0}1215\pm10$ $\phantom{0}364\pm\phantom{0}6$ $\phantom{0}151\pm\phantom{0}5$ $\phantom{0}120\pm\phantom{0}2$ ...
p-H$_2$O $4_{04}\rightarrow3_{13}$ 125.354 $\phantom{00}332\pm\phantom{0}8$ $\phantom{00}98\pm\phantom{0}5$ ... $\phantom{00}47\pm\phantom{0}2$ ...
p-H$_2$O $3_{31}\rightarrow3_{22}$ 126.714 $\phantom{000}43\pm\phantom{0}7$ ... ... ... ...
o-H$_2$O $7_{25}\rightarrow7_{16}$ 127.884 $\phantom{000}45\pm\phantom{0}5$ ... ... ... ...
$^{13}$CO $J=21\rightarrow20$ 129.891 $\phantom{000}21\pm\phantom{0}4$ ... ... ... ...
CO $J=20\rightarrow19$ 130.369 $\phantom{0}1203\pm\phantom{0}8$ $\phantom{0}322\pm\phantom{0}3$ $\phantom{0}105\pm\phantom{0}2$ $\phantom{0}118\pm\phantom{0}2$ $\phantom{0}26\pm\phantom{0}2$
o-H$_2$O $4_{23}\rightarrow4_{14}$ 132.408 $\phantom{00}106\pm\phantom{0}6$ $\phantom{00}69\pm\phantom{0}3$ $\phantom{00}24\pm\phantom{0}3$ $\phantom{00}20\pm\phantom{0}3$ $\phantom{0}27\pm\phantom{0}3$
o-H$_2$O $5_{14}\rightarrow5_{05}$ 134.935 $\phantom{00}181\pm\phantom{0}5$ $\phantom{00}19\pm\phantom{0}3$ $\phantom{00}35\pm\phantom{0}1$ $\phantom{00}12\pm\phantom{0}1$ ...
o-H$_2$O $3_{30}\rightarrow3_{21}$ 136.496 $\phantom{00}237\pm\phantom{0}7$ $\phantom{00}81\pm\phantom{0}4$ $\phantom{00}12\pm\phantom{0}3$ $\phantom{00}26\pm\phantom{0}2$ $\phantom{00}9\pm\phantom{0}2$
CO $J=19\rightarrow18$ 137.196 $\phantom{0}1523\pm\phantom{0}8$ $\phantom{0}471\pm\phantom{0}5$ $\phantom{0}265\pm\phantom{0}4$ $\phantom{0}133\pm\phantom{0}2$ $\phantom{0}35\pm\phantom{0}3$
p-H$_2$O $3_{13}\rightarrow2_{02}$ 138.528 $\phantom{00}507\pm\phantom{0}7$ $\phantom{0}126\pm\phantom{0}6$ $\phantom{00}75\pm\phantom{0}4$ $\phantom{00}60\pm\phantom{0}3$ $\phantom{0}16\pm\phantom{0}3$
p-H$_2$O $4_{13}\rightarrow3_{22}$ 144.518 $\phantom{00}166\pm\phantom{0}6$ $\phantom{00}35\pm\phantom{0}4$ ... $\phantom{000}8\pm\phantom{0}2$ ...
CO $J=18\rightarrow17$ 144.784 $\phantom{0}1555\pm\phantom{0}6$ $\phantom{0}520\pm\phantom{0}5$ $\phantom{0}270\pm\phantom{0}4$ $\phantom{0}137\pm\phantom{0}2$ $\phantom{0}17\pm\phantom{0}2$
$^3$P$_0\rightarrow$ $^3$P$_1$ 145.525 $\phantom{0}1029\pm\phantom{0}7$ $\phantom{0}425\pm\phantom{0}5$ $\phantom{0}177\pm\phantom{0}3$ $\phantom{0}125\pm\phantom{0}2$ $\phantom{0}73\pm\phantom{0}3$
p-H$_2$O $4_{31}\rightarrow4_{22}$ 146.923 $\phantom{000}29\pm\phantom{0}6$ ... $\phantom{00}14\pm\phantom{0}3$ $\phantom{00}10\pm\phantom{0}3$ ...
$^{13}$CO $J=18\rightarrow17$ 151.431 $\phantom{000}30\pm\phantom{0}7$ $\phantom{00}11\pm\phantom{0}4$ ... ... ...
CO $J=17\rightarrow16$ 153.267 $\phantom{0}1800\pm\phantom{0}9$ $\phantom{0}578\pm\phantom{0}4$ $\phantom{0}309\pm\phantom{0}4$ $\phantom{0}161\pm\phantom{0}3$ ...
p-H$_2$O $3_{22}\rightarrow3_{13}$ 156.194 $\phantom{00}291\pm\phantom{0}7$ $\phantom{00}24\pm\phantom{0}3$ $\phantom{00}20\pm\phantom{0}4$ $\phantom{00}17\pm\phantom{0}2$ $\phantom{00}8\pm\phantom{0}2$
o-H$_2$O $5_{23}\rightarrow4_{32}$ 156.265
$^2$P$_{\frac{3}{2}}\rightarrow$ $^2$P$_{\frac{1}{2}}$ 157.741 $\phantom{00}528\pm\phantom{0}7$ $\phantom{0}379\pm\phantom{0}5$ $\phantom{0}137\pm\phantom{0}3$ $\phantom{0}140\pm\phantom{0}3$ $123\pm\phantom{0}2$
o-H$_2$O $5_{32}\rightarrow5_{23}$ 160.510 $\phantom{000}62\pm\phantom{0}5$ ... ... ... ...
CO $J=16\rightarrow15$ 162.812 $\phantom{0}2015\pm\phantom{0}8$ $\phantom{0}601\pm\phantom{0}4$ $\phantom{0}297\pm\phantom{0}2$ $\phantom{0}191\pm\phantom{0}2$ $\phantom{0}16\pm\phantom{0}3$
OH $^2\Pi_{1/2}(J=3/2+\rightarrow1/2-)$ 163.124 $\phantom{00}477\pm\phantom{0}5$ $\phantom{00}46\pm\phantom{0}3$ $\phantom{00}17\pm\phantom{0}2$ $\phantom{00}36\pm\phantom{0}1$ ...
OH $^2\Pi_{1/2}(J=3/2-\rightarrow1/2+)$ 163.396 $\phantom{00}343\pm\phantom{0}5$ $\phantom{00}56\pm\phantom{0}3$ ... $\phantom{00}36\pm\phantom{0}1$ ...
$^{13}$CO $J=16\rightarrow15$ 170.290 $\phantom{000}58\pm\phantom{0}5$ ... ... ... ...
CO $J=15\rightarrow14$ 173.631 $\phantom{0}1952\pm10$ $\phantom{0}636\pm\phantom{0}5$ $\phantom{0}379\pm\phantom{0}4$ $\phantom{0}193\pm\phantom{0}3$ $\phantom{0}49\pm\phantom{0}2$
o-H$_2$O $3_{03}\rightarrow2_{12}$ 174.626 $\phantom{00}568\pm\phantom{0}6$ $\phantom{0}192\pm\phantom{0}4$ $\phantom{00}88\pm\phantom{0}2$ $\phantom{00}78\pm\phantom{0}3$ ...
o-H$_2$O $2_{12}\rightarrow1_{01}$ 179.527 $\phantom{00}724\pm\phantom{0}8$ $\phantom{0}206\pm\phantom{0}5$ $\phantom{00}80\pm\phantom{0}2$ $\phantom{0}123\pm\phantom{0}3$ ...
o-H$_2$O $2_{21}\rightarrow2_{12}$ 180.488 $\phantom{00}275\pm\phantom{0}5$ $\phantom{00}36\pm\phantom{0}5$ ... $\phantom{000}7\pm\phantom{0}2$ ...
$^{13}$CO $J=15\rightarrow14$ 181.608 $\phantom{000}38\pm\phantom{0}7$ $\phantom{00}27\pm\phantom{0}4$ ... ... ...
CO $J=14\rightarrow13$ 185.999 $\phantom{0}2160\pm\phantom{0}8$ $\phantom{0}687\pm\phantom{0}5$ $\phantom{0}323\pm\phantom{0}3$ $\phantom{0}207\pm\phantom{0}5$ $\phantom{0}64\pm\phantom{0}3$
p-H$_2$O $4_{13}\rightarrow4_{04}$ 187.111 $\phantom{00}111\pm\phantom{0}6$ ... $\phantom{00}11\pm\phantom{0}2$ ... ...
$^{13}$CO $J=14\rightarrow13$ 194.546 $\phantom{000}30\pm\phantom{0}3$ $\phantom{00}27\pm\phantom{0}2$ ... ... ...
\[tab:herschel\_lineparams\]
------------------------------ -------------------------------------------------------- ------------ ---------------------------------- --------------------------------- --------------------------------- --------------------------------- -------------------------------- --
: Identified spectral lines in the *Herschel*/PACS data and deconvolved line fluxes in the continuum point sources. All errors are 1$\sigma$ of the rms. For undetected lines, the $3\sigma$ upper limit lies between $60\times10^{-18}$ W m$^{-2}$ and $10\times10^{-18}$ W m$^{-2}$ from 55[ m]{} to 195[ m]{}. Lines in the leakage regions are not tabulated.
Continuum maps {#app:contmaps}
==============
The PACS continuum maps and POMAC residuals are found in Figs. \[fig:contmaps7\]–\[fig:contmaps5\].
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{contmaps7.png} &
\includegraphics[width=0.48\linewidth]{contres7.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{contmaps5.png} &
\includegraphics[width=0.48\linewidth]{contres5.png} \\
\end{array}$
Line maps {#app:linemaps}
=========
The line flux maps and POMAC residual maps of the CO lines are found in Figs. \[fig:comaps7\]–\[fig:comaps5\], the OH maps in Figs. \[fig:ohmaps7\]–\[fig:ohmaps5\], the H$_2$O maps in Figs. \[fig:ph2omaps7\]–\[fig:oh2omaps5\], and the \[\] and \[\] maps in Figs. \[fig:atomicmaps7\]–\[fig:atomicmaps5\].
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{comaps7.png} &
\includegraphics[width=0.48\linewidth]{cores7.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{comaps5.png} &
\includegraphics[width=0.48\linewidth]{cores5.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{ohmaps7.png} &
\includegraphics[width=0.48\linewidth]{ohres7.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{ohmaps5.png} &
\includegraphics[width=0.48\linewidth]{ohres5.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{ph2omaps7.png} &
\includegraphics[width=0.48\linewidth]{ph2ores7.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{ph2omaps5.png} &
\includegraphics[width=0.48\linewidth]{ph2ores5.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{oh2omaps7.png} &
\includegraphics[width=0.48\linewidth]{oh2ores7.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{oh2omaps5.png} &
\includegraphics[width=0.48\linewidth]{oh2ores5.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{atomicmaps7.png} &
\includegraphics[width=0.48\linewidth]{atomicres7.png} \\
\end{array}$
$\begin{array}{c@{\hspace{0.0cm}}c@{\hspace{0.0cm}}c}
\includegraphics[width=0.48\linewidth]{atomicmaps5.png} &
\includegraphics[width=0.48\linewidth]{atomicres5.png} \\
\end{array}$
The POMAC deconvolution algorithm {#app:pomac}
=================================
POMAC (Poor Man’s CLEAN) is a modification of the CLEAN algorithm [@hogbom74], adapted to deconvolve data with a small number of data points, using previous knowledge about point-source positions as a restricting assumption. The user needs to define the positions of possible point sources (and point sources can be added or removed after running the script, iteratively refining the used set of point sources). The algorithm iteratively subtracts emission corresponding to the pre-defined point sources, finally leaving a residual map showing any emission not attributed to these point sources (extended emission or emission from previously unknown point sources).
More specifically, the script creates an $n\times n$ grid (we use $n=400$, giving a pixel size of $\sim0\farcs2$ for the IRS7 map) for each of the point sources, covering the whole field-of-view of the PACS footprint(s). For each of the grids, a unitary point source is convolved with a simulated telescope PSF. We use the simulated Herschel PSF, which is available for 60, 70, 80, 90, 100, 120, 140, 160, 180, and 200[ m]{}[^5] (see Fig. \[fig:pacsbeam\]). We linearly interpolate these for the intermediate wavelengths. The script iterates over each grid point to determine which PACS spaxel(s) the signal in this grid point will fall into (zero or one spaxels per PACS footprint). The result of this is one PACS instrument PSF (dirty beam) for each of the point sources in the field. We assume PACS spaxel sizes of 94$\times$94, centred at the coordinates given by the telescope data, oriented parallel to the PACS grid. This is, however, a simplified model of the true spaxel footprint, but the simplification will not affect the results more than the calibration accuracy of the instrument (20%).
![The *Herschel* simulated point-spread function (PSF) at 120[ m]{} plotted with logarithmic scaling to better show the side lobes.[]{data-label="fig:pacsbeam"}](pacsbeam.png){width="0.48\linewidth"}
The line flux measured in each of the spaxels are put in a residual flux matrix. Next, the script determines which spaxel is most nearby each of the point sources. The script then analyses the residual map flux in each of these spaxels, and based on the point sources’ distances to their nearest spaxels the brightest point source is determined. A user-defined CLEAN gain (we use 0.01) is multiplied to the calculated flux and added to the CLEAN flux of this point source. The convolution of the PACS footprint Gaussian and a point source with the CLEANed flux is then subtracted from the residual map. This is iterated until one of several possible stop criteria is met:
1. The strongest point-source residual corresponds to negative flux.
2. The residual flux in any one spaxel near a point-source position drops below $-3\sigma$.
3. The residual fluxes in any two spaxels drop below $-3\sigma$.
4. The residual flux drops below $1\sigma$ in a spaxel which is associated with a point source.
5. The average of the residual map drops below 0.
6. The set maximum for CLEAN iterations is reached (this criterium was not reached in any of our deconvolutions).
After this has been finished, the flux of each point source is found in the CLEAN flux vector, and the remaining residual map can be investigated to search for more point sources or extended emission (note, however, that the residual map is still convolved with the instrument and telescope PSF).
The algorithm was tested on well-centred PACS data of the disc source HD 100546 [@sturm10], which is assumed to behave like a point source in the continuum. The residuals after removing the central point source from the continuum data were found to be less than 5% of the extracted flux density across the whole spectrum.
It should be noted that in the case of extended emission across the whole field, the algorithm will extract too much flux to the point source, producing a hole in the residual emission. It is, however, impossible to predict the amount of the emission on the point source that should be attributed to the extended emission, and we have thus chosen the most simple approach. In the CrA data, the large-scale extended emission is weak in comparison with the point source emission, so this does not appear to affect the results more than 20%.
The algorithm works particularly well in the IRS7 field, where we have two overlapping PACS footprints (creating an almost-Nyquist sampled grid for most of the PACS bandwidth), but it is also useful for regions with only one PACS footprint, such as IRS5. The method is useful, not only to disentangle emission from several point sources, but also to establish whether a source is a point source or shows extended emission.
SED flux densities
==================
The flux densities from the literature used in the SED fits in Sect. \[sec:sed\_analysis\] are listed in Table \[tab:sedfluxes\].
[l l l l]{} Name & Wavelength & Flux density & Reference\
*Telescope* & \[m\] & \[Jy\] &\
IRS7A\
*Spitzer* & 3.6 & 0.0509 & @peterson11\
& 4.5 & 0.239\
& 5.8 & 0.546\
& 8.0 & 1.05\
SMM 1C\
*JCMT SCUBA* & 450 & 45 & @nutter05\
& 850 & 5.6\
IRS7B\
*Spitzer* & 3.6 & 0.0367 & @peterson11\
& 4.5 & 0.257\
& 5.8 & 0.697\
& 8.0 & 0.957\
*JCMT SCUBA* & 450 & 50 & @nutter05\
& 850 & 5.4\
R CrA\
*SAAO* & 0.36 & 0.0141 & @koen10\
& 0.45 & 0.0359\
& 0.555 & 0.0647\
& 0.67 & 0.116\
& 0.87 & 0.180\
*2MASS* & 1.25 & 2.68 & @2mass\
& 1.65 & 10.7\
& 2.2 & 47.9\
IRS5A\
*Spitzer* & 3.6 & 0.389 & @peterson11\
& 4.5 & 0.798\
& 5.8 & 1.27\
& 8.0 & 1.81\
IRS5N\
*Spitzer* & 3.6 & 0.00856 & @peterson11\
& 4.5 & 0.0262\
& 5.8 & 0.0528\
& 8.0 & 0.0809\
*JCMT SCUBA* & 450 & 12 &@nutter05\
& 850 & 1.8\
[^1]: Refer to the PACS manual v. 2.4 for a detailed description of the PSF correction factor and the PACS beam: <http://herschel.esac.esa.int/Docs/PACS/html/pacs_om.html>
[^2]: Poor Man’s CLEAN.
[^3]: <http://caravan.astro.wisc.edu/protostars/>
[^4]: Refer to the PACS manual v. 2.4, Figure 4.5: <http://herschel.esac.esa.int/Docs/PACS/html/pacs_om.html>
[^5]: Zemax modelled point spread functions, which are found to agree remarkably well with verification observations:\
<http://herschel.esac.esa.int/twiki/pub/Public/PacsCalibrationWeb/PACSPSF_PICC-ME-TN-029_v2.0.pdf>
|
---
author:
- |
K. Pavlovski$^{1,2}$, J. Southworth$^2$, E. Tamajo$^1$, and V. Kolbas$^1$\
\
$^1$ Department of Physics, University of Zagreb, Croatia\
$^2$ Astrophysics Group, Keele University, Staffordshire, UK
title: '**Observational approach to the chemical evolution of high-mass binaries**'
---
PS. @plain[mkbothoddheadoddfoot[“The multi-wavelength view of hot, massive stars”; 39$^{\rm th}$ Liège Int. Astroph. Coll., 12-16 July 2010 ]{}evenheadevenfootoddfoot]{} \#1
\#1
[ The complexity of composite spectra of close binaries makes the study of the individual stellar spectra extremely difficult. For this reason there exists very little information on the chemical composition of high-mass stars in close binaries, despite its importance for understanding the evolution of massive stars and close binary systems. A way around this problem exists: spectral disentangling allows a time-series of composite spectra to be decomposed into their individual components whilst preserving the total signal-to-noise ratio in the input spectra. Here we present the results of our ongoing project to obtain the atmospheric parameters of high-mass components in binary and multiple systems using spectral disentangling. So far, we have performed detailed abundance studies for 14 stars in eight eclipsing binary systems. Of these, V380Cyg, V621 Per and V453Cyg are the most informative as their primary components are evolved either close to or beyond the TAMS. Contrary to theoretical predictions of rotating single-star evolutionary models, both of these stars show no abundance changes relative to unevolved main sequence stars of the same mass. It is obvious that other effects are important in the chemical evolution of components in binary stars. Analyses are ongoing for further systems, including AHCep, CWCep and V478Cyg.]{}
Introduction
============
In the last decade theoretical stellar evolutionary models, particularly for higher masses, were improved considerably with the inclusion of additional physical effects beyond the standard ingredients. It was found that rotationally induced mixing and magnetic fields could cause substantial changes in the resulting predictions (Meynet & Maeder 2000, Heger & Langer 2000). Some of these concern evolutionary changes in the chemical composition of stellar atmospheres. Due to the CNO cycle in the core of high-mass stars some elements are enhanced (such as helium and nitrogen), some are depleted (e.g. carbon), whilst some (e.g. oxygen) are not affected at all. The rotational mixing predicted by stellar models is so efficient that changes in the atmospheric composition should be identifiable whilst the star is still on the main sequence.
On the observational side, substantial progress has also been made. The VLT/FLAMES survey (Evans et al. 2005) produced CNO abundances for a large sample of B stars in the Milky Way, and the Magellanic Clouds. This survey has opened new questions since a large population of slow rotators have shown an enhancement of nitrogen (Hunter et al. 2009). Also, important empirical constraints on models arose from the observational study performed by Morel et al. (2008) who found that magnetic fields have an important effect on the atmospheric composition of these stars.
{width="14.5cm"}
Detached eclipsing binaries are fundamental objects for obtaining empirical constraints on the structure and evolution of high-mass stars, and are the primary source of directly measured stellar properties. Accurate physical properties are available for fewer than a dozen high-mass binaries, and most have no observational constraints on their chemical composition (Torres, Andersen & Giménez 2010). The aim of our projects is to obtain a sample of high-mass binaries both with accurate parameters and, for the first time, with detailed abundance studies of the individual stars. We aim to gain insight into the chemical evolution of high-mass stars in close binary systems. The close proximity of the components leads to strong tidal forces, which may be an important additional effect on the internal and chemical structure of the stars, beside rotation and magnetic fields.
Sample and Method
=================
The complexity of the composite spectra of close binaries makes studying the spectra of the individual stars extremely difficult. For this reason there exists very little information on the chemical composition of high-mass stars in close binaries, despite its importance for understanding the evolution of both massive stars and close binaries. A way around this problem exists: spectral disentangling. This technique allows a time-series of composite spectra to be decomposed into their individual components whilst preserving the total signal-to-noise ratio in the input spectra, and without the use of template spectra (Simon & Sturm 1994). An overview of almost a dozen methods for spectral disentangling has been given by Pavlovski & Hensberge (2010). For our work we use the [fdbinary]{} Fourier-space code (Ilijić et al. 2004). Synthetic spectra are generated using [atlas9]{} with non-LTE model atoms (see Pavlovski & Southworth 2009 and Pavlovski et al. 2009 for details).
{width="14.2cm"}
A vital step in a spectroscopic abundance study is precise determination of the stellar atmospheric parameters (effective temperature, surface gravity, microturbulence velocity, etc). When reconstructing the separate spectra of the components their individual light contributions have to be obtained either from the disentangled spectra itself, or from some other source such as a complementary light curve analysis (c.f. Pavlovski & Hensberge 2010).
So far, we have performed detailed abundance studies for 14 components in eight eclipsing binaries. In many cases we have also reanalysed existing or new light curves. Of the systems studied, V380Cyg (Pavlovski et al. 2009), V453Cyg (Pavlovski & Southworth 2009, Southworth et al. 2004a) and V621Per (Southworth et al., 2004b, 2011 in prep.) are the most informative as their primary components are evolved either close to or beyond the terminal-age main sequence (TAMS). Other binaries studied include V578Mon (Pavlovski & Hensberge 2005, see also Hensberge et al. 2000), AHCep, CWCep, YCyg and V478Cyg \[helium abundances have also been measured from disentangled spectra for DHCep (Sturm & Simon 1994), YCyg (Simon, Sturm & Fiedler 1994) and DWCar (Southworth & Clausen 2007)\]. These objects mostly contain stars at the beginning of their main sequence lifetimes, so are important for calibrating theoretical models near their initial conditions.
The quest for surface helium enrichment
=======================================
Theoretical stellar evolutionary models which include rotational mixing predict an enrichment of helium at the stellar surface, even during a star’s main sequence lifetime. Extensive observational studies comprising B-type stars in the field (Lyubimkov et al. 2004), and in stellar clusters (Huang & Gies 2006) yield evidence for this enrichment, but with a very large scatter in the individual measurements. An unexpectedly large fraction of both helium-rich, and helium-weak stars were detected by Huang & Gies (2006), who included only three helium lines in their analysis.
The results of our detailed abundance determinations in the sample of 14 components of close binary stars are shown in Fig. 2 (red symbols). The results of a recent study of helium abundances in the sample of sharp-lined main sequence and BA supergiants (Przybilla et al. 2010) are also plotted (blue symbols). It is interesting that no helium abundance enrichment has been detected in these studies, either for single stars (Przybilla et al. 2010) or the components of close binaries (this work). The studies therefore do not support a large spread in helium abundance, as found by other authors, with the caveat that the sample of main sequence stars studied is limited.
{width="15cm"}
The evolution of nitrogen in high-mass binaries
===============================================
In close binary stars, both fast rotation and tidal forces due to the proximity of the components play an important role in stellar evolution. Tides spin up (or spin down) the stars until their rotation period synchronises with the orbital period. The effects of tides, rotational mixing and magnetic fields were studied by de Mink et al. (2009). Their model calculations indicate a significant dependence of the surface helium and nitrogen abundances for short-period systems ($P < 2$ days) for a considerable fraction of their MS lifetime. The best candidates for testing these concepts contain more massive components, in advanced phases of the core hydrogen-burning phase, with significantly less massive and less evolved companions. V380Cyg, V621Per and V453Cyg fit this bill well, but have longer orbital periods (3.9d to 25.5d) so are not predicted to show significant abundance enhancements. This is illustrated in Fig.3 in which the abundance ratio N/O is plotted against $\log g$, which is a good indicator of evolutionary stage. The N/O ratios for the evolved stars in our sample are consistent with ZAMS values, like many of the stars in the VLT/FLAMES sample of Hunter et al. (2009). The evolutionary enhancement of nitrogen is only clearly present in the sample of supergiants observed by Przybilla et al. (2010). On average the magnetic B-type stars (Morel et al. 2008) have large nitrogen abundances, but definitive conclusions on the role of magnetic fields on nitrogen enrichment are still not possible (Morel 2010). The large spread in nitrogen abundances for MS stars is obvious.
Since the enhancements of helium and nitrogen are larger at lower metallicity, the best candidates for detailed study would be close binaries in the Magellanic Clouds. However, these are challenging objects for accurate abundance determination due to their high rotational velocities (resulting in line blending) and relative faintness.
Acknowledgements {#acknowledgements .unnumbered}
================
KP acknowledges receipt of the Leverhulme Trust Visiting Professorship which enables him to work at Keele University, UK, where this work was performed. This research is supported in part by a grant to KP from the Croatian Ministry of Science and Education.
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Evans, C.J., Smartt, S.J., Lee, J.-K., et al., 2005, A&A, 437, 467
Heger, A., Langer, N., 2000, ApJ, 544, 1016
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Huang, W., Gies, D.R., 2006, ApJ, 648, 591
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|
---
abstract: 'The use and location of memory in integrated circuits plays a key factor in their performance. Memory requires large physical area, access times limit overall system performance and connectivity can result in large fan-out. Modern FPGA systems and ASICs contain an area of memory used to set the operation of the device from a series of commands set by a host. Implementing these settings registers requires a level of care otherwise the resulting implementation can result in a number of large fan-out nets that consume valuable resources complicating the placement of timing critical pathways. This paper presents an architecture for implementing and programming these settings registers in a distributed method across an FPGA and how the presented architecture works in both clock-domain crossing and dynamic partial re-configuration applications. The design is compared to that of a ‘global’ settings register architecture. We implement the architectures using Intel FPGAs Quartus Prime software targeting an Intel FPGA Cyclone V. It is shown that the distributed memory architecture has a smaller resource cost (as small as 25% of the ALMs and 20% of the registers) compared to the global memory architectures.'
author:
- 'Alexander E. Beasley[^1]'
bibliography:
- 'references.bib'
- 'references\_dpr.bib'
- 'references\_fpga.bib'
- 'references\_distributed\_memories.bib'
title: 'A distributed memory, local configuration technique for re-configurable logic designs'
---
Introduction
============
The use of memory, memory accessing, and memory mapping techniques has a large impact on system performance [@Tirri1995hardaddress; @Cordasco2007boundedcollision; @Rutgers2013manycores]. Efficient mapping techniques, reduction in communication overhead and the use of distributed memories can vastly increase the systems overall performance, particularly for intensive tasks such as loops and scalable graph operations [@Darte1993uniformloops; @Kontothanassis1996memorymap; @Lin2013fastscalablegraph; @Azad2016bipartite; @Lo1991graph]. The implementation of memories inside an embedded system comes with many research possibilities. Memory technologies are becoming denser and faster, allowing for higher density memory to be implemented close to its point of use. Despite this, memory still requires large, physical space.
Distributed memory, where the memory is close to the point at which it is used offers huge benefits, so long as the memories are kept coherent where necessary [@Choi1997distributed; @Klein2014billions; @Huang1996os].
Integrated circuits often require memory to store user defined settings that control the mode of operation. Such examples could be the sample rate or resolution of a ADC; the applied phase shift of an RF phase shifter; the gain of a variable gain amplifier and so on.
Field Programmable Gate Arrays (FPGAs) provide a flexible platform for designers to fabricate seemingly endless weird and wonderful systems. Quite often designers wish to make parameterisable systems where their operation can be controlled by based on a number of settings. One way to achieve this is by use of parameters [@VerilogParameters] (or generics — VHDL [@VhdlGenerics]) that can be set at compile time. Parameters (generics) are a very powerful tool available in hardware descriptive languages, to create re-useable code. However, each combination of settings must be compiled separately, introducing a large amount of processing overhead and leading to a separate image file per configuration. Alternatively, designers can implement an area of the FPGA as an array of registers in which settings can be stored and propagated across the design. These registers can be programmed by means of a connection with a host system (typically USB in a modern system). Implementing these features in a device create demand on resources, reducing the overall resources available to be used as functional logic.
FPGA place and route stages are complicated procedures, attempting to locate resources as close as possible to reduce routing complication and net delay [@Fobel2009Hardwareaccelerated; @Wrighton2003simulatedannealingplacement; @Gudise2004PSOplacement; @Haldar2000ParallelPlacement]. As the resources are fixed, this often results in trade-offs between quality of the fitter result and run-time of the fitter [@Mulpuri2001runtimetradoffs]. In addition, large fan out nets often take priority during the ‘fitting’ stage of an FPGAs compilation. Synthesis tools often attempt to insert extra resources or promote high fan-out nets to the clocking nets [@AN903TimingClosure; @XilinxTimingClosure]; reducing the available resources for timing critical pathways. This leads to more complicated designs that suffer from bottle-necking, manifesting itself as a reduction in the maximum operating frequency of a design.
FPGAs have a large amount of memory distributed throughout the device. This memory neatly lends itself for tasks where distributed memory, close to the point of use, such as loop operations and array intensive operations [@Pal2007distributedmemorysynthesis]. By extension, we can use the embedded memory blocks to create the sets of registers used to set up and control the FPGA. Distributing the settings across the FPGA to their point of use helps to reduce the required routing resources, limit the high fan-out nets and improve timing closure.
In this paper we explore how a typical ‘global’ register map expands with the number of required settings for a design and the width of each of these settings. The global register map is connected to modules in which a varying number of entries in the global register map are used to help us model the resource requirements when fanning out the register map. A second architecture is presented that removes the global register map and distributes the settings across the design to the points at which they are used. Discussions are had as to how these architectures deal with the common problem of clock domain crossing and a more recent problem of how to deal with dynamic partial reconfiguration — the process by which a small portion of a design is changed at run-time without effecting the operation of the rest of the device.
The rest of this paper is organised as per the following. Section \[sec:architecture\] presents architectures for creating and distributing settings using a ‘global’ register map and an architecture for distributing the settings across the device in a method that is robust to multiple clocking domains and partial dynamic re-configuration. Metrics for the architectures are presented in Sect. \[sec:results\]. Finally, conclusions are drawn in Sect. \[sec:conclusions\].
Architecture {#sec:architecture}
============
The settings registers are usually considered as an area of memory, in which the stored values represent modes of operation for a design. These stored values are used throughout the design to influence operation. There are a number of ways to achieve the desired behaviour, the seemingly obvious is to simply reference the values, stored in a global memory location, throughout the respective parts of the design, leading to a routing as in Fig. \[fig:global\_arch\].
![A global copy of the memory map is populated via the host controller, respective settings are routed to the appropriate modules on multi-bit busses.[]{data-label="fig:global_arch"}](global_memory_map.pdf){width="50.00000%"}
Alternatively, distributing the memory map throughout the design moves the settings closer to where they are used. The result of which is to reduce the complexity of the routing, but not necessarily reduce the overall resource requirement. Local copies of the register map close to the point of which they are used allows the designer to safely register the values into the appropriate clock domains. The additional flip-flop stages play an important part in breaking up the total routed path into smaller elements, the shorter the path, the easier it is for a design to meet timing closure. However, the additional flip-flops used increase the overall resource cost for a design. An example of such a design can be seen in Fig. \[fig:global\_arch\_local\_copies\].
![Entries from the global memory map are copied to where they are required locally.[]{data-label="fig:global_arch_local_copies"}](global_memory_map_local_copies.pdf){width="50.00000%"}
Distributing the memory map across the design can be achieved without the need for a increasing the routing complexity. Designing the distributed memory map with a common bus interface for its configuration, Fig. \[fig:distributed\_arch\_1\], reduces the overall resource cost and significantly reduces the required routing resource.
![Bus connects elements of the design to a decoding module that distributes memory map information across the device. Uniform bus allows connection of partially dynamically re-configurable modules into the memory map bus.[]{data-label="fig:distributed_arch_1"}](distributed_settings_bus_1.pdf){width="\textwidth"}
The common bus interface has a number of benefits: reduced routing complexity, safe crossing into different clock domains, reduction in global memory resources, connection into dynamically partially re-configurable logic space.
Clock domain crossing
---------------------
It is not uncommon for a modern digital system to use multiple clocks [@Ragheb2018multipleclockdomains], in which data are moved from one clock domain to another and memories are connected to different clock domains. Moving from one clock domain to another requires the use of safe clock domain crossing domains - which in themselves are a large research field [@Li2010synchtech; @Bartik2018clockdomaincrossing; @Matsuda2011cdc; @Preetam2015CDC] - however they require using up yet more valuable resources. Typically configuration data would be set in a slow clock domain and moved into much faster domains - potentially as very wide, parallel busses.
In addition to the increase in resources required for crossing clock domains, multi-clock systems lack determinism which causes problems for the verification process. Rectifying the non-deterministic nature of such systems and providing verification techniques (both stand-alone and built-in) is a rich source of research [@Su2010multiclock; @Leong2010cdc]. Additionally, frameworks for performing timing analyses and signal integrity in a CDC application [@Matsuda2011cdc; @Preetam2015CDC] have been proposed.
The architecture presented here, fig. \[fig:distributed\_arch\_1\], exports a ‘Ready’ signal from each of the subsystems. The ‘Ready’ signal is used to indicate that the logic has been moved to a safe state in which the local memory map may be written to using the configuration bus. No changes are made to the local configuration memories while logic is operating, hence there is no danger of the registers being sampled while they are transitioning and the clock domains are safely crossed.
Dynamic partial reconfiguration
-------------------------------
Dynamic reconfiguration and Dynamic Partial Reconfiguration (DPR) is rapidly growing in popularity as it enables FPGA designs to be changed at run-time to better meet changing systems demands [@Lie2009dpr; @Di2012DPRflow]. The use of DPR is rapidly gaining popularity over a number of sectors including: fault recovery [@Alkady2015dprfaultrecovery], memory controllers [@Salah2017dprmemorycontroller], real-time signal processing [@Feilen2011dprrealtimesigprocessing], software defined radio [@Sadek2015dprsdr; @Hosny2018dprsdr; @Hassan2015dprsdr], cognitive radio [@Lie2009dprcognitiveradio], bandwidth reduction [@Najmabadi2016dprbandwidthreduction], video filters [@Khraisha2010dprvideofilter], and RADAR signal processing [@Zhang2016dprradarprocessing] to name a few.
DPR designs contain a mix of static logic and re-configurable logic. Between the elements of the design a common interconnect is implemented, Fig. \[fig:reconfig\]. The interconnect fabric contains the signals required for the configuration bus. When a module(s) in a re-configurable portion of the FPGA is changed, the configuration bus is connected into the new module along with all other data-path signals. Any settings registers inside partially configured module are then set over the configuration bus.
![Partially re-configurable design showing the common programming interface in the interconnect logic between static and re-configurable logic[]{data-label="fig:reconfig"}](reconfig_memory_map.pdf){width="50.00000%"}
Results {#sec:results}
=======
Example designs of the above architectures were written using SystemVerilog (IEEE 1800) and processed using Intel FPGA Quartus Prime 19.1.0 (Build 670); target device for compilation is a Cyclone V (5CSXFC6D6F31C8). Synthesis metrics — Adaptive Logic Modules (ALMs), registers, combinatorial Adaptive Look Up Tables (ALUTs) and maximum operating frequency — are presented for each architecture. Implementations are given for a variety of memory depths and widths.
Global configuration - no targets {#sec:global}
---------------------------------
Figures \[fig:global\_no\_target\_alm\] to \[fig:global\_no\_target\_fmax\] show key metrics for an implementation of a global memory system. The global memory system contains the decoding logic for writing to the memory, the memory, and the output stage that would be connected to the rest of the design. These figures do not include the resource consumption of slave modules where the settings would be used and any clock domain crossing logic that may be implemented.
ALMs (Intel) — similar to Configurable Logic Blocks (CLB) (Xilinx) — contain a number of resources, typically (A)LUTs, adders, multiplexers, routing logic, and registers [@IntelALMWP]. From fig. \[fig:global\_no\_target\_alm\] it is shown that adding a register stage to the output of the memory significantly increases the number of ALMs needed for implementation; for instance, in this case, 128 512-bit words with a final register stage require just over 10,000 (10,292.6) more ALMs for implementation — approximately an extra 40%. Similarly, the number of dedicated registers (fig. \[fig:global\_no\_target\_reg\]) requires an extra 65,536 dedicated logic registers — an approximately 100% increase in resource. Again, the number of ALUTs, fig. \[fig:global\_no\_target\_alut\], has also increased by approximately 40%. This is to be expected since the implementation shown in subfigures (a) of figs. \[fig:global\_no\_target\_alm\] to \[fig:global\_no\_target\_fmax\] have an extra register stage per bit of the memory map at the output.
This is an obvious draw back in terms of resource consumption. However, the accompanying benefits of the extra register stage is that the length of the routing between the memory and the target can now be broken down using the extra register stage. This manifests itself in an increase in operating frequency for the design. Figure \[fig:global\_no\_target\_fmax\] shows the maximum operating frequency of the implementation that uses an extra register. While synthesising just the memory module itself we are unable to provide $f_{\text{max}}$ figures when there is no additional output register because there are no valid paths (paths between two flip-flops) for which the timing analyzer (TimeQuest) can operate.
![Maximum operating frequency (die temperature 85$^{\circ}$C) for global memory module only. Data only given for global memory with registered output.[]{data-label="fig:global_no_target_fmax"}](global_map_reg_fmax85.pdf){width="80.00000%"}
Global configuration with targets {#sec:globaltargets}
---------------------------------
In sect. \[sec:global\] the resource consumption for the memory decode logic and memory itself are shown. However, this is only half the story for a design that uses a global set of memory where entries are propagated out to other areas of the design. In this section we take a global memory system that a global memory of 256 32-bit words and propagates these out to a slave module with a varying number of configuration registers in the slave module. In addition, designs that use a combination of output registers on the global memory map, clock domain crossing registers (synchronisation chain length is 2 registers) and final location registers are examined.
Figure \[fig:global\_targets\_alms\] is the after fitting ALM requirements, fig. \[fig:global\_targets\_reg\] is the after fitting register requirements, and fig. \[fig:global\_targets\_aluts\] is the after fitting ALUT requirements for each configuration of the global memory map architecture. As is expected, increasing the number of target registers linearly increases the requirement of each resource. Designs with a greater number of register stages (post global map register, clock domain crossing synchronisation chain registers and destination registers) significantly increases the resource requirements compared to design with fewer register stages. 10099.1ALMs, 38146registers, and 1925ALUTs for a design with 226 configuration registers and the maximum number of routing register stages compared to 2710.5ALMs, 8258registers, and 1913ALUTs for a design with the same number of configuration registers but no register stages to break down the length of the routing. The more crowded a design becomes, the greater the impact of removing the routing registers has on the maximum speed of a path.
![ALM consumption of global memory architecture with a single slave module using a variety of configuration registers and routing registers.[]{data-label="fig:global_targets_alms"}](global_map_targets_all_regs_alms_final.pdf){width="80.00000%"}
![Register consumption of global memory architecture with a single slave module using a variety of configuration registers and routing registers.[]{data-label="fig:global_targets_reg"}](global_map_targets_all_regs_registers.pdf){width="80.00000%"}
![ALUT consumption of global memory architecture with a single slave module using a variety of configuration registers and routing registers.[]{data-label="fig:global_targets_aluts"}](global_map_targets_all_regs_comb_aluts.pdf){width="80.00000%"}
Distributed configuration
-------------------------
The resources required for the distributed configuration memory architecture, shown in figs. \[fig:distri\_targets\_alms\] to \[fig:distri\_targets\_aluts\] are considerably lower than the global memory architecture. The graphs shown here are for implementations with a number of slave modules (1 to 4) each implementation varies the number of configurations per slave. For comparisons numbers from the ‘1 slave’ implementations can be mapped to the results given in sect. \[sec:globaltargets\].
The resources used for a distributed configuration memory implementation using 226 target registers per slave are: 2556.0ALMs, 7499registers, and 1887ALUTs. That is 25% of the ALMs, 20% of the registers used in the global design with maximum routing register. A significant cost saving. Increasing the number of slaves in the design has a linear effect on the resource cost.
![ALM consumption of distributed memory architecture with 1 to 4 slave module(s) and a variety of configuration registers.[]{data-label="fig:distri_targets_alms"}](distributed_map_no_reg_alms_final.pdf){width="80.00000%"}
![Register consumption of distributed memory architecture with 1 to 4 slave module(s) and a variety of configuration registers.[]{data-label="fig:distri_targets_reg"}](distributed_map_no_reg_registers.pdf){width="80.00000%"}
![ALUT consumption of distributed memory architecture with 1 to 4 slave module(s) and a variety of configuration registers.[]{data-label="fig:distri_targets_aluts"}](distributed_map_no_reg_comb_aluts.pdf){width="80.00000%"}
Operating frequency
-------------------
Figure \[fig:targets\_fmax\] shows that the maximum operating frequency of a design is also influenced by the topology of the configuration architecture. A global memory architecture achieves a maximum $f_{\text{max}}$ of just shy of 140MHz compared to the approximate 210MHz of the distributed memory architecture.
Conclusions {#sec:conclusions}
===========
In this paper it has been shown that there are a number of ways to achieve the implementation of configuration registers in an FPGA design. In this paper we proposed a global memory architecture and a distributed memory architecture, for completeness the global memory architecture was presented with combinations of register stages and clock domain crossing registers. It has been shown that the distributed architecture has a much lower resource cost for ALMs and registers (as small as 25% and 20% respectively for a design using 226 32-bit configuration registers). It has further been shown that there is a disparity in the maximum operating frequency between the designs with the distributed memory architecture achieving a higher maximum operating frequency.
Aside from the reduction in resource cost between the different architectures, the distributed memory architecture uses a common configuration bus that is independent of the number of target registers and their width. The uniformity of the configuration bus opens up the ability to implement the configuration system in a partially re-configurable FPGA design, where the configuration bus can be connected to any re-configurable design without penalty. Similarly, the architecture of the configuration bus is not liable to mis-sampling when crossing clock domains. It is set only when the slave module reports it is safe to change the settings.
[^1]: corresponding author: Alexander Beasley, [email protected]
|
---
abstract: 'We review some recent results on the connection between CP violation at low energies and Leptogenesis in the framework of specific flavour structures for the fundamental leptonic mass matrices with zero textures.'
address: |
Departamento de F' isica and Centro de F' isica Te' orica de Part' iculas (CFTP)\
Instituto Superior T' ecnico (IST), Av. Rovisco Pais, 1049-001 Lisboa, Portugal\
CERN, Department of Physics, Theory Division, CH-1211 Gen\` eve 23, Switzerland\
$^*$Presently at CERN on sabbatical leave from IST.\
E-mail: [email protected] and [email protected]
author:
- 'M. N. Rebelo$^*$'
title: Leptonic CP Violation and Leptogenesis
---
Introduction {#aba:sec1}
============
Neutrinos have masses which are much smaller than the other fermionic masses and there is large mixing in the leptonic sector. The Standard Model (SM) of electroweak interactions cannot accommodate the observed neutrino masses and leptonic mixing since in the Standard Model neutrinos are strictly massless: the absence of righthanded components for the neutrino fields does not allow one to write a Dirac mass term; the fact that the lefthanded components of the neutrino fields are part of a doublet of $SU(2)$ rules out the possibility of introducing Majorana mass terms since these would violate gauge symmetry; finally, in the SM, $B-L$ is exactly conserved, therefore Majorana mass terms cannot be generated neither radiatively in higher orders nor nonperturbatively. Therefore, neutrino masses require physics beyond the SM. At present, this is the only direct evidence for physics beyond the SM. The origin of neutrino masses remains an open question. It is part of a wider puzzle, the flavour puzzle, with questions such as whether or not there is a connection between quarks and leptons explaining the different patterns of flavour mixing in each sector and the different mass hierarchies. In the seesaw framework [@Minkowski:1977sc; @Yanagida:1979as; @Levy:1980ws; @VanNieuwenhuizen:1979hm; @Mohapatra:1979ia] the explanation of the observed smallness of neutrino masses is related to the existence of heavy neutrinos with masses that can be of the order of the unification scale and have profound implications for cosmology. Mixing in the leptonic sector leads to the possibility of leptonic CP violation both at low and at high energies. CP violation in the decay of heavy neutrinos may allow for the explanation of the observed baryon asymmetry of the Universe (BAU) through leptogenesis [@Fukugita:1986hr]. Neutrino physics may also be relevant to the understanding of dark matter and dark energy as well as galaxy-cluster formation. Recent detailed analyses of the present theoretical and experimental situation in neutrino physics and its future, can be found in Refs. and .
In this work the possibility that BAU may be generated via leptogenesis through the decay of heavy neutrinos is discussed. Leptogenesis requires CP violation in the decays of heavy neutrinos. However, in general it is not possible to establish a connection between CP violation required for leptogenesis and low energy CP violation [@Branco:2001pq; @Rebelo:2002wj]. This connection can only be established in specific flavour models. The fact that in this framework the masses of the heavy neutrinos are so large that they cannot be produced at present colliders and would have decayed in the early Universe shows the relevance of flavour models in order to prove leptogenesis. In what follows it will be shown how the imposition of texture zeros in the neutrino Yukawa couplings may at the same time constrain physics at low energies and lead to predictions for leptogenesis.
Framework and Notation
======================
The work described here is done in the seesaw framework, which provides an elegent way to explain the smallness of neutrino masses, when compared to the masses of the other fermions.
In the minimal seesaw framework, the SM is extended only through the inclusion of righthanded components for the neutrinos which are singlets of $SU(2) \times U(1)$. Frequently, one righthanded neutrino component per generation is introduced. This will be the case in what follows, unless otherwise stated. In fact, neutrino masses can be generated without requiring the number of righthanded and lefthanded neutrinos to be equal. Present observations are consistent with the introdution of two righthanded components only. In this case one of the three light neutrinos would be massless.
With one righthanded neutrino component per generation the number of fermionic degrees of freedom for neutrinos equals those of all other fermions in the theory. However neutrinos are the only known fermions which have zero electrical charge and this allows one to write Majorana mass terms for the singlet fermion fields. After spontaneous symmetry breakdown (SSB) the leptonic mass term is of the form: $$\begin{aligned}
{\cal L}_m &=& -[\overline{{\nu}_{L}^0} m_D \nu_{R}^0 +
\frac{1}{2} \nu_{R}^{0T} C M_R \nu_{R}^0+
\overline{l_L^0} m_l l_R^0] + h. c. = \nonumber \\
&=& - [\frac{1}{2} n_{L}^{T} C {\cal M}^* n_L +
\overline{l_L^0} m_l l_R^0 ] + h. c.
\label{lrd}\end{aligned}$$ with the $6 \times 6$ matrix $\cal M $ given by: $$\begin{aligned}
{\cal M}= \left(\begin{array}{cc}
0 & m_D \\
m^T_D & M_R \end{array}\right) \label{calm}\end{aligned}$$ the upperscript $0$ in the neutrino ($\nu$) and charged lepton fields ($l$) is used to indicate that we are still in a weak basis (WB), i.e., the gauge currents are still diagonal. The charged current is given by: $${\cal L}_W = - \frac{g}{\sqrt{2}} W^+_{\mu} \
\overline{l^0_L} \ \gamma^{\mu} \ \nu^0_{L} +h.c.
\label{16}$$ Since the Majorana mass term is gauge invariant there are no constraints on the scale of $M_R$. The seesaw limit consists of taking this scale to be much larger than the scale of the Dirac mass matrices $m_D$ and $m_l$. The Dirac mass matrices are generated from Yukawa couplings after SSB and are therefore at most of the electroweak scale. As a result the spectrum of the neutrino masses splits into two sets, one consisting of very heavy neutrinos with masses of the order of that of the matrix $M_R$ and the other set with masses obtained, to a very good approximation, from the diagonalisation of an effective Majorana mass matrix given by: $$m_{eff} = - m_D \frac{1}{M_R} m^T_D \label{meff}$$ This expression shows that the light neutrino masses are strongly suppressed with respect to the electroweak scale. There is no loss of generality in choosing a WB where $m_l$ is real diagonal and positive. The diagonalization of ${\cal M}$ is performed via the unitary transformation: $$V^T {\cal M}^* V = \cal D \label{dgm}$$ where ${\cal D} ={\rm diag} (m_1, m_2, m_3,
M_1, M_2, M_3)$, with $m_i$ and $M_i$ denoting the physical masses of the light and heavy Majorana neutrinos, respectively. It is convenient to write $V$ and $\cal D$ in the following block form: $$\begin{aligned}
V=\left(\begin{array}{cc}
K & G \\
S & T \end{array}\right) ; \qquad
{\cal D}=\left(\begin{array}{cc}
d & 0 \\
0 & D \end{array}\right) . \label{matd}\end{aligned}$$ The neutrino weak-eigenstates are then related to the mass eigenstates by: $$\begin{aligned}
{\nu^0_i}_L= V_{i \alpha} {\nu_{\alpha}}_L=(K, G)
\left(\begin{array}{c}
{\nu_i}_L \\
{N_i}_L \end{array} \right) \quad \left(\begin{array}{c} i=1,2,3 \\
\alpha=1,2,...6 \end{array} \right)
\label{15}\end{aligned}$$ and the leptonic charged current interactions are given by: $${\cal L}_W = - \frac{g}{\sqrt{2}} \left( \overline{l_{iL}}
\gamma_{\mu} K_{ij} {\nu_j}_L +
\overline{l_{iL}} \gamma_{\mu} G_{ij} {N_j}_L \right) W^{\mu}+h.c.
\label{phys}$$ with $K$ and $G$ being the charged current couplings of charged leptons to the light neutrinos $\nu_j$ and to the heavy neutrinos $N_j$, respectively.
In the seesaw limit the matrix $K$ coincides to an excellent approximation with the unitary matrix $U_{\nu}$ that diagonalises $m_{eff}$ of Eq. (\[meff\]): $$-U_{\nu}^\dagger \ m_D \frac{1}{M_R} m^T_D \ U_{\nu}^* =d \label{14}$$ and the matrix $G$ verifies the exact relation: $$G=m_D T^* D^{-1} \label{exa}$$ and is therefore very suppressed.
In a general framework, with ${\cal M}$ symmetric, without the zero block present in Eq. (\[calm\]) the $3 \times 6$ physical matrix $(K, G)$ of the $6 \times 6$ unitary matrix $V$ would depend on six independent mixing angles and twelve independent CP violating phases [@Branco:1986my]. This would be possible with a further extention of the SM including a Higgs triplet. The presence of the zero block reduces the number of independent CP violating phases to six [@Endoh:2000hc]. In the seesaw framework massive neutrinos lead to the possibility of CP violation in the leptonic sector both at low and at high energies. CP violation at high energies manifests itself in the decays of heavy neutrinos and is sensitive to phases appearing in the matrix $G$.
Low Energy Leptonic Physics
===========================
The light neutrino masses are obtained from the diagonalisation of $m_{eff}$ defined by Eq. (\[meff\]) which is an effective Majorana mass matrix. The unitary matrix $U_{\nu}$ that diagonalises $m_{eff}$ in the WB where the charged lepton masses are already diagonal real and positive is known as the Pontecorvo, Maki, Nakagawa, Sakata (PMNS) matrix [@pmns], and can be parametrised as [@Yao:2006px]:
$$\begin{aligned}
U_{\nu} =\left(
\begin{array}{ccc}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \delta} \\
-s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta}
& \quad c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta} \quad
& s_{23} c_{13} \\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta}
& -c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta}
& c_{23} c_{13}
\end{array}\right) \ \cdot \ P
\label{std}\end{aligned}$$
with $P = \mathrm{diag} \ (1,e^{i\alpha}, e^{i\beta})$, $\alpha $ and $\beta$ are phases associated to the Majorana character of neutrinos. There are three CP violating phases in $U_{\nu}$.
Experimentally it is not yet known whether any of the three CP violating phases of the leptonic sector is different from zero. The current experimental bounds on neutrino masses and leptonic mixing are [@Yao:2006px]: $$\begin{aligned}
\Delta m^2_{21} & = & 8.0 ^{+0.4}_{-0.3} \times 10^{-5}\ {\rm eV}^2 \\
\sin^2 (2 \theta_{12}) & = & 0.86 ^{+0.03}_{-0.04} \\
|\Delta m^2_{32}| & = & (1.9 \ \ \mbox{to} \ \ 3.0) \times 10^{-3}\
{\rm eV}^2 \\
\sin ^2 ( 2 \theta_{23}) & > & 0.92 \\
\sin ^2 \theta_{13} & < & 0.05 \end{aligned}$$ with $\Delta m^2_{ij} \equiv m^2_j - m^2_i$. The angle $ \theta_{23} $ may be maximal, meaning $45^{\circ}$, whilst $ \theta_{12} $ is already known to deviate from this value. At the moment, there is only an experimental upper bound on the angle $ \theta_{13}$.
It is also not yet known whether the ordering of the light neutrino masses is normal, i.e, $m_1<m_2<m_3$ or inverted $m_3<m_1<m_2$. The scale of the neutrino masses is also not yet established. Direct kinematical limits from Mainz [@Kraus:2004zw] and Troitsk [@Lobashev:1999tp] place an upper bound on $m_{\beta}$ defined as: $$m_{\beta} \equiv \sqrt{\sum_{i} |U_{ei}|^2 m^2_i}$$ given by $m_{\beta} \leq 2.3$ eV (Mainz), $m_{\beta} \leq 2.2$ eV (Troitsk). The forthcoming KATRIN experiment [@Osipowicz:2001sq] is expected to be sensitive to $m_{\beta} > 0.2$ eV and to start taking data in 2010 [@Valerius:2007fw].
It is possible to obtain information on the absolute scale of neutrino masses from the study of the cosmic microwave radiation spectrum together with the study of the large scale structure of the universe. For a flat universe, WMAP combined with other astronomical data leads to [@Spergel:2006hy] $\sum_{i} m_i \leq 0.66 $ eV ($95\%$ CL).
Neutrinoless double beta decay can also provide information on the absolute scale of the neutrino masses. In the present framework, in the absence of additional lepton number violating interactions, it provides a measurement of the effective Majorana mass given by: $$m_{ee} = \left| m_1 U_{e1}^2 + m_2 U_{e2}^2 + m_3 U_{e3}^2 \right|$$ The present upper limit is $m_{ee} \leq 0.9$ eV [@Fogli:2004as] from the Heidelberg-Moskow [@KlapdorKleingrothaus:2000sn] and the IGEX [@Aalseth:2002rf] experiments. There is a claim of discovery of neutrinoless double beta decay by the Heidelberg-Moscow collaboration [@KlapdorKleingrothaus:2004wj]. Interpreted in terms of a Majorana mass of the neutrino, this implies $m_{ee}$ between 0.12 eV to 0.90 eV. This result awaits confirmation from other experiments and would constitute a major discovery.
It was shown that the strength of CP violation at low energies, observable for example through neutrino oscillations can be obtained from the following low energy WB invariant [@Branco:1986gr]: $$Tr[h_{eff}, h_l]^3= - 6i \Delta_{21} \Delta_{32} \Delta_{31}
{\rm Im} \{ (h_{eff})_{12}(h_{eff})_{23}(h_{eff})_{31} \} \label{trc}$$ where $h_{eff}=m_{eff}{m_{eff}}^{\dagger} $, $ h_l = m_l m^\dagger_l $, and $\Delta_{21}=({m_{\mu}}^2-{m_e}^2)$ with analogous expressions for $\Delta_{31}$, $\Delta_{32}$. The righthand side of this equation is the computation of this invariant in the special WB where the charged lepton masses are real and diagonal. In the case of no CP violation of Dirac type in the leptonic sector this WB invariant vanishes; on the other hand, it is not sensitive to the presence of Majorana phases. This quantity can be computed in any WB and therefore is extremely useful for model building since it enables one to investigate whether a specific ansatz leads to Dirac type CP violation or not, without the need to go to the physical basis. It is also possible to write WB invariant conditions sensitive to the Majorana phases. The general procedure was outlined in Ref. where it was applied to the quark sector. For three generations it was shown that the following four conditions are sufficient [@Branco:1986gr] to guarantee CP invariance: $$\begin{aligned}
{\rm Im \ tr } \left[ h_l \; (m_{eff} \; m^*_{eff}) \;
( m_{eff} \; h^*_l \; m^*_{eff})\right] & = & 0 \label{41} \\
{\rm Im \ tr } \left[ h_l \; (m_{eff} \; m^*_{eff})^2 \;
( m_{eff} \; h^*_l \; m^*_{eff}) \right] & = & 0 \label{42} \\
{\rm Im \ tr } \left[ h_l \; (m_{eff} \; m^*_{eff})^2 \
( m_{eff} \; h^*_l \; m^*_{eff}) \; (m_{eff} \; m^*_{eff})\right]
& = & 0 \label{43} \\
{\rm Im \ det } \left[ ( m^*_{eff} \; h_l \; m_{eff})
+ (h^*_l \; m^*_{eff} \; m_{eff} )\right] & = & 0 \label{44} \end{aligned}$$ provided that neutrino masses are nonzero and nondegenerate (see also Ref. ). In Ref. alternative WB invariant conditions necessary to guarantee CP invariance in the leptonic sector under less general circumstances are given.
Leptogenesis
============
The observed baryon asymmetry of the universe (BAU) is given by [@Bennett:2003bz]: $$\frac{n_{B}-n_{\overline B}}{n_{\gamma}}= (6.1 ^{+0.3}_{-0.2}) \times 10^{-10}.$$ It is already established that this observation requires physics beyond the SM in order to be explained. One of the most plausibe explanations is Leptogenesis [@Fukugita:1986hr] where out-of-equilibrium L-violating decays of heavy Majorana neutrinos generate a lepton asymmetry which is partially converted through sphaleron processes [@Kuzmin:1985mm] into a baryon asymmetry. The lepton number asymmetry $\varepsilon _{N_{j}}$, thus produced was computed by several authors [@Liu:1993tg; @Flanz:1994yx; @Covi:1996wh; @Pilaftsis:1997jf; @Buchmuller:1997yu]. Summing over all charged leptons one obtains for the asymmetry produced by the decay of the heavy Majorana neutrino $N_j$ into the charged leptons $l_i^\pm$ ($i$ = e, $\mu$, $\tau$):
$$\begin{aligned}
\varepsilon _{N_{j}} = & \nonumber \\
= \frac{g^2} {{M_W}^2} & \sum_{k \ne j} \left[
{\rm Im} \left((m_D^\dagger m_D)_{jk} (m_D^\dagger m_D)_{jk} \right)
\frac{1}{16 \pi} \left(I(x_k)+ \frac{\sqrt{x_k}}{1-x_k} \right)
\right]
\frac{1}{(m_D^\dagger m_D)_{jj}} = \nonumber \\
= \frac{g^2}{{M_W}^2} & \sum_{k \ne j} \left[ (M_k)^2
{\rm Im} \left((G^\dagger G)_{jk} (G^\dagger G)_{jk} \right)
\frac{1}{16 \pi} \left(I(x_k)+ \frac{\sqrt{x_k}}{1-x_k} \right)
\right]
\frac{1}{(G^\dagger G)_{jj}} \nonumber \\
\label{rmy}\end{aligned}$$
where $M_k$ denote the heavy neutrino masses, the variable $x_k$ is defined as $x_k=\frac{{M_k}^2}{{M_j}^2}$ and $ I(x_k)=\sqrt{x_k} \left(1+(1+x_k) \log(\frac{x_k}{1+x_k}) \right)$. From Equation (\[rmy\]) it can be seen that, when one sums over all charged leptons, the lepton-number asymmetry is only sensitive to the CP-violating phases appearing in $m_D^\dagger m_D$ in the WB, where $M_R $ is diagonal. Weak basis invariants relevant for leptogenesis were derived in [@Branco:2001pq]: $$\begin{aligned}
I_1 \equiv {\rm Im Tr}[h_D H_R M^*_R h^*_D M_R]=0 \\
I_2 \equiv {\rm Im Tr}[h_D H^2_R M^*_R h^*_D M_R] = 0 \\
I_3 \equiv {\rm Im Tr}[h_D H^2_R M^*_R h^*_D M_R H_R] = 0 \end{aligned}$$ with $h_D = m^\dagger_D m_D$ and $H_R = M^\dagger_R M_R$. These constitute a set of necessary and sufficient conditions in the case of three heavy neutrinos. See also [@Pilaftsis:1997jf].
The simplest realisation of thermal leptogenesis consists of having hierarchical heavy neutrinos. In this case there is a lower bound for the mass of the lightest of the heavy neutrinos [@Davidson:2002qv; @Hamaguchi:2001gw]. Depending on the cosmological scenario, the range for minimal $M_1$ varies from order $10^7$ Gev to $10^9$ Gev [@Buchmuller:2002rq; @Giudice:2003jh]. Furthermore, an upper bound on the light neutrino masses is obtained in order for leptogenesis to be viable. With the assumption that washout effects are not sensitive to the different flavours of charged leptons into which the heavy neutrino decays this bound is approximately $0.1$ ev [@Buchmuller:2003gz; @Hambye:2003rt; @Buchmuller:2004nz; @Buchmuller:2004tu]. However, it was recently pointed out [@Barbieri:1999ma; @Endoh:2003mz; @Fujihara:2005pv; @Pilaftsis:2005rv; @Vives:2005ra; @Abada:2006fw; @Nardi:2006fx; @Abada:2006ea; @Blanchet:2006be] that there are cases where flavour matters and the commonly used expressions for the lepton asymmetry, which depend on the total CP asymmetry and one single efficiency factor, may fail to reproduce the correct lepton asymmetry. In this cases, the calculation of the baryon asymmetry with hierarchical righthanded neutrinos must take into consideration flavour dependent washout processes. As a result, in this case, the previous upper limit on the light neutrino masses does not survive and leptogenesis can be made viable with neutrino masses reaching the cosmological bound of $\sum_{i} m_i \leq 0.66 $ eV. The lower bound on $M_1$ does not move much with the inclusion of flavour effects. Flavour effects bring new sources of CP violation to leptogenesis and the possibility of having a common origin for CP violation at low energies and for leptogenesis [@Pascoli:2006ie; @Branco:2006hz; @Branco:2006ce; @Uhlig:2006xf].
There are very interesting alternative scenarios to the minimal leptogenesis scenario briefly mentioned here. It was pointed out at this conference that an SU(2)-singlet neutrino with a keV mass is a viable dark matter candidate [@Kusenko:2007ay]. Some leptogenesis scenarios are compatible with much lower heavy neutrino masses than the values required for minimal leptogenesis.
Implications from Zero neutrino Yukawa Textures
===============================================
The general seesaw framework contains a large number of free parameters. The introduction of zero textures and/or the reduction of the number of righthanded neutrinos to two, allows to reduce the number of parameters. In this work only zero textures imposed in the fundamental leptonic mass matrices are considered and, in particular, zero textures of the Dirac neutrino mass matrix, $m_D$ in the WB where $M_R$ and $m_l$ are real and diagonal. Zero textures of the low energy effective neutrino mass matrix are also very interesting phenomenologically [@Frampton:2002yf]. The physical meaning of the zero textures that appear in most of the leptonic mass ans" atze was analysed in a recent work [@Branco:2007nn] where it is shown that some leptonic zero texture ans" atze can be obtained from WB transformations and therefore have no physical meaning.
In general, zero textures reduce the number of CP violating phases, as a result some sets of zero textures imply the vanishing of certain CP-odd WB invariants [@Branco:2005jr]. This is an important fact since clearly zero textures are not WB invariant, therefore in a different WB the zeros may not be present making it difficult to recognise the ansatz. Furthermore, it was also shown [@Branco:2005jr] that starting from arbitrary leptonic mass matrices, the vanishing of certain CP-odd WB invariants, together with the assumption of no conspiracy among the parameters of the Dirac and Majorana mass terms, one is automatically lead to given sets of zero textures in a particular WB.
Frampton, Glashow and Yanagida have shown [@Frampton:2002qc] that it is possible to uniquely relate the sign of the baryon number of the Universe to CP violation in neutrino oscillation experiments by imposing two zeros in $m_D$, in the seesaw framework with only two righthanded neutrino components. Two examples were given by these authors: $$\begin{aligned}
m_D =
\left(
\begin{array}{cc}
a & 0 \\
a^\prime & b \\
0 & b^\prime
\end{array}\right) \qquad \mbox{or} \qquad
m_D =\left(
\begin{array}{cc}
a & 0 \\
0 & b \\
a^\prime & b^\prime
\end{array}\right)
\label{fgy}\end{aligned}$$ The two zeros in $m_D$ eliminate two CP violating phases, so that only one CP violating phase remains. This is the most economical extension of the standard model leading to leptogenesis and at the same time allowing for low energy CP violation. Imposing that the model accommodates the experimental facts at low energy strongly constrains its parameters.
In Ref. minimal scenarios for leptogenesis and CP violation at low energies were analysed in some specific realisations of seesaw models with three righthanded neutrinos and four zero textures in $m_D$, where three of the zeros are in the upper triangular part of the matrix. This latter particular feature was motivated by the fact that there is no loss of generality in parametrising $m_D$ as: $$m_D = U\,Y_{\triangle}\,, \label{mDtri}$$ with $U$ a unitary matrix and $Y_\triangle$ a lower triangular matrix, i.e.: $$Y_{\triangle}= \left(\begin{array}{ccc}
y_{11} & 0 & 0 \\
y_{21}\,e^{i\,\phi_{21}} & y_{22} & 0 \\
y_{31}\,e^{i\,\phi_{31}} & y_{32}\,e^{i\,\phi_{32}} & y_{33}
\end{array}
\right)\,, \label{Ytri1}$$ where $y_{ij}$ are real positive numbers. Choosing $U=1$ reduces the number of parameters in $m_D$. Moreover, $U$ cancels out in the combination $m^\dagger_D m_D$ relevant in the case of unflavoured leptogenesis, whilst it does not cancel in $m_{eff}$. Therefore choosing $U=1$ allows for a connection between low energy CP violation and leptogenesis to be established since in this case the same phases affect both phenomena. The nonzero entries of $m_D$ were written in terms of powers of a small parameter a la Frogatt Nielsen [@Froggatt:1978nt] and chosen in such a way as to accommodate the experimental data. Viable leptogenesis was found requiring the existence of low energy CP violating effects within the range of sensitivity of the future long baseline neutrino oscillation experiments under consideration.
In order to understand how the connection between CP violation required for leptogenesis and low energy physics is established in the presence of zeros in the matrix $m_D$, the following relation derived from Eq. (\[14\]) in the WB where $M_R$ and $m_l$ are real positive and diagonal is important: $$m_D = i U_{\nu} {\sqrt d} R {\sqrt D_R}
\label{udr}$$ with $R$ an orthogonal complex matrix, ${\sqrt D_R }$ a diagonal real matrix verifying the relation ${\sqrt D_R } {\sqrt D_R }= D_R $ and ${\sqrt d }$ a real matrix with a maximum number of zeros such that ${\sqrt d} {\sqrt d}^T = d $. This is the well known Casas and Ibarra parametrisation [@Casas:2001sr]. From this equation it follows that: $$m^\dagger_D m_D = {\sqrt D_R} R^{\dagger}{\sqrt d}^T {\sqrt d} R
{\sqrt D_R} \label{drr}$$ Since the CP violating phases relevant for leptogenesis in the unflavoured case are those contained in $m^\dagger_D m_D$, it is clear that leptogenesis can occur even if there is no CP violation at low energies i.e. no Majorana- or Dirac- type CP phases at low energies [@Rebelo:2002wj]. Unflavoured leptogenesis requires the matrix $R$ to be complex. In flavoured leptogenesis the separate lepton $i$ family asymmetry generated from the decay of the $k$th heavy Majorana neutrino depends on the combination [@Fujihara:2005pv] Im$\left( (m_D^\dagger m_D)_{k k^\prime}(m_D^*)_{ik} (m_D)_{ik^\prime}\right) $ as well as on Im$\left( (m_D^\dagger m_D)_{k^\prime k}(m_D^*)_{ik}
(m_D)_{ik^\prime}\right) $. The matrix $U_\nu$ does not cancel in each of these terms and it was shown that it is possible to have viable leptogenesis even in the case of real $R$, with CP violation in the PMNS matrix as the source of CP violation required for leptogenesis.
From Eq. (\[udr\]) it is clear that one zero in $(m_{D})_{ij}$ corresponds to having an orthogonality relation between the ith row of the matrix $U_{\nu }\sqrt{d}$ and the jth column of the matrix $R$: $$(m_{D})_{ij}=0~:\qquad (U_{\nu })_{ik}\sqrt{d}_{kl}R_{lj}=0
\label{orto}$$ Ibarra and Ross [@Ibarra:2003up] showed that, in the seesaw case with only two righthanded neutrinos, a single zero texture, has the special feature of fixing the matrix $R$, up to a reflection, without imposing any further restriction on light neutrino masses and mixing. The predictions from models with two zero textures in $m_D$ were also analysed in detail in their work, including the constraints on leptogenesis and lepton flavour violating processes. The number of all different two texture zeros is fifteen. Two zeros imply two simultaneous conditions of the type given by Eq. (\[orto\]). Compatibility of these two conditions implies restrictions on $U_\nu$ and $\sqrt{m_i}$. Only five of these cases turned out to be allowed experimentally, including the two cases of Eq. (\[fgy\]) in this reference.
All of these two zero texture ansätze satify the following WB invariant condition [@Branco:2005jr]: $$I_1 \equiv \mbox{tr} \left[ m_D M^\dagger_R M_R m^\dagger_D, h_l
\right] ^3 = 0
\label{inv}$$ with $h_l = m_l m^\dagger_l$, as before. It was also shown [@Branco:2005jr] that for arbitrary complex leptonic mass matrices, assuming that there are no special relations among the entries of $M_R$ and those of $m_D$ this condition automatically leads to one of the two zero anz" atze classified in Ref. . The assumption that $M_R$ and $m_D$ are not related to each other is quite natural, since $m_D$ and $M_R$ originate from different terms of the Lagrangian.
There are other CP-odd WB invariants which vanish for all of the two zero textures just mentioned, even if they arise in a basis where $M_R$ is not diagonal. An example is the following WB invariant condition [@Branco:2005jr]: $$I^\prime \equiv \mbox{tr} \left[ m_D m^\dagger_D, h_l
\right] ^3 =0
\label{wbi}$$ which is verified for any texture with two zeros in $m_D$ in a WB where $m_l$ is diagonal, while $M_R$ is arbitrary.
The case of zero textures with three righthanded neutrinos was also considered in Ref . In this case the WB invariant $I_1$ always vanishes for three zero textures in $m_D$ with two orthogonal rows, which implies that one row has no zeros. The case of three zeros corresponding to two orthogonal columns of $m_D$, which in this case implies that one column has no zeros leads to the vanishing of a new invariant $I_2$, defined by: $$I_{2}\equiv \mathrm{ tr \ }\left[ M_{R}^{\dagger }M_{R}\ ,\ \ m_{D}^{\dagger
}m_{D}\right]^3 \label{i2}$$
Four zero textures in the context of seesaw with three righthanded neutrinos are studied in detail in Ref. . It is shown that four is the maximum number of zeros in textures compatible with the observed leptonic mixing and with the additional requirement that none of the neutrino masses vanishes. It is also shown that such textures lead to important constraints both at low and high energies, and allow for a tight connection between leptogenesis and low energy parameters. It is possible in all cases to completely specify the matrix $R$ in terms of light neutrino masses and the PMNS matrix. These relations are explicitly given in Ref. .
Acknowledgements {#acknowledgements .unnumbered}
================
The author thanks the Organizers of the Sixth International Heidelberg Conference on Dark Matter in Astro and Particle Physics which took place in Sydney, Australia for the the warm hospitality and the stimulating scientifical environment provided. This work was partially supported by Fundaç\~ ao para a Ci\^ encia e a Tecnologia (FCT, Portugal) through the projects PDCT/FP/63914/2005, PDCT/FP/63912/2005 and CFTP-FCT UNIT 777 which are partially funded through POCTI (FEDER).
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abstract: |
Image alignment tasks require accurate pixel correspondences, which are usually recovered by matching local feature descriptors. Such descriptors are often derived using supervised learning on existing datasets with ground truth correspondences. However, the cost of creating such datasets is usually prohibitive. In this paper, we propose a new approach to align two images related by an unknown 2D homography where the local descriptor is learned from scratch from the images and the homography is estimated simultaneously. Our key insight is that a siamese convolutional neural network can be trained jointly while iteratively updating the homography parameters by optimizing a single loss function. Our method is currently weakly supervised because the input images need to be roughly aligned.
We have used this method to align images of different modalities such as RGB and near-infra-red (NIR) without using any prior labeled data. Images automatically aligned by our method were then used to train descriptors that generalize to new images. We also evaluated our method on RGB images. On the HPatches benchmark \[2\], our method achieves comparable accuracy to deep local descriptors that were trained offline in a supervised setting.
author:
- 'Jing Dong$^{1,2}$ Byron Boots$^{1}$ Frank Dellaert$^{1}$ Ranveer Chandra$^{2}$ Sudipta N. Sinha$^{2}$'
bibliography:
- 'ref.bib'
nocite: '[@*]'
title: Learning to Align Images using Weak Geometric Supervision
---
|
---
abstract: 'We study the behaviour of heavy inertial particles in the flow field of two like-signed vortices. In a frame co-rotating with the two vortices, we find that stable fixed points exist for these heavy inertial particles; these stable frame-fixed points exist only for particle Stokes number $St<St_{cr}$. We estimate $St_{cr}$ and compare this with direct numerical simulations, and find that the addition of viscosity increases the $St_{cr}$ slightly. We also find that the fixed points become more stable with increasing $St$ until they abruptly disappear at $St=St_{cr}$. These frame-fixed points are between fixed points and limit cycles in character.'
author:
- 'S. Ravichandran'
- Prasad Perlekar
- Rama Govindarajan
bibliography:
- '2vor\_paper.bib'
title: Attracting fixed points for heavy particles in the vicinity of a vortex pair
---
Introduction\[sec:Introduction\]
================================
Vortices are building blocks of turbulent fluid flow. During their evolution, vortices stretch, rotate and interact with other vortices. Energy in turbulent flows is transferred to larger and smaller scales by vortex mergers and stretching respectively. In the Earth’s atmosphere, in industrial processes and in water bodies, turbulent flows often carry particles such as dust or aerosols with them. Such particles are typically heavier than the fluid which carries them, and this paper is devoted to the effect of vortices on heavy particles. A large number of simulations and experiments [@Davilla2001; @Benczik2002; @Bec2003; @Chen2006; @Derevyanko2007; @Goto2008; @Tallapragada2008; @Toschi2009; @Eidelman2010; @Perlekar2011; @Gibert2012] have shown that the transport of inertial particles in two- and three-dimensional turbulence is not like the transport of inertialess particles. In particular, inertial particles cluster. The primary reason for this in a turbulent flow is vorticity, and the tendency of heavy inertial particles to leave the neighborhood of a vortex. What happens when there is more than one vortex? Could heavy particles in fact have long residence times near vortices?
We choose the simplest flow with more than one vortex, namely the flow generated by two identical vortices of the same sign at a distance $2R$ from each other. Kinematics dictates that these vortices, when there is no viscosity, will cause each other to rotate around the origin at an angular velocity $\Omega$. This flow, for particles of a given small Stokes number, is shown below to display ‘fixed’ points in a moving frame of reference. An important consequence of this is that particles cluster in a location of low vorticity, but close to the vortices. These cluster points rotate with the flow. The location of cluster is different for different Stokes number, and vanishes beyond a critical Stokes number. The behavior is first analyzed in an inviscid framework, and followed by viscous simulations to show that the same behavior is displayed there as well.
The rest of the paper is organized as follows. We first describe our system and approach, and then present an analytical investigation of heavy particles in the like-signed vortex-pair flow. This is followed by a presentation of our point vortex simulations and a linear analysis. We end with a discussion of our direct numerical simulations, and conclusions.
Elliptic Fixed Points in Lab and Rotating Frames of Reference
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We begin by describing the system and the equations we use for heavy particles – first in the lab-frame and then in a frame rotating with the vortices – and then discuss why elliptic fixed points in a lab-fixed frame of reference are very different from those in a rotating frame of reference. Heavy particles cannot cluster in the vicinity of the former as proved in Ref.[@Sapsis2010] but we show that they can and do cluster in the vicinity of the latter.
In the lab frame, we have two like-signed point vortices, each of circulation $\Gamma$ and placed at a distance of $2R$ from the other, describing motion on a circle of radius $R$, with a time period $T=8\pi^{2}R^{2}/\Gamma$. The vortices remain in antiphase from each other. The fluid velocity at a point $\mathbf{r}$ is given by: $$\begin{aligned}
\mathbf{u}_{\mbox{lab}} & = & \frac{d\mathbf{r}}{dt}=\frac{\Gamma}{2\pi}\mathbf{e}_{z}\times\left(\frac{\mathbf{r}-\mathbf{R}}{\left|\mathbf{r}-\mathbf{R}\right|^{2}}+\frac{\mathbf{r}+\mathbf{R}}{\left|\mathbf{r}+\mathbf{R}\right|^{2}}\right)\mbox{,}\label{velocity field, lab frame}\end{aligned}$$ where the vortices are at $\pm\mathbf{R=}\left(\pm X,\pm Y\right)$, and $\mathbf{e}_{z}$ is the unit vector pointing out of the page. The subscript ‘${\mbox{lab}}$’ denotes that something is measured in the lab-fixed frame of reference.
The motion of rigid, inertial point particles is modeled by using the Maxey-Riley equations [@Maxey1983] in the lab frame: $$\begin{aligned}
\frac{d\mathbf{\mathbf{x}}}{dt} & = & \mathbf{\mathbf{v}}\nonumber \\
\frac{d\mathbf{\mathbf{v}}}{dt} & = & -\frac{1}{St}(\mathbf{v}-\mathbf{u})+\beta\frac{D\mathbf{u}}{Dt}\label{eq:particle motion}\end{aligned}$$ where $\beta=3\rho_{f}/(2\rho_{p}+\rho_{f})$, $\rho_{p}$ and $\rho_{f}$ are the particle and fluid densities respectively, $\mathbf{v}$ and $\mathbf{u}$ are the particle and fluid velocities respectively, and $\mathbf{x}$ the location of a particle, $St=\tau/T$ is the Stokes number, $\tau=\frac{2}{9}\frac{a^{2}}{\nu}\frac{\rho_{p}}{\rho_{f}}$ is the relaxation time of the particle, and $T$ is a characteristic time scale of the flow, taken here to be the time period of rotation. Light particles ($\rho_{p}\ll\rho_{f}$ i.e., $\beta\approx3$) cluster in the regions of vortices whereas, heavy particles, such as aerosols, $\rho_{p}\gg\rho_{f}$ ($\beta=0$) are expelled from vortical regions [@Sapsis2010].
In the frame of reference rotating with the same angular velocity as the vortices [\[]{}$\Omega=\Gamma/(4\pi R^{2})$[\]]{}, the flow field is divided into water-tight compartments by the separatices – invariant manifolds, see e.g. [@Rom-Kedar1990; @Newton2001] – shown in figure \[fig:invariant manifolds\]. The equations of motion may be transformed to this rotating frame and are given below. Quantities with a $\hat{}$ are measured in the rotating frame.
$$\begin{aligned}
\frac{d\hat{\mathbf{x}}}{d\hat{t}} & =\hat{\mathbf{v}}\nonumber \\
\frac{d\hat{\mathbf{v}}}{d\hat{t}} & =\frac{\hat{\mathbf{u}}-\hat{\mathbf{v}}}{St}-2\bm{\Omega}\times\mathbf{\hat{v}}+\Omega^{2}\mathbf{\hat{r}}\label{eq:Particle_motion_rotating_frame}\end{aligned}$$
For $St=0$, the system has two fixed points (centers) at $(0,\pm\sqrt{3}R)$ in the rotating frame of reference. The elliptic nature of these fixed points may be seen by linearizing the velocity field at these points [\[]{}see section . These ‘fixed-points’ are actually rotating at the same angular velocity as the vortices in the lab-fixed frame, so what we have are moving fixed points!
Incidentally we would like to distinguish our rotating frame of reference from the one that is normally used to simulate, for instance, Earth’s rotation. In the latter, the rotation of the lab is imposed as an additional forcing. Our lab is not rotating, but our flow is steady in a non-inertial frame that rotates with the angular velocity of the vortex pair. We also note that the points at $(0,\pm\sqrt{3}R)$ are fixed points for fluid particles and tracer particles, and not necessarily for inertial particles.
![Invariant manifolds in the rotating frame in the flow around two identical point vortices. The vortices are indicated by the small circles at $\left(\pm0.5,0\right)$, and the elliptic fixed points of our interest by the filled circles at $\left(0,\pm\sqrt{3}/2\right)$.\[fig:invariant manifolds\]](01a_hamiltonian){width="50.00000%"}
As previously mentioned, elliptic fixed points in the rotating frame of reference are qualitatively different from elliptic fixed points in a lab fixed frame of reference. Sapsis & Haller [@Sapsis2010] show that the fixed points in a region of the flow consisting of elliptic streamlines cannot be attractors for heavy particles of small Stokes number. Their argument is that since $\beta\simeq0$ for heavy particles, at $O\left(St\right)$ $\mathbf{u}$ may be used to replace $\mathbf{v}$ on the left hand side of eq. (\[eq:particle motion\]), to get $$\mathbf{v}=\mathbf{u}-St\frac{D\mathbf{u}}{Dt}+\mathcal{O}\left(St^{2}\right),\label{eq: inertial equation}$$ where $\frac{d\mathbf{v}}{dt}$ was replaced with $\frac{D\mathbf{u}}{Dt}$ in the second term. This is called the ‘inertial equation’. The divergence of $\mathbf{v}$, since $\mathbf{u}$ is divergence-free in incompressible flow, becomes $$\nabla\cdot\mathbf{v}=-St\nabla\cdot\left(\mathbf{u}\cdot\nabla\mathbf{u}\right)=-St\frac{\partial u_{i}}{\partial x_{j}}\frac{\partial u_{j}}{\partial x_{i}}=StQ\mbox{,}\label{eq:Okubo-Weiss}$$ where $Q=\left(\omega^{2}-s^{2}\right)$ is the Okubo-Weiss parameter, $s$ and $\omega$ being the symmetric and anti-symmetric parts of the strain-rate tensor respectively. A word of caution: in a general turbulent flow we must remember that particle velocities do not form a field, in the sense that there can be two very different particle velocities at the same location, so strictly speaking we may not define a quantity called divergence for particle velocity. However, at small Stokes numbers, the inertial equation allows $\mathbf{v}$ to be approximated to a field, and thus lets us relate this quantity to particle clustering for negative divergence and to particles leaving the neighborhood at positive divergence. At an elliptic fixed point in a lab-fixed frame, we must have $Q>0$, and Liouville’s theorem may be applied to a small volume in its vicinity to show that it cannot attract heavy particles. There is thus no attracting elliptic fixed point for small Stokes number heavy particles in a fixed frame of reference.
How about in a rotating frame of reference? We saw in the flow under consideration that there are two elliptic fixed points for tracer particles in the rotating frame. We next ask whether there are fixed points for inertial particles as well, i.e., are there locations where an inertial particle would remain forever. This may be done by solving for fixed points of eq. \[eq:Particle\_motion\_rotating\_frame\], giving $$\frac{\hat{\mathbf{u}}}{St}=-\Omega^{2}\mathbf{r}.\label{eq:fixed points exact}$$ Equation (\[eq:fixed points exact\]) has to be solved numerically. This was done using the MATLAB function minimisation routine “fsolve”. Solutions for eq. \[eq:fixed points exact\] exist for $St<St_{cr}=0.04264543$. For $St>St_{cr}$, the minima of $\hat{\mathbf{u}}+St\Omega^{2}\hat{\mathbf{r}}$ are greater than zero; i.e., there exist no solutions. Solutions of eq. \[eq:fixed points exact\] lying in the top half-plane is shown in figure \[fig:Fixed-point-comparison\]. A symmetrically placed fixed point exists with $x<0,y<0$. We see that the fixed points start on the y-axis at $\sqrt{3}R$ for particles of $St=0$, and move progressively away from the vertical as $St$ increases. Thus particles of increasing inertia display fixed points which drift further and further away from the axis of symmetry. One may imagine that with inertia, the particles find it harder to “keep up” with the rotating frame and drift in the opposite direction. Note that the fixed points lie within the region of the flow where the streamlines form closed orbits. We therefore refer to them as particle fixed points in the elliptic region. We also alternatively refer to them as the finite $\mathbf{r}$ fixed points. Now that we have the exact solution for the fixed points, we may assess how good the $\mathcal{O}\left(St\right)$ approximation is, by substituting $\mathbf{v}$ from eq. (\[eq: inertial equation\]) into (\[eq:fixed points exact\]). We obtain fixed points, also shown in figure \[fig:Fixed-point-comparison\], that are accurate to $\mathcal{O}\left(St\right)$, as expected.
![\[fig:Fixed-point-comparison\]Locations of fixed points (in radians) for inertial particles with different Stokes number obtained from eq. (circles). We also show the location of fixed points obtained by using the small Stokes number ($\mathcal{O}(St)$) approximation (cross). There are no fixed points for $St>St_{cr}$. [\[]{}Here and in all figures to follow, simulations were done with $\Gamma=2\pi$. [\]]{}](02a_radius_vs_Stokes "fig:"){width="0.5\linewidth"}![\[fig:Fixed-point-comparison\]Locations of fixed points (in radians) for inertial particles with different Stokes number obtained from eq. (circles). We also show the location of fixed points obtained by using the small Stokes number ($\mathcal{O}(St)$) approximation (cross). There are no fixed points for $St>St_{cr}$. [\[]{}Here and in all figures to follow, simulations were done with $\Gamma=2\pi$. [\]]{}](02b_angle_vs_Stokes "fig:"){width="0.5\linewidth"}
Point vortex simulations\[sec:Point-vortex-simulations\]
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![\[fig:Poincare-sections\] (Left) Particle locations after $100T$ for $St=1/100\pi$ (black) and $1/7.5\pi$ (red). The fixed points are the filled circle and the crosses respectively. The phase chosen is when the point vortices, indicated by blue squares, lie on the horizontal axis. (In the figure, the thickness of outer ring-clusters is exaggerated.) (Right) Particle density versus radius after 10 time periods. The density profiles narrow with time. At a higher Stokes number, the particles are more sharply clustered. Also as Stokes increases, the nonlinear wave propagates outwards faster, so the second peak is seen at a bigger radius.](03_Poincare_100_7d5 "fig:"){width="0.4\linewidth"} ![\[fig:Poincare-sections\] (Left) Particle locations after $100T$ for $St=1/100\pi$ (black) and $1/7.5\pi$ (red). The fixed points are the filled circle and the crosses respectively. The phase chosen is when the point vortices, indicated by blue squares, lie on the horizontal axis. (In the figure, the thickness of outer ring-clusters is exaggerated.) (Right) Particle density versus radius after 10 time periods. The density profiles narrow with time. At a higher Stokes number, the particles are more sharply clustered. Also as Stokes increases, the nonlinear wave propagates outwards faster, so the second peak is seen at a bigger radius.](03_particle_densities_T10 "fig:"){width="0.5\linewidth"}
We perform inviscid numerical simulations for a range of Stokes numbers. We start with $5\times10^{4}$ particles uniformly distributed over a region encompassing the invariant manifolds. We use $\Gamma=2\pi$ unless otherwise mentioned. Figure \[fig:Poincare-sections\] shows the particle locations after $100$ time periods of rotation, for two Stokes numbers. In each case there are three fixed points at which heavy particles cluster: the fixed point at $|\mathbf{r}|=\infty$ (which exists because any nonzero inertia in a heavy particle centrifuges the particle out from a region of rotating motion), as well as the symmetric pair we have discussed. This pair, lying within the elliptic flow region, is indicated in the figure, and although not apparent, a very large number of particles are clustered at these. The new fixed points for inertial particles are thus attracting fixed points. The other fixed point, $|\mathbf{r}|=\infty$, attracts all particles which begin at a large distance away from the origin, since for them the system may effectively be replaced by a single vortex of twice the circulation, i.e., $2\Gamma$. The evolution of these particles would take the form of a nonlinear wave [@Raju1997], with a clumping that depends on Stokes number. In addition, the $|\mathbf{r}|=\infty$ fixed point has a basin of attraction within our region of interest. Particles which will travel towards $|\mathbf{r}|=\infty$ are seen as a circle of clustered points in figure \[fig:Poincare-sections\](left). The radius of this cluster is a slowly increasing function of time, going as $r\sim t^{1/4}St$ at large $r$. This is because, at large $r$, the two vortices act predominantly act as one larger vortex of twice the strength and the equation of motion in radial coordinates for particles around a single vortex is $dr/dt\sim St/r^{3}$ (see, e.g., [@Raju1997]). That particles form dense clusters at the fixed points is seen in the radial density profile of figure \[fig:Poincare-sections\](right), at a particular instant of time. Two regions of clumping are evident. The inner one corresponding to the elliptic-region fixed points remains there, but the outer spike slowly moves towards $|\mathbf{r}|=\infty$. With time, the clumps become narrower and taller. With increasing inertia too, the clumps become sharper, but not necessarily taller. At higher inertia, the profile increasingly resembles a double spike. The total number of particles clustered at the inner fixed points decreases with inertia, as we shall discuss soon, and at some Stokes number we no longer have any fixed points at finite $r$.
Applying the inertial equation (eq.\[eq: inertial equation\]) to this problem, rather than the complete force balance, has the effect of omitting the second time derivative of the radial location of a particle, and for a single vortex Raju & Meiburg [@Raju1997] show that this approximation becomes exact at long times.
Basin boundaries of each fixed point are shown in figure \[fig:Basin-boundaries\]. Particles which begin their lives inside the regions shown in red are attracted to one of the fixed points at finite radius and particles in the other regions drift away to infinity. As the Stokes number increases, the basin of attraction of the finite $r$ fixed points shrinks in size. These basins of attraction disappear completely for $St>St_{cr}$, along with the fixed points themselves. At very low inertia, the spirals are very tightly wound. It may be argued from the radial velocity of an inertial particle moving around a single vortex that the spacing between two crossings must be proportional to $St$. Thus two particles which begin life very close to each other, but in the basin boundaries of different attractors, have vastly different fortunes. One gets trapped forever in the vicinity of the vortices while the other is slowly centrifuged out to infinity.
![Basins of attraction of the fixed points for $St=1/300\pi$ (top left), $1/100\pi$ (top right), $1/20\pi$ (bottom left), and $1/7.5\pi$ (bottom right). The red region shows the basin of attraction of one of the finite $\mathbf{r}$ fixed points. Particles in the blue region escape to infinity. \[fig:Basin-boundaries\]](04_2vor_rest_1e+04_320_by_160_fac300_ub0\lyxdot 0_T500_500 "fig:"){width="6.4cm"} ![Basins of attraction of the fixed points for $St=1/300\pi$ (top left), $1/100\pi$ (top right), $1/20\pi$ (bottom left), and $1/7.5\pi$ (bottom right). The red region shows the basin of attraction of one of the finite $\mathbf{r}$ fixed points. Particles in the blue region escape to infinity. \[fig:Basin-boundaries\]](04_2vor_rest_1e+04_320_by_160_fac100_ub0\lyxdot 0_T500_500 "fig:"){width="6.4cm"} ![Basins of attraction of the fixed points for $St=1/300\pi$ (top left), $1/100\pi$ (top right), $1/20\pi$ (bottom left), and $1/7.5\pi$ (bottom right). The red region shows the basin of attraction of one of the finite $\mathbf{r}$ fixed points. Particles in the blue region escape to infinity. \[fig:Basin-boundaries\]](04_2vor_rest_1e+04_320_by_20_fac20_ub0\lyxdot 0_T100_100 "fig:"){width="6.4cm"} ![Basins of attraction of the fixed points for $St=1/300\pi$ (top left), $1/100\pi$ (top right), $1/20\pi$ (bottom left), and $1/7.5\pi$ (bottom right). The red region shows the basin of attraction of one of the finite $\mathbf{r}$ fixed points. Particles in the blue region escape to infinity. \[fig:Basin-boundaries\]](04_2vor_rest_1e+04_320_by_20_fac7\lyxdot 5_ub0\lyxdot 0_T100_100 "fig:"){width="6.4cm"}
In figure \[fig:n\_inside\], we plot the number of particles starting from a uniform grid over a domain $\left(-1.6,1.6\right)\times\left(-1.6,1.6\right)$ that are attracted to one of the two fixed points. Because we started with uniformly distributed points, this is proportional to the area of its basin of attraction. It is seen that close to the critical Stokes number, the rate of decrease of the number of particles is very high, the graph being practically vertical, indicative of a finite-Stokes singularity. The reason for this singular behavior is not clear to us at this point.
![The number of particles in each simulation that are attracted to the finite $\mathbf{r}$ fixed points. This number is proportional to the area of the basin of attraction. A quadratic fit is provided just to guide the eye. The vertical line indicates $St=St_{cr}$.\[fig:n\_inside\]](05_N_inside_semilogx){width="9cm"}
Linear analysis near the particle fixed points\[sub:Linear-analysis\]
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Behavior in the vicinity of the inertial particle fixed points is obtained by linearising the particle dynamics in the rotating frame (eq. ) along the lines of [@Bec2005]. At the fixed point, we have $\hat{\mathbf{v}}_{fp}=0$ and $\hat{\mathbf{u}}_{fp}=-St\mbox{\ensuremath{\left(\Omega^{2}\hat{\mathbf{r}}\right)}}$. This gives, for the linear perturbations,
$$\begin{aligned}
\frac{d\left(\delta\hat{\mathbf{x}}\right)}{d\hat{t}} & =\delta\hat{\mathbf{v}}\nonumber \\
\frac{d\left(\delta\hat{\mathbf{v}}\right)}{d\hat{t}} & =\frac{\delta\hat{\mathbf{u}}-\delta\hat{\mathbf{v}}}{St}-2\bm{\Omega}\times\left(\delta\mathbf{\hat{v}}\right)+\Omega^{2}\delta\mathbf{\hat{r}}\label{eq:Linearised_particle_dynamics}\end{aligned}$$
The derivatives on the LHS of eq. are derivatives along particle trajectories. The perturbation fluid velocities in the RHS are calculated by differentiating the Biot-Savart law. The eigenvalues of the linearised velocity equations are plotted in figure (\[fig:fixed-point-bifurcation\]). One of the eigenvalues crosses zero at the crititcal Stokes number. It is also worth noting that the slopes of the eigenvalue curves diverge at $St=St_{cr}.$
![Real and imaginary parts of the eigenvalues of eq. \[eq:Linearised\_particle\_dynamics\]. The two vertical dashed lines are at values of $St$ where a) the two pairs complex conjugate eigenvalues become wholly real, and b) at $St=St_{cr}$. The values for these Stokes numbers may be found in the text. The figures at the bottom are zoomed-in near $St_{cr}$. The real part crosses zero (at an infinite slope) at $St_{cr}$. \[fig:fixed-point-bifurcation\]](06_real "fig:"){width="8cm"} ![Real and imaginary parts of the eigenvalues of eq. \[eq:Linearised\_particle\_dynamics\]. The two vertical dashed lines are at values of $St$ where a) the two pairs complex conjugate eigenvalues become wholly real, and b) at $St=St_{cr}$. The values for these Stokes numbers may be found in the text. The figures at the bottom are zoomed-in near $St_{cr}$. The real part crosses zero (at an infinite slope) at $St_{cr}$. \[fig:fixed-point-bifurcation\]](06_imag "fig:"){width="8cm"}
![Real and imaginary parts of the eigenvalues of eq. \[eq:Linearised\_particle\_dynamics\]. The two vertical dashed lines are at values of $St$ where a) the two pairs complex conjugate eigenvalues become wholly real, and b) at $St=St_{cr}$. The values for these Stokes numbers may be found in the text. The figures at the bottom are zoomed-in near $St_{cr}$. The real part crosses zero (at an infinite slope) at $St_{cr}$. \[fig:fixed-point-bifurcation\]](06_real_zoomed "fig:"){width="8cm"}![Real and imaginary parts of the eigenvalues of eq. \[eq:Linearised\_particle\_dynamics\]. The two vertical dashed lines are at values of $St$ where a) the two pairs complex conjugate eigenvalues become wholly real, and b) at $St=St_{cr}$. The values for these Stokes numbers may be found in the text. The figures at the bottom are zoomed-in near $St_{cr}$. The real part crosses zero (at an infinite slope) at $St_{cr}$. \[fig:fixed-point-bifurcation\]](06_imag_zoomed "fig:"){width="8cm"}
![\[fig:Spiral Nodes\]Particle trajectories for representative Stokes numbers of $St=1/50\pi$ (left) and $St=1/10\pi$ (right) showing that the fixed points in the rotating frame are spiral nodes. The crosses indicate starting positions for the particles. Trajectories in blue leave the vicinity of the vortices and centrifuge out to infinity. Trajectories in red fall into one of the fixed points.](07_Fac50_trajectories "fig:"){width="8cm"} ![\[fig:Spiral Nodes\]Particle trajectories for representative Stokes numbers of $St=1/50\pi$ (left) and $St=1/10\pi$ (right) showing that the fixed points in the rotating frame are spiral nodes. The crosses indicate starting positions for the particles. Trajectories in blue leave the vicinity of the vortices and centrifuge out to infinity. Trajectories in red fall into one of the fixed points.](07_Fac10_trajectories "fig:"){width="8cm"}
At $\mathcal{O}\left(St\right)$, a negative Okubo-Weiss parameter is a necessary condition for the existence of an attracting fixed point. We have seen from the argument of Sapsis & Haller [@Sapsis2010] that this cannot happen in a neighborhood of closed streamlines in a fixed frame of reference. Things are different in a rotating frame: we may have a negative divergence of particle velocity in a neighborhood of streamlines that are closed paths in the rotating frame. The velocity in the rotating frame is (quantities with $\hat{}$s are measured in the rotating frame) $$\hat{\mathbf{v}}=\hat{\mathbf{u}}-\tau\left(\frac{d\hat{\mathbf{u}}}{d\hat{t}}+2\vec{\Omega}\times\hat{\mathbf{u}}-\Omega^{2}\hat{\mathbf{r}}\right)\mbox{,}$$
giving, for the divergence of the particle velocity,
$$\hat{\nabla}\cdot\hat{\mathbf{v}}=-\tau\left(\frac{\partial\hat{u}_{i}}{\partial\hat{x}_{j}}\frac{\partial\hat{u}_{j}}{\partial\hat{x}_{i}}+2\hat{\nabla}\cdot\left(\bm{\Omega}\times\hat{\mathbf{u}}\right)-2\Omega^{2}\right)=-\tau\left(\frac{\partial\hat{u}_{i}}{\partial\hat{x}_{j}}\frac{\partial\hat{u}_{j}}{\partial\hat{x}_{i}}-2\bm{\Omega}\cdot\hat{\bm{\omega}}-2\Omega^{2}\right)$$ $$\hat{\nabla}\cdot\hat{\mathbf{v}}=-\tau\left(\frac{\partial\hat{u}_{i}}{\partial\hat{x}_{j}}\frac{\partial\hat{u}_{j}}{\partial\hat{x}_{i}}+2\Omega^{2}\right)\mbox{, }$$ where $\hat{\bm{\omega}}=-2\Omega\mathbf{e}_{z}$ is the vorticity in the rotating frame and the summation convection has been used. The first product on the right hand side may be shown to be equal to $S_{ij}S_{ij}-2\Omega^{2}$, making the divergence of particle velocity equal to $$\nabla\cdot\mathbf{\hat{v}}=St{\hat{Q}},$$ where $\hat{Q}$ is the Okubo-Weiss parameter in the rotating frame.
The quantity $(S_{ij}S_{ij})$ is plotted in figure \[fig:rotational-okubo-weiss\], and seen to be positive in the vicinity of the attracting fixed point, making the Okubo-Weiss parameter negative; we thus satisfy the necessary condition for the existence of attracting fixed points, in a neighborhood where fluid particles follow elliptical streamlines.
![Contour-plot of $S_{ij}S_{ij}$. The negative of this quantity, $\hat{Q}$, is the Okubo-Weiss parameter, which is proportional to the divergence of particle velocity. \[fig:rotational-okubo-weiss\]](08_SijSij){width="9cm"}
Navier-Stokes Simulations
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We now substantiate our results by studying the above flow in a more realistic setting: by including viscosity and beginning with Lamb-Oseen vortices, in contrast to the point vortices of section III. The flow obeys the two-dimensional Navier-Stokes equations $$D_{t}\omega=\nu\nabla^{2}\omega\mbox{ , }-\nabla^{2}\psi=\omega,\label{eq:NS}$$ where $\omega$ is the vorticity, $\nu$ is the kinematic viscosity, $\psi$ is the streamfunction, i.e. ${\bf u}=(-\partial_{y}\psi,\partial_{x}\psi)$, and $D_{t}\equiv\partial_{t}+{\bf u}\cdot\nabla$ is the material derivative. Particles obey eq. (\[eq:particle motion\]). We use a square domain with each side of length $L=2\pi$ and employ periodic boundary conditions. Eqs., and the continuity equation are numerically integrated using a pseudo-spectral method. Time-advancement is done using an exponential Adams-Bashforth scheme. Space is discretized with $N^{2}$ collocation grid points. We have verified (not shown here) that grid-convergence is achieved by conducting simulations with varying grid resolutions $N=128$ and $N=256$ and obtaining the same results. In what follows, we have used $N=256$ unless stated otherwise.
We initialize the simulation with two Gaussian vortices positioned at $(x_{1},y_{1})$ and $(x_{2},y_{2})$, of vorticity $$\begin{aligned}
\omega_{1}(x,y) & = & \omega_{0}\exp[-(r_{1}/r_{0})^{2}]\ {\rm {and}}\\
\omega_{2}(x,y) & = & \omega_{0}\exp[-(r_{2}/r_{0})^{2}].\end{aligned}$$ Here, $r_{0}$ is the width of the vortex, $\omega_{0}$ denotes the vortex amplitude, $r_{1}^{2}=(x-x_{1})^{2}+(y-y_{1})^{2}$ and $r_{2}^{2}=(x-x_{2})^{2}+(y-y_{2})^{2}$ and the separation between the centers of the vortices $\ell\equiv(y_{2}-y_{1})^{2}+(x_{2}-x_{1})^{2}$ is kept fixed at $\ell=0.9818$ (we obtain this value because we choose an integer number of grid-spacings). We fix $r_{0}=\pi/32$, $\omega_{0}=2^{11}/\pi\approx652$ (these values are tuned so that the timeperiod of rotation is $T\approx1$), $\nu=10^{-4}$ giving $Re=10^{4}$, and vary $St$ from $0$ to $0.05$. We start the simulation with $N_{p}=10^{3}$ particles distributed randomly over the entire domain and study how particle distribution evolves.
An identical same-signed vortex pair will ultimately merge, In the first stage of this process, the vortices go around each other in a fashion very similar to the inviscid case. During this phase, they also diffuse out slowly. When the radius of the vortices reaches a quarter of the separation between them, a second stage in the merger process begins, with a significant and sudden increase in the radial component of the velocity of the vortices, drawing them towards each other. The Reynolds number in this simulation is $Re=10^{4}$, which means, given the initial radius, that the vortices complete about 90 complete cycles, up to a time of $t=86$ before the second stage of the merger process begins. We note that particles with $St\sim\mathcal{O}\left(1/T\right)$, (where $T=86$ timeperiods) would have fallen into the fixed points before the merger process begins. The cluster at the fixed point thus has a long life before the merger process is completed, after which it begins a slow drift away from the merged vortex. We are interested here in what happens to the cluster during the first phase of the merger process. Figure \[fig:Particles\_vorticity\_t10\] shows a snapshot of the particle positions and contours of the vorticity at $t=10$ for $St=0.027$. The cluster at the fixed point is remarkably similar to the one seen in the inviscid case, as is evident when one compares this figure with fig. \[fig:Poincare-sections\]. Figure \[fig:radial\_location\_viscous\_fixed\_point\] shows the radial distance at which particles cluster (measured from the point halfway between the vortex centres) as a function of time, showing that, for this Stokes number$=0.03$, (which is below $St_{cr},$) the cluster at the fixed points is remarkably stable for a long time. Figure \[fig:Comparison\_viscous\_inviscid\] compares the fixed points obtained from the viscous and inviscid simulations. The critical Stokes number is slightly higher in viscous simulations.
![LEFT: Plot of particle positions and vorticity contours at $t=10$ for $St=0.027$. The fixed point can be seen at ($\pi\pm0.32$,$\pi\pm0.8)$.\
RIGHT: Plot of particle density showing versus radius.\[fig:Particles\_vorticity\_t10\]](09_vorticity_particles_St_0\lyxdot 027_t=10 "fig:"){width="9cm"}![LEFT: Plot of particle positions and vorticity contours at $t=10$ for $St=0.027$. The fixed point can be seen at ($\pi\pm0.32$,$\pi\pm0.8)$.\
RIGHT: Plot of particle density showing versus radius.\[fig:Particles\_vorticity\_t10\]](09_viscous_particle_density "fig:"){width="9cm"}
![The radial location of the fixed point time for a viscous simulation. Inset in the figure are the contours of vorticity at $t=10$ and $t=86$ for $St=0.03$. The maxima of vorticity are at $\omega=452$ and $\omega=132$ respectively. The dashed line is $r=\sqrt{3}/2$ \[fig:radial\_location\_viscous\_fixed\_point\]. Since we are interested to look at times equal to and larger than merger time, we use $N^{2}=1024^{2}$ collocation points.](10_R_fp_vs_t_tau=0\lyxdot 030){width="9cm"}
![Fixed point predictions (at $t=50$). We plot the angle the line joining the fixed points makes with the line joining the centres of the vortices. The dots are exact solutions. The diamonds are point-vortex solutions. The circles and squares are DNS results for grid resolutions of $256^{2}$ and $512^{2}$ respectively. \[fig:Comparison\_viscous\_inviscid\]](11_t=50_angles){width="9cm"}
Conclusion\[sec:conclusion\]
============================
We have shown that apart from fixed points in the lab frame, clustering can be governed by fixed points in a rotating frame. Fixed points for heavy particles of small Stokes number do not coincide with those for tracer particles, but lie in the vicinity. The location of these fixed points is within a region where fluid particles follow elliptical trajectories. Contrary to our understanding of elliptic fixed points in the lab-fixed frame, these fixed points in a rotating frame can be attractive, which is the reason for clustering. Note that these moving fixed points are not limit cycles, because the phase is fixed.
The study here is on a simple model flow, but has relevance to particle dynamics in turbulence. Persistence times for particles near vortices are important in turbulent flows. With attracting fixed points in the vicinity of point vortices, the persistence times are in principle infinite. In cloud dynamics, for instance, we believe these fixed points in rotating frames could contribute to droplet clustering and therefore change the droplet size distributions. Our study indicates that regions of particle clustering may have to be calculated for frames of reference that are themselves not fixed. These frame-fixed points may be expected to have the same effect on the carrier flow as fixed points in the lab frame.
The authors wish to thank the two anonymous referees who made very useful suggestions. The authors wish to thank Jeremie Bec for the suggestion that led to the stability analysis in section IIIA.
|
---
address: 'University College Cork, Cork, Ireland'
author:
- Benjamin McKay
bibliography:
- 'introduction-to-exterior-differential-systems.bib'
date:
- 5 September 2016
-
---
Introduction
============
We assume that the reader is familiar with elementary differential geometry on manifolds and with differential forms. These lectures explain how to apply the Cartan–Kähler theorem to problems in differential geometry. We want to decide if there are submanifolds of a given dimension inside a given manifold on which given differential forms vanish. The Cartan–Kähler theorem gives a linear algebra test: if the test passes, such submanifolds exist. I will not give a proof or give the most general statement of the theorem, as it is difficult to state precisely.
For a proof of the Cartan–Kähler theorem, see [@Cartan:1945], which we will follow very closely, and also the canonical reference work [@BCGGG:1991] and the canonical textbook [@Ivey/Landsberg:2003]. The last two also give proof of the Cartan–Kuranishi theorem, which we will only briefly mention.
Expressing differential equations using differential forms
==========================================================
Take a differential equation of second order $0=f{\ensuremath{\!{\ensuremath{\left(x,u,u_x,u_{xx}\right)}}}}$. To write it as a first order system, add a new variable $p$ to represent $u_x$, and a new equation: $$\begin{aligned}
u_x &= p, \\
0 &= f{\ensuremath{\!{\ensuremath{\left(x,u,p,p_x\right)}}}}.\end{aligned}$$ It is easy to generalize this to any number of variables and equations of any order: reduce any system of differential equation to a first order system.
To express a first order differential equation $0=f{\ensuremath{\!{\ensuremath{\left(x,u,u_x\right)}}}}$, add a variable $p$ to represent the derivative $u_x$, let $\vartheta=du-p \, dx$ on the manifold $$M=\Set{(x,u,p)|0=f{\ensuremath{\!{\ensuremath{\left(x,u,p\right)}}}}}$$ (assuming it is a manifold). A submanifold of $M$ of suitable dimension on which $0=\vartheta$ and $0 \ne dx$ is locally the graph of a solution. It is easy to generalize this to any number of variables and any number of equations of any order.
The Cartan–Kaehler theorem
==========================
An *integral manifold* of a collection of differential forms is a submanifold on which the forms vanish. An *exterior differential system* is an ideal ${\ensuremath{\mathcal{I}}}\subset \Omega^*$ of smooth differential forms on a manifold $M$, closed under exterior derivative, which splits into a direct sum $${\ensuremath{\mathcal{I}}}={\ensuremath{\mathcal{I}}}^0 \oplus {\ensuremath{\mathcal{I}}}^1 \oplus \dots \oplus {\ensuremath{\mathcal{I}}}^n$$ of forms of various degrees: ${\ensuremath{\mathcal{I}}}^p {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{\mathcal{I}}}\cap \Omega^p$. Any collection of differential forms has the same integral manifolds as the exterior differential system it generates. An exterior differential system is *analytic* if it is locally generated by real analytic differential forms.
Some trivial examples: the exterior differential system generated by
1. $0$,
2. $\Omega^*$,
3. the pullbacks of all forms via a submersion,
4. $dx^1 \wedge dy^1 + dx^2 \wedge dy^2$ in ${\ensuremath{\mathbb{R}^{4}}}$,
5. $dy-z \,dx$ on ${\ensuremath{\mathbb{R}^{3}}}$.
What are the integral manifolds of our trivial examples?
The elements of ${\ensuremath{\mathcal{I}}}^0$ are 0-forms, i.e. functions. All ${\ensuremath{\mathcal{I}}}$-integral manifolds lie in the zero locus of these functions. Replace our manifold $M$ by that zero locus (which might not be a manifold, a technical detail we will ignore); henceforth we add to the definition of *exterior differential system* the requirement that ${\ensuremath{\mathcal{I}}}^0=0$.
An *integral element* at a point $m \in M$ of an exterior differential system ${\ensuremath{\mathcal{I}}}$ is a linear subspace $E \subset T_m M$ on which all forms in ${\ensuremath{\mathcal{I}}}$ vanish. Every tangent space of an integral manifold is an integral element, but some integral elements of some exterior differential systems don’t arise as tangent spaces of integral manifolds.
What are the integral elements of our trivial examples?
The *polar equations* of an integral element $E$ are the linear functions $$w \in T_m M \mapsto \vartheta{\ensuremath{\!{\ensuremath{\left(w,e_1,e_2,\dots,e_k\right)}}}}$$ where $\vartheta \in {\ensuremath{\mathcal{I}}}^{k+1}$ and $e_1, e_2, \dots, e_k \in E$. They vanish on a vector $w$ just when the span of $\set{w} \cup E$ is an integral element. If an integral element $E$ is contained in another one, $E \subset F$, then all polar equations of $E$ occur among those of $F$: larger integral elements have more (or at least the same) polar equations.
What are the polar equations of the integral elements of our trivial examples?
A *partial flag* ${\ensuremath{E}_{\bullet}}$ is a sequence of nested linear subspaces $$E_0 \subset E_1 \subset E_2 \subset \dots \subset E_p$$ in a vector space. The *increments* of a partial flag are the integers measuring how the dimensions increase: $$\begin{array}{@{}r@{}ll@{}}
\dim E_0&, \\
\dim E_1&{} - \dim E_0, \\
\dim E_2&{}- \dim E_1, \\
&{} \ \, \vdots \\
\dim E_p&{}- \dim E_{p-1}.
\end{array}$$ A *flag* is a partial flag $$E_0 \subset E_1 \subset E_2 \subset \dots \subset E_p$$ for which $\dim E_i=i$. Danger: most authors require that a flag have subspaces of all dimensions; we *don’t*: we only require that the subspaces have all dimensions $0,1,2,\dots,p$ up to some dimension $p$. In particular, the increments of any flag are $0,1,1,\dots,1$.
The polar equations of a flag ${\ensuremath{E}_{\bullet}}$ of integral elements form a partial flag in the cotangent space. The *characters* $s_0, s_1, \dots, s_p$ of ${\ensuremath{E}_{\bullet}}$ are the increments of its polar equations, i.e. the numbers of linearly independent polar equations added at each increment in the flag.
What are the characters of the integral flags of our trivial examples?
The rank $p$ *Grassmann bundle* of a manifold $M$ is the set of all $p$-dimensional linear subspaces of tangent spaces of $M$.
Recall how charts are defined on the Grassmann bundle. Prove that the Grassmann bundle is a fiber bundle over the underlying manifold.
The integral elements of an exterior differential system form a subset of the Grassmann bundle. Let us inquire whether this subset is a submanifold of the Grassmann bundle; if so, let us predict its dimension. We say that a flag of integral elements *predicts* the dimension $\dim M + s_1+2s_2+\dots+ps_p$; an integral element *predicts* the dimension predicted by the generic flag inside it.
Every integral element predicts the dimension of a submanifold of the Grassmann bundle containing all nearby integral elements.
An integral element $E$ *correctly* predicts dimension if the integral elements near $E$ form a manifold of dimension predicted by $E$. An integral element which correctly predicts dimension is *involutive*.
There is an integral manifold tangent to every involutive integral element of any analytic exterior differential system.
If an integral element is involutive, then all nearby integral elements are too, as the nonzero polar equations will remain nonzero. An exterior differential system is *involutive* if its generic maximal dimensional integral element is involutive.
The Frobenius theorem in this language: on a manifold $M$ of dimension $p+q$, take an exterior differential system ${\ensuremath{\mathcal{I}}}$ locally generated by $q$ linearly independent 1-forms together with all differential forms of degree more than $p$: ${\ensuremath{\mathcal{I}}}^k=\Omega^k$ for $k>p$. Prove that ${\ensuremath{\mathcal{I}}}$ is involutive if and only if every $2$-form in ${\ensuremath{\mathcal{I}}}$ is a sum of terms of the form $\xi \wedge \vartheta$ where $\vartheta$ is a 1-form in ${\ensuremath{\mathcal{I}}}$. Prove that this occurs just when the $(n-k)$-dimensional ${\ensuremath{\mathcal{I}}}$-integral manifolds form the leaves of a foliation $F$ of $M$. Prove that then ${\ensuremath{\mathcal{I}}}^1$ consists precisely of the 1-forms vanishing on the leaves of $F$.
Example: Lagrangian submanifolds
================================
Let $$\vartheta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}dx^1 \wedge dy^1 +
dx^2 \wedge dy^2 +
\dots
+
dx^n \wedge dy^n.$$ Let ${\ensuremath{\mathcal{I}}}$ be the exterior differential system generated by $\vartheta$ on $M{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{\mathbb{R}^{2n}}}$. The integral manifolds of ${\ensuremath{\mathcal{I}}}$ are called *Lagrangian manifolds*. Let us employ the Cartan–Kähler theorem to prove the existence of Lagrangian submanifolds of complex Euclidean space. Writing spans of vectors in angle brackets, $$\begin{array}{@{}r@{}c@{}lll@{}}
\toprule\multicolumn{3}{@{}l@{}}{\textrm{Flag}} &
\textrm{Polar equations} & \textrm{Characters}\\
\midrule
E_0&=&\set{0}&\set{0} & s_0=0 \\
E_1&=&{\ensuremath{\left<\partial_{x^1}\right>}}&{\ensuremath{\left<dy^1\right>}} & s_1=1 \\
&\vdotswithin{=}&&\vdotswithin{{\ensuremath{\left<dy^1\right>}}}&\vdotswithin{=} \\
E_n&=&{\ensuremath{\left<\partial_{x^1},\partial_{x^2},\dots,\partial_{x^n}\right>}}&{\ensuremath{\left<dy^1,dy^2,\dots,dy^n\right>}} & s_n=1 \\
\bottomrule
\end{array}$$ The flag predicts $$\dim M + s_1 + 2 \, s_2 + \dots + n \, s_n
=
2n + 1+2+\dots+n.$$ The nearby integral elements at a given point of $M$ are parameterized by $dy=a \, dx$, which we plug in to $\vartheta=0$ to see that $a$ can be any symmetric matrix. So the space of integral elements has dimension $$\dim M + \frac{n(n+1)}{2} = 2n + \frac{n(n+1)}{2},$$ correctly predicted. Therefore the Cartan–Kähler theorem proves the existence of Lagrangian submanifolds of complex Euclidean space, one (at least) through each point, tangent to each subspace $dy= a \, dx$, at least for any symmetric matrix $a$ close to $0$.
On a complex manifold $M$, take a Kähler form $\vartheta$ and a holomorphic volume form $\Psi$, i.e. closed forms expressed in local complex coordinates as $$\begin{aligned}
\vartheta &= \frac{\sqrt{-1}}{2} g_{\mu \bar\nu} dz^{\mu} \wedge dz^{\bar\nu}, \\
\Psi &= f(z) \, dz^1 \wedge dz^2 \wedge \dots \wedge dz^n,\end{aligned}$$ with $f(z)$ a holomorphic function and $g_{\mu \bar\nu}$ a positive definite self-adjoint complex matrix of functions. Prove the existence of *special Lagrangian manifolds*, i.e. integral manifolds of the exterior differential system generated by the pair of $\vartheta$ and the imaginary part of $\Psi$.
The last character
==================
In applying the Cartan–Kähler theorem, we are always looking for submanifolds of a particular dimension $p$. For simplicity, we can add the hypothesis that our exterior differential system contains all differential forms of degree $p+1$ and higher. In particular, the $p$-dimensional integral elements are maximal dimensional integral elements. The polar equations of any maximal dimensional integral element $E_p$ cut out precisely $E_p$, i.e. there are $\dim M - p$ independent polar equations on $E_p$. We encounter $s_0, s_1,\dots,s_p$ polar equations at each increment, so the number of independent polar equations is $s_0+s_1+\dots+s_p$. Our hypothesis helps us to calculate $s_p$ from the other characters: $$s_0+s_1+\dots+s_{p-1}+s_p=\dim M - p.$$ For even greater simplicity, we take this as a definition for the final character $s_p$, throwing out the previous definition. Now we can ignore any differential forms of degree more than $p$ when we test Cartan’s bound.
Example: harmonic functions
===========================
We will prove the existence of harmonic functions on the plane with given value and first derivatives at a given point. On $M={\ensuremath{\mathbb{R}^{5}}}_{x,y,u,u_x,u_y}$, let ${\ensuremath{\mathcal{I}}}$ be the exterior differential system generated by $$\begin{aligned}
\vartheta & {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}du-u_x \, dx - u_y \, dy, \\
\Theta &{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}du_x \wedge dy - du_y \wedge dx.\end{aligned}$$ Note that $$d\vartheta = -du_x \wedge dx -du_y \wedge dy$$ also belongs to ${\ensuremath{\mathcal{I}}}$ because any exterior differential system is closed under exterior derivative. An integral surface $X \subset M$ on which $0 \ne dx \wedge dy$ is locally the graph of a harmonic function $u=u(x,y)$ and its derivatives $u_x = \pderiv{u}{x}$, $u_x = \pderiv{u}{x}$.
Each integral plane $E_2$ (i.e. integral element of dimension 2) on which $dx \wedge dy \ne 0$ is given by equations $$\begin{aligned}
du_x &= u_{xx} dx + u_{xy} dy, \\
du_y &= u_{yx} dx + u_{yy} dy, \\\end{aligned}$$ for a unique choice of 4 constants $u_{xx}, u_{xy}, u_{yx}, u_{yy}$ subject to the 2 equations $u_{xy}=u_{yx}$ and $0=u_{xx}+u_{yy}$. Hence integral planes at each point have dimension 2. The space of integral planes has dimension $\dim M + 2 = 5+2=7$.
Each vector inside that integral plane has the form $$v = {\ensuremath{\left(\dot{x},\dot{y},u_x \dot{x} + u_y\dot{y}, u_{xx} \dot{x}+u_{xy} \dot{y}, u_{yx} \dot{x} + u_{yy}\dot{y}\right)}}$$ Each integral line $E_1$ is the span $E_1={\ensuremath{\left<v\right>}}$ of a nonzero such vector. Compute $$v {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\begin{pmatrix}
d\vartheta \\
\Theta
\end{pmatrix}
=
\begin{pmatrix}
\dot{x} du_x + \dot{y} du_y - {\ensuremath{\left(u_{xx} \dot{x} + u_{xy} \dot{y}\right)}} \, dx - {\ensuremath{\left(u_{yx} \dot{x} + u_{yy} \dot{y}\right)}} dy \\
\dot{y} du_x -\dot{x} du_y
+ {\ensuremath{\left(u_{xx} \dot{x} + u_{xy} \dot{y}\right)}} \, dy
-
{\ensuremath{\left(u_{yx} \dot{x} + u_{yy} \dot{y}\right)}} \, dx
\end{pmatrix}.$$ and $$\begin{array}{@{}r@{}c@{}lll@{}}
\toprule\multicolumn{3}{@{}l@{}}{\textrm{Flag}} &
\textrm{Polar equations} & \textrm{Characters}\\
\midrule
E_0&=&\set{0}&{\ensuremath{\left<\vartheta\right>}} & s_0=1 \\
E_1&=&{\ensuremath{\left<v\right>}}&{\ensuremath{\left<\vartheta, v{\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}d\vartheta, v{\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\Theta\right>}} & s_1=2 \\
\bottomrule
\end{array}$$ Since we are only interested in finding integral surfaces, we compute the final character from $$s_0+s_1+s_2=\dim M - 2.$$ The Cartan characters are ${\ensuremath{\left(s_0,s_1,s_2\right)}}={\ensuremath{\left(1,2,0\right)}}$ with predicted dimension $\dim M + s_1 + 2 s_2 = 5 + 2 + 2 \cdot 0 = 7$: involution. We see that harmonic functions exist near any point of the plane, with prescribed value and first derivatives at that point.
Generality of integral manifolds
================================
The proof of the Cartan–Kähler theorem (which we will not give) constructs integral manifolds inductively, starting with a point, then building an integral curve, and so on. The choice of the initial data at each inductive stage consists of $s_0$ constants, $s_1$ functions of 1 variable, and so on. Different choices of this initial data give rise to different integral manifolds in the final stage. In this sense, the integral manifolds depend on $s_0$ constants, and so on.
If one describes some family of submanifolds in terms of the integral manifolds of an exterior differential system, someone else might find a different description of the same submanifolds in terms of integral manifolds of a different exterior differential system, with different Cartan characters.
For example, any smooth function $y=f(x)$ of 1 variable is equivalent information to having a constant $f(0)$ and a function $y'=f'(x)$ of 1 variable.
For example, immersed plane curves are the integral curves of ${\ensuremath{\mathcal{I}}}=0$ on $M={\ensuremath{\mathbb{R}^{2}}}$. Check that any integral flag $E_0=\set{0}, E_1={\ensuremath{\left<v\right>}}$ has ${\ensuremath{\left(s_0,s_1\right)}}={\ensuremath{\left(0,1\right)}}$. But immersed plane curves are also the integral curves of the ideal ${\ensuremath{\mathcal{I}}}$ generated by $$\vartheta {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\sin \phi \, dx - \cos \phi \, dy$$ on $M{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{\mathbb{R}^{2}}}_{x,y} \times S^1_{\phi}$. Here ${\ensuremath{\left(s_0,s_1\right)}}={\ensuremath{\left(1,1\right)}}$. The last nonzero character does not change. In general, we cannot expect all of the Cartan characters to stay the same for different descriptions of various submanifolds, but we can expect the last nonzero Cartan character to stay the same.
Lagrangian submanifolds of ${\ensuremath{\mathbb{C}^{n}}}$ depend on 1 function of $n$ variables. This count is correct: those which are graphs $y=y(x)$ are precisely of the form $$y = \pderiv{S}{x}$$ for some potential function $S(x)$, unique up to adding a real constant. On the other hand, the proof of the Cartan–Kähler theorem builds up each Lagrangian manifold from a choice of one function of one variable, one function of two variables, and so on. Similarly, harmonic functions depend on 2 functions of 1 variable. Summing up: we “trust” the last nonzero Cartan–Kähler $s_p$ to tell us the generality of the integral manifolds: they depend on $s_p$ functions of $p$ variables, but we don’t “trust” $s_0, s_1, \dots, s_{p-1}$.
Example: triply orthogonal webs
===============================
{width="6cm"} {width="6cm"} {width="6cm"} {width="6cm"}
On a 3-dimensional Riemannian manifold $X$, a *triply orthogonal web* is a triple of foliations whose leaves are pairwise perpendicular. We will see that these exist, locally, depending on 3 functions of 2 variables. Each leaf is perpendicular to a unique smooth unit length 1-form $\eta_i$, up to $\pm$, which satisfies $0=\eta_i \wedge d \eta_i$, by the Frobenius theorem. Let $M$ be the set of all orthonormal bases of the tangent spaces of $X$, with obvious bundle map $x \colon M \to X$, so that each point of $M$ has the form $m={\ensuremath{\left(x,e_1,e_2,e_3\right)}}$ for some $x \in X$ and orthonormal basis $e_1,e_2,e_3$ of $T_x X$. The *soldering 1-forms* $\omega_1, \omega_2, \omega_3$ on $M$ are defined by $$v {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\omega_i = \left<e_i,x_* v\right>.$$ Note: they are 1-forms on $M$, not on $X$. Let $$\omega
=
\begin{pmatrix}
\omega_1 \\
\omega_2 \\
\omega_3
\end{pmatrix}.$$ Define cross product $\alpha \times \beta$ on ${\ensuremath{\mathbb{R}^{3}}}$-valued 1-forms by $$\alpha \times \beta(u,v)=\alpha(u) \times \beta(v) - \alpha(v) \times \beta(u).$$ By the fundamental lemma of Riemannian geometry, there is a unique ${\ensuremath{\mathbb{R}^{3}}}$-valued 1-form $\gamma$ for which $
d \omega = \frac{1}{2} \gamma \times \omega
$. Our triply orthogonal web is precisely a section $X \to M$ of the bundle map $M \to X$ on which $0=\omega_i \wedge d\omega_i$ for all $i$, hence an integral 3-manifold of the exterior differential system ${\ensuremath{\mathcal{I}}}$ on $M$ generated by the 3-forms $$\omega_1 \wedge d \omega_1, \quad
\omega_2 \wedge d \omega_2, \quad
\omega_3 \wedge d \omega_3.$$ Using the equations above, ${\ensuremath{\mathcal{I}}}$ is also generated by $$\gamma_3 \wedge \omega_1 \wedge \omega_2, \quad
\gamma_1 \wedge \omega_2 \wedge \omega_3, \quad
\gamma_2 \wedge \omega_3 \wedge \omega_1.$$ The 3-dimensional ${\ensuremath{\mathcal{I}}}$-integral manifolds on which $$0\ne \omega_1 \wedge \omega_2 \wedge \omega_3$$ are locally precisely the triply orthogonal webs. The 3-dimensional integral elements on which $$0\ne \omega_1 \wedge \omega_2 \wedge \omega_3$$ are precisely given by $$\gamma=
\begin{pmatrix}
0 & p_{12} & p_{13} \\
p_{21} & 0 & p_{23} \\
p_{31} & p_{32} & 0
\end{pmatrix}
\omega$$ for any $p_{ij}$, hence $6+6=12$ dimensions of integral elements. Since the system is generated by 3-forms, on any integral flag in this integral element, $0=s_0=s_1$. Count out ${\ensuremath{\left(s_0,s_1,s_2,s_3\right)}}={\ensuremath{\left(0,0,3,0\right)}}$, predicting 12 dimensions of integral elements: involution.
We conclude: for any orthonormal basis at a point of any real analytic Riemannian 3-manifold, there are infinitely many real analytic triply orthogonal webs, depending on 3 functions of 2 variables, defined near that point, with the tangent spaces of the leaves perpendicular to those basis vectors.
Example: isometric immersion
============================
Take a surface $P$ with a Riemannian metric. Naturally we are curious if there is an isometric immersion $f \colon P \to {\ensuremath{\mathbb{R}^{3}}}$, i.e. a smooth map preserving the lengths of all curves on $P$. For example, these surfaces
{width="\linewidth"}
are isometric immersions of a piece of this paraboloid
{width="1cm"}
More generally, take a Riemannian manifold ${\boldsymbol{P}}$ of dimension 3. We ask if there is an isometric immersion $f \colon P \to {\boldsymbol{P}}$.
On the orthonormal frame bundle ${\ensuremath{FP}}$, denote the soldering forms as ${\omega}={\omega}_1+i{\omega}_2$. By the fundamental lemma of Riemannian geometry there is a unique 1-form (the connection 1-form) ${\gamma}$ so that $d{\omega}=i{\gamma}\wedge {\omega}$ and $d{\gamma}=(i/2)K{\omega}\wedge \bar{\omega}$. As above, on ${\ensuremath{F{\boldsymbol{P}}}}$ there is a soldering 1-form ${{\boldsymbol{\omega}}}$ and a connection 1-form ${{\boldsymbol{\gamma}}}$ so that $d{{\boldsymbol{\omega}}}= \frac{1}{2} {{\boldsymbol{\gamma}}}\times {{\boldsymbol{\omega}}}$ and This ensures that $$d{{\boldsymbol{\gamma}}}= \frac{1}{2} {{\boldsymbol{\gamma}}}\times {{\boldsymbol{\gamma}}}+ \frac{1}{2}{\ensuremath{\left(\frac{s}{2} - R\right)}} {{\boldsymbol{\omega}}}\times {{\boldsymbol{\omega}}}.$$ with Ricci curvature $R_{ij}=R_{ji}$ and scalar curvature $s=R_{ii}$. If there is an isometric immersion $f \colon P \to {\boldsymbol{P}}$, then let $X{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}X_f \subset M {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{FP}} \times {\ensuremath{F{\boldsymbol{P}}}}$ be its *adapted frame bundle*, i.e. the set of all tuples $${\ensuremath{\left(p,e_1,e_2,{\boldsymbol{p}},{\boldsymbol{e}}_1,{\boldsymbol{e}}_2,{\boldsymbol{e}}_3\right)}}$$ where $p \in P$ with orthonormal frame $e_1, e_2 \in T_p P$ and ${\boldsymbol{p}} \in {\boldsymbol{P}}$ with orthonormal frames ${\boldsymbol{e}}_1, {\boldsymbol{e}}_2, {\boldsymbol{e}}_3 \in T_{{\boldsymbol{p}}} {\boldsymbol{P}}$, so that $f_* e_1={\boldsymbol{e}}_1$ and $f_* e_2={\boldsymbol{e}}_2$. Let ${\ensuremath{\mathcal{I}}}$ be the exterior differential system on $M$ generated by the 1-forms $$\begin{pmatrix}
\vartheta_1 \\
\vartheta_2 \\
\vartheta_3
\end{pmatrix}
{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\begin{pmatrix}
{{\boldsymbol{\omega}}}_1-{\omega}_1 \\
{{\boldsymbol{\omega}}}_2-{\omega}_2 \\
{{\boldsymbol{\omega}}}_3
\end{pmatrix}.$$ Along $X$, all of these 1-forms vanish, while the 1-forms ${\omega}_1, {\omega}_2, {\gamma}$ remain linearly independent. Conversely, we will eventually prove that all ${\ensuremath{\mathcal{I}}}$-integral manifolds on which ${\omega}_1, {\omega}_2, {\gamma}$ are linearly independent are locally frame bundles of isometric immersions. For the moment, we concentrate on asking whether we can apply the Cartan–Kähler theorem.
Compute: $$d
\begin{pmatrix}
\vartheta_1 \\
\vartheta_2 \\
\vartheta_3
\end{pmatrix}
=
-
\left( \ \begin{matrix}
0 & {{\boldsymbol{\gamma}}}_3 - {\gamma}& 0 \\
{-{\ensuremath{\left({{\boldsymbol{\gamma}}}_3 - {\gamma}\right)}}} & 0 & 0 \\
{{{\boldsymbol{\gamma}}}_2} & -{{{\boldsymbol{\gamma}}}_1} & 0
\end{matrix} \ \right)
\wedge
\begin{pmatrix}
{\omega}_1 \\
{\omega}_2 \\
{\gamma}\end{pmatrix}
\mod{\vartheta_1, \vartheta_2, \vartheta_3}.$$
We count $s_1=2, s_2=1, s_3=0$. Each 3-dimensional integral element has ${{\boldsymbol{\omega}}}={\omega}$, so is determined by the linear equations giving ${{\boldsymbol{\gamma}}}_1, {{\boldsymbol{\gamma}}}_2, {{\boldsymbol{\gamma}}}_3$ in terms of ${\omega}_1, {\omega}_2, {\gamma}$ on which $d\vartheta=0$: $$\begin{pmatrix}
{{\boldsymbol{\gamma}}}_1 \\
{{\boldsymbol{\gamma}}}_2 \\
{{\boldsymbol{\gamma}}}_3 - {\gamma}\end{pmatrix}
=
\begin{pmatrix}
a & b \\
c & -a \\
0 & 0
\end{pmatrix}
\begin{pmatrix}
{\omega}_1 \\
{\omega}_2
\end{pmatrix}.$$ Therefore there is a 3-dimensional space of integral elements at each point. But $s_1+2s_2=4>3$: no integral element correctly predicts dimension, so we can’t apply the Cartan–Kähler theorem.
What to do? On every integral element, we said that $$\begin{pmatrix}
{{\boldsymbol{\gamma}}}_1 \\
{{\boldsymbol{\gamma}}}_2 \\
{{\boldsymbol{\gamma}}}_3 - {\gamma}\end{pmatrix}
=
\begin{pmatrix}
a & b \\
c & -a \\
0 & 0
\end{pmatrix}
\begin{pmatrix}
{\omega}_1 \\
{\omega}_2
\end{pmatrix}.$$ Make a new manifold $M' {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}M \times {\ensuremath{\mathbb{R}^{3}}}_{a,b,c}$, and on $M'$ let ${\ensuremath{\mathcal{I}}}'$ be the exterior differential system generated by $$\begin{aligned}
\begin{pmatrix}
\vartheta_4 \\
\vartheta_5 \\
\vartheta_6
\end{pmatrix}
&{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\begin{pmatrix}
{{\boldsymbol{\gamma}}}_1 \\
{{\boldsymbol{\gamma}}}_2 \\
{{\boldsymbol{\gamma}}}_3-{\gamma}\end{pmatrix}
-
\begin{pmatrix}
a & b \\
c & -a \\
0 & 0
\end{pmatrix}
\begin{pmatrix}
{\omega}_1 \\
{\omega}_2
\end{pmatrix}.\end{aligned}$$
Prolongation
============
Take an exterior differential system ${\ensuremath{\mathcal{I}}}$ on a manifold $M$. What should we do if there are no involutive integral elements? Let $M'$ be the set of all pairs $(m,E)$ consisting of a point $m$ of $M$ and an ${\ensuremath{\mathcal{I}}}$-integral element $E \subset T_m M$. So $M'$ is a subset of the Grassmann bundle over $M$. Locally on $M$, take a local basis $\omega,\vartheta,\pi$ of the 1-forms, with $\vartheta$ a basis for the 1-forms in ${\ensuremath{\mathcal{I}}}$. We can write each integral element on which $\omega$ has linearly independent components as the solutions of the linear equations $0=\vartheta,\pi=a\omega$ for some constants $a$. On an open subset of $M'$, $a$ is a function valued in some vector space. Pull back the 1-forms $\vartheta, \omega, \pi$ to $M'$ via the map $(m,E) \in M' \mapsto m \in M$. On $M'$, let $\vartheta' {\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\pi-a\omega $. The exterior differential system ${\ensuremath{\mathcal{I}}}'$ on $M'$ generated by $\vartheta'$ is the *prolongation* of ${\ensuremath{\mathcal{I}}}$. Inductively, let $M^{(1)}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}M'$, ${\ensuremath{\mathcal{I}}}^{(1)}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{\mathcal{I}}}'$, $M^{(k+1)}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{\left(M^{(k)}\right)}}'$, ${\ensuremath{\mathcal{I}}}^{(k+1)}{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}{\ensuremath{\left({\ensuremath{\mathcal{I}}}^{(k)}\right)}}'$, if defined.
If each $M^{(k)}$ is a submanifold of the Grassmann bundle over $M^{(k-1)}$, with finitely many connected components, and if each $M^{(k)} \to M^{(k-1)}$ is a submersion, then all but finitely many ${\ensuremath{\mathcal{I}}}^{(k)}$ are involutive.
Back to isometric immersion
===========================
Returning to our example of isometric immersion of surfaces, we have prolongation given by $$\begin{aligned}
\begin{pmatrix}
\vartheta_4 \\
\vartheta_5 \\
\vartheta_6
\end{pmatrix}
&{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\begin{pmatrix}
{{\boldsymbol{\gamma}}}_1 \\
{{\boldsymbol{\gamma}}}_2 \\
{{\boldsymbol{\gamma}}}_3-{\gamma}\end{pmatrix}
-
\begin{pmatrix}
a & b \\
c & -a \\
0 & 0
\end{pmatrix}
\begin{pmatrix}
{\omega}_1 \\
{\omega}_2
\end{pmatrix}.\end{aligned}$$ Note that $0=d\vartheta_1, d\vartheta_2, d\vartheta_3$ modulo $\vartheta_4,\vartheta_5,\vartheta_6$, so we can forget about them.
Calculate the exterior derivatives: $$d
\begin{pmatrix}
\vartheta_4 \\
\vartheta_5 \\
\vartheta_6
\end{pmatrix}
=
-
\left( \ \begin{matrix}
Da & Db \\
Dc & -Da \\
0 & 0
\end{matrix} \ \right)
\wedge
\begin{pmatrix}
{\omega}_1 \\
{\omega}_2
\end{pmatrix}
+
\begin{pmatrix}
0 \\
0 \\
t {\omega}_1 \wedge {\omega}_2
\end{pmatrix}
\mod{\vartheta_1,\dots,\vartheta_6}.$$ where $$\begin{pmatrix}
Da \\
Db \\
Dc
\end{pmatrix}
{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\begin{pmatrix}
da + (b+c) {\gamma}- R_{23} {\omega}_1 \\
db - 2a {\gamma}+ R_{13} {\omega}_1 \\
dc - 2a {\gamma}\end{pmatrix}$$ and the *torsion* is $$t{\mathrel{\vcenter{\baselineskip0.5ex \lineskiplimit0pt
\hbox{\scriptsize.}\hbox{\scriptsize.}}} =}\frac{s}{4}-R_{33}-K-a^2-bc.$$ This torsion clearly has to vanish on any 3-dimensional ${\ensuremath{\mathcal{I}}}'$-integral element, i.e. every 3-dimensional ${\ensuremath{\mathcal{I}}}'$-integral element lives over the subset of $M'$ on which $$0=\frac{s}{4}-R_{33}-K-a^2-bc.$$ To ensure that this subset is a submanifold, we let $M'_0 \subset M'$ be the set of points where this equation is satisfied and at least one of $a,b,c$ is not zero. Clearly $M'_0 \subset M'$ is a submanifold, on which we find $Da, Db, Dc$ linearly dependent. On $M'_0$, every 3-dimensional ${\ensuremath{\mathcal{I}}}'$-integral element on which $\omega_1, \omega_2, \gamma$ are linearly independent has $s_1=2$, $s_2=0$ and 2 dimensions of integral elements at each point. Therefore the exterior differential system is in involution: there is an integral manifold through each point of $M'_0$, and in particular above every point of the surface. The prolongation exposes the hidden necessary condition for existence of a solution: the relation $t=0$ between the curvature of the ambient space, that of the surface, and the shape operator.
We won’t prove the elementary:
Take any smooth 3-dimensional integral manifold $X$ of the linear Pfaffian system constructed above. Suppose that on $X$, $0\ne \omega_1\wedge\omega_2\wedge\gamma$. Every point of $X$ lies in some open subset $X_0 \subset X$ so that $X_0$ is an open subset of the adapted frame bundle of an isometric immersion $P_0 \to {\boldsymbol{P}}$ of an open subset $P_0 \subset P$.
To sum up:
Take any surface $P$ with real analytic Riemannian metric, with chosen point $p_0 \in P$ and Gauss curvature $K$. Take any 3-manifold ${\boldsymbol{P}}$ with real analytic Riemannian metric, with chosen point ${\boldsymbol{p}}_0$, and a linear isometric injection $F \colon T_{p_0} P \to T_{{\boldsymbol{p}}_0} {\boldsymbol{P}}$. Let $\nu$ be a unit normal vector to the image of $F$. Let $R$ be the Ricci tensor on that 3-manifold and $s$ the scalar curvature. Pick a nonzero quadratic form $q$ on the tangent plane $T_{p_0} P$ so that $$\det q = K + R(\nu,\nu) -\frac{s}{4}.$$ Then there is a real analytic isometric immersion $f$ of some neighborhood of $p_0$ to ${\boldsymbol{P}}$, so that $f'{\ensuremath{\!{\ensuremath{\left(p_0\right)}}}}=F$ and so that $f$ induces shape operator $q$ at $p_0$.
For further information
=======================
For proof of the Cartan–Kähler theorem see [@Cartan:1945], which we followed very closely, and also the canonical reference work [@BCGGG:1991] and the canonical textbook [@Ivey/Landsberg:2003]. The last two also give proof of the Cartan–Kuranishi theorem. For more on triply orthogonal webs in Euclidean space, and orthogonal webs in Euclidean spaces of all dimensions, see [@Darboux:1993; @DeTurck/Yang:1984; @Terng/Uhlenbeck:1998; @Zakharov:1998]. For more on isometric immersions and embeddings see [@Han/Hong:2006].
Linearization {#section:linearization}
=============
Take an exterior differential system ${\ensuremath{\mathcal{I}}}$ on a manifold $M$, and an integral manifold $X \subset M$. Suppose that the flow of a vector field $v$ on $M$ moves $X$ through a family of integral manifolds. So the tangent spaces of $X$ are carried by the flow of $v$ through integral elements of ${\ensuremath{\mathcal{I}}}$. Equivalently, the pullback by that flow of each form in ${\ensuremath{\mathcal{I}}}$ also vanishes on $X$. So ${\ensuremath{\EuScript L}}_v \vartheta$ vanishes on $X$ for each $\vartheta \in {\ensuremath{\mathcal{I}}}$.
Prove that all vector fields $v$ tangent to $X$ satisfy this equation.
More generally, suppose that $E \subset T_m M$ is an integral element of ${\ensuremath{\mathcal{I}}}$. If a vector field $v$ on $M$ carries $E$ through a family of integral elements, then $0=\left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E$ for each $\vartheta \in {\ensuremath{\mathcal{I}}}$. Take local coordinates $x,y$, say $x={\ensuremath{\left(x^1,x^2,\dots,x^p\right)}}$ and $y={\ensuremath{\left(y^1,y^2,\dots,y^q\right)}}$, where $E$ is the graph of $dy=0$. We use a multiindex notation where if $$I={\ensuremath{\left(i_1,i_2,\dots,i_m\right)}}$$ then $$dx^I=dx^{i_1} \wedge dx^{i_2} \wedge \dots \wedge dx^{i_m}.$$ Allow the possibility of $m=0$, no indices, for which $dx^I=1$. Let $(-1)^I$ mean $(-1)^m$.
Take an integral element $E \subset T_m M$ for an exterior differential system ${\ensuremath{\mathcal{I}}}$ on a manifold $M$. Take local coordinates $x,y$, say $x={\ensuremath{\left(x^1,x^2,\dots,x^p\right)}}$ and $y={\ensuremath{\left(y^1,y^2,\dots,y^q\right)}}$, where $E$ is the graph of $dy=0$. For any $\vartheta \in {\ensuremath{\mathcal{I}}}$, if we write $\vartheta = c_{IA} dx^I$ then $$\left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E =
\pderiv{v^a}{x^j} c_{Ia} dx^{Ij} + v^a \pderiv{c_I}{y^a} dx^I .$$
Expand out $$\vartheta = c_{IA}(x,y) dx^I \wedge dy^A.$$ and $$v = v^j \pderiv{}{x^j} + v^b \pderiv{}{y^b}.$$ Note that $$v {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}c_I dx^I = (-1)^J v^i c_{JiK} dx^{JK}.$$ Commuting with exterior derivative, $$\begin{aligned}
{\ensuremath{\EuScript L}}_v dx^i &= \pderiv{v^i}{x^j} dx^j + \pderiv{v^i}{y^b} dy^b, \\
{\ensuremath{\EuScript L}}_v dy^a &= \pderiv{v^a}{x^j} dx^j + \pderiv{v^a}{y^b} dy^b.\end{aligned}$$ By the Leibnitz rule, $$\begin{aligned}
{\ensuremath{\EuScript L}}_v \vartheta
&=
v^i \pderiv{c_{IA}}{x^i} dx^I \wedge dy^A
+
v^a \pderiv{c_{IA}}{y^a} dx^I \wedge dy^A
\\
&\qquad
+ c_{JiKA} dx^J \wedge {\ensuremath{\left(\pderiv{v^i}{x^j} dx^j + \pderiv{v^i}{y^b} dy^b\right)}} \wedge dx^K \wedge dy^A
\\
&\qquad
+ c_{IBaC} dx^I \wedge dy^B \wedge {\ensuremath{\left(\pderiv{v^a}{x^j} dx^j + \pderiv{v^a}{y^b} dy^b\right)}} \wedge dy^C.\end{aligned}$$ On $E$, $dy=0$ so $$\begin{aligned}
\left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E
&=
v^i \pderiv{c_I}{x^i} dx^I
+
v^a \pderiv{c_I}{y^a} dx^I
\\
&\qquad
+ c_{JiK} dx^J \wedge \pderiv{v^i}{x^j} dx^j \wedge dx^K
\\
&\qquad
+ c_{Ia} dx^I \wedge \pderiv{v^a}{x^j} dx^j.\end{aligned}$$ Write the tangent part of $v$ as $$v' = v^i \pderiv{}{x^i}.$$ Let $A$ be the linear map $A \colon E \to E$ given by $$A^i_j = \pderiv{v^i}{x^j}$$ and apply this by derivation to forms on $E$, $$(A\xi)(v_1,\dots,v_k)=\xi(Av_1,v_2,\dots,v_k)-\xi(v_1,Av_2,v_3,\dots,nv_k)+\dots.$$ Then $$\left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E
=
\left.v' {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}d\vartheta\right|_E
+
v^a \pderiv{c_I}{y^a} dx^I
+ \left.A\vartheta\right|_E
+ c_{Ia} dx^I \wedge \pderiv{v^a}{x^j} dx^j.$$ Since $0=\left.\vartheta\right|_E=\left.d\vartheta\right|_E$, we find $$\left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E
=
v^a \pderiv{c_I}{y^a} dx^I
+ c_{Ia} dx^I \wedge \pderiv{v^a}{x^j} dx^j.$$
In our coordinates, take any submanifold $X$ which is the graph of some functions $y=y(x)$. Suppose that the linear subspace $E=(dy=0)$ at $(x,y)=(0,0)$ is an integral element. Pullback the forms from the ideal: $$\left.\vartheta\right|_X = \sum c_{IA}(x,y) dx^I \wedge
\pderiv{y^{a_1}}{x^{j_1}} dx^{j_1} \wedge \dots \wedge \pderiv{y^{a_{\ell}}}{x^{j_{\ell}}} dx^{j_{\ell}}.$$ The right hand side, as a nonlinear first order differential operator on functions $y=y(x)$, has linearization $\vartheta \mapsto \left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E$. That linearization is applied to sections of the normal bundle $\left.TM\right|_X/TX$, which in coordinates are just the functions $v^a$. The linearized operator at the origin of our coordinates depends only on the integral element $E=T_m X$, not on the choice of submanifold $X \subset M$ tangent to $E$.
Compute the linearization of $u_{xx}=u_{yy} + u_{zz} + u_x^2$ around $u=0$ by setting up this equation as an exterior differential system.
Take sections $v$ of the normal bundle of $X$ which vanish at $m$. Then the linearization applied to these sections is $$\left.{\ensuremath{\EuScript L}}_v \vartheta\right|_E = c_{Ia} dx^I \wedge \pderiv{v^a}{x^j} dx^j = \left.A \vartheta\right|_E,$$ where $A$ is the linear map $$A=\pderiv{v^a}{x^j},$$ the linearization of $v$ around the origin, and $A\vartheta$ as usual means the derivation action of linear maps $A$ on differential forms $\vartheta$: $$A\vartheta(u_1,u_2,\dots,u_k)=
\vartheta(Au_1,u_2,\dots,u_k)
-
\vartheta(u_1,Au_2,\dots,u_k)
+\dots \pm
\vartheta(u_1,u_2,\dots,Au_k)$$ if $\vartheta$ is a $k$-form.
The leading order terms of the linearization of an exterior differential system form the *tableau*: $$\vartheta_m \in {\ensuremath{\mathcal{I}}}_m , A \in E^* \otimes (T^*M/E) \mapsto \left.A\vartheta_m\right|_E,$$ where ${\ensuremath{\mathcal{I}}}_m$ is the set of all values $\vartheta_m$ of differential forms in ${\ensuremath{\mathcal{I}}}$.
The characteristic variety
==========================
Take an integral manifold $X$ of an exterior differential system ${\ensuremath{\mathcal{I}}}$ on a manifold $M$. The characteristic variety $$\Xi_x \subset \mathbb{P}E$$ of the linearization of the exterior differential system at any integral element $E$ is precisely the set of hyperplanes in $E$ which lie not only in the integral element $E$ but also in some other integral element different from $E$.
Take a submanifold $X$ tangent to $E$. Apply the linearization operator $\left.{\ensuremath{\EuScript L}}_v \vartheta \right|_E$ as above to sections $v$ of the normal bundle of $X$. We can also apply this operator formally to sections of the complexified normal bundle. In particular, we compute that for any smooth function $f$ on $X$, and real constant $\lambda$ $$e^{-i \lambda f} {\ensuremath{\EuScript L}}_{\, e^{i \lambda f} v} \vartheta
=
{\ensuremath{\EuScript L}}_v \vartheta + i\lambda df \wedge (v {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\vartheta).$$ In particular, the symbol of the linearization is $$\sigma(\xi)v = \xi \wedge (v {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\vartheta).$$ The linearization of ${\ensuremath{\mathcal{I}}}$ is just the sum of the linearizations for any spanning set of forms $\vartheta \in {\ensuremath{\mathcal{I}}}$. The characteristic variety $\Xi$ at a point $x \in X$ is therefore precisely the set of $\xi \in E^*$ for which there is some section $v$ of the normal bundle of $X$ with $v(x)\ne 0$ and $$\xi \wedge (v {\ensuremath{\mathbin{ \hbox{\vrule height1.4pt
width4pt depth-1pt \vrule height4pt width0.4pt depth-1pt}}}}\vartheta)|_E=0$$ for every $\vartheta \in {\ensuremath{\mathcal{I}}}$. If we look at the hyperplane $(0=\xi) \subset E$, the vector $v$ can be added to that hyperplane to make an integral element enlarging the hyperplane. So the characteristic variety $\Xi_x$ of ${\ensuremath{\mathcal{I}}}$ is precisely the set of hyperplanes in $E$ for which there is more than one way to extend that hyperplane to an integral element: you can extend it to become $E$ or to become this other extension containing $v$.
|
---
abstract: 'In this paper we consider three different 1D parabolic-parabolic systems of chemotaxis. For these systems we obtain the exact analytical solutions in terms of traveling wave variables.'
author:
- Maria Shubina
title: 'Exact traveling wave solutions of 1D parabolic-parabolic models of chemotaxis'
---
Introduction {#intro}
============
In this paper we consider a number of different systems of nonlinear partial differential equations, which describe a directed cells (bacteria or other organisms) movement up or down a chemical concentration gradient (chemotaxis). The aim of this paper is to obtain exact analytical solutions of these models. For 1D parabolic-parabolic systems under consideration we present these solutions in explicit form in terms of traveling wave variables. Of course, not all of the solutions obtained can have appropriate biological interpretation since the biological functions must be nonnegative in all domain of definition. However some of these solutions are positive and bounded and their analysis requires further investigation.
Chemotaxis plays an important role in many biological and medical fields such as embryogenesis, immunology, cancer growth. The macroscopic classical model of chemotaxis was proposed by Patlak in 1953 [@P] and by Keller and Segel in the 1970s [@KS1]-[@KS3]. This model describes the space-time evolution of a cells density $ u(t,\overrightarrow{r}) $ and a concentration of a chemical substance $ v(t,\overrightarrow{r}) $. The general form of this model is: {
u\_[t]{}-(\_[1]{} u - \_[1]{} u (v)) & =0\
v\_[t]{}-\_[2]{} \^[2]{} v - f(u, v) & =0,
. where $ \delta_{1} > 0 $ and $ \delta_{2} \geq 0 $ are cells and chemical substance diffusion coefficients respectively, $ \eta_{1} $ is a chemotaxis coefficient; when $ \eta_{1} > 0 $ this is an attractive chemotaxis (“positive taxis”), and when $ \eta_{1} < 0 $ this is a repulsive (“negative”) one [@Ni], . The function $ \phi (v) $ is the chemosensitivity function and $ f(u, v) $ characterizes the chemical growth and degradation. These functions are taken in different forms that corresponds to some variations of the original Keller–Segel model. We follow the reviews of T. Hillen and K. Painter and of Z.-A. Wang [@Wang] and consider models presented therein.
This paper is concerned with one-dimensional simplified models when the coefficients $ \delta_{1} $, $ \delta_{2} $ and $ \eta_{1} $ are positive constants, $ x \in \Re, t \geq 0 $, $ u=u(x,t) $, $ v=v(x,t) $.
Signal-dependent sensitivity model {#sec:1}
==================================
Let us start with a model that allows nonnegative bounded solutions that may be of interest from a biological point of view. Now consider the “logistic” model, one of versions of signal-dependent sensitivity model with the chemosensitivity function $ \phi (v) = (1 + \textit{b}) \ln(v + \textit{b}) $, $ \textit{b} = const $ and $ f(u, v)= \tilde{\sigma} u - \tilde{\beta} v $. In the review [@H1] one can see a mathematical analysis of this model. When $ \textit{b} = 0 $ and $ \tilde{\beta} = 0 $ the existence of traveling waves were established in , . The replacement $ t \rightarrow \delta_{1} t $, $ u \rightarrow \sigma \dfrac{\tilde{\sigma} }{\delta_{1}} u $ gives $\delta_{1} = 1 $, $ \alpha = \dfrac{\delta_{2}}{\delta_{1}} $, $ \beta = \dfrac{\tilde{\beta}}{\delta_{1}} $, $ \sigma = \pm 1 $. We also set $ \eta = \dfrac{\eta_{1} (1 + \textit{b})}{\delta_{1}} $, $ 1 + \textit{b} > 0 $, as well as $ \phi (v) = \ln|v + \textit{b}| $. It should be noted that a sign of $ \sigma $ may affect on mathematical properties of the system. So, $ \sigma = 1 $ corresponds to an increase of a chemical substance, proportional to cells density, whereas $ \sigma= - 1 $ corresponds to its decrease. And as we shall see later, various solutions correspond to these two cases.
After above replacements the model reads: \[eq:1\] {
u\_[t]{}- u\_[xx]{} + (u )\_[x]{} & =0\
v\_[t]{}-v\_[xx]{} - u + v & =0.\
. If we introduce the function $ \upsilon = v + \textit{b} $, in terms of traveling wave variable $ y = x - ct $, $ c = const $ this system has the form: {
u\_[y]{} + c u - u (( ))\_[y]{}+ & =0\
\_[yy]{} + c \_[y]{} - + *b* + u & =0,\
. where $ u=u(y) $, $ \upsilon = \upsilon (y) $ and $ \lambda $ is an integration constant.
In this paper we will consider the case of $ \lambda = 0 $. Then the first equation in ($1*$) gives u = C\_[u]{} e\^[-cy]{} \^, $ C_{u} $ is a constant and we will examine the following equation for $ \upsilon $: \_[yy]{} + c \_[y]{} - + *b* + C\_[u]{} e\^[-cy]{} \^ = 0. Since $ \eta $ is a positive constant we consider two cases: $ \eta = 1 $ and (3) is linear nonhomogeneous equation, and $ \eta \neq 1 $.
$\eta = 1$
----------
Let us begin with $ \eta = 1 $. We introduce the new variable $ z $ and the new function $ w $: z & = & ( ) \^e\^[- ]{}\
w & = & ( ) \^ e\^ and equation (3) becomes: z\^[2]{} w\_[zz]{} +z w\_[z]{} + w ( z\^[2]{} - \^[2]{}) = z\^[- ]{}, where $ \nu^{2} = \dfrac{1}{\alpha^{2}} (1 + \dfrac{4 \alpha \beta }{c^{2}} ) $, $ \Lambda = - \dfrac{4 \beta \textit{b} }{\alpha c^{2}}\left( \dfrac{ 4 \sigma C_{u}}{\alpha c^{2}} \right)^{\frac{1}{4}} $. Equation (5) is the Lommel differential equation , [@Watson] with $ \mu = -1 - \frac{1}{\alpha} $. For $ \sigma C_{u} > 0 $ its general solution has the form: w(z) = C\_[J]{} J\_ (z) + C\_[Y]{} Y\_ (z) + S\_[, ]{} (z), where $ C_{J} $, $ C_{Y} $ are constants, $ J_{\nu} (z) $ and $ Y_{\nu} (z) $ are Bessel functions and S\_[, ]{} (z) & = & s\_[, ]{} (z) + 2\^[- 1]{} ( ) ( ) ,\
s\_[, ]{} (z) & = & \_[1]{}F\_[2]{}( 1; , ; - ) are Lommel functions, $ _{1}F_{2} $ is generalized hypergeometric function , [@Watson]. Further, substituting of the initial variable $ y $ and the function $ v $ (see (4)) into (6) we obtain a formal solution.
### $\textit{b} = 0$
We first consider the case $ \textit{b} = 0$. Then $ \upsilon = v \geq 0 $ and $ C_{u} > 0 $. Equation (5) becomes homogeneous and for $ \sigma = 1 $ its general solution is w(z) = C\_[J]{} J\_ (z) + C\_[Y]{} Y\_ (z). However one can check that the function $ u=u(y) $ diverges as $ c y \rightarrow - \infty $ for all $ \nu $.
Consider now $ \sigma = - 1 $. For $ v(y) $ be real let $ \alpha = 2 $. Then (5) becomes the modified Bessel equation; the analysis of solutions behavior at $ \pm \infty $ leads to suitable solutions for $ v(y) $ and $ u(y)$: v(y) & = & e\^[-]{}K\_(e\^[-]{})\
u(y) & = & C\_[u]{} e\^[-]{}K\_(e\^[-]{}) with restrictions $ \nu \leq \dfrac{1}{2} $ and $ \beta \leq 0 $. So on can see that $ v(y) \rightarrow 0 $ as $ cy \rightarrow - \infty $ for all $ \nu \leq \dfrac{1}{2} $; $ v(y) \rightarrow 0 $ for $ \nu < \dfrac{1}{2} $ and $ v(y) \rightarrow \sqrt[4]{\dfrac{\pi^{2} c^{2}}{8 C_{u}}} $ for $ \nu = \dfrac{1}{2} $ as $ cy \rightarrow \infty $ and $ u(y)\rightarrow 0 $ as $ y \rightarrow \pm \infty $ for all $ \nu \leq \dfrac{1}{2} $. The curves of these functions are presented in Fig.1–Fig.2. Thus, the solution obtained may be considered as biologically appropriated one and this requires further investigation.
### $\textit{b} > 0$
Let us return to equation (5) with $ \Lambda \neq 0 $. The analysis of solutions asymptotic forms at $ \pm \infty $ , [@Watson] gives the following expressions for $ v(y) $ and $ u(y)$: v(y) + *b* & = & - ( )\^e\^[- ]{} S\_[, ]{} (e\^[- ]{})\
u(y) & = & - C\_[u]{} ( )\^e\^[- cy ( 1 + )]{} S\_[, ]{} (e\^[- ]{}) with $ \sigma C_{u} > 0 $ and for $ \nu < \dfrac{1}{\alpha} $. The latter condition leads to the requirement $ -\dfrac{c^{2}}{4\alpha} \leq \beta < 0 $. The $ v(y) \rightarrow -\textit{b} $ and $ u(y)\rightarrow -\dfrac{\beta \textit{b}}{\sigma} $ as $ cy \rightarrow - \infty $ and $ v(y) \rightarrow 0 $, $ u(y)\rightarrow 0 $ as $ cy \rightarrow \infty $. Thus, one can see that for $ \textit{b} > 0$, $ \sigma = 1 $ and $ C_{u} > 0 $ $ u(y) \geq 0 $ is satisfied but $ v(y) < 0 $. These functions are presented in Fig.3–Fig.4. It should be noted that $ \nu \neq \dfrac{1}{\alpha} $, or $ \beta \neq 0 $ because of pole in $ \Gamma $ - function.
### $\textit{b} < 0$
Using the analysis of (10) one can see that the condition $ \textit{b} < 0$ along with $ \sigma = - 1 $ and $ C_{u} < 0 $ ($ \sigma C_{u} > 0 $) leads to the fact that the function $ u(y) $ has not changed, but $ v(y) $ becomes positive on all domain of definition. This function is presented in Fig.5.
$\eta \neq 1$
-------------
Let us return to equation (3) and rewrite it in terms of the variable $ \xi = e^{-\frac{cy}{\alpha}} $: \^[2]{} \_ - + \^ \^ = - . To integrate this equation we use the Lie group method of infinitesimal transformations [@Olver]. We find a group invariant of a second prolongation of one–parameter symmetry group vector of (11) and with its help we transform equation (11) into an equation of the first order. It turns out that nontrivial symmetry group requires some conditions: = 0,\
= and we consider the case $ \textit{b} =0 $. Thus, $ \upsilon = v $ and for z & = &\
w & = & v\_[y]{}v\^[- ]{} we obtain the Abel equation of the second kind: w\_[z]{}\[(1 - ) w - z\] + (+ - 1) z\^[-1]{} w\^[2]{} + z ( - + z\^[-]{}) = 0. Then we find solutions of equation (14) in parametric form with the parameter $ t $. Now we consider the case $ 2 \alpha + \eta \neq 1$. A combination of substitutions leads to: z & = & ( -) \^\
w & = & z\^ ( t + z\^ ) + z, where we take (t) > 0 (2+ -1) \_[t]{}(t) < 0, and equation (14) becomes an equation for the function $ \vartheta(t) $. Solving it, for $ \sigma C_{u} > 0 $ we obtain: (t) = ( ) \^[-]{} t\_[2]{}F\_[1]{} ( , ; ; - t\^[2]{} ) + C\_, where $\tilde{C_{\vartheta}} $, $ C_{\vartheta} $ are constants and $ _{2}F_{1} $ is the hypergeometric Gauss function. Further we obtain the solutions of initial equations (2)–(3) in parametric form: y(t) & = & - ( (t) )\
v(t) & = & (-)\^ ( ( + 1) t\^[2]{} + ) \^[-]{} ( (t) )\^\
u(t) & = & C\_[u]{} (-)\^ ( ( + 1) t\^[2]{} + ) \^[-]{} ( (t) )\^ where the constant $ \tilde{C_{\vartheta}} $ is chosen so that $ (2\alpha + \eta -1) \tilde{C_{\vartheta}} < 0 $, what is consistent with (16). Using the asymptotic representation of hypergeometric Gauss function as $ t \rightarrow \pm \infty $ we can take C\_ > | | ( ) \^[-]{} in order for $ y, v $ and $ u $ be real. Then one can see that all functions (18) are continuous bounded ones for $ t \in \Re $ and $ v, u $ are positive. Hence, one may try biologically interpret the functions $ v(y) $ and $ u(y) $ and this requires further investigation. In Fig.6 one may see the different curves $ v(y) $ for $ \eta = 0.1 $ and different $ \alpha $. Fig.7 demonstrates $ v(y) $ and $ u(y) $ for two $ \eta < 1 $. Further, for larger values of $ \alpha $ and $ \eta $ it seems more convenient to present curves $ y(t)$, $ v(t) $ and $ u(t) $ to analyze them, see Fig.8–Fig.10. One can see from (12) that $ \beta \gtrless 0 $ when $ \alpha \gtrless 2 $, and the case of $ \beta = 0 $, $ \alpha = 2 $ is presented in Fig.11.
Logarithmic sensitivity {#sec:2}
=======================
The model with logarithmic chemosensitivity function $ \phi (v) \sim \ln v $ is also studied. For the case of $f(u, v) = - v^{m} u + \tilde{\beta} v $, $ \tilde{\beta} = const $ an extensive analysis is performed in [@Wang]. This survey is focused on different aspects of traveling waves solutions. When $ m = 0 $ this model coincides with (1) for $ \textit{b} = 0 $. When $ \tilde{\beta} = 0 $ and $ m = 1 $ the system was studied in [@Nossal], [@Rosen]. The complete analysis for $ \tilde{\beta} = 0 $ is performed in [@Wang]. An existence of global solution is established in [@W].
Now we consider the system with $ \phi (v) = \ln v $ and $ f(u, v)= \tilde{\sigma} v u - \tilde{\beta} v $. Similarly, a replacement $ t \rightarrow \delta_{1} t $, $ u \rightarrow \sigma \dfrac{\tilde{\sigma} }{\delta_{1}} u $ gives $\delta_{1} = 1 $, $ \eta = \dfrac{\eta_{1}}{\delta_{1}} $, $ \alpha = \dfrac{\delta_{2}}{\delta_{1}} $, $ \beta = \dfrac{\tilde{\beta}}{\delta_{1}} $, $ \sigma = \pm 1 $. Then the model has the form: \[eq:20\] {
u\_[t]{}- u\_[xx]{} + (u )\_[x]{} & =0\
v\_[t]{}-v\_[xx]{} - v u + v & =0.\
. Let us rewrite system (20) in terms of function $ \upsilon (x, t) = \ln v(x, t) $: {
u\_[t]{}- u\_[xx]{} + (u \_[x]{})\_[x]{} & = 0\
\_[t]{}- \_[xx]{} - (\_[x]{})\^[2]{} + - u & = 0,\
. then in terms of traveling wave variable $ y = x - ct $, $ c = const $, (20$'$) has the form: {
u\_[y]{} + c u - u \_[y]{}+ & =0\
\_[yy]{} + (\_[y]{})\^[2]{} + c \_[y]{} - + u & =0,\
. where $ u=u(y) $, $ \upsilon = \upsilon (y) $ and $ \lambda $ is an integration constant. To integrate (20$'*$) we tested this system on the Painlevé ODE test. One can show that for $ \eta > 0 $ it passes this test only if $ \alpha = 2 $ with the additional condition $ \lambda = - \sigma c \beta \left( 1 + \dfrac{\eta}{2} \right) $ [@MSh_ArXiv]. If we express $ u(y) $ as $ \upsilon(y) $ from (20$'*$), we obtain an equation only for $ \upsilon (y) $; for $ \alpha = 2 $ it has the form: 2\_[yyy]{} + 3c \_[yy]{} + (c\^[2]{} + ) \_[y]{} + 2(2 - )\_[y]{} \_[yy]{} + 2(2 - ) (\_[y]{})\^[2]{} -2(\_[y]{})\^[3]{} - c- = 0. For $ \lambda = - \sigma c \beta \left( 1 + \dfrac{\eta}{2} \right) $ this equation can be linearized. It becomes equivalent to the following linear equation for $ F $: F\_[y]{} + c F = 0, F(y) = e\^[2 ]{} ( 2 \_[yy]{} + c \_[y]{} - (\_[y]{})\^[2]{} + ) that gives the equation for $ \upsilon (y) $: 2 \_[yy]{} + c \_[y]{} - (\_[y]{})\^[2]{} + = C\_[F]{} e\^[-2 -cy]{}, $ C_{F} = const $. If we rewrite (23) in terms of the variable $ \xi = e^{-\frac{cy}{2}} $ for the function $ \Psi (\xi) = e^{- \frac{\eta}{2} \upsilon} $ we obtain an equation similar to (11) with zero right-hand side: \^[2]{} \_ - + \^[2]{} \^[ + 1]{} = 0. Using the result of the symmetry group analysis of (11) we can write solution for $ \beta = 0 $ (see (18)): y(t) & = & - ( (t) )\
v(t) & = & ( t\^[2]{} + ) \^\
where $ \vartheta (t) $ is given in (17) and $ u(y) $ may be expressed from (20$'*$). However one may see that $ v \rightarrow \infty $ as $ t \rightarrow \pm \infty $ and this solution is unacceptable as a biological function.
Another possibility to solve this equation exactly is to put $ C_{F}$ equal to zero. When $ C_{F} = 0 $, that means $ F(y) = 0 $, and $ \beta \neq 0 $ equation (24) can be linearized by $ \xi = e^{\tau} $ . Its solution has three forms according to a sign of the expression $ D = \dfrac{2 \eta^{2} \beta}{c^{2}} + 1$. Since $v$ should be nonnegative and bounded function as $ cy \rightarrow \pm \infty $ the only suitable solution is v(y) & = & e\^[ y]{} ( C\_[-]{}e\^[- y]{} + C\_[+]{} e\^[ y]{} ) \^[-]{} where $ C_{\pm} $ are positive constants and $ \beta > 0 $. Unfortunately, the corresponding solution for $ u(y) $ is alternating and has the form: u(y) & = & - ( C\_[-]{}\^[2]{} (1 + ) e\^[- y]{} + C\_[+]{}\^[2]{} (1 - ) e\^[ y]{}\
& - & C\_[-]{} C\_[+]{} ) ( C\_[-]{}e\^[- y]{} + C\_[+]{} e\^[ y]{} ) \^[-]{} It is easy to see what $ \sigma u(y) \rightarrow \frac{c^{2}(\eta + 2)} {2 \eta ^{2}} (-1 \pm \sqrt{D}) $ as $ cy \rightarrow \pm \infty $. These functions are presented in Fig.12–Fig.13.
Linear sensitivity {#sec:3}
==================
Let us consider the system with linear function $ \phi (v) \sim v $. When $ f(u, v) = u - v $ the system is called the minimal chemotaxis model following the nomenclature of . This model is often considered with $ f(u, v) = \tilde{\sigma} u - \tilde{\beta} v $ ($ \tilde{\sigma} $ and $ \tilde{\beta} $ are constants) and it is studied in many papers. As was proved in , the solutions of this system are global and bounded in time for one space dimension. The case of positive $ \tilde{\sigma} $ and nonnegative $ \tilde{\beta} $ is studied in -[@F]. As we noted earlier, a sign of $ \tilde{\sigma} $ may effect on mathematical properties of the system, what changes its solvability conditions [@TF]. The review article [@H1] summarizes different mathematical results.
Now we consider the linear chemosensitivity function $ \phi (v) = v $ and $ f(u, v)= \tilde{\sigma} u - \tilde{\beta} v $. The replacement $ t \rightarrow \delta_{1} t$, $ v \rightarrow \dfrac{\eta_{1}}{\delta_{1}} v $, $ u \rightarrow \sigma \dfrac{\tilde{\sigma} \eta_{1}}{\delta_{1}^{2}} u $ leads to $\delta_{1} = \eta_{1} = 1 $, $ \alpha = \dfrac{\delta_{2}}{\delta_{1}} $, $ \beta = \dfrac{\tilde{\beta}}{\delta_{1}} $, $ \sigma = \pm 1 $. Then the system has the form: \[eq:123\] {
u\_[t]{}- u\_[xx]{} + (u v\_[x]{})\_[x]{} & =0\
v\_[t]{}-v\_[xx]{} + v - u & =0.\
.
This system reduces to system of ODEs in terms of traveling wave variable $ y = x - ct $, $ c = const $: {
u\_[y]{}+cu-u v\_[y]{}+ & =0\
v\_[yy]{}+cv\_[y]{} - v + u & =0,\
. where $ u=u(y) $, $ v=v(y) $ and $ \lambda $ is an integration constant. As shown in [@MSh] this system passes the Painlevé ODE test only if $ \alpha = 2 $ and $ \beta = 0 $. Consequently, in this case we can solve ($28*$) and the exact solution has the form [@MSh]: v & = & -\
u & = & - ( (v\_[y]{})\^[2]{} - \^[2]{}e\^[-cy]{} + ) , , $ \kappa >0 $, $ A $ and $ B $ are arbitrary constants. The functions $ I_{\nu} $ and $ K_{\nu} $ are Infeld’s and Macdonald’s functions respectively (Bessel’s functions of imaginary argument). This solution is not satisfactory from the biological point of view, since $ v(y) $ is an alternating function for any $ \nu $. However it seems interesting because of the following: in the case of $ \nu = \dfrac{1}{2} $ and $ B = \frac{2 + \pi}{2 \pi} $ in terms of $ e^{-\frac{cy}{2}} $ its form coincides with the well-known Korteweg-de Vries soliton e\^[v(e\^[-]{})]{} = sech\^[2]{}( e\^[-]{} + ). For $ \nu = \dfrac{1}{2} $ and arbitrary $ B $ the function $ u(y) $ is u(y) = . One can see that for $ \sigma = 1 $ (an increase of a chemical substance) the cells density $ u(y) \geq 0 $ for $ B \geq \dfrac{1}{\pi} $, and that for $ B > 0 $ $ u(y) $ is the solitary continuous solution vanishing as $ y \rightarrow \pm \infty $, whereas for $ B < 0 $ $ u(y) $ has a point of discontinuity. One can say that when $ B < 0 $ we obtain “blow up” solution in the sense that it goes to infinity for finite $ y $, and this is true for different $ \nu $. The functions (29) for $ \nu = \dfrac{1}{2} $ are presented in Fig.14–Fig.15.
Conclusion {#sec:5}
==========
We investigate three different one-dimensional parabolic-parabolic Patlak-Keller-Segel models. For each of them we obtain the exact solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. It seems interesting to consider the latter for the experimental values of the parameters and see their correspondence with experiment. This question requires a further investigations.
[26]{}
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---
abstract: 'Resources of a multi-user system in multi-processor online scheduling are shared by competing users in which fairness is a major performance criterion for resource allocation. Fairness ensures equality in resource sharing among the users. According to our knowledge, fairness based on the user’s objective has neither been comprehensively studied nor a formal fairness model has been well defined in the literature. This motivates us to explore and define a new model to ensure algorithmic fairness with quantitative performance measures based on optimization of the user’s objective. In this paper, we propose a new model for fairness in *Multi-user Multi-processor Online Scheduling Problem(MUMPOSP)*. We introduce and formally define quantitative fairness measures based on user’s objective by optimizing makespan for individual user in our proposed fairness model. We also define the unfairness of deprived users and absolute fairness of an algorithm. We obtain lower bound results for the absolute fairness for $m$-identical machines with equal length jobs. We show that our proposed fairness model can serve as a framework for measuring algorithmic fairness by considering various optimality criteria such as flow time and sum of completion times.'
author:
- Debasis Dwibedy
- Rakesh Mohanty
title: ' A New Fairness Model based on User’s Objective for Multi-user Multi-processor Online Scheduling'
---
Introduction {#sec:Introduction}
============
Supercomputers, grids and clusters in High Performance Computing(HPC) and web servers in client-server networking have gained immense practical significance as Service Oriented System(SOS) in modern day computation \[1\]. Unlike the traditional computing system such as personal computer, the *SOS* supports multiple users. The users compete for system’s resources for execution of their jobs. The most popular cluster scheduler *MAUI* \[2\] and the well-known *BOINC* platform \[3\] deal with a number of competing users, where each user submits a set of jobs simultaneously and seeks for a *minimum time of completion(makespan)* for their respective submissions. A non-trivial challenge for the *SOS* is to ensure fair scheduling of jobs for multiple users while optimizing their respective makespans.\
In this paper, we define and quantify fairness as a performance measure based on user’s objective for any resource allocation algorithm in multi users *HPC* systems. Particularly, we consider processors as the resources to be shared and makespan as the user’s objective in the *Multi-user Multi-processor Online Scheduling Problem(MUMPOSP)*. We formally define the *MUMPOSP* as follows.\
\
**Multi-user Multi-processor Online Scheduling Problem(MUMPOSP)**
- **Inputs:** We are given a set of $m$ identical processors, i.e. $M$=$\{M_1, M_2,..,M_m\}$ and a set of $n$ jobs, where $m\geq 2$ and $n>>>m$. Let $U_r$ represents a *user*, where $1\leq r\leq k$ and $J^{r}$ is the sequence of jobs requested by *user* $U_r$, where $J^{r}$=$(J^{r}_i|1\leq i\leq n_r)$ such that $J$=$\bigcup_{r=1}^{k}{J^r}$, $\sum_{r=1}^{k}{n_r}$=$n$ and $J^x\cap J^y$=$\phi$, where $x\neq y$ and $1\leq (x, y)\leq k$. The processing time of job $J^{r}_i$ is $p^{r}_i$, where $p^{r}_i\geq 1$.
- **Output:** Generation of a Schedule($S$) in which makespan for each user ($U_r$) is denoted by $C^{r}_{max}$=$\max\{c^{r}_i|1\leq i\leq n_r\}$, where $c^{r}_i$ is the completion time of job $J^{r}_{i}$
- **Objective:** Minimize $C^{r}_{max}$, $\forall U_r$.
- **Constraint**: The scheduler can receive a batch of at most $r$ jobs at any time step and the jobs must be irrevocably scheduled before the arrival of next batch of jobs, where $1\leq r\leq k$.
- **Assumption.** The jobs are independent and are requested from $k$ parallel users, where $k\geq 2$
**Illustration of MUMPOSP.** For simplicity and basic understanding of the readers, we illustrate an instance of *MUMPOSP* for scheduling of $n$ jobs that are submitted by $k$ users($U_r$) as shown in *Figure* \[fig:mumposp.png\]. Here, $M_1$, $M_2$,...,$M_m$ represent $m$ identical machines and $U_1$, $U_2$,...,$U_{k-1}$, $U_k$ denote job sequences for $k$ users. Here, each user has $\frac{n}{k}$ jobs. Jobs are submitted in batches online, where a batch is constructed after receiving exactly one job from each user(as long as an user has an unscheduled job). A batch consists of at least one job. Therefore, we have at least $1$ batch, where $k$=$n$ and at most $n-k+1$ batches, where any one of the users $U_r$ has $n_r$=$n-k+1$ and remaining users have exactly one job each. Each user($U_r$) seeks to obtain a minimum value for its makespan($C^{r}_{max}$) as the output, rather than the overall makespan($C_{max}$) of the system. Hence, it is indispensable for the scheduler to be fair while optimizing the $C^{r}_{max}$ for each user.\
![Illustratopn of MUMPOSP for k Users with Equal Number of Jobs[]{data-label="fig:mumposp.png"}](mumposp.PNG)
**Representation of MUMPOSP.** By following general framework $\alpha|\beta|\gamma$ of Graham et al.\[16\], we represent *MUMPOSP* as $MUMPOSP(k, P_m|C^{r}_{max})$, where $P_m$ denotes $m$-identical machines and $k$ is the number of users.\
\
**Perspectives of Fairness.** Fairness has been considered as a major performance criterion for the scheduling algorithms in multi-user systems \[4, 5\]. Fairness has been studied from two perspectives such as allocation of resources to the users and user’s objective. Fairness of an algorithm with respect to resource allocation, guarantees uniform allocation of resources to the competing users \[6\]. The resources to be shared are application dependent. For example, in client-server networking, the shared resources may be throughput or network delay or specific time quantum \[7, 8\], whereas in distributed systems, the resources may be processors or memory or time slice \[9, 10\].\
Algorithmic fairness based on user’s objective is evaluated by the objective values achieved for respective users. An equality in the obtained objective values for a user ensures fairness of a scheduling algorithm. It is important for a fairness measure to define the equality for quantifying how far an achieved objective value is far from the defined equality.\
\
**Related Work.** Fairness as a quantitative performance measure based on resource allocation was comprehensively studied by Jain et al. \[6\]. A set of properties for an ideal fairness measure was defined and a fairness index $F(x)$ was proposed for resource allocation schemes. If any scheduling algorithm assigns resources to $k$ competing users such that $r^{th}$ user gets an allocation of $x_r$. then $F(x)$ is defined as follows.\
\
$F(x)$=$\frac{[{\sum_{r=1}^{k}{x_r}}]^2}{\sum_{r=1}^{k}{{x_r}^2}}$, where $x_r\geq 0$.\
\
The value of $F(x)$ is bounded between $0$ and $1$ to meaningfully show percentage of fairness and discrimination of a resource allocation scheme for each user. Fairness based on sharing of resources such as processors, memory, system clock and system bus in multi-programmed multi-user system was well studied in \[9-11\]. Some recent works with contributions on algorithmic fairness of online scheduling can be found in \[17-18\]. According to our knowledge, study of fairness based on user’s objective has not been exhaustively studied in the literature.\
In \[12-15\], *stretch* matrix has been considered as a user’s objective based fairness measure for resource scheduling algorithms in multi-user system. Stretch($d^{r}_{A}$) is defined as a degradation factor in the objective value obtained by any algorithm *A* for each user $U_r$. Let us consider $V^{r}_{A}$ is the objective value achieved by algorithm *A* and $V^{r}_{OPT}$ is the optimum objective value for respective $U_r$. Now, stretch can be defined as\
$d^{r}_{A}$=$\frac{V^{r}_{A}}{V^{r}_{OPT}}$\
\
The objective of any scheduling algorithm is to incur an equal stretch for each $U_r$ to ensure fairness. Stretch matrix guarantees fairness based on equality in achieved objective values. However, it fails to bound the fairness and unable to show the exact value of fairness per user as well as overall fairness of a scheduling algorithm. Further, the discrimination of any scheduling algorithm for the deprived users can not be captured by the stretch matrix. Therefore, it is quintessential to define a formal fairness measure based on user’s objective.\
\
**Our Contribution.** In our work, we propose a new model for fairness in *Multi-user Multi-processor Online Scheduling Problem(MUMPOSP)*. We introduce and formally define quantitative fairness measures based on user’s objective by optimizing makespan for individual user in our proposed fairness model. We also define the unfairness of deprived users and absolute fairness of an algorithm. We obtain lower bound results for the absolute fairness for $m$-identical machines with equal length jobs. We show that our proposed fairness model can serve as a framework for measuring algorithmic fairness by considering various optimality criteria such as flow time and sum of completion times.
Our Proposed Fairness Model
===========================
We develop a new fairness model in which, we define five quantitative measures to ensure algorithmic fairness. Instead of considering the resource allocation at the input level, the model considers the achieved value of user’s makespan at the output level to determine the fairness of a scheduling algorithm. The model captures the issues of relative and global parameters for fairness by a *Fairness Index(FI)*. The issues of unfairness is captured by a *Discrimination Index(DI)*. The *FI* includes fairness measures such as *Relative Fairness(RF)* and *Global Fairness(GF)*. Higher value of any fairness measure indicates more fair algorithm. The *DI* includes unfairness measures such as *User Discrimination Index(UDI)*, *Global Discrimination Index(GDI)* and *Relative Discrimination Index(RDI)*. Lower value of any unfairness measure indicates more fair algorithm. Before defining the fairness measures, we illustrate our proposed fairness model and discuss the characteristics of a good fairness model as follows.\
\
**Illustration of our Proposed Fairness Model.** We illustrate our proposed fairness model as shown in Figure \[fig:fairnessmodel3.png\]. The model considers the *MUMPOSP* and captures the fairness of online scheduling algorithm by considering the makespan($C^{r}_{max}$) of individual user at the output.\
\
\
![A Fairness Model based on User’s Objective[]{data-label="fig:fairnessmodel3.png"}](fairnessmodel3.PNG)
\
\
Characteristics of a Good Fairness Model {#subsec:Characteristics of a Good Fairness Model }
----------------------------------------
Motivated by the seminal work of Jain et al. \[6\] for characterization of a good fairness model, we capture the following essential properties in our model to develop our fairness measures.
- *Independent of Input Size*. A model must be applicable to any number of users irrespective of the number of jobs offered by the users and availability of any number of machines.
- *Independent of Scale*. The model should be independent of unit or scale of measurement. The model should be able to measure fairness irrespective of the fact that processing time of the jobs are given in seconds or micro-seconds or nano-seconds. The measuring unit must be uniform or inter convertible.
- *Finitely Bounded*. The model must bound the value of fairness measure within a finite range, preferably between $0$ and $1$ such that percentage of fairness for respective users can easily be determined.
- *Consistent*. If any change in the scheduling policy results in different makespan for at least one user, then the change in the fairness measure must be reflected to the concerned users as well as to the overall fairness of the policy.
In addition to the above mentioned properties from the literature, we also consider relative and overall fairness as en essential feature to develop our fairness measures. The model must represent *relative equality* among achieved objective values for the users to show user’s fairness of an algorithm. For example, the users may not seek equal makespan as a gesture of fairness, however, they require to obtain an equal ratio between the *desired makespan(optimum value)* to the achieved makespan for all users. The value obtained by an algorithm for relative equality leads to *relative fairness* with respect to each user. Also, the model should show *overall fairness* of the algorithm with respect to all users.
Our Proposed Fairness Measures
------------------------------
By considering the above mentioned desirable properties, we now define formal measures of fairness and unfairness for *MUMPOSP* as follows.\
If any algorithm *A*, schedules jobs of $k$ competing users on $m$ parallel machines such that $r^{th}$ user obtains a makespan of $C^{r}_{A}$, then we define the following measures.
The **Relative Fairness(RF)** obtained by algorithm *A* for any user $U_r$ is defined by\
${RF}(C^{r}_{A})$=$\frac{C^{r}_{OPT}}{C^{r}_{A}}$, where, $C^{r}_{OPT}$=$\frac{\sum_{i=1}^{n_r}{p^{r}_{i}}}{m}$ (1)
The **Relative Fairness Percentage(RFP)** for any user $U_r$ obtained by algorithm *A* is defined by\
${RFP}(C^{r}_{A})$=$R(C^{r}_{A})\cdot 100$(2)
The **Global Fairness(GF)** of algorithm *A* for $k$ users is defined by\
${GF}(C^{r}_{A}, k)$=$\frac{1}{k}\cdot \sum_{r=1}^{k}{(RF(C^{r}_{A}))}$ (3)
The **Global Fairness Percentage(GFP)** of any algorithm *A* for $k$ users is defined by\
${GFP}(C^{r}_{A}, k)$=$GF(C^{r}_{A}, k)\cdot 100$(4)
Again, if any algorithm *A*, schedules jobs of $k$ competing users such that $r^{th}$ user obtains a makespan of $C^{r}_{A}$, then we define **Fairness Index** for algorithm *A* represented by $2$-tuple with two parameters such as $RF$ and $GF$ as follows\
\
$FI(C_A)$=$<\{RF(C^{r}_{A})|1\leq r\leq k\}, GF(C^{r}_{A}, k)>$(5)\
\
**Example 1:** Let us consider $3$ users $U_1$, $U_2$ and $U_3$ with jobs $U_1$=$\{J^{1}_{1}/1, J^{1}_{2}/2 \}$, $U_2$=$\{J^{2}_{1}/3, J^{2}_{2}/4 \}$ and $U_3$=$\{J^{3}_{1}/5, J^{3}_{2}/6 \}$ respectively. Suppose any algorithm *A* schedules the jobs of $U_1$, $U_2$ and $U_3$ such that $C^{1}_{A}$=$11$, $C^{2}_{A}$=$9$ and $C^{3}_{A}$=$10$, then we have, ${RF}(C^{1}_{A})$=$\frac{1.5}{11}$=$0.13$ and ${RFP}(C^{1}_{A})$=$13\%$, ${RF}(C^{2}_{A})$=$\frac{3.5}{9}$=$0.38$ and ${RFP}(C^{2}_{A})$=$38\%$, ${RF}(C^{3}_{A})$=$\frac{5.5}{10}$=$0.55$ and ${RFP}(C^{3}_{A})$=$55\% $. Therefore, we have ${GF}(C^{r}_{A}, 3)$=$0.35$ and ${GFP}(C^{r}_{A}, 3)$=$35\%$.\
The **Unfairness** of any algorithm *A* for *MUMPOSP* with respect to each user $U_r$ is defined by **User Discrimination Index** as\
${UDI}_{A}^{r}$=$1- {RF}(C^{r}_{A})$(6)
The **Overall Unfairness** of algorithm *A* for $k$ users is defined by **Global Discrimination Index** as\
${GDI}(C^{r}_{A}, k)$=$1-{GF}(C^{r}_{A}, k)$ (7)
The **Realtive Discrimination Index(RDI)** of any algorithm *A* for *MUMPOSP* with respect to each user $U_r$ is defined as\
${RDI}_{A}^{r}$=${GF}(C^{r}_{A}, k)- {RF}(C^{r}_{A})$, if ${RF}(C^{r}_{A})< {GF}(C^{r}_{A}, k)$\
${RDI}_{A}^{r}$=$0$, otherwise (8)
Again, if any algorithm *A*, schedules jobs of $k$ competing users such that $r^{th}$ user obtains a makespan of $C^{r}_{A}$, then we define **Discrimination Index** for algorithm *A* as $3$-tuple with three parameters such as $UDI$, $GDI$ and $RDI$ as follows.\
\
${DI}(C_A)$=$<\{{UDI}^{r}_{A}|1\leq r\leq k\}, {GDI}(C^{r}_A, k), \{{RDI}_{A}^{r}|1\leq r\leq k\}>$(9)\
\
**Example 2:** Let us consider any algorithm *A* results relative fairness for $U_1$, $U_2$, $U_3$ and $U_4$ as $0.6$, $0.6$, $0.6$ and $0.2$ respectively. We now have ${GF}(C^{r}_{A}, 4)$=$0.5$. Therefore,${UDI}_{A}^{1}$=$1-0.6$=$0.4$, ${UDI}_{A}^{2}$=$1-0.6$=$0.4$, ${UDI}_{A}^{3}$=$1-0.6$=$0.4$, ${UDI}_{A}^{4}$=$1-0.2$=$0.8$, ${GDI}(C^{r}_{A}, k)$=$1-0.5$=$0.5$ and ${RDI}_{A}^{4}$=$0.5-0.2$=$0.3$.
Absolute Fairness and Lower Bound Results {#sec:Absolute Fairness and Lower Bound Results}
=========================================
We define absolute fairness as a quantitative measure and provide lower bound results for characterization of the same as follows.
Any algorithm *A* achieves **Absolute Fairness** if ${RF}(C^{r}_{A})$ is the same $\forall U_r$.
*Lemma 1. If any algorithm A incurs ${RDI}^{r}_{A}$=$0$, $\forall U_r$, then it achieves absolute fairness.*\
*Proof.* If ${RDI}^{r}_{A}$=$0$, $\forall U_r$, then by Eq. (8), we have\
${RF}(C^{r}_{A})\geq {GF}(C^{r}_{A})$(10)\
By Eqs. (3) and (10), we can infer that ${RF}(C^{r}_{A})$=${GF}(C^{r}_{A})$, $\forall U_r$\
Therefore Lemma 1 holds true. $\Box$
Any Algorithm *A* is **$b$-fair**, if it achieves ${RF}(C^{r}_{A})$=$b$ for all $U_r$, where $0<b\leq 1$.
**Theorem 1. Any algorithm that achieves absolute fairness for $MUMPOSP(k, P_2|C^{r}_{max})$ must be at least $(\frac{1}{k})$-fair, where $k\geq 2$ and $1\leq r\leq k$.**\
*Proof.* Let us consider an instance of $MUMPOSP(k, P_2|C^{r}_{max})$, where $k$=$2$. We analyze two cases based on $n_r$ as follows.\
**Case $1$:** $n_1\neq n_2$.\
Case $1$.(a): If the first job pair($J^{1}_{1}, J^{2}_{1}$) are scheduled on different machines. Let us consider the following instance $U_1:<J^{1}_2/2, J^{1}_1/1>$, $U_2:<J^{2}_{1}/1>$, where each job is specified by its processing time. Assigning $J^{1}_{1}/1$ and $J^{2}_{1}/1$ to machines $M_1$ and $M_2$ respectively followed by the assignment of $J^{1}_{2}/2$ to either of the machines such that $C^{1}_{A}$=$3$ and $C^{2}_{A}$=$1$, where $C^{1}_{OPT}\geq 1.5$ and $C^{2}_{OPT}\geq 0.5$. Therefore, we have $\frac{C^{1}_{OPT}}{C^{1}_A}\geq \frac{1}{2}$ and $\frac{C^{2}_{OPT}}{C^{2}_A}\geq \frac{1}{2}$\
Case $1$.(b): If the first job pair($J^{1}_{1}, J^{2}_{1}$) are scheduled on the same machine. Let us consider the following instance $U_1:<J^{1}_3/2, J^{1}_2/1, J^{1}_1/1>$, $U_2:<J^{2}_{2}/2, J^{2}_{1}/1>$. If the first job pair($J^{1}_{1}/1, J^{2}_{1}/1$) are scheduled either on machine $M_1$ or on $M_2$, then by assigning the next pair of jobs ($J^{1}_{2}, J^{2}_{2}$) to the same or different machines, followed by the assignment of job $J^{1}_{3}/2$ such that $C^{1}_{A}$=$4$ and $C^{2}_{A}$=$3$, where $C^{1}_{OPT}\geq 2$ and $C^{2}_{OPT}\geq 1.5$. Therefore, we have $\frac{C^{1}_{OPT}}{C^{1}_A}\geq \frac{1}{2}$ and $\frac{C^{2}_{OPT}}{C^{2}_A}\geq \frac{1}{2}$.\
**Case 2: $n_1$=$n_2$.**\
Case 2.(a): If the first job pair($J^{1}_{1}, J^{2}_{1}$) are scheduled on different machines. Let us consider the following instance $U_1:<J^{1}_3/2, J^{1}_2/1, J^{1}_1/1>$, $U_2:<J^{2}_{3}/2, J^{2}_{2}/2, J^{2}_{1}/1>$. Assigning jobs $J^{1}_1/1$ and $J^{2}_{1}/1$ to machines $M_1$ and $M_2$ respectively, followed by the assignment of the subsequent jobs as shown in Figure \[fig:lowerbound.png\].(a), such that $C^{1}_{A}$=$4$ and $C^{2}_{A}$=$5$, where $C^{1}_{OPT}\geq 2$ and $C^{2}_{OPT}\geq 2.5$. Therefore, we have $\frac{C^{1}_{OPT}}{C^{1}_A}\geq \frac{1}{2}$ and $\frac{C^{2}_{OPT}}{C^{2}_A}\geq \frac{1}{2}$.\
Case 2.(b): If the first job pair($J^{1}_{1}, J^{2}_{1}$) are assigned to the same machine. We consider the same instance of Case 2.(a). Assigning $J^{1}_{1}/1$ and $J^{2}_{1}$ on either machine $M_1$ or on $M_2$, followed by the assignment of the subsequent jobs as shown in Figure\[fig:lowerbound.png\].(b) such that $C^{1}_{A}$=$4$ and $C^{2}_{A}$=$5$. Therefore, we have $\frac{C^{1}_{OPT}}{C^{1}_A}\geq \frac{1}{2}$ and $\frac{C^{2}_{OPT}}{C^{2}_A}\geq \frac{1}{2}$. $\Box$
![Illustration of Case 2[]{data-label="fig:lowerbound.png"}](lowerbound.PNG)
Results on Absolute Fairness in MUMPOSP with $m$ Identical Machines for Equal Length Jobs
-----------------------------------------------------------------------------------------
For ease of understanding, we analyze the lower bound of absolute fairness for any algorithm in a generic *MUMPOSP* setting, where each user has equal number of jobs and all jobs have equal processing time of $x$ unit, where $x\geq 1$. The objective of each user is to obtain a minimum $C^{r}_{max}$. We formally denote the problem as *MUMPOSP($k, P_m|p^{r}_{i}$=$x|C^{r}_{max}$)*.\
\
*Lemma 2. In MUMPOSP($k, P_m|p^{r}_{i}$=$x|C^{r}_{max}$) with $k$=$b\cdot m$, any algorithm *A* obtains $C^{r}_{A}\leq b\cdot \sum_{i=1}^{n_r}{p^{r}_{i}}$, for each $U_r$ respectively, where $1\leq r\leq k$, $m\geq 2$ and $b\geq 1$.*\
*Proof.* We proof Lemma 2 by method of induction on number of jobs per user ($n_r$) as follows.\
*Induction Basis:* Let us consider $k$=$m$=$2$, $n_1$=$n_2$=$1$ and $p^{1}_1$=$p^{2}_1$=$1$. Clearly, $C^{r}_{A}$=$1 \leq b\cdot 1\cdot 1$, where $r$=$1, 2$ and $b\geq 1$.\
*Induction Hypothesis:* Let us consider $k$=$(b\cdot m)$, $n_r$=$\frac{n}{k}$=$y$, where $y\geq 1$, $b\geq 1$ and $n$=$\sum_{r=1}^{k}{n_r}$\
We assume that $C^{r}_{A} \leq b\cdot \sum_{i=1}^{n_r}{p^{r}_{i}}\leq b\cdot x\cdot y$ (11) *Inductive Step:* For $n_r$=$y+1$ with $p^{r}_{i}$=$x$, $\forall J^{r}_{i}$. We have to show that $C^{r}_{A} \leq (y+1)\cdot b\cdot x$.\
By Eq. (11), we have $C^{r}_{A}$=$y\cdot b\cdot x$ with $n_r$=$y$. When we add extra one job to each user, we have by *Induction Basis* $C^{r}_{A}$=$b\cdot x\cdot y+(b\cdot x)$=$(y+1)\cdot b\cdot x$. Therefore, Lemma 2 holds true. $\Box$\
\
*Lemma 3. Any algorithm *A* is $\frac{1}{k}$-fair for MUMPOSP($P_m|p^{r}_{i}$=$x|C^{r}_{max}$) with $k$=$b\cdot m$, where $m\geq 2$ and $b\geq 1$.*\
*Proof.* By Lemma 2, we have $C^{r}_{A} \leq b\cdot \sum_{i=1}^{n_r}{p^{r}_{i}}$, $\forall U_r$ (12)\
We have the fair optimum bound as $C^{r}_{OPT}\geq \frac{\sum_{i=1}^{n_r}{p^{r}_{i}}}{m}$, $\forall U_r$ (13)\
By Eqs. (12) and (13), we have $\frac{C^{r}_{OPT}}{C^{r}_{A}} \geq \frac{1}{k}$, $\forall U_r$.\
Therefore, *Lemma 3* holds true. $\Box$\
\
*Lemma 4. In MUMPOSP($P_m|p^{r}_{i}$=$x|C^{r}_{max}$) with $k> m$, any algorithm *A* obtains $C^{r}_{A}\leq \lceil\frac{n}{m}\rceil \cdot x$, for each $U_r$ respectively, where $k\neq m\cdot b$ for $b\geq 1$.*\
*Proof.* The correctness of Lemma 4 is shown by method of induction on $n_r$ as follows.\
*Induction Basis:* Let us consider $m$=$2$, $k$=$3$, $n_r$=$1$ and $p^{r}_{i}$=$1$. Now, we have $n$=$n_r\cdot k$=$3$. Clearly, $C^{r}_{A}\leq 2$=$\lceil\frac{n}{2}\rceil \cdot 1$\
*Induction Hypothesis:* Let us consider $n_r$=$\frac{n}{k}$=$y$, $p^{r}_{i}$=$x$ and $k> m$ with $k\neq m\cdot b$ for $b\geq 1$. We assume that $C^{r}_{A}\leq \lceil\frac{n}{m}\rceil \cdot x$, $\forall U_r$.\
*Inductive Step:* We show that $C^{r}_{A}\leq \lceil\frac{n+k}{m}\rceil \cdot x$ for $n_r$=$y+1$, $\forall U_r$.\
By our Induction Basis, for one extra job of each user $U_r$, where $1\leq r\leq k$, algorithm *A* incurs an additional time of $\lceil\frac{k}{m}\rceil \cdot x$ for each $U_r$.\
Therefore, $C^{r}_{A}\leq \lceil\frac{n}{m}\rceil \cdot x + \lceil\frac{k}{m}\rceil \cdot x \leq \lceil\frac{n+k}{m}\rceil \cdot x$\
Thus, Lemma 4 holds true. $\Box$\
\
**Theorem 2. Any Algorithm *A* is $\frac{1}{k}$-fair for MUMPOSP($k, P_m|p^{r}_{i}$=$x|C^{r}_{max}$), where $k\geq m$ and $m\geq 2$.**\
*Proof.* Theorem 2 holds true by Lemma 3 for $k$=$m\cdot b$, where $b\geq 1$.\
By Lemma 4, we have $C^{r}_{A}\leq \lceil\frac{n}{m}\rceil \cdot x$ (14)\
By Eq. (13), we have $C^{r}_{OPT}\geq \frac{\frac{n}{k}\cdot x}{m}$\
Implies $C^{r}_{OPT}\geq \frac{n\cdot x}{k\cdot m}$ (15)\
By Eqs. (14) and (15), we have $\frac{C^{r}_{OPT}}{C^{r}_{A}}\geq \frac{\frac{n\cdot x}{k\cdot m}}{\frac{n\cdot x}{m}}$\
$\geq \frac{n\cdot x\cdot m}{n\cdot k \cdot m \cdot x}\geq \frac{1}{k}$ $\Box$
Fairness Measure using Flow Time and Completion Time as User’s Objective
========================================================================
We show that our proposed Fairness Index can be served as a framework for measuring fairness of any algorithm based on well-known user’s objectives such as *sum of completion times($S^{r}$)* or weighted sum of completion times($W^{r}$) or sum of flow times($SF^{r}$). Selection of an user’s objective is application dependent. For instance, users of interactive systems require optimized value for respective flow time $f^{r}$, where $f^{r}_{i}$ of any $J^{r}_{i}$ is the difference between its completion time $c^{r}_{i}$ and arrival time($t^{r}_{i}$). We now define relative fairness measures based on the above mentioned user’s objectives respectively by our proposed *FI*.
- **Sum of Completion Times($S^{r}$):** Here, the objective for each $U_r$ is to obtain a minimum $S^{r}$=$\sum_{i=1}^{n_r}{c^{r}_{i}}$. The relative fairness for any $U_r$, obtained by any algorithm *A* based on $S^{r}$ is defined as\
$R_{A}(S^{r}_{A})$=$\frac{S^{r}_{OPT}}{S^{r}_{A}}$, where $S^{r}_{OPT}$ is the optimum value for $S^{r}$.\
- **Weighted Sum of Completion Times($W^{r}$):** Here, the $c^{r}_{i}$ is associated with certain positive weight $w^{r}_{i}$. The objective for each $U_r$ is to obtain a minimum $W^{r}$=$\sum_{i=1}^{n_r}{w^{r}_{i}\cdot c^{r}_{i}}$. The relative fairness for any $U_r$ obtained by algorithm *A* based on $W^{r}$ is defined as\
$R_{A}(W^{r}_{A})$=$\frac{W^{r}_{OPT}}{W^{r}_{A}}$ where, $W^{r}_{OPT}$ is the optimum value for $W^{r}$.\
- **Sum of Flow Times(${SF}^{r}$):** Here, each $U_r$ wants a minimum value for respective ${SF}^{r}$=$\sum_{i=1}^{n_r}{f^{r}_{i}}$, where $f^{*r}_{i}$ is the desired value of $f^{r}_{i}$ and ${SF}^{r}_{OPT}$=$\sum_{i=1}^{n_r}{f^{*r}_{i}}$. The relative fairness for any $U_r$ obtained by algorithm *A* based on ${SF}^{r}$ is defined as\
$R_{A}({SF}^{r}_{A})$=$\frac{{SF}^{r}_{OPT}}{{SF}^{r}_{A}}$.
Concluding Remarks and Scope of Future Work
============================================
In our work, we addressed the non-trivial research challenge of defining a new fairness model with quantitative measures of algorithmic fairness for *Multi-users Multi-processor Online Scheduling Problem(MUMPOSP)* based on user’s objective. We formally presented the *MUMPOSP* with an illustration followed by perspectives on fairness. We proposed a new fairness model and defined five quantitative measures to ensure algorithmic fairness by considering minimization of makespan as the user objective. By considering the properties of a good fairness model, our fairness measures were formally defined. We defined absolute fairness and obtained lower bound results for *MUMPOSP* with identical machines for equal length jobs. We show that our proposed fairness measure can serve as a framework for measuring algorithmic fairness based on other well-known user’s objective such as flow time and completion time.\
**Scope of Future Work.** We assumed a theoretical bound for the value of $C^{r}_{OPT}$ for each $U_r$. It is still open to find a realistic bound for the value of $C^{r}_{OPT}$. It is a non-trivial research challenge to compare the fairness of any two algorithms $A$ and $B$, when global fairness of algorithm $A$ and algorithm $B$ are same and the relative fairness of $A$ is more than the relative fairness of $B$ for some users or vice versa. In this scenario, it is interesting to make a trade-off by considering the number of users and individual relative fairness of each user for comparison of fairness of two different algorithms.
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abstract: |
We consider a quantum many-body problem in one-dimension described by a Jastrow type wavefunction, characterized by an exponent $\lambda$ and a parameter $\gamma$. In the limit $\gamma=0$ the model becomes identical to the well known $1/r^2$ pair-potential model; $\gamma$ is shown to be related to the strength of a many body correction to the $1/r^2$ interaction. Exact results for the one-particle density matrix are obtained for all $\gamma$ when $\lambda=1$, for which the $1/r^2$ part of the interaction vanishes. We show that with increasing $\gamma$, the Fermi liquid state (at $\gamma=0$) crosses over to distinct $\gamma$-dependent non-Fermi liquid states, characterized by effective “temperatures”.
PACS No. 71.10 +x
$^{\dag}$Permanent address: Physics Department, University of Florida, Gainesville, FL 32611, U.S.A.
author:
- |
Y. Chen and K. A. Muttalib$^{\dag}$\
Department of Mathematics\
Imperial College, London SW7 2BZ, U. K.
title: 'Crossover from Fermi Liquid to Non-Fermi Liquid Behavior in a Solvable One-Dimensional Model'
---
-.5in 0.1cm 16 cm 24 cm
A special class of Jastrow type wave functions \[1\] $$\Psi(u_1,\ldots,u_N)=C\prod_{1\leq a<b\leq N}|u_a-u_b|^{\lambda}
\;\prod_{c=1}^{N}{\rm e}^{-V(u_c)},$$ appear frequently in quantum many-body problems. In one dimension, many-body Hamiltonians with pair-potentials of the form $1/r^2$ (or its periodic equivalent $1/{\rm sin}^2r$), where $r=(u_a-u_b)$, have exact ground state wave functions of the Jastrow form \[2-5\]. It has been proposed that variational wave functions of this type give reasonably good descriptions of models for strongly interacting fermions \[6-11\]. It is therefore of interest to investigate the properties of such wave functions as exactly as possible. Of particular interest is the question whether such a wave function describes how interaction can change a Fermi liquid into a non Fermi liquid state.
In the solution of the $1/r^2$ model, the parameter $\lambda$ in Eq. (1) is related to the strength of the $1/r^2$ pair potential; in particular, $\lambda=1$ corresponds to the free fermion case. Unfortunately, the exact density matrix for this model can be obtained only for a few special values of $\lambda$ \[2,12\], so the question of the nature of crossover from the free fermion to the interacting non-Fermi liquid state can not be addressed exactly in this model. In the present work, we consider a one-dimensional wave function of the above form which can be considered as a generalization of the wavefunction corresponding to the $1/r^2$ model. In addition to the parameter $\lambda$, our model contains an additional parameter $\gamma$ which we show to be related to the strength of a many body correction to the $1/r^2$ interaction. We obtain the one-particle density matrix for this model exactly for all $\gamma$, for the particular case of $\lambda=1$. Thus $\gamma=0$ represents the free fermion case. We show that with increasing $\gamma$, the Fermi liquid state is destroyed by an effective non-zero “temperature” induced by interaction.
The exact density matrix of the $1/r^2$ model was obtained \[2\] by exploiting the analogy of the wave function with the eigenvalue distribution of random matrices \[13\]. The wavefunction we consider is motivated by our recent generalization of the conventional Wigner-Dyson-Mehta random matrices, for describing transport in disordered systems \[14\]. We consider the wavefunction given by Eq.(1) with $$V(u)={1\over 2\gamma}\left[{\rm sinh}^{-1}[(\gamma
\omega)^{1\over 2} u]\right]^{2}
+{1\over 2}{\rm ln}
\left[\vartheta_4\left({\pi\over \gamma}
{\rm sinh}^{-1}[(\gamma
\omega)^{1\over 2} u];p\right)\right],$$ characterized by a single parameter $\gamma$. Here $\vartheta_4(x;p)$ is the Jacobi Theta function \[15\], $p={\rm e}^{-\pi^2/\gamma}$, and $\omega$ has the dimension of $1/[{\rm length}]^2$. For $\gamma=0,$ $V(u)=
{1\over 2}\omega u^2,$ and the wavefunction reduces to the well known solution of the $1/r^2$ pair potential. The case $\lambda=1$ then represents a free Fermion problem (with the choice of Fermi statistics), with the one-particle density matrix given by ${{\rm sin}[\pi Dr]/(\pi r)},$ $D$ being the density of particles. The parameter $\gamma$ “deforms” the harmonic well into a weakly confining $[{\rm ln}u]^2$ term for large enough $u$, as shown in figure 1. We will show that this deformation leads to a qualitative change in the density matrix; for $\lambda=1$ this corresponds to a change from a Fermi liquid to a non-Fermi liquid state.
In order to understand the role of the parameter $\gamma$, let us concentrate for the moment on the small $\gamma (\ll \pi^2)$ limit where the second term in Eq. (2) can be neglected, and the Hamiltonian corresponding to the above wave function has a simple form. In this limit, to leading order in $\gamma$, the Schrödinger equation (in units where $\hbar^2/2m=1$) can be written as, $${1\over \Psi}\sum_{k}{\partial^2\over \partial u_k^2}\Psi
=\lambda(\lambda-1)\sum_{j\neq k}{1\over (u_k-u_j)^2}$$ $$+{\omega}^2[1+2\gamma]\sum_k{(u_k)^2}
-\omega[N+\lambda N(N-1)]$$ $$+{2\over 3}\lambda\gamma{\omega}^2\sum_{j\neq k}{1\over
u_k-u_j}[(u_k)^3-(u_j)^3]
-{4\over 3}\gamma{\omega}^3\sum_k{(u_k)^4}$$ Note that for $\gamma=0$, the Hamiltonian reduces to the $1/(u_k-u_j)^2$ pair potential, with energy $E=\omega[N+\lambda
N(N-1)]$ \[2,3\]. For $\lambda=1,$ with the choice of Fermi statistics, this becomes a free Fermion problem. However, for $\gamma\neq 0$, a many-body correction term survives even for $\lambda=1$. Nevertheless, the density matrix for $\lambda=1$ can still be found exactly for all $\gamma$. We will show that it corresponds to a non-Fermi liquid state, $\gamma$ playing the role of an effective temperature.
The exact density matrix corresponding to the wave function defined by eqns. (1) and (2) for $N\rightarrow \infty$ and for all $\gamma$, as obtained by exploiting the analogy of the present problem with the random matrix model recently constructed for describing transport in disordered systems \[14\], is given by $$\rho(u,v)=f(\gamma){\cal Q}(\mu,\nu)
{\vartheta_1\left({\pi(\mu-\nu)\over 2\gamma};p\right)\over u-v},$$ where $${\cal Q}(\mu,\nu)={\vartheta_4\left({\pi(\mu+\nu)\over 2\gamma};p\right)
\over \vartheta_4^{1/2}\left(\frac{\pi\mu}{\gamma};p\right)
\vartheta_4^{1/2}\left({\pi\nu\over \gamma};p\right)},\;
\mu={\rm sinh}^{-1}(\sqrt {\gamma
\omega} u),\;\;\;\;\nu={\rm sinh}^{-1}(\sqrt {\gamma
\omega} v),$$ $\vartheta_1(x;p)$ is a Jacobi Theta function \[15\] and $f(\gamma)$ is a known function of $\gamma$. For $\gamma\ll \pi^2,$ a simpler form for the density matrix is obtained $$\rho(u,v)\approx {1\over \pi}
{{\rm sin}\left[{\pi\over 2\gamma}\left(
{\rm sinh}^{-1}\left[(\omega\gamma)^{1/2}u\right]
-{\rm sinh}^{-1}\left[(\omega\gamma)^{1/2}v\right]\right)\right]\over
u-v}.$$ In the limit $\gamma\rightarrow 0$, in terms of the density at the origin $D_0=\frac{1}{2}(\frac{\omega}{\gamma})^{1/2},$ this reduces to the free Fermion density matrix $\rho(u-v)={{\rm sin}[D_0\pi(u-v)]/\pi(u-v)}.$ These oscillations are the characteristic signature of a normal Fermi system; its Fourier transform—the momentum distribution— is the familiar step function. On the other hand for increasing $\gamma$, these oscillations begin to die out, destroying the Fermi liquid behavior. The Fourier transform of $\rho(u,v)$ would involve two external momenta and this makes comparison with the Fermi liquid momentum distribution difficult. We shall instead consider the Fourier transform of $\rho(u,0),$ $n_k=\int_{-\infty}^{+\infty}du\; {\rm e}^{iku}\rho(u,0)$. With the change of variable $y={\rm sinh}^{-1}(2\gamma D_0 u),$ we have $$n_k=\frac{1}{\pi}\int_{0}^{\infty}dy\;{\rm coth}y
\left[{\rm sin}\frac{\pi}{2\gamma}\left(\frac{|k|}{\pi D_0}{\rm
sinh}y+y\right)
-{\rm sin}\frac{\pi}{2\gamma}\left(\frac{|k|}{\pi D_0}
{\rm sinh}y-y\right)\right].$$ For small enough $\gamma$ we can replace ${\rm sinh}y$ by $y$ within the $sine$ function, and ${\rm coth}y$ by $1/{\rm sinh}y$. The resulting distribution is given by $$n_{k}=n\left[\frac{\pi}{\gamma}\left(\frac{|k|}{D_0}-\pi\right)\right]-
n\left[\frac{\pi}{\gamma}\left(\frac{|k|}{D_0}+\pi\right)\right],$$ where $n[x]=1/[{\rm e}^x+1]$ is the Fermi function. As expected, this reduces to the step of the free fermions when $\gamma=0.$ Moreover, we find from the explicit expression (8) that $\gamma$ plays the role of an effective “temperature”.
In summary, we have considered a one-dimensional quantum many-body problem described by a Jastrow type wave function characterized by an exponent $\lambda$ and a parameter $\gamma$. We show that our model is a generalization of the $1/r^2$ pair-potential model considered by Calogero and Sutherland, which is obtained in the limit $\gamma=0.$ We obtain the exact one-particle density matrix for all $\gamma$ for the case $\lambda=1$ where the $1/r^2$ interaction vanishes. For $\gamma=0$, and the choice of Fermi statistics this becomes a free fermion problem, and we recover the step function for the momentum distribution. For $\gamma\neq 0$, an interaction term survives for the case $\lambda=1$ and the resulting momentum distribution is smeared out, destroying the Fermi Liquid. The explicit expression for the momentum distribution as a function of $\gamma$ for small $\gamma$ shows that the destruction of the Fermi liquid state occurs as increasing interaction induces an increase in the effective “temperature” in this regime.
One of us (K. A. M.) should like to thank the Science and Engineering Research Council, UK, for the award of a Visiting Fellowship, and the Mathematics Department of Imperial College for its kind hospitality. We should also like to thank A. C. Hewson for discussion, and J. R. Klauder for valuable comments on the manuscript.
Fig.1: $V(u)$ characterizing the model wave function considered, as given by Eq.(2), for various values of $\gamma$. The parameter $\omega$ has been set equal to unity.
[99]{} R. Jastrow, Phys. Rev. [**98**]{}, 1479 (1955). B. Sutherland, Phys. Rev. [**A4**]{}, 2019 (1971); J. Math. Phys. [**12**]{}, 246 (1971);Phys. Rev. [**A5**]{}, 1371 (1971); J. Math. Phys. [**12**]{}, 250 (1971). F. Calogero, J. Math. Phys. [**10**]{}, 2191 (1969); J. Math. Phys. [**10**]{}, 2197 (1969); J. Math. Phys. [**12**]{}, 419 (1970). Sutherland wrote down explicitly the Jastrow type wave function; this form was implicit in Calogero. F. D. M. Haldane, Phys. Rev. Lett. [**60**]{}, 635 (1988). B. S. Shastry, Phys. Rev. Lett. [**60**]{}, 639 (1988). M. C. Gutzwiller, Phys. Rev. Lett. [**10**]{}, 159 (1963). C. Gros, R. Joynt and T. M. Rice, Phys. Rev. [**B36**]{}, 381 (1987). P. W. Anderson, B. S. Shastry, and D. Hristopulos, Phys. Rev. [**B40**]{} 8939 (1989). C. S. Hellberg and E. J. Mele, Phys. Rev. Lett. [**67**]{} 2080 (1991). B. Sutherland, Phys. Rev. [**B45**]{}, 907 (1992). R. Valenti and C. Gros, Phys. Rev. Lett. [**68**]{}, 2402 (1992). T. Nagao and M. Wadati, J. Phys. Soc. Japan, [**62**]{}, 480 (1993). See e.g. M. L. Mehta, $Random\; Matrices$ (Academic Press, New York, 1991), 2nd ed. K. A. Muttalib, Y. Chen, M. E. H. Ismail and V. N. Nicopoulos, Phys. Rev. Lett. [**71**]{}, 471 (1993). E. T. Whittaker and G. N. Watson, $A\;Course\;of
\;Modern\;Analysis$ (Cambridge University Press, Cambridge, 1927), 4th ed.
|
---
abstract: 'The production site of gamma-rays in blazar jet is an unresolved problem. We present a method to locate gamma-ray emission region in the framework of one-zone emission model. From measurements of core-shift effect, the relation between the magnetic field strengths ($B''$) in the radio cores of jet and the distances ($R$) of these radio cores from central supermassive black hole (SMBH) can be inferred. Therefore once the magnetic field strength in gamma-ray emission region ($B''_{\rm diss}$) is obtained, one can use the relation of $B''$-$R$ to derive the distance ($R_{\rm diss}$) of gamma-ray emission region from SMBH. Here we evaluate the lower limit of $B''_{\rm diss}$ by using the criteria that the optical variability timescale $t_{\rm var}$ should be longer or equal to the synchrotron radiation cooling timescale of the electrons that emit optical photons. We test the method with the observations of PSK 1510-089 and BL Lacertae, and derive $R_{\rm diss}<0.15\delta_{\rm D}^{1/3}(1+A)^{2/3}\ $pc for PSK 1510-089 with $t_{\rm var}\sim$ a few hours, and $R_{\rm diss}<0.003\delta_{\rm D}^{1/3}(1+A)^{2/3}\ $pc for BL Lacertae with $t_{\rm var}\sim$ a few minutes. Here $\delta_{\rm D} $ is the Doppler factor and $A$ is the Compton dominance (i.e., the ratio of the Compton to the synchrotron peak luminosities).'
author:
- 'Dahai Yan, Qingwen Wu, Xuliang Fan, Jiancheng Wang, Li Zhang'
bibliography:
- 'ApJ.bib'
title: A Method for Locating High Energy Dissipation Region in Blazar
---
INTRODUCTION {#sec:intro}
============
Blazars are one class of radio-loud active galactic nuclei (AGNs), pointing their relativistic jets at us. According to the features of optical emission lines, blazars are usually divided into two classes: BL Lac objects (BL Lacs) with weak or even no observed optical emission lines and flat spectrum radio quasars (FSRQs) with strong optical emission lines [@urry]. Multi-wavelength radiations spanning from radio, optical to TeV gamma-ray energies have been observed from blazars. Blazar emission is generally dominated by non-thermal radiation from relativistic jet. The broadband spectral energy distribution (SED) of a blazar has two distinct humps. It is generally believed that the first hump is the synchrotron radiation of relativistic electrons in the jet, however the origin of the $\gamma$-ray hump is uncertain.
Leptonic and hadronic models have been proposed to produce the second bump [see, e.g., @bhk12 for review]. In leptonic models, $\gamma$ rays are produced through inverse Compton (IC) scattering of high energy electrons, including synchrotron-self Compton scattering [i.e., SSC; e.g., @Maraschi1992; @Tavecchio98; @Finke08; @Yan14] and external Compton (EC) scattering [e.g., @Dermer93; @Sikora1994; @bl; @Dermer09; @Paliya]. $\gamma$ rays in hadronic models can be attributed to the processes including proton- or pion-synchrotron radiation [@mann92; @Aharonian2000; @mucke2003] and $p\gamma$ interactions induced cascade [@bottcher09; @bottcher13; @murase14; @masti13; @Weidinger2015; @Yan15; @2015MNRAS.448..910C]. In general, both leptonic and hadronic models are able to reproduce the SEDs well, but they require quite different jet properties. For instance the hadronic models require extremely high jet powers for the most powerful blazars, FSRQs [@bottcher13; @Zdziarski].
In blazar jet physics, an open question is the location of gamma-ray emission region, which controls the radiative cooling processes in both leptonic and hadronic models. The location also means the place where the bulk energy of the jet is converted to an energy distribution of high energy particles. Because the gamma-ray emission region is usually compact, it cannot be directly resolved by current detectors. Many methods have been proposed to constrain the location of the gamma-ray emission region [e.g., @liu06; @Tavecchio2010; @Yan12; @Dotson; @Nalewajko14; @Jorstad01; @Jorstad10; @Agudo; @bottcher16]. One popular method is to model the SEDs of FSRQs, and the location of the gamma-ray emission region (i.e., the distance from central back hole to the gamma-ray emission region) is treated as a model parameter [e.g., @Yan15b].
Recently some methods independent of SED modeling are proposed to locate the gamma-ray emission region. @Dotson suggested that the energy dependence of the decay times in flare profiles could reflect the property of IC scattering. If the decay times depend on gamma-ray energies, it indicates that IC scattering happens in the Thomson region where the electron cooling time due to IC scattering depends on the energy of the electron. This situation will occur when the gamma-ray emission region locates in dust torus where the seed photons for IC scattering have the mean energy of $\sim0.1\ \rm eV$ [@Dotson; @Dotson15; @Yan16] . Moreover, the variability timescales of gamma-ray emissions also provide hints for the location of the gamma-ray emission region. For instance fast gamma-ray variability indicates that the emission region is very compact, which is usually thought to be close to the central black hole [e.g., @Tavecchio2010].
Currently there is no consensus on the location of the high energy dissipation region. The results given by the methods mentioned above are very inconsistent, from 0.01 pc to tens of pc [@Nalewajko14].
As suggested by [@Wu], we also use the relation of $B'$-$R$ derived in the measurements of radio core-shift effect [e.g., @Os; @Sokolovsky; @zam], to constrain the location of high energy dissipation in blazars. In @Wu, the magnetic field strength in gamma-ray emission region was derived in modeling SED. Here we use optical variability timescale to constrain the magnetic field strength in gamma-ray emission region, and therefore our method is fully independent of SED modeling.
Method and results
==================
Method
------
The variabilities of synchrotron radiations (e.g., variability timescale and time delay between emissions in different bands ) have been suggested to estimate comoving magnetic field [e.g., @Takahashi; @bottcher03].
Optical emission with fast variability from blazar is believed to be synchrotron radiation of relativistic electrons. If electron cooling is dominated by synchrotron cooling, the cooling timescale of electron in comoving frame is given by [e.g., @Tavecchio98] $$t'_{\rm cooling} = \frac{3}{4} \frac{m_ec^2}{\sigma_T c}(\gamma u_B)^{-1}
= \frac{6\pi m_ec}{\sigma_T\gamma B'^2_{\rm diss}}\ ,
\label{tcool}$$ where $u_B = B'^2_{\rm diss}/8\pi$ is the energy density of the magnetic field in comoving frame, $m_e$ is the mass of electron, $\sigma_T$ is the cross section of Thomson scattering, $\gamma$ is the electron energy. Meanwhile, the observational synchrotron frequency is written as $$\nu_{\rm syn} = \frac{4}{3} \nu_L \gamma^2 \frac{\delta_{\rm D}}{1+z}
\approx3.7\times10^6 \gamma^2 \frac{B'_{\rm diss}}{1\ \rm G} \frac{\delta_{\rm D}}{1+z}\ \rm Hz\ ,
\label{nu}$$ where $\nu_L = 2.8\times10^6 (B'_{\rm diss}/1\ \rm G)\ \rm s^{-1}$ is the Larmor frequency, and $z$ is redshift.
The observational variability timescale $t_{\rm var}$ can be taken as the upper limit for the cooling timescale in observer frame $t_{\rm cooling}=t'_{\rm cooling}(1+z)/\delta_{\rm D}$, i.e., $t_{\rm var} \geq t_{\rm cooling}$. Then we can get the lower limit for magnetic field strength from equation (\[tcool\]) and (\[nu\]), i.e., $$B'_{\rm diss} \geq 1.3\times 10^8 t_{\rm var}^{-2/3} \nu_{\rm syn}^{-1/3} \delta_{\rm D}^{-1/3} (1+z)^{1/3}\ \rm G\ ,
\label{estimatemag}$$ where $t_{\rm var}$ is in unit of second and $\nu_{\rm syn}$ in unit of Hz.
If electron cooling is dominated by EC cooling in the Thomson regime, the cooling timescale of the electron in equation (\[tcool\]) should be modified by a factor of $(1+k)^{-1}$ [@bottcher03], and then the lower limit of $ B'_{\rm diss}$ in equation (\[estimatemag\]) is modified by a factor of $(1+k)^{-2/3}$. Here $k$ is the ratio of the energy densities between an external photon field and the magnetic field in the comoving frame. $k$ can be replaced with Compton dominance $A$, i.e., the ratio of IC to synchrotron peak luminosity [@Finke13].
In analogy to the calculation of $A$ in @Nalewajko17[^1], $A$ can be calculated as $A=L_{\rm 1-100\ \rm GeV}/L_{\rm optical}$, where $L_{\rm 1-100\ \rm GeV}$ is the luminosity between 1 GeV to 100 GeV and $L_{\rm optical}$ is the optical luminosity.
Radio telescopes have the capability to resolve the structure of blazar jet on $\sim$pc scale. Very long baseline interferometry (VLBI) observations showed that core-shift effect (the frequency-dependent position of the VLBI cores) is common in AGNs. The core-shift effect is caused by synchrotron self absorption [e.g., @Blandford]. Under the condition of the equipartition between the jet particle and magnetic field energy densities, core-shift effect can be used to evaluate the magnetic field strength along the jet, and a relation between the magnetic field strength ($B'$, in units of Gauss) and the distance along the jet ($R$, in unit of pc) was found, i,e, **$B'\propto (R/1\ \rm pc)^{-1}\ \rm G$** [e.g., @Os; @zam].
Assuming that this relation still holds on in the sub-pc scale of the jet, we then can use it and the lower limit for $B'_{\rm diss}$ derived by using optical variability to constrain the distance of gamma-ray emission region from SMBH ($R_{\rm diss}$).
Results: testing the method with PKS 1510-089 and BL Lacertae
-------------------------------------------------------------
Here we use observations of two blazars to test the feasibility of our method.
PKS 1510-089 ($z$=0.361) is a TeV FSRQ. Using four bright gamma-ray flares detected by *Fermi*-LAT in 2009, [@Dotson15] located its high energy dissipation region in dust torus (DT). In May 2016, H.E.S.S. and MAGIC detected a very high energy (VHE) flare from PKS 1510-089 [@1510]. During the VHE flare, its optical emission also presented activity. The R band (the frequency is $4.5\times10^{14}\ $Hz) flux decreased from $1.4\times10^{-11}\ \rm erg\ cm^{-2}\ s^{-1}$ to $1.1\times10^{-11}\ \rm erg\ cm^{-2}\ s^{-1}$ within $\sim$ 2 hr [see Fig. 2 in @1510]. We adopt the variability timescale $t_{\rm var}\approx2\ $hr.
In hadronic models, the electron cooling in a FSRQ is dominated by synchrotron cooling [e.g., @bottcher13; @Diltz]. Using equation (\[estimatemag\]) we derive $B'_{\rm diss} \geq5\delta_{\rm D}^{-1/3}\ $G. With this magnetic field strength, we can constrain the distance $R_{\rm diss}$ using the relation of $B'\approx 0.73\cdot (R/1\ \rm pc)^{-1}\ $G provided by @zam, and derive $R_{\rm diss}<0.15\delta_{\rm D}^{1/3}\ $pc.
In leptonic models, the electron cooling in a FSRQ is dominated by EC cooling. Then we have $B'_{\rm diss} \geq5\delta_{\rm D}^{-1/3}(1+A)^{-2/3}\ $G, and $R_{\rm diss}<0.15\delta_{\rm D}^{1/3}(1+A)^{2/3}\ $pc.
For [*Fermi*]{}-LAT FSRQs, $A$ is in the range from 0.1 to 30 [@Finke13; @Nalewajko17]. @Dermer14 showed that for 3C 279 the value of $k$ varies from $\sim$3 to 20 from low states to high states. For PKS 1510-089, @Saito derived $k\sim20$ from the SED during a $\gamma$-ray flare in March 2009.
Taking $\delta_{\rm D}=30$ and $A\sim k=20$, we derive $R_{\rm diss}<0.5\ $pc for hadronic models, and $R_{\rm diss}<3.5\ $pc for leptonic models. Note that $A$ is sensitive to observing time for FSRQs, hence an $A$ obtained from simultaneous observation should be used in practice.
The sizes of broad line region (BLR) and dust torus can be estimated with the disk luminosity $L_{\rm disk}$ [@ghisellini09; @ghisellini14]: $$r_{\rm BLR}=10^{17}(L_{\rm disk}/10^{45}\rm \ erg\ s^{-1})^{1/2}\ \rm cm\ ,$$ $$r_{\rm DT}=10^{18}(L_{\rm disk}/10^{45}\rm \ erg\ s^{-1})^{1/2}\ \rm cm\ .$$ For PKS 1510-089, $L_{\rm disk}\approx5.9\times10^{45}\ \rm erg\ s^{-1}$ [@cas], we obtain $r_{\rm BLR}\approx0.1\ $pc and $r_{\rm DT}\approx0.8\ $pc.
Because BLR photons will attenuate gamma-ray photons above $\sim30/(1+z)\ $GeV[^2], the detection of VHE photons from PKS 1510-089 indicates that its gamma-ray emission region should be outside the BLR. Our results support the scenario that the gamma-rays of PKS 1510-089 are produced in dust torus. This is consistent with the result in @Dotson15.
This method is also used to constrain the location of gamma-ray emission region in BL Lacs. @Os derived a relation of $B'\approx 0.14\cdot (R/1\ \rm pc)^{-1}\ $G for BL Lacertae (2200+420; $z$=0.069). @Covino reported a very fast optical variability for this source. On 2012 September 1 the R-band flux decayed by a factor of about 3 in $5\ $min. This fast variability requires $B'_{\rm diss} \geq31.3\delta_{\rm D}^{-1/3}\ $G for hadronic models, and $B'_{\rm diss}\geq31.3\delta_{\rm D}^{-1/3}(1+A)^{-2/3}\ $G for leptonic models. Then we have $R_{\rm diss}<0.004\delta_{\rm D}^{1/3}\ $pc for hadronic models and $R_{\rm diss}<0.004\delta_{\rm D}^{1/3}(1+A)^{2/3}\ $pc for leptonic models.
For [*Fermi*]{}-LAT BL Lacs, $A$ is in the range from 0.1 to 3 [@Finke13; @Nalewajko17]. @Abdo showed that from a low state to a flare state, $k$ varies from $\sim0.1$ to 3 for BL Lacertae. Using $\delta_{\rm D}=30$ and $k=3$, we have $R_{\rm diss}<0.01\ $pc for hadronic models and $R_{\rm diss}<0.02\ $pc for leptonic models.
@Raiteri estimated that the accretion disk luminosity $L_{\rm disk}$ is $6\times10^{44}\rm \ erg\ s^{-1}$ for BL Lacertae. The energy density of the photon field attributed to accretion disk radiation at $R_{\rm diss}=0.02\ $pc is $\sim$0.4$\ \rm erg\ cm^{-3}$ which is much greater than the energy density of BLR photon field of $\sim$0.01$\ \rm erg\ cm^{-3}$ [e.g., @ghisellini09; @Hayashida]. However this situation prohibits the production of VHE photons because of $\gamma$-$\gamma$ absorption by accretion disk photons and BLR photons. Therefore the detection VHE photons [e.g., @Arlen] cannot be accompanied with a fast optical variability with $t_{\rm var}\sim 5\ $min.
Here we just aim to present the constraints on $R_{\rm diss}$ given by using various optical variability timescales, i.e., the feasibility of our method. From the above descriptions, one can find that our method is very effective, especially for the source having fast optical variability. In a specific study it is better to choose simultaneous optical flare respect to gamma-ray emission to obtain variability timescales and Compton dominance, and the definition of the variability timescale should be clarified.
Discussion
==========
Our method for locating the gamma-ray emission region relays on two assumptions:
\(1) optical and gamma-ray emissions are produced in the same region;
\(2) the relation of $B'$-$R$ obtained from radio core-shift measurements can be extrapolated into sub-pc scale of jet.
In general, the first assumption still works in the current blazar science, although a class of orphan gamma-ray flares seems challenge the one-zone emission model [e.g., @MacDonald]. For the second assumption, @Os extended the relation to the distance of $10^{-5}\ $pc at the SMBH (very close to the black hole jet-launching distance), and found that the extrapolated magnetic field strengths are in general consistent with that expected from theoretical models of magnetically powered jets [e.g., @Blandford]. So far, the above assumptions are reliable.
Besides the two assumptions, our constraint slightly relies on the value of Doppler factor, $\propto\delta_{\rm D}^{1/3}$; while it depends on Compton dominance $A$ in leptonic models, $\propto(1+A)^{2/3}$. In addition to the relation of $B'$-$R$, our method only requires simultaneous $\gamma$-ray and optical observations.
The stringency of our constraint mainly depends on the precision of the measurement for the relation of $B'$-$R$. @Pushkarev and @zam derived this relation for over 100 blazars by measuring the core-shift effect. Combining the measurement of this relation and optical variability timescale, one can independently constrain the location of gamma-ray emission region in blazar.
We use two TeV blazars, PSK 1510-089 and BL Lacertae, to test our method. Using the R-band variability with $t_{\rm var}\approx2\ $hr for PKS 1510-089, we derive $R_{\rm diss}<0.15\delta_{\rm D}^{1/3}(1+A)^{2/3}\ $pc for leptonic models and $R_{\rm diss}<0.15\delta_{\rm D}^{1/3}\ $pc for hadronic models. Using a typical value $\delta_{\rm D}=30$ and a large enough value $A=20$, we derive $R_{\rm diss}<0.5\ $pc for hadronic models and $R_{\rm diss}<3.5\ $pc for leptonic models. For BL Lacertae, we use a very short optical variability timescale of $t_{\rm var}\approx5\ $min reported in @Covino to estimate the lower limit for magnetic field strength, and derive $R_{\rm diss}<0.003\delta_{\rm D}^{1/3}\ $pc for hadronic models and $R_{\rm diss}<0.003\delta_{\rm D}^{1/3}(1+A)^{2/3}\ $pc for leptonic models. Using a typical value $\delta_{\rm D}=30$ and a large enough value $A=3$, we have $R_{\rm diss}<0.01\ $pc for hadronic models and $R_{\rm diss}<0.02\ $pc for leptonic models. One can see that with various optical variability timescales from minutes to a few hours, the high energy emission region can be located within pc or subpc scale from central black hole in the framework of one-zone emission model. The lower limit for the distance of the emission region from SMBH can be estimated by the absorption of GeV-TeV photons from low energy photons around jet [e.g., @liu06; @bai; @bottcher16].
By modeling blazar SED, one can determine emission mechanisms and physical properties of the relativistic jets [e.g., @ghisellini10; @ghisellini14; @Kang; @zhang15]. In the previous studies, $B'_{\rm diss}$, $R_{\rm diss}$, and other model parameters are fitted together. There are degeneracies between model parameters [see @Yan13; @Yan15b for correlations between model parameters given by Markov Chain Monte Carlo fitting technique]. Our method provide independent constraints for $B'_{\rm diss}$ and $R_{\rm diss}$, and break degeneracies between model parameters. This will lead to better understandings of emission mechanisms and physical properties of the relativistic jets.
It should be noted that the relation of $B'$-$R$ is derived under the assumption of the equipartition between electron and magnetic field energy densities [e.g., @Os]. On the aspect of SED modeling, it is found that the SEDs of FSRQs and low-synchrotron-peaked BL Lacs (LBLs) can be successfully fitted at the condition of ([*near*]{}-)equipartition between electron and magnetic field energy densities in leptonic models [e.g., @Abdo; @Yan16b; @Hu], while the leptonic modeling results for the SEDs of high-synchrotron-peaked BL Lacs (HBLs) are far out of equipartition [e.g., @Dermer15; @Zhu]. Therefore, for consistency, our method is applicable for FSRQs and LBLs in the framework of leptonic models. The jet equipartition condition in hadronic models is rather complex. The modeling results are inconsistent [e.g., @bottcher13; @Diltz]. It is unknown whether the ([*near*]{})-equipartition condition could be achieved in hadronic models.
Conclusion
==========
We presented an effective method for constraining the location of gamma-ray emission region in blazar jet in the framework of one-zone emission model. Our method uses the relation of $B'$-$R$ derived in the VLBI core-shit effect in blazar jet. The lower limit for magnetic field strength in gamma-ray emission region is estimated by utilizing the fact that the optical variability timescale should be longer or equal to the synchrotron radiation cooling timescale of the electrons that produce optical emission. Then the upper limit for the location of gamma-ray emission region is derived with the relation of $B'$-$R$. Our method is applicable for LBLs and FSRQs.
$\\$ We thank the anonymous referee for helpful comments which significantly improved the paper. We acknowledge the financial supports from the National Natural Science Foundation of China (NSFC-11573026, NSFC-11573060, NSFC-1161161010, NSFC-11673060, NSFC-U1738124). D. H. Yan is also supported by the CAS “Light of West China" Program.
[^1]: They defined Compton dominance as the ratio of Fermi-LAT luminosity above one GeV to the [*WISE*]{} luminosity at $3.6\ \mu$m.
[^2]: Assuming that the BLR radiation field is dominated by $\rm Ly\alpha$ line photons with the mean energy of $\approx10\ $eV.
|
---
abstract: 'We explore the possibility of retrieving Auger spectra with FEL radiation. Using a laser pulse of 260 eV photon energy, we study the interplay of photo-ionization and Auger processes following the initial formation of a 2p inner-shell hole in Ar. Accounting for the fine structure of the ion states we demonstrate how to retrieve the Auger spectrum of $\mathrm{Ar^{+}\rightarrow Ar^{2+}}$. Moreover, considering two electrons in coincidence we also demonstrate how to retrieve the Auger spectrum of $\mathrm{Ar^{2+}\rightarrow Ar^{3+}}$.'
author:
- 'A. O. G. Wallis'
- 'L. Lodi'
- 'A. Emmanouilidou'
bibliography:
- 'AOGW\_all.bib'
title: 'Auger spectra following inner-shell ionization of Argon by a Free-Electron Laser'
---
Introduction
============
The response of atoms to intense extreme ultraviolet (XUV) and X-ray Free-Electron-Lasers (FEL) is a fundamental theory problem. In addition, understanding FEL-driven processes is of interest for accurate modeling of laboratory and astrophysical plasmas. The fast progress in generating intense FEL pulses of femtosecond duration renders timely the study of FEL driven processes in atoms. Such processes include the formation of inner-shell vacancies by photo-absorption and the subsequent Auger decays. Exploring the interplay of photo-ionization and Auger processes is a key to understanding the rich electron dynamics underlying the formation of highly charged ions [@Sorokin:2007; @Young:Ne:2010; @Doumy:nonlinear:2011] and hollow atoms [@Young:Ne:2010; @Fukuzawa:2013; @Frasinski:2013].
Auger spectra have attracted a lot of interest over the years with early studies involving the formation of an inner-shell hole following the impact of a particle, such as an electron [@mehlhorn1985atomic; @Mcguire:nobelgas:1975; @Mcguire:argon:1975; @Mehlhorn:1968; @Werme:1973]. From the early 80s, synchrotron radiation has largely replaced particle impact as a triggering mechanism of Auger processes [@vonBusch:1994; @Alkemper:1997; @vonBusch:satellite:1999; @Lablanquie:multi:2011]. Such studies include the detailed Auger spectrum following the decay of Ar$^+(2p^{-1})$ [@Pulkkinen:1996; @Lablanquie:2007]. The reason for using synchrotron radiation is that it is monochromatic and allows for well defined initial excitations in the soft and hard X-ray regime. A recent study with synchrotron radiation [@Huttula:2013] involves the measurement of Auger spectra following the decay of the $\mathrm{Ar}^{2+}(2p^{-1}v^{-1})$ ionic states; $v^{-1}$ is a hole in a valence orbital and $\mathrm{Ar}^{2+}(2p^{-1}v^{-1})$ is formed by single-photon double ionization.
In this work, we explore the feasibility of obtaining detailed Auger spectra using FEL radiation. FEL radiation allows for well-defined initial excitations. It also allows for the creation of multiple inner-shell holes resulting in multiple Auger decays; generally the Auger spectra thus generated have larger yields than those generated from synchrotron radiation. The increasing availability of FEL sources provides an additional motivation for the current study. We explore the interplay of photo-ionization and Auger processes in $\mathrm{Ar}$ interacting with a 260 eV FEL pulse, a photon energy sufficient to ionize a single inner-shell $\mathrm{2p}$ electron in Ar. We compute the ion yields due to Auger and photo-ionization processes and study the ion yields dependence on the FEL pulse parameters. To do so we solve a set of rate equations [@Rohringer:2007; @Makris:2009]. Initially, in the rate equations we only account for the electronic configuration of the ion states. This simplification allows us to gain insight into the processes involved and explore the optimal parameters for observing Auger spectra. We next proceed to fully account for the fine structure of the ion states in the rate equations. We subsequently obtain the detailed Auger spectrum of $\mathrm{Ar^{+}\rightarrow Ar^{2+}}$. Moreover, we demonstrate how the detailed Auger spectrum of $\mathrm{Ar^{2+}\rightarrow Ar^{3+}}$ can be observed in an FEL two-electron coincidence experiment.
Auger and Ion yields excluding fine structure
==============================================
We model the response of Ar to a 260 eV FEL pulse by formulating and solving a set of rate equations for the time dependent populations of the ion states [@Rohringer:2007; @Makris:2009]. Our first goal is to gain insight into how the ion and Auger yields depend on the duration and intensity of the laser pulse. To do so, in this section, we simplify the theoretical treatment by accounting only for the electronic configuration, i.e, $(1s^a, 2s^b, 2p^c, 3s^d, 3p^e)$ of the ion states and not the fine structure of these states. By fine structure we refer to all possible $^{2S+1}L_J$ states for a given electronic configuration, accounting for spin-orbit coupling. To compute the Auger transition rates between different electron configurations we use the formalism introduced by Bhalla [*et al.*]{} [@Bhalla:1973] and refer to these transition rates as Auger group rates in accord with [@Bhalla:1973].
 Ionization pathways between different electronic configurations of Ar, up to Ar$^{4+}$, accessible with sequential single-photon ($\mathrm{\hbar\omega=260 eV})$ absorptions and Auger decays. The label $\mathrm{(a,b,c,d,e)}$ stands for the electronic configuration $\mathrm{(1s^a, 2s^b, 2p^c, 3s^d, 3p^e)}$. The red and green lines indicate photo-ionization and Auger transitions, respectively.](Fig_Arlevels260){width="0.99\linewidth"}
Rate equations
--------------
In the rate equations we account for single-photon ionization and Auger transitions. For the ion states considered the X-ray fluorescence widths are typically three orders of magnitude smaller than the Auger decay widths [@Chen:1974]; we can thus safely neglect the former. In [Fig. \[fig:Ar\_levels\]]{}, accounting for states up to Ar$^{4+}$, we illustrate the photo-ionization and Auger transitions between states with different electron configurations that are allowed for a laser pulse of 260 eV photon energy. This photon energy is sufficient for creating a single inner-shell $\mathrm{2p}$ hole and multiple valence holes in Ar. In the rate equations we include all possible ion states accessible by a 260 eV laser-pulse; the highest ion state is $\mathrm{Ar^{9+}(1s^2,2s^2,2p^5,3s^0,3p^0)}$. The rate equations describing the population $\mathcal{I}_i^{(q)}$ of an ion state $\mathrm{i}$ with charge $\mathrm{q}$ take the form
$$\begin{aligned}
\label{eqn:rate1}
\frac{d}{dt}&\mathcal{I}_j^{(q)}(t) =
\sum_i \left(\sigma_{i\rightarrow j} J(t)+ \Gamma_{i\rightarrow j} \right) \mathcal{I}_i^{(q-1)}(t)
\\ \nonumber
& - \sum_k \left( \sigma_{j\rightarrow k} J(t) + \Gamma_{j\rightarrow k} \right) \mathcal{I}_j^{(q)}(t)
\\ \nonumber
\frac{d}{dt} &\mathcal{A}^{(q)}_{i\rightarrow j} = \Gamma_{i\rightarrow j} \mathcal{I}_i^{(q-1)}(t),\\ \nonumber
\frac{d}{dt}&\mathcal{P}^{(q)}_{i\rightarrow j}= \sigma_{i\rightarrow j} J(t) \mathcal{I}_i^{(q-1)}(t), \end{aligned}$$
where $\mathrm{\sigma_{i\rightarrow j}}$ and $\mathrm{\Gamma_{i\rightarrow j}}$ are the single-photon absorption cross section and Auger decay rate from initial state $\mathrm{i}$ to final state $\mathrm{j}$, respectively. $\mathrm{J(t)}$ is the photon flux. Atomic units are used in this work. The temporal form of the FEL flux is modelled with a Gaussian function [@Rohringer:2007] which is given by $$J(t) = 1.554\times 10^{-16} \frac{I_0\textrm{[W cm}^{-2}]}{\hbar\omega\textrm{[eV]}} \exp\left\{ -4\ln2 \left(\frac{t}{\tau_X}\right)^2 \right\}$$ with $\mathrm{\tau_X}$ the full width at half maximum and $\mathrm{I_0}$ the peak intensity. The first term in [Eq. (\[eqn:rate1\])]{} accounts for the formation of the state $\mathrm{j}$ with charge $\mathrm{q}$ through the single-photon ionization and Auger decay of the state $\mathrm{i}$ with charge $\mathrm{q-1}$. The second term in [Eq. (\[eqn:rate1\])]{} accounts for the depletion of state $\mathrm{j}$ by single-photon ionization and Auger decay to the state $\mathrm{k}$ with charge $\mathrm{q+1}$. In [Eq. (\[eqn:rate1\])]{}, we also solve for the Auger yield $\mathcal{A}^{(q)}_{i\rightarrow j}$ from an initial state $\mathrm{i}$ with charge $\mathrm{q-1}$ to a final state $\mathrm{j}$ with charge $\mathrm{q}$. In addition, we solve for the photo-ionization yield $\mathcal{P}^{(q)}_{i\rightarrow j}$ from an initial state $\mathrm{i}$ with charge $\mathrm{q-1}$ to a final state $\mathrm{j}$ with charge $\mathrm{q}$. These yields provide the probability for observing an electron with energy corresponding to the transition $\mathrm{i\rightarrow j}$. The total Auger and photo-ionization yields for the transition from any state with charge $\mathrm{q-1}$ to any state with charge $\mathrm{q}$ are given by $$\begin{aligned}
\mathcal{A}^{(q)} = \sum_{i,j} \mathcal{A}^{(q)}_{i\rightarrow j}, & &
\mathcal{P}^{(q)} = \sum_{i,j} \mathcal{P}^{(q)}_{i\rightarrow j} .\end{aligned}$$ To find the total ion yield of a state with charge $\mathrm{q}$, i.e., the ion yield for Ar$^{q+}$ we sum over the populations of all ion states with charge $\mathrm{q}$ $$\mathcal{I}^{(q)} = \sum_i \mathcal{I}_i^{(q)}.$$ All yields are computed long after the end of the pulse.
As we show later in the paper, it is also of interest to compute the Auger and photo-ionization yields along a pathway $\mathrm{i \rightarrow j \rightarrow k}$. These yields provide the probability for observing in a two-electron coincidence experiment two electrons with energies corresponding to the transitions $\mathrm{i\rightarrow j}$ and $\mathrm{j\rightarrow k}$. If there is only one state $\mathrm{i}$ leading to state $\mathrm{j}$ then the probability for observing the electron emitted in the transition $\mathrm{i\rightarrow j}$ and the electron emitted in the transition $\mathrm{j\rightarrow k}$ is simply the Auger $\mathcal{A}^{(q)}_{j\rightarrow k}$ or the photo-ionization $\mathcal{P}^{(q)}_{j\rightarrow k}$ yield. However, it can be the case that we have multiple states leading to state $\mathrm{j}$, for example, $\mathrm{i \rightarrow j \rightarrow k}$ and $\mathrm{i' \rightarrow j \rightarrow k}$. Then to compute the probability $\mathcal{P}^{(q)}_{j(i) \rightarrow k}$ or $\mathcal{A}^{(q)}_{j(i) \rightarrow k}$ for observing the electron emitted in the transition $\mathrm{i\rightarrow j}$ and the electron emitted in the transition $\mathrm{j\rightarrow k}$ we need to solve separately for the contribution of state $\mathrm{i}$ to the population of state $\mathrm{j}$: $$\begin{aligned}
\label{eqn:coincidence}
\frac{d}{dt} \mathcal{I}^{(q-1)}_{j(i)}(t) =
& (\sigma_{i\rightarrow j}J(t)+\Gamma_{i\rightarrow j})\mathcal{I}^{(q-2)}_{i}(t) \\ \nonumber
&- \sum_{k'}(\sigma_{j\rightarrow k'}J(t)+\Gamma_{j\rightarrow k'})\mathcal{I}^{(q-1)}_{j(i)}(t) \\ \nonumber
\frac{d}{dt} \mathcal{P}^{(q)}_{j(i) \rightarrow k} = & \sigma_{j\rightarrow k}J(t)\mathcal{I}^{(q-1)}_{j(i)}(t) \\ \nonumber
\frac{d}{dt} \mathcal{A}^{(q)}_{j(i) \rightarrow k} = & \Gamma_{j\rightarrow k}\mathcal{I}^{(q-1)}_{j(i)}(t). \end{aligned}$$
Auger group rates
-----------------
To compute the Auger group rates $\Gamma_{i\rightarrow j}$ we use the formulation of Bhalla et al. [@Bhalla:1973]. For each electron configuration included in the rate equations, we obtain the energy and bound atomic orbital with a Hartree-Fock (HF) calculation. These calculations are performed with the *ab initio* quantum chemistry package <span style="font-variant:small-caps;">molpro</span> [@MOLPRO_brief_2009.1] using the split-valence 6-311G basis set. To compute the continuum orbital that describes the outgoing Auger electron we use the Hartree-Fock-Slater (HFS) one-electron potential that is obtained using an updated version of the Herman Skillman atomic structure code [@HermanSkillman:1963; @hermsk:program]. This one electron potential is expressed in terms of an effective nuclear charge $\mathrm{Z_\text{HFS}(r)}$. The resulting radial HFS equation is of the form $$\label{eqn:HFSwavefnc}
\left[ -\frac{d^2}{dr^2} +\frac{l(l+1)}{r^2} -\frac{Z_\text{HFS}(r)}{r} \right]P_{nl}(r) = E P_{nl}(r),$$ where the orbital wavefunction is given by $\mathrm{\psi_{nlm}(\mathbf{r})= r^{-1}P_{nl}(r)Y_{lm}(\hat{r})}$. We solve equation [Eq. (\[eqn:HFSwavefnc\])]{} for the continuum orbital ($\mathrm{E>0}$) using the modified Numerov method [@Numerov:1933; @Melkanoff:1966]. We match the solution to the appropriate asymptotic boundary conditions for energy normalized continuum wave functions [@Child:1974]. In Table I we list our results for the Auger group rates $\mathrm{Ar^{+}(2p^{-1})\rightarrow Ar^{2+}(3s^{-1}3p^{-1})}$, $\mathrm{Ar^{+}(2p^{-1})\rightarrow Ar^{2+}(3s^{-2})}$ and $\mathrm{Ar^{+}(2p^{-1})\rightarrow}$ $\mathrm{Ar^{2+}(3p^{-2})}$ and compare them with two other calculations that employ the HFS method [@McGuire:1971] and the HF method [@Dyall:1982] both for the bound and the continuum orbitals. As expected, our results lie between the results of these two calculations. For reference, we also list in Table I the results from a Configurational Interaction (CI) calculation [@Dyall:1982]. In Table II we list our results for all the Auger group rates involved in the rate equations for Ar for a 260 eV FEL pulse.
--- -------- --- --- --- --------------------- -------- -------- -------- -------
method
a b c d e $3s3s$ $3s3p$ $3p3p$ total
2 2 5 2 6 HFS [@McGuire:1971] 0.77 12.85 47.90 61.52
HF [@Dyall:1982] 0.28 15.74 56.97 72.99
CI [@Dyall:1982] 0.47 9.54 54.74 64.75
this work 0.45 15.60 51.67 67.72
--- -------- --- --- --- --------------------- -------- -------- -------- -------
: Auger group rates for a transition from an initial state $\mathrm{(1s^a,2s^b,2p^c,3s^c,3p^e)}$ to a final state where the electron filling in the $\mathrm{2p}$ hole in the initial state and the electron escaping to the continuum occupy $\mathrm{nl}$ and $\mathrm{n'l'}$ orbitals. We also list the Auger rates obtained in [@McGuire:1971] using the Hartree-Fock-Slater (HFS) method, in [@Dyall:1982] using a Hartree-Fock (HF) method, and in [@Dyall:1982] using a CI calculation. The rates are given in 10$^{-4}$ a.u.
--- --- --- --- --- -------- -------- -------- --------
a b c d e $3s3s$ $3s3p$ $3p3p$ total
2 2 5 2 6 0.450 15.598 51.665 67.713
2 2 5 2 5 0.502 9.615 25.457 35.575
2 2 5 1 6 - 9.244 58.693 67.937
2 2 5 2 4 0.568 9.429 20.324 30.321
2 2 5 1 5 - 5.780 29.273 35.053
2 2 5 0 6 - - 68.708 68.708
2 2 5 2 3 0.638 7.973 11.680 20.291
2 2 5 1 4 - 5.631 23.952 29.583
2 2 5 0 5 - - 33.761 33.761
2 2 5 2 2 0.710 5.845 4.349 10.905
2 2 5 1 3 - 4.650 13.337 17.986
2 2 5 0 4 - - 23.946 23.946
2 2 5 2 1 0.778 2.843 - 3.621
2 2 5 1 2 - 3.374 4.909 8.283
2 2 5 0 3 - - 14.309 14.309
2 2 5 2 0 0.863 - - 0.863
2 2 5 1 1 - 1.612 - 1.612
2 2 5 0 2 - - 5.168 5.168
--- --- --- --- --- -------- -------- -------- --------
: As in Table I for results obtained in this work for all Auger group rates included in the rate equations. \[tab:HFgroupRates\]
Results for Auger and Ion yields
--------------------------------
![(Color online) Total ion $\mathcal{I}^{(q)}$ (solid lines) and Auger $\mathcal{A}^{(q)}$ (dashed lines) yields as a function of intensity for pulse duration of 5 fs (top) and 50 fs (bottom). \[fig:Ar\_GroupAugerIonYields\] ](Fig_GroupAugerIon_5fs "fig:"){width="0.9\linewidth"} ![(Color online) Total ion $\mathcal{I}^{(q)}$ (solid lines) and Auger $\mathcal{A}^{(q)}$ (dashed lines) yields as a function of intensity for pulse duration of 5 fs (top) and 50 fs (bottom). \[fig:Ar\_GroupAugerIonYields\] ](Fig_GroupAugerIon_50fs "fig:"){width="0.9\linewidth"}
For the photo-ionization cross sections we use the Los Alamos National Laboratory atomic physics codes [@LANL:APC] that are based on the HF routines of R. D. Cowan [@Cowan:1981]. Assuming that the initial state is the neutral Ar, we solved numerically [@NumRec:2007] the set of first order differential rate equations in [Eq. (\[eqn:rate1\])]{}. In [Fig. \[fig:Ar\_GroupAugerIonYields\]]{} we show our results for the total ion $\mathcal{I}^{(q)}$ and Auger $\mathcal{A}^{(q)}$ yields as a function of the pulse intensity for pulse durations of 5 fs and 50 fs. From [Fig. \[fig:Ar\_GroupAugerIonYields\]]{} we observe that $\mathcal{A}^{(q)}$ can be very similar to $\mathcal{I}^{(q)}$ for $q \ge 2$ depending on the pulse intensity and duration. Indeed, the formation of $\mathrm{Ar^{q+}}$ occurs from a sequence of transitions where the final step involves either the single-photon ionization or the Auger decay of $\mathrm{Ar^{(q-1)+}}$. For high pulse intensities, independent of the pulse duration, both final steps are likely and thus $\mathcal{A}^{(q)}$ is different than $\mathcal{I}^{(q)}$. For small pulse intensities, if the pulse is short then the formation of $\mathrm{Ar^{q+}}$ through the Auger decay of $\mathrm{Ar^{(q-1)+}}$ is favored; if the pulse is long multi-photon absorption is highly likely making possible formation of $\mathrm{Ar^{q+}}$ also through single-photon ionization of $\mathrm{Ar^{(q-1)+}}$. Thus, generally, for small pulse intensities, if the pulse is short $\mathcal{A}^{(q)} \approx \mathcal{I}^{(q)}$ while if the pulse is long $\mathcal{A}^{(q)} \ne \mathcal{I}^{(q)}$.
Truncation of the number of states included in the rate equations
-----------------------------------------------------------------
![(Color online) Total ion yields $\mathcal{I}^{(q)}$ for $q =$ 0,1,2,3,4 when ion states up to Ar$^{9+}$ (dashed lines) and ion states up to Ar$^{4+}$ (solid lines) are included as a function of pulse intensity for pulse durations 5 fs (top) and 50 fs (bottom). \[fig:truncation\]](Fig_Truncation_5fs "fig:"){width="0.9\linewidth"} ![(Color online) Total ion yields $\mathcal{I}^{(q)}$ for $q =$ 0,1,2,3,4 when ion states up to Ar$^{9+}$ (dashed lines) and ion states up to Ar$^{4+}$ (solid lines) are included as a function of pulse intensity for pulse durations 5 fs (top) and 50 fs (bottom). \[fig:truncation\]](Fig_Truncation_50fs "fig:"){width="0.9\linewidth"}
[Fig. \[fig:Ar\_GroupAugerIonYields\]]{} shows that appropriate tuning of the laser parameters can result in large Auger yields even for high ion states. Regarding Auger spectra this is an advantage of FEL radiation compared to synchrotron radiation. However, discerning the Auger spectra produced by the FEL pulse is a challenging task since many photo-ionization and Auger electrons escape to the continuum. In the next section we focus on the Auger electron spectra resulting from ion states up to Ar$^{3+}$. To accurately describe these spectra we need to account for the fine structure of the ion states included in the rate equations. However, such an inclusion results in a very large increase of the number of ion states that need to be accounted for in the rate equations. For instance, when considering states up to Ar$^{4+}$ the number of ions states in the rate equations increases from 21(no fine structure) to 186 (with fine structure). We thus truncate the number of ion states we consider. In [Fig. \[fig:truncation\]]{} we compare $\mathcal{I}^{(q)}$, for $\mathrm{q =}$ 1,2,3,4, when we include ion states up to Ar$^{9+}$ and up to Ar$^{4+}$. We find that the truncation affects only $\mathcal{I}^{(4)}$ while $\mathcal{I}^{(1)}$, $\mathcal{I}^{(2)}$ and $\mathcal{I}^{(3)}$ are unaffected. Since the focus of the current work is the Auger electron spectra up to Ar$^{3+}$, in what follows we truncate to include only ion states up to Ar$^{4+}$. Moreover, comparing [Fig. \[fig:truncation\]]{} with [Fig. \[fig:Ar\_GroupAugerIonYields\]]{}, we find that a pulse duration of 5 fs is short enough for $\mathcal{A}^{(q)} \approx \mathcal{I}^{(q)}$ to be true for intensities up to roughly $10^{16}$ W cm$^{-2}$. This guarantees that less photo-ionization electrons are ejected to the continuum making it easier to discern the Auger electrons. We also find that for pulse intensities around $10^{15}$-$10^{16}$ W cm$^{-2}$ both $\mathcal{A}^{(2)}$ and $\mathcal{A}^{(3)}$ yields have significant values. Thus, a laser pulse with duration of 5 fs and intensity of $5\times10^{15}$ W cm$^{-2}$ is optimal for the experimental observation of the Auger electron spectra up to Ar$^{3+}$.
Auger spectra
==============
Computation of fine structure ion states
----------------------------------------
We next describe the method we use to compute the fine structure states of each electron configuration that is included in the truncated rate equations. To obtain the fine structure ion states we use the <span style="font-variant:small-caps;">grasp2k</span> package [@grasp2k:2013] and the <span style="font-variant:small-caps;">relci</span> extension [@RELCI:2002] provided in the <span style="font-variant:small-caps;">ratip</span> package [@RATIP:2012]. These packages are used to perform relativistic calculations within the Multi-Configuration Dirac-Hartree-Fock (MCDHF) formalism [@Grant:book:2006]. The photo-ionization cross sections and Auger decay rates between fine structure states are then calculated using the <span style="font-variant:small-caps;">photo</span> and <span style="font-variant:small-caps;">auger</span> components of the <span style="font-variant:small-caps;">ratip</span> package. Since <span style="font-variant:small-caps;">grasp2k</span> utilizes the Dirac equation the calculations are performed in the $j$-$j$ coupling scheme. We briefly outline the steps we follow to obtain the fine structure states for a given electron configuration of Ar; where appropriate we illustrate using Ar$^+(1s^2,2s^2,2p^5,3s^2,3p^6)$.
1\) We identify the fine structure states for the electron configuration at hand; in our example these states are $\mathrm{^{2}P_{1/2}}$ and $\mathrm{^{2}P_{3/2}}$. We identify the configurational state functions (CSFs) that can be constructed out of the possible $\mathrm{nlj}$ orbitals; in our example the possible CSFs are $$1. (1s_{1/2}^2,2s_{1/2}^2,2p_{1/2}^{1},2p_{3/2}^4,3s_{1/2}^2,3p_{1/2}^2,3p_{3/2}^4); J^P=\frac{1}{2}^{-}$$ $$2. (1s_{1/2}^2,2s_{1/2}^2,2p_{1/2}^{2},2p_{3/2}^3,3s_{1/2}^2,3p_{1/2}^2,3p_{3/2}^4); J^P=\frac{3}{2}^{-}$$ Each fine structure state is a linear combination of the CSFs that have the same total angular momentum $J$ and parity $P$; in our example $\mathrm{^2P_{1/2}}$ is expressed in terms of the first CSF and $\mathrm{^{2}P_{3/2}}$ in terms of the second CSF. A Self-Consistent-Field (SCF) DHF calculation is now performed for all the CSFs. This calculation optimizes the $\mathrm{nlj}$ orbitals and the coefficients in the expansion of each fine structure state in terms of CSFs.
2\) To account for electron correlation, as a first step, we include the additional orbitals $\mathrm{3d_{3/2}}$ and $\mathrm{3d_{5/2}}$. A new set of CSFs is generated from the single and double excitations of the step-1 CSFs, while keeping the occupation of the $\mathrm{1s}$, $\mathrm{2s}$ and $\mathrm{2p}$ orbitals frozen. A new MCDHF calculation is then performed with the new set of CSFs keeping the step-1 $\mathrm{nlj}$ orbitals frozen and only optimizing the newly added ones.
3\) As a second step in accounting for electron correlation, we include all orbitals up to $\mathrm{4d_{3/2}}$, $\mathrm{4d_{5/2}}$. Again, as for step-2, a new set of CSFs is generated from the single and double excitations of the step-1 CSFs, while keeping the occupation of the $1s$, $2s$ and $2p$ orbitals frozen. Another MCDHF calculation is performed optimizing only the newly added, compared to step-2, orbitals. Introducing correlation orbitals layer by layer as described in steps 1-3 is the recommended procedure in the <span style="font-variant:small-caps;">grasp2k</span> manual in order to achieve convergence of the SCF calculations.
4\) Finally, using the orbitals generated in steps 1-3 we perform a CI calculation that optimizes the coefficients that express each fine structure state in terms of all the CSFs generated in steps 1-3.
[c c | c c | cc c]{} &&\
& & $E$ & $I$ & $E$ & $\Gamma$ & $I$\
\
$3p3p$ &$^3P_2$&207.39&76&207.57&2.37&64\
&$^3P_1$&207.25&176&207.44&5.11&138\
&$^3P_0$&207.20&60&207.38&2.13&58\
&$^1D_2$&205.65&404&205.64&11.78&318\
&$^1S_0$&203.26&100&203.35&3.70&100\
$3s3p$ &$^3P_2$&-&-&193.25&0.02&1\
&$^3P_1$&193.13&24&193.12&1.12&30\
&$^3P_0$&193.07&18&193.06&0.58&16\
&$^1P_1$&189.50&39&188.66&1.85&50\
$3s3s$ &$^1S_0$&176.43&6&175.36&0.62&17\
\
$3p3p$ &$^3P_2$&205.24&261&205.43&7.58&240\
&$^3P_1$&205.10&73&205.30&2.77&88\
&$^3P_0$&205.08&26&205.24&0.73&23\
&$^1D_2$&203.50&390&203.50&11.01&348\
&$^1S_0$&201.11&100&201.22&3.16&100\
$3s3p$ &$^3P_2$&191.09&77&191.11&1.52&48\
&$^3P_1$&190.95&11&190.98&0.35&11\
&$^3P_0$&-&-&190.92&0&0\
&$^1P_1$&187.39&71&186.52&1.79&57\
$3s3s$ &$^1S_0$&174.27&13&173.22&0.61&19\
In Table \[tab:Ar2p\_CI\] we list the energies and Auger rates we obtain using the method described above for the fine structure states of Ar$^{+}(2p^{-1})$. To directly compare with the experimental results in [@Pulkkinen:1996] we define the intensity for an Auger decay from an initial state $\mathrm{i}$ to a final state $\mathrm{j}$ as $$I_{i\rightarrow j} = \frac{ \Gamma_{i \rightarrow j} }{ \sum_j \Gamma_{i \rightarrow j} },$$ and is scaled such that the intensity for the transition $\mathrm{Ar^+(2p^{-1})\rightarrow Ar^{2+}(3p^{-2}; ^1S_0)}$ is equal to 100 in accord with [@Pulkkinen:1996]. It can be seen that our calculated results are in good agreement with the experimental results of Pulkkinen [*et al*]{}. [@Pulkkinen:1996].
Results for Auger and Ion yields including fine structure
---------------------------------------------------------
![(Color online) Total ion $\mathcal{I}^{(q)}$ (solid lines) and Auger $\mathcal{A}^{(q)}$ yields (dashed lines) for $q=2,3,4$ as a function of intensity for a pulse duration of 5 fs. These yields are calculated with fine structure included in the rate equations. \[fig:FS\_AugerIon\]](Fig_FS_auger_ion_5fs){width="0.9\linewidth"}
In [Fig. \[fig:FS\_AugerIon\]]{} we show the total ion $\mathcal{I}^{(q)}$ and Auger $\mathcal{A}^{(q)}$ yields accounting for fine structure for a pulse duration of 5 fs. We find that these yields are very similar to the yields obtained in the previous section where fine structure was neglected. Thus our conclusions in the previous section regarding optimal laser parameters for observing the Auger electron spectra up to $\mathrm{Ar^{3+}}$ still hold. Also in [Fig. \[fig:FS\_Auger\]]{} we plot the Auger yields $\mathcal{A}_{i\rightarrow j}^{(2)}$ and $\mathcal{A}_{\i \rightarrow j}^{(3)}$ for all possible $\mathrm{i}$, $\mathrm{j}$ fine structure states.
![\[fig:FS\_Auger\] (Color online) The Auger yields $\mathcal{A}^{(2)}_{i\rightarrow j}$ (blue, solid lines) and $\mathcal{A}^{(3)}_{i\rightarrow j}$ (red, dashed lines) as a function of intensity for a pulse duration of 5 fs. These yields are calculated with fine structure included in the rate equations.](Fig_FS_AllAugerYields_5fs.pdf){width="0.9\linewidth"}
Auger spectra including fine structure
---------------------------------------
### One-electron Auger spectra
![\[fig:FS\_fullspectrum\] (Color online) The electron spectra for a 5 fs, 260 eV pulse with an intensity of $5\times10^{15}$ W cm$^{-2}$ for energies between 150 and 250 eV (a) and energies between 155 and 177 eV (b). For clarity the plot range of (b) is highlighted in yellow in (a). The peaks are convoluted by 0.37 eV FWHM Gaussian functions. The peaks of the photo-ionization electrons emitted during transitions from the initial states Ar (black) and Ar$^+$ (green) are in the energy ranges denoted by A, B, and C (see Table IV). The peaks of the Auger electrons emitted during the transitions $\mathrm{Ar^+(2p^{-1})\rightarrow Ar^{2+}}$ (red) are in the energy ranges denoted by D, E, and F, and the ones emitted during the transitions $\mathrm{Ar^{2+}(2p^{-1}v^{-1})\rightarrow Ar^{3+}}$ (blue) are in the energy ranges denoted by E, F, and G (see Table IV). ](Fig_FS_spectra_5e15_5fs){width="0.98\linewidth"}
![\[fig:FS\_fullspectrum\] (Color online) The electron spectra for a 5 fs, 260 eV pulse with an intensity of $5\times10^{15}$ W cm$^{-2}$ for energies between 150 and 250 eV (a) and energies between 155 and 177 eV (b). For clarity the plot range of (b) is highlighted in yellow in (a). The peaks are convoluted by 0.37 eV FWHM Gaussian functions. The peaks of the photo-ionization electrons emitted during transitions from the initial states Ar (black) and Ar$^+$ (green) are in the energy ranges denoted by A, B, and C (see Table IV). The peaks of the Auger electrons emitted during the transitions $\mathrm{Ar^+(2p^{-1})\rightarrow Ar^{2+}}$ (red) are in the energy ranges denoted by D, E, and F, and the ones emitted during the transitions $\mathrm{Ar^{2+}(2p^{-1}v^{-1})\rightarrow Ar^{3+}}$ (blue) are in the energy ranges denoted by E, F, and G (see Table IV). ](Fig_FS_spectra_5e15_5fs_zoom){width="0.98\linewidth"}
In [Fig. \[fig:FS\_fullspectrum\]]{} we compute the electron spectra for a 260 eV FEL pulse with $5\times10^{15}$ W cm$^{-2}$ intensity and 5 fs duration. Both the Auger $\mathcal{A}^{(q)}_{i\rightarrow j}$ and photo-ionization $\mathcal{P}_{i\rightarrow j}^{(q)}$ yields for charges up to $q=4$ contribute to the peaks in these electron spectra. To account for the energy uncertainty of a 5 fs pulse, which is 0.37 eV, we have convoluted the peaks in [Fig. \[fig:FS\_fullspectrum\]]{} with Gaussian functions of 0.37 eV FWHM. We find that the energies of the photo-ionization electrons ejected in the transition $\mathrm{Ar}^{+}\rightarrow \mathrm{Ar}^{2+}$ (peak height $\mathcal{P}_{i\rightarrow j}^{(2)}$) are well separated from the energies of the Auger electrons ejected in the transitions $\mathrm{Ar}^{+}\rightarrow \mathrm{Ar}^{2+}$ (peak height $\mathcal{A}^{(2)}_{i\rightarrow j}$) and $\mathrm{Ar}^{2+}\rightarrow \mathrm{Ar}^{3+}$ (peak height $\mathcal{A}^{(3)}_{i\rightarrow j}$); the photo-ionization peaks are above 210 eV while the Auger peaks are below 210 eV. In [Fig. \[fig:FS\_fullspectrum\]]{} and Table \[tab:peaks\], the energy range of the photo-ionization electrons is denoted by A, B, C; the energy range of the Auger electrons emitted during transitions from the initial states Ar$^{+}$ and Ar$^{2+}$ are denoted by D, E, F, and E, F, and G, respectively. In [Fig. \[fig:FS\_fullspectrum\]]{} we see that the Auger yields $\mathcal{A}^{(2)}_{i\rightarrow j}$ (D, E, F) are much larger than all other Auger yields in the same energy range. They can thus be discerned and measured for the laser parameters under consideration. The Auger yields $\mathcal{A}^{(3)}_{i\rightarrow j}$ (E, F, G) are smaller but still visible, while the Auger yields $\mathcal{A}^{(4)}_{i\rightarrow j}$ are too small to be discerned in [Fig. \[fig:FS\_fullspectrum\]]{}. However, except for the energy region below 170 eV, the Auger electron spectra resulting from the transitions $\mathrm{Ar}^{2+}\rightarrow \mathrm{Ar}^{3+}$ overlap with the Auger electron spectra resulting from the transitions $\mathrm{Ar}^{+}\rightarrow \mathrm{Ar}^{2+}$. Thus, in order to discern and be able to experimentally observe the latter Auger electron spectra we need to consider spectra of two electrons in coincidence. We do so in what follows.
Region Transitions
--------- -----------------------------------------------------------------------------
$A$ Ar$+\hbar\omega\rightarrow$Ar$^+(3p^{-1}) +e^-_{\rm P}$
$B$ Ar$+\hbar\omega\rightarrow$Ar$^+(3s^{-1}) +e^-_{\rm P}$
$B$ Ar$^+(u^{-1}) +\hbar\omega\rightarrow$Ar$^{2+}(u^{-1}3p^{-1}) +e^-_{\rm P}$
$C$ Ar$^+(u^{-1}) +\hbar\omega\rightarrow$Ar$^{2+}(u^{-1}3s^{-1}) +e^-_{\rm P}$
$D$ Ar$^+(2p^{-1}) \rightarrow$Ar$^{2+}(3p^{-2}) + e^-_{\rm A}$
$E$ Ar$^+(2p^{-1}) \rightarrow$Ar$^{2+}(3s^{-1}3p^{-1}) + e^-_{\rm A}$
$F$ Ar$^+(2p^{-1}) \rightarrow$Ar$^{2+}(3s^{-2}) + e^-_A$
$E,F,G$ Ar$^{2+}(2p^{-1}v^{-1}) \rightarrow$Ar$^{3+}(v^{-3})+ e^-_A$
: Labeling of energy regions in the electron spectrum shown in [Fig. \[fig:FS\_fullspectrum\]]{}. $e^-_{\rm p}$ and $e^-_{\rm A}$ stand for photo-ionization and Auger electrons, respectively. $u^{-1}$ represents a hole in any of the $2p$, $3s$ or $3p$ orbitals. \[tab:peaks\]
### Two-electron coincidence Auger spectra
We now consider the electron spectra resulting from the transitions: $$\mathrm{Ar} + \hbar\omega\rightarrow \mathrm{Ar}^{+}(2p^{-1})+e^-_{P} \rightarrow \mathrm{Ar}^{3+} + e^-_{P} + e^-_{B} + e^-_{C}
\label{transition}$$ The photo-ionization electron $\mathrm{e_{P}^{-}}$ has an energy of 12.3 eV for $\mathrm{Ar}^{+}(2p^{-1}_{3/2})$ and 10.2 eV for $\mathrm{Ar}^{+}(2p^{-1}_{1/2})$. This energy is very different from the energies of electrons $\mathrm{e_{B}^{-}}$ and $\mathrm{e_{C}^{-}}$. It thus suffices to plot in coincidence the energies of electrons $\mathrm{e_{B}^{-}}$ and $\mathrm{e_{C}^{-}}$. We note that many coincidence experiments have been performed with synchrotron radiation [@Alkemper:1997; @Lablanquie:2007; @Lablanquie:multi:2011; @Huttula:2013]. While some coincidence experiments have been performed with FEL radiation [@Kurka:2009; @Rudenko:2010] the low repetition rate poses a challenge. Advances in FEL sources should overcome such challenges in the near future.
![\[fig:FS\_coincidence1\] (Color online) Two-electron coincidence spectra for Ar$^+(2p_{3/2}^{-1})\rightarrow$Ar$^{3+}(v^{-3})$ generated by a 5 fs, $5\times10^{15}$ W cm$^{-2}$ FEL pulse. We show the spectrum below the $E_B=E_C$ line for the PPP (green squares), PAP (blue triangles) and PPA (red circles) transition sequences, see text for details. The Ar$^{3+}(3p^{-3})$ final fine structure states are labeled as 1:$^4S$, 2:$^2D$, 3:$^2P$, the Ar$^{3+}(3s^{-1}3p^{-2})$ states as 4:$^4P$, 5:$^2D$, 6:$^2S$, 7:$^2P$, and the Ar$^{3+}(3s^{-2}3p^{-1})$ state as 8:$^2P$. The energy range of the PPA transition sequences $\mathrm{Ar^{+}(2p^{-1}) \rightarrow Ar^{2+}(2p^{-1}3s^{-1}) \rightarrow Ar^{3+}}$ and $\mathrm{Ar^{+}(2p^{-1}) \rightarrow Ar^{2+}(2p^{-1}3p^{-1}) \rightarrow Ar^{3+}}$ are highlighted by yellow. ](Fig_3e_color_2P3_5e15_5fs){width="1.\linewidth"}
In [Fig. \[fig:FS\_coincidence1\]]{} we plot in coincidence the energies of electrons $\mathrm{e^-_{B}}$ and $\mathrm{e^-_{C}}$. Specifically, [Fig. \[fig:FS\_coincidence1\]]{} corresponds to the $\mathrm{ Ar^{+}(2p^{-1}_{3/2})}$ fine structure state in [Eq. (\[transition\])]{}. We only show the spectrum that lies below the line $E_B=E_{C}$ (black solid line), with $E_{B}$ the energy of electron $\mathrm{e_{B}^{-}}$ and $E_C$ the energy of electron $\mathrm{e_{C}^{-}}$. Since the two electrons are indistinguishable, the remaining spectrum can be obtained by a reflection with respect to the line $E_B=E_{C}$ of the spectrum shown in [Fig. \[fig:FS\_coincidence1\]]{}. From [Eq. (\[transition\])]{} it follows that each line with $E_{B}+E_{C}=constant$, grey lines in [Fig. \[fig:FS\_coincidence1\]]{}, scans the spectra of electrons emitted from transitions in [Eq. (\[transition\])]{} through any possible fine structure state of Ar$^{2+}$ to the same fine structure state of Ar$^{3+}$. The spectra of the electrons emitted from the transitions in [Eq. (\[transition\])]{} can be labelled according to the sequence of photo-ionization (P) and Auger processes (A) involved while transitioning from Ar to Ar$^{3+}$: PPA (red in [Fig. \[fig:FS\_coincidence1\]]{}), PPP (green) and PAP (blue). Our goal is to retrieve the Auger electron spectra corresponding to the transitions $\mathrm{ Ar^{2+} \rightarrow Ar^{3+}}$. These latter spectra are the ones labelled as PPA in [Fig. \[fig:FS\_coincidence1\]]{}; we highlight the energy range of the $\mathrm{e_{B}^{-}}$ and $\mathrm{e_{C}^{-}}$ electrons emitted in the PPA transition sequences $\mathrm{Ar^{+}(2p^{-1}) \rightarrow Ar^{2+}(2p^{-1}3s^{-1}) \rightarrow Ar^{3+}}$ and $\mathrm{Ar^{+}(2p^{-1}) \rightarrow Ar^{2+}(2p^{-1}3p^{-1}) \rightarrow Ar^{3+}}$. Thus to be able to retrieve the Auger electron spectra associated with the transitions $\mathrm{ Ar^{2+} \rightarrow Ar^{3+}}$ we must be able to discern the PPA from the PPP and the PAP transition sequences. We see that in the highlighted area in [Fig. \[fig:FS\_coincidence1\]]{} there is some small overlap of the PPA with the PPP and PAP sequences. However, we find that the height of the peaks of the PPA transition sequences are much larger than the height of the peaks of the PPP and PAP transition sequences. Specifically, the total Auger yield $\mathcal{A}^{(3)}$ associated with the PPA transition sequences is roughly 5 times larger than the photo-ionization yield $\mathcal{P}^{(3)}_{PAP}$ corresponding to the PAP transition sequences and 10 times larger than the photo-ionization yield $\mathcal{P}^{(3)}_{PPP}$ corresponding to the PPP transition sequences, with $\mathcal{P}^{(3)}_{PAP}+\mathcal{P}^{(3)}_{PPP}=\mathcal{P}^{(3)}$. To show that this is indeed the case we show in [Fig. \[fig:FS\_coincidence2\]]{} the contour plot of the two-electron coincidence spectra associated with the highlighted area in [Fig. \[fig:FS\_coincidence1\]]{} corresponding to the transitions $\mathrm{Ar^{+}(2p^{-1}) \rightarrow Ar^{2+}(2p^{-1}3s^{-1}) \rightarrow Ar^{3+}}$. Note that the height of the peaks in [Fig. \[fig:FS\_coincidence2\]]{} is given by $\mathcal{A}^{(3)}_{j\rightarrow k}$ or $\mathcal{A}^{(3)}_{j(i)\rightarrow k}$ (see discussion in section IIA) for the PPA transition sequences while the height is $\mathcal{P}^{(3)}_{j\rightarrow k}$ or $\mathcal{P}^{(3)}_{j(i)\rightarrow k}$ for the PPP and PAP transition sequences. Each coincidence peak has been convoluted by a 0.37 eV FWHM Gaussian function. We find that all except one of the observable peaks in [Fig. \[fig:FS\_coincidence2\]]{} are due to PPA transition sequences; the small height peak at $(E_C=211.7,E_B=190)$ is due to a PAP sequence. We have thus demonstrated that we can retrieve from the two-electron coincidence spectra the Auger electron spectra associated with the transitions $\mathrm{ Ar^{2+} \rightarrow Ar^{3+}}$. We note that a similar discussion and conclusions hold for the Auger spectra corresponding to the $\mathrm{ Ar^{+}(2p^{-1}_{1/2})}$ fine structure state in [Eq. (\[transition\])]{}.
![\[fig:FS\_coincidence2\] (Color online) Two-electron coincidence spectra for $\mathrm{Ar^{+}(2p^{-1}) \rightarrow Ar^{2+}(2p^{-1}3s^{-1}) \rightarrow Ar^{3+}}$ for the $\mathrm{Ar^{2+}(2p^{-1}3s^{-1})}$ fine structure states $\mathrm{^1P_1}$, $\mathrm{^3P_1}$ and $\mathrm{^3P_2}$. The peaks of the spectra corresponding to the Ar$^{2+}(2p^{-1}3s^{-1};^1P_1)$ fine structure state, to the left of the vertical dashed-white line, are much smaller than the rest of the spectra and we have thus multiplied them by a factor of 10 so that they are visible. The coincidence peaks have been convoluted by 0.37 eV FWHM Gaussian functions.](2p3s_matlab.pdf){width="1.\linewidth"}
Finally we note that our calculations neglect satellite structure. That is, we do not account for Auger transitions where one electron fills in the 2p hole, another one escapes to the continuum while a third one is promoted to an excited state. The main (larger) satellite Auger yields we are neglecting are most likely due to the Ar$^+(2p^{-1})\rightarrow$Ar$^{2+}(3s^{-1}3p^{-1})$ transition[@Pulkkinen:1996]. However, these satellite yields are smaller than the main Auger yields for this transition. In addition, these satellite Auger yields would only contribute to the part of the spectrum corresponding to PAP transition sequences in the energy region $E_B=$170 -180 eV and $E_C=$210-220 eV. But as we discussed above the contribution to the electron spectra from PAP transition sequences is smaller than the contribution from the PPA transition sequences. Thus our approximation is justified.
Conclusions
===========
We have explored the interplay of photo-ionization and Auger transitions in Ar when interacting with a 260 eV FEL pulse. Solving the rate equations we have explored the dependence of the ion and Auger yields on the laser parameters accounting, at first, only for the electron configuration of the ion states. We have found that an FEL pulse of roughly 5 fs duration and $5\times 10^{15}$ Wcm$^{-2}$ intensity is optimal for retrieving Auger electron spectra up to Ar$^{3+}$. Secondly, we have account for the fine structure of the ionic states and have truncated the rate equations to include states only up to $\mathrm{Ar^{4+}}$. We have shown how the Auger electron spectra of $\mathrm{Ar^{+}\rightarrow Ar^{2+}}$ can be retrieved. We have also shown that the Auger electron spectra of $\mathrm{Ar^{2+}\rightarrow Ar^{3+}}$ can also be retrieved when two electrons are considered in coincidence. We have thus demonstrated that interaction with FEL radiation is a possible route for retrieving Auger electron spectra. We believe that our work will stimulate further theoretical and experimental studies along these lines.
Acknowledgments
===============
The authors are grateful to Prof. P. Lambropoulos for initial motivation and valuable discussions. A.E. acknowledges support from EPSRC under Grant No. H0031771 and J0171831 and use of the Legion computational resources at UCL.
|
---
abstract: 'A dynamically affine map is a finite quotient of an affine morphism of an algebraic group. We determine the rationality or transcendence of the Artin-Mazur zeta function of a dynamically affine self-map of ${\ensuremath{{\mathbb{P}}}}^1(k)$ for $k$ an algebraically closed field of positive characteristic.'
address: |
Andrew Bridy\
Department of Mathematics\
University of Wisconsin-Madison\
Madison, WI 53706, USA
author:
- Andrew Bridy
bibliography:
- 'new\_bib.bib'
nocite: '[@*]'
title: 'The Artin-Mazur Zeta Function of a Dynamically Affine Rational Map in Positive Characteristic'
---
The Artin-Mazur Zeta Function
=============================
Let $X$ be a set and let $f:X\to X$ define a dynamical system. Let $\operatorname{Per}_n(f)=\{x\in X:f^n(x)=x\}$, where $f^n$ denotes the composition of $f$ with itself $n$ times. The Artin-Mazur zeta function of this dynamical system is the formal power series given by $$\zeta(f,X;t)=\exp\left( \sum_{n=1}^\infty \#\operatorname{Per}_n(f)\frac{t^n}{n} \right).$$ Assume that $\#\operatorname{Per}_n(f)<\infty$ for all $n$, as otherwise $\zeta$ is not defined. The power series $\zeta(f,X;t)$ has rational coefficients, and it is not hard to show that $\zeta(f,X;t)\in{\ensuremath{{\mathbb{Z}}}}[[t]]$ by means of the product formula $$\zeta(f,X;t)=\prod_{\text{cycles }C}\left(1-t^{|C|}\right)^{-1}.$$
This zeta function was introduced by Artin and Mazur in [@ArtinMazur], where it is studied for $X$ a manifold and $f$ a diffeomorphism. In this setting, only the *isolated* periodic points are counted. This will not be an important distinction for our purposes.
This paper continues the study of the following question, introduced in [@BridyActa] for polynomials, but just as easily phrased for rational functions.
For which $f\in{\overline{{\ensuremath{{\mathbb{F}}}}}}_p(x)$ is $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(\overline{{\ensuremath{{\mathbb{F}}}}}_p);t)$ rational?
The purpose of this paper is to answer this question for rational maps that are dynamically affine. These are maps that, loosely speaking, come from endomorphisms of algebraic groups; a precise definition will be given in section \[sec: affine\]. There are five families of dynamically affine maps in one dimension: power maps, Chebyshev polynomials, Lattès maps, additive polynomials, and subadditive polynomials. (To be precise, this classification only holds up to conjugacy by $\operatorname{Aut}({\ensuremath{{\mathbb{P}}}}^1)$, but $\zeta$ is a conjugacy invariant.) We determine the rationality of the zeta function for each of these families, and we show that when it fails to be rational, it is transcendental.
Let $k$ be an arbitrary algebraically closed field of characteristic $p$. Our main results are the following.
\[thm: main\] Let $f\in k(x)$ be a separable power map, Chebyshev polynomial, or Lattès map. Then $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$ is transcendental over ${\ensuremath{{\mathbb{Q}}}}(t)$.
\[thm: main2\] Let $f\in k[x]$ be a separable additive or subadditive polynomial. If $f'(0)$ is algebraic over ${\ensuremath{{\mathbb{F}}}}_p$, then $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$ is transcendental over ${\ensuremath{{\mathbb{Q}}}}(t)$. If $f'(0)$ is transcendental over ${\ensuremath{{\mathbb{F}}}}_p$, then $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$ is rational.
These theorems can be seen as the broadest possible generalization of the work in [@BridyActa], as the maps considered there are very specific cases of one-dimensional dynamically affine maps. In order to handle the new cases, we need to study the arithmetic of the endomorphism rings of one-dimensional algebraic groups, which can be somewhat complicated in the case of an elliptic curve.
Inseparable maps are excluded from the above theorems because their zeta functions are trivially rational. Let $f\in k(x)$ be inseparable and of degree $d$. The derivative of $f$ is identically zero, so $f^n(x)-x$ has distinct roots for every $n$ and $\#\operatorname{Per}_n(f)=d^n+1$. Therefore $$\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)=\exp\left( \sum_{n=1}^\infty\frac{(d^n+1)t^n}{n}\right) = \frac{1}{(1-t)(1-dt)}.$$ For a general $f\in k(x)$, it is not always the case that $\#\operatorname{Per}_n(f)=d^n+1$, but it is certainly true that $\#\operatorname{Per}_n(f)\leq d^n+1$. Therefore if we consider $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$ as a function of a complex variable, it converges to a holomorphic function in a positive radius around the origin.
In higher dimensions, the formula $\deg(f^n)=(\deg f)^n$ is not necessarily true. This complicates the above calculation of rationality. See, for example, [@Bellon1999], [@Hasselblatt2007], and [@SilvermanEntropy] for a discussion of this phenomenon.
In characteristic zero, the situation is very different. Hinkkanen shows that every $f\in{\ensuremath{{\mathbb{C}}}}(x)$ has a rational zeta function [@Hinkkanen]. The proof relies on the fact that there are only finitely many $x\in{\ensuremath{{\mathbb{P}}}}^1({\ensuremath{{\mathbb{C}}}})$ such that $(f^n)'(x)$ is a root of unity. This argument fails catastrophically in positive characteristic because every element of $\overline{{\ensuremath{{\mathbb{F}}}}}_p$ is a root of unity. Nevertheless, it is peculiar that the conclusion of Hinkkanen’s theorem holds in our setting almost exclusively when $f$ is inseparable, which is a phenomenon that cannot occur in characteristic 0.
The rest of the paper will prove Theorems \[thm: main\] and \[thm: main2\]. In Section \[sec: affine\] we define dynamically affine maps, and in Sections \[sec: G\_m\], \[sec: G\_a\], and \[sec: Lattes\] we classify all dynamically affine maps of ${\ensuremath{{\mathbb{P}}}}^1$ and establish some facts about their periodic points. A crucial and interesting feature of these maps is that we can count their periodic points by studying the arithmetic of certain endomorphism rings. Section \[sec: lifting the exponent\] provides some algebraic lemmas that are useful in this direction. Our proof also employs the theory of sequences generated by finite automata. Section \[sec: Automatic Sequences\] sketches the necessary background in this area. Sections \[sec: Proof 1.2\] and \[sec: Proof 1.3\] finish the proof of Theorems \[thm: main\] and \[thm: main2\].
The results of this paper suggest the following conjecture.
\[conj: separable\] If $f\in \overline{{\ensuremath{{\mathbb{F}}}}}_p(x)$ is separable, $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(\overline{{\ensuremath{{\mathbb{F}}}}}_p);t)$ is transcendental over ${\ensuremath{{\mathbb{Q}}}}(t)$.
If $f$ is not dynamically affine, the size of $\operatorname{Per}_n(f)$ can vary wildly as $n$ increases, making it difficult to determine the algebraic structure of the zeta function. Even the low degree map $f(x)=x^2+1$ behaves very irregularly in its periodic point counts (for $p\notin\{2,3\})$, so the nature of $\zeta$ is unclear. However, note that by Theorem \[thm: main2\] the above conjecture is false if we replace $\overline{{\ensuremath{{\mathbb{F}}}}}_p$ by an algebraically closed field $k$ that is transcendental over ${\ensuremath{{\mathbb{F}}}}_p$.
Overview of Dynamically Affine Maps {#sec: affine}
===================================
The following definitions are taken from [@SilvermanADS Ch 6.8].
Let $G$ be a commutative algebraic group. An *affine morphism* of $G$ is a map $\psi:G\to G$ that can be written as a composition of a finite endomorphism of degree at least 2 and a translation.
Let $V$ be a variety. A morphism $f:V\to V$ is *dynamically affine* if there exist a connected commutative algebraic group $G$, an affine morphism $\psi:G\to G$, a finite subgroup $\Gamma\subseteq\operatorname{Aut}(G)$ and a morphism $\pi:G\to G/\Gamma$, and a morphism that identifies $G/\Gamma$ with a Zariski dense open subset of $V$ such that the following diagram commutes: $$\label{eqn: dynamically affine}
\xymatrix{
G \ar[r]^\psi\ar[d]^\pi & G \ar[d]^\pi \\
G/\Gamma \ar[r]\ar[d] & G/\Gamma \ar[d] \\
V \ar[r]^f & V
}$$
It is well known that the only dynamically affine maps of ${\ensuremath{{\mathbb{P}}}}^1({\ensuremath{{\mathbb{C}}}})$ are power maps, Chebyshev polynomials, and Lattès maps (up to conjugacy by fractional linear transformations) [@SilvermanADS p. 378]. These arise when $G$ is either the multiplicative group ${\ensuremath{{\mathbb{G}}}}_m$ or an elliptic curve.
In characteristic $p$, there are two additional families of dynamically affine maps, both of which arise from the additive group ${\ensuremath{{\mathbb{G}}}}_a$. These are additive polynomials, which are maps such as $f(x)=x^p-x$ that distribute over addition, and subadditive polynomials such as $f(x)=x(x-1)^{p-1}$, which arise as the maps induced by additive polynomials on the quotient of ${\ensuremath{{\mathbb{G}}}}_a$ by a group of roots of unity.
We elaborate on these families in the sections that follow. The only one-dimensional algebraic groups are ${\ensuremath{{\mathbb{G}}}}_m$, ${\ensuremath{{\mathbb{G}}}}_a$, and elliptic curves. By considering all of the possibilities for the group $\Gamma$, we show that the maps listed above are all of the dynamically affine maps of ${\ensuremath{{\mathbb{P}}}}^1$. First, however, we establish a lemma that counts $\operatorname{Per}_n(f)$ in terms of the kernels of endomorphisms of $G$.
\[lem: periodic count\] Let $f:V\to V$ be dynamically affine. Assume that the affine morphism $\psi:G\to G$ is surjective. Write $\psi$ as $\psi(g)=\sigma(g)+h$, where $\sigma\in\operatorname{End}(G)$, $h\in G$ and the group law of $G$ is written additively. Then $$\#\operatorname{Per}_n(f)=\#(\operatorname{Per}_n(f) \setminus (G/\Gamma)) + \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\#\ker(\sigma^n-\gamma).$$
Recall that $G/\Gamma$ is identified with a Zariski open subset of $V$. The equation above claims that the $n$-periodic points that lie in this set are counted by the formula $\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\#\ker(\sigma^n-\gamma)$.
Suppose that $z\in\operatorname{Per}_n(f)\cap G/\Gamma$. By diagram \[eqn: dynamically affine\], there exists $g\in \pi^{-1}(z)$ and $\gamma\in\Gamma$ such that $\psi^n(g)=\gamma(g)$, and every such choice of $g$ and $\gamma$ gives some $z\in\operatorname{Per}_n(f)\cap G/\Gamma$. Therefore $$\label{eqn: Fix Count}
\operatorname{Per}_n(f)\cap G/\Gamma =\pi(\{g\in G:\psi^n(g)=\gamma(g)\text{ for some }\gamma\in\Gamma\}).$$
By slight abuse of notation, let $\ker(\psi^n-\gamma)=(\psi^n-\gamma)^{-1}(0)$. Define $$S=\bigcup_{\gamma\in\Gamma}\ker(\psi^n-\gamma),$$ so that $\operatorname{Per}_n(f)\cap G/\Gamma=\pi(S)$. We claim that $\Gamma$ acts on $S$.
Let $g\in S$, so that $\psi^n(g)=\delta(g)$ for some $\delta\in\Gamma$. Let $\gamma\in\Gamma$. Observe that $$\pi(\psi^n(\gamma(g))) = f^n(\pi(\gamma(g)))=f^n(\pi(g))=\pi(\psi^n(g)).$$ As $\psi^n(\gamma(g))$ and $\psi^n(g)$ have the same image under $\pi$, there exists some $\delta'\in\Gamma$ such that $\psi^n(\gamma(g))=\delta'(g)$, and therefore $\gamma(g)\in S$. (Somewhat surprisingly, $\delta'$ depends only on $\gamma$ and not on $g$, but we do not need this for our purposes. See [@SilvermanADS Prop 6.77].)
If $z\in\operatorname{Per}_n(f)\cap G/\Gamma$, the set $\pi^{-1}(z)$ is a $\Gamma$-orbit in $S$, so there is a bijection between $\operatorname{Per}_n(f)\cap G/\Gamma$ and the set of orbits $S/\Gamma$. Let $\Gamma_g$ be the subgroup of $\Gamma$ that fixes $g\in S$, and let $\delta$ be such that $\psi^n(g)=\delta(g)$. Then $$\#\{\gamma\in\Gamma: g\in\ker(\psi^n-\gamma)\} = \#\{\gamma\in\Gamma: g =\gamma^{-1}\delta(g)\}= |\Gamma_g|$$ By the orbit-stabilizer theorem [@Isaacs Cor 4.10], $$\begin{aligned}
\#S/\Gamma & = \sum_{g\in S}\frac{|\Gamma_g|}{|\Gamma|} = \frac{1}{|\Gamma|}\sum_{g\in S}\#\{\gamma\in\Gamma: g\in\ker(\psi^n-\gamma)\}\\
& = \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\#\{g\in S: g\in\ker(\psi^n-\gamma)\}=\frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\#\ker(\psi^n-\gamma).\end{aligned}$$ Recall that $\psi(g)=\sigma(g)+h$. So $\psi^n(g)=\sigma^n(g)+h_n$ for some $h_n\in G$, and $$\ker(\psi^n-\gamma)=(\psi^n-\gamma)^{-1}(0)=(\sigma^n-\gamma)^{-1}(-h_n).$$ We assumed $\psi$ is surjective, so $\ker(\psi^n-\gamma)$ is nonempty. Therefore $$\#(\sigma^n-\gamma)^{-1}(-h_n)=\#\ker(\sigma^n-\gamma),$$ completing the proof.
Maps from ${\ensuremath{{\mathbb{G}}}}_m$: Power Maps and Chebyshev Polynomials {#sec: G_m}
===============================================================================
Let ${\ensuremath{{\mathbb{G}}}}_m$ be the multiplicative group. The endomorphism ring $\operatorname{End}({\ensuremath{{\mathbb{G}}}}_m)$ is isomorphic to ${\ensuremath{{\mathbb{Z}}}}$, where the integer $d$ corresponds to the power map $x\mapsto x^d$. So every affine morphism $\psi:{\ensuremath{{\mathbb{G}}}}_m\to{\ensuremath{{\mathbb{G}}}}_m$ has the form $\psi(x)=ax^d$. The automorphism group is $\operatorname{Aut}({\ensuremath{{\mathbb{G}}}}_m)\cong{\ensuremath{{\mathbb{Z}}}}^\times=\{\pm 1\}$. There are only two subgroups $\Gamma\subseteq\operatorname{Aut}(E)$: either $\Gamma$ is trivial or $\Gamma=\{x,x^{-1}\}$.
The underlying scheme of ${\ensuremath{{\mathbb{G}}}}_m$ is $\operatorname{Spec}k[x,x^{-1}]\cong{\ensuremath{{\mathbb{A}}}}^1\setminus\{0\}$, which is Zariski open in ${\ensuremath{{\mathbb{P}}}}^1$. If $\Gamma$ is trivial, then a power map arises from the following commutative diagram: $$\xymatrix{
{\ensuremath{{\mathbb{G}}}}_m \ar[r]^{x\mapsto ax^d}\ar[d]^\wr & {\ensuremath{{\mathbb{G}}}}_m \ar[d]^\wr \\
{\ensuremath{{\mathbb{A}}}}^1\setminus\{0\}\ar [d]\ar[r] & {\ensuremath{{\mathbb{A}}}}^1\setminus\{0\}\ar[d]\\
{\ensuremath{{\mathbb{P}}}}^1 \ar[r]^f & {\ensuremath{{\mathbb{P}}}}^1
}$$ There are many choices for the inclusion map ${\ensuremath{{\mathbb{A}}}}^1\setminus\{0\}\hookrightarrow{\ensuremath{{\mathbb{P}}}}^1$. The most obvious is the map that extends to the identity map on ${\ensuremath{{\mathbb{P}}}}^1$, in which case $f$ is the “affine power map" $f(x)=ax^d$. Other choices give such maps conjugated by fractional linear transformations. Over an algebraically closed field, we can always conjugate by a linear polynomial $x\mapsto cx$ in order to make $f$ monic, so we may assume $f(x)=x^d$. (Recall that $\zeta$ is a conjugacy invariant.)
There are only two points in ${\ensuremath{{\mathbb{P}}}}^1$ that lie outside ${\ensuremath{{\mathbb{A}}}}^1\setminus\{0\}$, namely, 0 and $\infty$. If $d>0$, then $f$ fixes these two points, and if $d<0$, then $f$ swaps them. The group law of ${\ensuremath{{\mathbb{G}}}}_m$ is written multiplicatively, so if $d>0$, Lemma \[lem: periodic count\] gives $$\#\operatorname{Per}_n(f) = 2 + \#\ker(x^{d^n-1}),$$ and if $d<0$, then $$\#\operatorname{Per}_n(f) = \left\{
\begin{array}{ll}
2+\#\ker(x^{d^n-1}) & : d\text{ even}\\
\#\ker(x^{d^n-1}) & : d\text{ odd}
\end{array}
\right.$$
If we let $\Gamma=\{x,x^{-1}\}$, then ${\ensuremath{{\mathbb{G}}}}_m/\Gamma\cong{\ensuremath{{\mathbb{A}}}}^1$, and the quotient can be realized by the map $\pi(x)=x+x^{-1}$. There exists a polynomial $f$ such that the following diagram commutes [@SilvermanADS Prop 6.6]. $$\xymatrix{
{\ensuremath{{\mathbb{G}}}}_m \ar[r]^{x\mapsto ax^d}\ar[d]^\pi & {\ensuremath{{\mathbb{G}}}}_m \ar[d]^\pi \\
{\ensuremath{{\mathbb{A}}}}^1\ar[d]\ar[r] & {\ensuremath{{\mathbb{A}}}}^1\ar[d]\\
{\ensuremath{{\mathbb{P}}}}^1 \ar[r]^f & {\ensuremath{{\mathbb{P}}}}^1
}$$ If the inclusion ${\ensuremath{{\mathbb{A}}}}^1\hookrightarrow{\ensuremath{{\mathbb{P}}}}^1$ extends to the identity map and $a=1$, then $f$ is the $d$th Chebyshev polynomial $T_d(x)$ and satisfies $$f(x+x^{-1})=x^d+x^{-d}.$$ As with power maps, choosing other inclusions or $a\neq 1$ simply results in fractional linear conjugates of Chebyshev polynomials. Because of the symmetry in the definition, positive and negative $d$ give rise to the same $f$. Here the count of Lemma \[lem: periodic count\] is $$\#\operatorname{Per}_n(f) = 1 + \frac{1}{2}\left(\#\ker(x^{d^n-1})+\#\ker(x^{d^n+1})\right).$$
The kernel of the endomorphism $x\mapsto x^m$ is the set of $|m|$th roots of unity in $k$. If $(p,|m|)=1$, there are $|m|$ of these, and in general $$\#\ker(x^m)=\frac{|m|}{p^{v_p(|m|)}}$$ because there are no nontrivial $p$th roots of unity.
Therefore, for a power map $f\in k(x)$ associated to the endomorphism $\sigma(x)=x^d$, if $d>0$ we have the formula $$\label{eqn: periodic count power map}
\#\operatorname{Per}_n(f) = 2 + \frac{d^n-1}{p^{v_p(d^n-1)}},$$ and if $d<0$, $$\label{eqn: periodic count power map negative}
\#\operatorname{Per}_n(f) = \left\{
\begin{array}{ll}
2 + \frac{|d|^n-1}{p^{v_p(|d|^n-1)}} & : d\text{ even} \\
\frac{|d|^n+1}{p^{v_p(|d|^n+1)}} & : d\text{ odd}
\end{array}
\right.$$ For a Chebyshev polynomia $f$, the formula reads $$\label{eqn: periodic count Chebyshev polynomial}
\#\operatorname{Per}_n(f) = 1 + \frac{1}{2}\left(\frac{|d|^n-1}{p^{v_p(|d|^n-1)}}+\frac{|d|^n+1}{p^{v_p(|d|^n+1)}}\right).$$
Maps from ${\ensuremath{{\mathbb{G}}}}_a$: Additive and Subadditive Polynomials {#sec: G_a}
===============================================================================
Let ${\ensuremath{{\mathbb{G}}}}_a$ be the additive group. In characteristic zero, all endomorphisms of ${\ensuremath{{\mathbb{G}}}}_a$ are of the form $x\mapsto cx$, so $\operatorname{End}({\ensuremath{{\mathbb{G}}}}_a)\cong k$. In positive characteristic, the Frobenius map $\phi(x)=x^p$ and its iterates are also endomorphisms, and $\operatorname{End}({\ensuremath{{\mathbb{G}}}}_a)$ is the noncommutative polynomial ring $ k\langle\phi\rangle$ with the multiplication rule $\phi c=c^p\phi$ for $c\in k$.
The only automorphisms of ${\ensuremath{{\mathbb{G}}}}_a$ are the nonzero maps $x\mapsto cx$, as Frobenius is a bijection on $k$-valued points ($k$ is algebraically closed) but is not an isomorphism on the underlying scheme, which is ${\ensuremath{{\mathbb{A}}}}^1$. Therefore $\operatorname{Aut}({\ensuremath{{\mathbb{G}}}}_a)\cong k^\times$. The finite subgroups $\Gamma\subseteq\operatorname{Aut}({\ensuremath{{\mathbb{G}}}}_a)$ are all cyclic by a basic fact of field theory [@Isaacs Lem 17.12], so there is some $d$ such that $\Gamma\cong\mu_d$, the group of $d$th roots of unity.
Let $\psi:{\ensuremath{{\mathbb{G}}}}_a\to{\ensuremath{{\mathbb{G}}}}_a$ be an affine morphism, that is, an element of $k\langle\phi\rangle$ composed with a translation. If $\Gamma=\{1\}$, then $\pi:{\ensuremath{{\mathbb{G}}}}_a\to G/\Gamma\cong{\ensuremath{{\mathbb{A}}}}^1$ is the identity morphism on the underlying scheme. There always exists an $f$ that fits into the following diagram: $$\xymatrix{
{\ensuremath{{\mathbb{G}}}}_a \ar[r]^{\psi}\ar[d]^\pi & {\ensuremath{{\mathbb{G}}}}_a \ar[d]^\pi \\
{\ensuremath{{\mathbb{A}}}}^1\ar[d]\ar[r] & {\ensuremath{{\mathbb{A}}}}^1\ar[d]\\
{\ensuremath{{\mathbb{P}}}}^1 \ar[r]^f & {\ensuremath{{\mathbb{P}}}}^1
}$$ There are many inclusions ${\ensuremath{{\mathbb{A}}}}^1\hookrightarrow{\ensuremath{{\mathbb{P}}}}^1$, but as with power maps, $f$ is determined up to conjugacy. Therefore we may assume that the inclusion extends to the identity, in which case $f$ fixes $\infty$ and is a polynomial. We call $f$ an additive polynomial, as $f(x+y)=f(x)+f(y)$ for all $x,y\in k$.
If $\Gamma\cong\mu_d$ for $d>1$ and $(p,d)=1$, then the map $\pi:{\ensuremath{{\mathbb{G}}}}_a\to{\ensuremath{{\mathbb{G}}}}_a/\mu_d\cong {\ensuremath{{\mathbb{A}}}}^1$ can be taken to be $\pi(x)=x^d$. In this case there exists an $f$ to make the diagram commute if and only if $\psi$ satisfies $\psi(\omega_d x) = \omega_d\psi(x)$ for a primitive $d$th root of unity $\omega_d$. (This happens if and only if $\psi(x)$, written as a polynomial, has terms whose degrees are all 1 mod $d$.) If there is such an $f$, we call it a subadditive polynomial.
Let $\sigma\in k\langle\phi\rangle$ be an endomorphism of ${\ensuremath{{\mathbb{G}}}}_a$. The size of $\ker\sigma$ depends on the divisibility of $\sigma$ by $\phi$, that is, $$\#\ker\sigma = \frac{\deg\sigma}{p^{v_\phi(\sigma)}}.$$ Here $v_\phi(\sigma)$ denotes the largest power of the two-sided maximal ideal $(\phi)=\phi k\langle\phi\rangle$ that contains $\sigma$.
Let $\psi(x)=\sigma(x)+c$ for some $c\in{\ensuremath{{\mathbb{G}}}}_a$. For an additive or subadditive polynomial $f$, Lemma \[lem: periodic count\] and the above observation yield $$\label{eqn: periodic count additive polynomial}
\#\operatorname{Per}_n(f) = 1 + \frac{1}{d}\sum_{\omega\in\mu_d}\#\ker(\sigma^n-\omega) = 1 + \frac{1}{d}\sum_{\omega\in\mu_d}\frac{(\deg\sigma)^n}{p^{v_\phi(\sigma^n-\omega)}}.$$ Note that $\deg(\sigma^n-\omega)=\deg(\sigma^n)$ because $\omega:{\ensuremath{{\mathbb{G}}}}_a\to{\ensuremath{{\mathbb{G}}}}_a$ is the linear polynomial $\omega(x)=\omega x$, and $\deg\sigma=\deg\psi\geq 2$ by assumption.
Maps from Elliptic Curves: Lattès Maps {#sec: Lattes}
======================================
Let $E$ be an elliptic curve and let $\psi:E\to E$ be an affine morphism. The endomorphism ring $\operatorname{End}(E)$ can be identified with either ${\ensuremath{{\mathbb{Z}}}}$, an order in an imaginary quadratic field, or a maximal order in a quaternion algebra [@SilvermanEllipticCurves Thm V.3.1]. There are only six possibilities for $\operatorname{Aut}(E)$: it may be a cyclic group of order 2,3,4, or 6, or a certain nonabelian group of order 12 or 24 [@SilvermanEllipticCurves Thm III.10.1]. Let $\Gamma$ be a nontrivial subgroup of $\operatorname{Aut}(E)$. We say that $f$ is a Lattès map if the diagram commutes: $$\xymatrix{
E \ar[r]^{\psi}\ar[d]^\pi & E \ar[d]^\pi \\
E/\Gamma\ar[d]^\wr\ar[r] & E/\Gamma\ar[d]^\wr\\
{\ensuremath{{\mathbb{P}}}}^1 \ar[r]^f & {\ensuremath{{\mathbb{P}}}}^1
}$$ As $E$ is projective, the curve $E/\Gamma$ is isomorphic to ${\ensuremath{{\mathbb{P}}}}^1$, unlike in the cases coming from ${\ensuremath{{\mathbb{G}}}}_m$ and ${\ensuremath{{\mathbb{G}}}}_a$. For a given choice of $\psi$ and $\Gamma$, there is not necessarily an $f$ that makes the diagram commute.
A Lattès map is often defined to be $f\in k(x)$ such that there exists a morphism $\psi:E\to E$ with $\deg\psi\geq 2$ and a finite separable cover $\pi:E\to{\ensuremath{{\mathbb{P}}}}^1$ such that the following diagram commutes. $$\xymatrix{
E \ar[r]^{\psi}\ar[d]^\pi & E \ar[d]^\pi \\
{\ensuremath{{\mathbb{P}}}}^1 \ar[r]^f & {\ensuremath{{\mathbb{P}}}}^1
}$$ This is equivalent to our definition. Any self-morphism of an elliptic curve can be written as the composition of an isogeny and a translation [@SilvermanEllipticCurves p 75], so any morphism $\psi:E\to E$ with $\deg\psi\geq 2$ is affine. Also, if there exists a diagram as above, then there exists such a diagram with the same $f$ and a possibly different triple $(E',\psi',\pi')$ where the map $\pi'$ is the quotient of $E'$ by a subgroup of automorphisms. This result is due to Milnor over ${\ensuremath{{\mathbb{C}}}}$ [@MilnorLattes], and Ghioca and Zieve in arbitrary characteristic [@GhiocaZieve]. For a sketch of the Ghioca-Zieve proof, see [@SilvermanModuliSpaces pp. 54-56].
By a general fact about morphisms of elliptic curves [@SilvermanEllipticCurves Thm III.4.10(a)], $$\#\ker\sigma = \deg_s\sigma = \frac{\deg\sigma}{\deg_i\sigma}.$$ Here $\deg_s$ and $\deg_i$ denote separable and inseparable degrees. In the rest of this section we develop a formula for $\deg_i(\sigma)$ in terms of the arithmetic of $\operatorname{End}(E)$.
First we set some notation. Let $N,\operatorname{tr}:\operatorname{End}(E)\otimes{\ensuremath{{\mathbb{Q}}}}\to{\ensuremath{{\mathbb{Q}}}}$ denote the norm and trace maps, or the reduced norm and trace in the case that $\operatorname{End}(E)\otimes{\ensuremath{{\mathbb{Q}}}}$ is a quaternion algebra. Let $\phi_m:E\to E^{(p^m)}$ be the $p^m$th power Frobenius morphism. For an isogeny $\sigma:E_1\to E_2$, write $\hat{\sigma}:E_2\to E_1$ for the dual isogeny. The $j$-invariant of $E$ is denoted by $j(E)$.
\[prop: j(E) transcendental\] Suppose $j(E)$ is transcendental over ${\ensuremath{{\mathbb{F}}}}_p$. Let $\sigma\in\operatorname{End}(E)$. Then $\sigma\in{\ensuremath{{\mathbb{Z}}}}$ and $$\deg_i(\sigma)=p^{v_p(\sigma)}.$$
If $j(E)\notin\overline{{\ensuremath{{\mathbb{F}}}}}_p$, then $\operatorname{End}(E)\cong{\ensuremath{{\mathbb{Z}}}}$ [@SilvermanEllipticCurves p 145]. The multiplication by $p$ map $[p]:E\to E$ has inseparable degree $p$, because $[p]=\hat{\phi_1}\circ\phi_1$ and the dual isogeny $\hat{\phi_1}$ is separable [@SilvermanEllipticCurves Thm V.3.1]. The multiplication by $m$ map is separable if $(p,m)=1$. Therefore the isogeny $[m]:E\to E$ has inseparable degree equal to $p^{v_p(m)}$.
For the rest of this section, suppose that $j(E)$ is algebraic over ${\ensuremath{{\mathbb{F}}}}_p$. Up to isomorphism, $E$ is defined over a finite field, so the ring $\operatorname{End}(E)$ can be identified with an order in an imaginary quadratic field if $E$ is ordinary or a maximal order in a quaternion algebra if $E$ is supersingular [@SilvermanEllipticCurves Thm V.3.1].
\[prop: inseparable degree ordinary\] Let $E$ be ordinary and let $K=\operatorname{End}(E)\otimes{\ensuremath{{\mathbb{Q}}}}$. Let $\sigma\in\operatorname{End}(E)$. Let $\mathfrak{p}$ be the extension to $\mathcal{O}_K$ of the ideal in $\operatorname{End}(E)$ consisting of all inseparable isogenies. Then $\mathfrak{p}$ is prime and $$\deg_i(\sigma)=p^{v_\mathfrak{p}(\sigma)}.$$
By [@SilvermanEllipticCurves Cor II.2.12] we can write $\sigma=\lambda\circ\phi_{m}$ where $p^m=\deg_i(\sigma)$ and $\lambda:E^{(p^{m})}\to E$ is separable. It follows that $\deg_i(\sigma)\geq p^m$ if and only if $\sigma$ factors through $\phi_{m}:E\to E^{(p^{m})}$. (If $\sigma=0$, we set $\deg_i(\sigma)=\infty$.)
We know that $\operatorname{End}(E)$ is an order in $\mathcal{O}_K$ for some imaginary quadratic field $K$, and the conductor of $\operatorname{End}(E)$ is prime to $p$ [@Deuring]. Let $I_m$ be the $m$th *inseparable ideal* of $\operatorname{End}(E)$, defined as follows: $$I_m = \{\sigma\in\operatorname{End}(E): \deg_i(\sigma)\geq p^m\}.$$ It is routine to show that $I_m$ is an ideal. If $\sigma\tau\in I_m$, then $$\deg_i(\sigma\tau)=\deg_i(\sigma)\deg_i(\tau)\geq p^m.$$ If $\sigma\notin I_m$ then necessarily $\deg_i(\tau)\geq p$, so $\deg_i(\tau^m)\geq p^m$ and $\tau^m\in I_m$. For $m=1$ this shows that $I_1$ is prime, and for $m>1$ that $I_m$ is $I_1$-primary.
If $\operatorname{End}(E)$ were a Dedekind domain, it would follow that $I_m=(I_1)^m$ for each $m>1$, but orders of Dedekind domains are not in general Dedekind domains. Instead, consider the integral extension $\mathcal{O}_K/\operatorname{End}(E)$ and the prime ideal $\mathfrak{p}$ of $\mathcal{O}_K$ lying over $I_1$. The multiplication by $p$ map $[p]:E\to E$ is inseparable, so $p\in I_1$, and therefore $p\mathcal{O}_K\subseteq I_1\mathcal{O}_K\subseteq\mathfrak{p}$. So $I_1\mathcal{O}_K$ is either $\mathfrak{p}$ or $p\mathcal{O}_K=\mathfrak{p}\hat{\mathfrak{p}}$ (as $p$ splits in $\mathcal{O}_K$ [@Deuring]).
This shows that the ideal $I_1\mathcal{O}_K$ is prime to the conductor of $\operatorname{End}(E)$. For ideals in an order that are prime to the conductor, extension to $\mathcal{O}_K$ and contraction are inverses, and unique factorization holds [@Cox Prop 7.20]. Therefore $I_1\mathcal{O}_K=\mathfrak{p}$ and the $I_1$-primary ideal $I_m$ equals $(I_1)^m$ for each $m>1$. It follows easily that $(I_1)^m\mathcal{O}_K=(I_1\mathcal{O}_k)^m=\mathfrak{p}^m$. We conclude that $\deg_i(\sigma)=p^{v_\mathfrak{p}(\sigma)}$.
If $E$ is defined over ${\ensuremath{{\mathbb{F}}}}_p$, the $p$th power Frobenius morphism $\phi_1$ is an element of $\operatorname{End}(E)$. In this case, the ideal $\mathfrak{p}$ in Proposition \[prop: inseparable degree ordinary\] is the principal ideal $\phi_1\mathcal{O}_K$, and $v_\mathfrak{p}(\sigma)$ is simply the highest power of $\phi_1$ that divides $\sigma$ in $\mathcal{O}_K$. In general, the ideal $\mathfrak{p}$ need not be principal.
\[prop: inseparable degree supersingular\] Let $E$ be supersingular, so that $\operatorname{End}(E)$ can be identified with a maximal order $\mathcal{O}$ of the quaternion algebra $B$. There exists a two-sided maximal ideal $I$ of $\mathcal{O}$ such that $$\deg_i(\sigma)=p^{v_I(\sigma)}.$$
If we write $\sigma=\lambda\circ\phi_m$ where $\lambda$ is separable, then $\deg(\lambda)$ is not divisible by $p$. If it were, the map $\lambda\circ\hat{\lambda}=[N(\lambda)]=[\deg(\lambda)]:E\to E$ would factor through $[p]:E\to E$ and would be inseparable, so one of $\lambda$ or $\hat{\lambda}$ would be inseparable. If $\hat{\lambda}$ were inseparable it would factor through $\phi_1$, so $\lambda$ would factor through $\hat{\phi_1}$, which is inseparable [@SilvermanEllipticCurves Thm V.3.1], contradicting the fact that $\lambda$ is separable. Moreover, $\deg\phi_m$ is a $p$-power, and is therefore the largest power of $p$ that divides $\deg\sigma$. This shows that $$\deg_i\sigma=\deg\phi_m=p^{v_p(\deg\sigma)}=p^{v_p(N(\sigma))}.$$
We have $\operatorname{End}(E)\cong\mathcal{O}$, which is a maximal order of $B$, the unique quaternion algebra over ${\ensuremath{{\mathbb{Q}}}}$ ramified exactly at $p$ and $\infty$ [@Deuring]. Consider the localization $\mathcal{O}_p=\mathcal{O}\otimes{\ensuremath{{\mathbb{Z}}}}_p$, which is the unique maximal order of $B_p=B\otimes_{\ensuremath{{\mathbb{Q}}}}{\ensuremath{{\mathbb{Q}}}}_p$, and the inclusion $\mathcal{O}\hookrightarrow\mathcal{O}_p$.
In $\mathcal{O}_p$ there is a uniformizing element $\pi$ such that $\pi\mathcal{O}_p$ is the unique two-sided maximal ideal of $\mathcal{O}_p$, and every ideal of $\mathcal{O}_p$ is a power of $\pi\mathcal{O}_p$ [@Vigneras Ch 2, Thm 1.3]. In particular, $p\mathcal{O}_p=\pi^2\mathcal{O}_p$, so $v_{\pi\mathcal{O}_p}(p)=2$. Let $I=\pi\mathcal{O}_p\cap\mathcal{O}$. The ideal $I$ is maximal in $\mathcal{O}$ because locally it is either maximal or the unit ideal: $I_p=\pi\mathcal{O}_p$, and $I_\ell=\mathcal{O}_\ell$ for $\ell\neq p$ (this is because $p$, which lies in $I$, is invertible in $\mathcal{O}_\ell$). The ideal $I$ is also two-sided because it is two-sided locally [@Vigneras p. 84]. Therefore $v_I$ is a valuation on $\mathcal{O}$.
Any supersingular $E$ is defined over ${\ensuremath{{\mathbb{F}}}}_{p^2}$, and there exists an automorphism $i:E\to E$ such that $\phi_2=i\circ[p]$. As $v_I(p)=2$, it follows that $v_I(\sigma)=v_p(N(\sigma))$ for all $\sigma\in\mathcal{O}$ such that $v_p(N(\sigma))$ is even, i.e. such that $\sigma=\lambda\circ[p^n]$ for some separable $\lambda$. If $v_p(N(\sigma))$ is odd then $v_p(N(\sigma^2))$ is even, so $v_I(\sigma^2)=v_p(N(\sigma^2))$, and therefore $v_I(\sigma)=v_p(N(\sigma))$.
Let $\psi:E\to E$ be written $\psi(x) = \sigma(x) + P$, where $\sigma\in\operatorname{End}(E)$ and $P\in E$. The above propositions together with Lemma \[lem: periodic count\] prove the following formulas. If $j(E)$ is transcendental, then necessarily $\Gamma=\{\pm 1\}$, and $$\label{eqn: Lattes transcendental}
\#\operatorname{Per}_n(f) = \frac{1}{2}\left(\frac{|\sigma^n-1|}{p^{v_p(\sigma^n-1)}} + \frac{|\sigma^n+1|}{p^{v_p(\sigma^n+1)}}\right).$$ If $j(E)$ is algebraic and $E$ is ordinary, then there exists $\mathfrak{p}$ such that $$\label{eqn: Lattes ordinary}
\#\operatorname{Per}_n(f) = \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\frac{N(\sigma^n-\gamma)}{p^{v_{\mathfrak{p}}(\sigma^n-\gamma)}}.$$ If $j(E)$ is algebraic and $E$ is supersingular, then there exists $I$ such that $$\label{eqn: Lattes supersingular}
\#\operatorname{Per}_n(f) = \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\frac{N(\sigma^n-\gamma)}{p^{v_I(\sigma^n-\gamma)}}.$$
The sequence $n\mapsto N(\sigma^n-\gamma)$ that appears in the above equations satisfies a linear recurrence relation and is therefore periodic when reduced mod any prime $\ell$. We record for future reference the next proposition, which determines its possible periods.
\[prop: norm recurrent\] Let $j(E)\in\overline{{\ensuremath{{\mathbb{F}}}}}_p$ and let $a_n=N(\sigma^n-\gamma)$, where $\sigma,\gamma\in\operatorname{End}(E)$. For any prime $\ell$, the sequence $(a_n\pmod{\ell})$ is periodic of period dividing $(\ell-1)(\ell^2-1)\ell^A$ for some integer $A$.
Let $T=\operatorname{tr}(\sigma)$ and $N=N(\sigma)$. We compute $$\begin{aligned}
a_n=\widehat{(\sigma^n-\gamma)}(\sigma^n-\gamma)& = \hat{\sigma}^n\sigma^n - \sigma^n\hat{\gamma} - \hat{\sigma}^n\gamma + \hat{\gamma}\gamma = N^n - \operatorname{tr}(\sigma^n\hat{\gamma}) + N(\gamma).\end{aligned}$$ Let $b_n=N^n$, $c_n=\operatorname{tr}(\sigma^n\hat{\gamma})$, and $d_n=N(\gamma)$. Certainly $b_n=Nb_{n-1}$ and $d_n=d_{n-1}$; this shows that the linearly recurrent sequences $(b_n)$ and $(d_n)$ have characteristic polynomials $x-N$ and $x-1$ in the sense of [@LidlNiederreiter Ch 6].
Whether $\operatorname{End}(E)$ is an order in an imaginary quadratic field or an order in a quaternion algebra, any $\sigma\in\operatorname{End}(E)$ satisfies the Cayley-Hamilton identity $\sigma^2-T\sigma+N=0$. For $n\geq 2$, $$\begin{aligned}
c_n - T c_{n-1} + N c_{n-2} & = \operatorname{tr}(\sigma^n\hat{\gamma}) -T\operatorname{tr}(\sigma^{n-1}\hat{\gamma}) + N\operatorname{tr}(\sigma^{n-2}\hat{\gamma})\\
& = \operatorname{tr}((\sigma^2-T\sigma+N)\sigma^{n-2}\hat{\gamma}) = \operatorname{tr}(0)=0.\end{aligned}$$ Therefore $(c_n)$ is a linearly recurrent sequence, and its characteristic polynomial is $x^2-Tx+N$. It follows from [@LidlNiederreiter Thm 6.55] that $(a_n)$ is linearly recurrent with characteristic polynomial equal to $$g(x)=(x-1)(x-N)(x^2-Tx+N)\in{\ensuremath{{\mathbb{F}}}}_\ell[x].$$ Recall that if $g(0)\neq 0$, $\operatorname{ord}g(x)$ is defined to be the least $n$ such that $g(x)$ divides $x^n-1$. By [@LidlNiederreiter Thm 6.27], the least period of $(a_n\pmod{\ell})$ divides $\operatorname{ord}g(x)$, and by [@LidlNiederreiter Thm 3.11], there is some $A\geq 0$ such that $$\operatorname{ord}g(x)=LCM[\operatorname{ord}(x-1),\operatorname{ord}(x-N),\operatorname{ord}(x^2 - Tx+ N)]\ell^A.$$ The integer $A$ reflects the possible presence of repeated factors of $g(x)$. If $x-1$, $x-N$, and $x^2-Tx+N$ are coprime in ${\ensuremath{{\mathbb{F}}}}_\ell[x]$, then $A=0$.
Lifting the Exponent {#sec: lifting the exponent}
====================
The periodic point counts established in the previous sections all contain an expression of the form $v_P(x^n-\gamma)$, where $P$ is a prime ideal of an endomorphism ring $R$. In this section we develop formulas for writing these expressions in terms of $v_p(n)$. These resemble a result in elementary number theory popularly known as “lifting the exponent", which is related to (but does not follow from) Hensel’s Lemma.
\[lem: Lifting the Exponent Number Field\] Let $K$ be a number field. Let $\mathfrak{p}$ be a prime of $\mathcal{O}_K$ lying over the rational prime $p$, and let $e$ be the ramification index of $\mathfrak{p}$ over $p$. Let $x,y\in\mathcal{O}_K$ be such that $x,y\notin\mathfrak{p}$ and $x-y\in\mathfrak{p}$. If $e+1\geq p-1$, further assume that $v_\mathfrak{p}(x-y)\geq\frac{e+1}{p-1}$. Then $$v_\mathfrak{p}(x^n-y^n) = v_\mathfrak{p}(x-y) + ev_p(n).$$
The proof is by induction on $v_p(n)$. Assume that $x\neq y$; otherwise the proposition holds trivially.
First suppose that $v_p(n)=0$, which guarantees $v_\mathfrak{p}(n)=0$. We compute $$v_\mathfrak{p}(x^n-y^n) =v_\mathfrak{p}(x-y) + v_\mathfrak{p}(x^{n-1}+x^{n-2}y+\dots+xy^{n-2}+y^{n-1}).$$ As $x- y\in\mathfrak{p}$, we have $x^{n-1}+x^{n-2}y+\dots+xy^{n-2}+y^{n-1}\equiv nx^n\pmod{\mathfrak{p}}$, and $nx^n\notin\mathfrak{p}$. Therefore $v_\mathfrak{p}(x^n-y^n) = v_\mathfrak{p}(x-y)$, proving the proposition in this case.
If we show that the proposition holds for $n=p$, it follows for all $n$ by induction. Let $v=v_\mathfrak{p}(x-y)$, so that $x=y+z$ for some $z\in\mathfrak{p}^v\setminus\mathfrak{p}^{v+1}$. By the binomial theorem, $$x^p = \sum_{i=0}^p {p\choose i}z^{i}y^{p-i}\equiv y^n + pzy^{n-1} \pmod{\mathfrak{p}^{v+e+1}}.$$ For $i\geq 2$, the $i$th term of the expansion is in $\mathfrak{p}^{v+e+1}$. If $i\neq p$ this is because $p$ divides ${p\choose i}$, so ${p\choose i}\in\mathfrak{p}^e$ and ${p\choose i}z^iy^{n-i}$ lies in $\mathfrak{p}^{e+iv}\subseteq\mathfrak{p}^{v+e+1}$, as $v\geq 1$. If $i=p$, this is because we assumed that $v=v_\mathfrak{p}(z)\geq\frac{e+1}{p-1}$, so $v_\mathfrak{p}(z^p)=pv\geq v+e+1$ and $z^p\in\mathfrak{p}^{v+e+1}$. The $i=1$ term is not in $\mathfrak{p}^{v+e+1}$, as $v_\mathfrak{p}(pz)=e+1$. Therefore $x^p-y^p\in\mathfrak{p}^{v+e}\setminus\mathfrak{p}^{v+e+1}$, so $v_\mathfrak{p}(x^p-y^p)=v+e$ and we are done.
\[lem: Lifting the Exponent Quaternion Algebra\] Let $p$ be a prime. Let $B$ be the unique quaternion algebra over ${\ensuremath{{\mathbb{Q}}}}$ ramified precisely at $p$ and $\infty$, and let $\mathcal{O}$ be a maximal order of $B$. Let $\pi$ be a uniformizer for $\mathcal{O}_p$, and let $I=\pi\mathcal{O}_p\cap\mathcal{O}$. Let $x,y\in\mathcal{O}$ be such that $x,y\notin I$ and $x-y\in I$. If $p=3$, assume further that $v_I(x-y)\geq 2$, and if $p=2$, assume further that $v_I(x-y)\geq 3$. Then $$v_I(x^n-y^n) = v_I(x-y) + 2v_p(n).$$
The proof is essentially the same as the proof of Lemma \[lem: Lifting the Exponent Number Field\], and is omitted.
\[lem: Lifting the Exponent G\_a\] Let $k$ be an algebraically closed field of characteristic $p$, and let $k\langle\phi\rangle$ be the noncommutative polynomial ring with the multiplication rule $\phi c = c^p\phi$ for $c\in k$. Let $x\in k\langle\phi\rangle$ be such that $x-1\in\phi k\langle\phi\rangle$. Then $$v_\phi(x^n-1) = v_\phi(x-1)p^{v_p(n)}.$$
First assume that $v_p(n)=0$. Then $$\begin{aligned}
x^n-1 & = (1+(x-1))^n-1 \equiv na^{n-1}(x-1)\pmod{\phi^2 k\langle\phi\rangle},\end{aligned}$$ and $v_\phi(x^n-1)=v_\phi(x-1)$.
Next let $n=p$. As $x^p-1=(x-1)^p$, we have $v_\phi(x^p-1)=pv_\phi(x-1)$. The proposition follows by induction on $v_p(n)$.
Background from Automatic Sequences {#sec: Automatic Sequences}
===================================
This section contains several results from the theory of automatic sequences. A sequence $(a_n)$ is $k$-automatic if it can be produced as the output of a deterministic finite automaton that takes as input the base-$k$ expansion of the integer $n$. The theorems in this section are stated so that they can be used later without any specific knowledge of finite automata or automatic sequences. A good introduction to the theory can be found in [@AlloucheShallit].
The following two theorems underlie our proof of transcendence. Christol’s theorem gives a correspondence between automatic sequences and algebraic power series, and Cobham’s theorem shows that only eventually periodic sequences can be automatic with respect to multiplicatively independent bases.
The formal power series $\sum_{n=0}^\infty a_n t^n\in{\ensuremath{{\mathbb{F}}}}_p[[t]]$ is algebraic over ${\ensuremath{{\mathbb{F}}}}_p(t)$ if and only if the coefficient sequence $(a_n)$ is $p$-automatic.
[@AlloucheShallit Thm 12.2.5].
Let $p$ and $q$ be multiplicatively independent positive integers (i.e. $\log p/\log q\notin{\ensuremath{{\mathbb{Q}}}}$). If the sequence $(a_n)$ is both $p$-automatic and $q$-automatic, then it is eventually periodic.
[@AlloucheShallit Thm 11.2.2].
The converse to Cobham’s theorem is also true.
Let $(a_n)$ be eventually periodic. Then $(a_n)$ is $k$-automatic for every positive integer $k$.
[@AlloucheShallit Thm 5.4.2].
The following is a corollary to Christol’s theorem that will be used to derive the contradiction that shows that $\zeta_f$ is transcendental.
\[cor: ChristolCor\] If $\sum_{n=0}^\infty a_n t^n\in{\ensuremath{{\mathbb{Z}}}}[[t]]$ is algebraic over ${\ensuremath{{\mathbb{Q}}}}(t)$, then the reduced sequence $((a_n) \bmod{p})$ is $p$-automatic for every prime $p$.
[@AlloucheShallit Thm 12.6.1].
The next two propositions shows that the set of $p$-automatic sequences over a ring is closed under both the pointwise application of arithmetic operations and the operation of extracting subsequences indexed by arithmetic progressions.
\[prop: automatic closure\] Let $(a_n)$ and $(b_n)$ be $p$-automatic sequences with entries in the ring $R$, and let $c\in R$. The sequences $(a_n+b_n)$, $(a_nb_n)$, and $(ca_n)$ are $p$-automatic, as is the sequence $(a_n^{-1})$ if each $a_n$ is invertible. Also, the subsequence $(a_{mn+b})$ is $p$-automatic for any $m,b\in{\ensuremath{{\mathbb{Z}}}}_+$.
The closure properties under arithmetic operations are special cases of the general theorem that the set of $p$-automatic sequences with entries in the set $\Delta$ is closed under the pointwise application of any binary operation $(\cdot,\cdot):\Delta\times\Delta\to\Delta$ [@AlloucheShallit Cor 5.4.5] and the completely trivial theorem that it is closed under the pointwise application of any unary operation $(\cdot):\Delta\to\Delta$ (this follows directly from the definition of an automatic sequence). For the claim about the subsequences $(a_{mn+b})$, see [@AlloucheShallit Thm 6.8.1].
Propositions \[prop: v\_p automaticity\] and \[prop: v\_p automaticity G\_a\] are the major technical results of this section. They will be needed to produce a contradiction at key moments in the proof of Theorems \[thm: main\] and \[thm: main2\].
\[prop: v\_p automaticity\] Let $p$ and $\ell$ be distinct primes. Suppose $a\in{\ensuremath{{\mathbb{Z}}}}_+$, $a\not\equiv 1\pmod{\ell}$, and $(a,\ell)=1$. Also suppose $\alpha,\beta\in{\ensuremath{{\mathbb{Z}}}}$, $\alpha\neq 0$, and $v_p(\alpha)\leq v_p(\beta)$. Let the sequence $(a_n)$ with entries in ${\ensuremath{{\mathbb{Z}}}}/\ell{\ensuremath{{\mathbb{Z}}}}$ be defined by $$a_n=a^{v_p(\alpha n+\beta)}\pmod{\ell}.$$ The sequence $(a_n)$ is not $\ell$-automatic.
Let $d$ be the multiplicative order of $a$ mod $\ell$. It follows from the assumptions that $d$ exists and is greater than 1. The sequence $n\mapsto a^{v_p(n)}$ is a function of the equivalence class of $v_p(n)$ mod $d$, so it is $p$-automatic by [@BridyActa Lem 6]. Therefore the sequence $(a_n)$ is $p$-automatic by Proposition \[prop: automatic closure\].
Assume by way of contradiction that $(a_n)$ is $\ell$-automatic. Distinct primes are multiplicatively independent, so by Cobham’s theorem, $(a_n)$ is eventually periodic. Let $c$ be its eventual period, so that $a_{n+xc}=a_n$ for sufficiently large $n$ and every positive $x$. This means that $$a^{v_p(\alpha n+\beta)}\equiv a^{v_p(\alpha(n+xc)+\beta)}\pmod{\ell},$$ which implies that $$v_p(\alpha n+\beta)\equiv v_p(\alpha(n+xc)+\beta)\pmod{d}.$$
Let $\alpha'=\alpha/p^{v_p(\alpha)}$ and $\beta'=\beta/p^{v_p(\alpha)}$. It is clear that $(p,\alpha')=1$, and it follows from our assumption that $v_p(\alpha)\leq v_p(\beta)$ that $\beta'$ is an integer. We have $$\label{eqn: contradiction}
v_p(\alpha' n+\beta')\equiv v_p(\alpha'(n+xc)+\beta')\pmod{d}.$$ Let $m=v_p(c)$ so that $c'=c/p^m$ and $(p,c')=1$. We can solve the congruence $$\label{eqn: one}
\alpha'n\equiv -\beta' +p^m\pmod{p^{m+2}}$$ for $n$, and choose such an $n$ to be large enough so that the sequence $(a_n)$ is periodic at $n$. Therefore $v_p(\alpha' n+\beta')=m$. We can also solve $$\label{eqn: two}
\alpha' c'x\equiv p-1\pmod{p^{m+2}}$$ for $x$, and choose such an $x$ to be positive. Adding Equation \[eqn: one\] and $p^m$ times Equation \[eqn: two\] gives $$\alpha'(n+xc)\equiv -\beta'+ p^{m+1}\pmod{p^{m+2}}$$ and therefore $v_p(\alpha'(n+xc)+\beta')=m+1$. So by Equation \[eqn: contradiction\], $$m\equiv m+1\pmod{d}.$$ As $d>1$, this is a contradiction.
\[prop: v\_p automaticity G\_a\] Let $a\in{\ensuremath{{\mathbb{Z}}}}_+$ and let $p$ and $\ell$ be primes with $\ell>p^{ap^a}$. If $p$ is odd, also assume that $(p,\ell-1)=1$. If $p=2$, instead assume that $\ell\equiv 7\pmod{8}$. Let the sequence $(a_n)$ with entries in ${\ensuremath{{\mathbb{Z}}}}/\ell{\ensuremath{{\mathbb{Z}}}}$ be defined by $$a_n=p^{ap^{v_p(n)}}\pmod{\ell}.$$ The sequence $(a_n)$ is not $\ell$-automatic.
Let $d$ be the multiplicative order of $p^a$ mod $\ell$. Since $\ell>p^{ap^a}$, we have $d>p^a$. The sequence $a_n$ is a function of the equivalence class of $p^{v_p(n)}$ mod d.
First assume that $p$ is odd. Then $(p^a,\ell-1)=1$, and as $d$ divides $\ell-1$, also $(p^a,d)=1$. Let $e$ be the multiplicative order of $p^a$ mod $d$, and note that $e\geq 2$. So $a_n=a_m$ iff $p^{v_p(n)}\equiv p^{v_p(m)}\pmod{d}$ iff $v_p(n)\equiv v_p(m)\pmod{e}$. In particular, $a_n$ is a function of the equivalence class of $v_p(n)$ mod $e$. By [@BridyActa Lem 6], $(a_n)$ is $p$-automatic.
If instead $p=2$, then $\ell\equiv 7\pmod{8}$, so 2 is a quadratic residue mod $\ell$. It follows that $d$ divides $\frac{\ell-1}{2}$, so $d$ is odd and $(p,d)=1$. Let $e$ be the multiplicative order of $p$ mod $d$, and again note that $e\geq 2$, so again $a_n=a_m$ if and only if $v_p(n)\equiv v_p(m)\pmod{e}$. In particular, $(a_n)$ is $p$-automatic.
Assume that $(a_n)$ is $\ell$-automatic. By Cobham’s theorem, $(a_n)$ is eventually periodic. Let $c$ be its eventual period. Therefore, for $n$ large and $x>0$, $$v_p(n+xc)\equiv v_p(n)\pmod{e}.$$ This is impossible by the proof of Proposition \[prop: v\_p automaticity\]; simply set $\alpha=1$ and $\beta=0$.
Proof of Theorem 1.2 {#sec: Proof 1.2}
====================
Let $k$ be an algebraically closed field of characteristic $p$, and let $f\in k(x)$ be a separable power map, Chebyshev polynomial, or Lattès map. Let $\zeta=\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$. Assume by way of contradiction that $\zeta$ is algebraic over ${\ensuremath{{\mathbb{Q}}}}(t)$. Its derivative $\zeta'$ is algebraic, so its logarithmic derivative $\zeta'/\zeta$ is algebraic. We compute $$\zeta'/\zeta=(\log\zeta)'=\sum_{n\geq 1}\#\operatorname{Per}_n(f)t^{n-1}.$$ By Corollary \[cor: ChristolCor\], for any prime $\ell$, the reduction of the sequence $(\#\operatorname{Per}_n(f))$ mod $\ell$ is $\ell$-automatic. By carefully choosing $\ell$, we will produce a contradiction, showing that $\zeta$ is transcendental.
Power Maps and Chebyshev Polynomials
------------------------------------
Let $f$ be the power map $f(x)=x^d$, and note that $(p,d)=1$ because $f$ is separable. Let $m$ be even and such that $d^m\equiv 1\pmod{p}$. Let $\ell>p$ be a prime to be determined, and consider the sequence $(a_n)$ with entries in ${\ensuremath{{\mathbb{F}}}}_\ell$ given by $$a_n = \#\operatorname{Per}_{mn}(f)\pmod{\ell}.$$ By Proposition \[prop: automatic closure\], $(a_n)$ is $\ell$-automatic. As $mn$ is even, Equation \[eqn: periodic count power map\] and Proposition \[lem: Lifting the Exponent Number Field\] give $$a_n = 2+\frac{d^{mn}-1}{p^{v_p(d^{mn}-1)}}=2+(d^{mn}-1)(p^{-1})^{v_p(d^{m}-1)+v_p(n)}.$$
First suppose $p$ is odd. Note that $(d^{mn}-1)=d^m-1$ when $n\equiv 1\pmod{\ell-1}$, and that $d^m-1\not\equiv 0\pmod{\ell}$. Consider the subsequence $$b_n=(d^m-1)(p^{v_p(d^m-1)})^{-1}(a_{(\ell-1)n+1}-2)^{-1},$$ which is $\ell$-automatic by proposition \[prop: automatic closure\]. Then $$b_n= p^{v_p((\ell-1)n+1)}.$$ Choose $\ell$ such that $\ell\equiv 2\pmod{p}$. Then $v_p(\ell-1)=0=v_p(1)$, and by Proposition \[prop: v\_p automaticity\], $(b_n)$ is not automatic, which is a contradiction.
Now suppose $p=2$. Let $m=2$, so that $d^m\equiv 1\pmod{4}$ (the separability of $f$ forces $d$ to be odd). Let $b_n$ be given by $$b_n=(d^{2}-1)(p^{v_p(d^2-1)})^{-1}(a_{(\ell-1)n+2}-2)^{-1},$$ so that $$b_n= p^{v_p((\ell-1)n+2)}.$$ Choose $\ell$ such that $\ell\equiv 3\pmod{4}$ and $(\ell,d^2-1)=1$. Then $v_p(\ell-1)=1=v_p(2)$, and again Proposition \[prop: v\_p automaticity\] gives a contradiction.
Let $f$ be the $d$th Chebyshev polynomial. Again it must be true that $(p,d)=1$, because otherwise the $\psi(x)$ that fits into Diagram \[eqn: dynamically affine\] factors through the $p$th power map and is inseparable, so $f$ is also inseparable. Again let $m$ be such that $d^m\equiv 1\pmod{p}$, and let $a_n = \#\operatorname{Per}_{mn}(f)\pmod{\ell}$ for some $\ell$ to be determined. By equation \[eqn: periodic count Chebyshev polynomial\] and Proposition \[lem: Lifting the Exponent Number Field\], $$\begin{aligned}
a_n & = 1+\frac{1}{2}\left(\frac{d^{mn}-1}{p^{v_p(d^{mn}-1)}}+\frac{d^{mn}+1}{p^{v_p(d^{mn}+1)}}\right)\\
& =1+\frac{1}{2}\left((d^{mn}-1)(p^{-1})^{v_p(d^{m}-1)+v_p(n)}+(d^{mn}+1)(p^{-1})^{v_p(d^{mn}+1)}\right) .\end{aligned}$$
First suppose $p$ is odd. Then $v_p(d^{mn}+1)=0$ because $v_p(d^{mn}-1)>0$ by Proposition \[lem: Lifting the Exponent Number Field\]. The sequence $$n\mapsto(d^{mn}+1)$$ is periodic, and so is $\ell$-automatic. Let $b_n$ be defined by $$b_n=(2(a_{(\ell-1)n+1}-1)-(d^{m(\ell-1)n+1}))(d^m-1)^{-1}(p^{v_p(d^m-1)}).$$ As before, $b_n$ is $\ell$-automatic, but $$b_n^{-1}= p^{v_p((\ell-1)n+1)}.$$ Choosing $\ell$ such that $\ell \equiv 2\pmod{p}$ gives a contradiction by Proposition \[prop: v\_p automaticity\].
If $p=2$, then let $m=2$. In this case, $d^{2n}\equiv 1\pmod{4}$ for all $n$, so $v_p(d^{2n}+1)=1$. The sequence $$n\mapsto(d^{2n}+1)$$ is still eventually periodic, so using the same manipulations as above, the sequence $$b_n^{-1}= p^{v_p((\ell-1)n+2)}$$ is $\ell$-automatic. If we pick $\ell$ such that $\ell\equiv 3\pmod{4}$ and $(\ell,d^2-1)=1$, then $v_p(\ell-1)=1$ and again this is a contradiction by Proposition \[prop: v\_p automaticity\].
Lattès Maps
-----------
Let $f$ be a Lattès map associated to the elliptic curve $E$. If $j(E)$ is transcendental over $\overline{{\ensuremath{{\mathbb{F}}}}}_p$, then $\sigma\in{\ensuremath{{\mathbb{Z}}}}$, and as $f$ is separable we have $(p,\sigma)=1$ (otherwise $\sigma$ would be inseparable). By equation \[eqn: Lattes transcendental\], $$\#\operatorname{Per}_n(f) = \frac{1}{2}\left(\frac{|\sigma^n-1|}{p^{v_p(\sigma^n-1)}} + \frac{|\sigma^n+1|}{p^{v_p(\sigma^n+1)}}\right).$$ If $\sigma>0$, then this is the same as the periodic point count for the degree $|\sigma|$ Chebyshev polynomial $T_{|\sigma|}$, and we have already shown that there exists an $\ell$ such that $\#\operatorname{Per}_n(f)\pmod{\ell}$ is not $\ell$-automatic. In fact, the argument also goes through if $\sigma<0$, as we simply choose an even $m$ and use the subsequence $\#\operatorname{Per}_{mn}(f)\pmod{\ell}$, in which case $\sigma^{mn}$ is always positive.
Now suppose $j(E)\in\overline{{\ensuremath{{\mathbb{F}}}}}_p$ and $E$ is ordinary. The ring $\operatorname{End}(E)$ is an order in an imaginary quadratic field $K$, and $\operatorname{Aut}(E)$ is cyclic of order 2,4, or 6. Therefore $\Gamma$ is isomorphic to one of $\mu_2,\mu_3,\mu_4$, or $\mu_6$. By Proposition \[prop: inseparable degree ordinary\], there is a prime $\mathfrak{p}$ of $\mathcal{O}_K$ such that $$\#\operatorname{Per}_n(f)= \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma}\frac{N(\sigma^{n}-\gamma)}{p^{v_\mathfrak{p}(\sigma^{n}-\gamma)}},$$ and $\mathfrak{p}$ is the extension to $\mathcal{O}_K$ of the ideal in $\operatorname{End}(E)$ consisting of all inseparable isogenies, so $\sigma\notin\mathfrak{p}$.
Assume for the moment that $p\notin\{2,3\}$. Let $m$ be the multiplicative order of the image of $\sigma$ in the residue field $\mathcal{O}_K/\mathfrak{p}$, so that $\sigma^m-1\in\mathfrak{p}$. By Proposition \[lem: Lifting the Exponent Number Field\], $$v_\mathfrak{p}(\sigma^{mn}-1)=v_\mathfrak{p}(\sigma^m-1)+v_p(n).$$ In particular, $\sigma^{mn}-1\in\mathfrak{p}$ for $n\geq 1$. We argue that $\sigma^{mn}-\gamma\notin\mathfrak{p}$ for $\gamma\in\Gamma\setminus \{1\}$.
If $\sigma^{mn}-\gamma$ were in $\mathfrak{p}$, then $1-\gamma$ would be in $\mathfrak{p}$, and $p=N(\mathfrak{p})$ would divide $N(1-\gamma)$. We compute $$N(1-\gamma)=(1-\gamma)(1-\hat{\gamma})=1-\operatorname{tr}(\gamma)+N(\gamma)=2-\operatorname{tr}(\gamma).$$ As $\gamma$ is a root of the $k$th cyclotomic polynomial for some $k\in\{2,3,4,6\}$, it follows that $\operatorname{tr}(\gamma)\in\{-2,-1,0,1\}$. Therefore $N(1-\gamma)\in\{1,2,3,4\}$, so $v_p(N(1-\gamma))=0$ and $1-\gamma\notin\mathfrak{p}$, so $\sigma^{mn}-\gamma\notin\mathfrak{p}$. Therefore $$\#\operatorname{Per}_{mn}(f)= \frac{1}{|\Gamma|}\frac{N(\sigma^{mn}-1)}{p^{v_\mathfrak{p}(\sigma^m-1)+v_p(n)}} + \frac{1}{|\Gamma|}\sum_{\gamma\in\Gamma\setminus\{1\}}N(\sigma^{mn}-\gamma).$$
Let $\ell>p$ be a rational prime to be determined. For now, assume that $(\ell, N(\sigma))=(\ell,|\Gamma|)=1$. Each sequence $n\mapsto N(\sigma^{mn}-\gamma)\pmod{\ell}$ is periodic and therefore $\ell$-automatic. It follows that the sequence given by $$a_n=\frac{1}{|\Gamma|}\frac{N(\sigma^{mn}-1)}{p^{v_\mathfrak{p}(\sigma^m-1)+v_p(n)}}\pmod{\ell}$$ is $\ell$-automatic. By Proposition \[prop: norm recurrent\], the sequence $n\mapsto N(\sigma^{mn}-1)\pmod{\ell}$ is periodic of period dividing $(\ell-1)(\ell^2-1)\ell^A$ for some $A$. Let $c$ be this period. Choose $\ell\equiv 2\pmod{p}$ subject to the previous restrictions on $\ell$ and such that $(\ell,N(\sigma^m-1))=1$. Therefore $$N(\sigma^{m(cn+1)}-1)\equiv N(\sigma^m-1)\not\equiv 0\pmod{\ell}.$$ By the closure properties of Proposition \[prop: automatic closure\], the sequence given by $$n\mapsto p^{v_p(c n+1)}\pmod{\ell}$$ is $\ell$-automatic. We know $(p,\ell)=(p,\ell-1)=(p,\ell^2-1)=1$, so $v_p(c)=0$. So $v_p(c)\leq v_p(1)=1$, and this is a contradiction by Proposition \[prop: v\_p automaticity\].
If $p\in\{2,3\}$, then $\operatorname{Aut}(E)\cong\mu_2$, as the only possible larger automorphism groups in characteristic 2 or 3 are nonabelian and cannot be realized as subgroups of $\mathcal{O}_K^\times$ [@SilvermanEllipticCurves Appendix A]. Therefore $\Gamma=\operatorname{Aut}(E)=\{\pm 1\}$.
First assume that $p=3$. Everything in the above argument holds, except possibly that $c$, the period of $n\mapsto N(\sigma^{mn}-1)\pmod{\ell}$, might be divisible by $3$. Choose $\ell$ such that $\ell\equiv 2\pmod{9}$ and $(\ell,N(\sigma^{3m}-1))=1$. By Proposition \[prop: norm recurrent\], $v_3(c)\leq v_3((\ell-1)(\ell^2-1)\ell^A)=1$. The contradiction by Proposition \[prop: v\_p automaticity\] now comes from manipulating $(a_{cn+3})$ to produce the sequence $$n\mapsto 3^{v_3(c n+3)}\pmod{\ell}.$$
Now assume that $p=2$. As $\sigma\notin\mathfrak{p}$, the image of $\sigma$ in the finite ring $\mathcal{O}_K/\mathfrak{p}^2$ is invertible. Let $m$ be the multiplicative order of the image of $\sigma$, so that $\sigma^m-1\in\mathfrak{p}^2$. Using Lemma \[lem: Lifting the Exponent Number Field\] and following the above argument, we arrive at $$\#\operatorname{Per}_{mn}(f) = \frac{1}{2}\left(\frac{N(\sigma^{mn}-1)}{2^{v_\mathfrak{p}(\sigma^{mn}-1)}}+\frac{N(\sigma^{mn}+1)}{2^{v_\mathfrak{p}(\sigma^{mn}+1)}}\right)$$ As $v_\mathfrak{p}(\sigma^m-1)\geq 2$, we have $v_\mathfrak{p}(\sigma^{mn}-1)\geq 2$ for all $n$. By properties of valuations, $$v_\mathfrak{p}(\sigma^{mn}+1)= \min(v_\mathfrak{p}(\sigma^{mn}-1),v_\mathfrak{p}(2))=1,$$ where $v_\mathfrak{p}(2)=1$ because $2$ splits in $K$. Therefore $$\#\operatorname{Per}_{mn}(f) = \frac{1}{2}\left(\frac{N(\sigma^{mn}-1)}{2^{v_\mathfrak{p}(\sigma^{mn}-1)}}+\frac{N(\sigma^{mn}+1)}{2}\right).$$
Again the only difficulty is that $c$, the period of $n\mapsto N(\sigma^{mn}-1)\pmod{\ell}$, might be even. Choose $\ell\equiv 3\pmod{8}$ such that $(\ell,N(\sigma^{16m}-1))=1$. Now $v_2(c)\leq v_2((\ell-1)(\ell^2-1)\ell^A)=4$, and the same contradiction comes from the sequence $$n\mapsto 2^{v_2(c n+16)}\pmod{\ell}.$$
Now suppose that $E$ is supersingular. In this case $\operatorname{End}(E)$ can be identified with a maximal order $\mathcal{O}$ of a rational quaternion algebra $B$ that is ramified only at $p$ and $\infty$. Let $I$ be the two-sided maximal ideal of $\mathcal{O}$ from Proposition \[prop: inseparable degree supersingular\]. As $\sigma:E\to E$ is separable, $\sigma\notin I$. Let $m$ be the multiplicative order of $\sigma$ in the residue field $\mathcal{O}/I$, so that $v_I(\sigma^m-1)\geq 1$.
Assume first that $p\notin\{2,3\}$ so that $\Gamma\cong\mu_2$, $\mu_3$, $\mu_4$, or $\mu_6$. By Proposition \[prop: inseparable degree supersingular\] and Lemma \[lem: Lifting the Exponent Quaternion Algebra\], $$v_I(\sigma^{mn}-1)=v_I(\sigma^m-1) + 2v_p(n).$$ As in the case of $E$ ordinary, for $\xi\in\Gamma\setminus\{1\}$, $\xi$ is a root of the $k$th cyclotomic polynomial some for $k\in\{2,3,4,6\}$, so $N(1-\xi)\in\{1,2,3,4\}$. Norm considerations show that $\sigma^{mn}-\xi\notin I$ for $n\geq 1$, so $\#\ker(\sigma^{mn}-\xi)=N(\sigma^{mn}-\xi)$. Therefore $$\#\operatorname{Per}_{mn}(f) = \frac{1}{|\Gamma|}\frac{N(\sigma^{mn}-1)}{p^{v_I(\sigma^m-1)+2v_p(n)}} + \frac{1}{|\Gamma|}\sum_{\xi\in\Gamma\setminus\{1\}}N(\sigma^{mn}-\xi).$$ The same reasoning as in the ordinary case shows that there is a prime $\ell$ such that $\#\operatorname{Per}_{mn}(f)\pmod{\ell}$ is not $\ell$-automatic.
Now assume that $p\in\{2,3\}$. If $j(E)\neq 0$, then $\operatorname{Aut}(E)=\{\pm 1\}$, and the argument proceeds exactly as when $E$ is ordinary. Therefore assume that $j(E)=0$. For both $p=2$ and $p=3$, the maximal order $\mathcal{O}$ has trivial class group and so is the unique maximal order of $B$ up to conjugacy [@Vigneras Ch 1, Cor 4.11]. Therefore, for the purposes of identifying $\operatorname{End}(E)$ with $\mathcal{O}$, we may take $\mathcal{O}$ to be any maximal order of $B$.
First let $p=2$. In this case, $B$ is the Hamilton quaternions $\left(\frac{-1,-1}{{\ensuremath{{\mathbb{Q}}}}}\right)={\ensuremath{{\mathbb{Q}}}}(i,j)$ where $i^2=j^2=-1$ and $k=ij$. A maximal order of $B$ is given by the Hurwitz quaternions $$\mathcal{O} = {\ensuremath{{\mathbb{Z}}}}i+ {\ensuremath{{\mathbb{Z}}}}j + {\ensuremath{{\mathbb{Z}}}}k + {\ensuremath{{\mathbb{Z}}}}(1+i+j+k)/2.$$ Here $\operatorname{Aut}(E)\cong\operatorname{SL}_2({\ensuremath{{\mathbb{F}}}}_3)$ can be described explicitly as $$\operatorname{Aut}(E)\cong\mathcal{O}^\times=\{\pm 1, \pm i, \pm j, \pm k, (\pm 1\pm i\pm j\pm k)/2\}.$$ For $\gamma\in\operatorname{Aut}(E)$, explicit calculation of norms shows that $v_I(1-\gamma)=0$ unless $\gamma\in\{\pm 1,\pm i,\pm j, \pm k\}\cong Q_8$, which is the unique Sylow 2-subgroup of $\operatorname{Aut}(E)$. As $\sigma\notin I$, the image of $\sigma$ is invertible in $\mathcal{O}/I^3$. Pick $m$ such that $\sigma^m- 1\in I^3$, i.e. $v_I(\sigma^m-1)\geq 3$. Then for $\gamma\neq 1$, $$v_I(\sigma^{mn}-\gamma)=\min\{v_I(\sigma^{mn}-1),v_I(1-\gamma)\} = \left\{
\begin{array}{ll}
2 & : \gamma =-1 \\
1 & : \gamma= \pm i, \pm j, \pm k\\
0 & : \text{otherwise}
\end{array}
\right.$$ So for each $\gamma\in\Gamma$, there exists a constant $C(\gamma)$ such that $$\#\operatorname{Per}_{mn}(f) = \frac{1}{|\Gamma|}\left(\frac{N(\sigma^{mn}-1)}{2^{v_I(\sigma^m-1)+2v_2(n)}} + \sum_{\gamma\in\Gamma\setminus\{1\}}\frac{N(\sigma^{nm}-\gamma)}{C(\gamma)}\right)$$ Choosing $\ell$ appropriately, we can reduce this to the sequence $$n\mapsto 4^{v_2(cn + 16)}$$ and get a contradiction as before.
Now let $p=3$, so that $B=\left(\frac{-1,-3}{{\ensuremath{{\mathbb{Q}}}}}\right)={\ensuremath{{\mathbb{Q}}}}(i,j)$ where $i^2=-1$, $j^2=-3$, and $k=ij$. A maximal order $\mathcal{O}$ is given by $$\mathcal{O} = {\ensuremath{{\mathbb{Z}}}}+ {\ensuremath{{\mathbb{Z}}}}i + {\ensuremath{{\mathbb{Z}}}}(1+j)/2 + {\ensuremath{{\mathbb{Z}}}}(i+k)/2$$ and again $\operatorname{Aut}(E)\cong C_3\rtimes C_4$ has an explicit description as $$\operatorname{Aut}(E)\cong\mathcal{O}^\times = \{\pm 1, \pm i, (\pm 1\pm j)/2, (\pm i\pm k)/2 \}.$$ For $\gamma\in\operatorname{Aut}(E)$, another norm calculation shows that $v_I(1-\gamma)=0$ unless $\gamma\in\{1,\frac{-1\pm j}{2}\}\cong C_3$, the unique Sylow 3-subgroup of $\operatorname{Aut}(E)$. Pick $m$ such that $v_I(\sigma^m-1)\geq 2$. If $\gamma\neq 1$, $$v_I(\sigma^{mn}-\gamma)=\min\{v_I(\sigma^{mn}-1),v_I(1-\gamma)\} = \left\{
\begin{array}{ll}
1 & : \gamma =(-1\pm j)/2 \\
0 & : \text{otherwise}
\end{array}
\right.$$ As above, there are $C(\gamma)$ so that $$\begin{aligned}
\#\operatorname{Per}_{mn}(f) = \frac{1}{|\Gamma|}&\left( \frac{N(\sigma^{mn}-1)}{3^{v_I(\sigma^m-1)+2v_3(n)}} + \sum_{\gamma\in\Gamma\setminus\{1\}} \frac{N(\sigma^{mn}-\gamma)}{C(\gamma)} )\right).\end{aligned}$$ We can reduce this to the sequence $$n\mapsto 9^{v_3(c_n+3)}$$ and the same argument gives a contradiction.
Proof of Theorem 1.3 {#sec: Proof 1.3}
====================
Let $k$ be an algebraically closed field of characteristic $p$, and let $f\in k[x]$ be an additive or subadditive polynomial. The map $f$ fits into Diagram \[eqn: dynamically affine\] with $G={\ensuremath{{\mathbb{G}}}}_a$, $\Gamma\cong\mu_d$, and $\pi(x)=x^d$, where possibly $d=1$. Therefore $\psi(x)^d=f(x^d)$. As usual, let $\psi$ be the endomorphism $\sigma$ composed with a translation.
Let $a$ be the constant term of $\sigma\in k\langle\phi\rangle$, so that $\sigma=a+(\phi)$ (that is, $\sigma(x)=ax + g(x)$ for some $g\in(\phi)$). If $d=1$, then $f(x)$ is $\sigma(x)$ composed with a translation, so $f'(0)=a$. If $d\geq 2$, then $\psi(\omega_d x)=\omega_d\psi(x)$, so $\psi(0)=0$ and $\psi=\sigma$. So $\sigma^d=a^d + (\phi)$, and $f'(0)=a^d$. Therefore $f'(0)$ is algebraic if and only if $a$ is algebraic.
By equation \[eqn: periodic count additive polynomial\], $$\#\operatorname{Per}_n(f) = 1 + \frac{1}{d}\sum_{\omega\in\mu_d}\frac{(\deg\sigma)^n}{p^{v_\phi(\sigma^n-\omega)}}.$$ As $\sigma$ is separable, $v_\phi(\sigma)=0$, so $a\neq 0$.
If $f'(0)$ is transcendental, then so is $a$. The constant term of $\sigma^n-\omega$ is $a^n-\omega$, which is never zero (if it were, $a$ would be algebraic). Therefore $$\#\operatorname{Per}_n(f) = 1 + \frac{1}{d}\sum_{\omega\in\mu_d}(\deg\sigma)^n = 1+(\deg\sigma)^n.$$ It follows easily that $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$ is rational.
Suppose $f'(0)$ is algebraic over ${\ensuremath{{\mathbb{F}}}}_p$, so $a$ is algebraic and therefore is a root of unity. Aiming for a contradiction, assume that $\zeta(f,{\ensuremath{{\mathbb{P}}}}^1(k);t)$ is algebraic. There exists $m$ such that the image of $\sigma^m$ in $k\langle\phi\rangle/(\phi)$ is 1, and so $\sigma^m-1\in(\phi)$. By Lemma \[lem: Lifting the Exponent G\_a\], for $n\geq 1$, $$v_\phi(\sigma^{mn}-1)=v_\phi(\sigma^m-1)p^{v_p(n)}.$$ In particular, $\sigma^{mn}-1\in(\phi)$. Therefore, for $\omega\in\mu_d\setminus\{1\}$, we have $\sigma^{mn}-\omega\notin (\phi)$, because otherwise $1-\omega$ would be in $(\phi)$. But $1-\omega\in k\langle\phi\rangle$ represents the linear polynomial $x\mapsto (1-\omega)x$, which does not factor through $\phi:x\mapsto x^p$. Therefore $$\#\operatorname{Per}_{mn}(f) = 1 + \frac{1}{d}\left(\frac{(\deg\sigma)^n}{p^{v_\phi(\sigma^m-1)p^{v_p(n)}}} + \sum_{\omega\in\mu_d\setminus\{1\}}(\deg\sigma)^n\right).$$ Let $\ell$ be a large prime. If $p$ is odd, choose $\ell$ such that $\ell\equiv 2\pmod{p}$, and if $p=2$, choose $\ell\equiv 7\pmod{8}$. Let $a_n=\#\operatorname{Per}_{mn}\pmod{\ell}$. Note that $n\mapsto (\deg\sigma)^n$ is periodic and therefore $\ell$-automatic. By Proposition \[prop: automatic closure\], we can manipulate $a_n$ to arrive at the sequence $$b_n=(\deg\sigma)^n p^{-p^{\left(v_\phi(\sigma^m-1)p^{v_p(n)}\right)}},$$ which is $\ell$-automatic. The subsequence $b_{(\ell-1)n}$ is $\ell$-automatic, as is its reciprocal $$(b_{(\ell-1)n})^{-1} = p^{p^{\left(v_\phi(\sigma^m-1)p^{v_p(n)}\right)}}.$$ For a large enough $\ell$, the above sequence satisfies the assumptions of Proposition \[prop: v\_p automaticity G\_a\], so it is not $\ell$-automatic, which is a contradiction.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We would like to thank Michael Zieve for drawing our attention to the results in [@GhiocaZieve], Tonghai Yang for helpful advice regarding quaternion algebras, and Bjorn Poonen for pointing out the counterexample to Conjecture \[conj: separable\] in the case where $k$ contains transcendentals over ${\ensuremath{{\mathbb{F}}}}_p$. This research was partly supported by NSF Grant no. EMSW21-RTG and by the Wisconsin Alumni Research Foundation.
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abstract: 'We calculate Root Mean Square (RMS) deviations from equilibrium for atoms in a two dimensional crystal with local (e.g. covalent) bonding between close neighbors. Large scale Monte Carlo calculations are in good agreement with analytical results obtained in the harmonic approximation. When motion is restricted to the plane, we find a slow (logarithmic) increase in fluctuations of the atoms about their equilibrium positions as the crystals are made larger and larger. We take into account fluctuations perpendicular to the lattice plane, manifest as undulating ripples, by examining dual-layer systems with coupling between the layers to impart local rigidly (i.e. as in sheets of graphene made stiff by their finite thickness). Surprisingly, we find a rapid divergence with increasing system size in the vertical mean square deviations, independent of the strength of the interplanar coupling. We consider an attractive coupling to a flat substrate, finding that even a weak attraction significantly limits the amplitude and average wavelength of the ripples. We verify our results are generic by examining a variety of distinct geometries, obtaining the same phenomena in each case.'
author:
- 'D. J. Priour, Jr'
- James Losey
title: Melting and Rippling Phenomena in Two Dimensional Crystals with localized bonding
---
Introduction
============
Efforts to gain a quantitative microscopic understanding of melting have spanned more than a century. The Lindemann criterion developed in 1910 [@Lindemann] describes melting in terms of the Root Mean Square (RMS) deviation from the atomic equilibrium positions. Since long-range positional order stems from the periodic arrangement of atoms in crystalline solid, atomic deviations that are comparable to the separation between atomic species could obscure the regularity of the underlying crystal lattice with a concomitant loss of positional order. The Lindemann criterion specifies that melting has occurred if the RMS deviations reach on the order of a tenth of a lattice constant, and has proved to be a reasonably effective theory for three dimensional systems.
The Lindemann analysis does not take into account correlations of the motions of neighboring atoms. Correlations are more important at lower dimensions, and the process of melting is hence strongly dimensionally dependent. While three dimensional crystals exhibit long-range order below certain temperatures, statistical fluctuations play a significant role in one dimensional systems, precluding all but short-ranged local ordering for $T > 0$.
The process of melting in two dimensions is more subtle, and is understood in the modern context to occur in more than one stage. An initial continuous loss of positional order precedes the proliferation of lattice defects, which accumulate and eventually complete the melting process at sufficiently high temperatures by destroying even orientational order, where each atom has a fixed number of neighbors.
Thermally induced fluctuations in atomic positions can have an important effect on nano-engineered systems where structures may be on the atomic scale. Atomic clusters or quantum “dots” are mesoscopic assemblies of atoms where the scale is confined in all directions. Linear structures such as carbon nanotubes are essentially one dimensional objects (although having cross sections on the atomic scale) where the tube length may approach macroscopic scales. Finally, two dimensional systems with nanoscale thickness such as covalently bonded graphene sheets are genuine monolayers with thicknesses on the atomic scale, but spanning macroscopic areas.
The novel charge transport properties of graphene have been of intrinsic fundamental interest, and have also inspired scenarios for the use of graphene in semiconductor microprocessor applications. Technological uses for graphene will need a stable planar substrate for the implementation of nano-circuitry, and fundamental scientific research will also benefit from the minimization of the amplitude of random undulations in graphene layers.
We examine two dimensional crystals with properties that would generically be found in two dimensional covalently bonded crystals, including stiffness with respect to displacements perpendicular to the plane of the sheet. Although we do not consider temperature regimes capable of disrupting the lattice topology or number of neighbors for each atom (e.g. by thermal rupture of bonds between neighboring atomic species), we examine the loss of order caused by fluctuations of atomic positions about their equilibrium positions which nonetheless leave the bonding pattern intact.
If the motion of particles comprising the crystal is confined to the plane of the lattice, the gradual loss of long-range crystalline order with increasing system size has been understood as being in some respects similar to the destruction of ferromagnetic ordering in the $X$-$Y$ model (the motion of the spins are confined to the plane with a ferromagnetic coupling between them) by thermally excited spin waves. Nevertheless, on a detailed level the two systems differ. In the case of the $X$-$Y$ model, spin-spin correlation functions decay algebraically with the spatial separation between spins below the Kosterlitz-Thouless temperature for vortex unbinding. On the other hand, the RMS deviation in atomic positions in two dimensional crystals has been described as logarithmically divergent (i.e. varying as $\log[ \alpha(T) L ]$ where $\alpha(T)$ is a temperature dependent parameter) for any finite temperature [@Chaikin].
In Section II, we discuss theoretical techniques and the system geometries under consideration. Then, in section III we examine three dimensional lattices where we show directly for suitably rigid lattice geometries that the RMS deviations from equilibrium converge to a finite value in the thermodynamic limit, an anticipated property of three dimensional systems. Moreover, we determine a reference temperature threshold $T_{L}^{\textrm{3D}}$ where mean square fluctuations about equilibrium reach one tenth of a lattice constant, corresponding to the melting point according to the Lindemann criterion. This temperature will serve as a point of reference in the examination of two dimensional systems where thermal fluctuations disrupt long-range order for any finite temperature. However, although we find stable crystalline order in three dimensional geometries, we also discuss a significant caveat which applies for simple cubic lattices and other geometries which lack local stiffness. To a great extent, the lattice geometries we report on are based on the two dimensional examples shown in Fig. \[fig:Fig12\]. A square lattice pattern, and a triangular lattice structure are shown. The former lacks inherent rigidity, but the square lattice gains local rigidity through the activation of an extended coupling scheme in which both nearest neighbors and next-nearest neighbors interact. In the same way, a simple cubic lattice requires interactions between next-nearest as well as nearest neighbors to resist thermal fluctuations and maintain long-range crystalline order.
By considering two geometries and appropriate three dimensional generalizations which differ in significant ways (i.e. one base on a square pattern and the other assembled of triangles or tetrahedra joined at their corners), we identify generic thermodynamic characteristics common to both.
![\[fig:Fig12\] Square lattice extended coupling geometry with interactions with nearest and next-nearest neighbors (left) and the periodic triangular lattice and labeling scheme (right)](priourlatfig1.eps){width=".42\textwidth"}
In Section IV, we examine two dimensional lattices such as those shown in Fig. \[fig:Fig12\] with motion confined to the crystal plane, finding the very slow (i.e. logarithmic) loss of crystalline order anticipated for two dimensional crystals. On the other hand, for a two dimensional crystal embedded in three dimensions, it is important to consider transverse perturbations tending to push atoms out of the plane. We find that in the absence of binding to a substrate, two dimensional crystals are much less able to resist extraplanar distortions than fluctuations which are confined to the lattice plane.
In Section V, we examine dual-layer systems where coupling between the crystal planes imparts local stability with respect to extra-planar variations of atomic positions in a caricature of physical systems (e.g. sheets of graphene) which have a finite thickness, and would be imbued with local stiffness. We find for two distinct locally rigid dual-layer geometries similar rapid divergences of mean square displacements as the crystal is made larger, corresponding to thermally induced rippling of the crystal, and scaling linearly with the size of the system. Analysis of the density of vibrational states reveals that the length scale of the random undulations increases with the size of the system with strong long wavelength contributions. On the other hand coupling to a flat substrate, however weak, places an asymptotic upper bound on the ripple amplitudes and also limits the average wavelength of thermally induced undulations.
Calculation methods and Monte-Carlo Simulation Results
======================================================
We examine thermodynamic properties (e.g. the mean square deviations of atoms about equilibrium positions) for crystals with short range bonding in the regime where bonds remain intact and thermally induced lengthening and shortening of bonds is small relative to the unperturbed, or equilibrium, bond length. With individual bonds varying only slightly in length, it is appropriate to model the bonds as harmonic potentials so the couplings between neighboring atoms are effectively treated as springs connecting the two particles. It is important to note that although we neglect anharmonic effects from the bonds, the noncollinearity of bonds in the crystal geometry may in principle introduce anharmonic terms in the Hamiltonian. Nevertheless, at temperatures near and below the melting point, many scenarios are amenable to the harmonic approximation where the neglect of anharmonicities (whether intrinsic or geometric) has a small impact on the accuracy of the calculation. Analytical results obtained in the context of the harmonic approximation are validated in the cases we consider by good agreement with Monte-Carlo calculations where the anharmonic characteristics of the bonding stemming from peculiarities of the lattice geometry are rigorously taken into account.
The Hamiltonian is given by $$\mathcal{H} = \frac{1}{2} \sum_{i=1}^{N} \sum_{j = 1}^{m_{i}} \frac{K_{ij}}{2} \left( l_{ij} - l_{ij}^{0} \right)^{2}
\label{eq:eq1}$$ where $l_{ij}^{0}$ is the equilibrium energetically favored bond length and $l_{ij}$ is the instantaneous separation between atoms $i$ and $j$. The outer sum is over the atoms in the (finite) crystal, and the inner sum is over the neighbors associated with the atom indexed by the label $i$. The additional factor of 1/2 is included to compensate for double counting of bonds. The constant $K_{ij}$ is the second derivative of the interatomic potential $V_{ij}(r)$ at the equilibrium separation $l_{ij}^{0}$.
We develop the harmonic approximation directly from the bond length $l_{ij} = \sqrt{(x_{i} - x_{j})^{2} + (y_{i} - y_{j})^{2} + (z_{i} - z_{j})^{2} }$ For the $x$ coordinates, it is convenient to write, for example, $x_{i} = x_{i}^{0} + \delta_{i}^{x}$ where $x_{i}^{0}$ is the equilibrium coordinate and $\delta_{i}^{x}$ is the shift about equilibrium. We operate in the same way for the $y$ and $z$ coordinates, finding $$\begin{aligned}
l_{ij} = \sqrt{ \begin{array}{c} (\Delta_{ij}^{0x} + \delta_{i}^{x} - \delta_{j}^{x})^{2}
+ (\Delta_{ij}^{0y} + \delta_{i}^{y} - \delta_{j}^{y})^{2} \\ +
(\Delta_{ij}^{0z} + \delta_{i}^{z} - \delta_{j}^{z})^{2} \end{array}},\end{aligned}$$ where $\Delta_{ij}^{0x} \equiv (x_{i}^{0} - x_{j}^{0})$, $\Delta_{ij}^{0y} \equiv (y_{i}^{0} - y_{j}^{0})$, and $\Delta_{ij}^{0z} \equiv (z_{i}^{0} - z_{j}^{0})$. One may develop the harmonic approximation by expanding terms such as $(l_{ij} - l_{ij}^{0})^2$ to quadratic order in the shift differences $(\delta_{i}^{x} - \delta_{j}^{x})$, $(\delta_{i}^{y} - \delta_{j}^{y})$, and $(\delta_{i}^{z} - \delta_{j}^{z})$. The result will be $(l_{ij} - l_{ij}^{0}) \approx \left[ \hat{\Delta}_{ij} \cdot \left ( \vec{\delta}_{i} - \vec{\delta}_{j} \right ) \right]^{2}$, where $\hat{\Delta}_{ij}$ is a unit vector formed from $\vec{\Delta}_{ij} = (\Delta_{ij}^{0x},
\Delta_{ij}^{0y}, \Delta_{ij}^{0z})$.The terms $\vec{\delta}_{i}$ and $\vec{\delta}_{j}$ are vector atomic displacements such that, e.g., $\vec{\delta}_{i} = (\delta_{i}^{x},\delta_{i}^{y},\delta_{i}^{z})$. A salient characteristic of the bond energy is its dependence on the differences of the coordinate shifts (e.g. $\delta_{i}^{x} - \delta_{j}^{x}$ for the $x$ direction) instead of $\delta_{i}^{x}$, $\delta_{i}^{y}$, and $\delta_{i}^{z}$ by themselves, a condition which under many circumstances permits the neglect of anharmonicities due to bond non-collinearity.
In the harmonic approximation, the lattice energy due to deviations from equilibrium positions will be $$\begin{aligned}
\mathcal{H}^{\mathrm{Har}} = \frac{1}{2} \sum_{i=1}^{N} \sum_{j = 1}^{m_{i}} \frac{K_{ij}}{2}
\left[ \hat{\Delta}_{ij} \cdot \left ( \vec{\delta}_{i} - \vec{\delta}_{j} \right ) \right]^{2}\end{aligned}$$ On expanding, one obtains a quadratic expression mixing the displacements $$\begin{aligned}
\mathcal{H}^{\mathrm{Har}} \! = \! \sum_{i=1}^{N} \sum_{j=1}^{m_{i}}
\! \tfrac{K_{ij}}{4} \! \left[ \begin{array}{ccc} \delta_{i}^{x} &
\delta_{i}^{y} & \delta_{i}^{z} \end{array} \! \right] \! \! \!
\left[ \! \begin{array}{ccc} a_{xx} & a_{xy} & a_{xz} \\
a_{yx} & a_{yy} & a_{yz} \\ a_{zx} & a_{zy} & a_{zz} \end{array} \! \right] \! \! \! \!
\left[ \! \begin{array}{c} \delta_{j}^{x} \\
\delta_{j}^{y} \\ \delta_{j}^{z} \end{array} \! \right] \end{aligned}$$ Diagonalizing the appropriate matrix yields 3$N$ eigenvectors, taken to be normalized. Each of the set of 3N eigenvectors has a component for the individual degrees of freedom in the crystal lattice, permitting the lattice Hamiltonian to be written in decoupled form as $$\begin{aligned}
\mathcal{H}^{\mathrm{Har}} = \frac{K}{2} \sum_{\alpha = 1}^{3N} \lambda_{\alpha} c_{\alpha}^{2}
\end{aligned}$$ with eigenvector expansion coefficients $c_{\alpha}$ and eigenvalues $\lambda_{\alpha}$; the parameter $K$ is the “primary” harmonic constant, which is taken to be the nearest neighbor intra-planar coupling constant in schemes, such as extended models with multiple coupling constants. The eigen-modes are independently excited by thermal fluctuations, and thermodynamic equilibrium observables may be calculated by evaluating Gaussian integrals. As an example, the thermally averaged mean square fluctuation per atomic species $\langle \delta_{\textrm{RMS}} \rangle$ is (first moments of the coordinate shifts such as $\langle \delta_{i}^{x} \rangle$ vanish in the thermal average and do not appear in the expression below) $$\begin{aligned}
\langle \delta_{\textrm{RMS}} \rangle^{2} = \tfrac{1}{N} \sum_{i=1}^{N} \langle (\delta_{i}^{x})^{2}
+ (\delta_{i}^{y})^{2} + (\delta_{i}^{z})^{2} \rangle \end{aligned}$$ Indexing the eigenvectors with the label $\alpha$ and noting, e.g., that $\delta_{i}^{x} = \displaystyle{\sum_{\alpha = 1}^{3N}} c_{\alpha} v_{\alpha}^{ix}$, we see that the total square of the instantaneous fluctuations per particle is $$\begin{aligned}
\delta_{\mathrm{RMS}} = \tfrac{1}{N} \sum_{i=1}^{N} \!
\sum_{\alpha=1}^{3N} \sum_{\alpha^{'}=1}^{3N} \left [ c_{\alpha} c_{\alpha^{'}} ( v_{\alpha}^{ix}
v_{\alpha^{'}}^{ix} + v_{\alpha}^{iy} v_{\alpha^{'}}^{iy} +
v_{\alpha}^{iz} v_{\alpha^{'}}^{iz}) \right ]
\end{aligned}$$ In calculating the thermal average the term $c_{\alpha} c_{\alpha^{'}}$ will be as often negative as positive when $\alpha \neq \alpha^{'}$, and there will only be a non-zero contribution to $\langle \delta_{\mathrm{RMS}} \rangle^{2} $ if $\alpha = \alpha^{'}$. Hence, the double sum enclosed in square brackets will collapse to a single sum, and the calculation is reduced to the thermal average $$\begin{aligned}
\langle \delta_{\mathrm{RMS}} \rangle^{2} = \tfrac{1}{N} \sum_{\alpha=1}^{3N}
\langle c_{\alpha}^{2} \rangle \sum_{i=1}^{N}\left[ (v_{\alpha}^{ix})^{2} + (v_{\alpha}^{iy})^{2} +
(v_{\alpha}^{iz})^{2} \right]
\end{aligned}$$ The eigenvector normalization condition gives $$\begin{aligned}
\sum_{i=1}^{N} \left[ (v_{\alpha}^{ix})^{2} +
(v_{\alpha}^{iy})^{2} + (v_{\alpha}^{iz})^{2} \right] = 1,\end{aligned}$$ and hence $\langle \delta_{\mathrm{RMS}} \rangle^{2}$ appears simply as $$\begin{aligned}
\langle \delta_{\mathrm{RMS}} \rangle^{2} = \tfrac{1}{N} \sum_{\alpha=1}^{3N}
\langle ( c_{\alpha})^{2} \rangle\end{aligned}$$ The partition function $Z$ may be calculated with the aid of $\int_{-\infty}^{\infty} e^{-\sigma q^{2}} dq =
(\pi /\sigma)^{1/2}$, and one has a product of decoupled Gaussian integrals, which may be written as $$\begin{aligned}
Z = \prod_{\alpha=1}^{3N} \int_{-\infty}^{\infty} e^{-K \beta \lambda_{\alpha} c_{\alpha}^{2}/2 } d c_{\alpha}
\label{Eq:eq100}\end{aligned}$$ with $\beta = 1/k_{\mathrm{B}}$, $k_{\mathrm{B}}$ the Boltzmann constant, and the temperature $T$ is given in Kelvins. For the sake of convenience, units are chosen such that the lattice constant $a$ is equal to unity, and a reduced temperature is defined with $t \equiv k_{\mathrm{B}} T/K$. Evaluating the integrals in the product given in Eq. \[Eq:eq100\] yields for $Z$ $$\begin{aligned}
Z = \prod_{\alpha=1}^{3N} \left ( \frac{2 \pi t}{\lambda_{\alpha}} \right )^{1/2}\end{aligned}$$ The thermally averages mean square displacement may be written in terms of a thermal logarithmic derivative of $Z$, and in particular, one finds $$\begin{aligned}
\langle \delta_{\mathrm{RMS}} \rangle^{2} = t^{2} \frac{d}{dt} \textrm{Ln}(Z) = \sum_{\alpha = 1}^{3N} \lambda_{\alpha}^{-1} t\end{aligned}$$ Hence, the thermally averaged mean square deviation from equilibrium may be written as the square root of a sum over eigenvalue reciprocals. $$\begin{aligned}
\langle \delta_{\mathrm{RMS}} \rangle = t^{1/2} \sqrt{ \sum_{\alpha = 1}^{3N} \lambda_{\alpha}^{-1}}\end{aligned}$$ Zero eigenvalues would lead to a diverging expression, but eigenvalues which are strictly equal to zero are artifacts of periodic boundary conditions, correspond to global translations of the crystal lattice, and are excluded from the sum. The dependence on reduced temperature consists of a $t^{1/2}$ factor. To concentrate on characteristics specific to a lattice geometry and its coupling scheme, as well as trends with respect to system size $L$, the normalized mean square displacement $\delta_{\mathrm{RMS}}^{{n}}$ will often be discussed in lieu of the full temperature dependent quantity.
In the case of a periodic regular crystal lattice, it is useful exploit translational invariance, which will lead to exact expressions for the vibrational mode eigenstates and frequencies for periodic crystals (or at the very least yielding a small matrix which may be diagonalized analytically or by numerical means if necessary) if atomic displacements are written in terms of the corresponding Fourier components.
Using Monte Carlo calculations to sample thermodynamic quantities incorporates anharmonic effects in a rigorous manner, providing a means of assessing the validity of the harmonic approximation. We employ the Metropolis technique [@Metropolis] to introduce random displacements and sample the distribution corresponding to thermal equilibrium. We follow the standard Metropolis prescription, where an attempted random displacement with an associated energy shift $\Delta E$ is accepted with probability $e^{-\Delta E/k_{\mathrm{B}} T}$ if $\Delta E > 0$ and the Monte Carlo move is invariably accepted for cases in which $\Delta E < 0$.
In calculating thermodynamic quantities, we operate in terms of Monte Carlo sweeps where a sweep, on average, consists of an attempt to move each atom in the crystal with the acceptance of the move subject to the Metropolis condition. In the calculations, the sampling of thermodynamic quantities is postponed until the completion of the first 25% of the total number of sweeps to eliminate bias from the initial conditions, which are not typical thermal equilibrium configurations for the system. To reduce errors due to statistical fluctuations in the Monte Carlo simulation and obtain several digits of accuracy in the results, we conduct at least $5 \times 10^{5}$ sweeps. Figure \[fig:Fig1\] (for the square lattice with an extended coupling scheme) and Figure \[fig:Fig2\] (for the triangular lattice geometry) show mean square deviation curves for various temperatures ranging from temperatures an order of magnitude smaller than $T_{\mathrm{3D}}^{\mathrm{L}}$ to temperatures on par with the Lindemann criterion result for the melting temperature of the corresponding three dimensional system. The solid lines correspond to analytical results, while the symbols are RMS values obtained with Monte Carlo calculations.
The curves show very good agreement between the Monte Carlo data and analytical results over a wide range of temperatures and system sizes, and deviations are primarily mild statistical errors (on the order of one part in $10^{3}$) in the Monte Carlo calculations.
![\[fig:Fig1\] Graph of the RMS fluctuation about equilibrium versus systems size $L$ for various values of the reduced temperature $t$ for the square lattice with next-nearest neighbor couplings. The solid lines are analytic results obtained in the harmonic approximation, and symbols are results from Monte Carlo calculations.](priourlatfig2.eps){width=".48\textwidth"}
![\[fig:Fig2\] Graph of the RMS fluctuation about equilibrium versus systems size $L$ for various values of the reduced temperature $t$ for the triangular lattice. The solid lines are analytic results obtained in the harmonic approximation, and symbols are results from Monte Carlo calculations.](priourlatfig3.eps){width=".48\textwidth"}
![\[fig:Fig3\] Illustration of the simple cubic, nonrigid structure and rigidity gained by incorporating next nearest-neighbor couplings as shown in the image to the right.](priourlatfig4.eps){width=".35\textwidth"}
Rigid and Non-Rigid Three Dimensional lattices
==============================================
To establish a temperature scale for the two dimensional systems, where long-range crystalline order is not expected to exist at temperatures above 0K, we first examine three dimensional lattices, which may exhibit long-range positional order at finite temperature if the lattice is suitably rigid. As a preliminary step, we perform an analysis similar to the Lindemann treatment where an atom in a simple cubic geometry is coupled to six nearest neighbors. Since we do not take into account the motion of neighboring atoms, we take their displacements to be zero; certainly the excursions of neighboring atoms would average to zero, although to be precise, one would need to take into account cooperative effects of the atomic motions of the neighbors.
The lattice energy has the form $$\begin{aligned}
\hat{h}_{\mathrm{L}}^{\mathrm{3D}} = \frac{K}{2} \left ( \begin{array}{c} \Delta_{x_{+}}^{2} + \Delta_{x_{-}}^{2} + \Delta_{y_{+}}^{2} +
\Delta_{y_{-}}^{2} + \Delta_{z_{+}}^{2} + \Delta_{z_{-}}^{2} \end{array} \right) \end{aligned}$$ where, for example, $\Delta_{x_{+}}$ is the shift in length of the bond to the nearest neighbor in the positive $\hat{x}$ direction. Applying the harmonic approximation and taking the atomic shifts to be $\left \{ \delta_{x}, \delta_{y}, \delta_{z} \right \}$, the energy becomes $$\begin{aligned}
\hat{h}_{\mathrm{L}}^{\mathrm{3D}} = K \left[ \delta_{x}^{2} +
\delta_{y}^{2} + \delta_{z}^{2} \right]\end{aligned}$$ In the calculation of $\delta_{\mathrm{RMS}}$, the partition function has the form $$\begin{aligned}
Z = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} e^{-\tfrac{\delta_{x}^{2} +
\delta_{y}^{2} + \delta_{z}^{2}}{t}} d \delta_{x} d \delta_{y} d \delta_{z}\end{aligned}$$ From the Gaussian integration, we find $Z =
(\pi t)^{3/2}$. The RMS displacement will be $\langle r^{2} \rangle^{1/2}$ where $$\begin{aligned}
\langle r^{2} \rangle = t^{2} \tfrac{d}{dt} \textrm{Ln} (Z) = 3t/2 \end{aligned}$$ Hence, the thermally averaged mean square shift is $(3t/2)^{1/2}$. The Lindemann criterion places the melting temperature at a temperature high enough that the mean square deviation $\delta_{\mathrm{RMS}}$ reaches a tenth of a lattice constant, which corresponds to $t_{\mathrm{L}}^{\mathrm{3D}}
= \tfrac{2}{3}(10^{-2})$, a reduced temperature on the order of $0.01$.
If correlations among atoms are taken into account, next-nearest neighbor couplings become crucial to imparting local stiffness and maintaining long-range crystalline order. To see how rigidity is an important factor, we calculate the RMS displacements for a simple cubic lattice where only couplings between nearest neighbors are taken into account. The energy stored in the lattice will be $$\begin{aligned}
E = \frac{K}{2} \sum_{i,j,k = 0}^{n-1} \left[ \! \! \begin{array}{c} (\delta_{i+1jk}^{x} - \delta_{ijk}^{x})^{2} +
(\delta_{ij+1k}^{y} - \delta_{ijk}^{y})^{2} \\ + (\delta_{ijk+1}^{z}-\delta_{ijk}^{z})^{2} \end{array} \! \! \right]\end{aligned}$$ where a periodic geometry is assumed, and the counting factor of $1/2$ does not appear since the sum has been constructed to avoid redundancies. If we use the transformations $$\begin{aligned}
\delta_{ijk}^{x} = \sum_{k_{x}k_{y}k_{z}} \delta_{\bf{k}}^{x} e^{I(k_{x}i + k_{y}j + k_{z}k)}\end{aligned}$$ where $I$ is the imaginary unit, and similar expressions are used for $\delta_{ijk}^{y}$ and $\delta_{ijk}^{z}$. In terms of the Fourier components, the lattice energy may be written as $$\begin{aligned}
E = \frac{K}{2} \! \! \sum_{k_{x},k_{y},k_{z}} \left [ \! \begin{array}{c} (1- \cos k_{x}) | \delta_{\bf{k}}^{x}|^{2}
\\ + ( 1 - \cos k_{y}) | \delta_{\bf{k}}^{y}|^{2} \\ + (1 - \cos k_{z}) | \delta_{\bf{k}}^{z} |^{2}
\end{array} \! \right ] \end{aligned}$$ The $x$, $y$, and $z$ degrees of freedom $\delta_{\bf{k}}^{x}$, $\delta_{\bf{k}}^{y}$, and $\delta_{\bf{k}}^{z}$ automatically decouple. The normalized mean square deviation is $\sqrt{\sum_{\alpha} \lambda_{\alpha}^{-1}}$, where the sum is restricted to non-zero eigenvalues. We identify three eigenvalues, $\lambda_{\bf{k}}^{(1)} = 2(1 -\cos k_{x})$, $\lambda_{\bf{k}}^{(2)} = 2 (1 - \cos k_{y})$, and $\lambda_{\bf{k}}^{(3)} = 2 (1 - \cos k_{z})$ for each wave vector $\left \{ k_{x},k_{y},k_{z} \right \}$ As can be seen in Figure, the mean square fluctuation about equilibrium positions grows very rapidly with increasing system size. The divergence in the RMS displacements is a consequence of the lack of rigidity of the simple cubic geometry, which facilitates the destruction of long range crystalline order by thermal fluctuations. However, next-nearest neighbor couplings make the lattice rigid, and are very effective in suppressing fluctuations about equilibrium and establishing long-range crystalline order for the simple cubic lattice.
![\[fig:Fig4\] RMS displacements graphed with system size $L$ for various values of $K_{2}$, expressed in units of the nearest-neighbor coupling $K_{1}$. Panel (a) shows a closer view of the $\delta_{\mathrm{RMS}}$ curves over a smaller range of system sizes, and panel (b) is a graph with a broader range of system sizes included in the plot.](priourlatfig5.eps){width=".49\textwidth"}
![\[fig:Fig5\] RMS displacements graphed as a function of the ratio of next nearest neighbor to nearest neighbor coupling, $K_{2}/K_{1}$. Inset (a) is a graph of the normalized RMS deviation with respect to $\log_{10} (K_{2}/K_{1})$, and inset (b) shows the RMS fluctuations raised to the 1/4 power versus $\log_{10} (K_{2}/K_{1})$.](priourlatfig6.eps){width=".49\textwidth"}
The structure of the eigenvalue density states profile has informative characteristics particular to the lattice geometry from which it is obtained, and the density of states is calculated for many of the systems we report on. We achieve the thermodynamic in a genuine sense by not restricting $k_{x}$, $k_{y}$, and $k_{z}$ to discrete values as is done for finite systems. The density of states is built up by Monte Carlo sampling in which the wave-vector components are each generated independently from a uniform random distribution. To obtain good statistics, at least on the order of $2 \times 10^{8}$ eigenvalues are sampled in constructing the DOS. The same Monte Carlo sampling procedure is used to calculate the $\delta_{\mathrm{RMS}}$ values shown in Fig. \[fig:Fig5\], and thereby completely remove any bias from finite size effects. The density of states corresponding to the simple cubic system (shown in the graph in Fig. \[fig:Fig6\]) is consistent with the divergence of the RMS fluctuations with increasing system size. The bimodal structure is sharply peaked in the low and high eigenvalue regimes, with the former contributing to the steady rise of $\delta_{\mathrm{RMS}}$ with increasing system size $L$.
![\[fig:Fig6\] Normalized Eigenvalue Density of States for the simple cubic system for an extended coupling scheme with $K_{2} = \left \{ 0.0, 0.01, 0.02 \right \}$ with a sampling of $2.4 \times 10^{8}$ eigenvalues.](priourlatfig7.eps){width=".35\textwidth"}
![\[fig:Fig7\] Normalized Eigenvalue Density of States for the simple cubic system for an extended coupling scheme with $K_{2} = \left \{0.05, 0.10, 0.20 \right \}$ with a sampling of $2.4 \times 10^{8}$ eigenvalues.](priourlatfig8.eps){width=".45\textwidth"}
![\[fig:Fig8\] Normalized Eigenvalue Density of States for the simple cubic system with coupling for an extended coupling scheme with $K_{2} =
\left \{ 0.5, 0.7, 1.0 \right \}$ with a sampling of $2.4 \times 10^{8}$ eigenvalues.](priourlatfig9.eps){width=".45\textwidth"}
In the extended coupling scheme in the simple cubic geometry, the energy stored in the lattice is $$\begin{aligned}
E \! = \!
\frac{K_{1}}{2} \! \! \! \! \sum_{i,j,k=0}^{n-1} \! \left( \! \! \! \begin{array}{c}
\left \{ \! \! \! \begin{array}{l} [ \hat{x} \! \cdot \! (\vec{\delta}_{i+1jk} \! - \! \vec{\delta}_{ijk})]^{2}
\! + \! [\hat{y} \! \cdot \! (\vec{\delta}_{ij+1k} \! - \! \vec{\delta}_{ijk})]^{2} \\ \! + \!
[ \hat{z} \! \cdot \! (\vec{\delta}_{ijk+1} \! - \! \vec{\delta}_{ijk}) ]^{2} \end{array} \! \! \right \} \\
\! \! \! + \kappa_{2} \! \left \{ \! \! \! \begin{array}{l} \left[ \frac{1}{\sqrt{2}}(\hat{x} + \hat{y}) \cdot
(\vec{\delta}_{i+1j+1k} - \vec{\delta}_{ijk}) \right]^{2} \! + \! \\
\left[ \frac{1}{\sqrt{2}}( \hat{x} \!- \!\hat{y}) \! \cdot \! (\vec{\delta}_{i+1j-1k} \! - \! \vec{\delta}_{ijk} ) \right]^{2} \! \! + \! \\
\left[ \frac{1}{\sqrt{2}}( \hat{y} \!+ \!\hat{z}) \! \cdot \! (\vec{\delta}_{ij+1k+1} \!- \! \vec{\delta}_{ijk} ) \right]^{2} \! \! + \! \\
\left[ \frac{1}{\sqrt{2}}( \hat{y} \! - \! \hat{z}) \! \cdot \! (\vec{\delta}_{ij+1k-1} \!- \! \vec{\delta}_{ijk} ) \right]^{2} \! \! + \! \\
\left[ \frac{1}{\sqrt{2}}( \hat{x} \! + \! \hat{z}) \! \cdot \! (\vec{\delta}_{i+1jk+1} \!- \! \vec{\delta}_{ijk} ) \right]^{2} \! \! + \! \\
\left[ \frac{1}{\sqrt{2}}( \hat{x} \! - \! \hat{z}) \! \cdot \! (\vec{\delta}_{i+1jk-1} \!- \! \vec{\delta}_{ijk} ) \right]^{2}
\end{array}
%[ (\frac{\hat{x}}{\sqrt{2}} - \frac{\hat{y}}{\sqrt{2}}) \cdot
%(\vec{delta}_{i+1j-1k} - \vec{\delta}_{ijk})]^{2} \end{array}
\! \! \! \! \right \}
%E = \displaystyle{ \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} \sum_{k=0}^{n-1}} \\
%\begin{array}{l} \frac{K_{1}}{2} \left \{ \begin{array}{c} (\delta_{i+1jk}^{x} - \delta_{ijk}^{x})^{2} \\ +
%(\delta_{ij+1k}^{y} - \delta_{ijk}^{y})^{2} + (\delta_{ijk+1}^{z} - \delta_{ijk})^{2} \end{array} \right \} + \\
%\\
% \frac{K_{2}}{4} \left \{ \begin{array}{l}
%\left[ (\delta_{i+1j+1k}^{x} -
%\delta_{ijk}^{x}) + (\delta_{i+1j+1k}^{y} - \delta_{ijk}^{y}) \right ]^{2}
%\\ + \left[ (\delta_{i+1j-1k}^{x} -
%\delta_{ijk}^{x}) - (\delta_{i+1j-1k}^{y} - \delta_{ijk}^{y}) \right ]^{2}
%\\ + \left[ (\delta_{ij+1k+1}^{y} -
%\delta_{ijk}^{y}) + (\delta_{ij+1k+1}^{z} - \delta_{ijk}^{z}) \end{array} \right ]^{2}
%\\ + \left[ (\delta_{ij+1k-1}^{y} -
%\delta_{ijk}^{y}) - (\delta_{ij+1k-1}^{z} - \delta_{ijk}^{z}) \right ]^{2}
%\\ + \left[ (\delta_{i+1jk+1}^{x} -
%\delta_{ijk}^{x}) + (\delta_{i+1jk+1}^{z} - \delta_{ijk}^{z}) \right ]^{2} \\ +
% \left[ (\delta_{i+1jk-1}^{y} -
%\delta_{ijk}^{y}) - (\delta_{i+1jk-1}^{z} - \delta_{ijk}^{z}) \right ]^{2} \end{array}
% \right \}
\end{array}
\! \! \! \! \right)\end{aligned}$$ where $K_{1}$ is the coupling to nearest neighbors, $K_{2}$ is the coupling to next-nearest neighbors, and $\kappa_{2} \equiv K_{2}/K_{1}$ is the ratio of the next-nearest and nearest neighbor coupling constants. In terms of Fourier components, one has $$\begin{aligned}
E \! = \! \frac{K_{1}}{2} \! \! \sum_{\bf{k}} \! \!
\left( \! \! \!
\begin{array}{l}
\left \{ \! \! \begin{array}{l} \! (2 \! - \! 2 \cos k_{x}) \lvert \delta_{\bf{k}}^{x} \rvert^{2} \! + \!
(2 - 2 \cos k_{y}) \lvert \delta_{\bf{k}}^{y} \rvert ^{2} \\ \! + \! (2 \! - \! 2 \cos k_{z}) \lvert \delta_{\bf{k}}^{z} \rvert ^{2} \end{array}
\! \! \right \} \! + \! \\
\\
\! \kappa_{2} \! \left \{ \! \! \begin{array}{l} \left [ 2 \! - \! \cos k_{x} \cos k_{y} \!- \!
\cos k_{x} \cos k_{z} \right ] \! \lvert \delta_{\bf{k}}^{x} \rvert^{2} \! + \!\\
\left[ 2 \! - \! \cos k_{x} \cos k_{y} \! - \! \cos k_{y} \cos k_{z} \right]
\! \lvert \delta_{\bf{k}}^{y}\rvert^{2} \! + \! \\
\left[ 2 \! - \! \cos k_{y} \cos k_{z} \! - \! \cos k_{x} \cos k_{z} \right]
\! \lvert \delta_{\bf{k}}^{z} \rvert^{2} \! + \! \\
\sin k_{x} \sin k_{y} ( \delta_{\bf{k}}^{x} \delta_{\bf{k}}^{*y}
+ \delta_{\bf{k}}^{*x} \delta_{\bf{k}}^{y}) + \\
\sin k_{y} \sin k_{z} ( \delta_{\bf{k}}^{y} \delta_{\bf{k}}^{*z}
+ \delta_{\bf{k}}^{*y} \delta_{\bf{k}}^{z}) + \\
\sin k_{z} \sin k_{x} ( \delta_{\bf{k}}^{z} \delta_{\bf{k}}^{*x}
+ \delta_{\bf{k}}^{*z} \delta_{\bf{k}}^{x})
\end{array} \! \! \!
\right \}
\end{array} \! \! \! \! \!
\right )\end{aligned}$$ with $\bf{k}$ indicating the wave-vector with components $k_{x}$, $k_{y}$, and $k_{z}$, and again $\kappa_{2} = K_{1}/K_{2}$. The eigenvalues are hence obtained by diagonalizing the 3$\times$3 matrix $$\begin{aligned}
2 \! \! \left[ \! \! \!
\begin{array}{ccc}
\left( \! \! \begin{array}{c} 2 \! - \! \cos k_{x} \cos k_{y}\\ \! - \! \cos k_{x} \cos k_{z} \end{array} \! \! \! \right) \! \! & \! \!
\kappa_{2} \sin k_{x} \sin k_{y} \! \! \! & \! \! \! \kappa_{2} \sin k_{x} \sin k_{z} \\ \\
\kappa_{2} \sin k_{x} \sin k_{y} \! \! \! & \! \! \! \left( \! \! \begin{array}{c} 2 \! - \! \cos k_{y} \cos k_{z} \\
\! - \! \cos k_{z} \cos k_{x} \end{array} \! \! \! \right ) \! \! \! & \! \! \! \kappa_{2} \sin k_{y} \sin k_{z} \\ \\
\kappa_{2} \sin k_{z} \sin k_{x} \! \! \! & \! \! \! \kappa_{2} \sin k_{z} \sin k_{y} \! \! \! & \! \! \!
\left( \! \! \begin{array}{c} 2 \! - \! \cos k_{z} \cos k_{y} \\ \! - \!
\cos k_{z} \cos k_{x} \end{array} \! \! \! \right)
\end{array} \! \! \!
\right ]\end{aligned}$$ Although solving the cubic characteristic equation will yield analytical expressions for the eigenvalues, the result is cumbersome, and we instead use standard algorithms for the diagonalization of a symmetric matrix to efficiently obtain the eigenvalues numerically.
The eigenvalues determined in this manner are used to calculate the means square atomic fluctuations, and the results are shown in Figure \[fig:Fig4\], where $\delta_{\mathrm{RMS}}$ is graphed with respected to $L$ for a range of the next to nearest neighbor coupling strength ratio $\kappa_{2} = K_{2}/K_{1}$. Whereas the mean square displacement steadily rises with system size when $\kappa_{2} = 0$ (i.e with only nearest-neighbor couplings active), the curves behave very differently for nonzero $\kappa_{2}$, ultimately saturating with increasing $L$. The stabilization of $\delta_{\mathrm{RMS}}$ in the thermodynamic limit indicates the presence of intact long-range crystalline order. In the case of $\kappa_{2} = 0$, the mean square deviation steadily diverges with increasing $L$. The same divergence with only nearest-neighbor interactions taken into a account occurs whether one is considering the simple cubic structure, a square lattice, or a linear chain. Hence, in some lattice geometries, having a three dimensional structure may be insufficient to stabilize long-range order if an extended coupling scheme is not taken into consideration.
By switching on and varying the strength of the next-nearest coupling $K_{2}$, one sees the appearance of long-range crystalline order as the cubic system is made increasingly rigid. In Fig. \[fig:Fig5\] the mean square displacement is shown graphed versus the coupling ratio $K_{2}/K_{1}$. The tendency for atoms to be driven from their positions in the lattice does increase as $K_{2}$ is shut off, but the divergence occurs at a slow rate. Inset (a) is a graph of $\delta_{\textrm{RMS}}$ versus the logarithm of $K_{2}/K_{1}$. While the concavity of the curve indicates a somewhat more rapid than logarithmic divergence, a semi-logarithmic plot of $\delta_{\mathrm{RMS}}$ (i.e. as shown in inset (b) of Fig. \[fig:Fig5\]) shows an approximately linear scaling of $\delta_{\mathrm{RMS}}^{1/4}$ with the logarithm of the system size, still a relatively slow divergence, albeit somewhat more rapid than a simple logarithmic divergence. Hence, the next-nearest neighbor couplings in the extended coupling simple cubic model are very effective in restoring long-range crystalline order.
Trends in the eigenvalue density of states profile with increasing $K_{2}/K_{1}$ to next-nearest neighbors are shown in in Fig. \[fig:Fig6\], Fig. \[fig:Fig7\], Fig. \[fig:Fig8\]. The almost immediate retreat of the low and high frequency peaks toward the center is consistent with the effectiveness of an extended coupling scheme in stabilizing long-range crystalline order even for very small values of the ratio $\kappa_{2} = K_{2}/K_{1}$. The DOS profile has a simple structure for small $\kappa_{2} < 0.1$, while intermediate $\kappa_{2}$ values are associated with a richer density of states curve which changes rapidly as the coupling ratio is increased further.
![\[fig:Fig11\] Illustration of the tetrahedral lattice geometry](priourlatfig10.eps){width=".5\textwidth"}
As in the case of the cubic lattice with the extended coupling scheme, we may calculate the lattice energy in the harmonic approximation for the tetrahedral lattice, which is $$\begin{aligned}
E = \! \frac{K}{2} \! \! \sum_{i,j,k=0}^{n-1} \! \! \left ( \! \!
\begin{array}{l}
\left[ \vec{x} \! \cdot \! (\vec{\delta}_{i+1jk} \! - \! \vec{\delta}_{ijk}) \right]^{2} + \\
\left[ \! \left (\! \frac{1}{2} \hat{x} \! + \! \frac{\sqrt{3}}{2} \hat{y} \! \right ) \!
\cdot \! (\vec{\delta}_{ij+1k} \! - \! \vec{\delta}_{ijk}) \! \right]^{2} \! + \! \\
\left[ \! \left ( \! \frac{1}{2} \hat{x} \! - \! \frac{\sqrt{3}}{2} \hat{y} \! \right )
\! \cdot \! (\vec{\delta}_{i+1j-1k} \! - \! \vec{\delta}_{ijk}) \! \right]^{2} \! + \! \\
\left[ \! \left ( \! \frac{1}{2} \hat{x} \! + \! \frac{1}{2\sqrt{3}} \hat{y} \! + \! \sqrt{\frac{2}{3}} \hat{z} \! \right ) \! \cdot \!
(\vec{\delta}_{ijk+1} \! - \! \vec{\delta}_{ijk} ) \! \right]^{2} \! + \! \\
\left[ \! \left ( \! -\frac{1}{2} \hat{x} \! + \! \frac{1}{2 \sqrt{3}} \hat{y} \! + \! \sqrt{\frac{2}{3}} \hat{z} \! \right ) \! \cdot \!
(\vec{\delta}_{i-1jk+1} \! - \! \vec{\delta}_{ijk}) \! \right]^{2} \\
\! + \! \left[ \left ( -\frac{1}{\sqrt{3}} \hat{y} \! + \! \sqrt{\frac{2}{3}} \hat{z} \! \right )
\cdot (\vec{\delta}_{ij-1k+1} \! - \! \vec{\delta}_{ijk}) \! \right]^{2}
\end{array} \! \! \! \right )\end{aligned}$$ where there is only one coupling constant $K$ since bonds are considered between nearest neighbors only, the tedrahedral geometry being intrinsically rigid, and we have used the fact that the altitude of a tetrahedron is $\sqrt{\tfrac{2}{3}}$ times the lattice constant. The energy may be expressed in terms of Fourier components, and one has the task of diagonalizing the $3 \times 3$ matrix $$\begin{aligned}
\left[ \! \! \! \! \!
\begin{array}{ccc}
\left( \! \! \! \! \begin{array}{c} 4 - 2 \cos k_{x} - \\ \frac{1}{2} (\cos k_{y} \! + \! \cos k_{z}) \\
- \frac{1}{2} \cos (k_{y} \! - \! k_{x}) \\ - \frac{1}{2} \cos (k_{z}\! - \! k_{y} )
\end{array} \! \! \! \! \right)
\! \! \! \! \! \! \! &
\tfrac{\sqrt{3}}{2} \! \!
\left( \! \! \! \begin{array}{c} \cos (k_{y} \! - \! k_{x} ) \\ - \cos k_{y} \end{array} \! \! \! \right) \! \!
\! \! \! &
\sqrt{\tfrac{2}{3}} \! \! \left ( \! \! \! \begin{array}{c} \cos (k_{z} - k_{x}) \\ -
\cos k_{z} \end{array} \! \! \! \right) \\ \\
\tfrac{\sqrt{3}}{2} \! \!
\left ( \! \! \! \begin{array}{c} \cos (k_{y} \! - \! k_{x} ) \\ - \cos k_{y} \end{array} \! \! \!
\right ) \! \! \! \! \! \! \! & \left( \! \! \! \! \begin{array}{c} 4 - \tfrac{3}{2} \cos k_{y} \\
- \tfrac{3}{2} \cos (k_{y} \! - \! k_{x} ) \\ - \tfrac{1}{6} \cos k_{z} - \\
\tfrac{2}{3} \cos (k_{z} \! - \! k_{y} ) \\ - \tfrac{1}{6} \cos (k_{z}\! - \! k_{z} )
\end{array} \! \! \! \! \right) \! \! \! \! \!
& \tfrac{1}{\sqrt{3}} \! \! \left( \! \! \! \begin{array}{c} 2 \cos (k_{z} \! - \! k_{y} ) \\
- \cos k_{z} - \\ \cos ( k_{z} \! - \! k_{x} ) \end{array} \! \! \! \right) \\ \\
\sqrt{\tfrac{2}{3}} \! \! \left ( \! \! \! \begin{array}{c} \cos (k_{z} \! - \! k_{x}) \\ -
\cos k_{z} \end{array} \! \! \! \right) \! \! \! \! \! \! \! &
\tfrac{1}{\sqrt{3}} \! \! \left( \! \! \! \begin{array}{c} 2 \cos (k_{z} \! - \! k_{y} ) \\
- \cos k_{z} - \\ \cos ( k_{z} \! - \! k_{x} ) \end{array} \! \! \! \right) \! \! \! \! \! &
\tfrac{4}{3} \! \! \left( \! \! \! \! \begin{array}{c}
3 - \cos k_{z} - \\ \cos (k_{z} \! - \! k_{x}) - \\
\cos (k_{z} \! - \! k_{y} ) \end{array} \! \! \! \! \right )
\end{array} \! \! \! \!
\right ]\end{aligned}$$
![\[fig:Fig9\] Normalized root mean square (RMS) deviation shown versus $\log_{10}L$ for the three dimensional tetrahedral crystal. The inset is a graph of the normalized RMS deviations, again plotted with respect to $\log_{10}L$, with the horizontal line indicating the extrapolated $\delta_{\mathrm{RMS}}$ in the thermodynamic limit.](priourlatfig11.eps){width=".49\textwidth"}
The three dimensional tetrahedral lattice is locally stiff even with only nearest-neighbor couplings taken into account, and the rigidity inherent in the tetrahedral lattice geometry is sufficient to preserve long-range crystalline order, as may be seen in Figure \[fig:Fig9\] which displays the normalized mean square deviation versus the system size $L$. The inset is a semi-logarithmic plot with the horizontal axis extending over three decades of system sizes. The saturation of the normalized RMS displacement with increasing $L$ is evident in both of the graphs, and in the thermodynamic limit, is in the vicinity of $1.12$. With temperature dependence included, one will have $\delta_{\mathrm{RMS}} = 1.12 t^{1/2}$. Hence, the Lindemann criterion would give $t_{3\mathrm{D}}^{\mathrm{L}} = 0.0080$, compatible with the previous estimate which neglected correlations of the atomic displacements from equilibrium.
In inset (a) of Fig. \[fig:Fig10\], the density of states is shown for the simple cubic lattice geometry with the extended coupling scheme, and for the tetrahedral lattice in inset (b). For both systems, while other details of the density of states profiles differ, the curves tend swiftly to zero in the small eigenvalue regime, a hallmark of intact long range crystalline order in rigid three dimensional lattices.
![\[fig:Fig10\] Normalized Eigenvalue Density of States for the cubic crystal with nearest and next-nearest neighbor couplings (a), and the density of states profile for the tetrahedral lattice (b).](priourlatfig12.eps){width=".49\textwidth"}
Intraplanar Motion
==================
We first examine the case where motion perpendicular to the plan is forbidden, and atomic deviations from equilibrium are confined to the lattice plane. We consider various geometries, but first we consider a square (effectively a face-centered system), illustrated in Figure \[fig:Fig1\], where coupling to the four next nearest neighbors is taken into account. We then consider a locally rigid triangular lattice where each atom interacts with six nearest neighbors. In both the face-centered square and triangular systems, we find a logarithmic divergence with increasing system size in the mean square fluctuations about equilibrium.
For the periodic square geometry with the coupling scheme extended to next-nearest neighbors, the lattice energy to quadratic order is $$\begin{aligned}
E = \frac{K}{2} \! \!
\sum_{i,j=0}^{n-1} \left ( \! \! \begin{array}{l} \left [\hat{x} \! \cdot \! \left( \vec{\delta}_{i+1j}^{x} \!-\!
\vec{\delta}_{ij}^{x} \right) \! \right]^{2} \! + \! \left[ \hat{y} \! \cdot \!
\left( \vec{\delta}_{ij+1}^{y} \! - \! \vec{\delta}_{ij}^{y} \right) \! \right]^{2}
\\ \! + \! \left [ \! \tfrac{1}{\sqrt{2}} ( \hat{x} \! + \! \hat{y} ) \! \cdot \!
\left( \vec{\delta}_{i+1j+1}^{x} \! - \! \vec{\delta}_{ij}^{x} \right) \! \right]^{2} \\
\! + \! \left[ \! \tfrac{1}{\sqrt{2}} ( \hat{x} \! - \! \hat{y} ) \! \cdot \!
\left( \vec{\delta}_{i+1j-1}^{x} \! - \! \vec{\delta}_{ij}^{x} \right) \! \right ]^{2}
\end{array} \! \! \right )\end{aligned}$$ Operating in reciprocal space, one diagonalizes the $2 \times 2$ matrix $$\begin{aligned}
\left[ \! \! \begin{array}{ll} \left( \! \! \begin{array}{c} 2 - \cos k_{x} \\- \cos k_{x} \cos k_{y}
\end{array} \! \! \right) & \sin k_{x} \sin k_{y} \\
\sin k_{x} \sin k_{y} &
\left( \! \! \begin{array}{c} 2 - \cos k_{y} \\- \cos k_{x} \cos k_{y}
\end{array} \! \! \right ) \end{array} \! \! \right ]\end{aligned}$$ yielding the eigenvalues $$\begin{aligned}
\lambda^{\pm}_{\bf{k}} = \left( \begin{array}{c} 4 - \cos k_{x} - \cos k_{y} - 2 \cos k_{x} \cos k_{y} \end{array}
\right) \\ \nonumber
\pm \sqrt{\begin{array}{c} (\cos k_{x} - \cos k_{y} )^{2} + 4 \sin^{2} k_{x} \sin^{2} k_{y}
\end{array} }\end{aligned}$$
In the case of the triangular lattice with six fold coordination, one may also obtain analytical expressions for the mean square deviations. In real space, the harmonic approximation for the energy stored in the lattice is $$\begin{aligned}
E = \frac{K}{2} \sum_{i,j=0}^{n-1} \left ( \! \! \begin{array}{l} \left[ \hat{x} \! \cdot \!
\left( \vec{\delta}_{i+1j} - \vec{\delta}_{ij} \right) \right]^{2} + \\
\left [(\frac{1}{2}\hat{x} + \frac{\sqrt{3}}{2} \hat{y} ) \! \cdot \! \left( \vec{\delta}_{ij+1} - \vec{\delta}_{ij} \right) \right]^{2} + \\
\left [(\frac{1}{2}\hat{x} - \frac{\sqrt{3}}{2} \hat{y} ) \! \cdot \! \left( \vec{\delta}_{i+1j-1} - \vec{\delta}_{ij} \right) \right]^{2}
\end{array} \! \! \right ) \end{aligned}$$ Expressing the displacements in terms of Fourier components, one decouples the $x$ and $y$ degrees of freedom by diagonalizing the matrix $$\begin{aligned}
\left [ \! \! \! \begin{array}{cc} \left( \! \! \begin{array}{c} 3 \! - \! 2 \cos k_{x} \! - \! \\
\tfrac{1}{2}\cos k_{y} \! - \! \tfrac{1}{2} \cos [k_{y} \! - \! k_{x}] \end{array} \! \! \right ) \! \! &
\tfrac{\sqrt{3}}{2} \left ( \cos [k_{y} \! - \! k_{x}] \! - \! \cos k_{y} \right) \\ \\
\tfrac{\sqrt{3}}{2} \left ( \cos [k_{y} \! - \! k_{x}] \! - \! \cos k_{y} \right) \! \! & \left( \! \!
\begin{array}{c} 3 \! - \! \tfrac{3}{2} \cos k_{y} \\ \! - \! \tfrac{3}{2} \cos [k_{y} \! - \! k_{x}] \end{array} \! \! \right)
\end{array} \! \! \! \right]\end{aligned}$$ yields the eigenvalues $$\begin{aligned}
\! \! \! \! \! \lambda_{\bf{k}}^{\pm} = \left[ 3 - \cos k_{x} - \cos k_{y} -
\cos (k_{y} - k_{x} ) \right] \\ \nonumber
\pm \sqrt{\begin{array}{c} \cos^{2} k_{x} \! + \! \cos^{2} k_{y} \! + \! \cos^{2} (k_{y} \! - \! k_{x})
\! - \!
\cos k_{x} \cos k_{y} \\ \! - \! \cos k_{y} \cos (k_{y} \! - \! k_{x}) \! - \!
\cos (k_{y} \! - \! k_{x} ) \cos k_{x} \end{array}}\end{aligned}$$ For convenience in comparison with the analytical results in the harmonic approximation, we consider periodic boundary conditions in the crystal plane. We have also examined anchored lattices, where atoms at the periphery are prevented from moving, while those in the interior are free to move. For both the free and fixed boundary conditions, as in the three dimensional case, we obtain qualitatively similar results, and the same physical phenomena.
In Fig. \[fig:Fig13\] and Fig. \[fig:Fig14\], the normalized mean square deviation $\delta^{{n}}_{\mathrm{RMS}}$ is graphed with respect to the system size $L$ for the square lattice in the extended scheme and the triangular lattice, respectively. The overall behavior of the mean square deviations from equilibrium is qualitatively the same for both lattice geometries. In both cases, the main graph is semi-logarithmic with $(\delta_{\mathrm{RMS}}^{{n}})^{2}$ on the ordinate. The traces are linear to a very good approximation for all regimes of $L$ (i.e. for small, moderate, and large) shown, and the linearity is maintained for four decades of system sizes ranging from several to on the order of a few times $10^{4}$ lattice constants.
In Fig. \[fig:Fig13\] and Fig. \[fig:Fig14\], inset (a) is a standard plot, and the apparent saturation of the $\delta_{\mathrm{RMS}}^{n}$ curve is a hallmark of the slow loss of long-range crystalline order best seen on a semi-logarithmic graph. Inset (b) in Fig. \[fig:Fig13\] and Fig. \[fig:Fig4\] contains as semi-logarithmic plot with $\delta_{\mathrm{RMS}}^{{n}}$ \[ instead of $(\delta_{\mathrm{RMS}}^{{n}})^{2}$ \] on the ordinate axis. The curves plotted in this manner are not linear, and it is evident that the divergence of the fluctuations about equilibrium is actually somewhat slower than logarithmic; instead, it is $(\delta_{\mathrm{RMS}}^{{n}})^{2}$ which scales as $\textrm{Ln}(L)$.
![\[fig:Fig13\] Square of the normalized root mean square (RMS) deviation shown versus $\log_{10}L$ for the square lattice system with extended couplings. The solid line encompassing the open circular symbols is a strictly linear fit. Inset (b) is a semi-logarithmic graph of the normalized RMS deviations, plotted with respect to $\log_{10}L$.](priourlatfig13.eps){width=".49\textwidth"}
![\[fig:Fig14\] Square of the normalized root mean square (RMS) deviation shown versus $\log_{10}L$ for the triangular lattice. The solid line encompassing the open circular symbols is a strictly linear fit. Inset (a) is a standard plot of the RMS deviation with respect to system size $L$, while inset (b) is a semi-logarithmic graph of the normalized mean square deviations, plotted with respect to $\log_{10}L$.](priourlatfig14.eps){width=".49\textwidth"}
As in the case of the three dimensional systems, it is informative to examine the density of states, shown in the graph of Fig. \[fig:Fig15\] for the square lattice in the extended coupling scheme in panel (a) and the triangular lattice in panel (b) of Fig. \[fig:Fig15\]. Again, while details of the density of states profiles shown are peculiar to the lattice under consideration, the behavior in the regime of low eigenvalues is quite similar, and both curves tend to a finite value instead of dropping swiftly to zero as in the density of states for the rigid three dimensional lattices. The failure of the density of states to vanish in the small eigenvalue limit contributes to the slow divergence of $\delta_{\mathrm{RMS}}$ in $L$.
![\[fig:Fig15\] Normalized Eigenvalue Density of States for the face centered square lattice with motion confined to the lattice plane, depicted in panel (a), and for the triangular lattice in panel (b).](priourlatfig15.eps){width=".49\textwidth"}
Extra-planar Motion
===================
The locally stiff face-centered square and triangular lattices show the anticipated slow logarithmic divergence in system size. However since laboratory systems often are not vertically constrained, it is important to examine a scenario where motion perpendicular to the plane of the lattice may be considered. There is an important difficulty with single layer systems, in that motion perpendicular to the plane is not hindered since there are no restraining bonds with a directional component transverse to the plane of the layer.
However, by considering dual-layer geometries, it is possible to incorporate local stiffness with respect to perturbations that would push atoms above or below the lattice. We examine analogs of the simple cubic lattice, where we again use an extended coupling scheme to create local stiffness. On the other hand, we also consider a dual-layer tetrahedral lattice. Although the two lattice geometries achieve local stiffness in different ways, the similarities we find in thermodynamic behavior of the mean square atomic fluctuations suggest these characteristics would appear in the generic case as well.
Fig. \[fig:Fig16\] illustrates the structure of the dual-layer square lattice with an extended coupling scheme; the additional couplings between next nearest neighbors impart local stiffness to the system with respect to perturbations perpendicular to the planes of the square lattices.
Fig. \[fig:Fig17\] shows how the dual-layer systems is constructed as a caricature of the graphene lattice. The image labeled (a) is a schematic illustration of a single hexagonal cell in a graphene monolayer. The bonding shares similarities with that in a benzene ring with delocalized $\pi$ orbitals forming honeycomb networks of charge density above and below the plane occupied by the carbon atomic nuclei. The superimposed lattice work is a rigid network compatible with the symmetries of the graphene layer and set up to capture the rigidity of the hexagonal cells making up a sheet of graphene. With the honeycomb graphene pattern removed, the remaining lattice geometry and the labeling scheme for the crystal members is shown in Fig. \[fig:Fig18\].
![\[fig:Fig16\] Illustration of the periodic dual-layer square lattice with nearest and next-nearest neighbor coupling and labeling scheme; blue indices refer to the upper layer, while red indices pertain to the lower plane.](priourlatfig16.eps){width=".5\textwidth"}
![\[fig:Fig17\] Schematic representation of a coarse-grained super-structure for a graphene sheet](priourlatfig17a.eps "fig:"){width=".12\textwidth"} ![\[fig:Fig17\] Schematic representation of a coarse-grained super-structure for a graphene sheet](priourlatfig17b.eps "fig:"){width=".30\textwidth"}
![\[fig:Fig18\] Illustration of the periodic dual-layer triangular lattice and labeling scheme; blue indices refer to the upper layer, while red indices pertain to the lower plane.](priourlatfig18.eps){width=".5\textwidth"}
With the superscript I representing the lower plane and II indicating the upper plane, the lattice energy in the harmonic approximation is $$\begin{aligned}
E \! = \! \frac{K}{2} \! \sum_{i,j=0}^{n-1} \! \left( \! \!
\begin{array}{l}
\! \displaystyle \sum_{\alpha = \mathrm{I}}^{\mathrm{II}} \!
\left \{ \! \! \! \begin{array}{l} \left [ \hat{x} \! \cdot \! (\vec{\delta}_{i+1j}^{\alpha}
- \vec{\delta}_{ij}^{\alpha} ) \! \right]^{2} \! \! + \! \left[ \hat{y}
\! \cdot \! (\vec{\delta}_{ij+1}^{\alpha} -\vec{\delta}_{ij}^{\alpha}) \! \right]^{2}
\\ + \left[ \frac{1}{\sqrt{2}}(\hat{x} + \hat{y}) \! \cdot \! (\vec{\delta}_{i+1j+1}^{\alpha}
- \vec{\delta}_{ij}^{\alpha}) \! \right]^{2} + \\
\left[ \frac{1}{\sqrt{2}} ( \hat{x} - \hat{y} ) \cdot (\vec{\delta}_{i+1j-1}^{\alpha} \! - \! \vec{\delta}_{ij}^{\alpha} ) \! \right ]^{2}
\end{array} \! \! \right \} \\
+ \kappa_{z} \left \{ \! \! \begin{array}{l} \left[ \hat{z} \cdot \left ( \vec{\delta}_{ij}^{\mathrm{II}} -
\vec{\delta}_{ij}^{\mathrm{I}} \right ) \right ]^{2} \\ \! + \!
\left[ \tfrac{1}{\sqrt{2}}(\hat{x} \! + \! \hat{z} ) \! \cdot \!
\left( \vec{\delta}_{i+1j}^{\mathrm{II}} \! -\! \vec{\delta}_{ij}^{\mathrm{I}} \right) \right]^{2} \\
\! + \! \left[ \tfrac{1}{\sqrt{2}}(-\hat{x} \! + \! \hat{z} ) \! \cdot \!
\left( \vec{\delta}_{i-1j}^{\mathrm{II}} \! -\! \vec{\delta}_{ij}^{\mathrm{I}} \right) \! \right]^{2} \\
\! + \! \left[ \tfrac{1}{\sqrt{2}}(\hat{y} \! + \! \hat{z} ) \! \cdot \!
\left( \vec{\delta}_{ij+1}^{\mathrm{II}} \! -\! \vec{\delta}_{ij}^{\mathrm{I}} \right) \! \right]^{2} \\
\! + \! \left[ \tfrac{1}{\sqrt{2}}(-\hat{y} \! + \! \hat{z} ) \! \cdot \!
\left( \vec{\delta}_{ij-1}^{\mathrm{II}} \! -\! \vec{\delta}_{ij}^{\mathrm{I}} \right) \! \right]^{2} \end{array}
\! \! \right \}
\end{array}
\! \!
\! \! \right)\end{aligned}$$
The corresponding complex Hermitian 6$\times$6 matrix to be diagonalized has the form $$\begin{aligned}
\left[ \! \begin{array}{cccccc}
a_{\mathrm{I} x \mathrm{I} x} & a_{\mathrm{I} x \mathrm{I} y} &
a_{\mathrm{I} x \mathrm{I} z} & a_{\mathrm{I} x \mathrm{II} x} &
a_{\mathrm{I} x \mathrm{II} y} & a_{\mathrm{I} x \mathrm{II} z} \\
a_{\mathrm{I} y \mathrm{I}x} & a_{\mathrm{I} y \mathrm{I} y} &
a_{\mathrm{I} y \mathrm{I} z} & a_{\mathrm{I} y \mathrm{II} x} &
a_{\mathrm{I} y \mathrm{II} y} & a_{\mathrm{I} y \mathrm{II} z} \\
a_{\mathrm{I} z \mathrm{I} x} & a_{\mathrm{I} z \mathrm{I} y} &
a_{\mathrm{I} z \mathrm{I} z} & a_{\mathrm{I} z \mathrm{II} x} &
a_{\mathrm{I} z \mathrm{II} y} & a_{\mathrm{I} z \mathrm{II} z} \\
a_{\mathrm{II} x \mathrm{I}x} & a_{\mathrm{II} x \mathrm{I} y} &
a_{\mathrm{II} x \mathrm{I} z} & a_{\mathrm{II} x \mathrm{II} x} &
a_{\mathrm{II} x \mathrm{II} y} & a_{\mathrm{II} x \mathrm{II} z} \\
a_{\mathrm{II} y \mathrm{I} x} & a_{\mathrm{II} y \mathrm{I} y} &
a_{\mathrm{II} y \mathrm{I} z} & a_{\mathrm{II} y \mathrm{II} x} &
a_{\mathrm{II} y \mathrm{II} y} & a_{\mathrm{II} y \mathrm{II} z} \\
a_{\mathrm{II} z \mathrm{I}x} & a_{\mathrm{II} z \mathrm{I} y} &
a_{\mathrm{II} z \mathrm{I} z} & a_{\mathrm{II} z \mathrm{II} x} &
a_{\mathrm{II} z \mathrm{II} y} & a_{\mathrm{II} z \mathrm{II} z}
\end{array} \! \right ] = \left[ \! \begin{array}{cc} \hat{A} & \hat{B} \\
\hat{B}^{\dagger} & \hat{A} \end{array} \! \right]\end{aligned}$$ where $\hat{A}$ and $\hat{B}$ are $3 \times 3$ matrices and $\hat{B}^{\dagger}$ is the Hermitian conjugate of $\hat{B}$. The sub-matrices $\hat{A}$ and $\hat{B}$ are given by $$\begin{aligned}
\hat{A} = \! K \left[ \! \! \! \! \begin{array}{ccc} \left(
\! \! \begin{array}{c} \kappa_{z} \! + \! 4 \! - \! 2 \cos k_{x} \\ - 2 \cos k_{x} \cos k_{y}
\end{array} \! \! \! \right) \! \! \! \! & \! \! \! \! 2 \sin k_{x} \sin k_{y} \! \! \! \!
& \! \! \! \! 0
\\ \\ 2 \sin k_{x} \sin k_{y} \! \! \! \! & \! \! \! \! \left(
\! \! \begin{array}{c} \kappa_{z}\! + \! 4 \! - \! 2 \cos k_{y} \\ - 2 \cos k_{x} \cos k_{y}
\end{array} \! \! \! \right) \! \!\! \! & \! \! \! \! 0 \\ \\
0 \! \! \! \! & \! \! \! \! 0 \! \! \! \! & 3 \kappa_{z} \end{array} \! \! \! \right]\end{aligned}$$ where $\kappa_{z} \equiv K_{z}/K$ for $\hat{A}$ and $$\begin{aligned}
\hat{B} = \! K_{z} \left[ \! \! \! \begin{array}{ccc} -\cos k_{x} \! & \! 0 \! & \!
\! -i\sin k_{x} \\ \\
0 \! & \! - \cos k_{y} \! & \! -i \sin k_{y} \\ \\ -i \sin k_{x} \! & \!
-i \sin k_{y} \! & \! - (1 \! + \! \cos k_{x} \! + \! \cos k_{y} ) \end{array} \! \! \right]\end{aligned}$$ for $\hat{B}$ where the six eigenvalues for each wave-number pair are calculated numerically with code available in the EISPACK linear algebraic library for diagonalizing complex Hermitian matrices.
On the other hand a locally stiff dual-layer system which may be regarded as a section of the three dimensional tetrahedral lattice such that the upper and lower layers are triangular lattices with connections between the layers. The dual-layer lattice structure based on the tetrahedral geometry is illustrated in Figure \[fig:Fig18\]. The vertices of the upper layer are positioned above the centers of the triangles in the lower layer with bonds extending from atoms in the upper layer to each of the corners of the triangle below such that each atom in the dual-layer system is a member of a rigid tetrahedron; the result is a locally stiff layer, as in the dual-layer square lattice extended model, but with a very different geometric structure.
The lattice energy for the dual-layer tetrahedral system to quadratic order in the displacements $\vec{\delta_{ij}}^{\mathrm{I}}$ and $\vec{\delta_{ij}}^{\mathrm{II}}$ has the form $$\begin{aligned}
E = \frac{K}{2} \! \! \sum_{i,j=0}^{n-1} \! \left( \! \! \!
\begin{array}{l} \\ \! \displaystyle {\sum_{\alpha = \mathrm{I}}^{\mathrm{II}}}
\left \{ \! \! \!
\begin{array}{c} [ \hat{x} \! \cdot \! (\vec{\delta}_{i+1j}^{\alpha} \! - \! \vec{\delta}_{ij}^{\alpha} ) \! ]^{2} + \\
\left [ \left( \tfrac{\hat{x}}{2} \! + \!
\tfrac{ \sqrt{3} \hat{y}}{2} \right) \! \cdot \! ( \vec{\delta}_{ij+1}^{\alpha} -
\vec{\delta}_{ij}^{\alpha} ) \right ]^{2} + \\
\left[ \left( \tfrac{\hat{x}}{2} - \tfrac{ \sqrt{3} \hat{y}}{2} \right) \cdot ( \vec{\delta}_{i+1j-1}^{\alpha}
\! - \! \vec{\delta}_{ij}^{\alpha} ) \right]^{2} \end{array} \! \! \right \}
\!+ \! \\ \\ \! \kappa_{z} \!
\left \{ \! \! \! \begin{array}{c} \left[ \left( \tfrac{\hat{x}}{2} \! + \! \tfrac{\hat{y}}{2 \sqrt{3}}
\! + \!
\sqrt{\tfrac{2}{3}} \hat{z} \right) \cdot ( \vec{\delta}_{ij}^{\mathrm{II}}
- \vec{\delta}_{ij}^{\mathrm{I}} ) \right]^{2} + \\
\left[ \! \left(\! - \tfrac{\hat{x}}{2} \! + \! \tfrac{\hat{y}}{2 \sqrt{3}} \! + \!
\sqrt{\tfrac{2}{3}} \hat{z}
\right) \! \cdot \! ( \vec{\delta}_{ij}^{\mathrm{II}} \! - \! \vec{\delta}_{i+1j}^{\mathrm{I}})
\! \right]^{2} \! + \! \\
\left[ \! \left( \! -\tfrac{\hat{y}}{\sqrt{3}} \! + \! \sqrt{\tfrac{2}{3}} \hat{z} \right)
\! \cdot \! ( \vec{\delta}_{ij}^{\mathrm{II}} \! - \! \vec{\delta}_{ij+1}^{\mathrm{I}} )
\! \right]^{2} \end{array}
\! \! \! \right \} \end{array} \! \! \! \! \right)\end{aligned}$$
Expressing the lattice energy in terms of Fourier components leads to a $6 \times 6$ matrix to be diagonalized, which may again be written in terms of $3 \times 3$ submatrices as $\left [ \begin{array}{cc} \hat{A} & \hat{B} \\
\hat{B}^{\dagger} & \hat{A} \end{array} \right]$, where $$\begin{aligned}
\hat{A} = K \! \left[ \! \! \! \begin{array}{ccc} \left( \! \!
\begin{array}{c} \tfrac{\kappa_{z}}{2} \! + \! 3
\!- \! 2 \cos k_{x} \\ -\tfrac{1}{2} \cos (k_{y} \! - \! k_{x} )
\\ - \tfrac{1}{2} \cos k_{y}
\end{array} \! \! \! \right) \! \! & \! \! \tfrac{\sqrt{3}}{2} \left( \! \! \begin{array}{c}
\cos (k_{y} \! - \! k_{x}) \\ - \cos k_{y} \end{array} \! \! \right) \! \! & \! \! 0 \\ \\
\tfrac{\sqrt{3}}{2} \left( \! \! \begin{array}{c}
\cos (k_{y} \! - \! k_{x}) \\ - \cos k_{y} \end{array} \! \! \right) \! \!
& \! \! \left( \! \!
\begin{array}{c} \tfrac{\kappa_{z}}{2} \! + \! 3 \! - \! \tfrac{3}{2} \cos (k_{y} ) \\
- \tfrac{3}{2} \cos (k_{y} -k_{x} ) \end{array} \! \! \!\right) \! \! & \! \! 0 \\ \\
0 \! \! & \! \! 0 \! \! & \! \! 2 \kappa_{z} \end{array} \! \! \right]\end{aligned}$$ where again $\kappa_{z} \equiv K_{z}/K$ for the sub-matrix $\hat{A}$ and $$\begin{aligned}
\hat{B} \! = \! \tfrac{K_{z}}{4} \! \! \left[ \! \! \! \begin{array}{ccc} -(1 \!+ \! e^{-i k_{x}} \!) &
\tfrac{1}{\sqrt{3}} \! (e^{-ik_{x}}\! - \! 1 \! ) \! \! & \! \! \sqrt{\tfrac{8}{3}} (e^{-ik_{x}} \! - \! 1 \!
) \\ \\
\tfrac{1}{\sqrt{3}} (e^{-ik_{x}} \! - \! 1 \!) & -\tfrac{1}{3} \! \! \left ( \! \!
\begin{array}{c} 1 \! + \! e^{-ik_{x}} \\ + 3 e^{-ik_{y}} \end{array} \! \! \! \right )
\! \! & \! \! \tfrac{\sqrt{8}}{3} \! \! \left ( \! \! \begin{array}{c} 2 e^{-ik_{y}} - \\ e^{-i k_{x}} -1
\end{array} \! \! \! \right ) \\ \\ \sqrt{\tfrac{8}{3}} (e^{-i k_{x}} \! - \! 1 \!) & \frac{\sqrt{8}}{3}
\! \left( \! \! \begin{array}{c} 2 e^{-i k_{y}} - \\ e^{-i k_{x}} \! - \! 1 \end{array} \! \! \! \right)
\! \! & \! \! -\tfrac{8}{3} \! \! \left( \! \! \begin{array}{c} 1 \! + \!
e^{-ik_{z}} \\ - \! e^{-ik_{y}} \end{array} \! \! \! \right )
\end{array} \! \! \! \! \right]\end{aligned}$$ for the sub-matrix $\hat{B}$.
The use of a dual-layer lattice geometry to provide resistance to transverse deviations is not sufficient to prevent a rapid divergence in $\delta_{\mathrm{RMS}}$ with increasing system size $L$. Whereas thermally averaged mean square fluctuations grew very slowly (i.e. logarithmically) when the atomic motions are confined to the lattice plane, $\delta_{\mathrm{RMS}}$ for the dual-layer systems increases linearly with $L$; ultimately, it is not difficult for the sheet to bend and flex in the presence of thermal fluctuations in spite of its locally stiff characteristics. The diverging mean square deviations from equilibrium and other thermodynamic characteristics of the the dual-layer square lattice in the extended coupling scheme and its counterpart based on a tetrahedral geometry are examined, with consideration given to the effects of increasing $L$ and variations in the inter-layer coupling strength.
The graph of the normalized RMS displacement, shown in Fig. \[fig:Fig20\] for the dual-layer square lattice with an extended coupling pattern shows a dependence on systems size which is an asymptotically linear growth in $L$. $\delta_{\mathrm{RMS}}$ curves for several values of the interlayer coupling $K_{z}$ are shown; the intra-layer coupling is taken to be unity, so $K_{z}$ is effectively expressed in units of $K$. Although the relative interlayer couplings range over three orders magnitude, there is little variation of the curves, especially for $K_{z} =
\left \{ 0.1, 1.0, 10.0 \right \}$. Similarly, the $\delta_{\mathrm{RMS}}$ curves ultimately vary linearly in the system size $L$ with little dependence on the relative magnitude of $K_{z}$, which again is expressed in units of $K$.
Again, it is useful to examine the density of states for the eigenvalues in the case of the locally rigid dual-layer systems, which are richer than the density of states profiles corresponding to rigid three dimensional lattices or those of the single layer geometries with atomic fluctuations confined to intra-planar motion. Although details of the density of states profiles for the two geometries differ, the both curves show a divergence of the density of states with decreasing eigenvalue magnitude, whereas the density states remained constant in the case of the planar systems with exclusively intra-planar motion and vanished altogether for the rigid three dimensional systems. Inset (a) of Fig. \[fig:Fig19\] show the DOS for the dual-layer square lattice, while inset (b) is a graph of the density of states for the locally stiff tetrahedral system. The DOS cusp for both lattices at the zero eigenvalue point is responsible for the rapid divergence of $\delta_{\mathrm{RMS}}$ with increasing $L$.
![\[fig:Fig20\] Normalized mean square displacements for $K_{z}$ = 0.01, 0.03, 0.1, 1.0, and 10.0 for the dual-layer square lattice with an extended coupling scheme, where $K_{z}$ is in units of the intralayer coupling $K$. The inset is a closer view of the RMS curves.](priourlatfig19.eps){width=".49\textwidth"}
![\[fig:Fig21\] Normalized mean square displacements for $K_{z}$ = 0.01, 0.1, 1.0, and 10.0 for the dual-layer tetrahedral lattice where $K_{z}$ is in units of the triangular lattice intra-layer coupling $K$. The inset is a closer view of the RMS curves.](priourlatfig20.eps){width=".49\textwidth"}
![\[fig:Fig19\] Normalized Eigenvalue Density of States for the dual-layer cubic system with with nearest and next-nearest neighbor interactions (a) and for the dual-layer locally stiff lattice based on a tetrahedral lattice geometry (b) for system size $L = 5001$.](priourlatfig21.eps){width=".49\textwidth"}
While adjusting the interaction between the layers to enhance the resistance to local transverse perturbations has little effect on the mean square fluctuations for large values of $L$, the eigenvalue density of states evolves as the interplanar to intraplanar coupling ratio $\kappa_{z} = K_{z}/K$ is modified. Density of states profiles for $\kappa_{z}$ values ranging from $\kappa_{z} = 0.1$ to $\kappa_{z} = 3.0$ are shown for the dual-layer square system with and extended coupling pattern in Fig. \[fig:Fig22\] and for the tetrahedral counterpart in Fig. \[fig:Fig23\].
Density of states profiles are shown for strong ($\kappa_{z} = 3.0$) and moderate ($\kappa_{z} = 1.0$) values of the the coupling ratio in insets (a) and (b) of Fig. \[fig:Fig22\] and Fig. \[fig:Fig23\], and there is little change in the DOS curve in the low eigenvalue regime. On the other had, as $\kappa_{z}$ decreases further and the interplanar coupling begins to fall below parity with that in the plane, the eigenvalue density of states profiles begin to change more drastically, as may be seen in panels (b) and (c) of Fig. \[fig:Fig22\] for the dual-layer square system and Fig. \[fig:Fig23\] for the dual-layer tetrahedrally based geometry; the distribution in both cases rapidly grows narrower with decreasing $\kappa_{z}$. Although the two lattice geometries are very distinct, similar (and likely generic to locally rigid dual-layer lattices) trends may be seen in the DOS profiles in the regime of low eigenvalues as $\kappa_{z}$ is reduced.
![\[fig:Fig22\] Normalized density of states profiles for $K_{z}$ = 3.0, 1.0, 0.3, and 0.1 for the dual-layer square lattice with an extended coupling scheme where $K_{z}$ is in units of the triangular lattice intra-layer coupling $K$.](priourlatfig22.eps){width=".49\textwidth"}
![\[fig:Fig23\] Normalized density of states profiles for $K_{z}$ = 3.0, 1.0, 0.3, and 0.1 for the dual-layer tetrahedral lattice where $K_{z}$ is in units of the triangular lattice intra-layer coupling $K$.](priourlatfig23.eps){width=".49\textwidth"}
Coupling to a Substrate
=======================
We incorporate an attractive interaction with a flat substrate by including an additional harmonic potential acting on the lower members of the tetrahedral and square extended coupling dual-layer systems. We take the attraction to depend only on the atomic shift $\delta^{\mathrm{I}z}_{ij}$ above the planar system, and the additional term hence has the form $\frac{K_{s}}{2} \left( \delta_{ij}^{Iz}
\right)^{2}$
Figure \[fig:Fig24\] shows the effect of the substrate coupling on $\delta_{\mathrm{RMS}}$ in the case of the dual-layer square system with an extended coupling scheme. The graph, which shows mean square deviation curves for a wide range of $K_{s}$ values, indicates the capacity of even a very mild substrate coupling to suppress thermally induced undulations in the dual-layer sheet. Similarly, for the tetrahedrally based dual-layer lattice geometry, an attractive interaction with a substrate considerably reduces fluctuations transverse to the lattice planes, preventing a rapid divergence of $\delta_{\mathrm{RMS}}$. The mean square deviation curves are shown in Figure \[fig:Fig25\].
We also examine the effect of an attractive substrate coupling on the density of states profiles, and results are displayed in Fig. \[fig:Fig26\] for a range of substrate coupling constants $K_{s}$. With increasing $K_{s}$, a salient trend is the opening of a separation between the sharp cusp and the zero eigenvalue mark on the abscissa. The migration of the maximum formerly at the zero eigenvalue point to a peak at a larger eigenvalue is associated with a sharp reduction of the mean square fluctuations about equilibrium, and the lattice is better able to withstand transverse fluctuations.
The presence of a flat substrate plays a very important role in dictating the overall structure and amplitude of ripples in the dual-layer geometries we report on here. This result is in accord with recent experiments on graphene sheets deposited on cleaved mica substrates [@Lui], where the careful preparation of flat substrates significantly dampens the ripple amplitude, whereas much larger undulations are seen with sheets attached to substrates with poorer control over the flatness.
To determine which length scales are associated with the strongest contributions to the thermally averaged mean square deviations about equilibrium, we have prepared histograms showing the relative contribution to $\delta_{\mathrm{RMS}}$ versus inverse wave-vector magnitude, with the latter providing a length scale. Apart from a significant diminution in the height of thermally excited undulations in the dual-layered sheets, we also find a considerable reduction in their typical wavelength. In figure \[fig:Fig27\], for the dual-layer tetrahedral lattice in the absence of a substrate couplings, the dominant contribution to $\delta_{\mathrm{RMS}}$ comes from large length scales comparable to the scale of the lattice. However the picture changes with the activation of a finite substrate coupling as may be seen in the inset with the peak height at the minimal wave-number decreasing with increasing $K_{s}$. Moreover, as may be seen in Figure \[fig:Fig28\], introducing even a weak anchoring to the foundation below immediately creates a strong peak in the short wavelength regime, skewing the size of thermally induced ripples toward smaller length scales.
![\[fig:Fig24\] For the dual-layer square lattice in the extended scheme, the main figure and the inset are graphs of mean square fluctuations versus system size $L$ for a range of substrate couplings $K_{s}$, with $K_{s}$ given in units of the inter-planar and intra-planar coupling constant (both equal to $K$). The symbol legend on the main plot also pertains to the inset](priourlatfig24.eps){width=".49\textwidth"}
![\[fig:Fig25\] For the dual-layer tetrahedral lattice, the main figure and the inset are graphs of mean square fluctuations versus system size $L$ for a range of substrate couplings $K_{s}$, with $K_{s}$ given in units of the inter-planar and intra-planar coupling constant (both equal to $K$). The symbol legend on the main plot also pertains to the inset](priourlatfig25.eps){width=".49\textwidth"}
![\[fig:Fig26\] Normalized Density of States for the Dual layer system for various substrate coupling strengths $K_{s}$. Panels (a), (b), (c), and (d) show the density of states for $K_{s}$ equal to $0.0$, $0.1$, $0.5$, and $1.0$ respectively.](priourlatfig26.eps){width=".49\textwidth"}
![\[fig:Fig27\] Normalized Eigenvalue Density of States for the dual-layer triangular system with system size $L = 5001$.](priourlatfig27.eps){width=".48\textwidth"}
![\[fig:Fig28\] Normalized Eigenvalue Density of States for the dual-layer triangular system with system size $L = 5001$.](priourlatfig28.eps){width=".48\textwidth"}
In conclusion, we have examined thermally induced fluctuations about equilibrium in two and three dimensional crystalline solids with a local bonding scheme. While long-range crystalline order may exist in three dimensional crystal lattices, some geometries (e.g. the simple cubic lattice) are not rigid when only nearest-neighbor couplings are taken into account, and an extended coupling scheme is needed to prevent the divergence of mean square fluctuations with increasing system size $L$.
In two dimensional lattices, we find RMS fluctuations to increase at a very slow (logarithmic) rate when motion is confined to the lattice plane. On the other hand, when transverse motion is permitted, thermal fluctuations are very effective in bringing about significant vertical displacements of particles which contribute to rapidly growing deviations from equilibrium, and $\delta_{\mathrm{RMS}}$ ultimately diverges at a linear rate in $L$. The asymptotically linear divergence in the mean square deviations from equilibrium is insensitive to the strength of the interlayer coupling; $\delta_{\mathrm{RMS}}$ values appear to converge and eventually show identical behavior with increasing system size whether the coupling $K_{z}$ established between the layers to provide local stiffness is quite weak or very strong relative to the bonding $K$ between atoms in the same layer.
Introducing a coupling $K_{s}$ to a flat substrate very effectively hinders transverse fluctuations in two dimensional crystal lattices, even in the coupling is very weak, and reflects the importance of a substrate in shaping the characteristics of ripples set up by thermal fluctuations by inhibiting transverse deviations. An attractive coupling to a fixed substrate also reduces the typical lateral length scale or wavelength of thermally excited undulations in lattices bound to a substrate. These tendencies are consistent with recent experimental observations that control over the flatness of the underlying surface is directly related to the amplitude and length scale of thermally induced ripples.
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---
abstract: 'Simultaneous Localization And Mapping (SLAM) is a fundamental problem in mobile robotics. While point-based SLAM methods provide accurate camera localization, the generated maps lack semantic information. On the other hand, state of the art object detection methods provide rich information about entities present in the scene from a single image. This work marries the two and proposes a method for representing generic objects as quadrics which allows object detections to be seamlessly integrated in a SLAM framework. For scene coverage, additional dominant planar structures are modeled as infinite planes. Experiments show that the proposed points-planes-quadrics representation can easily incorporate Manhattan and object affordance constraints, greatly improving camera localization and leading to semantically meaningful maps. The performance of our SLAM system is demonstrated in <https://youtu.be/dR-rB9keF8M>.'
author:
- Mehdi Hosseinzadeh
- Yasir Latif
- Trung Pham
- Niko Suenderhauf
- Ian Reid
bibliography:
- 'refs.bib'
title: 'Structure Aware SLAM using Quadrics and Planes[^1]'
---
[^1]: Supported by the ARC Laureate Fellowship FL130100102 to IR and the ACRV CE140100016.
|
---
abstract: 'Our analysis of archival VLBI data of PSR 0834+06 revealed that its scintillation properties can be precisely modelled using the inclined sheet model [@2014MNRAS.442.3338P], resulting in two distinct lens planes. These data strongly favour the grazing sheet model over turbulence as the primary source of pulsar scattering. This model can reproduce the parameters of the observed diffractive scintillation with an accuracy at the percent level. Comparison with new VLBI proper motion results in a direct measure of the ionized ISM screen transverse velocity. The results are consistent with ISM velocities local to the PSR 0834+06 sight-line (through the Galaxy). The simple 1-D structure of the lenses opens up the possibility of using interstellar lenses as precision probes for pulsar lens mapping, precision transverse motions in the ISM, and new opportunities for removing scattering to improve pulsar timing. We describe the parameters and observables of this double screen system. While relative screen distances can in principle be accurately determined, a global conformal distance degeneracy exists that allows a rescaling of the absolute distance scale. For PSR B0834+06, we present VLBI astrometry results that provide (for the first time) a direct measurement of the distance of the pulsar. For targets where independent distance measurements are not available, which are the cases for most of the recycled millisecond pulsars that are the targets of precision timing observations, the degeneracy presented in the lens modelling could be broken if the pulsar resides in a binary system.'
author:
- |
Siqi Liu$^{1,3}$[^1], Ue-Li Pen$^{1,2}$[^2], J-P Macquart$^{4}$[^3], Walter Brisken$^{5}$[^4], Adam Deller$^{6}$[^5]\
$^1$ Canadian Institute for Theoretical Astrophysics, University of Toronto, M5S 3H8 Ontario, Canada\
$^2$ Canadian Institute for Advanced Research, Program in Cosmology and Gravitation\
$^3$ Department of Astronomy and Astrophysics, University of Toronto, M5S 3H4, Ontario, Canada\
$^4$ ICRAR-Curtin University of Technology, Department of Imaging and Applied Physics, GPO Box U1978, Perth, Western Australia 6102, USA\
$^5$ National Radio Astronomy Observatory, P.O. Box O, Socorro, NM 87801, USA\
$^6$ ASTRON, the Netherlands Institute for Radio Astronomy, Postbus 2, 7990 AA, Dwingeloo, The Netherlands\
bibliography:
- 'distance.bib'
title: Pulsar lensing geometry
---
\[firstpage\]
Pulsars: individual (B0834+06) – scattering – waves – magnetic: reconnection – techniques: interferometric – ISM: structure
Introduction
============
Pulsars have long provided a rich source of astrophysical information due to their compact emission and predictable timing. One of the least well constrained parameters for most pulsars is their distance. For some pulsars, timing parallax or VLBI parallax has resulted in direct distance determination. For most pulsars, the distance is a major uncertainty for precision timing interpretations, including mass, moment of inertia [@2006Sci...314...97K; @2012hpa..book.....L], and gravitational wave direction [@boyle2012].
Direct VLBI observations of PSR B0834+06 show multiple images lensed by the interstellar plasma. Combining the angular positions and scintillation delays, the authors [@2010ApJ...708..232B] (hereafter B10) published the derived effective distance (defined in Section \[21\]) of $1168\pm 23$ pc for apexes on the main scattering axis. This represents a precise measurement compared to all other attempts to derive distances to this pulsar. This effective distance is a combination of pulsar-screen and earth-screen distances, and does not allow a separate determination of the individual distances. A binary pulsar system would in principle allow a breaking of this degeneracy [@2014MNRAS.442.3338P]. One potential limitation is the precision to which the lensing model can be understood.
In this paper, we examine the geometric nature of the lensing screens. In B10, VLBI astrometric mapping directly demonstrated the highly collinear nature of a single dominant lensing structure. First hints of single plane collinear dominated structure had been realized in @2001ApJ...549L..97S. While the nature of these structures is already mysterious, for this pulsar, in particular, the puzzle is compounded by an offset group of lenses whose radiation arrive delayed by 1 ms relative to the bulk of the pulsar flux. The mysterious nature of lensing questions any conclusions drawn from scintillometry as a quantitative tool [@2014MNRAS.440L..36P].
Using archival data we demonstrate in this paper that the lensing screen consists of nearly parallel linear refractive structures, in two screens. The precise model confirms the one dimensional nature of the scattering geometry, and thus the small number of parameters that quantify the lensing screen.
The paper is structured as follows. Section \[sec:lensing\] overviews the inclined sheet lensing model, and its application to data. Section \[sec:astrometry\] presents new VLBI proper motion and distance measurements to this pulsar. Section \[sec:discussions\] describes the interpretation of the lensing geometry and possible improvements on the observation. We conclude in Section \[sec:conclusions\].
Lensing {#sec:lensing}
=======
In this section, we map the archival data of PSR 0834+06 onto the grazing incidence sheet model. The folded sheet model is qualitatively analogous to the reflection of a light across a lake as seen from the opposite shore. In the absence of waves, exactly one image forms at the point where the angle of incidence is equal to the angle of reflection. In the presence of waves, one generically sees a line of images above and below the unperturbed image. The grazing angle geometry simplifies the lensing geometry, effectively reducing it from a two dimensional problem to one dimension. The statistics of such reflections is sometimes called glitter, and has many solvable properties [@LonguetHiggins1960]. This is illustrated in Fig. \[fig:water\_reflection\].
{width="1.0\linewidth"}
A similar effect occurs when the observer is below the surface. Two major distinctions arise: 1. the waves can deform the surface to create caustics in projection. Near caustics, Snell’s law can lead to highly amplified refraction angles[@2006ApJ...640L.159G]. 2. due to the odd image theorem, each caustic leads to two images. In an astrophysical context, the surface could be related to magnetic reconnection sheets [@2015MNRAS.450.3201B], which have finite widths to regularize these singularities. Diffusive structures have Gaussian profiles, which were analysed in @2012MNRAS.421L.132P. The lensing details differ for convergent (under-dense) vs divergent (over-dense) lenses, first considered by @1998ApJ...496..253C.
The typical interstellar electron density $\sim 0.02$ cm$^{-3}$ is insufficient to deflect radio waves by the observed $\sim$ mas bending angles. At grazing incidence, Snell’s law results in an enhanced bending angle, which formally diverges. Magnetic discontinuities generically allow transverse surface waves, whose restoring force is the difference in Alfvén speed on the two sides of the discontinuity. This completes the analogy to waves on a lake: for sufficiently inclined sheets the waves will appear to fold back onto themselves in projection on the sky. At each fold caustic, Snell’s law diverges, leading to enhanced refractive lensing. The divergence is cut off by the finite width of the sheet. The generic consequence is a series of collinear images. Each projected fold of the wave results in two density caustics. Each density caustic leads to two geometric lensing images, for a total of 4 images for each wave inflection. The two geometric images in each caustic are separated by the characteristic width of the sheet. If this is smaller than the Fresnel scale, the two images become effectively indistinguishable. The geometry of the inclined refractive lens is shown in Fig. \[fig:sheetgeom\].
{width="100.00000%"}
A large number of sheets might intersect the line of sight to any pulsar. Only those sufficiently inclined would lead to caustic formation. Empirically, some pulsars show scattering that appears to be dominated by a single sheet, leading to the prominent inverted arclets in the secondary spectrum of the scintillations [@2001ApJ...549L..97S].
Archival data of B0834+06 {#21}
-------------------------
Our analysis is based on the apex data selected from the secondary spectrum of pulsar B0834+06 in B10, which was observed as part of a 300 MHz global VLBI project on 2005 November 12, with the GBT (GB), Arecibo (AR), Lovell and Westerbork (WB) telescopes. The GB-AR and AR-WB baselines are close to orthogonal and of comparable lengths, resulting in relatively isotropic astrometric positions. Information from each identified apex includes delay $\tau$, delay rate (differential frequency $f_{\rm D}$), relative Right Ascension, $\Delta\alpha$, and relative declination, $\Delta\delta$. Data of each apex are collected from four dual circular polarization $8$ MHz wide sub-bands spanning the frequency range $310.5$–$342.5$ MHz. As described in B10, the inverse parabolic arclets were fitted to positions of their apexes, resulting in a catalogue of apexes in each sub-band, each with delay and differential frequency. In this work, we first combine the apexes across sub-bands, resulting in a single set of images. We focus on the southern group with negative differential frequency: this grouping appears as a likely candidate for a double-lensing screen. However, two groups (with negative differential frequency) appear distinct in both the VLBI angular positions and the secondary spectra. We divide the apex data with negative $f_{\rm D}$ into two groups: in one group, time delays range from $0.1$ ms to $0.4$ ms, which we call the $0.4$ ms group; and in the other group, time delay at about $1$ ms, which we call the $1$ ms group. In summary, the $0.4$ ms group contains $10$ apexes in the first two sub-bands, and $14$ apexes in the last two sub-bands; the $1$ ms group, contains $5$, $6$, $5$ and $4$ apexes in the four sub-bands subsequently, with center frequency of each band $f_{\rm band}=314.5, 322.5, 330.5$ and $
338.5$ MHz. The apex positions in the secondary spectrum are shown in Fig. \[fig:apex\_pos\].
![Distribution of the apex positions in the sub-band centred at 314.5 MHz. The apexes that belong to the 1 and 0.4 ms groups are marked.[]{data-label="fig:apex_pos"}](apex_pos.pdf){width="\linewidth"}
We select the equivalent apexes from four sub-bands. To match the same apexes in different sub-bands, we scale the differential frequency in different sub-bands to $322.5$ MHz, by $f_{\rm D} (322.5/f_{\rm band})$ MHz. A total of $9$ apexes from the $0.4$ ms group and $5$ apexes from the $1$ ms group, were mapped. This results in an estimation for the mean referenced frequency $f=322.5$ MHz and a standard deviation among the sub-bands, listed in Table \[table:apex\]. The $f_{\rm D}$, $\tau$, $\Delta\alpha$ and $\Delta\delta$ are the mean values of $n$ sub-bands ($n=3$ for points 4 to 6 and points $1^\prime$, $2^\prime$ and $4^\prime$, while 4 for the remainder of the points), listed in Table \[table:apex\].
label $\theta_{\parallel}$(mas) $f_{\rm D}$(mHz) $\tau$(ms) $\Delta\alpha$(mas) $\Delta\delta$(mas) $t_0$(day)
------- --------------------------- ------------------ ------------ --------------------- --------------------- ------------
1 $-17.22$ $-26.1(4)$ 0.3743(6) 6.2 $-11.9$ $-107$
2 $-16.36$ $-24.9(4)$ 0.3378(3) 8.0(4) $-14.5(8)$ $-101$
3 $-16.08$ $-24.6(4)$ 0.327(3) 7.2(6) $-13.9(4)$ $-99.0$
4 $-14.45$ $-22.3(5)$ 0.2633(3) 6.1(4) $-13.1(7)$ $-88.1$
5 $-13.68$ $-21.6(6)$ 0.236(2) 5.1(4) $-12.7(5)$ $-83.3$
6 $-13.27$ $-20.4(5)$ 0.222(3) 5.8(4) $-11.8(1)$ $-81.4$
7 $-12.21$ $-18.9(2)$ 0.188(2) 5.5(6) $-10.8(6)$ $-74.2$
8 $-10.58$ $-16.8(3)$ 0.1412(9) 3.9(6) $-10.0(4)$ $-62.8$
9 $-8.18$ $-12.9(2)$ 0.0845(5) 2.8(3) $-8.6(4)$ $-48.7$
1’ $ \cdots$ $-43.1(4)$ 1.066(5) $-8(3)$ $-24(2)$ $-185$
2’ $\cdots$ $-41.3(5)$ 1.037(3) $-14(1)$ $-23(3)$ $-188$
3’ $\cdots$ $-40.2(6)$ 1.005(8) $-14(1)$ $-22.3(5)$ $-187$
4’ $\cdots$ $-38.3(6)$ 0.9763(9) $-14(1)$ $-20.6(3)$ $-190$
5’ $\cdots$ $-35.1(5)$ 0.950(2) $-15(1)$ $-21(1)$ $-202$
We estimate the error of time delay $\tau$, differential frequency $f_{\rm D}$, $\Delta\alpha$ and $\Delta\delta$ listed in Table \[table:apex\] from their band-to-band variance: $$\sigma^2_{\rm \tau, f_{\rm D},\Delta\alpha, \Delta\delta} = \frac{1}{n}\sum^{n}_{i=1}\frac{(x_i-\bar{x})^2}{n-1},$$ and $n$ is the number of sub-bands. The outer $1/n$ accounts for the expected variance of a mean of $n$ numbers.
One-lens model
--------------
### Distance to the lenses
In the absence of a lens model, the fringe rate, delay and angular position cannot be uniquely related. To interpret the data, we adopt the lensing model of @2014MNRAS.442.3338P. In this model, the lensing is due to projected fold caustics of a thin sheet closely aligned to the line of sight. We will list the parameters in this lens model in Table \[tab:parameters\].
-------------- -----------------------------------------
$D_{1\rm e}$ Effective Distance of $0.4$ group data
$D_{2\rm e}$ Effective Distance of $1$ ms group data
$D_1$ Distance of lens 1
$D_2$ Distance of lens 2
$\gamma$ Scattering axis angle of $0.4$ ms group
$\phi$ Angle of the velocity of the pulsar
$\theta$ Angular offset of the object
-------------- -----------------------------------------
: Parameters for double-lens model
[The angle is measured relative to the longitude and east is the positive direction.]{}
\[tab:parameters\]
We define the [*effective distance*]{} $D_{\rm e}$ as $$D_{\rm e} \equiv \frac{2c\tau}{\theta^2}.
$$ The differential frequency is related to the rate of change of delay as $f_{\rm D} =-f\frac{\rm d\tau}{\dif t}$. In general, $D_{\rm e}={D_{\rm p} D_{\rm s}}/({D_{\rm p} - D_{\rm
s}})$ for a screen at $D_{\rm s}$. The effective distance corresponds to the pulsar distance $D_{\rm p}$, if the screen is exactly halfway. Fig. \[fig:Singledegeneracy\] shows two sets of $D_{\rm p}$ and $D_{\rm s}$ with common $D_{\rm e}$.
![A single refracted light path showing the distance degeneracy. The primed and un-primed geometries result in the same observables: delay $\tau$ and angular offset $\theta$. $O$ denotes the observer; $A$ and $A'$ denote the positions of the pulsar; $D$ and $D'$ denote the positions of the refracted images on the interstellar medium. The un-primed geometry corresponds to a pulsar distance $D_{\rm p} = |AO| = 620$ pc, while the primed geometry has the same $D_{\rm e}$ but twice the $D_{\rm p}.$[]{data-label="fig:Singledegeneracy"}](single_degeneracy.pdf){width="3.9in"}
![${\theta}$ vs ${\sqrt{\tau}}$. Two separate lines through the origin were fitted to the points sampled among the $0.4$ ms group and $1$ ms group. The solid line is the fitted line of the $0.4$ ms positions, where $D_{1\rm e}=1044\pm 22$ pc. The dashed line is the fitted line of the $1$ ms position, where $D_{2\rm e}=1252\pm 49$ pc. []{data-label="thetatau"}](Theta_tau.pdf){width="1.0\linewidth"}
When estimating the angular offset of each apex, we subtract the expected noise bias: ${\theta}^2=({\Delta\alpha}\cos(\delta))^2+({\Delta\delta})^2-\sigma^2_{\Delta\alpha}-\sigma^2_{\Delta\delta}$. We plot the $\theta$ vs square root of $\tau$ in Fig. \[thetatau\]. A least-square fit to the distance results in $D_{1{\rm e}}=1044\pm 22$ pc for the $0.4$ ms group, which we call lens 1 (point 1 is excluded since VLBI astrometry was only known for one sub-band, thus we cannot obtain the variance nor weighted mean for that point), and $D_{2{\rm e}} = 1252 \pm 49$ pc for the $1$ ms group, hereafter lens 2. The errors, and uncertainties on the error, preclude a definitive interpretation of the apparent difference in distance. However, at face value, this indicates that lens 2 is closer to the pulsar, and we will use this as a basis for the model in this paper. The distances are slightly different from those derived in B10, which is partly due to a different subset of arclets analysed. We discuss consequences of alternate interpretations in Section \[sec:degeneracy\]. The pulsar distance was directly measured using VLBI parallax to be $D_{\rm p} = 620 \pm 60$ pc, described in more detail in Section \[sec:astrometry\]. We take $D_{1{\rm e}}=1044$ pc, and the distance of lens 1 $D_{1}$, where $0.4$ ms group scintillation points are refracted, as $389$ pc. Similarly, for $1$ ms apexes, the distance of lens 2 is taken as $D_2=415$ pc, slightly closer to the pulsar. For the 0.4 ms group, we adopt the geometry from B10, assigning these points along line $AD$ as shown in Fig. \[Doublelens\] based solely on their delay, which is the best measured observable. The line $AD$ is taken as a fixed angle of $\gamma=-25\degree.2$ east north. We use this axis to define ${\parallel}$ direction and define ${\bot}$ by a $90\degree$ direction clockwise rotation.
{width="7.5in"}
### Discussion of one-lens model {#222}
The $0.4$ ms group lens solution appears consistent with the premise of the inclined sheet lensing model [@2014MNRAS.442.3338P], which predicts collinear positions of lensing images. The time in the last column of Table \[table:apex\], which we denote as $t_0
=-2{\tau}f/{f_{\rm D}}$, corresponds to the time required for the arclet to drift in the secondary spectrum through a delay of zero.
The collinearity can be considered a post-diction of this model. The precise positions of each image are random, and with 9 images no precision test is possible. The predictive power of the sheet model becomes clear in the presence of a second, off-axis screen, which will be discussed below.
Double-lens model {#doublelensmodel}
-----------------
The apparent offset of the $1$ ms group can be explained by a second lens screen. The small number of apexes at $1$ ms suggests that the second lens screen involves a single caustic at a different distance. One expects each lens to re-image the full set of first scatterings, resulting in a number of apparent images equal to the product of number of lenses in each screen. In the primary lens system, the inclination appears such that typical waves form caustics. For the sake of discussion, we consider an inclination angle for lens 1 $\iota_1 = 0.1^o$, and a typical slope of waves $\sigma_\iota = \iota_1$. Each wave of gradient larger than 1-$\sigma$ will form a caustic in projection. The number of sheets at shallower inclination increases as the square of this small angle. A 3 times less-inclined $\iota_2=0.3^o$ sheet occurs 9 times as often. For the same amplitude waves on this second surface, they only form caustics for 3-$\sigma$ waves, which occur two hundred times less often. Thus, one expects such sheets to only form isolated caustics, which we expect to see occasionally. Three free parameters describe a second caustic: distance, angle, and angular separation. We fix the distances from the effective VLBI distance ($D_1$ and $D_2$), and fit the angular separations and angles with the 5 delays of the $1$ ms group.
### Solving the double-lens model
Apexes 1’–5’ share a similar 1 ms time delay, suggesting they are lensed by a common structure. We denote the position of the pulsar point as point $A$, the positions of the lensed image on lens 2 as point $H$, positions of the lensed image on lens 1 as point $B$, position of the observer as point $O$, and the nearest point on lens 2 to the pulsar as point $J$. The lines $AJ\perp HJ$ intersect at point $J$, $HF\perp BD$ intersect at the point $F$, and $BG\perp HJ$ intersect at the point $G$. A 3D schematic of two plane lensing by linear caustics is shown in Fig. \[fig:3D\_image\].
{width="9in"}
First, we calculate the position of $J$. We estimate the distance of $J$ from the $1$ ms $\theta$–$\sqrt{\tau}$ relation (see Fig. \[thetatau\]). We determine the position of $J$ by matching the time delays of point 4’ and point 1’, which is marked in Fig. \[Doublelens\]. The long dash dotted line on the right side of Fig. \[Doublelens\] denotes the inferred geometry of lens 2, and by construction vertical to $AJ$. The second step is to find the matched pairs of those two lenses. By inspection, we found that the 5 furthest points in $0.4$ ms group match naturally to the double-lens images. These five matched lines are marked with cyan dash dotted lines in Fig. \[Doublelens\] and their values are listed in the second column in Table \[table:double\_lens\_compare\].
label $\theta_{\parallel}$ (mas) $\tau_2$(ms) $\sigma_{\rm \tau}$(ms) $\tau_{\rm M}$(ms) $f_{\rm D}$(mHz) $\sigma_{f}$(mHz) $f_{\rm M}$(mHz) $t_1$(day)
------- ---------------------------- -------------- ------------------------- -------------------- ------------------ ------------------- ------------------ ------------
1’ $-17.22$ 1.0663 0.0050 1.0663\* $-43.08$ 0.84 $-42.26$ $-78$
2’ $-16.36$ 1.0370 0.0059 1.0362 $-41.27$ 0.88 $-41.04$ $-73$
3’ $-16.08$ 1.005 0.011 1.027 $-40.17$ 0.87 $-40.64$ $-72$
4’ $-14.45$ 0.9763 0.00088 0.9763\* $-38.31$ 0.64 $-38.31\dagger$ $-63$
5’ $-13.68$ 0.9495 0.0094 0.9550 $-35.06$ 0.78 $-37.21$ $-59$
They are the located at a distance $389$ pc away from us. Here we define three distances: $$\begin{aligned}
D_{\rm p2}&=620~{\rm pc}-415~{\rm pc} =205~{\rm pc},\\
D_{21}&=415~{\rm pc}-389~{\rm pc} =26~{\rm pc},
\end{aligned}$$ where $D_{\rm p2}$ is the distance from the pulsar to lens 2, and $D_{21}$ is the distance from lens 2 to lens 1. Fig. \[first\_reflect\] and Fig. \[second\_reflect\] are examples of how light is refracted on the first lens plane and the second lens plane. We specifically choose the point with $\theta_{\parallel}=-17.22$ mas, which refer point 1’ on lens 1 as an example. Equality of the velocity of the photon parallel to the lens plane before and after refraction implies the relation: $$\begin{aligned}
\frac{JH}{D_{\rm p2}}&=\frac{HG}{D_{21}},\\
\frac{FB}{D_{21}}&=\frac{BD}{D_{1}}.
\end{aligned}$$
![Refraction on lens 2. $A$ is the position of the pulsar. $H$ is the lensed image on lens 2. $B$ is the lensed image on lens 1. $AJ\bot HJ$ and $BG\bot HJ$. We illustrate the scenario for point 1’.[]{data-label="first_reflect"}](First_reflection.pdf){width="1.0\linewidth"}
![Refraction on lens 1. $H$ is the lensed image on lens 2. $B$ is the lensed image on lens 1. $O$ is the position of the observer. $HF\bot DF$ and $OD\bot HJ$. As in the previous figure, we illustrate the scenario for point 1’. []{data-label="second_reflect"}](Second_reflection.pdf){width="1.0\linewidth"}
We plot the solved positions in Fig. \[Doublelens\], and list respective time delays and differential frequencies in Table \[table:double\_lens\_compare\]. We take the error of the time delay $\tau$ in the double-lens model as $$\begin{aligned}
(\frac{\sigma_{\tau_i}}{\tau_{2i}})^2 = (\frac{\sigma_{\tau1i}}{\tau_{1i}})^2+(\frac{\sigma_{\tau2i}}{\tau_{2i}})^2 + (\frac{\sigma_{\tau2j}}{\tau_{2j}})^2,
\end{aligned}$$ where $\tau_1$ and $\sigma_{\tau1}$ represent the time delay and its error from the $0.4$ ms group on lens $1$, and $\tau_2$ and $\sigma_{\tau2}$ represent the time delay and its error from the $1$ ms group on lens 2. And $\tau_{2j}$ is the $\tau_2$ for the nearest reference point in Table \[table:double\_lens\_compare\] with error $\sigma_{\tau2j}$. Specifically, for point $i=5'$ and $3'$, $j=4'$ is the nearest reference point; while for point $i=2'$, $j=1'$ is the nearest reference point. The reference points are marked with star symbols in the fifth column in Table \[table:double\_lens\_compare\]. For the error of differential frequency $f_{\rm D}$, we add the error of the reference point (point 4’) to the error of each other measured point: $$\begin{aligned}
(\frac{\sigma_{f_i}}{f_{{\rm D}i}})^2=(\frac{\sigma_{f_{{\rm D}i}}}{f_{{\rm D}i}})^2+(\frac{\sigma_{f_{{\rm D}4'}}}{f_{{\rm D}4'}})^2
\end{aligned}$$ where $f_{{\rm D}4'}$ and $\sigma_{f_{{\rm D}4'}}$ are the differential frequency and we list its error of the point in the fourth row in Table \[table:double\_lens\_compare\].
### Comparing with observations
In order to compare $\tau$, we calculate model time delays $\tau_{\rm M}$ for these five points, and list the results in Table \[table:double\_lens\_compare\]. For points 4’ and 1’, they fit by construction since we use these to calculate the position of $J$; for the remaining three points, all of the results are within 3-$\sigma$ of the observed time delays.
To compare differential frequency $f_{\rm D}$, we need to calculate the velocity of the pulsar and the velocity of the lens. We take the lenses to be static, and solve the velocity of the pulsar relative to the lens (in geocentric coordinates). The pulsar has two velocity components, and the two 1-D lenses effectively determine one component each. For $v_{\parallel}$, we derive the velocity $172.4 \pm 2.4$ km $\rm s^{-1}$, which is $58.7$ mas/yr in a geocentric system, from $f_{\rm D}$ of point $1$ in $0.4$ ms group. The direct observable is the time to crossing of each caustic, denoted $t_0$ in Table \[table:apex\]. To calculate $v_{\bot}$, we choose the point 4’, which has the smallest errorbar of differential frequency. This gives a value of $67.9\pm 2.8$ km $\rm s^{-1}$ for $v_{\perp}$, which is $21.4$ mas/yr in geocentric system, with an angle $\phi=-3\degree.7 \pm 0\degree.8$ west of north. This represents the pulsar-screen velocity relative to the Earth. We can further transform this into the local standard of rest (LSR) frame to interpret the velocities in a Galactic context. The model derived and observed velocities (heliocentric and LSR) are listed in Table \[Table:velocity\]. The direction of the model velocity is marked on the top of the star in Fig. \[Doublelens\].
Parameter $\mu_{\alpha*}$(mas yr$^{-1}$) $\mu_{\delta}$(mas yr$^{-1}$) $\mu_{l*}$(mas yr$^{-1}$) $\mu_b$ (mas yr$^{-1}$) $v_{l*}$ (km $\rm s^{-1}$) $v_b$ (km $\rm s^{-1}$)
------------------------------ -------------------------------- ------------------------------- --------------------------- ------------------------- ---------------------------- -------------------------
model pulsar-screen velocity $-5.30 \pm 1.11$ $61.97 \pm 1.11$ $-56.45$ 22.23 $\cdots$ $\cdots$
VLBI pulsar proper motion $2.16 \pm 0.19$ $51.64 \pm 0.13$ $-46.69$ 28.02 -137.24 82.34
Screen motion $\cdots$ $\cdots$ 9.76 5.79 18.00 10.68
With this velocity of the pulsar, we calculate the model differential frequency $f_{\rm M}$ of points 5’,3’,2’ and 1’. Results are listed in Table \[table:double\_lens\_compare\]. The calculated results all lie within the 3-$\sigma$ error intervals of the observed data.
The reduced ${\chi}^2$ for time delay $\tau$ is $1.5$ for $3$ degrees of freedom and $2.2$ for $f_{\rm D}$ for $4$ degrees of freedom. This is consistent with the model.
Within this lensing model, we can test if the caustics are parallel. Using the lag error range of double-lensed point 4 (the best constrained), we find a 1-$\sigma$ allowed angle of 0.4 degrees from parallel with the whole lensing system. This lends support to the hypothesis of a highly inclined sheet, probably aligned to better than 1 per cent.
### Discussion of double-lens model {#subsec:doublelens}
For the $1$ ms group, lens 2 only images a subset of the lens 1 images. This could happen if lens 1 screen is just under the critical inclination angle, such that only 3-$\sigma$ waves lead to a fold caustic. If the lens 2 was at a critical angle, the chance of encountering a somewhat less inclined system is of order unity. More surprising is the absence of a single-refracted image of the pulsar, which is expected at position $J$. This could happen if the maximum refraction angle is just below critical, such that only rays on the appropriately aligned double refraction can form images. We plot the refraction angle $\beta$ in the direction that is transverse to the first lens plane in Fig. \[vtrans\]. The fractional bandwidth of the data is about 10 per cent, making it unlikely that single lens image $J$ would not be seen due to the larger required refraction angle. Instead, we speculate that the fold caustic terminates near double-lensed image 5’, and thus only intersections with the closer lens plane caustic south of image 5’ are doubly-lensed.
This is a generic outcome of a swallowtail catastrophe [@Arnold1990]. In this picture, the sheet just starts folding near point 5’. North of point 5’, no fold appears in projection. Far south of point 5’, a full fold exhibits two caustics emanating from the fold cusp. Near the cusp the magnification is the superposition of two caustics, leading to enhanced lensing and higher likelihood of being observed.
We denote $t_1$ the time for the lensed image on lens 2 to move from point $H$ to point $J$. From our calculation, we predict that on 2005 September 14, which is 59 days before the observation, the lensed image would have appeared overlayed on point $H_5$; and on 2005 August 26, which is 78 days before the observation, the lensed image would have appeared overlayed on point $H_1$. The model predicts the presence of a singly-lensed image refracted at these points, in addition to the doubly-lensed images.
![Deflection angle $\beta=\pi-\protect\angle AHB$ on lens 2. Point $J$ denotes the expected position to form a single refraction image, which was not observed. The small change in angle relative to the observed images precludes a finite refraction cut-off, since the data spans 10 per cent bandwidth, with a 20 per cent change in refractive strength. We propose a swallowtail caustic as the likely origin for the termination of the second lens sheet. []{data-label="vtrans"}](Reflection_angle.pdf){width="1.0\linewidth"}
The generic flux of a lensed image is the ratio of the lens transverse size to maximum impact parameter [@2012MNRAS.421L.132P]. Near the caustic, the lensed flux can become very high. The 1 ms group is about a factor of 4 fainter than the 0.4 ms group. The high flux of the second caustic suggests it to be relatively wide, perhaps a fraction of an AU. Due to the odd image theorem, one generically expects two distinct set of double lensed arcs. We only see one (generically the outer one), which places an upper bound on the brightness of the inner image. In a divergent lens[@1998ApJ...496..253C], the inner image is generically much fainter, so perhaps not surprising. For a convergent Gaussian lens, the two images are of similar brightness, but a more cuspy profile will also result in a faint inner image. In gravitational lensing, the odd image theorem is rarely seem to hold, generally thought to be due to one lens being very faint.
One can try to estimate the chance of accidental agreement between model and data. We show the data visually in Fig. \[fig:tau-fD\].
![Model comparison. Points with subscripts are derived from the double-lens model, see Table \[table:double\_lens\_compare\]. Rectangles mark the 1-$\sigma$ error region. Points 1’ (only for $\tau$) and 4’ were used to fit the model, and thus do not have an error region. The rectangles cover $10^{-3}$ of the area in the dotted region bounded by the parabolic arc and the data points. We interpret this precise agreement between model and data is unlikely to be a random coincidence. []{data-label="fig:tau-fD"}](tau-fD.pdf){width="1.0\linewidth"}
To estimate where points might lie accidentally, we conservatively compare the area of the error regions to the area bounded by the parabola and the data points, as shown by dotted lines. This results in about 10$^{-3}$, suggesting that the model is unlikely to be an accidental fit.
Distance degeneracies {#sec:degeneracy}
---------------------
With two lens screens, the number of observables increases: in principle one could observe both single refraction delays and angular positions, as well as the double reflection delays and angular positions. Three distances are unknown, equal to the number of observables. Unfortunately, these measurements are degenerate, which can be seen as follows. From the two screens $i=1,2$, the two single deflection effective distance observables are $D_{i{\rm e}} \equiv 2c\tau_i/\theta_i^2=D_i^2(1/D_i+1/D_{{\rm p}i})$. A third observable effective distance is that of screen 2 using screen 1 as a lens, $D_{21{\rm e}}=D_1^2(1/D_1+1/D_{21})$, within the triangle that is formed by lens 1, lens 2 and the observer. That is also algebraically derivable from the first two relations: $D_{{21}{\rm e}}=D_{1{\rm e}}D_{2{\rm e}}/(D_{2{\rm e}}-D_{1{\rm e}})$. We illustrate the light path in Fig. \[fig:double\_degeneracy\].
{width="9in"}
In this archival data set, the direct single lens from the further plane at position $J$ is missing. It would have been visible $59$ days earlier. The difference in time delays to image $J$ and the double reflection images would allow a direct determination of the effective distance to lens plane 2. Due to the close to $90 \degree$ angle $\angle~DAJ$ between lenses, the effect would be about a factor of 10 ill conditioned. With sufficiently precise VLBI imaging one could distinguish if the doubly-refracted images are at position $B$ (if lens 1 is closer to the observer) or position $H$ (if lens 2 is closer to the observer). As described above, we interpret the effective distances to place screen 2 further away.
VLBI astrometry {#sec:astrometry}
===============
The model kinematics can be compared to direct measurements of pulsar proper motion to infer the absolute motion of the lensing screen.
PSR B0834+06 was observed 8 times with the Very Long Baseline Array (VLBA), under the project code BB269, between 2009 May and 2011 January. Four 16 MHz bands spread across the frequency range 1406 – 1692 MHz were sampled with 2 bit quantization in both circular polarizations, giving a total data rate of 512 Mbps per antenna. The primary phase calibrator was J0831+0429, which is separated from the target by 2.1 degrees, but the target field also included an in-beam calibrator source J083646.4+061108, which is separated from PSR B0834+06 by only 5 . The cycle time between primary phase calibrator and target field was 5 minutes, and the total duration of each observation was 4 hours.
Standard astrometric data reduction techniques were applied [e.g., @deller12b; @deller13a], using a phase calibration solution interval of 4 minutes for the in-beam calibrator source J083646.4+061108. J083646.4+061108 is weak (flux density $\sim$4 mJy) and its brightness varied on the level of tens of percent. The faintness leads to noisy solutions, and the variability indicates that source structure evolution (which would translate to offsets in the fitted target position) could be present. Together, these two effects lead to reduced astrometric precision compared to that usually obtained with VLBI astrometry using in-beam calibration, and the results presented here could be improved upon if the observations were repeated using the wider bandwidths and higher sensitivity now available with the VLBA, potentially in conjunction with additional in-beam background sources.
While a straightforward fit to the astrometric observables yields a pulsar distance with a formal error $<$1 per cent, the reduced $\chi^2$ of this fit is $\sim$40, indicating that the formal position errors greatly underestimate the true position errors, and that systematic effects such as the calibrator effects discussed above as well as residual ionospheric errors dominate. Accordingly, the astrometric parameters and their errors were instead obtained by bootstrap sampling [@efron91a]. These results are presented in Table \[tab:vlbi\].
--------------------------------------- -------------------
Reference right ascension (J2000) 08:37:5.644606(9)
Reference declination (J2000) 06:10:15.4047(1)
Position epoch (MJD) 55200
$\mu_{\mathrm{R.A.}}$ (mas yr$^{-1}$) 2.16(19)
$\mu_{\mathrm{Dec}}$ (mas yr$^{-1}$) 51.64(13)
Parallax (mas) 1.63(15)
Distance (pc) 620(60)
$v_{\mathrm T}$ (km s$^{-1}$) 150(15)
--------------------------------------- -------------------
: Fitted and derived astrometric parameters for PSR B0834+06.
[The errors quoted here are from the astrometric fit only and do not include the $\sim$1 mas position uncertainty transferred from the in–beam calibrator’s absolute position.]{}
\[tab:vlbi\]
Discussions {#sec:discussions}
===========
Interpretation
--------------
The relative motion between pulsar and lens is directly measured by the differential frequency, and not sensitive to details of this model. B10 derived similar motions. This motion is in broad agreement with direct VLBI proper motion measurement, requiring the lens to be moving slowly compared to the pulsar proper motion or the LSR. The lens is $\sim200$ pc above the Galactic disk. Matter can either be in pressure equilibrium, or in free-fall, or some combination thereof. In free fall, one expects substantial motions. These data rule out retrograde or radially Galactic orbits: the lens is co-rotating with the Galaxy. In pressure equilibrium, gas rotates slower as its pressure scale height increases, which appears consistent with the observed slightly slower than co-rotating motion. The modest lens velocities also appear consistent with the general motion of the ISM, perhaps driven by Galactic fountains [@1976ApJ...205..762S] at these latitudes above the disk. In the inclined sheet model, the waves move at Alfvénic speed, but due to the high inclination, will move less than one percent of this speed in projection on the sky, and thus be completely negligible compared to other sources of motion.
Alternative models, for example, evaporating clouds [@1998ApJ...498L.125W] or strange matter [@2013PhLB..727..357P], do not make clear predictions. One would expect higher proper motions from these freely orbiting sources, and larger future scintillation samples may constrain these models.
In order to incline one sheet randomly to better than 1 per cent requires of order $10^4$ randomly placed sheets, i.e. many per parsec. This sheet extends for $\sim$ 10 AU in projection, corresponding to a physical scale greater than $1000$ AU. These two numbers roughly agree, leading to a physical picture of magnetic domain boundaries every $\sim 0.1$ pc. B0834+06 has had noted arcs for multiple years, perhaps suggesting this dominant lens plane is larger than typical. One might expect to reach the end of the sheet within decades.
A generic prediction of the inclined sheets model is a change in rotation measure across the scattering length. Over 1000 AU, one might expect a typical RM (rotation measure) change of $10^{-3}$ rad/m$^2$. At low frequencies, for example in LOFAR[^6] or GMRT[^7], the size of the scattering screen extends another order of magnitude in angular size, and the RM in different lensed images are different, increasing to $\sim 0.01$, which is plausibly measurable. Even for an un-polarized source, the left and right circularly polarized (LCP, RCP) dynamic spectra will be slightly different. Usually a secondary spectrum (SS) is formed by Fourier transforming the dynamic spectrum and multiplying by its complex conjugate. To measure the RM, one multiplies the Fourier transform of the LCP dynamic spectrum by the complex conjugate of the RCP Fourier transform. This will display a phase gradient along the Doppler frequency axis. In the SS, each pixel is the sum of correlations of pairs of scattering points with corresponding lag and Doppler velocity. The velocity is typically linear in the pair separation, which is also the case for differential RM. This statistic is analogous to the cross gate secondary spectrum as applied in @2014MNRAS.440L..36P.
Possible improvements
---------------------
We discuss several strategies which can improve on the solution accuracy. The single biggest improvement would be to monitor the speckle pattern over several months, as the pulsar crosses each individual lens, including both lensing systems. This allows a direct comparison of single lens to double-lens arclets.
Angular resolution can be improved using longer baselines, for example adding a GMRT-GBT baseline doubles the resolution. Observing at multiple frequencies over a longer period allows for a more precise measurement: when the pulsar is between two lenses, the refraction angle $\beta$ is small, and one expects to see the lensing at higher frequency, where the resolution is higher, and distances between lens positions can be measured to much higher accuracy.
Holographic techniques [@2008MNRAS.388.1214W; @2014MNRAS.440L..36P] may be able to measure delays, fringe rates, and VLBI positions substantially more accurately. Combining these techniques, the interstellar lensing could conceivably achieve distance measurements an order of magnitude better than the current published effective distance errors. This could bring most pulsar timing array targets into the coherent timing regime, enabling arc minute localization of gravitational wave sources, lifting any potential source confusion.
Ultimately, the precision of the lensing results would be limited by the fidelity of the lensing model. In the inclined sheet model, the images move along fold caustics. The straightness of these caustics depends on the inclination angle, which in turn depends on the amplitude of the surface waves. This analysis concludes a high degree of inclination, and thus high fidelity for geometric pulsar studies.
Conclusions {#sec:conclusions}
===========
We have applied the inclined sheet model [@2014MNRAS.442.3338P] to archival apex data of PSR B0834+06. The data is well-fit by two linear lensing screens, with nearly plane-parallel geometry. The second screen provides a precision test with 10 observables (5 time delays and 5 differential frequencies) and 3 free parameters (the marked points in Table \[table:double\_lens\_compare\]). The model fits the data to $\sim$ percent accuracy on each of 7 data points. This natural consequence of very smooth reconnection sheets is an unlikely outcome of ISM turbulence. These results, if extrapolated to multi-epoch observations of binary systems, might result in accurate distance determinations and opportunities for removing scattering induced timing errors. This approach also opens the window to measuring precise transverse motions of the ionized ISM outside the Galactic plane.
Acknowledgements
================
We thank NSERC for support. We acknowledge helpful discussions with Peter Goldreich and M. van Kerkwijk. We thank Michael Williams for photography help. Siqi Liu thanks Robert Main and JD Emberson for helpful discussions on improving the expression of the content. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: E-mail: [email protected]
[^3]: E-mail: [email protected]
[^4]: Email: [email protected]
[^5]: E-mail: [email protected]
[^6]: http://www.lofar.org/
[^7]: http://gmrt.ncra.tifr.res.in/
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---
abstract: 'We study a composition operator on Lorentz spaces. In particular we provide necessary and sufficient conditions under which a measurable mapping induces a bounded composition operator.'
address: |
Sobolev Institute of Mathematics\
4 Acad. Koptyug avenue\
630090 Novosibirsk\
Russia
author:
- Nikita Evseev
date: 'January 31, 2017'
title: Bounded composition operator on Lorentz spaces
---
[^1]
Introduction
============
Lorentz spaces $L_{p,q}$ are a generalization of ordinary Lebesgue spaces $L_p$, and they coincide with $L_p$ when $q=p$. Some references as to basics on Lorentz spaces may be found in [@SW; @M; @G]. A composition operator induced by map $\varphi$ on some function space is quite a natural object which is defined as $C_\varphi f = f\circ\varphi$. Depending on the structure of particular function space various properties of a compositions operator are under interest e.g. boundedness, compactness, inevitability and so on. The study of composition operators may be divided into three directions.
The first one could be referred to as classical and goes back to Littlewood’s Subordination Principle (1925). This principle states that a holomorphic self-mapping of the unit disk $D\subset C$ preserving $0$ induces a contractive composition operator on Hardy space $H^p(D)$, as well on Bergman and Dirichlet spaces. However, it is believed that the systematic study of composition operators induced by holomorphic maps started with the paper [@N] by E. A. Nordgren in the mid 1960’s. Afterwards the study of composition operator developed at the juncture of analytic function theory and operator theory. We refer the reader to book [@S] by J. Shapiro.
The second direction has a more operator flavor. Researchers raised all the questions about composition operators which could be posed regarding operators on normed spaces. One may find an exhaustive survey on the topic in the book [@SM] by R. K. Singh, J. S. Manhas and also in the proceedings [@SonCO].
The survey on composition operators on Sobolev spaces was motivated by the question, *what change of variables does preserve a Sobolev class?* Therefore the research was primarily focused on analytic and geometric properties of mappings, whereas operator theory was involved to a lesser extent. The first results in this area are due to S. L. Sobolev (1941), V. G. Maz’ya (1961), F. W. Gehring (1971). Subsequently S. K. Vodopjanov and V. M. Gold[š]{}te[ĭ]{}n (1975-76) studied a lattice isomorphism on Sobolev spaces. Later on many more mathematicians contributed to this research, see details in [@V2005; @V2012] and recent results on the subject in [@VN2015; @V2016]. We also mention here recent paper [@HKM] on composition operator on Sobolev-Lorentz space.
Our work belongs to the second of the described directions. As of right now composition operators on $L_p$ have been investigated thoroughly enough (see [@CJ; @SM; @VU]). In the case of Lorentz spaces most of the research has been concerned with composition operators from $L_{p,q}$ to $L_{p,q}$, domain and image spaces having the same parameters (see [@KK; @ADV; @KADA]). Here we initiate the study of a composition operator from $L_{r,s}$ to $L_{p,q}$, where the parameters may differ. The principal result of the paper is as follows.
\[theorem:principal\] A measurable mapping $\varphi:X\to Y$ satisfying $\mathcal N^{-1}$-property induces a bounded composition operator $$C_\varphi:L_{r,s}(Y)\to L_{p,q}(X), \qquad s\leq q$$ if and only if $$\int\limits_BJ_{\varphi^{-1}}(y)\, d\nu(y) \leq K^p\big(\nu(B)\big)^{\frac{p}{r}}$$ for some constant $K<\infty$ and any measurable set $B$.
We prove the theorem above in section \[comp\], while the range of composition operator and the case when composition operator is an isomorphism are studied in sections \[image\] and \[iso\].
Lorentz spaces
==============
Let $(X,\mathcal A, \mu)$ be a $\sigma$-finite measurable space. The Lorentz space $L_{p,q}(X,\mathcal A, \mu)$ is the set of all measurable functions $f:X\to\mathbb{C}$ for which $$\|f\|_{p,q} = \Bigg(\frac{q}{p}\int\limits_{0}^{\infty}\big(t^{\frac{1}{p}}f^*(t)\big)^q\,\frac{dt}{t}\Bigg)^{\frac{1}{q}} < \infty, \quad \text{ if } 1<p<\infty, 1\leq q<\infty,$$ or $$\|f\|_{p,\infty} = \sup\limits_{t>0}t^{\frac{1}{p}}f^*(t)<\infty, \quad \text{ if } 1<p<\infty, q=\infty.$$ The *non-increasing rearrangement* $f^*(t)$ of a function $f(x)$ is defined as $$f^*(t) = \inf\{\lambda>0 : \mu_{f}(\lambda)\leq t\},$$ where $$\mu_{f}(\lambda) = \mu\{x\in X : |f(x)|>\lambda\}$$ is the *distribution function* of $f(x)$.
Note that $\|\cdot\|_{p,q}$ is a norm if $1<q\leq p$ and a quasi-norm if $p<q$. We will refer to $\|\cdot\|_{p,q}$ as the Lorentz norm. For brevity we will use $L_{p,q}(X)$ instead of $L_{p,q}(X,\mathcal A, \mu)$.
In what follows we will need the next properties of Lorentz spaces.
The Lorentz norm can be computed via distribution: $$\label{eq:norm_distr}
\|f\|_{p,q} = \Bigg(\frac{q}{p}\int\limits_{0}^{\infty}\big(t^{\frac{1}{p}}f^*(t)\big)^q\,\frac{dt}{t}\Bigg)^{\frac{1}{q}}
= \Bigg(q\int\limits_{0}^{\infty}\left(\lambda\mu_{f}^{\frac{1}{p}}(\lambda)\right)^q\,\frac{d\lambda}{\lambda}\Bigg)^{\frac{1}{q}}$$ and $$\label{eq:norm_distr_infty}
\|f\|_{p,\infty} = \sup\limits_{\lambda>0}\lambda\big(\mu_{f}(\lambda)\big)^{\frac{1}{p}}.$$
Let $E\subset X$ be a measurable set. The Lorentz norm of its indicator is $$\label{eq:lemma_indicator_norm}
\|\chi_E\|_{p,q} = (\mu(E))^{\frac{1}{p}}.$$
Observe that $\mu_{\chi_E}(\lambda) = \mu(E)\cdot\chi_{(0,1)}(\lambda)$. If $q<\infty$ we apply formula : $$\|\chi_E\|_{p,q} = \Bigg(q\int\limits_{0}^{1}\big(\lambda\mu(E)^{\frac{1}{p}}\big)^q\,\frac{d\lambda}{\lambda}\Bigg)^{\frac{1}{q}} = (\mu(E))^{\frac{1}{p}}.$$
If $q=\infty$, we infer from that $$\|\chi_E\|_{p,\infty} = \sup\limits_{0<t<1}t\cdot (\mu(E))^{\frac{1}{p}} = (\mu(E))^{\frac{1}{p}}.$$
Suppose that $f\in L_{p,q_1}$ and $q_1\leq q_2$, then $\|f\|_{p,q_2}\leq \|f\|_{p,q_1}$.
Composition operator {#comp}
====================
Let $(X,\mathcal A, \mu)$ and $(Y,\mathcal B, \nu)$ be $\sigma$-finite measurable spaces and $\varphi:X\to Y$ be a measurable mapping.
\[lemma:principal\] Let $s\leq q$ and $f\circ\varphi \in L_{p,q}(X)$ for all $f\in L_{r,s}(Y)$, then the following two statements are equivalent
1\. $\|f\circ\varphi\|_{p,q}\leq K\|f\|_{r,s}$ for any $f\in L_{r,s}(Y)$;
2\. $(\mu(\varphi^{-1}(B)))^{\frac{1}{p}} \leq K(\nu(B))^{\frac{1}{r}}$ for any set $B\in\mathcal B$.
Let $B\in\mathcal B$ and $\nu(B)<\infty$. Plugging the indicator function $\chi_B(y)$ into statement 1 and using property , we obtain 2. If $\nu(B)=\infty$ the claim is trivial.
Suppose now that statement 2 holds. Let $f\in L_{r,s}(Y)$. First we find the expression for the distribution of the composition $f\circ\varphi$: $$\mu_{f\circ\varphi}(\lambda) = \mu(\{x\in X : |f(\varphi(x))|>\lambda\})
=\mu(\varphi^{-1}(\{y\in Y : |f(y)|>\lambda\})).$$ Denote $E_\lambda = \{y\in Y : |f(y)|>\lambda\}$, then $\nu_f(\lambda) = \nu(E_\lambda)$ and $\mu_{f\circ\varphi}(\lambda) = \mu(\varphi^{-1}(E_\lambda))$. From the inequality of statement 2 deduce $$\big(\mu(\varphi^{-1}(E_\lambda))\big)^{\frac{1}{p}} \leq K\big(\nu(E_\lambda)\big)^{\frac{1}{r}}$$ and thus $$\big(\mu_{f\circ\varphi}(\lambda)\big)^{\frac{1}{p}} \leq K\big(\nu_f(\lambda)\big)^{\frac{1}{r}}.$$ Consequently, $$\begin{gathered}
\|f\circ\varphi\|_{p,q}
= \Bigg(q\int\limits_{0}^{\infty}\left(\lambda\big(\mu_{f\circ\varphi}(\lambda)\big)^{\frac{1}{p}}\right)^q
\,\frac{d\lambda}{\lambda}\Bigg)^{\frac{1}{q}}\\
\leq
\Bigg(q\int\limits_{0}^{\infty}\left(\lambda K\big(\nu_{f}(\lambda)\big)^{\frac{1}{r}}\right)^q
\,\frac{d\lambda}{\lambda}\Bigg)^{\frac{1}{q}}
=K\|f\|_{r,q} \leq K\|f\|_{r,s}\end{gathered}$$ if $s<\infty$, and $$\|f\circ\varphi\|_{p,\infty}
= \sup\limits_{\lambda>0}\lambda\big(\mu_{f\circ\varphi}(\lambda)\big)^{\frac{1}{p}}
\leq K\sup\limits_{\lambda>0}\lambda\big(\nu_{f}(\lambda)\big)^{\frac{1}{r}}
= K\|f\|_{r,\infty}$$ as desired.
Assuming $p=r, q=s$ obtain [@KK Theorem 1] and [@ADV Theorem 2.1] as consequences of lemma \[lemma:principal\].
A mapping $\varphi$ induces a *composition operator* on Lorentz spaces $$\label{composition_operator}
C_\varphi:L_{r,s}(Y)\to L_{p,q}(X) \quad \text{ by the rule } C_\varphi f = f\circ\varphi$$ whenever $f\circ\varphi\in L_{p,q}(X)$.
Clearly that $C_\varphi$ is a linear operator between two vector spaces.
A composition operator $C_\varphi$ is bounded if $$\label{eq:bounded_operator}
\|C_\varphi f\|_{p,q}\leq K\|f\|_{r,s}$$ for every function $f\in L_{r,s}(Y)$, the constant $K$ being independent of the choice of $f$.
Similarly, $C_\varphi$ is bounded below if $$\label{bounded-below}
\|C_\varphi f\|_{p,q} \geq k\|f\|_{r,s}.$$
\[cor:LuzinN-1\] If a measurable mapping $\varphi$ induces a bounded composition operator, then $\varphi$ enjoys Luzin $\mathcal N^{-1}$-property (which means that $\mu(\varphi^{-1}(S))=0$ whenever $\nu(S)=0$).
In particular, corollary \[cor:LuzinN-1\] guarantees that if functions $f_1, f_2$ coincide a.e. on $Y$ then the images $C_\varphi f_1(x)$, $C_\varphi f_2(x)$ coincide a.e. on $X$. On the other hand the a priori assumption of $\mathcal N^{-1}$-property enables us to consider as an operator on equivalence classes.
Suppose we are given a measurable mapping $\varphi:X\to Y$ satisfying Luzin $\mathcal N^{-1}$-property. Then the measure $\mu\circ\varphi^{-1}$ is absolutely continuous with respect to $\nu$. Thus the Radon–Nikodym theorem guarantees the existence of a measurable function $J_{\varphi^{-1}}(y)$ (the Radon–Nikodym derivative) such that $$\label{eq:rd1}
\mu(\varphi^{-1}(E)) = \int\limits_E J_{\varphi^{-1}}(y)\, d\nu(y).$$ On account of , theorem \[theorem:principal\] follows immediately from lemma \[lemma:principal\].
Let $X$ and $Y$ be subsets of $R^n$ with Lebesgue measure ${|\cdot|}$. Consider a mapping $\varphi:X\to Y$ such that the Jacobian is bounded $J(x,\varphi)<M<\infty$ and the Banach indicatrix[^2] is bounded $N(y,\varphi, X) < N$ as well. Therefore $$\frac{N}{M} < J_{\varphi^{-1}}(y).$$ Suppose that $\varphi$ induces a bounded operator from $L_{r,s}(Y)$ to $L_{p,q}(X)$ then by theorem \[theorem:principal\] and by the inequality above we obtain $$0<\frac{N}{M}|B| < \int\limits_BJ_{\varphi^{-1}}(y)\, dy \leq K^p|B|^{\frac{p}{r}}$$ and $$\frac{N}{MK^p} < |B|^{\frac{p}{r} - 1}.$$ If we take a sequence of sets $B_k$ such that $|B_k|\to 0$ we will derive the necessary condition $p\leq r$, which is usually taken for granted.
Now let $X,Y\subset\mathbb R^2$. Examine a mapping $\varphi:X\to Y$ such that $\varphi(x_1,x_2) = (\frac{n}{2}, \frac{m}{2})$, where $n-1<x_1<n$, $m-1<x_2<m$, $n,m\in \mathbb Z$. Let $\mu$ be the Lebesgue measure on $\mathbb R^2$ while $\nu$ be a discrete measure with atoms in $(\frac{n}{2}, \frac{m}{2})$, $n,m\in \mathbb Z$ and for the sake of simplicity we set $\nu((\frac{n}{2}, \frac{m}{2})) = 1$. Then $J_{\varphi^{-1}}(y)=1$. In this case the mapping $\varphi$ could induce a bounded composition operator from $L_{r,s}(Y, \mathcal B, \nu)$ to $L_{p,q}(X, \mathcal A, \mu)$, even if $r<p$.
Properties of the image {#image}
=======================
In this section we exploit ideas from [@ADV] to investigate the range of a composition operator. First we show that $J_{\varphi^{-1}}(y)$ may be assumed to be positive a.e. on $Y$. Let $$Z=\{y\in Y : J_{\varphi^{-1}}(y)=0\},$$ then $$\mu(\varphi^{-1}(Z)) = \int\limits_Z J_{\varphi^{-1}}(y)\, d\nu(y) = 0.$$ Thus, after redefining the map $\varphi$ on the set $\mu(\varphi^{-1}(Z))$ of measure zero we obtain the property $J_{\varphi^{-1}}(y)>0$ a.e on $Y$.
\[theorem:bounded\_ below\] A measurable mapping $\varphi$ satisfying $\mathcal N^{-1}$-property induces a bounded below composition operator $$C_\varphi:L_{r,s}(Y)\to L_{p,q}(X), \quad s\geq q$$ if and only if $$\label{eq:rd-below}
\int\limits_BJ_{\varphi^{-1}}(y)\, d\nu(y) \geq k^p\big(\nu(B)\big)^{\frac{p}{r}}$$ for any $B\in \mathcal B$.
Applying to the indicator function $\chi_B(y)$ and using , we derive $$\bigg(\int\limits_BJ_{\varphi^{-1}}(y)\, d\nu(y)\bigg)^{\frac{1}{p}} \geq k\big(\nu(B)\big)^{\frac{1}{r}}.$$
Suppose now holds. Then in view of $$\mu_{f\circ\varphi}(\lambda) = \int\limits_Y\chi_{E_\lambda}(y) J_{\varphi^{-1}}(y)\, d\nu(y)
\geq k^p\big(\nu(E_\lambda)\big)^{\frac{p}{r}} = k^p\big(\nu_f(\lambda)\big)^{\frac{p}{r}}.$$ Thus $\big(\mu_{f\circ\varphi}(\lambda)\big)^{\frac{1}{p}}\geq k\big(\nu_f(\lambda)\big)^{\frac{1}{r}}$ and $$\|C_\varphi f\|_{p,q} \geq k\|f\|_{r,q} \geq k\|f\|_{r,s}.$$
Let $s=q$. Making use of the well known fact from functional analysis, which says that a linear bounded operator between Banach spaces is bounded below if and only if it is one-to-one and has closed range, we arrive to the following assertion.
\[theorem:closed\_image\] A bounded composition operator $C_\varphi:L_{r,s}(Y)\to L_{p,s}(X)$ is injective and has the closed image if and only if there is a constant $k>0$ such that $$\int\limits_BJ_{\varphi^{-1}}(y)\, d\nu(y) \geq k^p\big(\nu(B)\big)^{\frac{p}{r}}$$ for any $B\in \mathcal B$.
Next we discuss where a bounded composition operator has dense image.
\[theorem:dense\_image\] The image of a bounded composition operator $C_\varphi:L_{r,s}(Y, \mathcal B, \nu)\to L_{p,q}(X, \mathcal A, \mu)$ is dense in $L_{p,q}(X, \varphi^{-1}(\mathcal B), \mu)$.
Let $\chi_A\in L_{p,q}(X, \varphi^{-1}(\mathcal B), \mu)$ be the indicator function of a set $A=\varphi^{-1}(B)$, $B\in\mathcal B$. It is easy to see that $\chi_A(x) = \chi_B(\varphi(x))$, though we cannot ensure $\chi_B(y)\in L_{r,s}(Y, \mathcal B, \nu)$.
Let $B = \bigcup B_k$, where $\{B_k\}$ is an increasing sequence of sets of finite measure. Then $\chi_{B_k}(y)\in L_{r,s}(Y, \mathcal B, \nu)$. Denote $f_k=C_\varphi\chi_{B_k}$. Obviously $f_k(x)\leq \chi_A(x)$ and $f_k(x)\to \chi_A(x)$ as $k\to \infty$ a.e. on $X$. The similar inequality and convergence take place for distributions ($\mu_{f_k}$ and $\mu_{\chi_A}$), therefore from the Lebesgue theorem $f_k(x)\to \chi_A(x)$ in $L_{p,q}(X)$. The same arguments work for simple functions.
It follows that every simple function from $L_{p,q}(X, \varphi^{-1}(\mathcal B), \mu)$ is the limit of images. Since the set of simple functions is dense in $L_{p,q}(X)$ we conclude that the image $C_\varphi(L_{r,s}(Y, \mathcal B, \nu))$ is dense in $L_{p,q}(X, \varphi^{-1}(\mathcal B), \mu)$.
Isomorphism {#iso}
===========
We will say that a mapping $\varphi:X\to Y$ induces an isomorphism of Lorentz spaces $L_{p,q}(Y, \mathcal B, \nu)$, $L_{p,q}(X, \mathcal A, \mu)$ whenever the composition operator $C_\varphi$ is bijective and the inequalities $$\label{bounded_below_above}
k\|f\|_{p,q} \leq \|C_\varphi f\|_{p,q}\leq K\|f\|_{p,q}$$ hold for every function $f\in L_{p,q}(Y, \mathcal B, \nu)$ and for some constants $0<k\leq K<\infty$ independent of the choice of $f$.
A measurable mapping satisfying $\mathcal N^{-1}$-property induces an isomorphism of Lorentz spaces $$C_\varphi:L_{p,q}(Y, \mathcal B, \nu)\to L_{p,q}(X, \mathcal A, \mu)$$ if and only if $$\label{rd_below_above}
k^p \leq J_{\varphi^{-1}}(y) \leq K^p \quad \text{ a.e. } y\in Y$$ and $\varphi^{-1}(\mathcal B) = \mathcal A$.
Let $\varphi$ induce an isomorphism. Thanks to theorems \[theorem:principal\], \[theorem:bounded\_ below\] inequalities are a straightforward consequence of .
Show that $\varphi^{-1}(\mathcal B) = \mathcal A$. Let $A\in\mathcal A$ and $\mu(A)<\infty$, then the indicator function $\chi_A\in L_{p,q}(X, \mathcal A, \mu)$. Because of the surjectivity there is a function $f\in L_{p,q}(Y, \mathcal B, \nu)$ such that $\chi_A = C_\varphi f$. Observe that the set $B=\{y\in Y : f(y) = 1\}$ is an element of $\mathcal B$ and $f = \chi_B$. This yields $\chi_A = C_\varphi \chi_B = \chi_{\varphi^{-1}(B)}$ and hence $A=\varphi^{-1}(B)$. Thus $\mathcal A = \varphi^{-1}(\mathcal B)$.
Now assume that $\mathcal A = \varphi^{-1}(\mathcal B)$ and holds. Again is equivalent to owing to theorems \[theorem:principal\], \[theorem:bounded\_ below\]. From theorem \[theorem:closed\_image\] we infer that the operator $C_\varphi$ is one-to-one and the image $C_\varphi(L_{p,q}(Y, \mathcal B, \nu))$ is closed, whereas theorem \[theorem:dense\_image\] implies the density of the image in $L_{p,q}(X, \mathcal A, \mu)$. Consequently $C_\varphi(L_{p,q}(Y, \mathcal B, \nu))=L_{p,q}(X, \mathcal A, \mu)$.
[1]{}
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[^1]: This research was carried out at the Peoples’ Friendship University of Russia and financially supported by the Russian Science Foundation (Grant 16-41-02004)
[^2]: $N(y,f,X) = \#\{x\in X \mid f(x) = y\}$ is the number of elements of $f^{-1}(y)$ in $X$.
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abstract: 'We apply the bosonization technique to derive the phase diagram of a balanced unit density two-component dipolar Fermi gas in a one dimensional lattice geometry. The considered interaction processes are of the usual contact and dipolar long-range density-density type together with peculiar correlated hopping terms which can be generated dynamically. Rigorous bounds for the transition lines are obtained in the weak coupling regime. In addition to the standard bosonization description, we derive the low energy phase diagram taking place when part of the interaction is embodied non-perturbatively in the single component Hamiltonians. In this case the Luttinger liquid regime is shown to become unstable with respect to the opening of further gapped phases, among which insulating bond ordered wave and Haldane phases, the latter with degenerate edge modes.'
address:
- '$^1$ Institute for condensed matter physics and complex systems, DISAT, Politecnico di Torino, I-10129, Italy'
- '$^2$ CNR-IOM DEMOCRITOS Simulation Center and SISSA, Via Bonomea 265, I-34136 Trieste, Italy'
- '$^3$ Dipartimento di Fisica e Astronomia “Galileo Galilei", Università di Padova, 35131 Padova, Italy'
author:
- 'S Fazzini$^1$, L Barbiero$^{2,3}$ and A Montorsi$^1$'
title: Low energy quantum regimes of 1D dipolar Hubbard model with correlated hopping
---
Introduction
============
Experiments with cold atoms have disclosed a new way of investigating strongly correlated systems [@Bloch2008]. The possibility to cool down both fermionic and bosonic atomic gases to very low temperature and to trap them onto optical lattices, together with the ability to tune the interactions and the dimensionality with high accuracy, has allowed to simulate a great variety of interacting many-body lattice Hamiltonians [@dutta]. Particular theoretical efforts have been devoted to identifying fermionic Hubbard-like systems [@dutta1; @BMR; @FMRB] which ground state exhibits exotic [@Haldane] and topological phases [@Wen], described by string-like order parameters [@dennijs]. Also particles with dipolar long range interaction like polar molecules and magnetic atoms [@review_Santos] are currently available in laboratories. This has allowed for the experimental realization of a paradigmatic model in condensed matter physics namely the extended Bose-Hubbard model [@Ferlaino_Zoeller].\
On the other hand, bosonization [@giam; @nersesian] is a well-established analytical technique to investigate the low energy regime of one dimensional interacting fermionic systems. It consists in linearizing the spectrum around the Fermi points, passing to the continuum limit, and finally expressing the fermionic operators in terms of bosonic fields. In this way one has an efficient and general setting to study the low energy excitations induced by different types of interaction. Despite the fact that in most cases further numerical analyses are needed in order to get the full zero temperature phase diagram, bosonization is very useful to understand the nature of many quantum phases of matter and evaluate the correlation functions which characterize them.\
Within bosonization approach, usually the interaction is included in a perturbative way, starting from two non-interacting single component Luttinger liquids (LLs). In fact, as noticed in [@giam], in some cases part of the interaction can be included non-perturbatively already in the single component Hamiltonian, as long as it remains in a LL regime. This possibility has been exploited for instance in [@DBRD] to predict the presence of a bond ordered wave (BOW) phase induced by dipolar interaction already within one-loop bosonization. Here we adopt it to derive the zero temperature phase diagram of a one dimensional Hubbard model in presence of correlated hopping – induced by a periodical modulation of the on-site interaction [@diliberto], as recently shown experimentally [@meinert]– and long-range dipole-dipole interaction.\
The paper is organized as follows. In section 2 we introduce the model and rewrite the Hamiltonian in normal ordered form. In section 3 we review its bosonization phase diagram, which coincides with that of two spin-charge decoupled sine- Gordon models, due to the peculiar $r^{-3}$ nature of the power law decay of the dipolar interaction. The different gapped and partly gapped phases are characterized in terms of string and parity [@BDGA; @MR] non-local orders. In section 4 we then revisit the phase diagram by including part of both dipolar and correlated hopping interaction non perturbatively in the single component Hamiltonians. In this case, the phase diagram obtained from the study of renormalization group (RG) flow equations exhibit a richer structure. In section 5 we give some conclusions.
The model
=========
We consider a balanced unit density two-component dipolar Fermi mixture. Once these particles are trapped in a one dimensional lattice, an accurate description of the system is given by the following Hamiltonian [@FMRB] $$H=-J\sum_{j,\sigma}Q_{j,j+1, \sigma}\left[1-X(n_{j,\bar{\sigma}}-n_{j+1,\bar{\sigma}})^2\right]
+U_0\sum_j n_{j,\uparrow}n_{j,\downarrow}+V\sum_{j,r>0}\frac{ n_j n_{j+r}}{r^3}$$ where $Q_{j,j+1,\sigma}=c_{j,\sigma}^{\dagger}c_{j+1,\sigma} + h.c.$; $\sigma=\uparrow,\downarrow$ is the index species and $\bar{\sigma}$ denotes its opposite, $c_{j,\sigma}^{\dagger}$ and $c_{j,\sigma}$ are the creation and annihilation operators, respectively, $n_{j,\sigma}$ counts the number of particles of species $\sigma$ and $n_j=\sum_{\sigma}n_{j,\sigma}$. The coupling coefficients $J,U_0,V,X$, independently tunable in the experiments [@Bloch2008; @Goral2003; @bartolo], describe the tunneling probability, on-site and dipolar interactions, and correlated hopping processes, respectively. Upon normal ordering of the operators, $Q_{j,j+1,\sigma}=:Q_{j,j+1,\sigma}:+\langle Q_{j,j+1,\sigma}\rangle$, $n_{j,\sigma}=:n_{j,\sigma}:+\langle n_{j,\sigma}\rangle$ (with $\langle Q_{j,j+1,\sigma}\rangle=2/\pi$, $\langle n_{j,\sigma}\rangle=1/2$), and omitting the constant terms, we get $$\label{HamNormOrd}
\begin{split}
& H=-\left(1-\frac{X}{2}\right)\sum_{j,\sigma}:Q_{j,j+1,\sigma}:+\sum_{j,\sigma}\sum_rV_{\parallel}(r):n_{j,\sigma}::n_{j+r,\sigma}:+\\
&+\sum_{j,\sigma}\sum_rV_{\perp}(r):n_{j,\sigma}::n_{j+r,\bar{\sigma}}:+U_0\sum_j:n_{j,\uparrow}::n_{j,\downarrow}:\\&
-2 X \sum_{j,\sigma}:Q_{j,j+1,\sigma}::n_{j,\bar{\sigma}}::n_{j+1,\bar{\sigma}}:
\end{split}$$ where we have set $J=1$ and have defined $V_{\parallel}(r)=\frac{V}{r^3}-\frac{4X}{\pi}\delta_{r,1}$, $V_{\perp}=\frac{V}{r^3}$.
Weak coupling phase diagram
===========================
In the standard bosonization approach, one starts from the non-interacting Hamiltonian and consider the effect of interactions in a perturbative manner. The first step is to perform the continuum limit: $
\sum_j\longrightarrow \frac{1}{a}\int dx $; $c_{j,\sigma}\longrightarrow\sqrt{a}\left[e^{{\mathrm{i}}k_Fx}\Psi_{R\sigma}(x)+e^{-{\mathrm{i}}k_Fx}\Psi_{L\sigma}(x)\right]
$ (with $x=ja$, $a\to 0$ being the lattice constant). Here $\Psi_{R\sigma}$ and $\Psi_{L\sigma}$ are the fermionic field operators for the right and left movers, respectively. As claimed before, we will consider the particular case in which the system is at half filling; hence $k_F=\pi/(2a)$. We finally write the fermionic fields $\Psi_{\chi\sigma}$ in terms of the bosonic ones $\phi_{\sigma}$ and $\theta_{\sigma}$: $$\label{Campo_fermionico}
\Psi_{\chi\sigma}(x)=\frac{\eta_{\chi\sigma}}{\sqrt{2\pi\alpha}}e^{{\mathrm{i}}\sqrt{\pi}\left[\chi\phi_{\sigma}(x)+\theta_{\sigma}(x)\right]}$$ where the generic index $\chi$ can denote right (R) or left (L) movers, with the usual convention that it assumes a positive sign in the first case and a negative sign in the second one. Here $\alpha\sim a$ is an ultraviolet cutoff and $\eta_{\chi\sigma}$ are the Klein factors, which guarantee proper anti-commutation relations. With this notation, the kinetic operator $:Q_{j,j+1,\sigma}:$ and the density operator $:n_{j,\sigma}:$ appearing in Hamiltonian (\[HamNormOrd\]) have the following expressions in terms of the bosonic fields $$\label{:Q:}
:Q_{j,j+1,\sigma}:= -\frac{2}{\pi}(-1)^j\cos{(2\sqrt{\pi}\phi_{\sigma}(x))}-a^2\left[\left(\nabla\phi_{\sigma}(x)\right)^2+\left(\nabla\theta_{\sigma}(x)\right)^2\right]+...$$ $$\label{:n:}
:n_{j,\sigma}:=a\left[\frac{1}{\sqrt{\pi}}\nabla\phi_{\sigma}(x)-\frac{(-1)^j}{\pi a}\sin{(2\sqrt{\pi}\phi_{\sigma}(x))}\right]+...$$ Here we have used dots to denote the higher order terms in expansion with respect to $a$, which will be neglected. Bosonization of two and three body-terms in the Hamiltonian (\[HamNormOrd\]) involves calculating the product of operators of the form (\[:Q:\]) and (\[:n:\]). When the latters act on different fermionic species, the calculation is straightforward. When acting on the same species instead, the operator product expansion is needed. In deriving the bosonized expression for $:n_{j,\sigma}::n_{j+r,\sigma}:$, we make use of the fusion rule $
\sin{(2\sqrt{\pi}\phi_{\sigma}(x))}\sin{(2\sqrt{\pi}\phi_{\sigma}(x+R))}=\frac{a^2}{2R^2}-\frac{1}{2}\cos{(4\sqrt{\pi}\phi_{\sigma}(x))}-\pi a^2\left(\nabla\phi_{\sigma}(x)\right)^2+...
$, with $R=ra$. Thus we get $$ \label{:nn_r:}
:n_{j,\sigma}::n_{j+r,\sigma}:\simeq a^2\left\{ \frac{1-(-1)^r}{\pi}\left(\nabla\phi_{\sigma}(x)\right)^2-\frac{(-1)^r}{2\pi^2a^2}\cos{(4\sqrt{\pi}\phi_{\sigma}(x))}+\frac{(-1)^r}{2\pi^2R^2} + (-1)^j...\right\}. $$ In the three-body term, also the oscillating part of (\[:nn\_r:\]) (with $r=1$) contributes. To compute it, we apply the following operator product expansion, $
\nabla\phi_{\sigma}(x)\sin{(2\sqrt{\pi}\phi_{\sigma}(x+a))}=-\sin{(2\sqrt{\pi}\phi_{\sigma}(x))\nabla\phi_{\sigma}(x+a)}=\frac{1}{\sqrt{\pi}a}\cos{(2\sqrt{\pi}\phi_{\sigma}(x))}+...
$. One obtains $ (-1)^j\frac{2}{\pi^2a^2}\cos\left(2\sqrt{\pi}\phi_{\sigma}(x)\right)$, from which the three body term contribution is evaluated. As customary, it is now convenient to introduce charge and spin field operators, defined as $
\phi_c(x)=\left(\phi_{\uparrow}(x)+\phi_{\downarrow}(x)\right)/\sqrt{2}$ and $\phi_s(x)=\left(\phi_{\uparrow}(x)-\phi_{\downarrow}(x)\right)/\sqrt{2}$, respectively. Similar relations hold for the dual fields $\theta_{\nu}$ ($\nu=c,s$) as functions of $\theta_{\sigma}$ ($\sigma=\uparrow,\downarrow$). In terms of these operators, the Hamiltonian can be separated into the sum of two independent Hamiltonians in the charge and spin sectors plus a coupling term: $$H=H_c+H_s+H_{cs} \quad .$$ In each sector $H_{\nu}$ has the the form of a sine-Gordon model: $$\label{H_separate}
H_{\nu}=\frac{v_{\nu}}{2}\int dx \left[\left(\sqrt{K_{\nu}}\nabla\theta_{\nu}\right)^2+\left(\frac{\nabla\phi_{\nu}}{\sqrt{K_{\nu}}}\right)^2\right]+\frac{m_{\nu}v_{\nu}}{2\pi a^2}\int dx \cos{(\sqrt{8\pi}\phi_{\nu})},\quad \nu=c,s;$$ whereas the coupling Hamiltonian $H_{cs}$ reads $H_{cs}=\frac{M_{cs}}{\pi a}\int dx \cos{(\sqrt{8\pi}\phi_c)\cos{(\sqrt{8\pi}\phi_s)}}$. The coefficients of the Luttinger and mass terms are defined as follows: $$\begin{aligned}
&K_{\nu}=1+\frac{1}{4\pi}\left[\frac{16X}{\pi}-s_{\nu}U_0-\frac{3\zeta(3)V}{2}-4\zeta(3)V\delta_{\nu,c}\right]\quad ;\quad v_{\nu}=2a\left[2-\frac{X}{2}-\frac{X}{\pi^2}-K_{\nu}\right]\\
&m_{\nu}=\frac{1}{2\pi}\left[\frac{16X}{\pi}-s_{\nu}\left(U_0-\frac{3\zeta(3)V}{2}\right)\right]\quad ;\quad
M_{cs}=\frac{1}{2\pi}\left[\frac{3\zeta(3)V}{2}-\frac{8X}{\pi}\right].
\end{aligned}$$ where $s_c=1$, $s_s=-1$ and $\zeta(n)$ denotes the Riemann zeta function. The competition of the kinetic and mass terms in (\[H\_separate\]) can be discussed by analyzing the RG flow equations as in [@BMR]. Apart from the gapless LL phase, two insulating charge gapped phases are associated to the pinning of solely the charge field to the two values $\phi_c=0,\sqrt{\frac{\pi}{8}}$. In a uniform unit-density background, the latter values of $\phi_c$ describe respectively a Mott insulating (MI) regime with localized holon-doublon fluctuations and a Haldane insulator (HI) phase [[@dallatorre]]{} with “dilute" hidden antiferromagnetic order of holons and doublons [in analogy to the $XXZ$ spin-1 chain behavior [@Haldane; @dennijs]]{}. When $\phi_c$ instead is unpinned, a metallic spin gapped phase is associated to the pinning of solely the spin field to the value $\phi_s=0$. It actually describes the Luther Emery (LE) [@LE] liquid phase where spin-up and spin-down single particle quantum fluctuations take place in a uniform background of condensed holons and doublons. The $\phi_s=0$ value also supports two insulating fully gapped regimes, occurring when the charge field is pinned as well. In this case for $\phi_c=0$ the bond ordered wave (BOW) phase takes place. Whereas for $\phi_c=\sqrt{\frac{\pi}{8}}$ a charge density wave (CDW) phase with holon-doublon antiferromagnetic ordering is obtained. The five gapped or partly gapped phases can be characterized by the non-vanishing of the appropriate parity or string non-local order parameter, defined respectively as $$O_P^\nu(j)=\langle\prod_{k=0}^{j-1}{\rm e}^{i\pi S_{z,k}^\nu}\rangle\sim \langle\cos(\sqrt{2\pi}\phi_{\nu})\rangle \hspace{4pt} ,\hspace{4pt} O_S^\nu(j)=\langle\left (\prod_{k=0}^{j-1}{\rm e}^{i\pi S_{z,k}^\nu}\right )S_{z,j}^\nu\rangle\sim \langle\sin(\sqrt{2\pi}\phi_{\nu})\rangle \hspace{4pt} .$$ This is reported in Table \[table1\], following the procedure outlined in [@MR; @BMR]. In particular, the HI phase turns out to have non-trivial topological properties, as the presence of degenerate edge modes [@MDIR].
[llllll]{} & $\sqrt{2\pi}\Phi_{c}$ & $\sqrt{2\pi}\Phi_{s}$ & $\Delta_{c}$ & $\Delta_{s}$ & LRO\
LL & $u$ & $u$ & 0 & 0 & none\
LE & $u$ & 0 & 0 & open & $O_{P}^{s}$\
MI & 0 & $u$ & open & 0 & $O_{P}^{c}$\
HI & $\pi/2$ & $u$ & open & 0 & $O_{S}^{c}$\
BOW & 0 & 0 & open & open & $O_{P}^{c},\: O_{P}^{s}$\
CDW & $\pi/2$ & 0 & open & open & $O_{S}^{c},\, O_{P}^{s}$\
Depending on the values of the coupling constants $J,U_0,V,X$ all the above regimes can be realized. For instance, as reported in Fig. 1, already at $V=0$ by varying $U_0$ and $X$ the LL, MI and LE regimes are achieved.
Including interaction non-perturbatively
========================================
To gain further insight into the zero temperature phase diagram, we may regard Hamiltonian (\[HamNormOrd\]) as the sum of two single-species Hamiltonians, already containing part of the interaction non perturbatively, plus an inter-species part: $$H=\sum_{\sigma}H_{\sigma}+H_{\uparrow\downarrow} \quad .$$ Here $\sum_{\sigma}H_{\sigma}$ is given by the first two terms in (\[HamNormOrd\]). Up to a multiplicative constant $(1-\frac{X}{2})$, the single-species Hamiltonian $H_{\sigma}$ is a long-range “t-V" model, which is known to have a gapless Luttinger liquid phase for small enough interaction strength. In this case the Luttinger parameter $K(V,X)$ can be evaluated numerically with high precision [@CiOr; @Ot], and analytically both for vanishing $X$ [@PuZo], and for vanishing $V$ [@giam]. Explicitly: $$K(V,0)= \left[1+\frac{6 \zeta(3)V}{\pi^2}\right]^{-1/2} \quad , \quad
K(0,X)=\left[\frac{2}{\pi} \arccos\frac{2 X}{\pi\left(1-\frac{X}{2}\right)}\right]^{-1} \quad .
\label{Kex}$$ Of course, $K$ can also be obtained within a bosonization approximation [@cap]. In this case $K(X,V)\simeq 1+\frac{1}{4\pi}\left[\frac{16X}{\pi}-\frac{7\zeta(3)V}{2}\right]$. At this point one can proceed to bosonization of the inter-species interaction $H_{\uparrow\downarrow}$. It can be added to the Luttinger liquid phase of the remaining part of the Hamiltonian, which Luttinger coefficients may be determined either analytically or numerically [@CiOr; @Ot]. In so doing one ends up with a Hamiltonian which is fully decoupled in the charge and spin fields. Explicitly: $$H=H_c'+H_s'$$ with $$H_{\nu}'=\frac{v_{\nu}'}{2}\int dx \left[K_{\nu}'\left(\partial_x\theta_{\nu}\right)^2+\frac{1}{K_{\nu}'}\left(\partial_x\phi_{\nu}\right)^2\right]+\frac{m_{\nu}v_{\nu}}{2\pi a^2}\int dx \cos\left(\sqrt{8\pi}\phi_{\nu}\right) \label{Hamp}$$ and $$K_{\nu}'=\sqrt{K}\left\{1-\frac{X}{4\pi^2}+\frac{K^2}{8\pi}\left[\frac{2X}{\pi}-s_{\nu}\left(U_0+2\zeta(3)V\right)\right]\right\}\quad , \quad v_{\nu}'=v_{\nu}\frac{K_\nu}{K_{\nu}'}$$ where $s_c=1$, $s_s=-1$. Now the study of RG flow equations can be done by inserting the non-perturbative dependence of $K$ on $V$ ($X$), while keeping to first order the remaining interaction in $K$, $K_{\nu}'$ and $m_{\nu}$. The case $X=0$ has already been treated in this way in [@DBRD]. At variance with what obtained with standard bosonization, the latter approach allows to identify a further BOW phase already within one-loop expansion, thanks to the decoupling of the opening of the spin gap from the Gaussian transition of the charge gap. As an application of our findings at generic $X$ and $V$, here we report in Fig. \[fig1\] the phase diagram obtained at $V=0$, where $K$ is given by (\[Kex\]). In this case, the comparison with the results obtained within the standard weak coupling approach of section 3 shows the appearance of a further non-trivial HI phase for $\frac{4\pi}{3}X\leq U_0\leq\frac{16}{\pi} X$. From Table \[table1\] it is seen that the latter corresponds to a non-vanishing string order parameter in the charge sector, which is known to amount to the presence of degenerate edge modes [@MDIR].
![Phase diagram at $V=0$ from bosonization analysis of section 4. In case of standard bosonization of section 3, the Haldane insulator phase HI would still be a LL.[]{data-label="fig1"}](fig4n.png)
Conclusions
===========
We have derived the bosonization phase diagram of a unit density balanced two-component Fermi gas with correlated hopping, on-site, and long-range dipolar interactions. The power law decay of the dipolar interaction with exponent greater than 1 allows to re-sum its contributions into an effective short range potential. Moreover, spin-charge coupling terms can be neglected having, in general, larger scaling dimension. The resulting bosonized Hamiltonian consists of two spin-charge separated sine Gordon models, which phase diagram can be discussed according to Table \[table1\], depending on mass and Luttinger parameters. We derived the sine-Gordon models both within standard bosonization, in which case the single component Hamiltonians were the non-interacting up and down spin models; and by including non-perturbatively part of the interaction already at the level of the single component Hamiltonians. In the latter case further features emerge in the phase diagram, noticeably a non-trivial Haldane charge gapped phase also in absence of dipolar interaction. We expect that other 3- and 4-body processes [@DM] could be included non-perturbatively within the LL regime of the single component Hamiltonian, possibly inducing further orders in the ground state phase diagram. The present results should be compared with numerical findings. For instance in [@FMRB] by means of a density matrix renormalization group [@white] analysis it was found that further exotic phases, not present in the classification given here (Table \[table1\]), appear. In such cases, one should go beyond one loop bosonization, including higher order harmonics which were neglected here. Finally it is relevant to underline that the previous quantum phases could be studied and probed [@endress] with the currently available experimental setups as proposed in [@FMRB].
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abstract: 'We present a new method for optimally extracting point-source time variability information from a series of images. Differential photometry is generally best accomplished by subtracting two images separated in time, since this removes all constant objects in the field. By removing background sources such as the host galaxies of supernovae, such subtractions make possible the measurement of the proper flux of point-source objects superimposed on extended sources. In traditional difference photometry, a single image is designated as the “template” image and subtracted from all other observations. This procedure does not take all the available information into account and for sub-optimal template images may produce poor results. Given $N$ total observations of an object, we show how to obtain an estimate of the vector of fluxes from the individual images using the antisymmetric matrix of flux differences formed from the $N(N-1)/2$ distinct possible subtractions and provide a prescription for estimating the associated uncertainties. We then demonstrate how this method improves results over the standard procedure of designating one image as a “template” and differencing against only that image.'
author:
- 'Brian J. Barris, John L. Tonry, Megan C. Novicki, and W. Michael Wood-Vasey'
title: The NN2 Flux Difference Method for Constructing Variable Object Light Curves
---
Introduction
============
The astronomical time domain provides unique insight into a range of astrophysical phenomena. Studies of variable stars yield information about stellar structure and evolution as well as help to set the extra-galactic distance scale. Active Galactic Nuclei (AGN) reveal the high-energy phenomena associated with the super-massive black holes that reside at the centers of most galaxies. Supernovae (SNe) and Gamma-Ray Bursts (GRBs) provide a glimpse of the fantastic energies released during the violent death throes of several types of stars. Type Ia supernovae (SNe Ia) are of particular interest because their use as “standard candles” has revealed the acceleration of the expansion of the universe from an inferred cosmological constant-like force [@riess98; @perlmutter99].
Studies of variable sources require specific analysis methods that are not necessary for non-variable sources. Since it is often difficult to detect variation in an object by simply inspecting images, the standard procedure is to subtract images taken at different times to remove objects with constant flux. Photometrically variable objects are then obvious. For the case of SNe, one typically obtains a pair of observations separated in time to allow for SNe not present in the first image to reach observable brightness in the second (see, e.g., @perlmutter95 and @schmidt98 for a description of the method). After detection, additional observations are made to obtain the complete light curve necessary for cosmological analysis [see @phillips93; @riess96]. In order to construct the light curve, it is necessary for at least one observation (the “template” image) to contain no SN flux. Often this is the initial image used during discovery of the SN. In many instances, however, the SN is present at a faint level in this image, so an additional observation, taken after the SN has faded from view, is required. The light curve is then calculated by measuring the flux levels in subtractions of each image from the designated template using, for example, the subtraction procedure described by @alard98. This, the “single-template method,” is the typical means of constructing light curves of SNe and other variable sources.
However, this method has certain drawbacks. The primary flaw is that the quality of any subtraction depends greatly upon the two images involved. If the template is of a poor quality caused, for instance, by poor seeing or a low signal-to-noise ratio (S/N), then $every$ subtraction will be degraded, with a corresponding increase in the measured flux uncertainty, even if all other images are of high quality. Any flaw in the template creates a systematic error for the entire light curve that is not detectable from internal consistency checks or through comparison with another SN light curve.
In order to alleviate this problem, we have developed a new method for constructing light curves of photometrically variable objects. Given $N$ observations there are a total of $N(N-1)/2$ pairs of images that can be subtracted together, only $N-1$ of which are performed in the single-template method. A matrix of flux differences can be constructed from these subtractions and used to determine the flux at each individual epoch. This process removes the dependence on any single observation, because all observations are treated equally as a “template.” We refer to this method as the “N(N-1)/2” method (hereafter abbreviated NN2; see @novicki00 for an initial description).
Section \[sec-math\] describes the mathematical underpinnings of NN2. In Section \[sec-example\] we demonstrate the efficacy of the method using simulated SNe inserted into images used during an actual high-redshift SN survey. Section \[sec-conclusions\] gives our conclusions.
Mathematical Basis of the NN2 Method {#sec-math}
====================================
We assume that we start with $N$ observations of an object, so that one may construct from all pairs of subtractions an $N\times N$ antisymmetric matrix $A$ of flux differences that we wish to analyze as a “vector-term difference.” In other words, we want to find an $N$-vector $V$ of fluxes such that $$A_{ij} = V_j - V_i.$$ We also assume that we have a symmetric $N\times N$ error matrix $E$ that expresses our uncertainty in each term of $A$. As we shall see, this matrix may not be easy to generate, and its interpretation may be somewhat ambiguous. However, one can imagine generating an error matrix by the following procedure.
In each of the difference images, we measure a flux for the object in question. In general this measurement consists of fitting a fixed point-spread function (PSF) profile at the location of the object by adjusting the amplitude of the PSF (both positive or negative) and the local background level. The PSF profile may be obtained from a suitable star in the original image while the location of the variable source may be determined by summing all the difference images (adjusted to keep the sign of the object positive) and fitting the location in this sum. Once we have a flux measurement, we can insert copies of the object at nearby empty regions of the difference image and repeat the procedure. The mean of the recovered fluxes indicates whether there is a bias in the measurement, and the scatter may be used as a term $E_{ij}$ in the error matrix.
The crux of the NN2 method is the distillation of the photometric measurements from the full set of $N(N-1)/2$ subtractions to a lightcurve, $V$, that represents our best understanding of the behavior of the object under consideration. As long as it is consistent, the exact procedure for measuring the flux on the difference images is not central to the NN2 method we present here.
In order to find an optimal $V$, we wish to minimize the quantity $$\chi^2 = \sum_{i,j;i<j} {(-A_{ij} + V_j - V_i)^2 \over E_{ij}^2}
\label{eq:chi2}$$ This construction may not be entirely appropriate depending on the errors in the flux differences $A_{ij}$. Ideally, if we possessed an extremely high-quality template with no SN flux present and applied an optimal subtraction procedure, the errors would be primarily due to photon counting statistics (see @alard98 for a discussion). We would expect these errors to be uncorrelated and would simply wish to employ the single-template method to construct the SN light curve. As mentioned in the introduction, in practice there are nearly always imperfections associated with the template image that remove us from this idealized regime. These template errors introduce correlations in the individual flux measurement errors that are difficult to quantify and are typically assumed to be negligible in SN light-curve analysis. The use of the NN2 method, however, will introduce further correlations as a result of the common images in the various subtractions (for instance, the error in $V_1-V_2$ will be anti-correlated with the error in $V_2-V_3$ due to the common error in $V_2$). Although we believe that the use of the NN2 method will improve the ability to accurately recover variable object light curves, one should recognize that the NN2 method is expected to introduce these additional correlations to the fluxes measured from the various subtraction images, and so the $\chi^2$ given above is not technically appropriate. Errors due to systematics in the subtraction procedure, such as those associated with template or software imperfections, would be expected to be effectively uncorrelated, and if they were dominant then Eq. \[eq:chi2\] would indeed represent the proper $\chi^2$. With these caveats in mind, we will proceed to use the definition of $\chi^2$ as given in Eq. \[eq:chi2\] as the basis of the NN2 method. Tests of its ability to recover accurate light-curve information in the following section demonstrate its effectiveness in practice.
However, we need to make one minor modification to our $\chi^2$ because the $\chi^2$ defined in Eq. \[eq:chi2\] is degenerate to the addition of a constant to the $V$ vector—geometrically, $\chi^2$ is constant along the line $\sum
\hat i$. In order to lift this degeneracy and permit us to solve for $V$, we add a term to $\chi^2$ that is quadratic in the degenerate direction, so that $$\chi^2 = \sum_{i,j;i<j} {(-A_{ij} + V_j - V_i)^2 \over E_{ij}^2} +
{(\sum_i V_i)^2 \over \langle E\rangle^2},
\label{eq:final_chi2}$$ where $\langle E\rangle$ is a suitable typical uncertainty; for example, $${1\over \langle E\rangle^2} = {2\over N(N-1)}\,\sum_{i,j;i<j} {1\over E_{ij}^2}.$$ Our solution will therefore have $\sum_i V_i = 0$. This construction explicitly forces one to determine an accurate zero flux level at a later stage. In the single-template method this zero flux level is generally implicitly determined by assuming that the object of interest has zero flux in the template image. This same assumption can similarly be used in the NN2 method, but more sophisticated methods involving comparisons of many different images can also be invoked. If the absolute brightness of the variation being studied is important, the NN2 method clearly cannot free one from the requirement of having a fiducial image to measure the zero flux level. This is a fundamental limitation of any differential photometry method as the information is simply not available without such a fiducial image. However, even in the absence of a fiducial image, the NN2 method will produce a sensible and accurate relative lightcurve.
We now seek to solve for our lightcurve vector $V$ by minimizing $\chi^2$ with respect to $V$: $$\begin{aligned}
0 & = & {\partial \chi^2 \over \partial V_k} \\
& = & 2\sum_{i,j;i<j} {(-A_{ij} + V_j - V_i) \over E_{ij}^2} (\delta_{jk} - \delta_{ik}) + 2\sum_i {V_i\over \langle E \rangle^2}.
\label{eq:chi2v}\end{aligned}$$ Exploiting the antisymmetry of $A$ and the symmetry of $E$ we can rewrite Eq. \[eq:chi2v\] as $$0 = 2\sum_{i;i\neq k} {(-A_{ik} + V_k - V_i) \over E_{ik}^2}
+ 2\sum_i {V_i \over \langle E \rangle^2}.$$ These $N$ equations can be solved for $V$ by inverting a matrix $C$: $$\sum_{i;i\neq k} {A_{ik} \over E_{ik}^2} = \sum_i C_{ik} V_i$$ where $$C_{ik} = {-1 \over E_{ik}^2} + \sum_j{1 \over E_{kj}^2} \delta_{ik} +
{1 \over \langle E \rangle^2}.$$
The inverse of this curvature (Hessian) matrix $C$ yields uncertainties for $V$ from the square root of the diagonal elements as well as covariances from normalizing the off-diagonal elements by the two diagonal terms (under the assumption that the error matrix truly does represent Gaussian, independent uncertainties for each of the terms of the antisymmetric difference matrix).
An alternative approach to calculating uncertainties in $V$ stems from assuming that there is a vector $\sigma$ such that $$E_{ij}^2 = \sigma_i^2 + \sigma_j^2.$$ Under this assumption, we seek to minimize $$\chi_e^2 = \sum_{i,j;i<j} \left(-E_{ij}^2 + \sigma_i^2 + \sigma_j^2 \right)^2.$$ The minimization condition is $$\begin{aligned}
0 & = & {\partial \chi_e^2 \over \partial \sigma_k^2 } \\
& = & 2\sum_{i,j;i<j} \left(-E_{ij}^2 + \sigma_i^2 + \sigma_j^2 \right)\left(\delta_{ik}+\delta_{jk}\right) \\
& = & 2\sum_{i;i\neq k} \left(-E_{ik}^2 + \sigma_i^2 + \sigma_k^2 \right).\end{aligned}$$ These $N$ equations are solved by inverting a matrix $D$ $$\sum_{i;i\neq k} E_{ik}^2 = \sum_i D_{ik} \sigma_i^2$$ where $$D_{ik} = 1 + (N-2) \delta_{ik}.$$
After solving for $V$ and $\sigma$, we can evaluate the quality of the fit by comparing $\chi^2$ to the number of degrees of freedom, $$N_{\mathrm dof} = {N(N-1)\over2} - (N-1).$$ This $N_{\mathrm dof}$ comes from the number of data points, $N(N-1)/2$, minus the number of model parameters, $N-1$. Recall that we’ve explicitly required $\sum_i V_i = 0$, so that the number of model parameters is $N-1$ rather than $N$.
Having outlined the basic method, we now discuss a fundamental uncertainty in the NN2 process. We can imagine two types of error that will cause $V$ to differ from the true flux values. The first, which we term “external error,” is intrinsic to the images themselves. For example, if the object has a positive statistical fluctuation in flux in one image or is corrupted by a cosmic ray that happens to coincide with the position of the object on the detector, this error will propagate through the entire differencing and analysis procedure. It is possible to obtain an antisymmetric difference matrix that is an exact vector-term difference ($\chi^2 = 0$), but the solution vector will still contain errors. The second type of error, which we call “internal error,” is caused by the procedure of generating the antisymmetric matrix. One might imagine a set of images that have no flux error whatsoever, but through errors in convolving, differencing, or flux fitting, an antisymmetric matrix may be created that is not a perfect vector-term difference and for which $\chi^2 > 0$.
Roughly speaking, one might expect that if the error matrix $E$ consists entirely of external errors the resulting $\sigma$ terms will all be approximately $E/\sqrt{2}$, since $E$ is the quadrature sum of two $\sigma$ terms. Alternatively, if the error matrix is purely internal error the $\sigma$ terms might be expected to be approximately $E/\sqrt{N}$, since each term in $V$ comes from comparison with $N-1$ other images. In the case of external errors, the uncertainties derived from the $\chi_e^2$ analysis are correct. In the internal error case the uncertainties obtained from the covariance matrix derived from the $\chi^2$ analysis are likewise appropriate.
It is not clear how to disentangle these different sorts of errors. The procedure suggested above of dropping copies of the object into each difference image and evaluating the scatter of the result will be sensitive to each sort of error, but it is possible to imagine cases where this procedure is unsatisfactory. We suggest that the errors provided in the $E$ matrix be interpreted as external errors and taken seriously as such. Thus, the vector $V$ is assigned an external uncertainty equal to the $\sigma$ vector. However, in order to handle a situation where $\chi^2 / N_{\mathrm dof}$ is much greater than 1 (i.e., where the antisymmetric matrix is simply [*not*]{} well represented as a vector-term difference), we suggest also creating an internal uncertainty vector $\tau$ that is obtained from the diagonal terms of the covariance matrix, scaled by $\chi^2 / N_{\mathrm dof}$:
$$\tau_k = \left( C^{-1}_{kk} \; {\chi^2 \over N_{\mathrm dof}} \right)^{1/2}$$
The total uncertainty is then the quadrature sum of $\sigma$ and $\tau$. Note that this approach implicitly assumes that the internal and external errors are uncorrelated and are proportional to one another as well as the provided $E$ matrix.
For problems where $\chi^2/N_{\mathrm dof}$ is near unity without adjustment, the $\tau$ vector will be smaller than the $\sigma$ vector by approximately $\sqrt{2/N}$ and will make a fairly small contribution to the total uncertainty. When $\chi^2/N_{\mathrm dof} \ll 1$ (i.e., the antisymmetric matrix is very closely represented by the vector-term difference), the $\tau$ vector will be negligible. However, when $\chi^2/N_{\mathrm dof} \gg 1$, the $\tau$ vector will act to correct $\chi^2/N_{\mathrm dof}$ to approximately unity, and this procedure will provide reasonable uncertainties, even though $E$ may be much too small.
Demonstration of Improved Accuracy in Recovering Variable Object Light Curves {#sec-example}
=============================================================================
The first extensive use of the NN2 method we have developed here occurred during the SN-search component of the IfA Deep Survey [@barris04a], although we also employed it to a limited extent in a previous SN survey by @tonry03. The IfA Deep Survey was undertaken primarily with Suprime-Cam [@miyazaki98] on the Subaru 8.2-m telescope and was supplemented with the 12K camera [@cuillandre99] on the Canada-France-Hawaii 3.6-m telescope. Scores of high-redshift SN candidates were discovered [@barris01; @barris02] with 23 confirmed as SNe Ia. We here present several tests we performed to demonstrate the improved performance of NN2 relative to the single-template method.
In order to make a controlled test of the effectiveness of NN2 vs. the single-template method, we inserted artificial SNe into the survey images. The light curves of these objects consisted of a linear ramp-up and ramp-down in brightness over the time period covered by the survey observations. The timing of the light-curve maxima were selected at random and could lie within or outside of the survey period. The simulated SNe were laid down in a regular grid across the survey area, and all pairs of images were subtracted. Object-detection software was then run on all subtraction images to detect photometrically variable objects (both real objects and the artificial SNe) and to construct the NN2 flux difference matrix. The positions of both real and artificial SNe were fit by the object-detection software as described in Section \[sec-math\]. Since we knew the true light-curve properties used to create the synthetic SNe, we could calculate the root-mean-square scatter (RMS) around this artificial light curve using both the NN2 flux calculation and the single-template method with every individual observation as the template (this latter is equivalent to taking the flux values from a single column of the NN2 flux difference matrix).
We inserted approximately 2000 simulated SNe into the $I$-band observations of each of four $\sim$0.5 square-degree fields from the IfA Deep Survey, spanning a peak magnitude range of approximately $m_I=21$–$25$. We used a predefined grid of positions to insert the simulated SNe, without taking into consideration the presence of actual objects nearby that would cause problems for detection and accurate photometric measurements. The small fraction that were so affected were accordingly not used in the final analysis.
Figure \[fig:fakehisto\] shows the percentage improvement in the cumulative distributions of the RMS (in flux units) from the NN2 method over the set of all RMS values from the single-template method from the four survey fields (RMS values are calculated from flux measurements scaled so that a value of $\mathrm{flux}=1$ corresponds to a magnitude of $25$). The cumulative fraction for the NN2 method is larger than that for the single-template method distribution at all values of RMS, indicating that the NN2 method does indeed tend to yield $smaller$ RMS values. The NN2 method more accurately recovers the actual light curve of these variable objects.
Since one could imagine that certain templates of very high quality could outperform the NN2 method while the collection of all single-template measurements, as shown in Fig. \[fig:fakehisto\], does not, we next examine the relationship between the NN2 RMS and the single-template RMS for individual observations. To illustrate this comparison, we will concentrate on only one of the survey fields, although details of our investigation of the entire survey area can be found in @barris04b. Table \[table:obsinfo-0438\] contains relevant information for the 16 observations of the selected field, f0438. This field is representative of the entire survey area, though it is notable that it contains an observation that was quite strongly affected by clouds (Observation 4), as seen by its unusually bright zero-point magnitude. Also noteworthy is Observation 11, taken in poor seeing conditions. We would expect the performance of the single-template method using these observations to be poor in comparison to the NN2 procedure.
In Table \[table:obsinfo-0438\] we demonstrate that the typical RMS obtained with the NN2 method for the set of 1775 simulated SNe is smaller than the single-template method RMS using $every$ observation of the selected field. The improvement is generally fairly small, ranging from $\sim5-10$%. We demonstrate below that these differences are statistically significant. The use of either Observations 4 and 11 as single templates, as expected, produces substantially worse results relative to the NN2 method than the other observations. For these observations the improvement due to the NN2 method is substantially larger than 10%. Figure \[fig:rmscumfake-0438\] shows graphically the percentage difference in the cumulative RMS distributions, similar to Figure \[fig:fakehisto\], for each individual observation of f0438.
Having demonstrated the improved performance of the NN2 method, we can test the statistical significance of the differences between the NN2 RMS values and those calculated via the single-template method and examine whether these differences indicate an actual difference in the distributions of the results from the two methods. To do so we use the non-parametric Kolmogorov-Smirnov (K-S) test, with results given in Table \[table:ksinfo-0438\]. For the sample of all single-template RMS values compared to NN2, the K-S probability value is $\lesssim5\times10^{-9}$, indicating with strong confidence that the distributions are different. We also divide the sample into magnitude bins, since the relative behavior of NN2 RMS to single-template RMS is expected to be sensitive to the object’s S/N and hence to the magnitude for a given sensitivity.
The K-S probability values for three approximately equal magnitude bins show that for each of the subsamples the difference between NN2 and the single-template method is statistically significant, increasingly so at fainter magnitudes. Finally, we compare each observation individually with the NN2 method and see again that the observed improvement in RMS with NN2 is highly significant for nearly all observations (the only obvious potential exception is observation 8). These K-S probability values demonstrate that the NN2 method is not distributed identically to the single-template method, and the differences in median values given in Table \[table:obsinfo-0438\] are indeed indicative of statistically significant differences in the distributions. This test confirms that the NN2 method truly does produce improved results in generating differential light curves.
Conclusions {#sec-conclusions}
===========
We have described the mathematical foundation of a new method for constructing the light curves of photometrically variable objects. This method uses all $N(N-1)/2$ possible subtractions involving $N$ images in order to calculate a vector of fluxes of the variable object and offers a powerful alternative to the single-template method that is in standard use for studying variable sources.
If one has a data set with a limited number of good fiducial observations, the NN2 method will outperform any single-template subtraction approach. For cases where a large number of fiducial observations are available to construct a deep template image, the NN2 method and the single-template approach using this deep template should yield comparable results. In this situation we would encourage the use of both methods to provide additional checks and constraints on the differential light curve.
We have tested the performance of the NN2 method by inserting artificial SNe into images from the IfA Deep Survey and comparing the RMS scatter from flux measurements using the two different methods. We find that the RMS from the NN2 method is better than the single-template RMS for the large majority (typically 65%-72%) of the SNe for every possible template. The median values for the ratio of NN2 RMS (in flux units) to single-template RMS measurements are typically $0.93$–$0.96$, demonstrating that the NN2 method results in a $\sim5$% improvement in the accuracy of the recovered light curve for these observations.
Using Kolmogorov-Smirnov statistics, we have demonstrated that these differences are significant, reflecting an actual difference between the performance of the two methods. We find extremely high probabilities that the NN2 RMS is distributed significantly differently from the single-template RMS values. This difference and improvement in RMS holds even for the very high quality templates that would be considered ideal for the single-template method.
We therefore make the following conclusions:
1\. For the IfA Deep Survey observations, use of the NN2 method typically results in a 5-10% improvement in the RMS of the recovered light curve in comparison to the single-template method.
2\. For observations that have a large external error, such as those taken under poor conditions, the NN2 method results in a substantial improvement ($\gg10\%$) over the single-template method.
3\. When working with high-quality observations, with small external error, the internal errors (such as those due to implementation of the subtraction process) dominate. If these errors are large, the NN2 method should outperform the single-template method to a large degree. If these errors are kept small, as we believe is possible based our extensive experience with SN surveys, then the NN2 method will result in a modest but significant improvement in accuracy of light-curve recovery.
In summary, the NN2 method we present here maximizes the time variability information contained in a series of observations by using the relative differences between all pairs of images to construct the optimal differential light curve. references
The source code for our implementation of the NN2 method presented here is available at <http://www.ctio.noao.edu/essence/nn2/>.
This work was supported in part by grant AST-0443378 from the United States National Science Foundation.
Alard, C., & Lupton, R. H. 1998, , 503, 325
Barris, B., et al. 2001, IAU Circ. 7745, 7755, 7767, 7768
Barris, B., et al. 2002, IAU Circ. 7801, 7802, 7805, 7806, 7849
Barris, B.J., et al. 2004, ApJ, 602, 571
Barris, B.J. 2004, Ph.D. thesis, University of Hawaii
Cuillandre, J-C., et al. 1999, CFH Bull. 40
Miyazaki, S., et al. 1998, in Proc. SPIE, 3355, 363
Novicki, M. C., & Tonry, J. 2000, , 32, 1576
Perlmutter, S., et al. 1995, in Thermonuclear Supernovae, ed. P. Ruiz-Lapuente, R. Canal, & J. Isern (NATO ASI Ser. C, 486) (Dordrecht: Kluwer), 74
Perlmutter, S., et al. 1999, ApJ, 517, 565
Phillips, M.M. 1993, ApJ, 413, L105
Riess, A.G., Press, W.H., & Kirshner, R.P. 1996, ApJ, 473, 88
Riess, A.G., et al. 1998, AJ, 116, 1009
Schmidt, B.P., et al. 1998, , 507, 46
Tonry, J.L., et al. 2003, , 594, 1
[cccccc]{} 1 & 52164.57 & 0.75 & 30.48 & 0.441 & 0.951\
2 & 52176.49 & 0.72 & 30.05 & 0.432 & 0.963\
3 & 52191.50 & 0.71 & 30.12 & 0.427 & 0.970\
4 & 52198.56 & 0.59 & 27.63 & 0.810 & 0.625\
5 & 52204.59 & 0.63 & 30.46 & 0.424 & 0.973\
6 & 52225.39 & 0.62 & 30.17 & 0.445 & 0.960\
7 & 52231.38 & 0.51 & 30.46 & 0.455 & 0.944\
8 & 52232.35 & 0.68 & 30.21 & 0.419 & 0.970\
9 & 52236.34 & 0.94 & 30.36 & 0.440 & 0.938\
10 & 52252.35 & 0.83 & 30.59 & 0.429 & 0.957\
11 & 52263.38 & 1.14 & 30.33 & 0.515 & 0.839\
12 & 52283.39 & 0.68 & 30.51 & 0.460 & 0.946\
13 & 52288.23 & 0.84 & 30.65 & 0.438 & 0.941\
14 & 52289.39 & 0.93 & 30.27 & 0.448 & 0.936\
15 & 52323.35 & 0.97 & 30.23 & 0.468 & 0.908\
16 & 52369.25 & 0.58 & 29.58 & 0.432 & 0.943\
\[table:obsinfo-0438\]
[rrccl]{} 1775 & 28400 & all & 0.0769 & 4.7430e-09\
482 & 7712 & $\ \ \ \ \ \ \ \ \ m < 23.0$ & 0.0809 & 4.9083e-03\
617 & 9872 & $23.0\leq m \leq24.5$ & 0.0897 & 1.5707e-04\
676 & 10816 & $24.5 < m \ \ \ \ \ \ \ \ \ $ & 0.1330 & 2.5150e-10\
1775 & 1775 & $\ $observation 1 & 0.0524 & 1.4698e-02\
1775 & 1775 & $\ $observation 2 & 0.0462 & 4.3869e-02\
1775 & 1775 & $\ $observation 3 & 0.0411 & 9.6888e-02\
1775 & 1775 & $\ $observation 4 & 0.3977 & $<$1.0e-30\
1775 & 1775 & $\ $observation 5 & 0.0445 & 5.7717e-02\
1775 & 1775 & $\ $observation 6 & 0.0608 & 2.6510e-03\
1775 & 1775 & $\ $observation 7 & 0.0845 & 5.6244e-06\
1775 & 1775 & $\ $observation 8 & 0.0332 & 2.7611e-01\
1775 & 1775 & $\ $observation 9 & 0.0518 & 1.6326e-02\
1775 & 1775 & observation 10 & 0.0417 & 8.9127e-02\
1775 & 1775 & observation 11 & 0.1639 & 2.5672e-21\
1775 & 1775 & observation 12 & 0.0794 & 2.4901e-05\
1775 & 1775 & observation 13 & 0.0575 & 5.4234e-03\
1775 & 1775 & observation 14 & 0.0710 & 2.4226e-04\
1775 & 1775 & observation 15 & 0.0885 & 1.6592e-06\
1775 & 1775 & observation 16 & 0.0541 & 1.0650e-02\
\[table:ksinfo-0438\]
|
---
abstract: 'Say that a permutation of $1,2,\ldots,n$ is *$k$-bounded* if every pair of consecutive entries in the permutation differs by no more than $k$. Such a permutation is *anchored* if the first entry is $1$ and the last entry is $n$. We give a explicit recursive formulas for the number of anchored $k$-bounded permutations of $n$ for $k=2$ and $k=3$, resolving a conjecture listed on the Online Encyclopedia of Integer Sequences (entry [[](http://oeis.org/A249665)]{}). We also pose the conjecture that the generating function for the enumeration of $k$-bounded anchored permutations is always rational, mirroring the known result on (non-anchored) $k$-bounded permutations due to Avgustinovich and Kitaev.'
---
Maria M. Gillespie[^1]\
Department of Mathematics\
University of California Davis\
Davis, CA 95616\
USA\
<[email protected]>\
\
Kenneth G. Monks\
Department of Mathematics\
University of Scranton\
Scranton, PA 18510\
USA\
<[email protected]>\
\
Kenneth M. Monks\
Department of Mathematics\
Front Range Community College\
Longmont, CO 80501\
USA\
<[email protected]>\
\
.2 in
Introduction
============
Suppose one starts on the first stair of a staircase with $n$ steps labeled $1,\ldots,n$ in order, and at each step one either steps forwards or backwards by at most $k$ steps, such that every stair is used exactly once and the climb ends on the $n$th stair. How many distinct such ways are there to climb the stairs?
This question can be stated more precisely as follows. For a positive integer $k$, define a **$k$-bounded** permutation of $[n]=\{1,2,\ldots,n\}$ to be a bijection $\pi:[n]\to [n]$ such that for all $i\in \{1,2,\ldots,n-1\}$ we have $$\left|\pi(i)-\pi(i+1)\right|\leq k.$$ We say that such a permutation is **anchored** if $\pi(1)=1$ and $\pi(n)=n$. We are interested in enumerating the $k$-bounded anchored permutations in terms of $k$ and $n$.
The permutation $1,4,2,3,6,5,7,8,9$ is a $3$-bounded anchored permutation of $\{1,2,\ldots,9\}$, since the first entry is $1$, the last entry is $9$, and no pair of consecutive entries differs by more than $3$.
The question of explicitly enumerating $3$-bounded anchored permutations was first posed on the Online Encyclopedia of Integer Sequences, entry [[](http://oeis.org/A249665)]{} [@OEIS]. Our results resolve the stated conjectures in this entry.
Several related questions have been studied previously. Positive stair climbing problems were studied by Goins and Washington [@Standard; @stair; @climbing], extending the well-known fact that the number of ways to climb a staircase of length $n$ using positive steps of $+1$ or $+2$ each time is the $n$th Fibonacci number.
Avgustinovich and Kitaev [@AK] studied “$k$-determined permutations”, which they show are equivalent to $(k-1)$-bounded, *non-anchored* permutations, as well as certain Hamiltonian paths in graphs. They resolve a conjecture of Plouffe [@Plouffe] by providing the generating function for $2$-bounded non-anchored permutations, which were originally defined as *key permutations* [@Page]. Avgustinovich and Kitaev further show that the generating function for any $k$ is always rational using the transfer-matrix method described by Stanley [@Stanley ch. 4].
Main results
------------
For $k=1$, there is clearly only one $1$-bounded anchored permutation for each $n$, namely the identity permutation. In this paper, we resolve the cases $k=2$ and $k=3$ completely, as well as the $k=2$ non-anchored setting.
Our main results can be summarized in the following two theorems.
\[thm:2-bounded\] Let $R_n$ be the number of $2$-bounded anchored permutations of $[n]$. Then the sequence $\left(R_n\right)_{n\geq1}$ is given by the recurrence $R_1=1$, $R_2=1$, $R_3=1$, and $$\label{eqn:recursion2}
R_n=R_{n-1}+R_{n-3}$$ for all $n\geq 4$. The generating function of the sequence is $$R(x)=\sum_{n=1}^\infty R_n x^n=\frac{x}{1-x-x^3}.$$
This sequence $R_n$ is also known as *Narayana’s cows sequence* [@OEIS-Narayana-Cows], and the particular interpretation as $2$-bounded anchored permutation is stated without proof (in a slightly different but equivalent form) in Flajolet and Sedgewick [@Flajolet p. 373]. We include a proof in this paper for completeness. Note the similarity to the Fibonacci recurrence. It is interesting that for steps of $+1$ and $+2$ only, the recurrence is precisely the Fibonacci sequence, and here, with the added steps of $-1$ and $-2$ where every step is reached, it is one index off of the Fibonacci recurrence.
\[thm:3-bounded\] Let $F_n$ be the number of $3$-bounded anchored permutations of $[n]$. Then the sequence $\left(F_n\right)_{n\geq1}$ is given by the recurrence $F_1=1$, $F_2=1$, $F_3=1$, $F_4=2$, $F_5=6$, $F_6=14$, $F_7=28$, $F_8=56$, and $$\label{eqn:recursion3}
F_n=2F_{n-1}-F_{n-2}+2F_{n-3}+F_{n-4}+F_{n-5}-F_{n-7}-F_{n-8}$$ for all $n\geq 9$. The generating function of the sequence is $$F(x)=\frac{x-x^2-x^4}{1-2x+x^2-2x^3-x^4-x^5+x^7+x^8}.$$
In Section \[sec:2-bounded\], we prove Theorem \[thm:2-bounded\], and in Section \[sec:3-bounded\] we prove Theorem \[thm:3-bounded\]. Interestingly, we do not know a direct combinatorial proof of the recursion (\[eqn:recursion3\]), and some open problems in this and other directions are posed in Section \[sec:conjectures\].
Notation
--------
We write our permutations $\pi:[n]\to [n]$ in list notation, where the $i$th entry of the list is $\pi(i)$.
A **gap** of a permutation $\pi$ is a difference $\pi(i+1)-\pi(i)$ between two consecutive entries. We will always write our gaps with a $+$ or $-$ sign in front to indicate the sign, even if the sign is clear, to distinguish gaps from entries. For instance, we would say that the first gap of the permutation $1,3,2,4$ is $+2$, and the second gap is $-1$. We sometimes refer to the gaps of a sequence that is not a permutation as well, defined in the same way as consecutive differences between entries.
A sequence whose gaps are all between $-k$ and $+k$ is said to be **blocked** or **stuck** at the end if the last entry $a$ has the property that $a\pm 1, \ldots, a\pm k$ all either occur in the sequence or are less than or equal to $0$. For instance, if $k=3$, the sequence $1,3,4,6,5,2$ is blocked at $2$; the next possible positive integer that has not been used is $7$, which is more than a gap of $k$ away.
The **graph** of a permutation of $\{1,\ldots,n\}$ is the plot of all points $(i,\pi(i))$ in the plane. The **main diagonal** is the line with equation $y=x$. Note that a point in the graph of a permutation is on the main diagonal if and only if it is a fixed point of the permutation.
Structure and enumeration for k=2 {#sec:2-bounded}
=================================
As in Theorem \[thm:2-bounded\], we define $R_n$ to be the number of $2$-bounded anchored permutations of $[n]$. To get a handle on these permutations, we first prove the following lemma. It is worth noting that a weaker version of the lemma suffices to prove recursion (\[eqn:recursion2\]), but the stronger statement explicitly describes the structure of a $2$-bounded permutation.
\[lem:structure2\] Let $\pi$ be an anchored $2$-bounded permutation of $[n]$. Then there exists a subset $I\subseteq \{2,\ldots,n-2\}$ such that
1. Any pair of numbers in $I$ differ by at least three, and
2. For all $i\in [n]$, $$\pi(i)=\begin{cases} i+1, & \text{if $i\in I$;} \\ i-1, & \text{if $i-1\in I$;} \\ i, & \text{otherwise.} \end{cases}$$
In other words, the graph of the permutation can only deviate from the diagonal $x=y$ in consecutive pairs, with an up-step of $2$ and a down-step of $1$, before returning to the diagonal with an up-step of $2$. (See Figure \[fig:2-bounded\].)
![\[fig:2-bounded\] The $2$-bounded permutation graphed above, $1,2,4,3,5,7,6,8$, has subset $I=\{3,6\}$ as the set of indices $i$ for which $\pi(i)=i+1$.](main-2)
The lemma is clearly true when $n=1$. We proceed by strong induction on $n$. Assume that the lemma holds for all positive integers $n'<n$, and let $\pi$ be a permutation of $[n]$.
If $\pi$ is the identity permutation then $I=\emptyset$ and we are done, so we may assume that $\pi$ is not the identity. Let $i$ be the smallest index for which $\pi(i)\neq i$. Note that $i\in\left\{2,\ldots,n-2\right\}$. Then since $\pi(j)=j$ for all $j<i$, the gap from $\pi(i-1)$ to $\pi(i)$ cannot be $-1$, $-2$, or $+1$. It therefore must be $+2$, and we have $$\pi(i)=\pi(i-1)+2=i-1+2=i+1.$$
Now, the next gap, from $\pi(i)$ to $\pi(i+1)$, can either be $-1$, $+1$, or $+2$. We claim that it is not $+1$ or $+2$. If the gap were $+1$, then $i+1$ and $i+2$ both occur, before the value $i$ appears in the permutation. So for some $j>i+1$, $\pi(j)=i$. But then the value of $\pi(j+1)$ must be at least $i+3$ (since all other possible values are already used), and this contradicts $2$-boundedness. Otherwise, if the gap between $\pi(i)$ and $\pi(i+1)$ is $+2$, so that $\pi(i+1)=i+3$, then the only way to reach $i$ in the permutation is via a $-2$ step from $i+2$, and the same argument shows a contradiction.
It follows that the gap at $i$ is $-1$, so $\pi(i+1)=i$. The only possible value for $\pi(i+2)$ is then $i+2$ (with a $+2$ step from the previous), which is on the diagonal again with all smaller numbers having occurred to the left of it. The remaining entries form a $2$-bounded, anchored permutation of $\{i+2,i+3,\ldots,n\}$, which has a corresponding subset $I'\subseteq \{i+3,i+4,\ldots,n-2\}$ that satisfies the conditions above by the inductive hypothesis. Since $i$ is at least $3$ less than any element of $I'$, we see that setting $I=\{i\}\cup I'$ gives a valid subset that corresponds to $\pi$.
We now can prove Theorem \[thm:2-bounded\].
It is easily checked that $R_1=R_2=R_3=1$. Let $n\geq 4$. Then any anchored $2$-bounded permutation $\pi$ of $[n]$ either starts with $1,2$ or $1,3$. In the former case, there are $R_{n-1}$ ways of completing the permutation, since any $2$-bounded way of completing it that ends at $n$ is an anchored permutation of $\{2,\ldots,n\}$.
In the latter case, by Lemma \[lem:structure2\], the first four entries of the permutation must be $1,3,2,4$, and then the remaining entries starting from $4$ form $2$-bounded anchored permutation of $\{4,5,\ldots,n\}$. It follows that there are $R_{n-3}$ possibilities if the permutation starts with $1,3$.
It follows that $R_n=R_{n-1}+R_{n-3}$.
The generating function now follows from a straightforward calculation. We have $$\begin{aligned}
R(x)-xR(x)-x^3R(x) & = \sum_{n=1}^\infty R_n x^n - \sum_{n=2}^\infty R_{n-1}x^n - \sum_{n=4}^\infty R_{n-3} x^n \\
&= x+x^2+x^3-(x^2+x^3)+\sum_{n=4}^\infty (R_n-R_{n-1}-R_{n-3})x^n \\
&= x + \sum_{n=4}^\infty 0 \cdot x^n \\
&= x,\end{aligned}$$ and it follows that $R(x)=x/(1-x-x^3)$.
Structure and enumeration for k=3 {#sec:3-bounded}
=================================
As in Theorem \[thm:3-bounded\], we define $F_n$ to be the number of $3$-bounded anchored permutations of $[n]$. In the $2$-bounded case, we saw that there is one possible pattern in which the permutations can veer from the identity, and used that to generate the recursion. Similarly, in the $3$-bounded case, we will need to single out a certain special sequence that interferes with an otherwise regular pattern that the permutations must follow.
The **Joker** is the sequence $3,1,4,2,5$. We say the Joker **appears** in a $3$-bounded permutation if for some $i$, the $i$th through $(i+4)$th entries of the permutation are $i+2,i,i+3,i+1,i+4$.
![\[fig:Joker\] The Joker appears in the above permutation, in its second through sixth entries.](main-3)
Aside from the Joker, the $3$-bounded permutations turn out to follow a predictable pattern in terms of runs of $+3$ and $-3$ steps. We will use this structure to devise a three-term recurrence for $F_n$.
Define $G_n$ to be the number of $3$-bounded permutations $\pi$ of $\{1,2,\ldots,n\}$ that start with either $\pi(1)=1$ or $\pi(1)=2$ (so they are not necessarily anchored) and end at $\pi(n)=n$.
Define $H_n$ to be the number of $3$-bounded permutations $\pi$ of $\{1,2,\ldots,n\}$ that start with $\pi(1)=3$, end with $\pi(n)=n$, and do not start with the Joker as the first five terms.
We claim that for all $n\geq 6$, the sequences $F_n$, $G_n$, $H_n$ satisfy the following recurrence relations: $$\begin{aligned}
F_n &= G_{n-1} + H_{n-1}+F_{n-5}, \\
G_n &= F_n + G_{n-2} + F_{n-3} + G_{n-4} + H_{n-2}, \\
H_n &= F_{n-3} + G_{n-3} + F_{n-4} + G_{n-5} + H_{n-3}. \end{aligned}$$ To prove these relations, we first prove the following structure lemma.
\[lem:Joker\] Suppose $\pi$ is a $3$-bounded anchored permutation of $[n]$, and that the first $i$ entries form a $3$-bounded anchored permutation of $[i]$, so that $\pi(1)=1$, $\pi(i)=i$, and the numbers $1,\ldots,i$ comprise the first $i$ entries of the permutation in some order. If the next step is a $+3$, then one of the following two patterns occurs starting at entry $i$:
1. \[option1\] The Joker appears as entries $i$ through $i+4$.
2. \[option2\] There is a positive integer $m$ and a gap $d\in \{\pm 1, \pm 2\}$ such that the sequence of gaps after $i$ is $$+3,+3,\ldots,+3,d,-3,-3,\ldots,-3,\overline{d},+3,+3,\ldots,+3$$ where the first run of $+3$’s has length $m$, the run of $-3$’s has length $m'$ where $m'=m-1$ if $d<0$ and $m'=m$ if $d>0$, the last run of $+3$’s has length $m'$ as well, and $$\overline{d}=
\begin{cases}
+1, & \text{if $d=1$ or $d=-2$;} \\
-1, & \text{if $d=2$ or $d=-1$.}
\end{cases}$$ We call such a pattern a **cascading $3$-pattern**.
![\[fig:cascading3\] An example of a cascading $3$-pattern, with $m=3$ and $d=-1$.](main-4)
First, note that since $\pi$ restricts to a permutation on $\{1,\ldots,i\}$, we can assume for simplicity that $i=1$. Now, suppose the next gap is $+3$, so $\pi(2)=4$.
Let $m$ be the length of the run of consecutive gaps of $+3$ starting from $1$ before a gap $d$ not equal to $+3$ occurs. Notice that $d$ cannot be $-3$ or else the same entry would occur twice in the permutation, and so $d\in \{\pm 1, \pm 2\}$. We will prove that one of the two possibilities above hold by induction on $m$.
**Base Case.** Suppose $m=1$. We consider several subcases based on the value of $d$.
If $d=-2$, then the first three entries of the sequence are $1,4,2$, and the next entry may be $5$ or $3$. If the next entry is $5$ and the fifth entry is larger than $5$, then the only way to reach $3$ later in the permutation is by a gap of $-3$ from $6$, in which case we would be stuck at $3$, having used $1$, $2$, $4$, $5$, and $6$ already. Thus, if $\pi$ starts with $1,4,2,5$ then it must continue $1,4,2,5,3,6$, which is the Joker. Otherwise, it starts $1,4,2,3$, which is a cascading $3$-pattern for $m=1$ and $d=-2$.
If $d=-1$, suppose for contradiction that the next gap is positive, so that the first four entries are either $1,4,3,5$ or $1,4,3,6$. Then $2$ must be reached from a gap of $-3$ from $5$, at which point the permutation is stuck. Thus the next gap must be $-1$ as well, and the permutation must start $1,4,3,2,5$, which is a cascading $3$-pattern for $m=1$ and $d=-1$.
If $d=+1$, suppose for contradiction that the next gap is positive, so that the first four entries are either $1,4,5,6$ or $1,4,5,7$ or $1,4,5,8$. Then to reach $2$ or $3$, there must be a gap of $-3$ from $6$, at which point the permutation is blocked by $4$, $5$, and $6$ and ends at $2$ or $3$, a contradiction. It follows that the next gap is $-2$ or $-3$, and in fact it must be $-3$ so as to reach the entry $2$ without being blocked. Thus, the first five entries are $1,4,5,2,3,6$, which is a cascading $3$-pattern with $m=1$ and $d=+1$.
Finally, if $d=+2$, suppose for contradiction that the next gap is positive or $-1$. Then as in the case above, the permutation becomes blocked once it reaches $2$ or $3$. So the next gap must be $-3$ and we have $1,4,6,3$ as the first four entries. We must then have $2$ as the fifth entry, or else the sequence would get blocked at $2$ later, so the first six entries are $1,4,6,3,2,5$, which is a cascading $3$-pattern with $m=1$ and $d=+2$.
**Induction step.** Suppose $m>1$ and assume the lemma holds for $m'=m-1$. Then $\pi$ starts with $1,4,7$. We claim that the entries $2$ and $3$ must be adjacent in $\pi$. Suppose they are not adjacent. If $3$ comes first, then the only way to reach $2$ is by a $-3$ gap from $5$ (since $1$ and $4$ are already used) at which point the permutation would be stuck at $2$, a contradiction. If $2$ comes first, then since $7$ comes after $4$ we must have reached the $2$ using a $-3$ gap from $5$. But then the only possible entry that can follow the $2$ is $3$, and they are in fact adjacent.
Now, consider the adjacent positions of the $2$ and $3$. Then the other entry adjacent to $2$ must be $5$, and $6$ must be adjacent to $3$ as well, so the $5$ and $6$ surround the $2$ and $3$. It follows that if we remove $2$, $3$, and $4$ from the permutation and shift all entries larger than $4$ down by $3$, we obtain a permutation $\pi'$ that starts at $1$ with a $+3$ gap to $4$ (which was the $7$ in $\pi$). Since the $5$ and $6$ surrounded the $2$ and $3$ in $\pi$, they become $2$ and $3$ and are adjacent in $\pi'$. All other pairs of adjacent entries in $\pi'$ still have a difference of at most $3$, because they did in $\pi$ and were both translated down by $3$. Thus, $\pi'$ is a $3$-bounded anchored permutation starting with $m-1$ gaps of $+3$, and by the induction hypothesis it must either start with the Joker or a cascading $3$-pattern.
Since the $2$ and $3$ are adjacent in $\pi'$ it cannot start with the Joker and so it must be of the second form. It follows that $\pi$ also starts with a cascading $3$-pattern, formed by inserting one more $+3$ and $-3$ and $+3$ into each of the runs of $3$’s that comprise the gaps of $\pi'$.
We now prove each of the recurrence relations as their own lemma.
\[lem:Fn-recurrence\] We have $F_n = G_{n-1} + H_{n-1} + F_{n-5}$.
Any $3$-bounded anchored permutation either starts with a gap of $+1$, $+2$, or $+3$. If it starts with $+1$ or $+2$, together the number of possibilities are equal to the number of $3$-bounded permutations of $\{2,\ldots,n\}$ that start with either $2$ or $3$, which is exactly $G_{n-1}$.
If it starts with $+3$, then by Lemma \[lem:Joker\] it either starts with the Joker sequence or is a cascading $3$-pattern. If it starts with the Joker, then $\pi(6)=6$ and the first six entries are a permutation of $[6]$, so the entries after the fifth form a $3$-bounded anchored permutation of $\{6,7,\ldots,n\}$. There are therefore $F_{n-5}$ possibilities in this case. Otherwise, the number of possibilities is equal to the number of $3$-bounded permutations of $\{2,\ldots,n\}$ that start with $4$ and end at $n$ but do not start with the Joker, which is exactly $H_{n-1}$. The recursion follows.
\[lem:Gn-recurrence\] We have $G_n=F_n + G_{n-2} + F_{n-3} + G_{n-4} + H_{n-2}$.
We now wish to enumerate the $3$-bounded permutations that start at either $1$ or $2$ and end at $n$. The number starting at $1$ is $F_n$, which is the first term in the recurrence.
For those starting at $2$, if the next entry is $1$ then the third entry can either be $3$ or $4$. We now wish to count $3$-bounded permutations of $\{3,\ldots,n\}$ that start at either $3$ or $4$ and end at $n$, which is exactly $G_{n-2}$.
If the first two entries are $2,3$, then if the next gap is positive it follows that the $1$ can only be reached by a gap of $-3$ from $4$, at which point the permutation is stuck. It follows that the next gap is negative, and it must be a gap of $-2$. So the first four entries are $2,3,1,4$, and the remaining entries starting from $4$ form a $3$-bounded anchored permutation of $\{4,\ldots,n\}$. Thus, there are $F_{n-3}$ possibilities in this case.
If the first two entries are $2,4$, then $1$ can either be reached from a gap of $-3$ from $4$, or later from a gap of $-2$ from $3$. But the latter option becomes stuck at $1$, and so there must be a gap of $-3$ from $4$ to $1$. It follows that the permutation starts $2,4,1,3$ and then continues with a $3$-bounded permutation of $\{5,\ldots,n\}$ that starts at either $5$ or $6$. There are therefore $G_{n-4}$ such possibilities.
Finally, if the first two entries are $2,5$, then the $1$ must occur at some point in $\pi$ and must be surrounded by $3$ and $4$. If we remove the $1$, then, we obtain a $3$-bounded permutation of $\{2,\ldots,n\}$ starting at $2$ and with a starting gap of $+3$, with the $3$ and $4$ adjacent. By Lemma \[lem:Joker\], the $3$ and $4$ will always be adjacent in such a permutation with a starting gap of $+3$ unless it starts with the Joker pattern, and so, removing the $1$ and the $2$, we see that there are exactly $H_{n-2}$ possibilities in this case.
Notice that the final step in the above proof was analogous to the final step of the proof of Lemma \[lem:Fn-recurrence\]. Deleting the $1$ from the permutation resulted in the $H_{n-1}$ term in the $F_n$ recurrence, just as deleting the $1$ and the $2$ from the permutation resulted in the $H_{n-2}$ term in the $G_n$ recurrence. We will use this trick once more below, deleting the $1$, $2$, and $3$, resulting in a $H_{n-3}$ term in the $H_n$ recurrence.
\[lem:Hn-recurrence\] We have $H_n=F_{n-3} + G_{n-3} + F_{n-4} + G_{n-5} + H_{n-3}$.
We wish to enumerate the $3$-bounded permutations that start at $3$ and end at $n$ but do not start with the Joker sequence $3,1,4,2,5$. The second entry can either be $1$, $2$, $4$, $5$, or $6$.
Notice that if we add a $0$ to the front of the permutation, we will get a $3$-bounded anchored permutation of $\{0,\ldots,n\}$ that starts with a gap of $+3$. By Lemma \[lem:Joker\], since the permutation does not start with the Joker, it must start with a cascading $3$-pattern.
Thus, if the first gap after the $3$ is not $+3$, then $d$ is determined and the $3$-pattern is determined as well. In particular, if the first two entries are $3,1$ then the permutation must start with $3,1,2$, and so the entries after the third form a $3$-bounded permutation of $\{4,\ldots,n\}$ that starts at either $4$ or $5$ and ends at $n$. There are exactly $G_{n-3}$ such entries in this case.
If the first two entries are $3,2$ then since the start is a cascading $3$-pattern, the first four entries are $3,2,1,4$. The entries starting at $4$ form a $3$-bounded permutation of $\{4,\ldots,n\}$ starting at $4$ and ending at $n$, giving us $F_{n-3}$ more possibilities.
If the first two entries are $3,4$, then by the cascading $3$-pattern the first five entries are $3,4,1,2,5$. The entries starting at $5$ form a $3$-bounded permutation of $\{5,\ldots,n\}$ starting at $5$ and ending at $n$, giving us $F_{n-4}$ more possibilities.
If the first two entries are $3,5$, the cascading $3$-pattern tells us that the first five entries are $3,5,2,1,4$, with the next entry either $6$ or $7$. The entries starting after the fifth form a $3$-bounded permutation of $\{6,\ldots,n\}$ starting at either $6$ or $7$ and ending at $n$, giving us $G_{n-5}$ more possibilities.
Finally, if the first two entries are $3,6$, then since it is a cascading $3$-pattern the $1$ and $2$ must be adjacent in $\pi$. Removing the $1$, $2$, and $3$ then gives a $3$-bounded permutation of $\{4,\ldots,n\}$ that starts at $6$ and ends at $n$ but avoids the Joker. There are $H_{n-3}$ such possibilities, and the proof is complete.
We can now eliminate $H_n$ from these recurrences to form a two-term recurrence. Putting $n-1$ in the recurrence for $G_n$, we have $G_{n-1}=F_{n-1}+G_{n-3} + F_{n-4} + G_{n-5} + H_{n-3}$, which nearly matches the recurrence for $H_n$. From this we conclude $H_n=F_{n-3}+G_{n-1}-F_{n-1}$. We can now substitute for the $H$ terms in the $F$ and $G$ recurrences to obtain the following relationships: $$\begin{aligned}
F_n &=G_{n-1}+F_{n-4}+G_{n-2}-F_{n-2}+F_{n-5}, \label{eqn:recursionF}\\
G_n &=F_n+G_{n-2}+G_{n-3}+G_{n-4}+F_{n-5}. \label{eqn:recursionG}
\end{aligned}$$ Notice that our proofs above actually show that these recursions hold for all $n$, even $n\leq 5$, where we set $F_j=G_j=0$ for any $j\leq 0$. Thus, we can unwind the recursions to find the first few values of $F_n$ and $G_n$, as follows.
$n$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$
------- ----- ----- ----- ----- ------ ------ ------ ------
$F_n$ $1$ $1$ $1$ $2$ $6$ $14$ $28$ $56$
$G_n$ $1$ $1$ $2$ $4$ $10$ $22$ $45$ $93$
We now have the tools to prove Theorem \[thm:3-bounded\].
We first find the generating function for $\{F_n\}$, and use this to find the single-term recurrence for the sequence.
Let $F(x)=\sum_{n=1}^\infty F_n x^n$ and $G(x)=\sum_{n=1}^\infty G_n x^n$. Then we have $$\begin{aligned}
F(x) & = x+x^2+x^3+2x^4+6x^5 + \sum_{n=6}^\infty F_n x^n \\
x^2F(x)&=\hspace{1.6cm}x^3+\phantom{2}x^4+\phantom{6}x^5 + \sum_{n=6}^\infty F_{n-2} x^n \\
x^4F(x)&=\phantom{x+x^2+x^3+2x^4+6}x^5+ \sum_{n=6}^\infty F_{n-4} x^n \\
x^5F(x) &=\hspace{4.8cm} \sum_{n=6}^\infty F_{n-5} x^n,
\end{aligned}$$ and $$\begin{aligned}
G(x) & = x+x^2+2x^3+4x^4+10x^5 + \sum_{n=6}^\infty G_n x^n \\
xG(x)& = \hspace{0.75cm}x^2+\phantom{2}x^3+2x^4+\phantom{1}4x^5 + \sum_{n=6}^\infty G_{n-1} x^n \\
x^2G(x)&=\phantom{x+x^2+2} x^3+\phantom{2}x^4+\phantom{1}2x^5 + \sum_{n=6}^\infty G_{n-2} x^n \\
x^3G(x)&=\hspace{3cm}x^4+\phantom{10}x^5 + \sum_{n=6}^\infty G_{n-3} x^n \\
x^4G(x)&=\hspace{4.3cm}x^5+ \sum_{n=6}^\infty G_{n-4} x^n. \\
\end{aligned}$$
We can now utilize the recursions (\[eqn:recursionF\]) and (\[eqn:recursionG\]) to make the infinite summations cancel and keep track of the smaller terms, obtaining the following two equations: $$\begin{aligned}
F(x)-xG(x)-x^2G(x)+x^2F(x)-x^4F(x)-x^5F(x)&=x,\\
G(x)-F(x)-x^2G(x)-x^3G(x)-x^4G(x)-x^5F(x)&=0.
\end{aligned}$$ Solving these two equations for $F(x)$ and $G(x)$ gives us that $$F(x)=\frac{x-x^2-x^4}{1-2x+x^2-2x^3-x^4-x^5+x^7+x^8}.$$ Finally, we can multiply both sides of the above relation by the denominator of the fraction, and we find that for $n\geq 8$, $F_n$ satisfies the recursion $$F_n=2F_{n-1}-F_{n-2}+2F_{n-3}+F_{n-4}+F_{n-5}-F_{n-7}-F_{n-8},$$ as desired.
Conjectures and open problems {#sec:conjectures}
=============================
As future work, a direct combinatorial proof of the eight-term recurrence for $F_n$, without relying on algebraic methods for simplification, may lend new insights into the structure of these permutations. In particular, it would be interesting if there was an intrinsic reason for why the recursion has depth $8$.
Along similar lines, for $k\geq 4$, one can ask whether there is always a linear recurrence relation of some depth for the number of $k$-bounded anchored permutations. Given Avgustinovich and Kitaev’s work [@AK] on non-anchored $k$-bounded permutations, and given the complicated recurrence that exists for $k=3$, it seems plausible that there would always exist such a recurrence. This conjecture can be stated in terms of generating functions as follows.
Let $F_{k,n}$ be the number of $k$-bounded anchored permutations of length $n$. Then the generating function $$\sum_{n=1}^\infty F_{k,n}x^n$$ is a rational function of $x$ for any $k\geq 1$.
[99]{} S. Avgustinovich and S. Kitaev, On uniquely $k$-determined permutations, *Discrete Math.* **308** (2008), 1500–1507. P. Flajolet and R. Sedgewick, *Analytic Combinatorics*, Cambridge University Press, 2009. E. Goins and T. Washington, On the generalized climbing stairs problem, *Ars Combin.* **117** (Oct 2014), 183–190. Online Encyclopedia of Integer Sequences, [[](http://oeis.org/A000930)]{}, <https://oeis.org/A000930>. Online Encyclopedia of Integer Sequences, [[](http://oeis.org/A249665)]{}, <https://oeis.org/A249665>. E. S. Page, Systematic generation of ordered sequences using recurrence relations, *Comput. J.* **14** (1971), 150–153. Simon Plouffe, *Approximations de séries génératrices et quelques conjectures*, Dissertation, Université du Québec à Montréal, 1992. R. Stanley, *Enumerative Combinatorics, Vol. 1*, Cambridge University Press, 2001.
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2010 [*Mathematics Subject Classification*]{}: Primary 05A15; Secondary 05A05.
*Keywords:* Permutation, enumeration, recurrence, generating function.
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(Concerned with sequences [[](http://oeis.org/A000930)]{}, [[](http://oeis.org/A249665)]{}.)
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Received ???; revised version received ???. Published in [*Journal of Integer Sequences*]{}, ???.
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Return to . .1in
[^1]: Partially supported by NSF grant PDRF 1604262.
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Introduction
============
The study of strongly correlated systems in low spatial dimensions is nowadays the center of an intense effort both theoretically and experimentally. Among these low-dimensional systems the ones containing two-leg ladders have received considerable attention. One of the original motivation for the study of ladders was the search for a simple mechanism for pairing in a strongly correlated model.[@drs] This possibility was later confirmed by a number of studies and the symmetry of the superconducting order parameter was found to be d$_{x^2-y^2}$ (Refs. ). A considerable interest in these theoretical predictions was renewed by the discovery of superconductivity in the ladder compound Sr$_{14}$Cu$_{24}$O$_{41}$ (14-24-41) after being doped with Ca and under a pressure of $\approx 3 \rm GPa$ (Ref. ).
However, the 14-24-41 compound, as well as many other compounds, like another cuprate SrCu$_2$O$_3$ and the vanadates CaV$_2$O$_5$ and NaV$_2$O$_5$, actually contain layers of two-leg ladders which are [*coupled*]{} by frustrated effective interactions in the so-called trellis lattice. The strength of these frustrating interladder couplings may be weak enough to consider the ladders as essentially isolated or strong enough to change radically the physical behavior of a single ladder.
The original experiments on Sr$_{0.4}$Ca$_{13.6}$Cu$_{24}$O$_{41}$ (Ref. ) reveal that the superconducting critical temperature reaches its maximum of $\rm 12 K$ at a pressure of $\approx 3 \rm GPa$ and then decreases as the pressure is further increased. Similar results were obtained for Sr$_{2.5}$Ca$_{11.5}$Cu$_{24}$O$_{41}$ (Ref. ). The application of pressure to this compound may change the strength of some couplings, or to additionally increase the doping of the ladder layers as holes are transfered from the chain layers also present in this compound. The main purpose of this paper is to analyze the first possibility, neglecting more radical changes in the crystallographic structure.[@pachot] In a highly simplified model for this compound, we consider three sets of couplings: along the ladder legs, along the rungs, and on the “zig-zag" interactions between the ladders.[@Note1] The effect of varying the rung interactions, keeping fixed the leg ones, on a single ladder, has been analyzed extensively.[@rpd] In the present study we will concentrate on the effects of varying the interladder couplings specially on the magnetic and pairing properties.
To this purpose we study the $t-J$ model which is appropriate to describe these cuprates and vanadates characterized by large on-site Coulomb repulsion and close to half-filling. We start our studies with undoped furstrated coupled ladders. For the trellis latttice, it was suggested that it is possible a transition from a spin liquid to a possible spiral order with incommensurate magnetic correlations as the interladder coupling (ILC) increases.[@normandmila] Then, we will consider specially the case of two-hole doping, i.e. the evolution of pairing as the interladder coupling is varied. Preformed hole pairs are already present in the uncoupled ladders[@drs; @noack; @magishi] and a small ILC could lead to superconductivity (SC) as a proximity effect between the ladders. However, our main concern in this work is not the onset of SC but rather the effect on pairing due to somewhat large ILC.
There are further motivations to study both experimentally the 14-24-41 compound and theoretically the $t-J$ model on the trellis lattice. It is in effect remarkable how a relatively small difference like the one between the square lattice of the Cu-O planes in high-T$_c$ cuprates and the trellis lattice in 14-24-41 leads to such a considerable difference in the superconducting properties of those materials. This difference is even more remarkable if we take into account the presence of stripes [@tranquada] in the underdoped regime and (at least) above the superconducting region of the high-T$_c$ cuprates. These stripes can be thought as metallic ladders separated by insulating antiferromagnetic ones, specially in the “bond-centered" stripes obtained from a numerical study of the 2D t-J model[@white].
In this sense, the $t-J$ model on the trellis lattice is a testing ground for the study of the competition between magnetic and kinetic energies which is at the core of the mechanism of micro-phase separation leading to the formation of stripes in the high-T$_c$ cuprates.[@white] Our main conclusion for the trellis lattice is that by increasing the interladder coupling, the balance between the magnetic energy in the ladders and the kinetic energy in the zig-zag chains in between the ladders is altered leading eventually to the destruction of the hole pairs initially formed and localized in the ladders. We also suggest the possibility that the hole pairs may go to the zig-zag chains in a process which represents a transition from a magnetic to a kinetic mechanism of pair binding.
The t-J model on the trellis lattice is given by the Hamiltonian: $$\begin{aligned}
{\cal H}={\cal H}_{leg} + {\cal H}_{rung} + {\cal H}_{inter}
\label{hamtrel}\end{aligned}$$ where $$\begin{aligned}
{\cal H}_\alpha =&-&t_\alpha \sum_{ \langle i j
\rangle,\sigma }({\tilde c}^{\dagger}_{i \sigma}
{\tilde c}_{ j \sigma} + h.c. ) \nonumber \\
&+&~J_\alpha \sum_{
\langle i j \rangle }( {\bf S}_{ i} \cdot
{\bf S}_{ j} -{\frac{1}{4}} n_{ i} n_{ j} )
\nonumber\end{aligned}$$ The couplings are $(t,J)$, $(t^\prime,J^\prime)$ and $(t_{inter},J_{inter})$ along the legs, along the rungs and between the ladders respectively (Fig. \[trelfig\]). The rest of the notation is standard. Periodic boundary conditions in both directions are considered except otherwise stated. We adopt $J=0.4 t$, a value usually taken to model high-T$_c$ cuprates, and in order to reduce the number of independent variables $J_\alpha=J(t_\alpha/t)^2$. Moreover, we take $t=1$. For the undoped compound, neutron scattering experiments for the 14-24-41 compound[@eccleston] suggest $J=2 J^\prime$. However, taking into account the analysis mentioned in Ref . most of our calculations have been done for the isotropic case.
We use various numerical techniques like quantum Monte Carlo (QMC), with a conventional world-line algorithm, and exact diagonalization with the Lanczos algorithm (LD), complemented by the continued fraction formalism to compute dynamical properties.
Frustrated coupled spin ladders {#undoped}
===============================
The first issue we want to address is the evolution of the magnetic order in the absence of doping as the interladder coupling is increased. To this end we have computed the static magnetic structure factor $\rm S(\bf q)$ on $\rm L \times L$ clusters using QMC. Due to the presence of frustration, the minus sign problem prevents us to reach low temperatures and to study larger clusters which would be necessary to perform a finite size scaling. The same problems were already faced in a previous QMC study of the susceptibility of this system.[@miyahara] In order to reduce the minus sign problem and to take advantage of the simple checkerboard decomposition,[@reger] we take a slightly modified lattice in which the interladder couplings are “perpendicular" ($J_{perp}$) and “diagonal" ($J_{diag}$) as shown in Fig. \[trelfig\]. This lattice contains just half of the diagonal interladder couplings than in the trellis lattice, and we have checked by exact diagonalization on small clusters that this difference does not change qualitatively the results. Even for this modified lattice the minus sign problem is severe as shown in Fig. \[signo\]. We recall that the average of any observable ${\cal O}$, in a system which presents this problem is computed as:[@wiese] $$\begin{aligned}
\langle {\cal O}\rangle= \frac{\langle {\cal O} Sign\rangle}
{\langle Sign \rangle}
\nonumber\end{aligned}$$ with respect to a modified partition function of the $2+1$-dimensional problem, ${\cal Z'}= \sum_s |\exp{S(s)}|$ where $S(s)$ is an effective action. In particular, the average sign is: $$\begin{aligned}
{\langle Sign \rangle } = \frac{1}{\cal Z'} \sum_s Sign(s) |\exp{S(s)}|,
\nonumber\end{aligned}$$ where $Sign(s)=sign(\exp{S(s)})$. Then ${\langle Sign \rangle }$ is the ratio of the original partition function ${\cal Z}= \sum_s \exp{S(s)}$ to ${\cal Z'}$. In practice, at each measurement step, $\exp{S(s)}$ is computed as the product of the transition elements of
all the cubes that makes up the $2+1$-dimensional lattice.[@reger] Nonetheless, although we cannot compute some quantities in the bulk limit, we can indicate qualitatively the behavior of the magnetic order as a function of $J_{inter}$ (we take $J$ as the unit of energies in this section).
Typical results are shown in Fig. \[stfac\] for coupled isotropic ($J^\prime=J$) ladders on the $8\times 8$ cluster. In Fig. \[stfac\](a) we show for $J_{perp}=J_{diag}=0.2$ the characteristic structure factor of isolated ladders with a peak at $(q_x,q_y)=(\pi,\pi)$ ($x$ ($y$) is the direction along (transversal) to the ladders). This peak becomes more pronounced as the temperature is lowered. On the other hand, keeping $J_{perp}=0.2$ and as $J_{diag}$ is increased, the peak starts to shift from $(\pi,\pi)$ to $(\pi,\pi/2)$ (Fig. \[stfac\](b,c,d)). A second interesting feature should be noticed in Fig. \[stfac\](d): the peak of $\rm S(\bf q)$ is located at $(\pi,\pi)$ at high temperature (in this case down to $\rm T \approx
0.8$, in units of $J$) and as the temperature is lowered it starts to shift away from $(\pi,\pi)$. At low temperatures (in this case below $\rm T \approx 0.4$ the peak is located at $(\pi,\pi/2)$. Since it is clear that this behavior is caused by the frustration of the interladder couplings, it will certainly be present in the original trellis lattice. As indicated in Fig. \[stfac\]c, it is possible that an incommensurate peak across the ladder direction could be present at intermediate values of $J_{diag}$ and intermediate temperatures.
The second point we want to examine is the behavior of the excitations of these systems, in particular the $S=1$ excitations as can be measured by neutron scattering experiments. For this purpose, using conventional LD with the standard continued fractions formalism,[@haas] we have computed the zero temperature dynamical structure function ($zz$ component) $S({\bf q},\omega)$. In this case, we have to limit ourselves to somewhat smaller clusters but we are confident that the qualitative features we found will survive in the bulk limit.
Results obtained for the $4 \times 4$ cluster are shown in Fig. \[dynamic\]. In the absence of frustration (Fig. \[dynamic\](a)) the peak in $S({\bf q},\omega)$ which corresponds besides to the lowest excitation, is located at $(\pi,\pi)$, as expected in the bulk limit for an AF order. As a frustrating ILC is increased (Fig. \[dynamic\](b,c,d)) it can be seen that considerable spectral weight is transferred to the peak at $(\pi,\pi/2)$, which becomes finally the lowest energy excitation. Similar results are also shown for the $4 \times 6$ cluster for ${\bf q}=(\pi,\pi)$ and $(\pi,\pi/2)$.
The results shown in Figs. \[stfac\] and \[dynamic\] are unequivocally due to frustration and are qualitatively similar to the ones previously obtained in a system of ferromagnetically (FM) coupled ladders.[@dalosto] Similar results have been obtained by LD on the $4\times4$ and $4\times6$ clusters of the real trellis lattice. In the case of the trellis lattice, as in the FM ILC case, we expect that the behavior above discussed will be present in the bulk limit for strong enough interladder couplings and low enough temperatures. The impossibility of assessing finite size effects prevents us to determine if this behavior is present for arbitrarily small values of $\rm J_{inter}$ or, on the contrary, only for values larger than a “critical" one.
We have not detected any sign of incommensurability along the ladder direction.[@normandmila] Such an incommensurability is expected in principle since a trellis lattice can be also considered as coupled $J_1-J_2$ chains ($J_1=J_{inter}$, $J_2=J$, in our notation) which are known to present a peak in $S(q)$ at a momentum which continuously varies from $\pi$ to $\pi/2$ (as defined on our modified trellis lattice) as $J_2 / J_1$ goes from $\infty$ to zero.[@whiteaffleck] However, notice that, as can be seen in Fig. \[signo\], values of $J_{inter} > 0.6$ at low enough temperatures cannot be reached in our simulations.[@Note3]
Doped trellis lattice. {#doped}
======================
We now analyze the hole pairing in the doped trellis lattice as the interladder couplings are increased. To gain some insight in this problem we start by considering an isolated building block of the trellis lattice. This minimal system is a three chain cluster consisting of a ladder and a zig-zag chain, (lines “1", “2" and ‘3’ in Fig. \[trelfig\]). The Hamiltonian is the one defined in Eq. (\[hamtrel\]). All the results in this section have been obtained by exact diagonalization. The justification of this study involving somewhat small clusters is based on an extensive body of similar studies of strongly correlated models which shows that an important part of the physics of these models is dominated by short range effects, which are appropriately captured in these small cluster calculations.
In this minimal trellis lattice already appears, upon doping with two holes, the main feature we want to emphasize. In Fig. \[holeleg\], the relative hole occupancy (or probability of finding a hole) on each chain is shown in the $3\times 6$ cluster with two holes for $t^\prime=0.75$, 1.0 and 1.5 as a function of $t_{inter}$. In the three cases, for small $t_{inter}$ the holes are almost completely located in the ladder legs. As $t_{inter}$ increases the probability of finding a hole in the outer ladder leg decreases while increases the occupancy of chain “3" which is connected to the ladder by the zig-zag interaction. There is a neat change of behavior, from a situation in which the occupancy of chain “3" is virtually unoccupied to a situation in which the unoccupied chain is “1". moreover, this crossover is rather abrupt, specially for $t^\prime=0.75$.
For relatively small $t_{inter}$, the hole distribution is typical of that of an isolated ladder[@rieradag], i.e. they form a bound pair with one hole on each leg. On the other hand, for larger values of $t_{inter}$ ($t_{inter} > 0.6$ for $t^\prime=1.0$ and $t_{inter} > 0.9$, for $t^\prime=1.5$, on the $3\times 8$ cluster) the holes have moved from the ladder to the two chains containing the zig-zag interactions. As we indicated in the previous section, these two chains with the zig-zag interactions form a chain with first and second neighbor interactions, in this case with $t-J$ couplings, i.e. a frustrated $t_1-t_2-J_1-J_2$ chain.[@ogata]
To understand the mechanism that produces this change in hole pairing, let us examine the contributions to the total energy from different terms of the Hamiltonian (\[hamtrel\]) as the interladder hopping is increased. As it can be seen in Fig. \[contr3x8h2\], the main differences between the contributions from the ladder and those of the zig-zag chain are (i) in the ladder the magnetic energy dominates (along the legs for $t^\prime=1.0$ or along the rungs for $t^\prime=1.5$) while in the zig-zag the kinetic energy is the most important, and (ii) the main gain in energy as $t_{inter}$ is increased comes from precisely the hopping term of the zig-zag ILC while the magnetic energy on ladders is the most strongly decreased.
The gain in kinetic energy on the frustrated chain with respect to the ladder can be explained by qualitative arguments as is shown schematically in Fig \[esquema\]. In the frustrated chain we have assumed an AF order of the spins along the chains which is expected for $J_2 > J_1$ ($t_2 > t_1$). When the hole moves, as in a simple $t-J$ chain, the hole leaves behind just a single frustrated (ferromagnetic) bond. Something similar occurs in the case of AF order along the zigzag chain ($J_1 \ge J_2$). In the case of ladders, we have assumed a magnetic background formed by spin singlets on the rungs. As a hole moves from its initial position, it leaves behind a string of higher energy singlets on the diagonals of the plaquettes. Hence there
is a cost in energy which increases roughly linearly with the distance traveled by the hole.
What are the possible consequences of this behavior found in the three-chain cluster for the trellis lattice? In this case, of course, any chain along the ladder direction belongs at the same time to a ladder and to a frustrated interladder chain. The question is if the holes, initially paired on a plaquette in isolated ladders, would tend to break the pairs and move independently on the frustrated chains as the ILC are increased. There is yet another interesting possibility that is that the holes form pairs on the frustrated chains. These pairs would have more kinetic energy than the ones formed on ladders and this change of pairing would imply a change from a magnetic binding on ladders to a “kinetic binding" on chains. In any case, taking into account the results from the three-chain cluster, we predict a loosening of the pairing on ladders. It is difficult to answer these questions by calculations on finite clusters. Exact diagonalization results for two holes on the $4\times4$ cluster give support to the above mentioned possibilities. In Fig. \[hh4xn\] the hole-hole correlation functions for $t^\prime=1.0$ at several distances as a function of $t_{inter}$ are shown. At small $t_{inter}$, the largest correlation corresponds to a pair of holes along the diagonal of a plaquette, which is typical of isolated ladders. Around $t_{inter}=0.7$ there is an abrupt change to a situation in which the largest correlations correspond to holes belonging to the same frustrated chain. The second hole is slightly more likely to be in the other chain of the same interladder zig-zag chain (site ‘4’ of Fig. \[hh4xn\]). Somewhat smaller is the correlation on the same chain but at the largest distance on this cluster (site ‘3’). This behavior is radically different to ladders AF coupled in a square lattice without frustration. In this case, the d-wave pair typical of a ladder evolves smoothly to the rotationally invariant d-wave pair of the square lattice.
Similar results are obtained for the $6\times4$ cluster with two holes. In Fig.\[hh6x4\], we show pictorially the most likely hole probability for $t^{\prime}=t_{inter}=1$. The area of the circles is proportional to the probability of finding a hole if there is a hole in a given site. In this case the largest probability corresponds to holes located at the maximum distance along the same chain. The next probability in decreasing order also corresponds to a hole in the frustrated zig-zag chain. On this cluster we found that as $t_{inter}$ is increased from zero the holes initially at a distance $\sqrt{2}$ starts to move away on the same ladder and finally they move to the same ILC chain.
Finally, for the sake of completeness, we show in Fig. \[ss4xn\] the largest nearest and next nearest neighbor spin-spin correlations for small and large values of $t_{inter}$ for the most likely position of the holes in each case (see Fig. \[hh4xn\]). The structure of these correlations globally agrees with the schematic picture of Fig. \[esquema\].
Conclusions
===========
In summary, we have performed numerical studies on strongly correlated electron systems, as described by the $t-J$ model, on frustrated coupled ladders, in particular on the trellis lattice. In the undoped case, QMC simulations, although hampered by the minus sign problem, allowed us to reach large enough clusters to detect meaningful changes in the magnetic properties of this system. In this case, for ladders coupled with frustrating interactions, we have shown that the peak of the magnetic structure factor shifts from $(\pi,\pi)$ to $(\pi,\pi/2)$ for low enough temperatures as the ILC is increased. Moreover, the peak at $(\pi,\pi/2)$ becomes also the lowest energy excitation. This behavior is very similar to the one previously found for FM coupled ladders[@dalosto]. We have shown also that this behavior appears due to the onset of frustration and hence we expect that it will appear in the trellis lattice as well. This behavior could be detected experimentally on a ladder compound like SrCu$_2$O$_3$[@azuma] or Sr$_{14}$Cu$_{24}$O$_{41}$ upon a suitable application of pressure. In fact, after the submission of this manuscript we became aware of an experimental study[@kiryukhin] on the similar ladder compound CaCu$_2$O$_3$. The neutron diffraction results reported in this manuscript indicate that the magnetic structure is incommensurate in the direction of the frustrated interladder interaction in the plane of the ladders, as suggested by the present study (see also Ref. ). The validity of the simpler FM coupled ladders model also explains the vanishing of the spin gap[@mayaffre] as due to increasing interladder couplings.
Next, we have analyzed the evolution of pairing when the ILC are increased. In this case the physics is governed by short range effects and so we studied small clusters with exact diagonalization. Our main result is that ILC suppresses pairing in ladders. Results on $4\times4$ and $6\times4$ clusters indicate that holes move to the chains with first and second neighbor interactions formed by the legs of neighboring ladders and the zig-zag interactions between them. We have identified the mechanism of this suppression of pairing as a gain of kinetic energy of the holes by moving on the frustrated chains. Even for these clusters we have noticed important size effects which unable us to determine if it appears an alternative pairing of holes on the frustrated chains or rather the pairing is completely lost when ILC are large enough and holes begin to move independently from each other. Finally, we would like to stress the radically different behavior found in this case with respect to that found in ladders coupled in a square lattice, where the d$_{x^2-y^2}$ pairing is preserved.
The authors acknowledge many interesting discussions with A. Dobry, C. Gazza, A. Greco, D. Poilblanc and A. Trumper. Part of the computations were performed at the Supercomputer Computations Research Institute (SCRI) and at the Academic Computing and Network Services at Tallahassee(Florida).
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---
abstract: 'Employing deep neural networks as natural image priors to solve inverse problems either requires large amounts of data to sufficiently train expressive generative models or can succeed with no data via untrained neural networks. However, very few works have considered how to interpolate between these no- to high-data regimes. In particular, how can one use the availability of a small amount of data (even $5-25$ examples) to one’s advantage in solving these inverse problems and can a system’s performance increase as the amount of data increases as well? In this work, we consider solving linear inverse problems when given a small number of examples of images that are drawn from the same distribution as the image of interest. Comparing to untrained neural networks that use no data, we show how one can pre-train a neural network with a few given examples to improve reconstruction results in compressed sensing and semantic image recovery problems such as colorization. Our approach leads to improved reconstruction as the amount of available data increases and is on par with fully trained generative models, while requiring less than $1 \%$ of the data needed to train a generative model.'
author:
- |
Oscar Leong[^1]\
Rice University\
`[email protected]`\
Wesam Sakla\
Lawrence Livermore National Laboratory\
`[email protected]`\
bibliography:
- 'low\_shot.bib'
title: Low Shot Learning with Untrained Neural Networks for Imaging Inverse Problems
---
Introduction
============
We study the problem of recovering an image ${\pmb{x}}_0 \in {\mathbb{R}}^n$ from $m$ linear measurements of the form $${\pmb{y}}_0 = {\pmb{A}}{\pmb{x}}_0 + {\pmb{\eta}}\in {\mathbb{R}}^m$$ where ${\pmb{A}}\in {\mathbb{R}}^{m \times n}$ is a known measurement operator and ${\pmb{\eta}}\in {\mathbb{R}}^m$ denotes the noise in our system. Problems of this form are ubiquitous in various domains ranging from image processing, machine learning, and computer vision. Typically, the problem’s difficulty is a result of its ill-posedness due to the underdetermined nature of the system. To resolve this ambiguity, many approaches enforce that the image must obey a natural image model. While traditional approaches typically use hand-crafted priors such as sparsity in the wavelet basis [@Dono2007], recent approaches inspired by deep learning to create such natural image model surrogates have shown to outperform these methods.
#### Deep Generative Priors:
Advancements in generative modelling have allowed for deep neural networks to create highly realistic samples from a number of complex natural image classes. Popular generative models to use as natural image priors are latent variable models such as Generative Adversarial Networks (GANs) [@Goodfellow2014] and Variational Autoencoders (VAEs) [@Kingma2014]. This is in large part due to the fact that they provide a low-dimensional parameterization of the natural image manifold that can be directly exploited in inverse imaging tasks. When enforced as a natural image prior, these models have shown to outperform traditional methods and provide theoretical guarantees in problems such as compressed sensing [@Boraetal2017; @Wuetal2019; @Hand2017; @Huangetal2018; @SH2018; @Hussein2019], phase retrieval [@Leong2018; @Shamshad2018; @Hyderetal2019], and blind deconvolution/demodulation [@Asimetal2018; @HJ2019]. However, there are two main drawbacks of using deep generative models as natural image priors. The first is that they require a large amount of data to train, e.g., hundreds of thousands of images to generate novel celebrity faces. Additionally, they suffer from a non-trivial representation error due to the fact that they model the natural image manifold through a low-dimensional parameterization.
#### Untrained Neural Network Priors:
On the opposite end of the data spectrum, recent works have shown that randomly initialized neural networks can act as natural image priors without any learning. [@Ulyanov2017] first showed this to be the case by solving tasks such as denoising, inpainting, and super-resolution via optimizing over the parameters of a convolutional neural network to fit to a single image. The results showed that the neural network exhibited a bias towards natural images, but due to the high overparameterization in the network, required early stopping to succeed. A simpler model was later introduced in [@HH2019] which was, in fact, underparameterized and was able to both compress images while solving various linear inverse problems. Both methods require no training data and do not suffer from the same representation error as generative models do. Similar to generative models, they have shown to be successful image priors in a variety of inverse problems [@HH2019; @Heckel2019; @VVetal2019; @JH2019].
Based on these two approaches, we would like to investigate how can one interpolate between these data regimes in a way that improves upon work with untrained neural network priors and ultimately reaches or exceeds the success of generative priors. More specifically, we would like to develop an algorithm that 1) performs just as well as untrained neural networks with no data and 2) improves performance as the amount of provided data increases.
#### Our contributions:
We introduce a framework to solve inverse problems given a few examples (e.g., $5-25$) drawn from the same data distribution as the image of interest (e.g., if the true image is of a human face, the examples are also human faces). Our main contributions are the following:
- We show how one can pre-train a neural network using a few examples drawn from the data distribution of the image of interest. Inspired by [@Bojanowski2017], we propose to jointly learn a latent space and parameters of the network to fit to the examples that are given and compare the use of an $\ell_2$ reconstruction loss and a kernel-based Maximum Mean Discrepancy (MMD) loss.
- We then propose to solve the inverse problem via a two-step process. We first optimize over the pre-trained network’s latent space. Once a solution is found, we then refine our estimate by optimizing over the latent space and parameters of the network jointly to improve our solution. [@Hussein2019] found this method to work well in the case when the network is a fully trained generative model, and we show here that even a pre-trained neural network from a small number of examples can benefit from such an approach.
- We show that our approach improves upon untrained neural networks in compressed sensing even with as few as $5$ examples from the data distribution and exhibits improvements as the number of examples increases. We also show that semantics can be learned from these few examples in problems such as colorization where untrained neural networks fail. With only $100$ examples, our model’s performance is competitive with fully trained generative models.
#### Related work:
We mention that there has been previous work [@VVetal2019] in investigating how to use a small amount of data to help solve the compressed sensing problem. The authors use an untrained neural network as a natural image prior and, when given a small amount of data, adopt a learned regularization term when solving the inverse problem. This term is derived by posing the recovery problem as a Maximum a Posteriori (MAP) estimation problem and by placing a Gaussian prior on the weights of the untrained network. While we have not compared our method to this learned regularization approach here, we aim to do so in a subsequent manuscript.
Low Shot Learning For Imaging Inverse Problems {#low_shot_sec}
==============================================
We consider the problem of recovering an image ${\pmb{x}}_0 \in {\mathbb{R}}^n$ from noisy linear measurements of the form ${\pmb{y}}_0 = {\pmb{A}}{\pmb{x}}_0 + {\pmb{\eta}}\in {\mathbb{R}}^m$ where ${\pmb{A}}\in {\mathbb{R}}^{m \times n}$ and $m \leqslant n$. We also assume that ${\pmb{x}}_0$ is drawn from a particular data distribution $\mathcal{D}$ and that we are given a low number of examples drawn from the same distribution, i.e., ${\pmb{x}}_0 \sim \mathcal{D}$ and we are given ${\pmb{x}}_i \sim \mathcal{D}$ where $i \in [S]$. Here and throughout this work, we refer to these examples drawn from $\mathcal{D}$ as *low shots*.
We propose using the range of a deep neural network as a natural image model. In particular, we model the image ${\pmb{x}}_0$ as the output of a neural network ${\mathcal{G}}({\pmb{z}}; {\pmb{\theta}})$, where ${\pmb{z}}\in {\mathbb{R}}^k$ is a latent code and ${\pmb{\theta}}\in {\mathbb{R}}^P$ are the parameters of the network.
#### Pre-training:
Prior to solving the inverse problem, we propose to first pre-train the network using the low shots that are given. More specifically, we fit the weights and input of the neural network to the low shots to provide a crude approximation to the data distribution underlying the image of interest. Given low shots $\{{\pmb{x}}_i\}_{i=1}^S$, we aim to find latent codes $\{{\pmb{z}}_i\}_{i=1}^S$ and parameters ${\pmb{\theta}}$ that solve $$\begin{aligned}
\min_{{\pmb{\theta}},{\pmb{z}}_1,\dots,{\pmb{z}}_S} \frac{1}{S} \sum_{i=1}^S \mathcal{L}(\mathcal{G}({\pmb{z}}_i; {\pmb{\theta}}), {\pmb{x}}_i). \label{general_training_objective}
\end{aligned}$$ where $\mathcal{L} : {\mathbb{R}}^n \times {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ is a loss function. We investigate the use of different loss functions in a later section. The resulting optimal parameters found are denoted by $\hat{{\pmb{\theta}}}$, $\hat{{\pmb{z}}}_1,\dots,\hat{{\pmb{z}}}_S$.
#### Solving the inverse problem:
Using the weights found via pre-training, we begin solving the inverse problem by first optimizing over the latent code space to find an approximate solution: $$\begin{aligned}
\min_{{\pmb{z}}} \frac{1}{2} \Big\| {\pmb{A}}\mathcal{G}({\pmb{z}};\hat{{\pmb{\theta}}}) - {\pmb{y}}_0 \Big\|^2_2. \label{solve_wrt_z}\end{aligned}$$ We investigated different ways to initialize the latent code and found that sampling from a multivariate Gaussian distribution fit using $\{\hat{{\pmb{z}}}_i\}_{i=1}^S$ was sufficient. Note here that we keep the parameters of the network fixed after training. The intuition is that we want to use the semantics regarding the data distribution learned via pre-training the network’s parameters and find the optimal latent code that corresponds to the image of interest. Once the optimal latent code $\hat{{\pmb{z}}}$ is found, we then refine our solution by solving $$\begin{aligned}
\min_{{\pmb{\theta}},{\pmb{z}}} \frac{1}{2} \Big\| {\pmb{A}}\mathcal{G}({\pmb{z}};{\pmb{\theta}}) - {\pmb{y}}_0 \Big\|^2_2 \label{solve_wrt_z_and_theta}\end{aligned}$$ with $\hat{{\pmb{\theta}}}$ and $\hat{{\pmb{z}}}$ as our initial iterates. The resulting parameters ${\pmb{\theta}}_0$ and ${\pmb{z}}_0$ give our final estimate: ${\pmb{x}}_0 \approx \mathcal{G}({\pmb{z}}_0; {\pmb{\theta}}_0)$.
#### Losses to learn the data distribution:
We discuss two loss functions that we considered in our experiments to learn semantics regarding the underlying data distribution. The first is a simple $\ell_2$ reconstruction loss to promote data fidelity, i.e., we pre-train our network by solving $$\begin{aligned}
\min_{{\pmb{\theta}},{\pmb{z}}_1,\dots,{\pmb{z}}_S} \frac{1}{S}\sum_{i=1}^S \left\|\mathcal{G}({\pmb{z}}_i; {\pmb{\theta}}) - {\pmb{x}}_i\right\|_2^2. \label{ell2_training_objective}\end{aligned}$$ While [@Bojanowski2017] used a combination of the Laplacian-L1 loss and $\ell_2$ loss, we found the $\ell_2$ loss to work well.
The second loss is an estimate of the kernel MMD for comparing two probability distributions using only finitely many samples [@Gretton2012]. In our case, given a kernel $k(\cdot,\cdot)$[^2] and low shots ${\pmb{x}}_j \sim \mathcal{D}$ for $j \in [S]$, we want to find parameters and $S$ inputs that solve the following: $$\begin{aligned}
\min_{{\pmb{\theta}},{\pmb{z}}_1,\dots,{\pmb{z}}_S} \frac{1}{\binom{S}{2}}\sum_{i \neq i'} k(\mathcal{G}({\pmb{z}}_i; {\pmb{\theta}}), \mathcal{G}({\pmb{z}}_{i'}; {\pmb{\theta}})) + \frac{1}{\binom{S}{2}}\sum_{j \neq j'} k({\pmb{x}}_j, {\pmb{x}}_{j'}) - \frac{2}{\binom{S}{2}} \sum_{i \neq j} k(\mathcal{G}({\pmb{z}}_i; {\pmb{\theta}}), {\pmb{x}}_j). \label{MMD_training_objective}\end{aligned}$$
We compare the success of these two loss functions in the following section.
Experiments
===========
We now consider solving inverse problems with our approach and compare to three different baselines: an untrained neural network, optimizing the latent space of a trained Wasserstein GAN [@WGAN2017] with gradient penalty [@WGANGP2017], and the image-adaptivity approach of [@Hussein2019] (IAGAN). Each method uses the same DCGAN architecture with a latent code dimension of $128$. In each problem, the image of interest is from a hold-out test set from the CelebA dataset [@CelebA]. The GAN was trained on a corpus of over $200,000$ $64 \times 64$ celebrity images and our low-shot models were trained on small ($5-100$ images) subsets of this.
#### Implementation details:
For training our low shot models, we used the Adam optimizer [@ADAM] with a learning rate of $10^{-3}$ for $50,000$ iterations. To solve the inverse problem, the latent space of the GAN was optimized using Adam for $1700$ iterations and a learning rate of $10^{-1}$. For IAGAN, the parameters and latent code were then jointly optimized for an additional $350$ iterations with a learning rate of $10^{-4}$. Our low shot models followed a similar procedure where we first optimized the latent space for $1250$ iterations with a learning rate of $5*10^{-2}$. Then the parameters and latent space were jointly optimized for $350$ iterations with a learning rate of $10^{-4}$. For the untrained neural network, we solely optimized the network’s parameters using the RMSProp optimizer [@RMSProp] with a learning rate of $10^{-3}$ and momentum of $0.9$ as in [@VVetal2019]. For low compression ratios (defined below) in compressed sensing ($\frac{m}{n} \leqslant 0.025$), we used $350$ iterations as we noticed overfitting in this noisy setting. When $0.025 < \frac{m}{n} \leqslant 0.5$ and $\frac{m}{n} > 0.5$, we used $500$ and $1000$ iterations, respectively.
#### Compressed Sensing:
We first consider the compressed sensing problem where we want to recover an image ${\pmb{x}}_0 \in {\mathbb{R}}^n$ from random Gaussian measurements of the form ${\pmb{y}}_0 = {\pmb{A}}{\pmb{x}}_0 \in {\mathbb{R}}^m$ where ${\pmb{A}}\in {\mathbb{R}}^{m \times n}$ has i.i.d. $\mathcal{N}(0,1)$ entries with $m \ll n$. We refer to amount of undersampling $\frac{m}{n}$ as the *compression ratio*. We trained our models using the two different loss functions proposed in the previous section for various numbers of shots $S \in [5,10,15,25,50, 100]$.
![Average PSNR over $50$ different test images for models trained with the MMD loss (left) and the $\ell_2$ loss (right).[]{data-label="fig:comparisons_cs"}](psnr_cs_comparison_ggplot_cropped.png){width="\textwidth"}
Figure \[fig:comparisons\_cs\] compares the average PSNR for each method at various compression ratios over $50$ different test images and different loss functions. We note that as the number of shots increases, our method continues to improve and we see comparable performance between our method with only $100$ shots and optimizing over the latent code space of a fully trained GAN. While the $\ell_2$ trained nets perform slightly better than the MMD trained nets for low numbers of shots, the MMD trained nets improve more steadily and consistently as the number of shots increases. While we expect IAGAN to be superior due to being trained with over $200,000$ images, the MMD trained model’s performance with $100$ images is not far behind. We note that for higher numbers of measurements, the untrained neural network’s performance surpasses that of our $5$ shot models. This is mainly due to the fact that we optimized the untrained neural network’s parameters for a longer period of time and with a higher learning rate than our low shot models in this easier setting.
#### Colorization:
We now consider an inverse problem that requires an understanding of the underlying data’s semantics: the colorization task. Here we want to recover an RGB image ${\pmb{x}}_0 \in {\mathbb{R}}^{64 \times 64 \times 3}$ from its grayscale version ${\pmb{y}}_0 = {\pmb{A}}{\pmb{x}}_0 \in {\mathbb{R}}^{64 \times 64}$. The operator ${\pmb{A}}$ mixes the color channels of the image via the ITU-R 601-2 luma transform (the same transform used by the Python Imaging Library (PIL)). Untrained neural networks clearly fail in solving this type of problem since they have no prior information regarding the data distribution. We compare our model trained with $10$ shots using the MMD loss to the various baselines in Figure \[fig:comparisons\_colorization\]. Note that even with only $10$ previous examples, our model does not fall prey to the usual issues with using untrained neural networks for colorization. Our algorithm provides faithful image reconstructions that are even on par with a trained GAN.
### Disclaimer {#disclaimer .unnumbered}
This document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.
[^1]: This work was performed at the Data Science Summer Institute (DSSI) under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
[^2]: We only consider the Gaussian kernel $k_{{\alpha}}({\pmb{x}}_1, {\pmb{x}}_2) := \exp(-\|{\pmb{x}}_1 - {\pmb{x}}_2\|_2^2 / {\alpha})$ in our experiments.
|
---
abstract: 'We show that the following group constructions preserve the semilinearity of the solution sets for knapsack equations (equations of the form $g_1^{x_1} \cdots g_k^{x_k} = g$ in a group $G$, where the variables $x_1, \ldots, x_k$ take values in the natural numbers): graph products, amalgamated free products with finite amalgamated subgroups, HNN-extensions with finite associated subgroups, and finite extensions. Moreover, we study the dependence of the so-called magnitude for the solution set of a knapsack equation (the magnitude is a complexity measure for semi-linear sets) with respect to the length of the knapsack equation (measured in number of generators). We investigate, how this dependence changes under the above group operations.'
address:
- 'Universit[ä]{}t Siegen, Germany'
- 'Max Planck Institute for Software Systems, Kaiserslautern, Germany'
author:
- Michael Figelius
- Markus Lohrey
- Georg Zetzsche
title: Closure properties of knapsack semilinear groups
---
[^1]
Introduction
============
The study of algorithmic problems has a long tradition in combinatorial group theory, going back to the work of Dehn [@Dehn11] on the word and conjugacy problem in finitely generated groups. Myasnikov, Nikolaev, and Ushakov initiated in [@MyNiUs14] the systematic investigation of a new class of algorithmic problems that have their origin in discrete optimization problems over the integers. One of these problems is the [*knapsack problem*]{}. Myasnikov et al. proposed the following definition for the knapsack problem in a finitely generated group $G$: The input is a sequence of group elements $g_1, \ldots, g_k, g \in G$ (specified by finite words over the generators of $G$) and it is asked whether there exist natural numbers $x_1, \ldots, x_k \in \mathbb{N}$ such that $g_1^{x_1} \cdots g_k^{x_k} = g$ in $G$. For the particular case $G = \mathbb{Z}$ (where the additive notation $x_1 \cdot g_1 + \cdots + x_k \cdot g_k = g$ is usually preferred) this problem is [NP]{}-complete if the numbers $g_1,\ldots, g_k,g \in \mathbb{Z}$ are given in binary notation [@Karp72; @Haa11].[^2] On the other hand, if $g_1,\ldots, g_k,g$ are given in unary notation, then the knapsack problem for the integers was shown to be complete for the circuit complexity class $\mathsf{TC}^0$ [@ElberfeldJT11]. Note that the unary notation for integers corresponds to the case where an integer is given by a word over a generating set $\{t,t^{-1}\}$. In on particular case, the knapsack problem was studied for a non-commutative group before the work of Myasnikov et al.: In [@BabaiBCIL96], it was shown that the knapsack problem for commutative matrix groups over algebraic number fields can be solved in polynomial time.
Let us give a brief survey over the results that were obtained for the knapsack problem in [@MyNiUs14] and successive papers:
- Knapsack can be solved in polynomial time for every hyperbolic group [@MyNiUs14]. In [@FrenkelNU15] this result was extended to free products of any finite number of hyperbolic groups and finitely generated abelian groups. Another further generalization was obtained in [@LOHREY2019], where the smallest class of groups that can be obtained from hyperbolic groups using the operations of free products and direct products with $\mathbb{Z}$ was considered. It was shown that for every group in this class the knapsack problem belongs to the parallel complexity class ${\sf LogCFL}$ (a subclass of ${\sf P}$).
- There are nilpotent groups of class $2$ for which knapsack is undecidable. Examples are direct products of sufficiently many copies of the discrete Heisenberg group $H_3(\mathbb{Z})$ [@KoenigLohreyZetzsche2015a], and free nilpotent groups of class $2$ and sufficiently high rank [@MiTr17].
- Knapsack for $H_3(\mathbb{Z})$ is decidable [@KoenigLohreyZetzsche2015a]. In particular, together with the previous point it follows that decidability of knapsack is not preserved under direct products.
- Knapsack is decidable for every co-context-free group [@KoenigLohreyZetzsche2015a], i.e., groups where the set of all words over the generators that do not represent the identity is a context-free language. Lehnert and Schweitzer [@LehSch07] have shown that the Higman-Thompson groups are co-context-free.
- Knapsack belongs to ${\sf NP}$ for all virtually special groups (finite extensions of subgroups of graph groups) [@LohreyZetzsche2016a]. The class of virtually special groups is very rich. It contains all Coxeter groups, one-relator groups with torsion, fully residually free groups, and fundamental groups of hyperbolic 3-manifolds. For graph groups (also known as right-angled Artin groups) a complete classification of the complexity of knapsack was obtained in [@LohreyZ18]: If the underlying graph contains an induced path or cycle on 4 nodes, then knapsack is ${\sf NP}$-complete; in all other cases knapsack can be solved in polynomial time (even in [LogCFL]{}).
- Decidability of knapsack is preserved under finite extensions, HNN-extensions over finite associated subgroups and amalgamated free products over finite subgroups [@LohreyZetzsche2016a].
For a knapsack equation $g_1^{x_1} \cdots g_k^{x_k} = g$ we may consider the set $\{ (n_1, \ldots, n_k) \in \mathbb{N}^k \mid g_1^{n_1} \cdots g_k^{n_k} = g \text{ in } G\}$. In the papers [@LOHREY2019; @KoenigLohreyZetzsche2015a; @LohreyZ18] it turned out that in many groups the solution set of every knapsack equation is a [*semilinear set*]{}. Recall that a subset $S \subseteq \mathbb{N}^k$ is semilinear if it is a finite union of linear sets, and a subset $L \subseteq \mathbb{N}^k$ is linear if there a vectors $v_0, v_1, \ldots, v_l \in \mathbb{N}^k$ such that $L = \{ v_0 + \lambda_1 v_1 + \cdots + \lambda_l v_l \mid \lambda_1, \ldots, \lambda_l \in \mathbb{N}\}$. Semilinear sets play a very important role in many areas of computer science and mathematics, e.g. in automata theory and logic. It is known that the class of semilinear sets is closed under Boolean operations and that the semilinear sets are exactly the Presburger definable sets (i.e., those sets that are definable in the structure $(\mathbb{N},+)$.
We say that a group is [*knapsack-semilinear*]{} if for every knapsack equation the set of all solutions is semilinear. Note that in any group $G$ the set of solutions on an equation $g_1^x = g$ is periodic and hence semilinear. Moreover, every finitely generated abelian group is semilinear (since solution sets of linear equations are Presburger definable). Nontrivial examples of knapsack-semilinear groups are hyperbolic groups [@LOHREY2019], graph groups [@LohreyZ18], and co-context free groups [@KoenigLohreyZetzsche2015a].[^3]
Obviously, every finitely generated subgroup of a finitely generated knapsack-semilinear group is knapsack-semilinear as well. Moreover, it was shown in [@GanardiKLZ18] that the class of knapsack-semilinear groups is closed under wreath products. In this paper we prove the closure of the class of knapsack-semilinear groups under
- finite extensions,
- graph products,
- amalgamated free products with finite amalgamated subgroups, and
- HNN-extensions with finite associated subgroups.
The operation of graph products interpolates between direct products and free products. It is specified by a finite graph $(V,E)$, where every node $v \in V$ is labelled with a group $G_v$. One takes the free product of the groups $G_v$ ($v \in V$) modulo the congruence that allows elements from adjacent groups to commute. Amalgamated free products and HNN-extensions are fundamental constructions in combinatorial group theory; see Section \[sec-HNN+amalgamated\] for references.
In order to get complexity bounds for the knapsack problem, the concept of knapsack-semilinearity is not useful. For this purpose, we need a quantitative measure for semilinear sets; see also [@chistikov_et_al:LIPIcs:2016:6263]: For a semilinear set $$L = \bigcup_{1 \leq i \leq n} \{ v_{i,0} + \lambda_1 v_{i,1} + \cdots + \lambda_{l_i} v_{i,l_i} \mid \lambda_1, \ldots, \lambda_{l_i} \in \mathbb{N}\}$$ we call the tuple of all vectors $v_{i,j}$ a [*semilinear representation*]{} for $L$. The [*magnitude*]{} of this semilinear representation is the largest number that occurs in some of the vectors $v_{i,j}$. Finally, the magnitude of a semilinear set $L$ is the smallest magnitude among all semilinear representations of $L$.
Our proofs showing that the above group constructions preserve knapsack-semilinearity also yield upper bounds for the magnitude of solution sets in terms of (i) the total length of the knapsack equation (measured in the total number of generators) and (ii) the number of variables in the knapsack equation. For this, we introduce a function $\mathsf{K}_G(n,m)$ that yields the maximal magnitude of a solution set for a knapsack equation over $G$ of total length at most $n$ and at most $m$ variables. Roughly speaking, it turns out that finite extensions, amalgamated free products with finite amalgamated subgroups, and HNN-extensions with finite associated subgroups only lead to a polynomial blowup for the function $\mathsf{K}_G(n,m)$ (actually, this function also depends on the generating set for $G$), whereas graph products can lead to an exponential blowup. On the other hand, if we bound the number of variables by a constant, then also graph products only lead to a polynomial blowup for the function $\mathsf{K}_G(n,m)$.
Words, monoids and groups
=========================
Fix a non-empty set $\Sigma$, which is also called an alphabet in the following. Its elements are also called symbols. A word over $\Sigma$ is a finite sequence $w = a_1 a_2 \ldots a_n$ of elements $a_1, \ldots, a_n \in \Sigma$. We write $|w|=n$ for the length $w$ and ${\mathsf{alph}}(w) = \{ a_1, a_2, \ldots, a_n\}$ for the set of symbols that occur in $w$. For $a\in \Sigma$, we write $|w|_{a}$ to denote the number of occurrences of $a$ in $w$. The [*free monoid*]{} $\Sigma^*$ consists of all finite words over $\Sigma$ and the monoid operation is the concatenation of words. The concatenation of words $u,v \in \Sigma^*$ is simply denoted with $uv$. The identity element of the free monoid $\Sigma^*$ is the empty word, which is usually denoted with $\varepsilon$. Here, we prefer to denote the empty word with $1$ according to the following convention:
For every monoid $M$ we denote the identity element of $M$ with the symbol $1$; even in cases where we deal with several monoids.
So intuitively, all monoids that we deal with share the same identity element $1$. This convention will simplify our notations.
For a set $\Omega$ we denote with $F(\Omega)$ the free group generated by $\Omega$. Formally, it can be defined as follows: Let $\Omega^{-1} = \{ a^{-1} \mid a \in \Omega\}$ be a disjoint copy of $\Omega$ (the set of formal inverses) and let $\Sigma = \Omega \cup \Omega^{-1}$. Then the [*free group*]{} $F(\Omega)$ can be identified with the set of all words $w \in \Sigma^*$ that do not contain a factor of the form $a a^{-1}$ or $a^{-1} a$ for $a \in \Omega$ (so called irreducible words). The product of two irreducible words $u,v$ is the unique irreducible word obtained from $uv$ by replacing factors of the form $a a^{-1}$ or $a^{-1} a$ ($a \in \Omega$) by the empty word as long as possible. For a set $R \subseteq \Sigma^*$ of irreducible words (the relators) we denote with $\langle \Omega \mid R \rangle$ the quotient group $F(\Omega)/N$, where $N$ is the smallest normal subgroup of $F(\Omega)$ that contains $R$. Every group is isomorphic to a group $\langle \Omega \mid R \rangle$. If $\Omega$ is finite, then $\langle \Omega \mid R \rangle$ is called [*finitely generated*]{}. In other words: a group $G$ is finitely generated if there exists a finite subset $\Sigma \subseteq G$ such that every element of $G$ is a product of elements of $\Sigma$. If for every $a \in \Sigma$ also $a^{-1}$ belongs to $\Sigma$ then $\Sigma$ is called a [*symmetric generating set*]{} for $G$.
Semilinear sets
===============
Fix a dimension $d \ge 1$. All vectors will be column vectors. For a vector $v=(v_1, \dots , v_d)^{\mathsf{T}}\in {\mathbb{Z}}^d$ we define its norm ${|\!| v |\!|}:=\max \{ |v_i| \mid 1 \leq i \leq d \}$ and for a matrix $M \in {\mathbb{Z}}^{c \times d}$ with entries $m_{i,j}$ ($1 \le i \le c$, $1 \le j \le d$) we define the norm ${|\!| M |\!|} = \max \{|m_{i,j}| \mid 1 \le i \le c, 1 \leq j \leq d \}$. Finally, for a finite set of vectors $A \subseteq {\mathbb{N}}^d$ let ${|\!| A |\!|} = \max \{ {|\!| a |\!|} \mid a \in A \}$.
We extend the operations of vector addition and multiplication of a vector by a matrix to sets of vectors in the obvious way. A linear subset of ${\mathbb{N}}^d$ is a set of the form $$L= L(b,P) := b + P \cdot {\mathbb{N}}^k$$ where $b \in {\mathbb{N}}^d$ and $P \in {\mathbb{N}}^{d \times k}$. We call a set $S\subseteq {\mathbb{N}}^d$ *semilinear*, if it is a finite union of linear sets. The class of semilinear sets is known be closed under boolean operations.
If a semilinear set $S$ is given as a union $\bigcup_{i=1}^k L(b_i,P_i)$, we call the tuple $\mathcal{R} = (b_1, P_1, \ldots, b_k, P_k)$ a *semilinear representation* of $S$. For a semilinear representation $\mathcal{R} = (b_1, P_1, \ldots, b_k, P_k)$ we define ${|\!| \mathcal{R} |\!|} = \max \{ {|\!| b_1 |\!|}, {|\!| P_1 |\!|} \ldots, {|\!| b_k |\!|}, {|\!| P_k |\!|}\}$. The *magnitude* of a semilinear set $S$, ${\mathsf{mag}}(S)$ for short, is the smallest possible value for ${|\!| \mathcal{R} |\!|}$ among all semilinear representations $\mathcal{R}$ of $S$.
For a linear set $L(b,P) \subseteq {\mathbb{N}}^d$ we can assume that all columns of $P$ are different. Hence, if the magnitude of $L(b,P)$ is bounded by $s$ then we can bound the number of columns of $P$ by $(s+1)^d$ (since there are only $(s+1)^d$ vectors in ${\mathbb{N}}^d$ of norm at most $s$). No better upper bound is known, but if we allow to split $L(b,P)$ into several linear sets, we get the following lemma from [@EisenbrandS06]:
\[lemma-number-periods\] Let $L = L(b,P) \subseteq {\mathbb{N}}^d$ be a linear set of magnitude $s = {\mathsf{mag}}(L)$. Then $L = \bigcup_{i\in I} L(b,P_i)$ such that every $P_i$ consists of at most $2 d \log(4ds)$ columns from $P$ (and hence, ${\mathsf{mag}}(L(b,P_i)) \leq s$).
We also need the following bound on the magnitude for the intersections of semilinear sets:
\[intersection-semilinear-sets\] Let $K = L(b_1,P_1)$ and $L = L(b_2,P_2) \subseteq {\mathbb{N}}^d$ be semilinear sets of magnitude at most $s \geq 1$. Then the intersection $K \cap L$ is semilinear and ${\mathsf{mag}}(K \cap L) \le (12 d^2 \log^2(4ds) d^{d/2} s^{d+1}+1)s
\le \mathcal{O}(d^{d/2+3} s^{d+3})$.
By Lemma \[lemma-number-periods\] we can write $K = \bigcup_{i \in I_1} L(b_1,P_{1,i})$ and $L = \bigcup_{i \in I_2} L(b_2,P_{2,i})$ where every $P_{1,i}$ ($P_{2,i}$) consists of at most $2 d \log(4ds)$ columns from $P_1$ ($P_2$). We have $K \cap L = \bigcup_{(i,j) \in I_1\times I_2} L(b_1,P_{1,i}) \cap L(b_2,P_{2,i})$. From [@Beier Theorem 4] we get the upper bound $(12 d^2 \log^2(4ds) d^{d/2} s^{d+1}+1)s$ for the magnitude of each intersection $L(b_1,P_{1,i}) \cap L(b_2,P_{2,i})$.
In the context of knapsack problems (which we will introduce in the next section), we will consider semilinear subsets as sets of mappings $f : \{x_1, \ldots, x_d\} \to {\mathbb{N}}$ for a finite set of variables $U = \{x_1, \ldots, x_d\}$. Such a mapping $f$ can be identified with the vector $(f(x_1), \ldots, f(x_d))^{\mathsf{T}}$. This allows to use all vector operations (e.g. addition and scalar multiplication) on the set ${\mathbb{N}}^U$ of all mappings from $U$ to ${\mathbb{N}}$. The pointwise product $f \cdot g$ of two mappings $f,g \in {\mathbb{N}}^U$ is defined by $(f \cdot g)(x) = f(x) \cdot g(x)$ for all $x \in U$. Moreover, for mappings $f \in {\mathbb{N}}^U$, $g \in {\mathbb{N}}^V$ with $U \cap V = \emptyset$ we define $f \oplus g \in {\mathbb{N}}^{U \cup V}$ by $(f \oplus g)(x) = f(x)$ for $x \in U$ and $(f \oplus g)(y) = g(y)$ for $y \in V$. All operations on ${\mathbb{N}}^U$ will be extended to subsets of ${\mathbb{N}}^U$ in the standard pointwise way. Note that ${\mathsf{mag}}(K \oplus L) \le \max\{ {\mathsf{mag}}(K), {\mathsf{mag}}(L)\}$ for semilinear sets $K,L$. If $L \subseteq {\mathbb{N}}^U$ is semilinear and $V \subseteq U$ then we denote with $L{\mathord\restriction}_{V}$ the semilinear set $\{ f{\mathord\restriction}_{V} \mid f \in L \}$ obtained by restricting every function $f \in L$ to the subset $V$ of its domain. Clearly, $L {\mathord\restriction}_{V}$ is semilinear too and ${\mathsf{mag}}(L{\mathord\restriction}_{V}) \le {\mathsf{mag}}(L)$.
Knapsack and exponent equations
===============================
Let $G$ be a finitely generated group with the finite symmetric generating set $\Sigma$. Moreover, let $X$ be a set of formal variables that take values from ${\mathbb{N}}$. For a subset $U\subseteq X$, we call a mapping $\sigma : U \to {\mathbb{N}}$ a *valuation* for $U$. An *exponent expression* over $\Sigma$ is a formal expression of the form ${e}= u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_k^{x_k} v_k$ with $k \geq 1$, words $u_i, v_i \in \Sigma^*$ and variables $x_1, \ldots, x_k \in X$. Here, we allow $x_i = x_j$ for $i \neq j$. The words $u_i$ are called the [*periods*]{} of ${e}$, and we can assume that $u_i \neq 1$ for all $1 \le i \le k$. If every variable in an exponent expression occurs at most once, it is called a *knapsack expression*. Let $X_{e}= \{ x_1, \ldots, x_k \}$ be the set of variables that occur in ${e}$. For a valuation $\sigma : U \to {\mathbb{N}}$ such that $X_{e}\subseteq U$ (in which case we also say that $\sigma$ is a valuation for ${e}$), we define $\sigma({e}) = u_1^{\sigma(x_1)} v_1 u_2^{\sigma(x_2)} v_2 \cdots u_k^{\sigma(x_k)} v_k \in \Sigma^*$. We say that $\sigma$ is a *$G$-solution* of the equation ${e}=1$ if $\sigma({e})$ evaluates to the identity element $1$ of $G$. With ${\mathsf{sol}}_G({e})$ we denote the set of all $G$-solutions $\sigma : X_{e}\to {\mathbb{N}}$ of ${e}$. We can view ${\mathsf{sol}}_G({e})$ as a subset of ${\mathbb{N}}^k$. The *length* of ${e}$ is defined as ${|\!| {e}|\!|} =\sum_{i=1}^k |u_i|+|v_i|$, whereas $k \le {|\!| {e}|\!|}$ is its *degree*, $\deg({e})$ for short. We define [*solvability of exponent equations over $G$*]{} as the following decision problem:
Input
: A finite list of exponent expressions ${e}_1,\ldots,{e}_n$ over $G$.
Question
: Is $\bigcap_{i=1}^n {\mathsf{sol}}_G({e}_i)$ non-empty?
The [*knapsack problem for $G$*]{} is the following decision problem:
Input
: A single knapsack expression ${e}$ over $G$.
Question
: Is ${\mathsf{sol}}_G({e})$ non-empty?
It is easy to observe that the concrete choice of the generating set $\Sigma$ has no influence on the decidability and complexity status of these problems.
One could also allow exponent expressions of the form ${e}= v_0 u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_k^{x_k} v_k$. Note that ${\mathsf{sol}}_G({e}) = {\mathsf{sol}}_G(v_0^{-1} {e}v_0)$. Moreover, we could also restrict to exponent expressions of the form ${e}= u_1^{x_1} u_2^{x_2} \cdots u_k^{x_k} v$: for ${e}= u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_k^{x_k} v_k$ and ${e}' = u_1^{x_1} (v_1 u_2 v_1^{-1})^{x_2} (v_1v_2 u_3 v_2^{-1} v_1^{-1})^{x_3} \cdots (v_1\cdots v_{k-1} u_3 v_{k-1}^{-1} \cdots v_1^{-1})^{x_k}
v_1 \cdots v_{k-1} v_k$ we have ${\mathsf{sol}}_G({e}) = {\mathsf{sol}}_G({e}')$.
Knapsack-semilinear groups
--------------------------
The group $G$ is called [*knapsack-semilinear*]{} if for every knapsack expression ${e}$ over $\Sigma$, the set ${\mathsf{sol}}_G({e})$ is a semilinear set of vectors and a semilinear representation can be effectively computed from ${e}$. Since semilinear sets are effectively closed under intersection, it follows that for every exponent expression ${e}$ over $\Sigma$, the set ${\mathsf{sol}}_G({e})$ is semilinear and a semilinear representation can be effectively computed from ${e}$. Moreover, solvability of exponent equations is decidable for every knapsack-semilinear group. As mentioned in the introduction, the class of knapsack-semilinear groups is very rich. An example of a group $G$, where knapsack is decidable but solvability of exponent equations is undecidable is the Heisenberg group ${H_3({\mathbb{Z}})}$ (which consists of all upper triangular $(3 \times 3)$-matrices over the integers, where all diagonal entries are $1$), see [@KoenigLohreyZetzsche2015a]. In particular, ${H_3({\mathbb{Z}})}$ is not knapsack-semilinear.
For a knapsack-semilinear group $G$ and a finite generating set $\Sigma$ for $G$ we define two growth functions. For $n,m \in {\mathbb{N}}$ with $m \leq n$ let $\mathsf{Exp}(n,m)$ be the finite set of all exponent expressions ${e}$ over $\Sigma$ such that (i) ${\mathsf{sol}}_G({e}) \neq \emptyset$, (ii) ${|\!| {e}|\!|} \leq n$ and (iii) $\deg({e}) \leq m$. Moreover, let $\mathsf{Knap}(n,m) \subseteq \mathsf{Exp}(n,m)$ be the set of all knapsack expressions in $\mathsf{Exp}(n,m)$. We define the mappings $\mathsf{E}_{G,\Sigma} : \{(n,m) \mid m,n \in {\mathbb{N}}, m \leq n\} \rightarrow {\mathbb{N}}$ and $\mathsf{K}_{G,\Sigma} : \{(n,m) \mid m,n \in {\mathbb{N}}, m \leq n\} \rightarrow {\mathbb{N}}$ as follows:
- $\mathsf{E}_{G,\Sigma}(n,m) = \max \{ {\mathsf{mag}}({\mathsf{sol}}_G({e})) \mid {e}\in \mathsf{Exp}(n,m) \}$,
- $\mathsf{K}_{G,\Sigma}(n,m) = \max \{ {\mathsf{mag}}({\mathsf{sol}}_G({e})) \mid {e}\in \mathsf{Knap}(n,m) \}$.
Clearly, if ${\mathsf{sol}}_G({e}) \neq \emptyset$ and ${\mathsf{mag}}({\mathsf{sol}}_G({e})) \le N$ then ${e}$ has a $G$-solution $\sigma$ such that $\sigma(x) \leq N$ for all variables $x\in X_{e}$. Therefore, if $G$ has a decidable word problem and we have a bound on the function $\mathsf{E}_{G,\Sigma}$ then we obtain a nondeterministic algorithm for solvability of exponent equations over $G$: given an exponent expression ${e}$ we can guess $\sigma : X_{e}\to {\mathbb{N}}$ with $\sigma(x) \le N$ for all variables $x$ and then verify (using an algorithm for the word problem), whether $\sigma$ is indeed a solution.
Let $\Sigma$ and $\Sigma'$ be two generating sets for the group $G$. Then there is a constant $c$ such that $\mathsf{E}_{G,\Sigma}(n,m) \le \mathsf{E}_{G,\Sigma'}(cn,m)$ and $\mathsf{K}_{G,\Sigma}(n,m) \le \mathsf{K}_{G,\Sigma'}(cn,m)$. To see this, note that for every $a\in \Sigma'$ there is a word $w_a\in \Sigma^*$ such that $a$ and $w_a$ are representing the same element in $G$. Then we can choose $c=\max \{ |w_a| \mid a\in \Sigma'\}$.
Part 1: Exponent equations in graph products {#sec-part1}
============================================
In this section we introduce graph products of groups (Section \[subsect:graph products\]) and show that the graph product of knapsack semilinear groups is again knapsack semilinear (Section \[sec-main\]). Our definition of graph products is based on trace monoids (also known as partially commutative monoids) that we discuss first.
Trace monoids {#sec-traces}
-------------
In the following we introduce some notions from trace theory, see [@Die90lncs; @DieRoz95] for more details. An [*independence alphabet*]{} is an undirected graph $(\Sigma,I)$ (without loops). Thus, $I$ is a symmetric and irreflexive relation on $\Sigma$. The set $\Sigma$ may be infinite. The *trace monoid* ${\mathbb{M}}(\Sigma,I)$ is defined as the quotient ${\mathbb{M}}(\Sigma,I)=\Sigma^*/\{ab=ba\mid (a,b)\in I\}$ with concatenation as operation and the empty trace $1$ as the neutral element. Its elements are called *traces*. We denote by $[w]_I$ the trace represented by the word $w\in \Sigma^*$. Let ${\mathsf{alph}}([w]_I) = {\mathsf{alph}}(w)$ and $|[w]_I| = |w|$. The [*dependence alphabet*]{} associated with $(\Sigma,I)$ is $(\Sigma,D)$, where $D=(\Sigma\times \Sigma)\setminus I$. Note that the relation $D$ is reflexive. For $a\in \Sigma$ let $I(a)=\{b\in\Sigma\mid (a,b)\in I\}$ be the letters that commute with $a$. For traces $u,v\in {\mathbb{M}}(\Sigma,I)$ we denote with $uIv$ the fact that ${\mathsf{alph}}(u)\times {\mathsf{alph}}(v)\subseteq I$. The trace $u$ is [*connected*]{} if we cannot write $u = v w$ in ${\mathbb{M}}(\Sigma,I)$ such that $v \neq 1 \neq w$ and $v I w$.
An [*independence clique*]{} is a subset $\Delta \subseteq \Sigma$ such that $(a,b) \in I$ for all $a,b \in \Delta$ with $a \neq b$. A [*finite*]{} independence clique $\Delta$ is identified with the trace $[a_1 a_2 \cdots a_n]_I$, where $a_1, a_2, \ldots, a_n$ is an arbitrary enumeration of $\Delta$.
The following lemma, which is known as Levi’s lemma, is one of the most fundamental facts for trace monoids, see e.g. [@DieRoz95]:
\[lemma-levi\] Let $u_1, \ldots, u_m, v_1, \ldots, v_n \in {\mathbb{M}}(\Sigma,I)$. Then $$u_1u_2 \cdots u_m = v_1 v_2 \cdots v_n$$ if and only if there exist $w_{i,j} \in {\mathbb{M}}(\Sigma,I)$ $(1 \leq i \leq m$, $1 \leq j \leq n)$ such that
- $u_i = w_{i,1}w_{i,2}\cdots w_{i,n}$ for every $1 \leq i \leq m$,
- $v_j = w_{1,j}w_{2,j}\cdots w_{m,j}$ for every $1 \leq j \leq n$, and
- $(w_{i,j}, w_{k,\ell})\in I$ if $1 \leq i < k \leq m$ and $n \geq j > \ell \geq 1$.
The situation in the lemma will be visualized by a diagram of the following kind. The $i$–th column corresponds to $u_i$, the $j$–th row corresponds to $v_j$, and the intersection of the $i$–th column and the $j$–th row represents $w_{i,j}$. Furthermore $w_{i,j}$ and $w_{k,\ell}$ are independent if one of them is left-above the other one.
[c"c|c|c|c|c|]{} $v_n$ & $w_{1,n}$ & $w_{2,n}$ & $w_{3,n}$ & …& $w_{m,n}$\
& & & & &\
$v_3$ & $w_{1,3}$ & $w_{2,3}$ & $w_{3,3}$ & …& $w_{m,3}$\
$v_2$ & $w_{1,2}$ & $w_{2,2}$ & $w_{3,2}$ & …& $w_{m,2}$\
$v_1$ & $w_{1,1}$ & $w_{2,1}$ & $w_{3,1}$ & …& $w_{m,1}$\
------------------------------------------------------------------------
height 1pt @a xhline
& $u_1$ & $u_2$ & $u_3$ & …& $u_m$
A consequence of Levi’s lemma is that trace monoids are cancellative, i.e., $usv=utv$ implies $s=t$ for all traces $s,t,u,v\in{\mathbb{M}}(\Sigma,I)$.
A *trace rewriting system* $R$ over $\mathbb{M}(\Sigma,I)$ is just a finite subset of $\mathbb{M}(\Sigma,I) \times
\mathbb{M}(\Sigma,I)$ [@Die90lncs]. We define the *one-step rewrite relation* $\to_R \;\subseteq
\mathbb{M}(\Sigma,I) \times \mathbb{M}(\Sigma,I)$ by: $x \to_R y$ if and only if there are $u,v \in \mathbb{M}(\Sigma,I)$ and $(\ell,r) \in
R$ such that $x = u\ell v$ and $y = u r v$. With $\xrightarrow{*}_R$ we denote the reflexive transitive closure of $\rightarrow_R$. The notion of a confluent and terminating trace rewriting system is defined as for other types of rewriting systems [@BoOt93]: A trace rewriting system $R$ is called *confluent* if for all $u,v,v'\in
\mathbb{M}(\Sigma,I)$ with $u \xrightarrow{*}_R v$ and $u\xrightarrow{*}_R v'$ there exists a trace $w$ with $v
\xrightarrow{*}_R w$ and $v'\xrightarrow{*}_R w$. It is called *terminating* if there does not exist an infinite chain $u_0\rightarrow_R u_1 \rightarrow_R u_2 \cdots$. A trace $u$ is *$R$-irreducible* if no trace $v$ with $u \to_R v$ exists. The set of all $R$-irreducible traces is denoted with ${\mathsf{IRR}}(R)$. If $R$ is terminating and confluent, then for every trace $u$, there exists a unique *normal form* ${\mathsf{NF}}_R(u) \in
{\mathsf{IRR}}(R)$ such that $u \xrightarrow{*}_R {\mathsf{NF}}_R(u)$.
Graph products {#subsect:graph products}
--------------
Let us fix a [*finite*]{} independence alphabet $(\Gamma,E)$ and finitely generated groups ${{G}_i}$ for $i\in \Gamma$. Let $\alpha$ be the size of a largest clique of the independence alphabet $(\Gamma, E)$. As usual $1$ is the identity element for each of the groups ${G}_i$. Let $\Sigma_i$ be a finite and symmetric generating set of ${G}_i$ such that $\Sigma_i \cap \Sigma_j = \emptyset$ for $i\neq j$. We define a (possibly infinite) independence alphabet as in [@DiLo08IJAC; @KuLo05ijac]: Let $$A_i={{G}_i}\setminus \{1\} \quad\text{and}\quad A =
\bigcup_{i \in \Gamma} A_i.$$ We assume that $A_i \cap A_j = \emptyset$ for $i\neq j$. We fix the independence relation $$I = \bigcup_{(i,j)\in E}A_i\times A_j$$ on $A$. The independence alphabet $(A,I)$ is the only independence alphabet in this paper, which may be infinite. Recall that for a trace $t \in {\mathbb{M}}(A,I)$, ${\mathsf{alph}}(t) \subseteq A$ is the set of symbols that occur in $t$. We definite the [*$\Gamma$-alphabet*]{} of $t$ as $${\mathsf{alph}}_\Gamma(t) = \{ i \in \Gamma \mid {\mathsf{alph}}(t) \cap A_i \neq \emptyset \}.$$ Note that whether $u I v$ (for $u,v \in {\mathbb{M}}(A,I)$) only depends on ${\mathsf{alph}}_{\Gamma}(u)$ and ${\mathsf{alph}}_{\Gamma}(v)$.
Every independence clique of $(A,I)$ has size at most $\alpha$ and hence can be identified with a trace from ${\mathbb{M}}(A,I)$. Let $C_1$ and $C_2$ be independence cliques. We say that $C_1$ and $C_2$ are compatible, if ${\mathsf{alph}}_{\Gamma}(C_1) = {\mathsf{alph}}_{\Gamma}(C_2)$. In this case we can write $C_1 = \{ a_1, \ldots, a_m \}$ and $C_2 = \{ b_1, \ldots, b_m \}$ for some $m \leq \alpha$ such that for every $1 \leq i \leq m$ there exists $j_i \in \Gamma$ with $a_i, b_i \in A_{j_i}$. Let $c_i = a_i b_i$ in the group ${G}_{j_i}$. If $c_i \neq 1$ for all $1 \leq i \leq m$, then $C_1$ and $C_2$ are [*strongly compatible*]{}. In this case we define the independence clique $C_1 C_2 = \{ c_1, \ldots, c_m\}$. Note that ${\mathsf{alph}}_{\Gamma}(C_1) = {\mathsf{alph}}_{\Gamma}(C_2) = {\mathsf{alph}}_{\Gamma}(C_1C_2)$.
We will work with traces $t \in {\mathbb{M}}(A,I)$. For such a trace we need two length measures. The ordinary length of $t$ is $|t|$ as defined in Section \[sec-traces\]: If $t = [a_1 \cdots a_k]_I$ with $a_j \in A$ then $|t|=k$. On the other hand, if we deal with computational problems, we need a finitary representations of the elements $a_j$. Assume that $a_j \in A_{i_j}$. Then, $a_j$ can be written as a word over the alphabet $\Sigma_{i_j}$. Let $n_j={|\!| a_j |\!|}$ denote the length of a shortest word over $\Sigma_{i_j}$ that evaluates to $a_j$ in the group ${G}_{i_j}$; this is also called the [*geodesic length*]{} of the group element $a_j$. Then we define ${|\!| t |\!|} = n_1 + n_2 + \cdots + n_k$.
A trace $a \in A$ (i.e., a generator of ${\mathbb{M}}(A,I)$) is also called [*atomic*]{}, or an [*atom*]{}. For an atom $a \in A$ that belongs to the group ${G}_i$, we write $a^{-1}$ for the inverse of $a$ in ${G}_i$; it is again an atom. On ${\mathbb{M}}(A,I)$ we define the trace rewriting system $$\label{system-R}
R=\bigcup_{i\in \Gamma} \bigg(\{([aa^{-1}]_I,1)\mid a\in A_i\} \cup
\{([ab]_I,[c]_I)\mid a,b,c\in A_i,ab=c\;\mathrm{in}\;{G}_i\} \bigg).$$ The following lemma was shown in [@KuLo05ijac]:
\[R-confluent\] The trace rewriting system $R$ is confluent.
Since $R$ is length-reducing, it is also terminating and hence defines unique normal forms. We define the [*graph product*]{} ${G}(\Gamma,E,({{G}_i})_{i\in \Gamma})$ as the quotient monoid $${G}(\Gamma,E,({{G}_i})_{i\in \Gamma}) = {\mathbb{M}}(A,I)/R.$$ Here we identify $R$ with the smallest congruence relation on ${\mathbb{M}}(A,I)$ that contains all pairs from $R$. In the rest of Section \[sec-part1\], we write ${G}$ for ${G}(\Gamma,E,({{G}_i})_{i\in \Gamma})$. It is easy to see that ${G}$ is a group. The inverse of a trace $t= [a_1 a_2 \cdots a_k]_I \in {\mathbb{M}}(A,I)$ with $a_i \in A$ is the trace $t^{-1} = [a_k^{-1} \cdots a_2^{-1} a_1^{-1}]_I$. Note that $t$ is well defined: If $[a_1 a_2 \cdots a_k]_I = [b_1 b_2 \cdots b_k]_I$ then $[a_k^{-1} \cdots a_2^{-1} a_1^{-1}]_I = [b_k^{-1} \cdots b_2^{-1} b_1^{-1}]_I$. We can apply this notation also to an independence clique $C$ of $(A,I)$ which yields the independence clique $C^{-1} = \{ a^{-1} \mid a \in C\}$.
Note that ${G}$ is finitely generated by $\Sigma = \bigcup_{i\in \Gamma}\Sigma_i$. If $E = \emptyset$, then ${G}$ is the free product of the groups ${G}_i$ ($i \in \Gamma$) and if $(\Gamma,E)$ is a complete graph, then ${G}$ is the direct product of the groups ${G}_i$ ($i \in \Gamma$). In this sense, the graph product construction generalizes free and direct products.
For traces $u,v \in {\mathbb{M}}(A,I)$ or words $u,v \in \Sigma^*$ we write $u =_{{G}} v$ if $u$ and $v$ represent the same element of the group ${G}$. The following lemma is important for solving the word problem in a graph product ${G}$:
\[rewriting=product\] Let $u,v\in {\mathbb{M}}(A,I)$. Then $u=_{{G}} v$ if and only if ${\mathsf{NF}}_R(u)={\mathsf{NF}}_R(v)$. In particular we have $u=_{{G}} 1$ if and only if ${\mathsf{NF}}_R(u)=1$.
The if-direction is trivial. Let on the other hand $u,v\in {\mathbb{M}}(A,I)$ and suppose that $u=v$ in ${G}$. By definition this is the case if and only if $u$ and $v$ represent the same element from ${\mathbb{M}}(A,I)/R$ and are hence congruent with respect to $R$. Since $R$ produces a normal form for elements from the same congruence class, this implies that ${\mathsf{NF}}_R(u)={\mathsf{NF}}_R(v)$.
Graph products of copies of $\mathbb{Z}$ are also known as graph groups or right-angled Artin groups. Graph products of copies of $\mathbb{Z}/2\mathbb{Z}$ are known as [*right-angled Coxeter groups*]{}, see [@Dro2003] for more details.
For the rest of the paper we fix the graph product ${G}= {G}(\Gamma,E,({{G}_i})_{i\in \Gamma})$. Moreover, $\Sigma_i$, $A_i$ ($i \in \Gamma$), $\Sigma$, $A$, $I$, and $R$ will have the meaning defined in this section.
Results from [@LohreyZ18]
-------------------------
In this section we state a small modification of results from [@LohreyZ18], where the statements are made for finitely generated trace monoids ${\mathbb{M}}(\Sigma,I)$. We need the corresponding statements for the non-finitely generated trace monoid ${\mathbb{M}}(A,I)$ from Section \[subsect:graph products\]. The proofs are exactly the same as in [@LohreyZ18], one only has to argue with the $\Gamma$-alphabet ${\mathsf{alph}}_{\Gamma}(t)$ instead the alphabet ${\mathsf{alph}}(t)$ of traces.
Note that all statements in this section refer to the trace monoid ${\mathbb{M}}(A,I)$ and not to the corresponding graph product ${G}$. In particular, when we write a product $t_1 t_2 \cdots t_n$ of traces $t_i \in {\mathbb{M}}(A,I)$ no cancellation occurs between the $t_i$. We will also consider the case that $E=\emptyset$ (and hence $I=\emptyset$), in which case ${\mathbb{M}}(A,I) = A^*$.
Let $s,t \in {\mathbb{M}}(A,I)$ be traces. We say that $s$ is a *prefix* of $t$ if there is a trace $r\in {\mathbb{M}}(A,I)$ with $sr=t$. Moreover, we define $\rho(t)$ as the number of prefixes of $t$. We will use the following statement from [@BeMaSa89].
\[lemma-prefixes\] Let $t \in {\mathbb{M}}(A,I)$ be a trace of length $n$. Then $\rho(t) \le \mathcal{O}(n^\alpha) \le \mathcal{O}(n^{|\Gamma|})$, where $\alpha$ is the size of a largest clique of the independence alphabet $(\Gamma, E)$.
It is easy to see that $\rho(t)=n+1$ if $E=\emptyset$.
\[lemma-simplify-factorization-power\] Let $u \in {\mathbb{M}}(A,I) \setminus \{1\}$ be a connected trace and $m \in \mathbb{N}$, $m \geq 2$. Then, for all $x \in \mathbb{N}$ and traces $y_1, \ldots, y_m$ the following two statements are equivalent:
(i) $u^x = y_1 y_2 \cdots y_m$.
(ii) There exist traces $p_{i,j}$ $(1 \leq j < i \leq m)$, $s_i$ $(1 \leq i \leq m)$ and numbers $x_i, c_j \in \mathbb{N}$ $(1 \leq i \leq m$, $1 \leq j \leq m-1)$ such that:
- ${\mathsf{alph}}(p_{i,j}) \cup {\mathsf{alph}}(s_i) \subseteq {\mathsf{alph}}(u)$,
- $y_i = (\prod_{j=1}^{i-1} p_{i,j}) u^{x_i} s_i$ for all $1 \leq i \leq m$,
- $p_{i,j} I p_{k,l}$ if $j < l < k < i$ and $p_{i,j} I (u^{x_k} s_k)$ if $j < k < i$,[^4]
- $s_m = 1$ and for all $1 \leq j < m$, $s_j \prod_{i=j+1}^m p_{i,j} = u^{c_j}$,
- $c_j \leq |\Gamma|$ for all $1 \leq j \leq m-1$,
- $x = \sum_{i=1}^m x_i + \sum_{i=1}^{m-1} c_i$.
The proof of this lemma is the same as for Lemma 6 in [@LohreyZ18], where the statement is shown for the case of a finite independence alphabet $(A,I)$. In our situation the independency between traces only depends on their $\Gamma$-alphabets. This allows to carry over the proof of [@LohreyZ18 Lemma 6] to our situation by replacing the alphabet ${\mathsf{alph}}(t)$ of a trace $t \in {\mathbb{M}}(A,I)$ by ${\mathsf{alph}}_{\Gamma}(u)$.
\[remark-simplify-factorization-power\] In Section \[sec-main\] we will apply Lemma \[lemma-simplify-factorization-power\] in order to replace an equation $u^x = y_1 y_2 \cdots y_m$ (where $x,y_1, \ldots, y_m$ are variables and $u$ is a concrete connected trace) by an equivalent disjunction. Note that the length of all factors $p_{i,j}$ and $s_i$ in Lemma \[lemma-simplify-factorization-power\] is bounded by $|\Gamma| \cdot |u|$ and that $p_{i,j}$ and $s_i$ only contain symbols from $u$. Hence, one can guess these traces as well as the numbers $c_j \leq |\Gamma|$ (the guess results in a disjunction). We can also guess which of the numbers $x_i$ are zero and which are greater than zero (let $K$ consists of those $i$ such that $x_i>0$). After these guesses we can verify the independencies $p_{i,j} I p_{k,l}$ ($j < l < k < i$) and $p_{i,j} I (u^{x_k} s_k)$ ($j < k < i$), and the identities $s_m = 1$, $s_j \prod_{i=j+1}^m p_{i,j} = u^{c_j}$ ($1 \leq j < m$). If one of them does not hold, the specific guess does not contribute to the disjunction. In this way, we can replace the equation $u^x = y_1 y_2 \cdots y_m$ by a disjunction of formulas of the form $$\exists x_i > 0 \; (i \in K):
x = \sum_{i\in K} x_i + c \wedge \bigwedge_{i \in K} y_i = p_i u^{x_i} s_i \wedge \bigwedge_{i \in [1,m] \setminus K} y_i = p_i s_i ,$$ where $K \subseteq [1,m]$, $c \leq |\Gamma| \cdot (m-1)$ and the $p_i, s_i$ are concrete traces of length at most $|\Gamma| \cdot (m-1) \cdot |u|$. The number of disjuncts in the disjunction will not be important for our purpose.
\[lemma-connected-star\] Let $p,q,u,v,s,t \in {\mathbb{M}}(A,I)$ with $u\neq 1$ and $v \neq 1$ connected. Let $m = \max\{ \rho(p), \rho(q), \rho(s), \rho(t) \}$ and $n = \max \{ \rho(u), \rho(v) \}$. Then the set $$L(p,u,s,q,v,t) := \{ (x,y) \in \mathbb{N} \times \mathbb{N} \mid p u^x s = q v^y t \}$$ is a union of $\mathcal{O}(m^8 \cdot n^{4 |\Gamma|})$ many linear sets of the form $\{ (a+bz, c+dz) \mid z \in \mathbb{N} \}$ with $a,b,c,d \le \mathcal{O}(m^8 \cdot n^{4 |\Gamma|})$. In particular, $L(p,u,s,q,v,t)$ is semilinear. If $|\Gamma|$ is a fixed constant, then a semilinear representation for $L(p,u,s,q,v,t)$ can be computed in polynomial time.
Again, the proof of Lemma \[lemma-connected-star\] is exactly the same as the proof of Lemma 11 in [@LohreyZ18]. One simply substitutes $|A|$ by $|\Gamma|$ and ${\mathsf{alph}}(x)$ by ${\mathsf{alph}}_{\Gamma}(x)$.
\[rem-2dim-words\] Let us consider again the case $E = I = \emptyset$ in Lemma \[lemma-connected-star\]. Let $m = \max\{ |p|, |q|, |s|, |t|, |u|, |v| \}$. We can construct an automaton for $p u^* s$ of size at most $3m$ and similarly for $q v^* t$. Hence, we obtain an automaton for $L := p u^* s \cap q v^* t$ of size $\mathcal{O}(m^2)$. We are only interested in the length of words from $L$. Let $\mathcal{A}$ be the automaton obtained from the automaton for $L$ by replacing every transition label by the symbol $a$. The resulting automaton $\mathcal{A}$ is defined over a unary alphabet. Let $P = \{ n \mid a^n \in L(\mathcal{A}) \}$. By [@To09ipl Theorem 1], the set $P$ can be written as a union $$P = \bigcup_{i=1}^r \{ b_i + c_i \cdot z \mid z \in \mathbb{N} \}$$ with $r \in \mathcal{O}(m^4)$ and $b_i, c_i \in \mathcal{O}(m^4)$. For every $1 \leq i \leq r$ and $z \in \mathbb{N}$ there must exist a pair $(x,y) \in \mathbb{N} \times \mathbb{N}$ such that $$b_i + c_i \cdot z = |ps| + |u| \cdot x = |qt| + |v| \cdot y.$$ In particular, $b_i \geq |ps|$, $b_i \geq |qt|$, $|u|$ divides $b_i-|ps|$ and $c_i$, and $|v|$ divides $b_i-|qt|$ and $c_i$. We get $$\begin{gathered}
L(p,u,s,q,v,t) = \bigcup_{i=1}^r \bigg\{ \bigg( \frac{b_i-|ps|}{|u|} + \frac{c_i}{|u|} \cdot z, \frac{b_i-|qt|}{|v|} + \frac{c_i}{|v|} \cdot z \bigg) \; \bigg| \bigg. \; z \in \mathbb{N}\bigg\}\end{gathered}$$ and all numbers that appear on the right-hand side are bounded by $\mathcal{O}(m^4)$.
Irreducible powers in graph products
------------------------------------
In this section, we study powers $u^n$ for an irreducible trace $u \in {\mathsf{IRR}}(R)$. We need the following definitions: A trace $u \in {\mathbb{M}}(A,I)$ is called *cyclically reduced* if $u \in {\mathsf{IRR}}(R)$ and there do not exist $a \in A$ and $v \in {\mathbb{M}}(A,I)$ such that $u = a v a^{-1}$. A trace $t \in {\mathbb{M}}(A,I)$ is called [*well-behaved*]{} if it is connected and $t^m \in {\mathsf{IRR}}(R)$ for every $m \geq 0$.
\[lemma-square\] Let $u \in {\mathsf{IRR}}(R)$. If $u^2 \in {\mathsf{IRR}}(R)$ then $u^m \in {\mathsf{IRR}}(R)$ for all $m \ge 0$.
Assume that $m\ge 3$ is the smallest number, such that $u^{m-1} \in {\mathsf{IRR}}(R)$ and $u^m \not\in {\mathsf{IRR}}(R)$. Hence we can write $u^m=xaby$ with $x,y \in {\mathsf{IRR}}(R)$ and $a,b \in A_i$ for some $i \in \Gamma$. Applying Levi’s lemma, we get factorizations $x=x_1x_2 \cdots x_m$ and $y=y_1y_2 \cdots y_m$ and the following diagram:
[c"c|c|c|c|c|]{} $y$ & $y_1$ & $y_2$ & …& $y_{m-1}$ & $y_m$\
$b$ & & & …& & $b$\
$a$ & $a$ & & …& &\
$x$ & $x_1$ & $x_2$ & …& $x_{m-1}$ & $x_m$\
------------------------------------------------------------------------
height 1pt @a xhline
& $u$ & $u$ & …& $u$ & $u$
This is in fact the only possibility for the positions of the atoms $a$ and $b$: If $a$ and $b$ would be in the same column then $u$ would contain the factor $ab$ and hence $u \notin {\mathsf{IRR}}(u)$. Also $a$ and $b$ are not independent, which means $b$ has to be top-right from $a$. If $a$ is not in the first column or $b$ is not in the last column, then $u^{m-1}$ is reducible, which contradicts the choice of $m$. Hence, we have $u = x_1 a y_1 = x_m b y_m$ with $a I x_m$, $y_1 I x_m$ and $b I y_1$. We get $u^2 = x_1 a y_1 x_m b y_m = x_1 a x_m y_1 b y_m = x_1 x_m a y_1 b y_m = x_1 x_m a b y_1 y_m$. Hence $u^2 \not\in {\mathsf{IRR}}(R)$, which is a contradiction.
\[well-behaved-iff\] A trace $u \in {\mathbb{M}}(A,I)$ is well-behaved if and only if it has the following properties:
- $u$ is irreducible,
- $u$ is not atomic,
- $u$ is connected, and
- one cannot write $u$ as $u = a v b$ such that $a,b \in A_i$ for some $i \in \Gamma$ (in particular, $u$ is cyclically reduced).
Clearly, if one the four conditions in the lemma is not satisfied, then $u$ is not well-behaved. Now assume that the four conditions hold for $u$. By Lemma \[lemma-square\], it suffices to show that $u^2 \in {\mathsf{IRR}}(R)$. Assume that $u^2=xaby$ with $a,b \in A_i$. Applying Levi’s lemma, and using $u \in {\mathsf{IRR}}(R)$ and $(a,b) \notin I$, we obtain the following diagram:
[c"c|c|]{} $y$ & $y_1$ & $y_2$\
$b$ & & $b$\
$a$ & $a$ &\
$x$ & $x_1$ & $x_2$\
------------------------------------------------------------------------
height 1pt @a xhline
& $u$ & $u$
From Levi’s lemma we also get $bIy_1$ and $aIx_2$. But $a$ and $b$ are in the same group, hence $aIy_1$ and $bIx_2$ also hold. The first property implies $u=va$ with $v=x_1y_1$ and the seconds property gives us $u=bw$ with $w=x_2y_2$. Since $u$ is not atomic, we have $v\neq 1 \neq w$. Now we apply Levi’s lemma to $va=bw$, which yields one of the following diagrams:
[c"c|c|]{} $w$ & $v=w$ &\
$b$ & & $a=b$\
------------------------------------------------------------------------
height 1pt @a xhline
& $v$ & $a$
[c"c|c|]{} $w$ & $w^\prime$ & $a$\
$b$ & $b$ &\
------------------------------------------------------------------------
height 1pt @a xhline
& $v$ & $a$
From the left diagram we get $a I v$. Hence $u = va$ is not connected, which is a contradiction. From the right diagram we get $u = va = b w' a$ for some trace $w^\prime$, which is a contradiction to our last assumption. This finally proves $u^2 \in {\mathsf{IRR}}(R)$, hence $u$ is well-behaved.
\[normalform-cyclic\] Let $u \in {\mathsf{IRR}}(R)$. Then there exist strongly compatible independence cliques $C, D$ and unique factorizations $u = p C v D p^{-1}$ such that every connected component of $v (DC)$ is atomic or well-behaved.
Assume that we can write $u$ as $a u_1 a^{-1}$ and $b u_2 b^{-1}$ for different atoms $a$, $b$. From Levi’s lemma it follows easily that $a I b$ and that there exists $u'$ with $u = ab u' b^{-1} a^{-1}$. It follows that there exists a unique factorization $u = p u' p^{-1}$ such that $u'$ cannot be written as $a u'' a^{-1}$ for an atom $a$.
Let $u' = a_1 \cdots a_k u_1 \cdots u_l$ be the factorization of $u'$ into connected components, where $a_1, \ldots, a_k$ are atoms and $u_1 \cdots u_l$ are non-atomic. Let $M_i = \min(u_i)$ be the set of maximal atoms of $u_i$ and $N = \min(u_i)$ the set of maximal atoms of $u_i$. Hence, we can write every $u_i$ as $u_i = [M_i]_I u'_i [N_i]_I$ for some $u'_i$. Define $C_i \subseteq M_i$ as the set of all atoms $a \in M_i$ such that $A_j \cap N_i \neq \emptyset$ for the unique $j \in \Gamma$ with $a \in A_j$. Similarly, define $D_i \subseteq N_i$ as the set of all atoms $a \in N_i$ such that $A_j \cap M_i \neq \emptyset$ for the unique $j \in \Gamma$ with $a \in A_j$. Then, $C_i$ and $D_i$ are strongly compatible (otherwise we could write $u'$ as $a u'' a^{-1}$ for an atom $a$. Let $v_i = [M_i \setminus C_i]_I u'_i [N_i \setminus D_i]_I$, $v = a_1 \cdots a_k v_1 \cdots v_l$, $C = \bigcup_{i=1}^l C_i$ and $D = \bigcup_{i=1}^l D_i$ This implies $$\begin{aligned}
p [C]_I v [D]_I p^{-1} &=&
p [C]_I a_1 \cdots a_k \prod_{i=1}^l ([M_i \setminus C_i]_I u'_i [N_i \setminus D_i]_I [D]_I) p^{-1} \\
&=& p a_1 \cdots a_k \prod_{i=1}^l ([M_i]_I u'_i [N_i]_I) p^{-1} \\
&=& p a_1 \cdots a_k u_1 \cdots u_l p^{-1} \\
&= & p u' p^{-1} \\
&= & u\end{aligned}$$ Since $p$ and $v$ are factors of $u$, they are irreducible too. We can write $v [DC]_I$ as $$\begin{aligned}
v [DC]_I &=& a_1 \cdots a_k v_1 \cdots v_l [DC]_I \\
& = & a_1 \cdots a_k \prod_{i=1}^l ([M_i \setminus C_i]_I u'_i [N_i \setminus D_i]_I) [DC]_I \\
&= & a_1 \cdots a_k \prod_{i=1}^l ([M_i \setminus C_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I\end{aligned}$$ We claim that the last line is the factorization of $v[DC]_I$ into its connected components. This follows from $${\mathsf{alph}}_{\Gamma}([M_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I) =
{\mathsf{alph}}_{\Gamma}([M_i]_I u'_i [N_i]_I) = {\mathsf{alph}}_{\Gamma}(u_i)$$ and the fact that $a_1 \cdots a_k u_1 \cdots u_l$ is the factorization of $u'$ into its connected components. Also note that the above identity shows that every trace $[M_i \setminus C_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I$ is non-atomic. Intuitively, $[M_i \setminus C_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I$ is obtained from $u_i = [M_i]_I u'_i [N_i]_I$ by moving the atoms from $C_i \subseteq M_i$ from the beginning to the end.
It remains to show that every $[M_i \setminus C_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I$ is well-behaved. Irreducibility follows from the fact that $u_i$ is irreducible. Now assume that we can write $[M_i \setminus C_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I = a w b$ where $a,b \in A_j$ for some $j \in \Gamma$. The set of minimal atoms of $[M_i \setminus C_i]_I u'_i [N_i \setminus D_i \cup D_i C_i]_I$ is $N_i \setminus D_i \cup D_i C_i$. Hence, we have $b \in N_i \setminus D_i \cup D_i C_i$. Moreover, either $a \in M_i \setminus C_i$ or $a \in \min(u'_i)$ and $a I (M_i \setminus C_i)$. This leads to for cases:
[*Case 1.*]{} $a \in M_i \setminus C_i$ and $b \in N_i \setminus D_i$. This contradicts the definition of $C_i$ and $D_i$ since $a$ and $b$ belong to the same $A_j$.
[*Case 2.*]{} $a \in M_i \setminus C_i$ and $b \in D_i C_i$ Again this contradicts the fact that $a$ and $b$ belong to the same $A_j$.
[*Case 3.*]{} $a \in \min(u'_i)$, $a I (M_i \setminus C_i)$ and $b \in N_i \setminus D_i$. From $b \in N_i \setminus D_i$ we get $b I D_i$ and thus $a I C_i$. Since $a I (M_i \setminus C_i)$ we have $a I M_i$. Moreover, since $a$ is a minimal atom of $u'_i$, it follows that $a$ is a minimal atom of $u_i = [M_i]_I u'_i [N_i]_I$. This contradicts the fact that $M_i$ is the set of minimal atoms of $u_i$.
[*Case 4.*]{} $a \in \min(u'_i)$, $a I (M_i \setminus C_i)$ and $b \in D_i C_i$. Since $b \in D_i C_i \cap A_j$, there exists $b' \in C_i \cap A_j$. With $a \in \min(u'_i)$ and $a I (M_i \setminus C_i)$ it follows that $u_i = [M_i]_I u'_i [N_i]_I$ contains the factor $b'a$. Since $a,b' \in A_j$ this contradicts the irreducibility of $u_i$.
\[power-presentation\] From a trace $u \in {\mathbb{M}}(A,I)$ one can compute (i) traces $s,t, v_1, \ldots, v_k \in {\mathsf{IRR}}(R)$, such that the following hold:
- every $v_i$ is either atomic or well-behaved,
- $u^m =_{{G}} s v_1^m \cdots v_k^{m} t$ for all $m \geq 0$,
- ${|\!| s |\!|} + {|\!| t |\!|} + \sum_{i=1}^k {|\!| v_i |\!|} \le 3 {|\!| u |\!|}$,
- $k \leq \alpha$, where $\alpha$ is the size of a largest clique in $(\Gamma,E)$.
Let $u \in {\mathbb{M}}(A,I)$. As an initial processing, we can replace every $u$ by ${\mathsf{NF}}_R(u) \in {\mathsf{IRR}}(R)$. So we can assume that $u$ is already irreducible. In the next step, we compute irreducible traces $s, w, t$, such that $u^m=_{{G}} sw^m t$ for all $m \ge 0$ and $w$ cannot be written as $w=aw' b$ with $a,b \in A_j$ for some $j \in \Gamma$. For this, we will inductively construct irreducible traces $s_i, u_i, t_i$ (with $0 \le i \le l$ for some $l$) such that $u^m=_{{G}}s_{i} u_{i}^m t_{i}$ for all $m \ge 0$. Moreover, if $0 \le i < l$ then $|u_{i}| > |u_{i+1}|$. We start with $u_0=u$ and $s_0=t_0=1$. Assume that after $i$ steps we already found irreducible traces $s_{i}, u_{i}, t_{i}$ with $u^m=_{{G}} s_{i} u_{i}^m t_{i}$ for all $m \ge 0$. If $u_{i}$ cannot be written in the form $au'b$ with $a,b \in A_j$ for some $j$, then we are done. Otherwise assume that $u_{i}=av_{i}b$ for some $a, b \in A_j$. Let $c \in A_j \cup \{ 1 \}$ such that $c = ba$ in the group ${G}_j$. So we get $u_{i}^m=_{{G}} a(v_{i}c)^m a^{-1}$ for all $m \ge 0$. This means $u^m=_{{G}} (s_{i}a) (v_{i}c)^m (a^{-1}t_{i})$. Hence, we can set $u_{i+1} :=v_ic$, $s_{i+1} := {\mathsf{NF}}_R(s_ia)$ and $t_{i+1} :={\mathsf{NF}}_R(a^{-1}t_i)$. Note that $|u_{i+1}| = |u_i|-1$, ${|\!| u_{i+1} |\!|} \leq {|\!| u_i |\!|}$, ${|\!| s_{i+1} |\!|} \leq {|\!| s_i |\!|} + {|\!| a |\!|}$, and ${|\!| t_{i+1} |\!|} \leq {|\!| t_i |\!|} + {|\!| a |\!|}$. This process is terminating after at most $|u|$ steps. Note also that each $u_{i+1}$ is irreducible. When our algorithm is terminating after step $l$, we set $v=u_l$, $s=s_l$ and $t=t_l$. We have $$\label{norm-bound}
{|\!| s |\!|}, {|\!| t |\!|}, {|\!| v |\!|} \leq {|\!| u |\!|} .$$ Finally, we split $v$ into its connected components, i.e., we write $v=v_1 \cdots v_{k}$, where every $v_j$ is connected and $v_i I v_j$ for $i \neq j$. We obtain for every $m \ge 0$ the identity $u^m =_{{G}} s v_{1}^m \cdots v_k^{m} t$ as described in the statement of the lemma. If a $v_j$ is not atomic then it cannot be written as $v_j=bv'_jc$ with $b, c \in A_i$ (otherwise the above reduction process would continue). Thus Lemma \[well-behaved-iff\] implies that the non-atomic $v_j$ are well-behaved. Finally, we have $\sum_{i=1}^k {|\!| v_i |\!|} = {|\!| v |\!|} \le {|\!| u |\!|}$ by .
\[remark-power-presentation\] If $E=\emptyset$ then we must have $k = 1$ in Lemma \[power-presentation\] since $\alpha=1$. Hence, we obtain $s,t,v$, where $v$ is either atomic or well-behaved, such that $u^m = s v^m t$ for every $m \geq 0$ and ${|\!| s |\!|} + {|\!| v |\!|} + {|\!| t |\!|} \le 3 {|\!| u |\!|}$.
Reductions to the empty trace
-----------------------------
For the normal form of the product of two $R$-irreducible traces we have the following lemma, which was shown in [@DiLo08IJAC] (equation (21) in the proof of Lemma 22) using a slightly different notation.
\[normalform\] Let $u,v \in {\mathbb{M}}(A,I)$ be $R$-irreducible. Then there exist strongly compatible independence cliques $C, D$ and unique factorizations $u = p C s$, $v = s^{-1} D t$ such that ${\mathsf{NF}}_R(uv) = p (CD) t$.
In the following, we consider tuples over ${\mathsf{IRR}}(R)$ of arbitrary length. We identify tuples that can be obtained from each other by inserting/deleting $1$’s at arbitrary positions. Clearly, every tuple is equivalent to a possibly empty tuple over ${\mathsf{IRR}}(R) \setminus \{1\}$.
\[def-reduction\] We define a reduction relation on tuples over ${\mathsf{IRR}}(R)$ of arbitrary length. Take $u_1, u_2, \ldots, u_m \in {\mathsf{IRR}}(R)$. Then we have
- $(u_1, u_2, \ldots, u_m) \to (u_1, \ldots, u_{i-1}, u_{i+1}, u_i, u_{i+2}, \ldots, u_m)$ if $u_i I u_{i+1}$ (a [*swapping step*]{}),
- $(u_1, u_2, \ldots, u_m) \to (u_1, \ldots, u_{i-1}, u_{i+2}, \ldots, u_m)$ if $u_i = u_{i+1}^{-1}$ in ${\mathbb{M}}(A,I)$ (a [*cancellation step*]{}),
- $(u_1, u_2, \ldots, u_m) \to (u_1, \ldots, u_{i-1}, a, u_{i+2}, \ldots, u_m)$ if there exists $j \in \Gamma$ with $u_i, u_{i+1}, a \in A_k$, and $a =_{{G}_j} u_i u_{i+1}$ (an [*atom creation step of type $j$*]{}).
Moreover, these are the only reduction steps. A concrete sequence of these rewrite steps leading to the empty tuple is a [*reduction*]{} of $(u_1, u_2, \ldots, u_m)$. If such a sequence exists, the tuple is called [*$1$-reducible*]{}.
A reduction of the tuple $(u_1, u_2, \ldots, u_m)$ can be seen as a witness for the fact that $u_1 u_2 \cdots u_m =_{{G}} 1$. On the other hand, $u_1 u_2 \cdots u_m =_{{G}}1$ does not necessarily imply that $(u_1, u_2, \ldots, u_m)$ has a reduction. For instance, the tuple $(a^{-1}, ab, b^{-1})$ has no reduction. But we can show that every sequence which multiplies to $1$ in ${G}$ can be refined (by factorizing the elements of the sequence) such that the resulting refined sequence has a reduction. We say that the tuple $(v_1, v_2, \ldots, v_n)$ is a [*refinement*]{} of the tuple $(u_1, u_2, \ldots, u_m)$ if there exists factorization $u_i = u_{i,1} \cdots u_{i,k_i}$ in ${\mathbb{M}}(A,I)$ such that $(v_1, v_2, \ldots, v_n) = (u_{1,1}, \ldots, u_{1,k_1}, \; u_{2,1}, \ldots, u_{2,k_2}, \; \ldots, u_{m,1}, \ldots, u_{m,k_m})$. In the following, if an independence clique $C$ appears in a tuple over ${\mathsf{IRR}}(R)$, we identify this clique with the sequence $a_1, a_2, \ldots, a_n$ which is obtained by enumerating the elements of $C$ in an arbitrary way. For instance, $( [abcd]_I, \{a,b,c\})$ stands for the tuple $([abcd]_I, a,b,c)$. Let us first prove the following lemma:
\[lemma-aux-eps-red\] Assume that the tuple $(v_1, v_2, \ldots, v_n)$ is $1$-reducible with at most $m$ atom creations of each type. For all $1 \leq i \leq n$ let $v_i = p_i D_i t_i$ be a factorization in ${\mathbb{M}}(A,I)$ where $D_i$ is an independence clique of $(A,I)$. By refining $p_1, t_1, \ldots, p_n, t_n$ into totally at most $4n + \sum_{i=1}^n |D_i|$ traces, we can obtain a refinement of $(p_1, D_1, t_1, p_2, D_2, t_2, \ldots, p_n, D_n, t_n)$ which is $1$-reducible with at most $m$ atom creations of each type.
Basically, we would like to apply to $(p_1, D_1, t_1, p_2, D_2, t_2, \ldots, p_n, D_n, t_n)$ the same reduction that reduces $(v_1, v_2, \ldots, v_n)$ to the empty tuple. If we do a swapping step $v_i, v_j \to v_j, v_i$ then we can swap also the order of $p_i, D_i, t_i$ and $p_j, D_j, t_j$ in several swapping steps. Also notice that if $v_i$ is an atom, then the subsequence $p_i, D_i, t_i$ is equivalent to the atom $v_i$. The only remaining problem are cancellation steps. Assume that $v_i$ and $v_j$ cancel, i.e., $v_i = v_j^{-1}$. The traces $t_i$ and $t_j$ do not necessarily cancel out, and similarly for $p_i$ and $p_j$ and the atoms in $D_i$ and $D_j$. Hence, we have to further refine $p_i, t_i, p_j, t_j$ using Levi’s lemma. Applied to the identity $p_i D_i t_i = t_j^{-1} D_j^{-1} p_j^{-1}$ it yields the following diagram: $$\label{diagramm-aux}
\begin{tabular}{c"c|c|c|}\hline
$p_j^{-1}$ & $x_{i,j}$ & $N_{i,j}$ & $z_{i,j}$ \\ \hline
$D_j^{-1}$ & $W_{i,j}$ & $C_{i,j}$ & $E_{i,j}$ \\ \hline
$t_j^{-1}$ & $w_{i,j}$ & $S_{i,j}$ & $y_{i,j}$ \\ { \noalign {\ifnum 0=`}\fi \hrule height 1pt
\futurelet \reserved@a \@xhline
}& $p_i$ & $D_i$ & $t_i$
\end{tabular}$$ Hence, we get factorizations $$\begin{aligned}
p_i &=& w_{i,j} W_{i,j} x_{i,j} \label{refine-pi} \\
t_i &=& y_{i,j} E_{i,j} z_{i,j} \label{refine-ti} \\
p_j &=& z_{i,j}^{-1} N_{i,j}^{-1} x_{i,j}^{-1} \label{refine-pj} \\
t_j &=& y_{i,j}^{-1} S_{i,j}^{-1} w_{i,j}^{-1} \label{refine-tj} .\end{aligned}$$ where $D_i = S_{i,j} \uplus C_{i,j} \uplus N_{i,j}$ and $D_j = E_{i,j}^{-1} \uplus C_{i,j}^{-1} \uplus W_{i,j}^{-1}$. Using these facts and the independencies obtained from the diagram shows that the tuple $$(w_{i,j}, W_{i,j}, x_{i,j}, D_i, y_{i,j}, E_{i,j}, z_{i,j}, z_{i,j}^{-1}, N_{i,j}^{-1}, x_{i,j}^{-1}, D_j, y_{i,j}^{-1}, S_{i,j}^{-1}, w_{i,j}^{-1})$$ is $1$-reducible. Hence, by refining $p_i$, $t_i$, $p_j$, and $t_j$ according to the factorizations , , , and , respectively, we obtain a $1$-reducible refinement of $(p_1, D_1, t_1, p_2, D_2, t_2, \ldots, p_n, D_n, t_n)$ . Note that $|W_{i,j} \cup E_{i,j}| \le |D_j|$ and $|N_{i,j} \cup S_{i,j}| \le |D_i|$. Hence, the $2n$ traces $p_1, t_1, \ldots, p_n, t_n$ are refined into totally at most $4n + \sum_{i=1}^n |D_i|$ traces.
As before, $\alpha$ denotes the size of a largest independence clique in $(A,I)$.
\[lemma-reduction\] Let $m \geq 2$ and $u_1, u_2, \ldots, u_m \in {\mathsf{IRR}}(R)$. If $u_1 u_2 \cdots u_m = 1$ in ${G}$, then there exists a $1$-reducible refinement of $(u_1, u_2, \ldots, u_m)$ that has length at most $(3\alpha+4)m^2 \leq 7 \alpha m^2$ and there is a reduction of that refinement with at most $m-2$ atom creations of each type $i \in \Gamma$.
The proof of the lemma will be an induction on $m$. For this we first assume that $m$ is a power of 2. We will show that there exist factorizations of the $u_i$ with totally at most $(\frac{3}{4}\alpha+1)m^2-(\frac{3}{2}\alpha+1)m$ factors such that the resulting tuple is $1$-reducible and has a reduction with at most $(m-2)$ atom creations of each type $i \in \Gamma$. This implies the lemma for the case that $m$ is a power of two.
The case $m=2$ is trivial (we must have $u_2 = u_1^{-1}$). Let $m = 2n \geq 4$. Then by Lemma \[normalform\] we can factorize $u_{2i-1}$ and $u_{2i}$ for $1\leq i \leq n$ as $u_{2i-1} = p_i C_{2i-1} s_i$ and $u_{2i} = s_i^{-1} C_{2i} t_i$ in ${\mathbb{M}}(A,I)$ such that $C_{2i-1}$ and $C_{2i}$ are strongly compatible independence cliques and $v_i := p_j (C_{2i-1}C_{2i}) t_i$ is irreducible. Define the independence clique $D_i = C_{2i-1}C_{2i}$. We have $v_1 v_2 \cdots v_n = 1$ in ${G}$. By induction, we obtain factorizations $p_i D_i t_i = v_i = v_{i,1} \cdots v_{i,k_i}$ ($1\leq i \leq n$) such that the tuple $$\label{sequence-IH}
(v_{i,1}, \ldots, v_{i,k_i})_{1 \le i \leq n}$$ is $1$-reducible. Moreover, $$\sum_{i=1}^{n} k_i \leq \left(\frac{3}{4}\alpha+1 \right)n^2 - \left(\frac{3}{2}\alpha +1\right)n$$ and there exists a reduction of the tuple with at most $n-2$ atom creations of each type. By applying Levi’s lemma to the trace identities $p_i D_i t_i = v_{i,1} v_{i,2} \cdots v_{i,k_i}$, we obtain factorizations $v_{i,j} = x_{i,j} D_{i,j} y_{i,j}$ in ${\mathbb{M}}(A,I)$ such that $D_i = \biguplus_{1 \leq j \leq k_i} D_{i,j}$, $p_i = x_{i,1} \cdots x_{i,k_i}$, $t_i = y_{i,1} \cdots u_{i,k_i}$, and the following independencies hold for $1 \leq j < l \leq k_i$: $y_{i,j} I x_{i,l}$, $y_{i,j} I a$ for all $a \in D_{i,l}$, $a I x_{i,l}$ for all $a \in D_{i,j}$. Note that $D_{i,j}$ can be the empty set.
Let us now define for every $1 \leq i \leq n$ the tuples $\overline{u}_{2i-1}$ and $\overline{u}_{2i}$ as follows:
- $\overline{u}_{2i-1} = (x_{i,1}, \ldots, x_{i,k_i}, C_{2i-1}, s_i)$
- $\overline{u}_{2i} = (s_i^{-1}, C_{2i}, y_{i,1}, \ldots, y_{i,k_i})$
Thus, the tuple $\overline{u}_i$ defines a factorization of the trace $u_i$ and the tuple $(\overline{u}_1, \overline{u}_2, \ldots, \overline{u}_{2n})$ is a refinement of $(u_1, \ldots, u_{2n})$ of length $2 \ell(n) + 2 n (\alpha + 1)$. This tuple can be transformed using $n$ cancellation steps (cancelling $s_i$ and $s_i^{-1}$) and $n$ atom creations of each type into the sequence $$(x_{i,1}, \ldots, x_{i,k_i}, D_i, y_{i,1}, \ldots, y_{i,k_i})_{1 \le i \le n} .$$ Using swappings, we finally obtain the sequence $$\label{tuple-xDy}
(x_{i,1},D_{i,1},y_{i,1}, \ldots, x_{i,k_1},D_{i,k_1},y_{i,k_1})_{1 \le i \le n} .$$ Recall that $v_{i,j} = x_{i,j} D_{i,j} y_{i,j}$. Hence, the tuple is a refinement of the $1$-reducible tuple . We are therefore in the situation of Lemma \[lemma-aux-eps-red\]. By further refining the totally at most $2 \ell(n)$ factors $x_{i,j}$ and $y_{i,j}$ of the traces $u_1, \ldots, u_{2n}$ we obtain a $1$-reducible tuple. The resulting refinements of $(u_1, \ldots, u_{2n})$ has length at most $$\begin{aligned}
&& 4 \sum_{i=1}^{n} k_i + \sum_{i=1}^n \sum_{j=1}^{k_i} |D_{i,j}| + 2n + 2n\alpha \\
&\le & 4 \left(\frac{3}{4}\alpha+1 \right)n^2 - 4 \left(\frac{3}{2}\alpha+1\right)n + \sum_{i=1}^n |D_i| + 2n + 2n\alpha \\
&\le& (3\alpha+4) n^2 - (6\alpha+4)n + (3\alpha+2) n \\
&=& (3\alpha+4) n^2 - (3\alpha+2) n \\
& = & \left(\frac{3}{4}\alpha+1\right) m^2 - \left(\frac{3}{2}\alpha+1\right) m\end{aligned}$$ ($\sum_{i=1}^{n} k_i + \sum_{i=1}^n \sum_{j=1}^{k_i} |D_{i,j}|$ traces from the refinement of the traces $x_{i,j}$ and $y_{i,j}$ by Lemma \[lemma-aux-eps-red\], $2n$ traces $s_i^{\pm 1}$, and $2n\alpha$ atoms from the independence cliques $C_i$). Finally, the total number of atom creations of a certain type is $n + n-2 = 2n-2 = m-2$.
In the general case, where $m$ is not assumed to be a power of two, we can naturally extend the sequence to $u_1, u_2, \dots , u_{l}$ by possibly adding $u_i=1$ for $i>m$ to the smallest power of 2. Hence $l\leq 2m$. Substituting $2m$ for $m$ yields the desired bound. Note that by this process, the number of atom creations will not increase. This concludes the proof of the lemma.
Since by this result we also get a $1$-reducible tuple with at most $\mathcal{O}(m^2)$ many elements for equations $u_1 u_2 \cdots u_m = 1$ over a graph group, this improves the result of [@LohreyZ18].
\[rem-atom-creation\] The atom creations that appear in a concrete reduction can be collected into finitely many identities of the form $a_1 a_2 \cdots a_k =_{{G}_i} b_1 b_2 \cdots b_l$ (or $a_1 a_2 \cdots a_k b_l^{-1} \cdots b_2^{-1} b_1^{-1} =_{{G}_i} 1$), where $a_1, a_2, \ldots, a_k, b_1, b_2,\ldots, b_l$ are atoms from the initial sequence that all belong to the same group ${G}_i$. The new atoms $a_1 a_2 \cdots a_k$ and $b_1 b_2 \cdots b_l$ are created by at most $m-2$ atom creations. Finally, the two resulting atoms cancel out. Note that $k-1+l-1 \leq m-2$, i.e., $k+l \leq m$.
In case $E = I = \emptyset$ the quadratic dependence on $m$ in Lemma \[lemma-reduction\] can be avoided:
\[lemma-reduction-free-product\] Let $m \geq 2$ and $u_1, u_2, \ldots, u_m \in {\mathsf{IRR}}(R)$. Moreover let $E=I=\emptyset$. If $u_1 u_2 \cdots u_m = 1$ in the free product ${G}$, then there exist a $1$-reducible refinement of the tuple $(u_1, u_2, \ldots, u_m)$ that has length at most $7m-12\leq 7m$ and there is a reduction of this refinement with at most $m-2$ atom creations.
We prove the lemma by induction on $m$. The case $m=2$ is trivial (we must have $u_2 = u_1^{-1}$). If $m \geq 3$ then for the normal form of $u_1 u_2$ there are two cases: either $u_1 u_2 \in {\mathsf{IRR}}(R)$ or $u_1=pas$ and $u_2=s^{-1}bt$ for atoms $a,b$ from the same group ${G}_i$ that do not cancel out. We consider only the latter case. Let $c := ab$ in ${G}_i$, i.e., $c \in A_i$. By the induction hypothesis, the tuple $(pct, u_3, \dots , u_m)$ has a $1$-reducible refinement $$\label{eq-refinement-free}
(v_1, \dots , v_k, w_1, \dots , w_l)$$ with $k+l \leq 7(m-1)-12$ and $pct=v_1 \cdots v_k$, where the latter is an identity between words from $A^*$. Moreover, there is a reduction of with at most $m-3$ atom creations. Since $pct=v_1 \cdots v_k$, one of the $v_j$ ($1\leq j\leq k$) must factorize as $v_j = v_{j,1} c v_{j,2}$ such that $p=v_1 \cdots v_{j-1}v_{j,1}$ and $t = v_{j,2} v_{j+1}\cdots v_k$, which implies $u_1 = v_1 \cdots v_{j-1}v_{j,1} a s$ and $u_2 = s^{-1}b v_{j,2} v_{j+1}\cdots v_k$. Therefore we have a $1$-reducible tuple of the form $$\label{eq-refinement-free2}
(v_1, \dots , v_{j-1}, v_{j,1},a ,s , s^{-1}, b, v_{j,2}, v_{j+1}, \dots , v_k , \tilde{w}_1, \dots , \tilde{w}_l),$$ where the sequence $\tilde{w}_i$ is $w_i$ unless $w_i$ cancels out with $v_j$ in our reduction of (there can be only one such $i$), in which case $\tilde{w}_i$ is $(v_{j,2})^{-1}, c^{-1}, (v_{j,1})^{-1}$. It follows that the tuple is a refinement of $(u_1, u_2, \ldots, u_m)$ with at most $7(m-1)-12+7=7m-12$ words, having a reduction with at most $m-2$ atom creations.
Graph products preserve knapsack semilinearity {#sec-main}
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In this section, we assume that every group ${G}_i$ ($i \in \Gamma$) is knapsack-semilinear. Recall that we fixed the symmetric generating set $\Sigma_i$ for ${G}_i$, which yields the generating set $\Sigma = \bigcup_{i \in \Gamma} \Sigma_i$ for the graph product ${G}$. In this section, we want to show that the graph product ${G}$ is knapsack-semilinear as well. Moroever, we want to bound the function $\mathsf{E}_{{G},\Sigma}$ in terms of the functions $\mathsf{K}_{{G}_i,\Sigma_i}$. Let $\mathsf{K} : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ be the pointwise maximum of the functions $\mathsf{K}_{{G}_i,\Sigma_i}(n,m)$. We will bound $\mathsf{E}_{{G}, \Sigma}$ in terms of $\mathsf{K}$.
Consider an exponent expression ${e}= u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_m^{x_m} v_m$, where $u_i, v_i$ are words over the generating set $\Sigma$. Let $g_i$ (resp., $h_i$) be the element of ${G}$ represented by $u_i$ (resp., $v_i$). We can assume that all $u_i$ and $v_i$ are geodesic words in the graph product ${G}$.[^5] We will make this assumption throughout this section. Moreover, we can identify each $u_i$ (resp., $v_i$) with the unique irreducible trace from ${\mathsf{IRR}}(R)$ that represents the group element $g_i$ (resp., $h_i$). In addition, for each atom $a \in A$ (say $a \in A_i$) that occurs in one of the traces $u_1, u_2, \ldots, u_m, v_1, \ldots, v_m \in {\mathsf{IRR}}(R)$ a geodesic word $w_a \in \Sigma_i^*$ that evaluates to $a$ in the group ${G}_i$ is given. This yields geodesic words for the group elements $g_1, \ldots, g_m, h_1, \ldots, h_m \in {G}$. The lengths of these words are ${|\!| u_1 |\!|}, \ldots, {|\!| u_m |\!|}, {|\!| v_1 |\!|}, \ldots, {|\!| v_m |\!|}$ and we have ${|\!| {e}|\!|} = {|\!| u_1 |\!|} + \cdots + {|\!| u_m |\!|} + {|\!| v_1 |\!|} + \cdots + {|\!| v_m |\!|}$.
We start with the following preprocessing step:
\[lemma-preproc\] Let ${e}$ be an exponent expression over $\Sigma$. From ${e}$ we can compute a knapsack expression ${e}'$ with the following properties:
- $X_{{e}} \subseteq X_{{e}'}$,
- ${|\!| {e}' |\!|} \le 3 {|\!| {e}|\!|}$,
- $\deg({e}') \le \alpha \cdot \deg({e})$,
- every period of ${e}'$ is either atomic or well-behaved, and
- ${\mathsf{sol}}_{{G}}({e}) = (K \cap {\mathsf{sol}}_{{G}}({e}')) {\mathord\restriction}_{X_{{e}}}$ for a semilinear set $K$ of magnitude one.
Let $u_1, \ldots, u_m \in \Sigma^*$ be the periods of ${e}$. We can view these words as traces $u_1, \ldots, u_m \in {\mathbb{M}}(A,I)$ that are moreover irreducible. We apply Lemma \[power-presentation\] to each power $u_i^{x}$ in ${e}$ and obtain an equivalent exponent expression $\tilde{e}$ of degree $n \le \alpha \cdot m$ and ${|\!| \tilde{e}|\!|} \leq 3{|\!| {e}|\!|}$. We have $X_{\tilde{e}} = X_{{e}}$ and ${\mathsf{sol}}_{{G}}({e}) = {\mathsf{sol}}_{{G}}(\tilde{e})$.
We now rename in $\tilde{e}$ the variables by fresh variables in such a way that we obtain a knapsack expression ${e}'$. Moreover, for every $x \in X_{{e}}$ we keep exactly one occurrence of $x$ in $\tilde{e}$ and do not rename this occurrence of $x$. This implies that there is a semilinear set $K \subseteq \mathbb{N}^{X_{\overline{a}}}$ of magnitude one such that ${\mathsf{sol}}_{{G}}({e}) = (K \cap {\mathsf{sol}}_{{G}}({e}')){\mathord\restriction}_{X_{{e}}}$.
In case $E = I = \emptyset$ and that ${e}$ is a knapsack expression, we can simplify the statement of Lemma \[lemma-preproc\] as follows:
\[rem-preproc-free-product\] Assume that $E = I = \emptyset$ and that ${e}$ is a knapsack expression as in Lemma \[lemma-preproc\]. By Remark \[remark-power-presentation\] we can compute from ${e}$ a knapsack expression ${e}'$ over $\Sigma$ with the following properties:
- ${|\!| {e}' |\!|} \leq 3{|\!| {e}|\!|}$,
- $\deg({e}') \le \deg({e})$,
- every period of ${e}'$ is either atomic or well-behaved, and
- ${\mathsf{sol}}_{{G}}({e}) = {\mathsf{sol}}_{{G}}({e}')$.
We now come to the main technical result of Section \[sec-part1\]. As before, we denote with $\alpha$ the size of a largest independence clique in $(\Gamma,E)$.
\[thm-main-technical\] Let $\mathsf{K} : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ be the pointwise maximum of the functions $\mathsf{K}_{{G}_i,\Sigma_i}(n,m)$ for $i \in \Gamma$. Then $\mathsf{E}_{{G},\Sigma}(n,m) \leq \max\{\mathsf{K}_1 , \mathsf{K}_2\}$ with $$\begin{aligned}
\mathsf{K}_1 & \le \mathcal{O}\big( (\alpha m)^{\alpha m/2+3} \cdot \mathsf{K}(6 \alpha m n,\alpha m)^{\alpha m+3}\big),\\
\mathsf{K}_2 & \le (\alpha m)^{\mathcal{O}(\alpha^2 m)} \cdot n^{\mathcal{O}(\alpha^2 |\Gamma| m)}.\end{aligned}$$
Consider an exponent expression ${e}= u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_m^{x_m} v_m$. Let us denote with $A({e}) = {\mathsf{alph}}(u_1 v_1 \cdots u_m v_m) \subseteq A$ the set of all atoms that appear in the traces $u_i,v_i$. Finally let $\mu({e}) = \max\{{|\!| a |\!|} \mid a \in A({e})\}$ and let $\lambda({e})$ be the maximal length $|t|$ where $t$ is one of the traces $u_1, u_2, \ldots, u_m, v_1, \ldots, v_m$. We clearly have $\mu({e}) \le {|\!| {e}|\!|}$ and $\lambda({e}) \le {|\!| {e}|\!|}$.
Let us first assume that ${e}$ is a knapsack expression (i.e., $x_i \neq x_j$ for $i \neq j$) where every period $u_i$ is either an atom or a well-behaved trace (see Lemma \[lemma-preproc\]).
In the following we describe an algorithm that computes a semilinear representation of ${\mathsf{sol}}_{{G}}({e})$ (for ${e}$ satisfying the conditions from the previous paragraph). At the same time, we will compute the magnitude of this semilinear representation. The algorithm transforms logical statements into equivalent logical statements (we do not have to define the precise logical language; the meaning of the statements should be always clear). Every statement contains the variables $x_1, \ldots, x_m$ from our knapsack expression and equivalence of two statements means that for every valuation $\sigma : \{x_1, \ldots, x_m\} \to {\mathbb{N}}$ the two statements yield the same truth value. We start with the statement ${e}= 1$. In each step we transform the current statement $\Phi$ into an equivalent disjunction $\bigvee_{i=1}^n \Phi_i$. We can therefore view the whole process as a branching tree, where the nodes are labelled with statements. If a node is labelled with $\Phi$ and its children are labelled with $\Phi_1, \ldots, \Phi_n$ then $\Phi$ is equivalent to $\bigvee_{i=1}^n \Phi_i$. The leaves of the tree are labelled with semilinear constraints of the form $(x_1, \ldots, x_m) \in L$ for semilinear sets $L$. Hence, the solution set ${\mathsf{sol}}_{{G}}({e})$ is the union of all semilinear sets that label the leaves of the tree. A bound on the magnitude of these semilinear sets yields a bound on the magnitude of ${\mathsf{sol}}_{{G}}({e})$. Therefore, we can restrict our analysis to a single branch of the tree. We can view this branch as a sequence of nondeterministic guesses. Some guesses lead to dead branches because the corresponding statement is unsatisfiable. We will speak of a bad guess in such a situation.
Let $N_a \subseteq [1,m]$ be the set of indices such that $u_i$ is atomic and let $N_{\overline{a}} = [1,m] \setminus N_a$ be the set of indices such that $u_i$ is not atomic (and hence a well-behaved trace). For better readability, we write $a_i$ for the atom $u_i$ in case $i \in N_a$. Define $X_a = \{ x_i \mid i \in N_a\}$ and $X_{\overline{a}} = \{ x_i \mid i \in N_{\overline{a}} \}$. For $i \in N_a$ let $\gamma(i) \in \Gamma$ be the index with $u_i \in A_{\gamma(i)}$.
[*Step 1: Eliminating trivial powers.*]{} In a first step we guess a set $N_1 \subseteq N_a$ of indices with the meaning that for $i \in N_1$ the power $a_i^{x_i}$ evaluates to the identity element of the group ${G}_{\gamma(i)}$. To express this we continue with the formula $$\label{formula-Phi}
\Phi[N_1] := (e[N_1] = 1) \wedge \bigwedge_{i \in N_1} a_i^{x_i} =_{{G}_{\gamma(i)}} 1,$$ where $e[N_1]$ is the knapsack expression obtained from $e$ by deleting all powers $a_i^{x_i}$ with $i \in N_1$. Note that the above constraints do not exclude that a power $u_i^{x_i}$ with $i \in [1,m] \setminus N_1$ evaluates to the identity element. This will not cause any trouble for the following arguments. Clearly, the initial equation $e=1$ is equivalent to the formula $\bigvee_{N_1 \subseteq N_a} \Phi[N_1]$.
In the following we transform every equation $e[N_1] = 1$ into a formula $\Psi[N_1]$ such that the following hold for every valuation $\sigma : \{ x_i \mid i \in [1,m] \setminus N_1 \} \to {\mathbb{N}}$:
(1) if $\Psi[N_1]$ is true under $\sigma$ then $\sigma(e[N_1]) =_{{G}} 1$,
(2) if $a_i^{\sigma(x_i)} \neq_{{G}_{\gamma(i)}} 1$ for all $i \in N_a \setminus N_1$ and $\sigma(e[N_1]) =_{{G}} 1$ then $\Psi[N_1]$ is true under $\sigma$.
This implies that $\bigvee_{N_1 \subseteq N_a} \Phi[N_1]$ (and hence $e=1$) is equivalent to the formula $$\bigvee_{N_1 \subseteq N_a} (\Psi[N_1] \wedge \bigwedge_{i \in N_1} a_i^{x_i} =_{{G}_{\gamma(i)}} 1).$$ [*Step 2: Applying Lemma \[lemma-reduction\].*]{} We construct the formula $\Psi[N_1]$ from the knapsack expression $e[N_1]$ using Lemma \[lemma-reduction\]. More precisely, we construct $\Psi[N_1]$ by nondeterministically guessing the following data:
(i) factorizations $v_i = v_{i,1} \cdots v_{i,l_i}$ in ${\mathbb{M}}(A,I)$ of all non-trivial traces $v_i$. Each factor $v_{i,j}$ must be nontrivial too.
(ii) “symbolic factorizations” $u_i^{x_i} = y_{i,1} \cdots y_{i,k_i}$ for all $i \in N_{\overline{a}}$. The numbers $k_i$ and $l_i$ must sum up to at most $28\alpha m^2$ (this number is obtained by replacing $m$ by $2m$ in Lemma \[lemma-reduction\]). The $y_{i,j}$ are existentially quantified variables that take values in ${\mathsf{IRR}}(R)$ and which will be eliminated later.
(iii) non-empty alphabets $A_{i,j} \subseteq {\mathsf{alph}}(u_i)$ for each symbolic factor $y_{i,j}$ ($i \in N_{\overline{a}}$, $1 \leq j \leq k_i$) with the meaning that $A_{i,j}$ is the alphabet of atoms that appear in $y_{i,j}$.
(iv) a reduction (according to Definition \[def-reduction\]) of the resulting refined factorization of $u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_m^{x_m} v_m$ with at most $2m-2$ atom creations of each type $i \in \Gamma$.
Note that every factor $a_i^{x_i}$ with $i \in N_{a} \setminus N_1$ evaluates (for a given valuation) either to an atom from $A_{\gamma(i)}$ or to the identity element. Hence, there is no need to further factorize such a power $a_i^{x_i}$. In our guessed reduction we treat $a_i^{x_i}$ as a symbolic atom (although it might happen that $a_i^{\sigma(x_i)} = 1$ for a certain valuation $\sigma$; but this will not make the above statements (1) and (2) wrong).
We can also guess $k_i = 0$ in (ii). In this case, we can replace $u_i^{x_i}$ in $e[N_1]$ by the empty trace and add the constraint $x_i = 0$ (note that a well-behaved trace $u_i \neq 1$ represents an element of the graph product ${G}$ without torsion). Hence, in the following we can assume that the $k_i$ are not zero. Some of the $y_{i,j}$ must be atoms since they take part in an atom creation in our guessed reduction. Such an $y_{i,j}$ is replaced by a nondeterministically guessed atom $a_{i,j}$ from the atoms in $u_i$.
The guessed alphabetic constraints from (iii) must be consistent with the independencies from our guessed reduction in (iv). This means that if for instance $y_{i,j}$ and $y_{k,l}$ are swapped in the reduction then we must have $A_{i,j} \times A_{j,k} \subseteq I$. Here comes a subtle point: Recall that each power $a^{x_i}$ ($i \in N_a \setminus N_1$) evaluates for a given valuation either to an atom from $A_{\gamma(i)}$ or to the identity element. When checking the consistency of the alphabetic constraints with the guessed reduction we make the (pessimistic) assumption that every $a^{x_i}$ evaluates to an atom from $A_{\gamma(i)}$. This is justified below.
For every specific guess in (i)–(iv) we write down the existentially quantified conjunction of the following formulas:
- the equation $u_i^{x_i} = y_{i,1} \cdots y_{i,k_i}$ from (ii) (every trace-variable $y_{i,j}$ is existentially quantified),
- all trace equations that result from cancellation steps in the guessed reduction,
- all “local” identities that result from the atom creations in the guessed reduction,
- all alphabetic constraints from (iii) and
- all constraints $x_i = 0$ in case we guessed $k_i = 0$ in (ii).
The local identities in the third point involve the above atoms $a_{i,j}$ and the powers $a_i^{x_i}$ for $i \in N_a \setminus N_1$. According to Remark \[rem-atom-creation\] they are combined into several knapsack expressions over the groups ${G}_i$.
The formula $\Psi[N_1]$ is the disjunction of the above existentially quantified conjunctions, taken over all possible guesses in (i)–(iv). It is then clear that the above points (1) and (2) hold. Point (2) follows immediately from Lemma \[lemma-reduction\]. For point (1) note that each of the existentially quantified conjunctions in $\Psi[N_1]$ yields the identity $e[N_1]=1$, irrespective of whether a power $a_i^{x_i}$ is trivial or not.
So far, we have obtained a disjunction of existentially quantified conjunctions. Every conjunction involves the equations $u_i^{x_i} = y_{i,1} \cdots y_{i,k_i}$ from (ii), trace equations that result from cancellation steps (we will deal with them in step 4 below), local knapsack expressions over the groups ${G}_i$, alphabetic constraints for the variables $y_{i,j}$ and constraints $x_i = 0$ (if $k_i = 0$). In addition we have the identities $a_i^{x_i} =_{{G}_{\gamma(i)}} 1$ ($i \in N_1$) from . In the following we deal with a single existentially quantified conjunction of this form.
[*Step 3: Isolating the local knapsack instances for the groups ${G}_i$.*]{} In our existentially quantified conjunction we have knapsack expressions ${e}_1, \ldots, {e}_q$ over the groups ${G}_j$ ($j \in \Gamma$). These knapsack expressions involve the atoms $a_{i,j}$ and the symbolic expressions $a_i^{x_i}$ with $i \in N_a$. Note that every identity $a_i^{x_i} =_{{G}_{\gamma(i)}} 1$ ($i \in N_1$) yields the knapsack expression $a_i^{x_i}$. Each of the expressions ${e}_j$ is built from at most $2m$ atom powers $a_i^{x_i}$ and atoms $a_{i,j}$ (since for every $j \in \Gamma$ there are at most $2m-2$ atom creations of type $j$) and its degree is at most $m$ (since there are at most $m$ atom powers $a_i^{x_i}$). All atoms $a_i$ and $a_{i,j}$ belong to $A({e})$. This yields the bound ${|\!| {e}_j |\!|} \le 2m \mu({e})$ for $1 \leq j \leq q$. We can assume that each expression ${e}_j$ contains at least one atom power $u_i^{x_i}$ (identities between the explicit atoms $a_{i,j}$ can be directly verified; if they do not hold, one gets a bad guess). Moreover, note that every atom power $a_i^{x_i}$ with $i \in N_a$ occurs in exactly one ${e}_j$. Assume that the knapsack expression ${e}_j$ is defined over the group $H_j \in \{ {G}_i \mid i \in \Gamma\}$. The solution sets ${\mathsf{sol}}_j := {\mathsf{sol}}_{H_j}({e}_j)$ of these expressions are semilinear by the assumption on the groups ${G}_i$. Each ${\mathsf{sol}}_j$ has some dimension $d_j \le m$ (which is the number of symbolic atoms in ${e}_j$), where $\sum_{j=1}^q d_j = |N_a|$ and the magnitude of ${\mathsf{sol}}_j$ is bounded by $\mathsf{K}(2m \mu({e}), m) \le \mathsf{K}(2m {|\!| {e}|\!|}, m)$. Finally, we can combine these sets ${\mathsf{sol}}_j$ into the single semilinear set $S_a := \bigoplus_{j=1}^q {\mathsf{sol}}_j \subseteq {\mathbb{N}}^{X_a}$ of dimension $|N_a|$ and magnitude at most $\mathsf{K}(2m {|\!| {e}|\!|},m)$. Recall that the sets ${\mathsf{sol}}_j$ refer to pairwise disjoint sets of variables. For the variables $x_i \in X_a$ we now obtain the semilinear contraint $(x_i)_{i \in N_a} \in S_a$.
[*Step 4: Reduction to two-dimensional knapsack instances.*]{} Let us now deal with the cancellation steps from our guessed reduction. From these reduction steps we will produce two-dimensional knapsack instances on pairwise disjoint variable sets.
If two explicit factors $v_{i,j}$ and $v_{k,l}$ (from (i) in step 2) cancel out in the reduction, we must have $v_{k,l} = v_{i,j}^{-1}$; otherwise our previous guess was bad. If a symbolic factor $y_{i,j}$ and an explicit factor $v_{k,l}$ cancel out, then we can replace $y_{i,j}$ by $v_{k,l}^{-1}$. Before doing this, we check whether ${\mathsf{alph}}(v_{k,l}^{-1}) = A_{i,j}$ and if this condition does not hold, then we obtain again a bad guess. Let $S$ be the set of pairs $(i,j)$ such that the symbolic factor $y_{i,j}$ still exists after this step. On this set $S$ there must exist a matching $M \subseteq \{ (i,j,k,l) \mid (i,j), (k,l) \in S \}$ such that $y_{i,j}$ and $y_{k,l}$ cancel out in our reduction if and only if $(i,j,k,l) \in M$. We have $(i,j,k,l) \in M$ if and only if $(k,l,i,j) \in M$.
Let us write the new symbolic factorization of $u_i^{x_i}$ as $u_i^{x_i} = \tilde{y}_{i,1} \cdots \tilde{y}_{i,k_i}$, where every $\tilde{y}_{i,j}$ is either the original symbolic factor $y_{i,j}$ (in case $(i,j) \in S$) or a concrete trace $v_{k,l}^{-1}$ (in case $y_{i,j}$ and $v_{k,l}$ cancel out in our reduction) or an atom $a_{i,j} \in {\mathsf{alph}}(u_i)$ (that was guessed in step 2). It remains to describe the set of all tuples $(x_1,\ldots, x_m)$ that satisfy a statement of the following form: there exist traces $y_{i,j}$ ($(i,j) \in S$) such that the following hold:
(a) $u_i^{x_i} = \tilde{y}_{i,1} \cdots \tilde{y}_{i,k_i}$ in ${\mathbb{M}}(A,I)$ for all $i \in N_{\overline{a}}$
(b) ${\mathsf{alph}}(y_{i,j}) = A_{i,j}$ for all $(i,j) \in S$,
(c) $y_{i,j} = y_{k,l}^{-1}$ in ${\mathbb{M}}(A,I)$ for all $(i,j,k,l) \in M$
(d) $(x_i)_{i \in N_a} \in S_a$
In the next step, we eliminate the trace equations $u_i^{x_i} = \tilde{y}_{i,1} \cdots \tilde{y}_{i,k_i}$ ($i \in N_{\overline{a}}$). We apply to each of these trace equations Lemma \[lemma-simplify-factorization-power\] (or Remark \[remark-simplify-factorization-power\]). For every $i \in N_{\overline{a}}$ we guess a subset $K_i \subseteq [1,k_i]$, an integer $0 \le c_i \leq |\Gamma| \cdot (k_i-1)$ and traces $p_{i,j}, s_{i,j}$ with ${\mathsf{alph}}(p_{i,j}) \subseteq {\mathsf{alph}}(u_i) \supseteq {\mathsf{alph}}(s_{i,j})$ and $|p_{i,j}|, |s_{i,j}| \leq |\Gamma| \cdot (k_i-1) \cdot |u_i| \leq |\Gamma| \cdot (k_i-1) \cdot \lambda({e})$, and replace $u_i^{x_i} = \tilde{y}_{i,1} \cdots \tilde{y}_{i,k_i}$ by the following statement: there exist integers $x_{i,j} > 0$ ($j \in K_i$) such that
- $x_i = c_i + \sum_{j \in K_i} x_{i,j}$,
- $\tilde{y}_{i,j} = p_{i,j} u_i^{x_{i,j}} s_{i,j}$ for all $j \in K_i$,
- $\tilde{y}_{i,j} = p_{i,j} s_{i,j}$ for all $j \in [1,k_i] \setminus K_i$.
At this point we can check whether the alphabetic constraints ${\mathsf{alph}}(y_{i,j}) = A_{i,j}$ for $(i,j) \in S$ hold (note that an equation $y_{i,j} = p_{i,j} s_{i,j}$ or $y_{i,j} = p_{i,j} u_i^{x_{i,j}} s_{i,j}$ with $x_{i,j} > 0$ determines the alphabet of $y_{i,j}$). Equations $\tilde{y}_{i,j} = p_{i,j} s_{i,j}$, where $\tilde{y}_{i,j}$ is an explicit trace can be checked and possibly lead to a bad guess. From an equation $\tilde{y}_{i,j} = p_{i,j} u_i^{x_{i,j}} s_{i,j}$, where $\tilde{y}_{i,j}$ is an explicit trace, we can determine a unique solution for $x_{i,j}>0$ (if it exists) and substitute this value into the equation $x_i = c_i + \sum_{j \in K_i} x_{i,j}$. Note that we must have $x_{i,j} \le |\tilde{y}_{i,j}| \leq \lambda({e})$, since $\tilde{y}_{i,j}$ is an atom or a factor of a trace $v_k^{-1}$. Similarly, an equation $y_{i,j} = p_{i,j} s_{i,j}$ with $(i,j) \in S$ allows us to replace the symbolic factor $y_{i,j}$ by the concrete trace $p_{i,j} s_{i,j}$ and the unique symbolic factor $y_{k,l}$ with $(i,j,k,l) \in M$ by the concrete trace $s_{i,j}^{-1} p_{i,j}^{-1}$. If we have an equation $y_{k,l} = p_{k,l} s_{k,l}$ then we check whether $s_{i,j}^{-1} p_{i,j}^{-1} = p_{k,l} s_{k,l}$ holds. Otherwise we have an equation $y_{k,l} = p_{k,l} u_k^{x_{k,l}} s_{k,l}$, and we can compute the unique non-zero solution for $x_{k,l}$ (if it exists). Note that $x_{k,l} \le |s_{i,j}^{-1} p_{i,j}^{-1}| \le 2 |\Gamma| \cdot (k_i-1) \cdot \lambda({e}) \in \mathcal{O}(|\Gamma|^2 \cdot m^2 \cdot \lambda({e}))$. We then replace $x_{k,l}$ in the equation $x_k = c_k + \sum_{l \in K_k} x_{k,l}$ by this unique solution.
By the above procedure, our statement (a)–(d) (with existentially quantified traces $y_{i,j}$) is transformed nondeterministically into a statement of the following form: there exist integers $x_{i,j} > 0$ ($i \in N_{\overline{a}}$, $j \in K'_i$) such that the following hold:
(a) $x_i = c'_i + \sum_{j \in K'_i} x_{i,j}$ for $i \in N_{\overline{a}}$,
(b) $p_{i,j} u_i^{x_{i,j}} s_{i,j} = s_{k,l}^{-1} (u_k^{-1})^{x_{k,l}} p_{k,l}^{-1}$ in ${\mathbb{M}}(A,I)$ for all $(i,j,k,l) \in M'$,
(c) $(x_i)_{i \in N_a} \in S_a$.
Here, $K'_i \subseteq K_i \subseteq [1,k_i]$ is a set of size at most $k_i \leq 28\alpha m^2$, $M' \subseteq M$ is a new matching relation (with $(i,j,k,l) \in M'$ if and only if $(k,l,i,j) \in M'$), and $c'_i \le |\Gamma| \cdot (k_i-1) + k_i \cdot \mathcal{O}(|\Gamma|^2 \cdot m^2 \cdot \lambda({e})) \le
\mathcal{O}(|\Gamma|^3 \cdot m^4\cdot \lambda({e}))$. [*Step 5: Elimination of two-dimensional knapsack instances.*]{} The remaining knapsack equations $p_{i,j} u_i^{x_{i,j}} s_{i,j} = s_{k,l}^{-1} (u_k^{-1})^{x_{k,l}} p_{k,l}^{-1}$ in (b) are two-dimensional and can be eliminated with Lemma \[lemma-connected-star\]. By this lemma, every trace equation $$p_{i,j} u_i^{x_{i,j}} s_{i,j} = s_{k,l}^{-1} (u^{-1}_k)^{x_{k,l}} p_{k,l}^{-1}$$ (recall that all $u_i$ are connected, which is assumed in Lemma \[lemma-connected-star\]) can be nondeterministically replaced by a semilinear constraint $$(x_{i,j}, x_{k,l}) \in \{ (a_{i,j,k,l} + b_{i,j,k,l} \cdot z, a_{k,l,i,j} + b_{k,l,i,j} \cdot z) \mid z \in \mathbb{N} \}.$$ For the numbers $a_{i,j,k,l}, b_{i,j,k,l}, a_{k,l,i,j}, b_{k,l,i,j}$ we obtain the bound $$a_{i,j,k,l}, b_{i,j,k,l}, a_{k,l,i,j}, b_{k,l,i,j} \in \mathcal{O}(\mu^8 \cdot \nu^{4 |\Gamma|}),$$ where, by Lemma \[lemma-prefixes\], $$\label{eq-mu}
\mu = \max\{ \rho(p_{i,j}), \rho(p_{k,l}), \rho(s_{i,j}), \rho(s_{k,l})\} \le \mathcal{O}(|\Gamma|^{2\alpha} \cdot 28^\alpha \cdot m^{2 \alpha } \cdot \lambda({e})^\alpha)$$ and $$\label{eq-nu}
\nu = \max\{ \rho(u_i), \rho(u_k) \} \le \mathcal{O}(\lambda({e})^\alpha).$$ Note that $\rho(t) = \rho(t^{-1})$ for every trace $t$. Moreover, note that we have the constraints $x_{i,j}, x_{k,l} > 0$. Hence, if our nondeterministic guess yields $a_{i,j,k,l} = 0$ or $a_{k,l,i,j} = 0$ then we make the replacement $a_{i,j,k,l} := a_{i,j,k,l} + b_{i,j,k,l}$ and $a_{k,l,i,j} := a_{k,l,i,j} + b_{k,l,i,j}$. If after this replacement we still have $a_{i,j,k,l} = 0$ or $a_{k,l,i,j} = 0$ then our guess was bad.
At this point, we have obtained a statement of the following form: there exist $z_{i,j,k,l} \in {\mathbb{N}}$ (for $(i,j,k,l) \in M'$) with $z_{i,j,k,l} = z_{k,l,i,j}$ and such that
(a) $x_i = c'_i+ \sum_{(i,j,k,l) \in M'} (a_{i,j,k,l} + b_{i,j,k,l} \cdot z_{i,j,k,l})$ for $i \in N_{\overline{a}}$, and
(b) $(x_i)_{i \in N_a} \in S_a$.
Note that the sum in (a) contains $|K'_i| \leq 28m^2\alpha$ many summands (since for every $j \in K'_i$ there is a unique pair $(k,l)$ with $(i,j,k,l) \in M'$). Hence, (a) can be written as $x_i = c''_i+ \sum_{(i,j,k,l) \in M'} b_{i,j,k,l} \cdot z_{i,j,k,l}$ with $$\begin{aligned}
c''_i & = & c'_i + \sum_{(i,j,k,l) \in M'} a_{i,j,k,l} \\
& \le & \mathcal{O}( |\Gamma|^3 \cdot m^4\cdot \lambda({e})) + \mathcal{O}(\alpha \cdot m^2 \cdot \mu^8 \cdot \nu^{4 |\Gamma|}) \\
& \le & \mathcal{O}(|\Gamma|^{16 \alpha +1} \cdot 28^{8 \alpha} \cdot m^{16\alpha +2} \cdot \lambda({e})^{8 \alpha + 4 \alpha |\Gamma|}) \\
& \le & \mathcal{O}(|\Gamma|^{16 \alpha +1} \cdot 28^{8 \alpha} \cdot m^{16\alpha +2} \cdot {|\!| {e}|\!|}^{8 \alpha + 4 \alpha |\Gamma|})\end{aligned}$$ (since $\lambda({e}) \leq {|\!| {e}|\!|}$). The bound in the last line is also an upper bound for the numbers $b_{i,j,k,l}$. Hence, we have obtained a semilinear representation for ${\mathsf{sol}}_{{G}}({e})$ whose magnitude is bounded by $\max\{ \mathsf{K}_1, \mathsf{K}_2 \}$, where $$\mathsf{K}_1 \le \mathsf{K}(2 m {|\!| {e}|\!|},m)$$ (this is our upper bound for the magnitude of the semilinear set $S_a$) and $$\mathsf{K}_2 \le \mathcal{O} \big(|\Gamma|^{16 \alpha +1} \cdot 28^{8 \alpha} \cdot m^{16\alpha +2} \cdot {|\!| {e}|\!|}^{8 \alpha + 4 \alpha |\Gamma|}\big).$$ [*Step 6: Integration of the preprocessing step.*]{} Recall that so far we only considered the case where ${e}$ is a knapsack expression having the form of the ${e}'$ in Lemma \[lemma-preproc\]. Let us now consider an arbitrary exponent expression ${e}$ of degree $m$. By Lemma \[lemma-preproc\] we have ${\mathsf{sol}}_{{G}}({e}) = (K \cap {\mathsf{sol}}_{{G}}({e}')) {\mathord\restriction}_{X_{{e}}}$ where $K$ is semilinear of magnitude one and ${e}'$ has degree at most $\alpha \cdot m$ and satisfies ${|\!| {e}' |\!|} \leq 3 {|\!| {e}|\!|}$. We can apply the upper bound shown so far to ${e}'$. Hence, the magnitude of ${\mathsf{sol}}_{{G}}({e}')$ is bounded by $\max\{ \mathsf{K}'_1, \mathsf{K}'_2 \}$, where $$\mathsf{K}'_1 \le \mathsf{K}(6 \alpha m {|\!| {e}|\!|},\alpha m)$$ and $$\mathsf{K}'_2 \le \mathcal{O}\big(|\Gamma|^{16 \alpha +1} \cdot 28^{8 \alpha} \cdot (\alpha m)^{16\alpha +2} \cdot (3 {|\!| {e}|\!|})^{8 \alpha + 4 \alpha |\Gamma|}\big).$$ It remains to analyze the influence of intersecting with $K$. For this, we can apply Proposition \[intersection-semilinear-sets\], which yields for the magnitude the upper bound $\max\{ \mathsf{K}_1, \mathsf{K}_2 \}$, where $$\begin{aligned}
\mathsf{K}_1 & \le & \mathcal{O}\big( (\alpha m)^{\alpha m/2+3} \cdot \mathsf{K}(6 \alpha m {|\!| {e}|\!|},\alpha m)^{\alpha m+3}\big) \end{aligned}$$ and $$\begin{aligned}
\mathsf{K}_2 & \le & \mathcal{O}\big( (\alpha m)^{\alpha m/2+3} \cdot \mathcal{O}\big(|\Gamma|^{32 \alpha +3} \cdot 28^{8 \alpha} \cdot (\alpha m)^{16\alpha +2} \cdot (3 {|\!| {e}|\!|})^{8 \alpha + 4 \alpha |\Gamma|}\big)^{\alpha m+3} \big) \\
& \le & (\alpha m)^{\mathcal{O}(\alpha^2 m)} \cdot {|\!| {e}|\!|}^{\mathcal{O}(\alpha^2 |\Gamma| m)} .\end{aligned}$$ This concludes the proof of the theorem.
\[rem:solution-set-GP\] Assume that $G$ is a fixed graph product (hence, $|\Gamma|$ is a constant). Consider again the case that ${e}$ is a knapsack expression (i.e., $x_i \neq x_j$ for $i \neq j$) where every period $u_i$ is either an atom or a well-behaved trace. Let $m = \deg({e})$. In the above proof, we show that the set of solutions ${\mathsf{sol}}_{{G}}({e})$ can be written as a finite union $${\mathsf{sol}}_{{G}}({e}) = \bigcup_{i=1}^p \bigoplus_{j=1}^{q_i} {\mathsf{sol}}_{H_{i,j}}({e}_{i,j}) \oplus L_i$$ such that the following hold for every $1 \leq i \leq p$:
- every $H_{i,j}$ is one of the groups $G_k$ and ${e}_{i,j}$ is a knapsack expression over the group $H_{i,j}$. The variable sets $X_{{e}_{i,j}}$ ($1 \leq j \leq q_i$) form a partition of the set $X_a$ (the variables corresponding to atomic periods).
- Every ${e}_{i,j}$ is a knapsack expression of size at most $2m {|\!| {e}|\!|}$ and degree at most $m$ (see step 3 in the above proof).
- The set $L_i$ is semilinear of magnitude $\mathcal{O} \big(|\Gamma|^{16 \alpha +1} 28^{8 \alpha} m^{16\alpha +2} {|\!| {e}|\!|}^{8 \alpha + 4 \alpha |\Gamma|}\big) =
\mathcal{O} \big(m^{16\alpha +2} {|\!| {e}|\!|}^{8 \alpha + 4 \alpha |\Gamma|}\big)$ (see step 5 in the above proof).
Moreover, given $i \in [1,p]$ (i.e., a specific guess), one can compute knapsack expressions ${e}_{i,j}$ ($1 \leq j \leq q_i$) and a semilinear representation of $L_i$ in polynomial time. This yields a nondeterministic reduction of the knapsack problem for the graph product $G$ to the knapsack problems for the groups $G_i$ ($i \in \Gamma$), assuming the input expression ${e}$ satisfies the above restriction. Recall that in general, direct products do not preserve decidability of the knapsack problem.
The reader might wonder, whether we can obtain a bound for the function $\mathsf{K}_{G,\Sigma}$ in terms of the function $\mathsf{K}_{G_i,\Sigma_i}$, which is better than the corresponding bound for $\mathsf{E}_{G,\Sigma}$ from Theorem \[thm-main-technical\]. This is actually not the case (at least with our proof technique): a power of the form $u^x$ where $u = u_1 u_2 \in {\mathbb{M}}(A,I)$ with $u_1 I u_2$ is equivalent to $u_1^x u_2^x$. Hence, powers $u^x$ with $u$ a non-connected trace naturally lead to a duplication of the variable $x$ (and hence to an exponent expression which is no longer a knapsack expression). This is the reason why we bounded the (in general faster growing) function $\mathsf{E}_{G,\Sigma}$ in terms of the functions $\mathsf{K}_{G_i,\Sigma_i}$ in Theorem \[thm-main-technical\].
An application of Theorem \[thm-main-technical\] is the following:
Let $G$ be a graph product of hyperbolic groups. Then solvability of exponent equations over $G$ belongs to [**NP**]{}.
For a hyperbolic group $H$ (with an arbitrary generating set $\Sigma'$) it was shown in [@LOHREY2019] that the function $\mathsf{K}_{H,\Sigma'}(n) := \mathsf{K}_{H,\Sigma'}(n,n)$ is polynomially bounded. Theorem \[thm-main-technical\] yields an exponential bound for the function $\mathsf{E}_{G,\Sigma}(n) := \mathsf{E}_{G,\Sigma}(n,n)$ (note that $|\Gamma|$ and $\alpha$ are constants since we consider a fixed graph product $G$). A nondeterministic polynomial time Turing machine can therefore guess the binary encodings of numbers $\sigma(x) \leq \mathsf{K}_{G,\Sigma}({|\!| {e}|\!|})$ for each variable $x$ of the input exponent expression $e$. Checking whether $\sigma$ is a $G$-solution of $e$ is an instance of the compressed word problem for $G$. By the main result of [@HoltLS19] the compressed word problem for a hyperbolic group can be solved in polynomial time and by [@HauLohHau13] the compressed word problem for a graph product of groups $G_i$ ($i \in \Gamma$) can be solved in polynomial time if for every $i \in \Gamma$ the compressed word problem for $G_i$ can be solved in polynomial time. Hence, we can check in polynomial time if $\sigma$ is a $G$-solution of $e$.
Let us now consider the special case where the graph product is a free product of two groups ${G}_1$ and ${G}_2$.
\[thm-cor-technical\] Let $\mathsf{K}(n,m)$ be the pointwise maximum of the functions $\mathsf{K}_{{G}_1,\Sigma_1}$ and $\mathsf{K}_{{G}_2,\Sigma_2}$. Then for ${G}= {G}_1 * {G}_2$ we have $\mathsf{K}_{{G},\Sigma}(n,m) \le \max\{\mathsf{K}_1, \mathsf{K}_2\}$ with $$\mathsf{K}_1 = \mathsf{K}(6 m n,m) \text{ and }
\mathsf{K}_2 \le \mathcal{O}(m n^4) .$$
The proof is similar to the one from Theorem \[thm-main-technical\]. We first consider the case where every period $u_i$ is either an atom or a well-behaved word (see Remark \[rem-preproc-free-product\]).
Let us go throw the six steps from the proof of Theorem \[thm-main-technical\]:
[*Step 1.*]{} This step is carried out in the same way as in the proof of Theorem \[thm-main-technical\].
[*Step 2.*]{} Here we can use Lemma \[lemma-reduction-free-product\] instead of Lemma \[lemma-reduction\], which yields the upper bound of $14m$ on the number of factors in our refinement of $u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_m^{x_m} v_m$ (where powers $u_i^{x_i}$ with $i \in N_1$ have been replaced by single atoms). The number of atom creations (of any type) is at most $2m-2$. We do not have to guess the atom sets $A_{i,j} \subseteq {\mathsf{alph}}(u_{i,j})$ since there are no swapping steps in Lemma \[lemma-reduction-free-product\].
[*Step 3.*]{} This step is copied from the proof of Theorem \[thm-main-technical\]. We obtain for the variables $x_i$ with $i \in N_a$ the semilinear constraint $(x_i)_{i \in N_a} \in S_a$ where $S_a$ is of magnitude at most $\mathsf{K}(2m {|\!| {e}|\!|},m)$.
[*Step 4.*]{} Also this step is analogous to the proof of Theorem \[thm-main-technical\]. Recall that we have the better bound $14 m$ on the number of factors in our refinement of $u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_m^{x_m} v_m$. Eliminating the equations $u_i^{x_i} = \tilde{y}_{i,1} \cdots \tilde{y}_{i,k_i}$ ($i \in N_{\overline{a}}$), which are interpreted in $A^*$, is much easier due to the absence of commutation. For every $i \in N_{\overline{a}}$ we obtain a disjunction of statements of the following form: there exist integers $x_{i,j} \geq 0$ ($1 \leq j \leq k_i$) such that
- $x_i = c_i + \sum_{j=1}^{k_i} x_{i,j}$,
- $\tilde{y}_{i,j} = p_{i,j} u_i^{x_{i,j}} s_{i,j}$ for all $1 \le j \le k_i$.
Here, every $p_{i,j}$ is a suffix of $u_i$, every $s_{i,j}$ is a prefix of $u_i$ and $c_i \leq k_i \leq 14 m$. Basically, $c_i$ is the number of factors $u_i$ that are split non-trivially in the factorization $u_i^{x_i} = \tilde{y}_{i,1} \cdots \tilde{y}_{i,k_i}$. We can then carry out the same simplifications that we did in the proof of Theorem \[thm-main-technical\]. If $\tilde{y}_{i,j}$ is an explicit word $v_{k,l}^{-1}$ then we determine the unique solution $x_{i,j}$ (if it exists) of $\tilde{y}_{i,j} = p_{i,j} u_i^{x_{i,j}} s_{i,j}$ and replace $x_{i,j}$ by that number, which is at most $\lambda({e})$. We arrive at a statement of the following form: there exist integers $x_{i,j} \geq 0$ ($i \in N_{\overline{a}}$, $j \in K_i$) such that the following hold:
(a) $x_i = c'_i + \sum_{j \in K_i} x_{i,j}$ for $i \in N_{\overline{a}}$,
(b) $p_{i,j} u_i^{x_{i,j}} s_{i,j} = s_{k,l}^{-1} (u_k^{-1})^{x_{k,l}} p_{k,l}^{-1}$ in $A^*$ for all $(i,j,k,l) \in M$,
(c) $(x_i)_{i \in N} \in S_a$.
Here, $K_i \subseteq [1,k_i]$ is a set of size at most $k_i \leq 14m$, $M$ is a matching relation (with $(i,j,k,l) \in M$ if and only if $(k,l,i,j) \in M$), and $c'_i \le 14m + k_i \cdot \lambda({e}) \leq \mathcal{O}(m \cdot \lambda({e}))$. The words $p_{i,j}$ and $s_{i,j}$ have length at most $\lambda({e})$.
[*Step 5.*]{} The remaining two-dimensional knapsack equations from point (b) are eliminated with Remark \[rem-2dim-words\]. Every equation $$p_{i,j} u_i^{x_{i,j}} s_{i,j} = s_{k,l}^{-1} (u^{-1}_k)^{x_{k,l}} p_{k,l}^{-1}$$ can be nondeterministically replaced by a semilinear constraint $$(x_{i,j}, x_{k,l}) \in \{ (a_{i,j,k,l} + b_{i,j,k,l} \cdot z, a_{k,l,i,j} + b_{k,l,i,j} \cdot z) \mid z \in \mathbb{N} \}.$$ where the numbers $a_{i,j,k,l}, b_{i,j,k,l}, a_{k,l,i,j}, b_{k,l,i,j}$ are bounded by $\mathcal{O}(\lambda({e})^4)$.
At this point, we have obtained a statement of the following form: there exist $z_{i,j,k,l} \in {\mathbb{N}}$ (for $(i,j,k,l) \in M$) with $z_{i,j,k,l} = z_{k,l,i,j}$ and such that
(a) $x_i = c'_i+ \sum_{(i,j,k,l) \in M} (a_{i,j,k,l} + b_{i,j,k,l} \cdot z_{i,j,k,l})$ for $i \in N_{\overline{a}}$, and
(b) $(x_i)_{i \in N_a} \in S_a$.
The sum in (a) contains $|K_i| \leq 14m$ many summands. Hence, (a) can be written as $x_i = c''_i+ \sum_{(i,j,k,l) \in M'} b_{i,j,k,l} \cdot z_{i,j,k,l}$ with $$\begin{aligned}
c''_i & = & c'_i + \sum_{(i,j,k,l) \in M} a_{i,j,k,l} \\
& \le & \mathcal{O}(m \cdot \lambda({e})) + 14 m \cdot \mathcal{O}(\lambda({e})^4) \\
& = & \mathcal{O}(m \cdot \lambda({e})^4) .\end{aligned}$$ We therefore obtained a semilinear representation for ${\mathsf{sol}}_{{G}}({e})$ whose magnitude is bounded by $\max\{ \mathsf{K}_1, \mathsf{K}_2 \}$, where $$\mathsf{K}_1 = \mathsf{K}(2m {|\!| {e}|\!|},m) \text{ and }
\mathsf{K}_2 \le \mathcal{O}(m {|\!| {e}|\!|}^4) .$$ [*Step 6.*]{} For the preprocessing we apply Remark \[rem-preproc-free-product\]. Hence, we just have to replace ${|\!| {e}|\!|}$ by $3 {|\!| {e}|\!|}$ in the above bounds, which yields the statement of the theorem.
By Theorem \[thm-cor-technical\], $\mathsf{K}_{{G},\Sigma}$ is polynomially bounded if $\mathsf{K}_{{G}_1,\Sigma}$ and $\mathsf{K}_{{G}_2,\Sigma}$ are polynomially bounded. This was also shown in [@LohreyZ18].
Analogously to Remark \[rem:solution-set-GP\], the above proof shows that the set of solutions ${\mathsf{sol}}_{{G}}({e})$ for $G = G_1*G_2$ can be written as a finite union $${\mathsf{sol}}_{{G}}({e}) = \bigcup_{i=1}^p \bigoplus_{j=1}^{q_i} {\mathsf{sol}}_{H_{i,j}}({e}_{i,j}) \oplus L_i$$ such that the following hold for every $1 \leq i \leq p$:
- every $H_{i,j}$ is either $G_1$ or $G_2$ and ${e}_{i,j}$ is a knapsack expression over the group $H_{i,j}$. The variable sets $X_{{e}_{i,j}}$ ($1 \leq j \leq q_i$) form a partition of the set $X_a$ (the variables corresponding to atomic periods).
- Every ${e}_{i,j}$ is a knapsack expression of size at most $6 m {|\!| {e}|\!|}$ and degree at most $m$.
- The set $L_i$ is semilinear of magnitude $\mathcal{O}(m n^4)$.
Moreover, given $i \in [1,p]$, one can compute the knapsack expressions ${e}_{i,j}$ ($1 \leq j \leq q_i$) and a semilinear representation of $L_i$ in polynomial time.
The above remark immediately yields the following complexity transfer result. A language $A$ is nondeterministically polynomial time reducible to a language $B$ if there exists a nondeterministic polynomial time Turing-machine $M$ that outputs on each computation path after termination a word over the alphabet of the language $B$ and such that $x \in A$ if and only if on input $x$, the machine $M$ has at least one computation path on which it outputs a word from $B$.
\[thm:free-product-NP-reduction\] The knapsack problem for $G_1*G_2$ is nondeterministically polynomial time reducible to the knapsack problems for $G_1$ and $G_2$.
Part 2: Knapsack in HNN-extensions and amalgamated products {#sec-HNN+amalgamated}
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The remaining transfer results concern two constructions that are of fundamental importance in combinatorial group theory [@LySch77], namely HNN-extensions and amalgamated products. In their general form, HNN-extensions have been used to construct groups with an undecidable word problem, which means they may destroy desirable algorithmic properties. We consider the special case of finite associated (resp. identified) subgroups, for which these constructions already play a prominent role, for example, in Stallings’ decomposition of groups with infinitely many ends [@Stal71] or the construction of virtually free groups [@DiDu89]. Moreover, these constructions are known to preserve a wide range of important structural and algorithmic properties [@AllGre73; @Bez98; @HauLo11; @KaSiSt06; @KaWeMy05; @KaSo70; @KaSo71; @LohSen06icalp; @LohSen08; @MeRa04].
HNN-extensions preserve knapsack semilinearity
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Suppose $G=\langle \Sigma\mid R\rangle$ is a finitely generated group with the finite symmetric generating set $\Sigma = \Omega \cup \Omega^{-1}$ and the set of relators $R \subseteq \Sigma^*$. Fix two isomorphic subgroups $A$ and $B$ of $G$ together with an isomorphism $\varphi\colon A\to
B$. Let $t \notin \Sigma$ be a fresh generator. Then the corresponding *HNN-extension* is the group $$H=\langle \Omega\cup \{t\} \mid R\cup \{t^{-1}a^{-1}t\varphi(a) \mid a\in A\}\rangle$$ (formally, we identify here every element $c \in A \cup B$ with a word over $\Sigma$ that evaluates to $c$). This group is usually denoted by $$\label{eq-HNN}
H=\langle G, t \mid t^{-1}at=\varphi(a)~(a\in A)\rangle .$$ Intuitively, $H$ is obtained from $G$ by adding a new element $t$ such that conjugating elements of $A$ with $t$ applies the isomorphism $\varphi$. Here, $t$ is called the *stable letter* and the groups $A$ and $B$ are the *associated subgroups*. A basic fact about HNN-extensions is that the group $G$ embeds naturally into $H$ [@HiNeNe49]. We assume that $(A \cup B) \setminus \{1\}$ is contained in the finite generating set $\Sigma$.
Here, we only consider the case that $A$ and $B$ are finite groups, so that we may assume that $A\cup B\subseteq\Sigma$. To exploit the symmetry of the situation, we use the notation $A(+1)=A$ and $A(-1)=B$. Then, we have $\varphi^{\alpha}\colon A(\alpha)\to A(-\alpha)$ for $\alpha\in\{+1,-1\}$. We will make use of the (possibly infinite) alphabet $\Gamma=G\backslash \{1\}$. By $h\colon (\Gamma \cup\{t,t^{-1}\})^*\to H$, we denote the canonical morphism that maps each word to the element of $H$ it represents.
A word $u\in (\Gamma \cup \{t,t^{-1}\})^*$ is called *Britton-reduced* if it does not contain a factor of the form $cd$ with $c,d\in \Gamma$ or a factor $t^{-\alpha} a t^\alpha$ with $\alpha\in\{-1,1\}$ and $a\in A(\alpha)$. A factor of the form $t^{-\alpha} a t^\alpha$ with $\alpha\in\{-1,1\}$ and $a\in A(\alpha)$ is also called a [*pin*]{}. Note that the equation $t^{-\alpha} a t^\alpha=\varphi^\alpha(a)$ allows us to replace a pin $t^{-\alpha}at^{\alpha}$ by $\varphi^{\alpha}(a)\in A(-\alpha)$. Since this decreases the number of $t$’s in the word, we can reduces every word to an equivalent Britton-reduced word. We denote the set of all Britton-reduced words in the HNN-extension by ${\mathsf{BR}}(H)$.
In this section, let $\gamma$ be the cardinality of $A$. A word $w\in {\mathsf{BR}}(H) \setminus \Gamma$ is called [*well-behaved*]{}, if $w^m$ is Britton-reduced for every $m \geq 0$. Note that $w$ is well-behaved if and only if $w$ and $w^2$ are Britton-reduced. Elements of $\Gamma$ are also called [*atomic*]{}.
The *length* of a word $w \in (\Gamma \cup \{t,t^{-1}\})^*$ is defined as usual and denoted by $|w|$. For a word $w = a_1 a_2 \cdots a_k$ with $a_i \in \Gamma \cup \{t,t^{-1}\}$ we define the the *representation length* of $w$ as ${|\!| w |\!|} = \sum_{i=1}^k n_i$, where $n_i = 1$ if $a_i \in \{t,t^{-1}\}$ and $n_i$ is the geodesic length of $a_i$ in the group $G$ if $a_i \in \Gamma$. Note that ${|\!| a |\!|} = 1$ for $a \in (A \cup B) \setminus \{1\}$.
The following lemma provides a necessary and sufficient condition for equality of Britton-reduced words in an HNN-extension (cf. Lemma 2.2 of [@HauLo11]):
\[boat\] Let $u=g_0 t^{\delta_1}g_1 \cdots t^{\delta_k}g_k$ and $v=h_0 t^{\varepsilon_1}h_1 \cdots t^{\varepsilon_l}h_l$ be Britton-reduced words with $g_0,\dots , g_k, h_0, \dots , h_l \in G$ and $\delta_1,\dots \delta_k,\varepsilon_1, \dots , \varepsilon_l\in \{ 1, -1 \}$. Then $u=v$ in the HNN-extension $H$ of $G$ if and only if the following hold:
- $k=l$ and $\delta_i=\varepsilon_i$ for $1\leq i \leq k$
- there exist $c_1, \dots, c_{2m} \in A\cup B$ such that:
- $g_ic_{2i+1}=c_{2i}h_i$ in $G$ for $0\leq i \leq k$ (here we set $c_0=c_{2k+1}=1$)
- $c_{2i-1}\in A(\delta_i)$ and $c_{2i}=\varphi^{\delta_i}(c_{2i-1})\in A(-\delta_i)$ for $1\leq i \leq k$.
The second condition of the lemma can be visualized by the diagram from Figure \[fig-van-kampen\] (also called a van Kampen diagram, see [@LySch77] for more details), where $k=l=4$. Light-shaded (resp. dark-shaded) faces represent relations in $G$ (resp. relations of the form $ct^\delta=t^\delta \varphi^\delta (c)$ with $c \in A(\delta)$). The elements $c_1, \dots , c_{2k}$ in such a diagram are also called *connecting elements*.
\(A) at (0,0) ; (R) at (14,0) ;
\(A) to\[bend right=20\] node\[pos=0.16\] (C) node\[pos=0.24\] (E) node\[pos=0.36\] (G) node\[pos=0.44\] (I) node\[pos=0.56\] (K) node\[pos=0.64\] (M) node\[pos=0.76\] (O) node\[pos=0.84\] (Q) (R);
\(A) to\[bend left=20\] node\[pos=0.16\] (B) node\[pos=0.24\] (D) node\[pos=0.36\] (F) node\[pos=0.44\] (H) node\[pos=0.56\] (J) node\[pos=0.64\] (L) node\[pos=0.76\] (N) node\[pos=0.84\] (P) (R);
(A.center) to (B.center) to (C.center) to (A.center); (B.center) to (C.center) to (E.center) to (D.center) to (B.center); (E.center) to (D.center) to (F.center) to (G.center) to (E.center); (F.center) to (G.center) to (I.center) to (H.center) to (F.center); (I.center) to (H.center) to (J.center) to (K.center) to (I.center); (J.center) to (K.center) to (M.center) to (L.center) to (J.center); (M.center) to (L.center) to (N.center) to (O.center) to (M.center); (N.center) to (O.center) to (Q.center) to (P.center) to (N.center); (Q.center) to (P.center) to (R.center) to (Q.center);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) to\[bend left=20\] (R); (A) to\[bend right=20\] (R);
\(A) – (B) node\[midway,above\] [$g_0$]{}; (A) – (C) node\[midway,below\] [$h_0$]{};
\(B) – (D) node\[midway,above\] [$t^{\delta_1}$]{}; (C) – (E) node\[midway,below\] [$t^{\delta_1}$]{}; (B) – (C) node\[midway,right\] [$c_1$]{};
\(D) – (F) node\[midway,above\] [$g_1$]{}; (E) – (G) node\[midway,sloped,below\] [$h_1$]{}; (D) – (E) node\[midway,right\] [$c_2$]{};
\(F) – (H) node\[midway,above\] [$t^{\delta_2}$]{}; (G) – (I) node\[midway,below\] [$t^{\delta_2}$]{}; (F) – (G) node\[midway,right\] [$c_3$]{};
\(H) – (J) node\[midway,above\] [$g_2$]{}; (I) – (K) node\[midway,below\] [$h_2$]{}; (H) – (I) node\[midway,right\] [$c_4$]{};
\(J) – (L) node\[midway,above\] [$t^{\delta_3}$]{}; (K) – (M) node\[midway,below\] [$t^{\delta_3}$]{}; (J) – (K) node\[midway,right\] [$c_5$]{};
\(L) – (N) node\[midway,above\] [$g_3$]{}; (M) – (O) node\[midway,below\] [$h_3$]{}; (L) – (M) node\[midway,right\] [$c_6$]{};
\(N) – (P) node\[midway,above\] [$t^{\delta_4}$]{}; (O) – (Q) node\[midway,below\] [$t^{\delta_4}$]{}; (N) – (O) node\[midway,right\] [$c_7$]{};
\(P) – (R) node\[midway,above\] [$g_4$]{}; (Q) – (R) node\[midway,below\] [$h_4$]{}; (P) – (Q) node\[midway,right\] [$c_8$]{};
\(A) circle (1pt); (B) circle (1pt); (C) circle (1pt); (D) circle (1pt); (E) circle (1pt); (F) circle (1pt); (G) circle (1pt); (H) circle (1pt); (I) circle (1pt); (J) circle (1pt); (K) circle (1pt); (L) circle (1pt); (M) circle (1pt); (N) circle (1pt); (O) circle (1pt); (P) circle (1pt); (Q) circle (1pt); (R) circle (1pt);
For our purposes, we will need the following lemma (cf. Lemma 2.3 of [@HauLo11]), which allows us to transform an arbitrary string over the generating set of an HNN-extension into a reduced one:
\[red uv\] Assume that $u=g_0 t^{\delta_1}g_1 \cdots t^{\delta_k}g_k$ and $v=h_0 t^{\varepsilon_1}h_1 \cdots t^{\varepsilon_l}h_l$ are Britton-reduced words ($g_i, h_j \in G$). Let $m(u,v)$ be the largest number $m\geq 0$ such that
(a) $A(\delta_{k-m+1})=A(-\varepsilon_m)$ (we set $A(\delta_{k+1})=A(-\varepsilon_0)=1$) and
(b) there is $c \in A(-\varepsilon_m)$ such that $$t^{\delta_{k-m+1}} g_{k-m+1} \cdots t^{\delta_k} g_k h_0 t^{\varepsilon_1} \cdots h_{m-1} t^{\varepsilon_m} =_H c$$ (for $m=0$ this condition is satisfied with $c=1$).
Moreover, let $c(u,v) \in A(-\varepsilon_m)$ be the element $c$ in (b) (for $m=m(u,v)$). Then $$g_0 t^{\delta_1}g_1 \cdots t^{\delta_{k-m(u,v)}} \gamma(u,v) t^{\varepsilon_{m(u,v)+1}}h_{m(u,v)+1} \cdots t^{\varepsilon_l}h_l$$ is a Britton-reduced word equal to $uv$ in $H$, where $\gamma(u,v) \in G$ such that $\gamma(u,v) =_G g_{k-m(u,v)} c(u,v) h_{m(u,v)}$.
The above lemma is visualized in Figure \[fig-peak-elimination\].
\(A) at (-10,0) ; (J) at (0,14) ; (S) at (10,0) ;
\(A) to\[out=0,in=270\] node\[pos=0.11\] (B) node\[pos=0.22\] (C) node\[pos=0.33\] (D) node\[pos=0.44\] (E) node\[pos=0.55\] (F) node\[pos=0.66\] (G) node\[pos=0.77\] (H) node\[pos=0.88\] (I) (J);
\(J) to\[out=270,in=180\] node\[pos=0.12\] (K) node\[pos=0.23\] (L) node\[pos=0.34\] (M) node\[pos=0.45\] (N) node\[pos=0.56\] (O) node\[pos=0.67\] (P) node\[pos=0.78\] (Q) node\[pos=0.89\] (R) (S);
(F.center) to (G.center) to (M.center) to (N.center) (H.center) to (I.center) to (K.center) to (L.center);
\(A) to\[bend right=6\] (B);
\(A) to\[out=0,in=270\] (J);
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\(J) to\[out=270,in=180\] (S);
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\(J) to\[out=270,in=180\] (S);
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\(J) to\[out=270,in=180\] (S);
\(A) – (B) node\[midway,above\] [$g_0$]{}; (B) – (C) node\[midway,above\] [$t^{\delta_1}$]{}; (C) – (D) node\[midway,above\] [$g_1$]{}; (D) – (E) node\[midway,sloped,above\] [$\cdots$]{}; (E) – (F) node\[midway,left\] [$g_{k-m}$]{}; (F) – (G) node\[midway,left\] [$t^{\delta_{k-m+1}}$]{}; (G) – (H) node\[midway,sloped,above\] [$\cdots$]{}; (H) – (I) node\[midway,left\] [$t^{\delta_k}$]{}; (I) – (J) node\[midway,left\] [$g_k$]{};
\(J) – (K) node\[midway,right\] [$h_0$]{}; (K) – (L) node\[midway,right\] [$t^{\varepsilon_1}$]{}; (L) – (M) node\[midway,sloped,above\] [$\cdots$]{}; (M) – (N) node\[midway,right\] [$t^{\varepsilon_m}$]{}; (N) – (O) node\[midway,right\] [$h_m$]{}; (O) – (P) node\[midway,sloped,above\] [$\cdots$]{}; (P) – (Q) node\[midway,above=2pt\] [$h_{l-1}$]{}; (Q) – (R) node\[midway,above\] [$t^{\varepsilon_l}$]{}; (R) – (S) node\[midway,above\] [$h_l$]{};
\(F) – (N) node\[midway,below\] [$c(u,v)$]{};
\(A) circle (1pt); (B) circle (1pt); (C) circle (1pt); (D) circle (1pt); (E) circle (1pt); (F) circle (1pt); (G) circle (1pt); (H) circle (1pt); (I) circle (1pt);
\(J) circle (1pt);
\(K) circle (1pt); (L) circle (1pt); (M) circle (1pt); (N) circle (1pt); (O) circle (1pt); (P) circle (1pt); (Q) circle (1pt); (R) circle (1pt); (S) circle (1pt);
\[Britton-reduced-powers\] From a given word $u\in {\mathsf{BR}}(H)$ we can compute words $s,p,v \in {\mathsf{BR}}(H)$ such that $u^m =_H s v^m p$ for every $m\geq 0$ and either $v \in G$ or $v$ is well-behaved and starts with $t^{\pm 1}$. Moreover, ${|\!| s |\!|}+{|\!| p |\!|}+{|\!| v |\!|} \leq 3{|\!| u |\!|}$.
Let $u\in {\mathsf{BR}}(H)$. Assume that $u$ is not atomic; otherwise we are done. Let us now consider the word $u^2$. If $u^2$ is not Britton-reduced, we can do the following: With Lemma \[red uv\] it is easy to compute a factorization $u=xyz$, such that $zx =_H c\in A \cup B$ and hence $u^2 =_H xycyz$, where $xycyz$ becomes Britton-reduced after multiplying successive symbols from $\Gamma$. We obtain the equality $u^m = (xyz)^m =_H x (yc)^m c^{-1} z$ for every $m \geq 0$. If $y \in G$ then we have $v = yc \in G$ and we can set $s = x$, $v = yc$, and $p = c^{-1} z$. Otherwise, assume that $y$ contains an occurrence of $t$ or $t^{-1}$ and let us write $y = g y' g'$ where $y'$ starts and ends with $t$ or $t^{-1}$ and $g, g' \in G$. We have $u^m =_H x (yc)^m c^{-1} z = x (g y' g' c)^m c^{-1} z = xg (y' g'cg)^m (cg)^{-1} z$. We now set $s = xg$, $v = y' (g'cg)$, and $p = (cg)^{-1} z$. It easy to observe that ${|\!| s |\!|}$, ${|\!| v |\!|}$, and ${|\!| p |\!|}$ are bounded by ${|\!| u |\!|}$.
\[2dim\] Let $u,v \in {\mathsf{BR}}(H) \backslash G$ be well-behaved, both starting with $t^{\pm 1}$, $a,b \in A \cup B$, $u'$ (resp., $v'$) be a proper suffix of $u$ (resp., $v$) and $u''$ (resp., $v''$) be a proper prefix of $u$ (resp., $v$). Let $\mu = \max\{ |u|, |v|\}$. Then the set $$L(a,u',u,u'',v',v,v'',b) := \{ (x,y) \in \mathbb{N} \times \mathbb{N} \mid a u' u^x u'' =_H v' v^y v'' b \}$$ is semilinear. Moreover, one can compute in polynomial time a semilinear representation whose magnitude is bounded by $\mathcal{O}(\gamma^{2} \mu^{4})$.
The proof is inspired by the proof of [@LOHREY2019 Lemma 8.3] for hyperbolic groups. The assumptions in the lemma imply that for all $x,y \in \mathbb{N}$ the words $a u' u^x u''$ and $v' v^y v'' b$ are Britton-reduced (possibly after multiplying $a$ and $b$ with neighboring symbols from $\Gamma$. For a word $w \in (\Gamma \cup \{t,t^{-1}\})^*$ we define $|w|_{t^{\pm 1}} = |w|_{t} + |w|_{t^{-1}}$ (the $t^{\pm 1}$-length of $w$). Clearly, for Britton-reduced words $w,w'$ with $w =_H w'$ we have $|w|_{t^{\pm 1}} = |w'|_{t^{\pm 1}}$.
We will first construct an NFA $\mathcal{A}$ over the unary alphabet $\{ a \}$ such that $$L(\mathcal{A}) = \{ a^\ell \mid \exists x,y \in \mathbb{N} : a u' u^x u'' =_H v' v^y v'' b, \ell = |a u' u^x u''|_{t^{\pm 1}} \}.$$ Moreover, the number of states of $\mathcal{A}$ is $\mathcal{O}(\mu^2 \cdot \gamma)$. Roughly speaking, the NFA $\mathcal{A}$ verifies from left to right the existence of a van Kampen diagram of the form shown in Figure \[fig-van-kampen\]. Thereby it stores the current connecting element (an element from $A \cup B$). By the assumptions on $u$ and $v$ we can write both words as $u = u_0 u_{1} \cdots u_{m-1}$, $v = v_{0} v_{1} \cdots v_{n-1}$ where $n$ and $m$ are even, $u_{i} \in \{t,t^{-1}\}$ for $i$ even and $u_{i} \in G$ for $i$ odd, and analogously for $v$. Let us write $u' = u_{p} u_{p+1} \cdots u_{m-1}$, $u'' = u_{0} u_{1} \cdots u_{q-1}$, $v' = v_{r} v_{r+1} \cdots v_{n-1}$, $v'' = v_{0} v_{1} \cdots v_{s-1}$. We set $p = m$ if $u'$ is empty and $q = 0$ if $u''$ is empty and similarly for $r$ and $s$. We will first consider the case that $p \equiv r {\operatorname{mod}}2$ and $q \equiv s {\operatorname{mod}}2$; other cases are just briefly sketched at the end of the proof.
The state set of the NFA $\mathcal{A}$ is $$Q = \{ (c,i,j) \mid c \in A \cup B, 0 \leq i < m, 0 \leq j < n, i \equiv j \bmod 2 \}.$$ The initial state is $(a,p \bmod m, r \bmod n)$ and the only final state is $(b, q {\operatorname{mod}}m, s {\operatorname{mod}}n)$. Finally, $\mathcal{A}$ contains the following transitions for $c_1, c_2 \in A \cup B$ such that $c_1 u_{i} =_H v_{j} c_2$ (in case $i$ and $j$ are odd, this must be an identity in $G$ since $u_i, v_j \in G$):
- $(c_1,i,j) \xrightarrow{a} (c_2, i+1 {\operatorname{mod}}m, j+1 {\operatorname{mod}}n)$ if $i$ and $j$ are even,
- $(c_1,i,j) \xrightarrow{1} (c_2, i+1 {\operatorname{mod}}m, j+1 {\operatorname{mod}}n)$ if $i$ and $j$ are odd.
The number of states of $\mathcal{A}$ is $\mathcal{O}(\gamma \cdot \mu^2)$. If $p \equiv r {\operatorname{mod}}2$ does not hold, then we have to introduce a fresh initial state $q_0$. Assume that for instance $p$ is odd and $q$ is even. Thus $u_p$ belongs to $G$ whereas $v_q$ is $t$ or $t^{-1}$. Then we add all transitions $q_0 \xrightarrow{1} (c,p+1 {\operatorname{mod}}m, q)$ for every $c \in A \cup B$ with $a u_{p} =_G c$. If $q \equiv s {\operatorname{mod}}2$ does not hold, then we have to add a fresh final state $q_f$.
The rest of the argument is the same as in Remark \[rem-2dim-words\], we only have to replace the length of words by the $t^{\pm 1}$-length. We obtain a semilinear representation of $L(a,u',u,u'',v',v,v'',b)$ of magnitude $\mathcal{O}(\gamma^2 \cdot \mu^4)$.
\(A) at (0,0) ; (B) at (0,1) ; (C) at (10,0) ; (D) at (10,1) ;
\(A) rectangle (D);
\(A) – (B) node\[midway,left\] [$a$]{}; (D) – (C) node\[midway,right\] [$b$]{};
\(B) to node\[pos=0.05,above\] (E) [$u'$]{} node\[pos=0.10\] (F) node\[pos=0.2,above\] (G) [$u$]{} node\[pos=0.3\] (H) node\[pos=0.4,above\] (I) [$u$]{} node\[pos=0.5\] (J) node\[pos=0.6,above\] (K) [$u$]{} node\[pos=0.7\] (L) node\[pos=0.8,above\] (M) [$u$]{} node\[pos=0.9\] (N) node\[pos=0.95,above\] (O) [$u''$]{} (D);
\(A) to node\[pos=0.03,below\] (P) [$v'$]{} node\[pos=0.06\] (Q) node\[pos=0.2,below\] (R) [$v$]{} node\[pos=0.33\] (S) node\[pos=0.47,below\] (T) [$v$]{} node\[pos=0.6\] (U) node\[pos=0.73,below\] (V) [$v$]{} node\[pos=0.86\] (W) node\[pos=0.93,below\] (X) [$v''$]{} (C);
\(A) circle (1pt); (B) circle (1pt); (C) circle (1pt); (D) circle (1pt);
\(F) circle (1pt); (H) circle (1pt); (J) circle (1pt); (L) circle (1pt); (N) circle (1pt);
\(Q) circle (1pt); (S) circle (1pt); (U) circle (1pt); (W) circle (1pt);
We now define $1$-reducible tuples for HNN-extensions similar to the case of graph products. Again we identify tuples that can be obtained from each other by inserting/deleting $1$’s at arbitrary positions.
\[def-reduction-HNN\] We define a reduction relation on tuples over ${\mathsf{BR}}(H)$ of arbitrary length. Take $u_1, u_2, \ldots, u_m \in {\mathsf{BR}}(H)$. Then we have
- $(u_1, u_2, \ldots, u_i, a, u_{i+1}, \ldots, u_m) \to (u_1, \ldots, u_{i-1},b, u_{i+2}, \ldots, u_m)$ if both $u_i$ and $u_{i+1}$ contain $t$ or $t^{-1}$ and $u_i a u_{i+1} =_H b$ for $a,b \in A \cup B$ (a [*generalized cancellation step*]{}),
- $(u_1, u_2, \ldots, u_m) \to (u_1, \ldots, u_{i-1}, g, u_{i+2}, \ldots, u_m)$ if $u_i, u_{i+1} \in \Gamma$ and $g =_G u_i u_{i+1} \in G$
If $g \neq 1$ then we call the last rewrite step an [*atom creation*]{}. A concrete sequence of these rewrite steps leading to the empty tuple is a [*reduction*]{} of $(u_1, u_2, \ldots, u_m)$. If such a sequence exists, the tuple is called [*$1$-reducible*]{}.
A reduction of a tuple $(u_1, u_2, \ldots, u_m)$ can be seen as a witness for the fact that $u_1 u_2 \cdots u_m =_H 1$. On the other hand, $u_1 u_2 \cdots u_m =_H 1$ does not necessarily imply that $u_1, u_2, \ldots, u_m$ has a reduction (as seen for graph products). But we can show that every sequence which multiplies to $1$ in $H$ can be refined (by factorizing the elements of the sequence) such that the resulting refined sequence has a reduction. We say that the tuple $(v_1, v_2, \dots, v_n)$ is a *refinement* of the tuple $(u_1, u_2, \dots, u_m)$ if there exist factorizations $u_i=u_{i,1} \cdots u_{i,k_i}$ in $(\Gamma \cup \{t,t^{-1}\})^*$ such that $(v_1, v_2, \dots, v_n)=(u_{1,1}, \dots, u_{1,k_1}, \dots, u_{m,1}, \dots, u_{m,k_m})$.
\[Britton-lemma-reduction\] Let $m \geq 2$ and $u_1, u_2, \ldots, u_m \in {\mathsf{BR}}(H)$. If $u_1 u_2 \cdots u_m = 1$ in $H$, then there exists a $1$-reducible refinement of $(u_1, u_2, \dots , u_m)$ that has length at most $7m-12\leq 7m$ and there is a reduction of this refinement with at most $4m-8$ atom creations.
We prove the lemma by induction on $m$. The induction is similar to the one of Lemma \[lemma-reduction-free-product\]. The case $m=2$ is trivial (we must have $u_2 = u_1^{-1}$). If $m \geq 3$ then by Lemma \[red uv\] we can factorize $u_1$ and $u_2$ in $(\Gamma \cup \{t,t^{-1}\})^*$ as $u_1=u'_1 g_1 r$ and $u_2= s g_2 u'_2$ such that $rs=_H c \in A \cup B$, $g_1, g_2 \in G$ and $u_1u_2 =_H u'_1 g u'_2\in {\mathsf{BR}}(H)$ for $g :=g_1c g_2 \in G$. The words $r$ and $s$ are either both empty (in which case we have $a=1$) or $r$ starts with some $t^{\varepsilon}$ and $s$ ends with $t^{-\varepsilon}$.
By induction hypothesis, for the tuple $(u'_1g u'_2, u_3, \dots, u_m)$ there is a $1$-reducible refinement $$\label{seq-HNN}
(v_1, \dots, v_k, u_{3,1}, \dots, u_{3,k_3}, \dots, u_{m,1}, \dots, u_{m,k_m}),$$ with $4(m-1)-8$ atom creations, where $k+\sum_{i=3}^m k_i \leq 7(m-1)-14$ and $u'_1gu'_2=v_1\cdots v_k$ in $(\Gamma \cup \{t,t^{-1}\})^*$. Since $g \in G$, there exists $1\leq i \leq k$, such that $v_i=v_{i,1} g v_{i,2}$, $u'_1 = v_1 \cdots v_{i-1} v_{i,1}$ and $u'_2 = v_{i,2} v_{i+1} \cdots v_k$. Now we replace $v_i$ by $v_{i,1}, g, v_{i,2}$ in the above refinement . If there exists $u_{j,l}$ such that $v_i$ and $u_{j,l}$ cancel out in a generalized cancellation step in the $1$-reduction of then there exist $a,b \in A \cup B$ such that $v_{i,1} g v_{i,2} a u_{j,l} = v_i a u_{j,l} =_H b$. The generalized cancellation replaces $v_i,a,u_{j,l}$ by $b$.
Recall that $v_i$ and $u_{j,l}$ are both Britton-reduced. By Lemma \[boat\] we can factorize $u_{j,l}$ in $(\Gamma \cup \{t,t^{-1}\})^*$ as $u_{j,l} = w_1 g' w_2$ such that there exist connecting elements $a',b' \in A \cup B$ with $v_{i,2} a w_1 =_H a'$, $v_{i,1} b' w_2 =_H b$, and $ga'g' =_G b'$; see Figure \[fig-ujl-vi\]. This yields the refined tuple $$(v_1, \ldots, v_{i-1}, v_{i,1}, g_1, r, s, g_2, v_{i,2}, v_{i+1}, \ldots, v_k, \tilde{u}_{3,1}, \dots, \tilde{u}_{3,k_3}, \dots, \tilde{u}_{m,1}, \dots, \tilde{u}_{m,k_m}),$$ of $(u_1, u_2, \dots , u_m)$, where $\tilde{u}_{j,l} = w_1, g', w_2$ and $\tilde{u}_{p,q} = u_{p,q}$ in all other cases. The length of this tuple is at most $k + 7 + \sum_{i=3}^m k_i \leq 7m-14$. The above tuple is also $1$-reducible: First, $r,s$ is replaced by $c$ in a generalized cancellation step. Then, after at most two atom creations we obtain the tuple $$(v_1, \ldots, v_{i-1}, v_{i,1}, g, v_{i,2}, v_{i+1}, \ldots, v_k, \tilde{u}_{3,1}, \dots, \tilde{u}_{3,k_3}, \dots, \tilde{u}_{m,1}, \dots, \tilde{u}_{m,k_m}) .$$ At this point, we can basically apply the fixed reduction of . The generalized cancellation $v_i,a,u_{j,l} \to b$ is replaced by the sequence $$v_{i,1}, g, v_{i,2}, a, w_1, g', w_2 \to v_{i,1}, g, a', g', w_2 \to v_{i,1}, ga', g', w_2 \to v_{i,1}, b', w_2 \to b$$ which contains at most two atom creations. Hence, the total number of atom creations is at most $4 + 4(m-1)-8 = 4m-8$. This concludes the proof of the lemma.
\(A) at (0,0) ; (B) at (0,1) ; (C) at (10,0) ; (D) at (10,1) ;
\(A) – (B) node\[midway,left\] [$a$]{}; (D) – (C) node\[midway,right\] [$b$]{};
\(B) to node\[pos=0.22,above\] (E) [$w_1$]{} node\[pos=0.44\] (F) node\[pos=0.5,above\] (G) [$g'$]{} node\[pos=0.56\] (H) node\[pos=0.78,above\] (I) [$w_2$]{} (D);
\(A) to node\[pos=0.22,below\] (J) [$v_{i,2}^{-1}$]{} node\[pos=0.44\] (K) node\[pos=0.5,below\] (L) [$g^{-1}$]{} node\[pos=0.56\] (M) node\[pos=0.78,below\] (N) [$v_{i,1}^{-1}$]{} (C);
\(K) – (F) node\[midway,left\] [$a'$]{}; (M) – (H) node\[midway,right\] [$b'$]{};
\(A) circle (1pt); (B) circle (1pt); (C) circle (1pt); (D) circle (1pt);
\(F) circle (1pt); (H) circle (1pt); (K) circle (1pt); (M) circle (1pt);
Now we can prove the next main theorem.
\[HNNbound\] For the HNN-extension $H$ of a group $G$ (with respect to the isomorphism $\varphi$) we have $\mathsf{K}_{H,\Sigma}(n,m) \le \max\{\mathsf{K}_1, \mathsf{K}_2\}$ with $$\mathsf{K}_1 = \mathsf{K}_{G,\Sigma}(24mn,m) \text{ and }
\mathsf{K}_2 \le \mathcal{O}(\gamma^{2} m n^{4}) .$$
We will follow the idea of the proof of Theorems \[thm-main-technical\] and \[thm-cor-technical\], respectively. We first consider the case where every period $u_i$ is either an atom or well-behaved and starts with $t^{\pm 1}$. Again we are going through the six steps. For simplicity, we write $\mathsf{K}$ instead of $\mathsf{K}_{G,\Sigma}$.
[*Step 1.*]{} This step is carried out in the same way as in the proof of Theorem \[thm-main-technical\].
[*Step 2.*]{} Here we can use Lemma \[Britton-lemma-reduction\] instead of Lemma \[lemma-reduction\], which yields the upper bound of $14m$ on the number of factors in our refinement of $u_1^{x_1} v_1 u_2^{x_2} v_2 \cdots u_m^{x_m} v_m$ (where powers $u_i^{x_i}$ with $i \in N_1$ have been removed). The number of atom creations is at most $8m-8$.
[*Step 3.*]{} This step is copied from the proof of Theorem \[thm-main-technical\]. We obtain for the variables $x_i$ with $i \in N_a$ the semilinear constraint $(x_i)_{i \in N_a} \in S_a$ where $S_a$ is of magnitude at most $\mathsf{K}(8m {|\!| {e}|\!|},m)$.
[*Step 4.*]{} This step is analogous to the proof of Theorem \[thm-cor-technical\]. The only difference is that the 2-dimensional knapsack instances are produced by the generalized cancellation steps from Definition \[def-reduction-HNN\]. We arrive at a statement of the following form: there exist integers $x_{i,j} \geq 0$ ($i \in N_{\overline{a}}$, $j \in K_i$) such that the following hold:
(a) $x_i = c'_i + \sum_{j \in K_i} x_{i,j}$ for $i \in N_{\overline{a}}$,
(b) $a_{i,j} p_{i,j} u_i^{x_{i,j}} s_{i,j} =_H s_{k,l}^{-1} (u_k^{-1})^{x_{k,l}} p_{k,l}^{-1} b_{i,j}$ for all $(i,j,k,l) \in M$,
(c) $(x_i)_{i \in N_a} \in S_a$.
Here, $K_i \subseteq [1,k_i]$ is a set of size at most $k_i \leq 14m$, $M$ is a matching relation (with $(i,j,k,l) \in M$ if and only if $(k,l,i,j) \in M$), $a_{i,j}, b_{i,j} \in A \cup B$, and $c'_i \le \mathcal{O}(m \cdot \lambda({e}))$. Every word $p_{i,j}$ is a suffix of $u_{i,j}$ and every $s_{i,j}$ is a prefix of $u_{i,j}$. In particular, $p_{i,j}$ and $s_{i,j}$ have length at most $\lambda({e})$.
[*Step 5.*]{} The remaining two-dimensional knapsack equations from point (b) are eliminated with Lemma \[2dim\]. Every equation $$a_{i,j} p_{i,j} u_i^{x_{i,j}} s_{i,j} =_H s_{k,l}^{-1} (u_k^{-1})^{x_{k,l}} p_{k,l}^{-1} b_{i,j}$$ can be nondeterministically replaced by a semilinear constraint for $x_{i,j}$ and $x_{k,l}$ of magnitude $\mathcal{O}(\gamma^{2} \lambda({e})^{4})$. By substituting these semilinear constraints in the above equations (a) for the $x_i$ (as we did in the proof of Theorem \[thm-main-technical\]), we obtain for the variables $x_i$ ($i \in N_{\overline{a}}$) a semilinear constraint of magnitude $\mathcal{O}(m \cdot \gamma^{2} \cdot \lambda({e})^{4})$. This leads to a semilinear representation for ${\mathsf{sol}}_{H}({e})$ of magnitude at most $\max\{ \mathsf{K}_1, \mathsf{K}_2 \}$, where $$\mathsf{K}_1 = \mathsf{K}(8m {|\!| {e}|\!|},m) \text{ and }
\mathsf{K}_2 \le \mathcal{O}(m \cdot \gamma^{2} \cdot {|\!| {e}|\!|}^{4}) .$$ [*Step 6.*]{} For the preprocessing we can apply Lemma \[Britton-reduced-powers\] to each period $u_i$. Hence, we just have to replace ${|\!| {e}|\!|}$ by $3 {|\!| {e}|\!|}$ in the above bounds, which yields the statement of the theorem.
The following result is obtained analogously to Theorem \[thm:free-product-NP-reduction\].
\[thm:HNN-NP-reduction\] The knapsack problem for $\langle G, t \mid t^{-1}at=\varphi(a)~(a\in A)\rangle$ (with $A$ finite) is nondeterministically polynomial time reducible to the knapsack problem for $G$.
Amalgamated products preserve knapsack semilinearity
----------------------------------------------------
Using our results for free products and HNN-extensions, we can easily deal with amalgamated products. For $i\in\{1,2\}$, let $G_i=\langle \Sigma_i \mid R_i\rangle$ be a finitely generated group with $\Sigma_1 \cap \Sigma_2 = \emptyset$ and let $A$ be a finite group that is embedded in each $G_i$ via the injective morphism $\varphi_i\colon A\to G_i$ for $i\in\{1,2\}$. Then, the *amalgamated product with identified subgroup $A$* is the group $$\langle \Sigma_1\uplus \Sigma_2 \mid R_1\uplus R_2\cup \{\varphi_1(a)=\varphi_2(a) \mid a\in A\}\rangle.$$ This group is usually written as $$\langle G_1*G_2 \mid \varphi_1(a)=\varphi_2(a)~(a\in A)\rangle.$$ or just $G_1 *_A G_2$. Note that the amalgamated product depends on the morphisms $\varphi_i$, although they are omitted in the notation $G_1 *_A G_2$.
From Theorem \[HNNbound\], we can easily deduce a similar result for amalgamated products:
Let $G_1$ and $G_2$ be finitely generated groups with a common subgroup $A$. Let $\mathsf{K}(n,m)$ be the pointwise maximum of the functions $\mathsf{K}_{G_1,\Sigma_1}$ and $\mathsf{K}_{G_2,\Sigma_2}$. Furthermore, let $\gamma=|A|$ and let $G$ be the amalgamated product $G_1 *_A G_2$. Then with $\Sigma=\Sigma_1\cup \Sigma_2$ we have $\mathsf{K}_{G,\Sigma}(n,m) \le \max\{\mathsf{K}_1, \mathsf{K}_2, \mathsf{K}_3\}$ where $$\mathsf{K}_1 = \mathsf{K}_{G,\Sigma}(36m^2n,m) ,
\mathsf{K}_2 \le \mathcal{O}(m^5 n^4) \text{ and }
\mathsf{K}_3 \le \mathcal{O}(m \cdot \gamma^{2} \cdot n^{4}) .$$
For the proof we will make use of the previous theorem and Theorem \[thm-cor-technical\] for free products.
It is well-known [@LySch77 Theorem 2.6, p. 187] that $G_1 *_A G_2$ can be embedded into the HNN-extension $$H = \langle G_1*G_2, t \mid t^{-1}\varphi_1(a)t=\varphi_2(a)~(a\in A) \rangle$$ by the morphism $\Phi\colon G_1 *_A G_2\to H$ with $$\Phi(g)=\begin{cases} t^{-1}gt & \text{if $g\in G_1$} \\ g & \text{if $g\in G_2$}. \end{cases}$$ Obviously we have $\mathsf{K}_{G,\Sigma}(n,m)\leq \mathsf{K}_{H,\Sigma}(n,m)$. Hence we can calculate the bound by first getting the bound for the free product $G_1*G_2$ and then proceeding with the HNN-extension. Theorem \[thm-cor-technical\] tells us that $\mathsf{J}(n,m):=\mathsf{K}_{G_1*G_2,\Sigma}(n,m)\leq \max \{ \mathsf{K}'_1,\mathsf{K}'_2 \}$, where $\mathsf{K}'_1=\mathsf{K}(6mn,m)$ and $\mathsf{K}'_2 \le \mathcal{O}(m n^4)$. To obtain $\mathsf{K}_{H,\Sigma}(n,m)$, we make use of Theorem \[HNNbound\]. We have $\mathsf{K}_{H,\Sigma}(n,m) \leq \max \{ \mathsf{J}_1,\mathsf{J}_2 \}$, where $\mathsf{J}_1=\mathsf{J}(6mn,m)$ and $\mathsf{J}_2\leq \mathcal{O}(\gamma^{2} m n^{4})$. Since the function $\mathsf{J}(n,m)$ appears in $\mathsf{J}_1$, we have to substitute by what we calculated before. More precisely we have to make the substitution $n \mapsto 6mn$ for the values $\mathsf{K}'_1$ and $\mathsf{K}'_2$. This yields $\mathsf{K}_{H,\Sigma}(n,m) \leq \max \{ \mathsf{K}_1,\mathsf{K}_2, \mathsf{K}_3 \}$, where $$\begin{aligned}
\mathsf{K}_1 &= \mathsf{K}(36m^2n,m) ,\\
\mathsf{K}_2 &\leq \mathcal{O}(m^5 n^4) ,\\
\mathsf{K}_3 &= \mathsf{J}_2 \leq \mathcal{O}(\gamma^{2} m n^{4}) .\end{aligned}$$ This finishes the proof of the theorem.
From Theorems \[thm:free-product-NP-reduction\] and \[thm:HNN-NP-reduction\] and the above embedding of $G_1 *_A G_2$ in $\langle G_1*G_2, t \mid t^{-1}\varphi_1(a)t=\varphi_2(a)~(a\in A) \rangle$ we obtain:
The knapsack problem for $G_1*_A G_2$ (with $A$ finite) is nondeterministically polynomial time reducible to the knapsack problems for $G_1$ and $G_2$.
Part III: Knapsack in finite extensions
=======================================
\[thm-finite-index\] Let $G$ be a finitely generated group with a finite symmetric generating set $\Sigma$ and let $H$ be a finite extension of $G$ (hence, it is finitely generated too) with the finite symmetric generating set $\Sigma' = \Sigma \cup (C \setminus \{1\}) \cup (C \setminus \{1\})^{-1}$, where $C$ is a set of coset representatives with $1 \in C$. Let $l = |C|$ be the index of $G$ in $H$. If $G$ is knapsack-semilinear then $H$ is knapsack-semilinear too and we have the bounds $$\begin{aligned}
\mathsf{E}_{H,\Sigma'}(n,m) &\leq & l \cdot \mathsf{E}_{G,\Sigma}(\mathcal{O}(l^2 n),m) +2l, \label{eq-finite-index-E} \\
\mathsf{K}_{H,\Sigma'}(n,m) & \leq & l \cdot \mathsf{K}_{G,\Sigma}(\mathcal{O}(l^2 n),m) +2l \label{eq-finite-index-K}.\end{aligned}$$
Suppose we are given an exponent expression $$e=u_1^{x_1}v_1\cdots u_m^{x_m}v_m \label{fi:input}$$ in $H$ where the $u_i$ and $v_i$ are words over $\Sigma'$. Let $n$ be the length of $e$. As a first step, we guess which of the variables $x_i$ assume a value smaller than $l$. For those that do, we can guess the value and merge the resulting power with the $v_i$ on the right. This increases the size of the instance by at most a factor of $l$, which is a constant. Hence, from now on, we only look for $H$-solutions $\sigma$ to $e=1$ where $\sigma(x_i) \ge l$ for $1\le i\le m$. At the end we will compensate this by applying the substitution $n\mapsto l n$.
Next we guess the cosets of the prefixes of $u_1^{x_1}v_1\cdots u_m^{x_m}v_m$, i.e., we guess coset representatives $c_1,d_1,\ldots,c_{m-1},d_{m-1},c_m \in C$ and restrict to $H$-solutions $\sigma$ to $e=1$ such that $u_1^{\sigma(x_1)}v_1\cdots u_i^{\sigma(x_i)}\in Gc_i$ and $u_1^{\sigma(x_1)}v_1\cdots u_i^{\sigma(x_i)}v_i\in Gd_i$ for $1\le i\le m$. Here, we set $d_m=1$. Equivalently, we only consider $H$-solutions $\sigma$ where $d_{i-1}u_i^{x_i}c_i^{-1}$ and $c_{i}v_id_i^{-1}$ all belong to $G$ for $1\le i\le m$. Here, we set $d_0 = 1$. We can verify in polynomial time that all $c_iv_id_i^{-1}$ ($1\le i\le m$) belong to $G$. It remains to describe the set of all $H$-solutions $\sigma$ for $e=1$ that fulfill the following constraints for all $1\le i\le m$: $$\label{constraints-finite-ext}
d_{i-1}u_i^{\sigma(x_i)}c_i^{-1}\in G \text{ and } \sigma(x_i) \ge l .$$ For $1 \leq i \leq m$ consider the function $f_i\colon C\to C$, which is defined so that for each $c\in C$, $f_i(c)$ is the unique element $d\in C$ with $cu_id^{-1}\in G$. Note that we can compute $f_i$ in polynomial time if $G$ and $H$ are fixed groups (all we need for this is a table that specifies for each $c \in C$ and $a \in \Sigma'$ the coset representative of $ca$; this is a fixed table that does not depend on the input). Then there are numbers $1\le k_i\le
l$ such that $f_i^{l+k_i}(d_{i-1})=f_i^l(d_{i-1})$. With this notation, we have $d_{i-1}u_i^{z}c_i^{-1}\in G$ if and only if $f_i^{z}(d_{i-1})=c_i$ for all $z \in {\mathbb{N}}$.
We may assume that there is an $z\ge l$ with $f_i^{z}(d_{i-1})=c_i$; otherwise, there is no $H$-solution for $e=1$ fulfilling the above constraints and we have a bad guess. Therefore, there is a $0\le r_i<k_i$ such that $f_i^{l+r_i}(d_{i-1})=c_i$. This means that for all $z \ge l$, we have $d_{i-1}u_i^{z}c_i^{-1}\in G$ if and only if $f_i^{z}(d_{i-1})=c_i$ if and only if $z=l+k_i\cdot
y+r_i$ for some $y\ge 0$. This allows us to construct an exponent expression over $G$.
Let $e_i=f_i^l(d_{i-1})$. Then, the words $d_{i-1}u_i^l e_i^{-1}$, $e_iu_i^{k_i}e_i^{-1}$, and $e_iu_i^{r_i}c_i^{-1}$ all represent elements of $G$. Moreover, for all $y_i \geq 0$ and $z_i=l+k_i\cdot y_i+r_i$ ($1 \leq i \leq m$), we have $$\begin{aligned}
u_1^{z_1}v_1\cdots u_m^{z_m}v_m &= \prod_{i=1}^m d_{i-1} u_i^{l+k_i\cdot y_i+r_i}c_i^{-1}c_i v_i d_i^{-1} \\
&= \prod_{i=1}^m (d_{i-1} u_i^l e_i^{-1})(e_iu_i^{k_i} e_i^{-1})^{y_i} (e_i u_i^{r_i} c_i^{-1}c_i v_i d_i^{-1})\end{aligned}$$ and each word in parentheses represents an element of $G$. Hence, we can define the exponent expression $$e' = \prod_{i=1}^m (d_{i-1} u_i^l e_i^{-1})(e_iu_i^{k_i} e_i^{-1})^{x_i} (e_i u_i^{r_i} v_i d_i^{-1})$$ over the group $G$. From the above consideration we obtain $$\begin{aligned}
{\mathsf{sol}}_H(e) \cap \{ \sigma : X_e \to H \mid \sigma \text{ satisfies the constraints } \eqref{constraints-finite-ext} \} & = \nonumber \\
\{ \sigma \mid \sigma(x_i) = k_i \cdot \sigma'(x_i) + (l+r_i) \text{ for some } \sigma' \in {\mathsf{sol}}_G(e') \} . & \label{sigma'-sigma}\end{aligned}$$ The set in is semilinear by assumption and since all $k_i$ and $r_i$ are bounded by $l$, we can bound its magnitude by $l \cdot {\mathsf{mag}}({\mathsf{sol}}_G(e')) + 2l$. Moreover, we have $\deg(e') = \deg(e)$. It remains to bound ${|\!| e' |\!|}$. For this, we first have to rewrite the words $d_{i-1} u_i^l e_i^{-1}$, $e_iu_i^{k_i} e_i^{-1}$, and $e_i u_i^{r_i} v_i d_i^{-1}$ (which represent elements of $G$) into words over $\Sigma$. This increases the length of the words only by a constant factor: for every $c \in C$ and every generator $a \in \Sigma'$ there exists a fixed word $w_{c,a} \in \Sigma^*$ and $d_{c,a} \in C$ such that $ca = w_{c,a} d_{c,a}$ holds in $H$. After this rewriting we have ${|\!| e' |\!|} \leq \mathcal{O}(l n)$, which implies ${\mathsf{mag}}({\mathsf{sol}}_G(e')) \leq \mathsf{E}_{G,\Sigma}(\mathcal{O}(l n),m)$. This yields the bound $l \cdot \mathsf{E}_{G,\Sigma}(\mathcal{O}(l n),m) + 2l$ for the magnitude of the semilinear set in . Applying the substitution $n \mapsto ln$ from the first step finally yields . The corresponding bound for knapsack expressions can be shown in the same way. Note that in the above transformation of $e$ into $e'$ we do not duplicate variables.
From the above proof we also obtain the following result (similarly to Theorem \[thm:free-product-NP-reduction\]).
\[thm:finite-NP-reduction\] The knapsack problem for a finite extension of a group $G$ is nondeterministically polynomial time reducible to the knapsack problem for $G$.
#### **Acknowledgement.**
The first and second author were supported by the DFG research project LO 748/12-1.
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[^1]: This work has been supported by the DFG research project LO 748/13-1
[^2]: Karp in his seminal paper [@Karp72] defined knapsack in a slightly different way. [NP]{}-completeness of the above version was shown in [@Haa11].
[^3]: Knapsack-semilinearity of co-context free groups is not stated in [@KoenigLohreyZetzsche2015a] but follows immediately from the proof for the decidability of knapsack.
[^4]: Note that since ${\mathsf{alph}}(p_{i,j}) \subseteq {\mathsf{alph}}(u)$, we must have $p_{i,j}=1$ or $x_k=0$ whenever $j < k < i$.
[^5]: Since the word problem for every ${G}_i$ is decidable, also the word problem for ${G}$ is decidable [@Gre90], which implies that one can compute a geodesic word for a given group element of ${G}$.
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author:
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Georg Engel$^a$, Christof Gattringer$^a$, Leonid Ya. Glozman$^a$, C. B. Lang$^a$, Markus Limmer$^a$, $^a$ and Andreas Schäfer$^b$\
Institut für Physik, FB Theoretische Physik, Universität Graz, A-8010 Graz, Austria\
Fakultät für Physik, Universität Regensburg, D-93040 Regensburg, Germany\
E-mail: , , , , , ,
bibliography:
- 'bibtex.bib'
title: 'Baryon axial charges from Chirally Improved fermions - first results'
---
Introduction
============
The axial charge of the nucleon, or more precisely the ratio $G_A(q^2=0)/G_v(q^2=0)$ has been determined to a high precision from neutron $\beta$ decay, with $G_A(0)/G_v(0)=1.2695(29)$. In general, the axial form factor $G_{A,BB^\prime}$ for an octet baryon is given by $$\begin{aligned}
\left<B^\prime|A_\mu(q)|B\right>&=\bar{u}_{B^\prime}(p^\prime)\left(\gamma_\mu\gamma_5G_{A,BB^\prime}(q^2)+\gamma_5q_\mu\frac{G_P(q^2)}{2M_B}\right)u_B(p)\mathrm{e}^{-iq\cdot
x}\label{ga_def}\ ,\end{aligned}$$ where $G_P$ is the induced pseudoscalar form factor. The axial charge is defined as the value of the axial form factor at zero momentum transfer $G_{A,BB^\prime}(q^2=0)$. In the following, we will omit the indices $B$ and $B^\prime$ when referring to the nucleon. For the nucleon in the chiral limit, the Goldberger-Treiman relation $G_A=F_\pi g_{\pi NN}/M_N$ connects the axial charge to the pion decay constant $F_\pi$, the pion-nucleon coupling constant $g_{\pi NN}$ and the nucleon mass $M_N$. Away from the chiral limit, this relation is still approximately fulfilled. Assuming the conservation of the vector current (which is the case for mass-degenerate light quarks $m_u=m_d$), the nucleon axial charge is also related to the polarized quark distributions in the proton: $G_A=\Delta u-\Delta d$ [@Sasaki:2003jh]. In an isovector combination, disconnected contributions cancel, making high-precision lattice computations feasible.
The Chiral Perturbation Theory ($\chi PT$) expressions relevant to the nucleon axial charge have been calculated in [@Beane:Detmold], where finite volume effects are taken into consideration. While a recent simulation with domain wall fermions [@Yamazaki:2008py] finds considerable finite volume effects and scaling in $M_\pi L$, volume effects calculated in $\chi PT$ lead to differing conclusions. Trying to attribute this difference to excited state contaminations arising from finite separation in Euclidean time, Tiburzi [@Tiburzi:2009zp] estimates the effects of such contaminations and obtains that they would lead to an over-estimation of $G_A$ rather than an under-estimation. He also suggests to study $G_A$ using the variational method. Lattice results for the nucleon axial charge have furthermore been presented in [@Syritsyn:Edwards:Takahashi]. For a recent review, please refer to the review by Renner [@renner_lat09].
So far, only one group has reported results for the axial couplings of sigma and cascade hyperons [@Lin:2007ap]. The corresponding $\chi$PT calculations can be found in [@Jiang:Tiburzi]. In [@Jiang:2009sf] input from experiment and lattice QCD is used to determine the unknown parameters in the $\chi PT$ expansion and predict the mass dependence and values of the axial charges in the chiral limit.
In the next section, we explain the setup for calculations of baryon axial charges using Chirally Improved (CI) lattice fermions and the variational method. We will then move on and present results from our calculations of the axial charges of the nucleon and of $\Sigma$ and $\Xi$ hyperons.
Details of our calculational setup
==================================
For our simulations we use CI fermions [@Gattringer:CI], which are approximate Ginsparg-Wilson fermions based on an expansion of Dirac operator terms on a hypercube. CI fermions have been tested extensively in quenched calculations [@Gattringer:2003qx] and results for the ground state spectrum of mesons and baryons from dynamical CI simulations have been presented recently [@Gattringer:2008vj].
Assuming mass-degenerate up- and down quarks it is sufficient to consider the following current insertion to extract the axial charge $$\begin{aligned}
\label{axial_current}
2A_\mu^3&=A_\mu^u-A_\mu^d\ ,\qquad A_\mu^u=\bar{u}\gamma_\mu\gamma_5 u\ ,\qquad
A_\mu^d=\bar{d}\gamma_\mu\gamma_5 d \ ,\end{aligned}$$ in which $u$ denotes an up quark and $d$ denotes a down quark. In the following, we will show how three-point functions with insertions like those in Equation \[axial\_current\] can be evaluated on the lattice.
For our calculations we use so-called *sequential propagators*. In [@Sasaki:2003jh], two methods for the calculation of sequential quark propagators are presented. Figure \[sequential\_possibilities\] illustrates these two approaches. On the left-hand side, sequential sources are built from specific diquark propagators, while in the middle, the propagators are calculated for each possible insertion separately. An illustration of the full baryon three-point function is provided on the right-hand side. Which approach is computationally cheaper depends on the physics objective. In our case, we want to use two different insertions and two widths of smearing for three different interpolator types. As we are using a rather coarse lattice, a small number of insertion timeslices should be enough. Therefore, even when just considering the nucleon case, the second approach using sequential sources from single quark propagators is slightly cheaper. Moreover, these propagators can subsequently be used for other hadrons and for the calculation of transition form factors.
To relate lattice operators, which receive a finite renormalization, to their continuum counterparts, we need to estimate the renormalization factors $Z_\Gamma$ of the bilinear currents in question. In general, we have to multiply the lattice result $G_\Gamma^{lat}$ by the appropriate renormalization factor to obtain values that can be compared with results extracted from experiments $$\begin{aligned}
G_\Gamma^{phys}=Z_\Gamma G_\Gamma^{lat},\end{aligned}$$ which are typically given in the *modified minimal subtraction* ($\overline{MS}$) renormalization scheme.
For dynamical CI fermions, these renormalization constants have been estimated using local bilinear quark field operators in [@philipp_renorm]. It would however be useful to have an independent estimation of these constants from a different method. In the case of the vector current, one can estimate the constant $Z_V$ by calculating the vector charge $G_V$ defined in analogy with via $$\begin{aligned}
\left<B^\prime|V_\mu(q)|B\right>&=\bar{u}_{B^\prime}(p^\prime)\left(\gamma_\mu G_V(q^2)+q_\nu\sigma_{\nu\mu}\frac{G_T(q^2)}{2M_B}\right)u_B(p)\mathrm{e}^{-iq\cdot
x}\ ,\end{aligned}$$ as $G_V(q^2=0)$. This quantity has to be 1 in the continuum, as it is related to the electric charge of the proton in the limit of equal quark masses [@Sasaki:2003jh].
For lattice fermions with exact chiral symmetry, the axial vector renormalization constant $Z_A$ and the vector renormalization constant $Z_V$ have to be equal. For lattice fermions which only fulfill the Ginsparg-Wilson relation approximately, there should be small deviations from this. To obtain an independent estimate of $Z_V$, we use a ratio of two-point over three-point functions [@Burch:2008qx] $$\begin{aligned}
R^{(k)}&=\frac{\sum_l\sum_m\psi_l^{(k)}C(t)_{lm}\psi_m^{(k)}}{\sum_i\sum_j\psi_i^{(k)}T_V(t,t^\prime)_{ij}\psi_j^{(k)}}=Z_V
\ ,\end{aligned}$$ where $C(t)$ is the matrix of two-point correlation functions and $T_V(t,t^\prime)$ is the matrix of three-point correlators with a vector insertion. The eigenvectors $\psi$ are the ones obtained from a variational analysis of $C(t)$. We then compare with the preliminary estimates from [@philipp_renorm]. The ensemble names are according to [@Gattringer:2008vj], where details of the run parameters are provided. For runs A and B the two methods agree within $2-3\%$. While a determination using local quark bilinears yields 0.818(2) for run A and 0.826(1) for run B, we find values for $Z_V$ of 0.803(2) and 0.792(2) respectively. For run C there is a rather large discrepancy and the method of [@philipp_renorm] leads to a value of 0.829(1) while we obtain 0.77(1). Notice also that two different methods for the determination of the renormalization constants are presented in [@philipp_renorm] which only agree after a chiral extrapolation of the results is performed. At the same time, the ratio $Z_A/Z_V$ determined from the values in [@philipp_renorm] is almost identical for both methods used and also stable under chiral extrapolation of the results.
In our determination of the axial charge from run C, we encounter what we suspect to be large finite volume effects. Notice that the value of $Z_V$ obtained from the nucleon three-point functions might be plagued by the same effects. As we cannot calculate $Z_A$ from baryon three-point functions, we therefore always use the ratio $Z_A/Z_V$ from [@philipp_renorm]. In the next section, we discuss in detail which ratios we measure on the lattice to obtain the renormalized axial charge $G_A$.
Nucleon axial charge from dynamical CI fermions
===============================================
The usual approach [@Sasaki:2003jh; @Yamazaki:2008py] is to extract the nucleon axial charge from ratios of $G_A$ over $G_V$ $$\begin{aligned}
G_A&=\frac{Z_A}{Z_V}\frac{T_A^3(t,t^\prime)}{T_V(t,t^\prime)}\ ,
\label{simple_ratio}\end{aligned}$$ using single correlation functions built from either smeared quarks or gauge fixed box or wall sources. This approach has the advantage that some of the systematic errors entering the lattice determination will cancel.
We instead use the variational method, which is commonly used to extract ground and excited state masses. It is based on a correlation matrix $C_{ij}(t)=\langle O_i(t) O^\dagger_j(0) \rangle$ where $O_i(t)$ are operators with the quantum numbers of the state of interest. The eigenvalues $\lambda_i$ of the generalized eigenvalue problem $C(t)v_i=\lambda_iC(t_0)v_i$ may be shown to behave as $\lambda_i(t)\propto\mathrm{e}^{-tE_i}\left(1+\mathcal{O}\left(\mathrm{e}^{-t\Delta
E_i}\right)\right)$, where $E_i$ is the energy of the $i$-th state. The approach may be generalized to three point functions. Following [@Burch:2008qx], we obtain an expression for $G_A$: $$\begin{aligned}
G_A&=\frac{Z_A}{Z_V}\frac{\sum_i\sum_j\psi_i^{(k)}T_A^3(t,t^\prime)_{ij}\psi_j^{(k)}}{\sum_l\sum_m\psi_l^{(k)}T_V(t,t^\prime)_{lm}\psi_m^{(k)}}
\ .\end{aligned}$$ Figure \[bh\_ga\] shows a typical plateau for the axial charge of the nucleon from run C extracted from such a ratio. The horizontal lines denote the results from a linear fit in the displayed range. Notice that we observe a plateau in the full range of points we calculated. For all three ensembles, we choose timeslice $t$ = 9 for the position of the sink. This corresponds to a source-sink separation of roughly $1.2\,\mathrm{fm}$. For run B, we currently only have data for insertion timeslices $t^\prime$ from 5 to 9. Instead of assuming that the central value at 5 is the physical one, we perform a linear fit in the range 5 to 7.
![ Example plot to illustrate typical plateaus observed with our variational basis for a source-sink separation of $\approx 1.2\,\mathrm{fm}$. The data is from run C. []{data-label="bh_ga"}](ga_full_matrix_C.eps){width="7cm"}
![ We compare our results for $G_A$ to a recent determination from domain wall fermions. Our data is labeled as “2 flavor CI”. Results from 2+1 flavor domain wall fermions are taken from Yamazaki et al. [@Yamazaki:2008py]. On the left-hand side, we plot the results over $M_\pi^2$. Towards lower quark masses finite volume effects are clearly visible. On the right-hand side we display our data in units of $M_\pi
L$. []{data-label="ga_compareplot_nucleon"}](ga_over_mpi_compare.eps "fig:"){height="5.6cm"} ![ We compare our results for $G_A$ to a recent determination from domain wall fermions. Our data is labeled as “2 flavor CI”. Results from 2+1 flavor domain wall fermions are taken from Yamazaki et al. [@Yamazaki:2008py]. On the left-hand side, we plot the results over $M_\pi^2$. Towards lower quark masses finite volume effects are clearly visible. On the right-hand side we display our data in units of $M_\pi
L$. []{data-label="ga_compareplot_nucleon"}](ga_over_mpiL_compare.eps "fig:"){height="5.6cm"}
We compare our data to recent results from domain wall fermions [@Yamazaki:2008py] in Fig. \[ga\_compareplot\_nucleon\]. The left-hand side plot shows the results for $G_A$ plotted over the square of the pion mass $M_\pi^2$. While results at large pion masses lead to values close to the experimental value, the result from run C deviates substantially from this behavior. The same is true for the domain wall data and this behavior seems to be a universal feature associated with finite volume effects [@Yamazaki:2008py; @renner_lat09]. On the right-hand side of the figure we therefore plot the results for $G_A$ over $M_\pi L$, where $L$ corresponds to the spatial extent of the lattice. This plot can be directly compared to Fig. 3 of [@Yamazaki:2008py].
Before we move on to calculations for hyperons, let us briefly comment on the sink-dependence of our results. While results from run A and B are rather insensitive to the sink location in the region explored (timeslices 9-13), a systematic shift upwards can be observed for run C when reducing the separation between the source and the sink from 8 to 6 timeslices which corresponds to distances of $1.2\,\mathrm{fm}$ and $0.9\,\mathrm{fm}$. We want to point out that this does not affect the quality of the plateau which still stretches over the entire region of insertion times. Taking a look at the nucleon two point functions, contributions from excited states to the ground state of the variational analysis are visible up to timeslice 4. This is an indication that excited states may indeed be responsible for measuring a larger value of $G_A$ if excited state contributions are not sufficiently suppressed. With just 50 configurations, the statistical errors from our preliminary dataset are by far too large to make a stronger and more quantitative statement.
Hyperon axial charges
=====================
![ Results for the axial charges of the $\Sigma$ (left-hand side) and $\Xi$ (right-hand side) hyperons compared to the mixed action results by Lin and Orginos [@Lin:2007ap]. []{data-label="ga_compareplot_sigmaxi"}](sigma_ga_over_mpi_compare.eps "fig:"){width="7.3cm"} ![ Results for the axial charges of the $\Sigma$ (left-hand side) and $\Xi$ (right-hand side) hyperons compared to the mixed action results by Lin and Orginos [@Lin:2007ap]. []{data-label="ga_compareplot_sigmaxi"}](xi_ga_over_mpi_compare.eps "fig:"){width="7.3cm"}
In this section we present results for a calculation of hyperon axial charges. For the $\Sigma$ and $\Xi$ hyperons we adopt the following definitions: $$\begin{aligned}
\left<\Sigma^{+}|A_\mu^3|\Sigma^{+}\right>-\left<\Sigma^{-}|A_\mu^3|\Sigma^{-}\right>=G_{\Sigma\Sigma}\;\bar{u}^\nu\gamma_\mu\gamma_5
u^\nu \ ,\\
\left<\Xi^{0}|A_\mu^3|\Xi^{0}\right>-\left<\Xi^{-}|A_\mu^3|\Xi^{-}\right>=G_{\Xi\Xi}\;\bar{u}^\nu\gamma_\mu\gamma_5
u^\nu \ .\end{aligned}$$ Again, no disconnected contributions appear in the isovector quantities and the calculation proceeds similar to the nucleon case. In particular, no additional sequential propagators are needed.
Figure \[ga\_compareplot\_sigmaxi\] shows our results for the axial charges of the $\Sigma$ and $\Xi$ hyperons. We compare our data to [@Lin:2007ap] and we can see a quantitative agreement in the full range of masses. Unlike for the nucleon, no significant decrease is observed towards the chiral limit in the plot for the $\Sigma$ (l.h.s.). The plot for the $\Xi$ (r.h.s.) also shows a nice agreement with the results from [@Lin:2007ap]. In this case our value corresponding to the smallest pion mass shows a slight decrease towards the chiral limit, but the error bars are large and this may as well be an effect of our limited statistics. Our purely statistical errors on the preliminary dataset of 50 configurations are still large but can be substantially reduced by using our full statistics.
\[ga\_outlook\]Summary and outlook
==================================
We have presented preliminary results from a calculation of baryon axial charges using a full variational basis to efficiently suppress contaminations from excited states. We used a basis of baryon interpolators with different Dirac structures and two different smearing widths for the quarks. The results are in good agreement with the literature and we obtain clear plateaus for ratios calculated with the method of [@Burch:2008qx]. Provided the signal for the states in question is strong enough, this method can also be applied to several other quantities of interest, among them the axial charge of the delta baryon and the $N$-$\Delta$ or $\Sigma$-$\Lambda$ transition.
In general, the method we use can also be applied to three-point functions involving excited states, provided that the signal is good enough to ensure the necessary separation between the source/sink and the current insertion.
The calculations have been performed on the SGI Altix 4700 of the Leibniz-Rechenzentrum Munich and on local clusters at ZID at the University of Graz. We thank these institutions for providing support. M.L. and D.M. are supported by the DK W1203-N08 of the "Fonds zur Förderung wissenschaflicher Forschung in Österreich”. D.M. acknowledges support by COSY-FFE Projekt 41821486 (COSY-105) and G.E. and A.S. acknowledge support by the DFG project SFB/TR-55.
|
---
abstract: 'We present a calibration component for the Murchison Widefield Array All-Sky Virtual Observatory (MWA ASVO) utilising a newly developed <span style="font-variant:small-caps;">PostgreSQL</span> database of calibration solutions. Since its inauguration in 2013, the MWA has recorded over thirty-four petabytes of data archived at the Pawsey Supercomputing Centre. According to the MWA Data Access policy, data become publicly available eighteen months after collection. Therefore, most of the archival data are now available to the public. Access to public data was provided in 2017 via the MWA ASVO interface, which allowed researchers worldwide to download MWA uncalibrated data in standard radio astronomy data formats (CASA measurement sets or UV FITS files). The addition of the MWA ASVO calibration feature opens a new, powerful avenue for researchers without a detailed knowledge of the MWA telescope and data processing to download calibrated visibility data and create images using standard radio-astronomy software packages. In order to populate the database with calibration solutions from the last six years we developed fully automated pipelines. A near-real-time pipeline has been used to process new calibration observations as soon as they are collected and upload calibration solutions to the database, which enables monitoring of the interferometric performance of the telescope. Based on this database we present an analysis of the stability of the MWA calibration solutions over long time intervals.'
author:
- 'M. Sokolowski$^{1}$[^1], C. H. Jordan$^{2,3}$, G. Sleap$^{2}$, A. Williams$^{2}$, R. B. Wayth$^{1,3}$, M. Walker$^{2}$, D. Pallot$^{4}$, A. Offringa$^{5,6}$, N. Hurley-Walker$^{1}$, T. M. O. Franzen$^{5}$, M. Johnston-Hollitt$^{1}$, D. L. Kaplan$^{7}$, D. Kenney$^{1}$, S. J. Tingay$^{1}$'
bibliography:
- 'caldb\_mwa\_asvo.bib'
title: 'Calibration database for the Murchison Widefield Array All-Sky Virtual Observatory'
---
astronomical databases: miscellaneous – virtual observatory tools – methods: data analysis – instrumentation: interferometers – techniques: interferometric
INTRODUCTION {#sec:intro}
============
The Murchison Widefield Array [MWA; @2013PASA...30....7T; @2018PASA...35...33W] is one of the low-frequency precursors of the Square Kilometer Array (SKA[^2]). It is located at the Murchison Radio-astronomy Observatory (MRO), the future site of the low-frequency component of the SKA (SKA-Low). The MWA began its operations in 2013 and since then has recorded over thirty-four petabytes of visibility (output from the MWA correlator) and Voltage Capture System [VCS; @2015PASA...32....5T] data, which are archived in the Pawsey Supercomputing Centre (PSC). Based on the MWA Data Access Policy[^3] the data become publicly available eighteen months after collection or immediately after collection for the members of the MWA collaboration. As a consequence, most of the archival visibility data (approximately 19.3PB[, representing 79% of the total]{}) are now (as of March 2020) publicly available. Public data have been available via the MWA All-Sky Virtual Observatory (MWA ASVO[^4]) since its initial pilot release in 2017. The pilot MWA ASVO interface enabled users to download raw MWA data in standard radio astronomy data formats such as CASA measurement sets [@casa] or UV FITS files [@uvfits]. The data sets returned to the end user are flagged for radio-frequency interference (RFI) using <span style="font-variant:small-caps;">aoflagger</span> software and averaged in time and frequency according to the requirements specified in the web interface or request file. However, these data are not calibrated and require several additional steps before sky images could be formed, which could have been difficult for users not familiar with MWA data calibration. Here we present a calibration extension of the MWA ASVO, opening a new avenue for any researcher worldwide without deep knowledge of the details of the MWA instrument and data processing to download calibrated visibility data in the aforementioned formats.
The calibration database have been populated with calibration solutions from the entire history of MWA observations. In order to do it we developed a dedicated calibration pipeline. The newly collected [calibration observations are automatically processed]{} in near real-time and the resulting calibration solutions are uploaded to the calibration database (CALDB). This enables us to monitor the stability of their phase and amplitude components, i.e. the inteferometric performance of the MWA, which allows the MWA Operations team to identify problems which may go undetected by other components of the Monitor and Control (M&C) software[^5].
This paper is organised as follows. In Section \[sec:calibration\_database\] we present the structure of the calibration database. In Section \[sec:automatic\_pipelines\] we describe software pipelines developed to populate the database with calibration solutions derived from archived calibrator observations since 2013, and a near-real-time version of this pipeline which is used to process new calibrator observations (collected just after sunset and before sunrise). We also present a system developed for handling requests for missing calibration solutions, and web services developed to download calibration solutions for a specific observation via web browser or command line (<span style="font-variant:small-caps;">wget</span> command). In Section \[sec:applications\] we describe the applications of the calibration database, such as the MWA ASVO, monitoring of the interferometric performance of the MWA, providing calibration solutions to the new MWA correlator, and potential future applications for transient and ionosphere monitoring. Finally, in Section \[sec:summary\] we make summarising remarks and discuss the importance of these developments in the context of the future SKA-Low telescope.
DATABASE OF CALIBRATION SOLUTIONS {#sec:calibration_database}
=================================
The overview of the current MWA ASVO system is shown in Figure \[fig\_asvo\_overview\]. The newly added calibration component consists of the calibration database, and several scripts and pipelines for: populating this database with calibration solutions, accessing the database, and applying solutions to uncalibrated data downloaded from the MWA archive at the PSC.
![General overview of the MWA ASVO system with the new calibration component.[]{data-label="fig_asvo_overview"}](Figures/calibration_system.png){width="\columnwidth"}
{width="\textwidth"}
DATABASE STRUCTURE {#subsec:database_structure}
------------------
The calibration database (CALDB) has been implemented as a <span style="font-variant:small-caps;">PostgreSQL</span> database[^6], which is an advanced open source relational [database]{} and has also been used for storing other monitor and control (M&C) MWA data. Presently the MWA can record up to 30.72MHz of bandwidth split into 24 coarse channels of 1.28MHz each. Figure \[fig\_phase\_vs\_freq\_fullband\_examples\] shows phase of calibration solutions in the frequency range 70 - 230MHz computed from Pictor A observations recorded between [10:32 and 10:48 UTC on the 24$^{th}$ March 2020]{}. This figure shows that phase as a function of frequency is very well modelled by a linear function as unaccounted delays (due to cables or fibres) are the main contributors to the MWA calibration terms. Therefore, fitted parameters of a linear function provide a compact and efficient way of storing phase of calibration solutions, which is also robust against any ripples caused by reflections in cables (e.g. Tile051 in Fig. \[fig\_phase\_vs\_freq\_fullband\_examples\] or Fig. \[fig\_fit\_example\]) or inaccuracy of the sky model used in the calibration process. This way four [double precision]{} values (two for each polarisation) are preserved for each tile[^7] instead of two times the number of fine channels (typically 768). Amplitude, on the other hand, can have complicated frequency structure related to the MWA tile bandpasses. However, locally (in a sufficiently narrow frequency band) it can also be approximated by a linear function; and the natural choice of the narrow band for the linear fit is the 1.28MHz MWA coarse channel. Therefore the database was designed to store parameters of low order (first order) polynomials fitted to amplitudes and phases of calibration solutions as a function of frequency. The original calibration solutions are stored on a hard drive and are not inserted into the database in order to keep the database compact. We envisage that this approach will likely continue in the future. The calibration database is a part of the MWA M&C schema and consists of the following three tables (Fig. \[fig\_caldb\_structure\]):
{width="\textwidth"}
- **<span style="font-variant:small-caps;">calibration\_fits</span>** : provides versioning of calibration solutions stored in the database. It enables uploading newer (possibly better) calibration solutions without the necessity of removing the older versions from the database. The table contains its own unique identifier (<span style="font-variant:small-caps;">fitid</span> field), a reference to observation identifier (<span style="font-variant:small-caps;">obsid</span> field); and a timestamp of calibration solution (<span style="font-variant:small-caps;">fit\_time</span> field).
- **<span style="font-variant:small-caps;">calibration\_solutions</span>** : each record of this table contains information about calibration solutions for both polarisations (X and Y) of a single MWA antenna. Besides references to <span style="font-variant:small-caps;">calibration\_fits</span> record (<span style="font-variant:small-caps;">fitid</span> field) and the observation identification field (<span style="font-variant:small-caps;">obsid</span>), it contains calibration fields for both polarisations with names differing by prefix in field names (x\_ or y\_). Slope (fields x\_delay\_m and y\_delay\_m) and intercept (fields x\_intercept and y\_intercept) fitted to the phase of calibration solutions are used to describe the phase of the calibration solution over the entire 30.72MHz of the MWA’s instantaneous bandwidth, which can be either continuous or non-continuous (the latter is commonly called “picket fence” mode). The fitted slopes are converted to length units using speed of light in vacuum and these values are stored in the database. First-order polynomial fits (thus two database fields) are sufficient to accurately fit the phase over the full observing band provided an accurate sky model is used in the calibration process (this will be described in Section \[subsec:calsols\_of\_archive\_data\]), which was verified during development and testing. Amplitudes of calibration solutions were also fitted with a linear function, but in this case the fit was performed over every 1.28MHz coarse channel. Therefore, they are stored as two arrays (for X and Y polarisations) of real values (typically 24, but the arrays are of variable size). This table also contains quality flags, which are real values in \[0,1\] range (<span style="font-variant:small-caps;">x\_phase\_fit\_quality</span> and <span style="font-variant:small-caps;">x\_gains\_fit\_quality</span> for X polarisation). These flags are calculated as ratios of the number of “good” channels, where the difference between the original value of a calibration solution (either phase or amplitude) and the fitted curve is smaller than 5 standard deviations ($5\sigma$), to the number of all channels. Ratio values above 0.6 are considered good quality calibration solutions.
- **<span style="font-variant:small-caps;">calsolution\_request</span>** : enables requests for missing calibration solutions. If the user requests calibrated data which do not have corresponding calibration solutions for the same frequency channels within 12hours in the calibration database, a new request record is inserted into the table <span style="font-variant:small-caps;">calsolution\_request</span> (if it is not already present there). Then an automatic script finds all the new records in this table, identifies corresponding calibration observations in the main MWA database, calibrates them, and uploads resulting calibration solutions into the calibration database. If the appropriate calibration observations cannot be found or the calibration procedure fails, the error message is stored in the field <span style="font-variant:small-caps;">error</span> and can be returned to the end user.
PRESENT STATUS OF THE DATABASE {#subsec:status_of_the_db}
------------------------------
Currently (as of March 2020) the database contains calibration solutions from around 11,200 calibration observations, which provides, on average, five calibration solutions per day; one for each of the primary MWA frequency bands (at centre frequencies of 88.32, 119.04, 154.88, 185.6, and 216.32MHz). The database grows every day as new calibrator observations are collected and the near-real time pipeline calibrates them and inserts calibration solutions into the database.
In order to populate the database with the historic and new calibration solutions, we developed a dedicated reduction pipeline, described in Section \[sec:automatic\_pipelines\].
AUTOMATIC CALIBRATION PIPELINES {#sec:automatic_pipelines}
===============================
Originally the pipeline used <span style="font-variant:small-caps;">CASA</span> software to calibrate calibrator observations in near-real time and create control images of the calibrator observations. In order to calibrate many archival calibrator observations, we developed a new pipeline using software more suited to the MWA observations. We are upgrading the current contents of the database with calibration solutions from the new pipeline in order to create a uniform database of calibration solutions resulting from the same data reduction pipeline, software and sky model. Both pipelines use the MWA ASVO interface to download uncalibrated <span style="font-variant:small-caps;">CASA</span> measurement sets of calibrator observations, which are produced on the MWA ASVO severs. As described in Section \[subsec:applying\_solutions\], the <span style="font-variant:small-caps;">cotter</span> program is used in the conversion process, which implies that RFI flagging is also applied at this stage.
Near real-time calibration of new calibration observations {#subsec:real_time_calibration}
----------------------------------------------------------
The CASA-based pipeline has been used to reduce newly collected MWA calibrator observations. Every day the MWA observes a calibrator source shortly after sunset and before sunrise at five standard frequency bands (at centre frequencies of 88.32, 119.04, 154.88, 185.6, and 216.32MHz) and in the so-called “picket fence” mode with 24 coarse channels spread regularly over the frequency range $78 - 240$MHz. The calibrator script continuously runs on one of the MWA servers, checks the MWA schedule database for new calibrator observations and whenever it detects that new calibrator observations were collected they are automatically downloaded, calibrated and control images of the calibrator field are formed at selected frequencies. Presently the observations are downloaded from [the PSC and processed]{} on a server at Curtin Institute of Radio Astronomy (CIRA), which introduces additional delay. In the future, this processing will be relocated to the MRO, and the pipeline will use “raw” visibility files as they are produced by the MWA correlator. If the quality of the resulting calibration solutions satisfies minimum requirements they are uploaded to the calibration database. [The requirements]{} for the new calibration solutions to be loaded to the database are the following: (i) they must be better than the ones already in the database (if there are any for this <span style="font-variant:small-caps;">obsid</span>) and (ii) more than half of antennas have acceptable calibration solutions, where “acceptable” means that more than 60% of the channels have a phase fit within $5\sigma$ from the data (this criteria is to avoid storing calibration solutions of very low quality). These near-real time calibration solutions are used for monitoring of the interferometric performance and enable us to examine the long term stability of the MWA (Sec. \[sec:applications\]).
Calibration solutions of archived data {#subsec:calsols_of_archive_data}
--------------------------------------
The calibration database has been populated with calibration solutions starting from the beginning of 2013. In order to achieve this we created a list of all calibrator observations and submitted them for processing by the calibration reduction pipeline <span style="font-variant:small-caps;">Heracles</span>[^8]. The pipeline is using <span style="font-variant:small-caps;">calibrate</span> software [@offringa-2016] upgraded with the newest 2016 MWA beam model [@beam2016] and sky model generated by PUMA[^9] [@puma2018]. It creates binary files with calibration solutions and control images of the field using the <span style="font-variant:small-caps;">WSCLEAN</span>[^10] program [@OffMcK14].
### HERACLES pipeline {#subsec:heracles}
Our initial attempt for calibration of the MWA archive used a single virtual machine on the cloud-based system “Nimbus” hosted by the PSC. However, our task quickly proved to be insufficient for the amount computing resources afforded by this system. For this reason, <span style="font-variant:small-caps;">heracles</span> was converted into a generalised and distributed system, which could be run on any free resources available (such as unused desktop computers in CIRA).
<span style="font-variant:small-caps;">heracles</span> is primarily utilised by a single executable which has two modes of operation: server and client. The server mode is primarily concerned with which observations need to be calibrated, based on a <span style="font-variant:small-caps;">SQLite</span> database. This database tracks which observations have not yet been calibrated, which have been calibrated, and which have failed. The <span style="font-variant:small-caps;">heracles</span> server must also coordinate with the state of any MWA ASVO data downloads (available, in progress, not available, etc.). So as to not flood the MWA ASVO with download requests, the <span style="font-variant:small-caps;">heracles</span> server uses a run-time setting to prepare a certain number of observations for download as the clients progress.
Once connected to a server, the operation of a <span style="font-variant:small-caps;">heracles</span> client follows a simple loop:
1. Request an observation to calibrate. If an observation is ready, move to step (ii), otherwise, wait for one minute before querying the server again;
2. Download the observation;
3. Operate upon the data with an executable (set at run-time, typically a <span style="font-variant:small-caps;">bash</span> script);
4. If the result of the executable was a success, the calibration solutions and any other useful products are transmitted to the server. Return to step (i); and
5. If any failure occurs in the loop, it is also reported to the server, before returning to step (i)
As the computational load of the server is negligible, clients may also be run on the same computer as the server.
{width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"} {width="\columnwidth"}
The mode of operation of <span style="font-variant:small-caps;">heracles</span> clients allows users to enable or disable clients dynamically, which allows otherwise unused computing resources to be utilised, and proved to be an effective, efficient solution for calibrating a large volume of data. Within a few months, we were able to download, calibrate, and image observations and insert solutions into the calibration database from nearly six years of MWA operations.
### CASA pipeline {#subsec:casa}
Originally, the <span style="font-variant:small-caps;">CASA</span>-based pipeline was used to calibrate evening and morning calibration scans and store the calibration solutions in the database. This pipeline used VLA Low-Frequency Sky Survey Redux (VLSSr; @2014MNRAS.440..327L) images of calibrator sources (such as Hydra A, 3C444, Hercules A and Pictor A) to derive calibration solutions. Since the creation of the new <span style="font-variant:small-caps;">heracles</span> pipeline, the <span style="font-variant:small-caps;">CASA</span> pipeline will be retired and the calibration solutions in the database are being superseded with the results from the new pipeline.
Uploading calibration solutions to the database {#subsec:uploading}
-----------------------------------------------
Phase and amplitude of calibration solutions resulting from the reduction pipeline are fitted with the first order polynomial as a function of frequency. Figure \[fig\_phase\_vs\_freq\_fullband\_examples\] shows that phase of calibration solutions is a linear function of frequency over a very wide band. The lowest and highest four 40kHz fine channels (160kHz) in each coarse [channel]{} as well as the fine channels flagged due to RFI (during the conversion process) are excluded from the fitting.
First, the phase of calibration solutions over the 30.72MHz band is “unwrapped”; phase values are not limited to $[-180,+180]$ degrees, but can range from minus infinity to plus infinity. Then the phase is fitted with a linear function over the entire observing band (30.72MHz in continuous observations) resulting in two fit parameters: slope and intercept (right column in Fig. \[fig\_fit\_example\]). These are sufficient to accurately describe the phase of calibration solutions as a function of frequency, provided that the sky model used in the calibration process is complete (this was verified in the development and testing stage). The fitted slope is converted to a corresponding time delay ($\Delta t$) and eventually the length $c \Delta t$ (in metres), where $c$ is speed of light in vacuum, is saved to the calibration database.
The amplitude of calibration solutions is also fitted with a first order polynomial and in this case the fit is performed over a single 1.28MHz coarse channel, resulting in different slopes and intercepts for each of the 24 coarse channels (left column in Fig. \[fig\_fit\_example\]). It was verified that linear fit is the optimal polynomial order to fit amplitudes over an MWA coarse channel as a parabola had only slightly lower $\chi^2$ values and nearly two times higher Bayesian Information Criterion value (BIC; @1978AnSta...6..461S), which proved that the linear fit is a more appropriate representation of data than the parabola.
If the fit satisfies quality requirements the resulting fitted parameters are stored in the database. The current quality requirement is that the ratio between number of good quality channels to all the channels in the calibrated observation is above 0.6 (Figure \[fig\_quality\_distribtion\]).
![Distribution of quality flag for calibration solutions in the MWA ASVO database for X and Y polarisations. Quality flag does not exceed $\approx$0.88 because fine channels at the edges of coarse channels are always flagged. The dual peak structure is due to the fact that calibration solutions in coarse channel 121 (154.88MHz), which is the optimal MWA operating frequency, are noticeably better than at other frequencies and they generate a peak at around 0.85. There are 419 observations with flag quality exceeding 0.75 for all antennas (in both polarisations) and these are observations of : 3C444 (224), Pictor A (59), Centaurus A (45), Hydra A (29), PKS2356-61 (27) and a drift scan field (11), where the numbers in brackets are the number of observations for the particular calibrator. This indicates that majority of best quality calibration solutions comes from observations of 3C444. The main peak at around 0.73 is due to good quality calibration solutions at other standard frequencies (88.32, 119.04, 185.6 and 216.32MHz).[]{data-label="fig_quality_distribtion"}](Figures/quality_flag_distribution.png){width="\columnwidth"}
Accessing calibration solutions in the database {#subsec:getting_solutions}
-----------------------------------------------
The calibration solutions in the database can be accessed via a web service with a standard <span style="font-variant:small-caps;">wget</span> command[^11]. The request is executed on an MWA server and if the appropriate (the same frequency band and within 24hours from the target observation) calibration solution exists in the database it is returned to the user in the same binary file format (“.bin” file) as produced by the calibration procedure developed by @2016MNRAS.458.1057O. If there is no suitable calibration solution an error message stored in a text file is returned to the end user.
Application of calibration solutions to data downloaded from the MWA ASVO {#subsec:applying_solutions}
-------------------------------------------------------------------------
The MWA ASVO website and API[^12] allow users to submit “conversion” jobs which, when run, retrieve the observation, pre-process the data, converting the raw MWA correlator visibility format into a standard CASA or UV fits format, and then make the data product available for download. The conversion/pre-processing steps expose the options available in the <span style="font-variant:small-caps;">cotter</span> pre-processing pipeline [@2015PASA...32....8O]. The MWA ASVO calibration option utilises a recently-added feature of <span style="font-variant:small-caps;">cotter</span>, allowing calibration solutions retrieved from the calibration database to be applied to the data before any data averaging takes place.
In a typical conversion job with the calibration option set, the requested observation is staged from the Pawsey Long Term Archive (LTA). The LTA has a Hierarchical Storage Management (HSM) system consisting of several different tiers of storage, ranging from 1.5 PB of spinning disk to an allocation of 40 PB of magnetic tape. Once the observation data are available on the disk cache they are copied to a scratch area. A web service call is made to retrieve the metafits file that contains much of the metadata associated with the requested observation - this is also stored with the observation files in the scratch area.
The calibration web service is called by the MWA ASVO to retrieve the best calibration solution for this observation (Sec. \[subsec:getting\_solutions\]). If the solution is found, then the calibration solution binary file is retrieved and stored with the rest of the observation files in the scratch area. If no calibration solution is suitable, then the job fails and a request record is added to the calibration database to produce a solution for this observation. The user is informed to try again once this is complete (usually within 24 - 48 hours).
With all of the files now available, the server then executes <span style="font-variant:small-caps;">cotter</span>, with the “–full-apply” command line argument, which applies the provided calibration solution to every integration before averaging (if requested). There is also another new <span style="font-variant:small-caps;">cotter</span> option “–apply” which applies provided calibration solutions after averaging integrations over a requested interval. Once <span style="font-variant:small-caps;">cotter</span> has produced the output data in a standard radio-astronomy data format, a download <span style="font-variant:small-caps;">url</span> is provided to the user via the website or API so the data can be retrieved. During the conversion process RFI flags (either pre-computed by <span style="font-variant:small-caps;">aoflagger</span> or calculated by <span style="font-variant:small-caps;">cotter</span>) are also applied to the data. Hence, the resulting data sets do not require any further pre-processing and initial sky images can be formed using standard radio astronomy software tools, such as for example <span style="font-variant:small-caps;">WSCLEAN</span>, <span style="font-variant:small-caps;">CASA</span> or <span style="font-variant:small-caps;">MIRIAD</span> [@1995ASPC...77..433S]. These initial images can be used in self-calibration procedure in order to improve calibration solutions and/or further processing steps, such as primary beam[^13] or ionospheric corrections[^14], may be applied depending on the requirements of the specific science case.
OTHER APPLICATIONS OF THE CALIBRATION PIPELINES AND DATABASE {#sec:applications}
============================================================
Besides the main application of the CALDB, which is to enable downloading of calibrated data in standard astronomy data formats via the MWA ASVO interface, there are several other benefits of having a complete database of calibration solutions, which will be described in this section.
Monitoring performance and stability of the MWA telescope {#subsec:monitoring_performance}
---------------------------------------------------------
{width="\textwidth"}
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The near real-time pipeline reduces daily calibrator observations, fits their phases and amplitudes with first order polynomials (Fig. \[fig\_fit\_example\]) and inserts the resulting cable delays and intercepts into the CALDB. These fitted cable delays can be plotted as a function of time to monitor the long-term stability of the instrument. If the system is stable, the slope should be approximately constant over long periods of time (timescales of weeks or even months). Figures \[fig\_mwa\_month\_stability\_extended\] and \[fig\_mwa\_month\_stability\_compact\] show fitted delays (in nanoseconds) as a function of time for selected sixteen MWA tiles in the extended and compact configurations respectively. It can be seen that the instrument remains very stable over many weeks. A compilation of such plots for all tiles is shown in Figures \[fig\_mean\_and\_stddev\_201903\_xy\] and \[fig\_mean\_and\_stddev\_201912\_xy\] where standard deviation of fitted delays is plotted against the antenna index enabling aggregation of the system stability in a single plot.
Routine monitoring of these plots enabled the identification of problems which can remain undetected in real time plots showing power spectra of all the MWA tiles. In particular it enables the monitoring of clock signals connected to the MWA receivers and in a few cases it identified the “drift” of a receiver clock due to a failure at the initialisation process, which was fixed by rebooting the receiver. It was also noticed that the phases of the calibration solutions can abruptly change when an MWA receiver is power cycled and the clock latches with an accuracy of 10ns resulting in a step-like change of slope (corresponding to less then 3metres of length using speed of light in vacuum). Since this is not a large delay, with insignificant impact on data quality, cable delays below 13ns (4metres) typically remain un-corrected, which results in less than 145degrees of phase difference over the $30.72$MHz band. However, if the delay exceeds 13ns the cable length in the instrument setup database is updated based on the fitted value. Usually, after re-configuration between the compact and extended configurations, several tiles need cable length adjustments in the instrument description database in order to avoid large, uncorrected cable delays (fast “phase wraps”) that, if uncorrected, reduce the MWA sensitivity. The calibration system also helped to identify situations when coaxial cables from two tiles were accidentally swapped at a receiver input during a re-configuration between the compact and extended MWA configurations causing large delays (due to cable length from a different tile being used to correct the phase).
The MWA instantaneous observing bandwidth is $30.72$MHz, which is typically placed between 50 and 350 MHz. Thus, we could not perform the fit of a straight line over the full band (starting from zero frequency). Moreover, the combination of sky and beam models used in the calibration are usually not a perfect representation of the sky and instrument. Therefore, we allowed the intercept to be a free parameter of the fit. We verified that the fitted values of the intercept are also very stable over time (excluding times when receivers are rebooted) and they are often close to zero or a multipliety of 360. Hence, with the future improvements in the sky model based on the recent extensions of the GaLactic Extragalactic All-sky MWA (GLEAM) catalogue [@2019PASA...36...47H] we will consider constraining the intercept value to be either zero or a multipliety of 360.
Providing calibration solutions to the new MWA correlator {#subsec:calsol_for_new_correlator}
---------------------------------------------------------
The near real-time pipeline will be used to provide calibration solutions for the new MWA “fringe-stopping” correlator, which is currently in development (Morrison et al. in preparation). The MWA telescope is very stable (Section \[subsec:monitoring\_performance\]) and hence one or two sets of calibrations per 24hour interval should be sufficient. However, if it turns out to be insufficiently accurate, the calibration solutions for the new correlator will be updated more often.
A dedicated calibration server will be deployed at the MRO which will enable immediate direct access to visibility files generated by the MWA correlator. This will significantly speed-up the calibration process by eliminating time required to transfer data from the MRO to the PSC archive making the pipeline a truly real-time one.
Monitoring of calibrator field images for transients and ionospheric quality {#subsec:monitoring_transients}
----------------------------------------------------------------------------
For the last three years of the MWA operation the near-real-time calibration pipeline produced control images of the standard MWA calibrators: Pictor A (537 images); Centaurus A (342 images); Hydra A (323 images); Hercules A (197 images); and 3C444 (214 images). There are even more archival images (before 2016) reduced when populating the MWA ASVO database with the archival data, which opens a possibility of radio-transient searches in these fields over a long time baseline similar to those performed by @2011ApJ...728L..14B with the VLA. Roughly - of these images are at the MWA optimal frequency of 154.88MHz. We have executed the <span style="font-variant:small-caps;">Aegean</span> source finder[^15] on these images in order to find transient candidates and catalogue sources to a PostgreSQL database. Analysis is on-going. We are also planning to extend the existing near real-time pipeline and look for transient candidates on a daily basis. The results of these searches will be reported in a separate publication. Finally, such a database populated soon after the calibrator field data are collected provides an excellent opportunity to calculate the mean offset of the sources from their nominal positions in the GLEAM catalogue [@gleam_nhw; @2015PASA...32...25W] or other catalogues and provide early information of the given night’s data quality. However, in such a case the database should be populated more densely with at least one observation every hour (or more if possible) as the ionosphere can change on timescales of hours during the night.
SUMMARY {#sec:summary}
=======
The MWA ASVO calibration component opens a new avenue for researchers worldwide to download calibrated MWA data, create sky images using standard radio-astronomy software packages and analyse these images for multiple purposes. The development of the calibration component of the MWA ASVO interface is a very important contribution to the astronomical community in Australia and beyond, providing access to the MWA data archive to every researcher without requiring a deep knowledge of the instrument. Using the recent sky models obtained from the MWA data [@2019PASA...36...47H; @gleam_nhw] it will be possible to further improve calibration solutions in the database and consequently improve the quality of the resulting images. We expect that this endeavour will [facilitate greater use of the system by]{} researchers from outside the MWA Collaboration using MWA data.
The development of the calibration database triggered the establishment of an automated data reduction pipeline, which has been used in near real-time for daily monitoring of the quality of the MWA calibration solutions and hence the interferometric performance of the telescope. The pipeline also produces sky images which can be used for monitoring the quality of the ionosphere and looking for transient objects on a daily basis.
Finally, this work has been a starting point to develop a database of calibration solutions for the upcoming low-frequency component of the Square Kilometre Array telescope. Based on the MWA experience, a similar calibration database was developed to store calibration solutions from the SKA-Low prototype stations Aperture Array Verification Systems (AAVS-1 and AAVS-2) and Engineering Development Array 2 (Wayth et. al. in preparation) already deployed at the MRO. This database will be further extended in order to handle more SKA-Low stations as they will soon be built at the MRO.
### People {#people .unnumbered}
We would like to thank the anonymous referee for the prompt review of our manuscript.
### Facilities {#facilities .unnumbered}
This scientific work makes use of the Murchison Radio-astronomy Observatory (MRO), operated by CSIRO. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site.
### Funding {#funding .unnumbered}
Support for the operation of the MWA is provided by the Australian Government (NCRIS), under a contract to Curtin University administered by Astronomy Australia Limited. Development of the MWA ASVO was funded via the Australian Research Data Commons (ARDC), administered by Astronomy Australia Limited. Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013. This work was further supported by resources provided by the Pawsey Supercomputing Centre with funding from the Australian Government and the Government of Western Australia. DLK was supported by NSF grant AST-1816492.
### Software {#software .unnumbered}
We acknowledge the work and support of the developers of the following following Python packages: Astropy [@astropy:2013; @astropy:2018], Numpy [@2011CSE....13b..22V], Scipy [@2020SciPy-NMeth], Matplotlib [@Hunter:2007] and AegeanTools [@2018PASA...35...11H]. We acknowledge developers of the MWA\_Tools library. This research has made use of NASA’s Astrophysics Data System.
[^1]: [email protected]
[^2]: https://www.skatelescope.org/
[^3]: http://www.mwatelescope.org/team/policies
[^4]: https://asvo.mwatelescope.org
[^5]: http://www.mwatelescope.org/telescope/monitor-control
[^6]: https://www.postgresql.org/
[^7]: MWA 16-dipoles antenna units are commonly referred to as “tiles”
[^8]: <https://gitlab.com/chjordan/heracles>
[^9]: <https://github.com/JLBLine/srclists>
[^10]: <https://sourceforge.net/p/wsclean/wiki/Home/>
[^11]: wget http://mro.mwa128t.org/calib/get\_calfile\_for\_obsid? obs\_id=OBSID&zipfile=1&add\_request=1 where <span style="font-variant:small-caps;">OBSID</span> should be replaced with the observation ID of the target or explicit calibrator observation.
[^12]: https://github.com/ICRAR/manta-ray-client
[^13]: https://github.com/MWATelescope/mwa\_pb
[^14]: https://github.com/nhurleywalker/fits\_warp
[^15]: https://github.com/PaulHancock/Aegean
|
---
abstract: 'We generalize our earlier work on the symplectic/Hamiltonian formulation of the dynamics of the Gaussian wave packet to non-Gaussian semiclassical wave packets. We find the symplectic forms and asymptotic expansions of the Hamiltonians associated with these semiclassical wave packets, and obtain Hamiltonian systems governing their dynamics. Numerical experiments demonstrate that the dynamics give a very good approximation to the short-time dynamics of the expectation values computed by a method based on Egorov’s Theorem or the Initial Value Representation.'
address: 'Department of Mathematical Sciences, The University of Texas at Dallas, 800 W Campbell Rd, Richardson, TX 75080-3021'
author:
- Tomoki Ohsawa
bibliography:
- 'SympSWPDyn.bib'
title: |
Symplectic Semiclassical Wave Packet Dynamics II:\
Non-Gaussian States
---
=.6in
Introduction
============
Dynamics of Gaussian and Semiclassical Wave Packets
---------------------------------------------------
The Gaussian wave function is one of the most ubiquitous wave functions in quantum mechanics. The most familiar form of Gaussian wave function appears as the ground state of the harmonic oscillator. Gaussians also appear in many forms and play significant roles in quantum dynamics or time-dependent quantum mechanics as well; see, e.g., @Ta2007. One of the most significant results regarding Gaussians in quantum dynamics is that the Gaussian wave packet $$\label{eq:chi_0}
\chi_{0}(q,p,\mathcal{A},\mathcal{B},\phi,\delta; x)
= \exp\braces{ \frac{\rmi}{\hbar}\brackets{ \frac{1}{2}(x - q)^{T}(\mathcal{A} + \rmi\mathcal{B})(x - q) + p \cdot (x - q) + (\phi + \rmi \delta) } }$$ gives an exact solution of the Schrödinger equation if the potential is quadratic and the parameters $(q,p,\mathcal{A},\mathcal{B},\phi,\delta)$ satisfy the following set of ordinary differential equations (see @He1975a [@He1976b; @He1981], @Ha1980 [@Ha1998], and @Li1986): $$\label{eq:Heller}
\begin{array}{c}
\DS
\dot{q} = \frac{p}{m},
\qquad
\dot{p} = -D_{q}V(q),
\medskip\\
\DS
\dot{\mathcal{A}} = -\frac{1}{m}(\mathcal{A}^{2} - \mathcal{B}^{2}) - D^{2}V(q),
\qquad
\dot{\mathcal{B}} = -\frac{1}{m}(\mathcal{A}\mathcal{B} + \mathcal{B}\mathcal{A}),
\medskip\\
\DS
\dot{\phi} = \frac{p^{2}}{2m} - V(q) - \frac{\hbar}{2m} \tr\mathcal{B},
\qquad
\dot{\delta} = \frac{\hbar}{2m} \tr\mathcal{A}.
\end{array}$$
The parameters $(q,p) \in T^{*}\R^{d}$ may be thought of as the position and momentum in the classical sense: In fact the first two equations are nothing but the classical Hamiltonian system and is decoupled from the rest; they also give the expectation values of the position and momentum operators with respect to the Gaussian if it is normalized, i.e., if $\norm{\chi_{0}}=1$ then $\ip{ \chi_{0} }{ \hat{x} \chi_{0} } = q$ and $\ip{ \chi_{0} }{ \hat{p} \chi_{0} } = p$, where $\ip{\,\cdot\,}{\,\cdot\,}$ is the standard inner product on $L^{2}(\R^{d})$, $\hat{x}$ is position operator, i.e., the multiplication by the position vector $x$, and $\hat{p} = -\rmi\hbar\tpd{}{x}$ is the momentum operator. The matrices $(\mathcal{A},\mathcal{B})$ quantify the uncertainties in the position and momentum, and live in the so-called Siegel upper half space [@Si1943] $$\label{eq:Sigma_d}
\Sigma_{d} \defeq
\setdef{ \mathcal{A} + {\rm i}\mathcal{B} \in \Mat_{d}(\mathbb{C}) }{ \mathcal{A}, \mathcal{B} \in \Mat_{d}(\R),\, \mathcal{A}^{T} = \mathcal{A},\, \mathcal{B}^{T} = \mathcal{B},\, \mathcal{B} > 0 },$$ i.e., the set of symmetric (in the real sense) $d \times d$ complex matrices with positive-definite imaginary parts; this guarantees that $\chi_{0}$ is an element in $L^{2}(\R^{d})$. The parameter $\phi \in \mathbb{S}^{1}$ is the phase factor and $\delta \in \R$ controls the norm of $\chi_{0}$ as the square of the norm of $\chi_{0}$ is given by $$\label{eq:N}
\mathcal{N}_{\hbar}(\mathcal{B},\delta) \defeq \norm{\chi_{0}}^{2} = \sqrt{ \frac{(\pi\hbar)^{d}}{\det \mathcal{B}} }\, \exp\parentheses{ -\frac{2\delta}{\hbar} }.$$
@Ha1980 [@Ha1998] came up with an orthonormal basis $\{\chi_{n}\}_{n\in\N_{0}^{d}}$ for $L^{2}(\R^{d})$ whose ground state with $n = 0$ is the normalized version of the Gaussian ; see Section \[ssec:Hagedorn\] below for a brief summary of its construction by ladder operators. It was also shown that each $\chi_{n}$ gives an exact solution of the Schrödinger equation with quadratic potential if the parameters evolve according to . Moreover, even with [*non-quadratic*]{} potentials, @Ha1998 gave, under some technical assumptions, an asymptotic error estimate as $\hbar \to 0$ of the approximations by certain linear combinations of the basis elements—each of which is evolving in time according to —to the solution of the Schrödinger equation.
Previous Work and Motivation
----------------------------
In our previous work [@OhLe2013], we followed @FaLu2006 and @Lu2008 to come up with the symplectic-geometric/Hamiltonian formulation of the dynamics of the Gaussian wave packet as follows: The parameters $(q,p,\mathcal{A},\mathcal{B},\phi,\delta)$ associated with the Gaussian live in the manifold $\mathcal{M} \defeq T^{*}\R^{d} \times \Sigma_{d} \times \mathbb{S}^{1} \times \R$. But then we can induce a natural symplectic structure $\Omega_{\mathcal{M}}^{(0)}$ and a Hamiltonian function $H^{(0)}$ on $\mathcal{M}$ by exploiting the Hamiltonian formulation of the Schrödinger equation (see Section \[ssec:Hamiltonian\_Schroedinger\] below). This results in the Hamiltonian system $\ins{X_{H^{(0)}}} \Omega_{\mathcal{M}}^{(0)} = \d{H^{(0)}}$, and gives almost the same set of equations as —the only difference being that the second equation is replaced by $$\label{eq:pdot-chi_0}
\dot{p} = -D_{q}V^{(0)}_{\hbar}(q,\mathcal{B}),$$ where the potential $V^{(0)}_{\hbar}$ has an $O(\hbar)$ correction term to the classical one: $$V^{(0)}_{\hbar}(q,\mathcal{B}) \defeq
V(q) + \frac{\hbar}{4}\tr\parentheses{ \mathcal{B}^{-1} D^{2}V(q) },$$ and hence the dynamics of $(q,p)$ does not satisfy the classical Hamiltonian system any more. Numerical experiments suggest that the $(q,p)$ dynamics of our system gives a better approximation than to the short time dynamics of the expectation values $$\exval{\hat{x}}(t) \defeq \ip{ \psi(t,\,\cdot\,) }{ \hat{x} \psi(t,\,\cdot\,) }
\quad\text{and}\quad
\exval{\hat{p}}(t) \defeq \ip{ \psi(t,\,\cdot\,) }{ \hat{p} \psi(t,\,\cdot\,) }$$ of the position and momentum with respect to the solution $\psi(t,x)$ to the initial value problem of the Schrödinger equation $$\label{eq:SchroedingerEq-coordinates}
\rmi\hbar\pd{}{t}\psi(t,x) = \hat{H}\psi(t,x)$$ with the Gaussian initial condition $$\psi(0,x) = \chi_{0}(q(0),p(0),\mathcal{A}(0),\mathcal{B}(0),\phi(0),\delta(0); x),$$ where $\hat{H}$ is the standard Schrödinger operator $$\label{eq:SchroedingerOp}
\hat{H} \defeq -\frac{\hbar^{2}}{2m} \Delta + V(x).$$
The main motivation for this work is to extend our approach to non-Gaussian elements of the semiclassical wave packets $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$, i.e., we would like to generalize our work [@OhLe2013] done for $n = 0$ (i.e., the Gaussian ) to those elements with $n \neq 0$. Since the semiclassical wave packets $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$ provide an orthogonal basis for $L^{2}(\R^{d})$, our extension opens the door to new semiclassical approximation methods for the Schrödinger equation, potentially offering improvements on the results obtained by @Ha1980 [@Ha1998].
The main difficulty in extending our approach to the non-Gaussian elements $\chi_{n}$ ($n \neq 0$) is that there is no known explicit formula for $\chi_{n}$ that is valid for any $n$. The difficulty is exacerbated in the multi-dimensional case, i.e., $d > 1$: Unlike the Hermite functions, the multi-dimensional semiclassical wave packets [*cannot*]{} be written as products of the one-dimensional components. In other words, the only practical way to come up with an explicit expression for $\chi_{n}$ for a given $n \in \N_{0}^{d}$ is to apply the associated raising operator $|n|$ times to the Gaussian $\chi_{0}$ for the given dimension $d$. This makes those calculations involving $\chi_{n}$ for an arbitrary multi-index $n \in \N_{0}^{d}$ particularly cumbersome. The calculations of the symplectic form $\Omega_{\mathcal{M}}^{(0)}$ and Hamiltonian $H^{(0)}$ performed in our previous work [@OhLe2013] were fairly straightforward because $\chi_{0}$ is a Gaussian. However, mimicking the same calculations for an arbitrary $n\in\N_{0}^{d}$ is not feasible because of the above difficulty in obtaining an explicit expression for $\chi_{n}$ with an arbitrary $n \in \N_{0}^{d}$.
Main Results
------------
We focus on those semiclassical wave packets $\{\chi_{n}\}_{n\in\N_{0}^{d}}$ that are parametrized by the same parameters $(q,p,\mathcal{A},\mathcal{B},\phi,\delta)$ as the Gaussian $\chi_{0}$, and circumvent the above difficulty by proving those recurrence relations that hold between the symplectic forms and Hamiltonians associated with $\{\chi_{n}\}_{n\in\N_{0}^{d}}$. Then the symplectic form $\Omega_{\mathcal{M}}^{(n)}$ and Hamiltonian $H^{(n)}$ for an arbitrary $n\in\N_{0}^{d}$ follow by induction; see Propositions \[prop:Omega\_M\] and \[prop:H\_M\]. As a result, we can formulate the Hamiltonian system associated with the semiclassical wave packet $\chi_{n}$ for an arbitrary $n \in \N_{0}^{d}$; see Theorem \[thm:Hamiltonian\_system\].
We also extend our previous results [@OhLe2013] on the symplectic reduction of the dynamics of the Gaussian wave packet $\chi_{0}$ to the dynamics of an arbitrary semiclassical wave packets $\chi_{n}$ with $n \in \N_{0}^{d}$. This results in a Hamiltonian system on the reduced symplectic manifold $T^{*}\R^{d} \times \Sigma_{d}$; see Theorem \[thm:ReducedSemiclassicalSystem\]. The reduced symplectic structure takes a much simpler and suggestive form that carries an $O(\hbar)$ correction term to the classical one, and the same goes with the reduced Hamiltonian; that is, it reveals the quantum correction as an $O(\hbar)$ perturbation to the classical Hamiltonian system.
Numerical experiments with a simple one-dimensional test case demonstrate that the these Hamiltonian systems provide very good approximations to the short-time dynamics of those expectation values $\exval{\hat{x}}(t)$ and $\exval{\hat{p}}(t)$ computed by Egorov’s Theorem [@Eg1969; @CoRo2012; @KeLaOh2016] or the Initial Value Representation (IVR) method [@Mi1970; @Mi1974b; @WaSuMi1998; @Mi2001] with $\chi_{n}$ as the initial wave functions for several $n$. The IVR is a popular method for computing such expectation values and is shown to have $O(\hbar^{2})$ asymptotic accuracy by Egorov’s Theorem.
Outline
-------
We start with a brief review of the semiclassical wave packets of @Ha1980 [@Ha1998] in Section \[sec:Hagedorn\_wave\_packets\]. We present two different parametrizations of the wave packets: One is that used by @Ha1998 and the other with the same set of parameters as ; we use the latter throughout the paper as in our earlier work [@OhLe2013]. In Section \[sec:Symplectic\_Structures\], we find the symplectic forms $\{ \Omega_{\mathcal{M}}^{(n)} \}_{n\in\N_{0}^{d}}$ associated with the semiclassical wave packets $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$. In Section \[sec:Hamiltonian\_Dynamics\_of\_SWP\], we find the semiclassical Hamiltonians $\{ H^{(n)} \}_{n\in\N_{0}^{d}}$ and the Hamiltonian systems associated with the semiclassical wave packets $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$. In Section \[sec:Symplectic\_Reduction\], we perform the symplectic reduction mentioned above to simplify the formulations. Finally, in Section \[sec:Numerical\_Results\], we show numerical results of a simple test case comparing our solutions with the classical solution and those obtained by an Egorov/IVR-type method.
The Semiclassical Wave Packets {#sec:Hagedorn_wave_packets}
==============================
Two Parametrizations and the Siegel Upper Half Space
----------------------------------------------------
@Ha1980 [@Ha1981; @Hagedorn1985; @Ha1998] uses a different parametrization for the elements $\mathcal{C} = \mathcal{A} + \rmi\mathcal{B}$ in the Siegel upper half space $\Sigma_{d}$ defined in . Namely the matrix $\mathcal{C}$ in the Gaussian wave packet is replaced by $P Q^{-1}$ to have $$\label{eq:chi_0-Hagedorn}
\chi_{0}(q,p,Q,P,\phi,\delta; x) = \exp\braces{ \frac{\rmi}{\hbar}\brackets{ \frac{1}{2}(x - q)^{T}P Q^{-1}(x - q) + p \cdot (x - q) + (\phi + \rmi\delta) } },$$ where $Q, P \in \Mat_{d}(\C)$, i.e., $d \times d$ complex matrices, that satisfy $$\label{eq:Q_P-Hagedorn}
Q^{T}P - P^{T}Q = 0,
\qquad
Q^{*}P - P^{*}Q = 2\rmi I_{d},$$ where $I_{d}$ is the $d \times d$ identity matrix. It is pointed out by @Lu2008 [Section V.1] that this is a parametrization of elements in the symplectic group $\Sp(2d,\R)$ in the following way: $$\Sp(2d,\R)
= \setdef{
\begin{bmatrix}
{\operatorname{Re}}Q & {\operatorname{Im}}Q \\
{\operatorname{Re}}P & {\operatorname{Im}}P
\end{bmatrix} \in \Mat_{2d}(\R)
}
{
\begin{array}{c}
Q, P \in \Mat_{d}(\C),\ Q^{T}P - P^{T}Q = 0, \smallskip\\
Q^{*}P - P^{*}Q = 2\rmi I_{d}
\end{array}
}.$$ In fact, one can show that if $(Q,P)$ satisfies then $Q$ is invertible and also $P Q^{-1} \in \Sigma_{d}$; see, e.g., [@Lu2008 Lemma V.1.1 on p. 124]. However, for a given $\mathcal{A} + \rmi\mathcal{B} \in \Sigma_{d}$, the corresponding $(Q, P)$ satisfying and $P Q^{-1} = \mathcal{A} + \rmi\mathcal{B}$ is not unique: For example, one finds that, by setting $$\begin{bmatrix}
{\operatorname{Re}}Q_{0} & {\operatorname{Im}}Q_{0} \\
{\operatorname{Re}}P_{0} & {\operatorname{Im}}P_{0}
\end{bmatrix}
=
\begin{bmatrix}
\mathcal{B}^{-1/2} & 0 \\
\mathcal{A}\mathcal{B}^{-1/2} & \mathcal{B}^{1/2}
\end{bmatrix},$$ one sees that $(Q_{0},P_{0})$ satisfies as well as $P_{0}Q_{0}^{-1} = \mathcal{A} + \rmi\mathcal{B}$. However, setting $$\begin{bmatrix}
{\operatorname{Re}}Q & {\operatorname{Im}}Q \\
{\operatorname{Re}}P & {\operatorname{Im}}P
\end{bmatrix}
=
\begin{bmatrix}
\mathcal{B}^{-1/2} & 0 \\
\mathcal{A}\mathcal{B}^{-1/2} & \mathcal{B}^{1/2}
\end{bmatrix}
\begin{bmatrix}
U & V \\
-V & U
\end{bmatrix}
=
\begin{bmatrix}
\mathcal{B}^{-1/2} U & \mathcal{B}^{-1/2} V \\
\mathcal{A}\mathcal{B}^{-1/2} U - \mathcal{B}^{1/2} V & \mathcal{A}\mathcal{B}^{-1/2} V + \mathcal{B}^{1/2} U
\end{bmatrix}$$ for any $U + {\rm i}V \in \U(d)$ (the unitary group of degree $d$) would do as well: $(Q,P)$ again satisfies as well as $P Q^{-1} = \mathcal{A} + \rmi\mathcal{B}$. Therefore one has $$\label{eq:Q_P-A_B}
Q = \mathcal{B}^{-1/2} \mathcal{U},
\qquad
P = (\mathcal{A} + \rmi\mathcal{B}) \mathcal{B}^{-1/2} \mathcal{U},$$ where $\mathcal{U} \defeq U + {\rm i}V \in \U(d)$. This is because $\Sigma_{d}$ is actually the homogeneous space $\Sp(2d,\R)/\U(d)$; see, e.g., @Si1943, @Fo1989 [Section 4.5], @McSa1999 [Exercise 2.28 on p. 48], and @Oh2015c for details.
The Hagedorn Wave Packets {#ssec:Hagedorn}
-------------------------
Upon normalizing and getting rid of the phase factor in , we have the ground state of the Hagedorn wave packets: $$\varphi_{0}(q,p,Q,P; x) = \frac{(\det Q)^{-1/2}}{(\pi\hbar)^{d/4}} \exp\braces{ \frac{{\rm i}}{\hbar}\brackets{ \frac{1}{2}(x - q)^{T}P Q^{-1}(x - q) + p \cdot (x - q) } }.$$ @Ha1998[^1] came up with the ladder operators $$\begin{aligned}
\mathscr{L}(q,p,Q,P) &= -\frac{\rmi}{\sqrt{2\hbar}} \brackets{ P^{T}(\hat{x} - q) - Q^{T}(\hat{p} - p) }, \\
\mathscr{L}^{*}(q,p,Q,P) &= \frac{\rmi}{\sqrt{2\hbar}} \brackets{ P^{*}(\hat{x} - q) - Q^{*}(\hat{p} - p) },\end{aligned}$$ that satisfy the same relationships that are satisfied by the ladder operators of the Hermite functions, i.e., $$\begin{array}{c}
[\mathscr{L}_{j}(q,p,Q,P), \mathscr{L}_{k}(q,p,Q,P)] = 0,
\medskip\\
{[\mathscr{L}_{j}^{*}(q,p,Q,P), \mathscr{L}^{*}_{k}(q,p,Q,P)] = 0},
\qquad
[\mathscr{L}_{j}(q,p,Q,P), \mathscr{L}^{*}_{k}(q,p,Q,P)] = \delta_{jk}.
\end{array}$$ Then these operators are used to define an orthonormal basis $\{ \varphi_{n}(q,p,Q,P;\,\cdot\,) \}_{n \in \N_{0}^{d}}$ for $L^{2}(\R^{d})$ recursively by applying the raising operator $\mathscr{L}^{*}$ repeatedly, i.e., for any multi-index $n = (n_{1}, \dots, n_{d}) \in \N_{0}^{d}$ and $j \in \{1, \dots, d\}$, $$\varphi_{n + e_{j}}(q,p,Q,P;\,\cdot\,) \defeq \frac{1}{\sqrt{n_{j} + 1}}\,\mathscr{L}^{*}_{j}(q,p,Q,P) \varphi_{n}(q,p,Q,P;\,\cdot\,),$$ where $e_{j} \in \R^{d}$ is the unit vector whose $j$-th entry is 1. One can also show that the lowering operator $\mathscr{L}$ satisfies $$\varphi_{n - e_{j}}(q,p,Q,P;\,\cdot\,) = \frac{1}{\sqrt{n_{j}}}\,\mathscr{L}_{j}(q,p,Q,P) \varphi_{n}(q,p,Q,P;\,\cdot\,).$$
Semiclassical Wave Packets
--------------------------
We would like to use the parametrization $(\mathcal{A},\mathcal{B})$ instead of $(Q,P)$ here. So we may first rewrite the above ladder operators in terms of $(\mathcal{A},\mathcal{B},\mathcal{U})$ instead of $(Q,P)$ using . But then the resulting operators define ladder operators for an arbitrary $\mathcal{U} \in \U(d)$; hence we set $\mathcal{U} = I_{d}$ to have—with an abuse of notation—the ladder operators
\[eq:ladder\_operators\] $$\begin{aligned}
\mathscr{L}(q,p,\mathcal{A},\mathcal{B})
&\defeq -\frac{\rmi}{\sqrt{2\hbar}}\, \mathcal{B}^{-1/2} \brackets{ (\mathcal{A} + \rmi\mathcal{B}) (\hat{x} - q) - (\hat{p} - p) },\\
\mathscr{L}^{*}(q,p,\mathcal{A},\mathcal{B})
&\defeq \frac{\rmi}{\sqrt{2\hbar}}\, \mathcal{B}^{-1/2} \brackets{ (\mathcal{A} - \rmi\mathcal{B})(\hat{x} - q) - (\hat{p} - p) },
\label{eq:raising_operator}
\end{aligned}$$
and generate an orthogonal basis $\{ \chi_{n}(q,p,\mathcal{A},\mathcal{B},\phi,\delta;\,\cdot\,) \}_{n \in \N_{0}^{d}}$ by setting $$\label{eq:chi_n-raised}
\chi_{n + e_{j}}(q,p,\mathcal{A},\mathcal{B},\phi,\delta;\,\cdot\,)
\defeq \frac{1}{\sqrt{n_{j} + 1}}\,\mathscr{L}^{*}_{j}(q,p,\mathcal{A},\mathcal{B}) \chi_{n}(q,p,\mathcal{A},\mathcal{B},\phi,\delta;\,\cdot\,)$$ starting with the ground state (without normalization; hence only orthogonal), whereas the lowering operator works as follows: $$\label{eq:chi_n-lowered}
\chi_{n - e_{j}}(q,p,\mathcal{A},\mathcal{B},\phi,\delta;\,\cdot\,)
= \frac{1}{\sqrt{n_{j}}}\,\mathscr{L}_{j}(q,p,\mathcal{A},\mathcal{B}) \chi_{n}(q,p,\mathcal{A},\mathcal{B},\phi,\delta;\,\cdot\,).$$
Setting $\mathcal{U} = I_{d}$ has the advantage of parametrizing the wave packets by $(\mathcal{A},\mathcal{B})$ as is done for the Gaussian, but results in a slightly less general form of wave packets than Hagedorn’s.
Note that the norms of these wave packets are all the same because, writing $\chi_{n} = \chi_{n}(y;\,\cdot\,)$ and $\mathscr{L}^{*} = \mathscr{L}^{*}(q,p,\mathcal{A},\mathcal{B})$ for brevity, $$\begin{aligned}
\norm{\chi_{n+e_{j}}}^{2}
&= \frac{1}{n_{j}+1} \ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{j} \chi_{n} } \\
&= \frac{1}{n_{j}+1} \ip{ \mathscr{L}_{j} \mathscr{L}^{*}_{j} \chi_{n} }{ \chi_{n} } \\
&= \ip{ \chi_{n} }{ \chi_{n} } \\
&= \norm{ \chi_{n} }^{2},\end{aligned}$$ and hence we have $\norm{ \chi_{n} }^{2} = \norm{ \chi_{0} }^{2} = \mathcal{N}_{\hbar}(\mathcal{B},\delta)$ for any $n\in\N_{0}^{d}$ by induction, where $\mathcal{N}_{\hbar}$ was defined in . As we shall see later in Proposition \[prop:Omega\_M-reduced\], $\mathcal{N}_{\hbar}$ is the Noether conserved quantity corresponding to the inherent phase symmetry of our Hamiltonian dynamics; see also Remark \[rem:N\] below. Therefore, if necessary, one may normalize the orthogonal basis $\{ \chi_{n}(q,p,\mathcal{A},\mathcal{B},\phi,\delta;\,\cdot\,) \}_{n \in \N_{0}^{d}}$ to obtain an orthonormal basis just like the Hagedorn wave packets $\{ \varphi_{n}(q,p,Q,P;\,\cdot\,) \}_{n \in \N_{0}^{d}}$. We will show later in Section \[sec:Symplectic\_Reduction\] that the normalization corresponds to the symplectic reduction by the phase symmetry with respect to the variable $\phi$.
Embeddings of Semiclassical Wave Packets and Symplectic Structures {#sec:Symplectic_Structures}
==================================================================
Let us write $y \defeq (q, p, \mathcal{A}, \mathcal{B}, \phi, \delta) \in \mathcal{M}$ for short. Our goal is to come up with the dynamics $y(t)$ of the parameters so that each wave packet $\chi_{n}(y(t);x)$ approximates the solution of the initial value problem of the Schrödinger equation with the initial condition $\psi(0,x) = \chi_{n}(y(0);x)$. Particularly we would like to obtain a Hamiltonian dynamics of the parameters $y$ that is naturally related to the Hamiltonian/symplectic structure associated with the Schrödinger equation. This amounts to finding the symplectic structure $\Omega_{\mathcal{M}}^{(n)}$ and Hamiltonian $H^{(n)}$ on $\mathcal{M}$ naturally associated with $\chi_{n}$, and results in the Hamiltonian system defined in terms of $\Omega_{\mathcal{M}}^{(n)}$ and $H^{(n)}$. Indeed, one can show that this gives the best approximation in some appropriate sense as we shall see below in Section \[ssec:BestApproximation\].
In our previous work [@OhLe2013] on the dynamics of the Gaussian $\chi_{0}$, we followed the approach by @FaLu2006 and @Lu2008 [Section II.1] and obtained the symplectic structure $\Omega_{\mathcal{M}}^{(0)}$ by regarding the Gaussian wave packet $\chi_{0}(y,\,\cdot\,)$ as the embedding $\iota_{0}\colon\mathcal{M} \hookrightarrow L^{2}(\R^{d})$ defined by $y \mapsto \chi_{0}(y,\,\cdot\,)$; see Section \[ssec:embedding\] below for more details. We would like to generalize the approach to $\chi_{n}$ for an arbitrary $n \in \N_{0}^{d}$.
Hamiltonian Formulation of the Schrödinger Equation {#ssec:Hamiltonian_Schroedinger}
---------------------------------------------------
Let us first briefly review the Hamiltonian formulation of the Schrödinger equation following @MaRa1999 [Section 2.2]. Let $\mathcal{H}$ be a complex Hilbert space—$\mathcal{H} = L^{2}(\R^{d})$ throughout the paper—equipped with a (right-linear) inner product $\ip{\cdot}{\cdot}$. Then $\mathcal{H}$ is a symplectic vector space with the symplectic structure $\Omega$ defined by $$\Omega(\psi_{1}, \psi_{2}) \defeq 2\hbar {\operatorname{Im}}\ip{\psi_{1}}{\psi_{2}}.$$ In fact, defining a one-form $\Theta$ on $\mathcal{H}$ by $$\label{eq:Theta}
\Theta(\psi) = -\hbar {\operatorname{Im}}\ip{\psi}{\mathbf{d}\psi},$$ one obtains $\Omega = -\mathbf{d}\Theta$. Given a Hamiltonian operator $\hat{H}$ on $\mathcal{H}$ (we proceed formally here without specifying the domain of definition of $\hat{H}$), we may write the expectation value of the Hamiltonian $\texval{\hat{H}}\colon \mathcal{H} \to \R$ as $$\texval{\hat{H}}(\psi) \defeq \texval{\psi, \hat{H}\psi}.$$ Now we think of $\texval{\hat{H}}$ as a Hamiltonian [*function*]{} on the symplectic vector space $\mathcal{H}$, and define the corresponding Hamiltonian vector field $X_{\exval{\hat{H}}}$ on $\mathcal{H}$ by the Hamiltonian system $$\label{eq:Schroedinger-HamiltonianSystem}
{\bf i}_{X_{\texval{\hat{H}}}} \Omega = \mathbf{d}\texval{\hat{H}}.$$ Writing the vector field $X_{\exval{\hat{H}}}$ as $X_{\exval{\hat{H}}}(\psi) = (\psi,\dot{\psi}) \in T\mathcal{H} \cong \mathcal{H} \times \mathcal{H}$, one obtains the Schrödinger equation $$\dot{\psi} = -\frac{\rmi}{\hbar}\hat{H} \psi.$$ For $\mathcal{H} = L^{2}(\R^{d})$ with the Schrödinger operator , the above equation gives .
Embeddings defined by Semiclassical Wave Packets {#ssec:embedding}
------------------------------------------------
We would like to exploit the above Hamiltonian approach to the Schrödinger equation in order to formulate Hamiltonian dynamics of the parameters $(q, p, \mathcal{A}, \mathcal{B}, \phi, \delta)$. First note that the parameters $y = (q, p, \mathcal{A}, \mathcal{B}, \phi, \delta)$ live in the space $$\mathcal{M} \defeq T^{*}\R^{d} \times \Sigma_{d} \times \mathbb{S}^{1} \times \R,$$ which is an even-dimensional manifold for any $d \in \N$ because the (real) dimension of $\Sigma_{d}$ is $d(d+1)$ and hence the dimension of $\mathcal{M}$ is $(d+1)(d+2)$. Then we may define a family of embeddings of $\mathcal{M}$ to $\mathcal{H} \defeq L^{2}(\R^{d})$ by $$\label{eq:iota_n}
\iota_{n}\colon \mathcal{M} \hookrightarrow \mathcal{H};
\quad
\iota_{n}(y) = \chi_{n}(y;\,\cdot\,)$$ for any $n \in \N_{0}^{d}$.
Can we naturally induce a symplectic structure on $\mathcal{M}$ from the symplectic structure $\Omega$ on $\mathcal{H}$? In our previous work [@OhLe2013 Proposition 2.1], we reformulated the work of @Lu2008 [Section II.1] and showed the following: Let $\iota\colon \mathcal{M} \hookrightarrow \mathcal{H}$ be an embedding of a manifold $\mathcal{M}$ in a complex Hilbert space $\mathcal{H}$ and suppose that $\mathcal{M}$ is equipped with an almost complex structure $J_{y}: T_{y}\mathcal{M} \to T_{y}\mathcal{M}$ that is compatible with the multiplication by the imaginary unit $\rmi$ in $\mathcal{H}$, i.e., $$\label{eq:iota-compatibility}
T_{y}\iota \circ J_{y} = \rmi\cdot T_{y}\iota$$ for any $y \in \mathcal{M}$; then $\mathcal{M}$ is a symplectic manifold with symplectic form defined by the pull-back $\Omega_{\mathcal{M}} \defeq \iota^{*}\Omega$. In [@OhLe2013], we worked out the Gaussian case, i.e., $\iota = \iota_{0}$ explicitly: We found that $$\begin{gathered}
\label{eq:J}
J_{y}\parentheses{ \dot{q}, \dot{p}, \dot{\mathcal{A}}, \dot{\mathcal{B}}, \dot{\phi}, \dot{\delta} }
\\
= \parentheses{
\mathcal{B}^{-1}(\mathcal{A}\dot{q} - \dot{p}),\,
(\mathcal{A}\mathcal{B}^{-1}\mathcal{A} + \mathcal{B})\dot{q} - \mathcal{A}\mathcal{B}^{-1}\dot{p},\,
-\dot{\mathcal{B}},\,
\dot{\mathcal{A}},\,
p^{T}\mathcal{B}^{-1}(\mathcal{A}\dot{q} - \dot{p}) - \dot{\delta},\,
-p \cdot \dot{q} + \dot{\phi}
},\end{gathered}$$ is an almost complex structure that satisfies $T_{y}\iota_{0} \circ J_{y} = \rmi\cdot T_{y}\iota_{0}$, and found the pull-back $\Theta_{\mathcal{M}}^{(0)} \defeq \iota_{0}^{*} \Theta$ of the canonical one-form $\Theta$ in . Setting $\Omega_{\mathcal{M}}^{(0)} \defeq -\d\Theta_{\mathcal{M}}^{(0)}$ gives a symplectic form on $\mathcal{M}$.
Hamiltonian Dynamics as the Best Approximation {#ssec:BestApproximation}
----------------------------------------------
Given an embedding $\iota\colon \mathcal{M} \hookrightarrow \mathcal{H}$ with $\iota(y) = \chi(y;\,\cdot\,)$ satisfying , one can also define a Hamiltonian function $H$ on $\mathcal{M}$ via the above embedding as $H \defeq \iota^{*}\texval{\hat{H}} = \bigl\langle \chi, \hat{H} \chi \bigr\rangle$. So one can formulate a Hamiltonian system on $\mathcal{M}$ as $\ins{X_{H}}{\Omega_{\mathcal{M}}} = \d{H}$. As shown in @Lu2008 [Section II.1.2] (see also [@OhLe2013 Proposition 2.4]), the Hamiltonian vector field $X_{H}$ gives the best approximation to the vector field $X_{\texval{\hat{H}}}$ of the Schrödinger dynamics in the following sense: $X_{H}$ is the least squares approximation—in terms of the norm in $L^{2}(\R^{d})$—among the vector fields on $\mathcal{M}$ to the vector field $X_{\exval{\hat{H}}}$ defined by the Schrödinger equation . More specifically, we have, for any $y \in \mathcal{M}$, $$\| X_{\texval{\hat{H}}}(\iota(y)) - T_{y}\iota(X_H(y)) \|
\le
\| X_{\texval{\hat{H}}}(\iota(y)) - T_{y}\iota(V_{y}) \|$$ for any $V_{y} \in T_{y}\mathcal{M}$, where the equality holds if and only if $V_{y} = X_{H}(y)$; see Fig. \[fig:BestApproximation\].
![The Hamiltonian vector field $X_{H}$ gives the best approximation among the vector fields on $\mathcal{M}$ to the Schrödinger dynamics $X_{\texval{\hat{H}}}$.[]{data-label="fig:BestApproximation"}](BestApproximation){width=".6\linewidth"}
Semiclassical Symplectic Structures
-----------------------------------
We would like to apply the above approach to the embedding $\iota_{n}\colon \mathcal{M} \hookrightarrow \mathcal{H}$ of the semiclassical wave packets $\chi_{n}(y;\,\cdot\,)$ with an arbitrary $n \in \N_{0}^{d}$. As the first step, let us find the symplectic form $\Omega^{(n)}_{\mathcal{M}} \defeq \iota_{n}^{*}\Omega$. As mentioned earlier, a concrete expression for $\chi_{n}$ with general $n \in \N_{0}^{d}$ is essentially out of reach, and so it is not feasible to calculate $\Theta_{\mathcal{M}}^{(n)} \defeq \iota_{n}^{*} \Theta$ directly. A way around it is to find a recurrence relation between the canonical one-forms $\{ \Theta_{\mathcal{M}}^{(n)} \}_{n\in\N_{0}^{d}}$.
\[prop:Omega\_M\] Let $\iota_{n}\colon \mathcal{M} \hookrightarrow \mathcal{H} = L^{2}(\R^{d})$ be the embedding defined by the semiclassical wave packet $\chi_{n}(y;\,\cdot\,)$ for an arbitrary multi-index $n = (n_{1}, \dots, n_{d}) \in \N_{0}^{d}$. Then the almost complex structure $J$ in satisfies $T_{y}\iota_{n} \circ J_{y} = \rmi\cdot T_{y}\iota_{n}$ for any $y \in \mathcal{M}$ and hence the pull-back $\Omega_{\mathcal{M}}^{(n)} \defeq \iota_{n}^{*}\Omega$ defines a symplectic structure on $\mathcal{M}$, and is given by $\Omega_{\mathcal{M}}^{(n)} = -\mathbf{d}\Theta_{\mathcal{M}}^{(n)}$ with $$\label{eq:Theta_M}
\Theta_{\mathcal{M}}^{(n)} \defeq \iota_{n}^{*}\Theta
= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
p_{i}\,\mathbf{d}q_{i} - \frac{\hbar}{4}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1} \mathbf{d}\mathcal{A} } - \mathbf{d}\phi
},$$ where $\mathcal{B}^{(n)}$ is the $d \times d$ positive-definite matrix defined as $$\label{eq:mathcalB_n}
\mathcal{B}^{(n)} \defeq \mathcal{B}^{1/2} (\Lambda^{(n)})^{-1} \mathcal{B}^{1/2},$$ and $\Lambda^{(n)}$ is the $d \times d$ diagonal matrix defined as $$\label{eq:Lambda_n}
\Lambda^{(n)} \defeq \diag(2n_{1}+1, \dots, 2n_{d}+1).$$ More explicitly, $$\label{eq:Omega_M}
\begin{split}
\Omega_{\mathcal{M}}^{(n)}
=\mathcal{N}_{\hbar}(\mathcal{B},\delta) \biggl\{&
\mathbf{d}q_{i} \wedge \mathbf{d}p_{i}
- \frac{p_{i}}{2} \mathbf{d}q_{i} \wedge \tr\parentheses{ (\mathcal{B}^{(n)})^{-1}\mathbf{d}\mathcal{B}^{(n)} }
- \frac{2p_{i}}{\hbar}\mathbf{d}q_{i} \wedge \mathbf{d}\delta \\
&- \frac{\hbar}{4}\d\mathcal{A}_{ij} \wedge \d(\mathcal{B}^{(n)})^{-1}_{ij} + \frac{\hbar}{8}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1}\mathbf{d}\mathcal{A} } \wedge \tr\parentheses{ (\mathcal{B}^{(n)})^{-1}\d\mathcal{B}^{(n)} } \\
&+ \frac{1}{2}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1}\d\mathcal{A} } \wedge \d\delta
- \frac{1}{2}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1}\d\mathcal{B}^{(n)} } \wedge \d\phi + \frac{2}{\hbar} \mathbf{d}\phi \wedge \mathbf{d}\delta
\biggr\}.
\end{split}$$
Let us first show that $T_{y}\iota_{n} \circ J_{y} = \rmi\cdot T_{y}\iota_{n}$ for any $n \in \N_{0}^{d}$ by induction. One can check that it holds for $n = 0$ by direct calculations. Now let $n \in \N_{0}^{d}$ and suppose that $\iota_{n}$ satisfies $T_{y}\iota_{n} \circ J_{y} = \rmi\cdot T_{y}\iota_{n}$, and let $e_{j} \in \R^{d}$ be the unit vector with 1 in the $j$-th component with $j \in \{1, \dots, d\}$. Then implies that the embeddings $\iota_{n}$ and $\iota_{n+e_{j}}$ are related as $$\iota_{n+e_{j}} = \frac{1}{\sqrt{n_{j} + 1}}\mathscr{L}_{j}^{*} \circ \iota_{n}
\quad
\text{or}
\quad
\begin{tikzcd}[row sep=7ex, column sep=7ex]
& L^{2}(\R^{d}) \\
\mathcal{M} \arrow[r,hook,swap,"\iota_{n}"] \arrow[ru,hook,"\iota_{n+e_{j}}"] & L^{2}(\R^{d}) \arrow[u,swap,"\frac{1}{\sqrt{n_{j} + 1}}\mathscr{L}_{j}^{*}"]
\end{tikzcd}.$$ However, since $\mathscr{L}_{j}^{*}$ is a linear operator, we have, for any $y \in \mathcal{M}$, $$T_{y}\iota_{n+e_{j}} = \frac{1}{\sqrt{n_{j} + 1}}\mathscr{L}_{j}^{*} \circ T_{y}\iota_{n}.$$ But then this implies that $$\begin{aligned}
T_{y}\iota_{n+e_{j}} \circ J_{y}
&= \frac{1}{\sqrt{n_{j} + 1}}\mathscr{L}_{j}^{*} \circ T_{y}\iota_{n} \circ J_{y} \\
&= \frac{\rmi}{\sqrt{n_{j} + 1}}\mathscr{L}_{j}^{*} \circ T_{y}\iota_{n} \\
&= \rmi\cdot T_{y}\iota_{n+e_{j}}.
\end{aligned}$$
The expression follows from the following recurrence relation that holds between the one-forms $\{ \Theta_{\mathcal{M}}^{(n)} \}_{n \in \N_{0}^{d}}$: Let $n \in \N_{0}^{d}$ and $e_{j}$ be as above; then, as we shall prove later, $$\label{eq:rec_rel-Theta}
\Theta_{\mathcal{M}}^{(n+e_{j})} = \Theta_{\mathcal{M}}^{(n)} - \frac{\hbar}{2} \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{ \mathcal{B}^{-1/2}\,\d\mathcal{A}\,\mathcal{B}^{-1/2} }_{jj},$$ where summation on the index $j$ is [*not*]{} assumed on the right-hand side, i.e., it is the $(j,j)$-entry of the matrix $\mathcal{B}^{-1/2}\,\d\mathcal{A}\,\mathcal{B}^{-1/2}$. But then direct calculations yield, as is done in [@OhLe2013], $$\Theta_{\mathcal{M}}^{(0)} \defeq \iota_{0}^{*}\Theta = \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
p_{i}\,\mathbf{d}q_{i} - \frac{\hbar}{4}\tr\parentheses{ \mathcal{B}^{-1/2}\,\mathbf{d}\mathcal{A}\,\mathcal{B}^{-1/2} } - \mathbf{d}\phi
},$$ and hence we obtain as follows: $$\begin{aligned}
\Theta_{\mathcal{M}}^{(n)}
&= \Theta_{\mathcal{M}}^{(0)} - \frac{\hbar}{2} \mathcal{N}_{\hbar}(\mathcal{B},\delta) \sum_{j=1}^{d} n_{j} (\mathcal{B}^{-1/2}\,\d\mathcal{A}\,\mathcal{B}^{-1/2})_{jj} \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
p_{i}\,\mathbf{d}q_{i} - \frac{\hbar}{4}\tr\parentheses{ \mathcal{B}^{-1/2}\Lambda^{(n)}\mathcal{B}^{-1/2}\,\mathbf{d}\mathcal{A} } - \mathbf{d}\phi
} \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
p_{i}\,\mathbf{d}q_{i} - \frac{\hbar}{4}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1} \mathbf{d}\mathcal{A} } - \mathbf{d}\phi
},
\end{aligned}$$ where $\mathcal{B}^{(n)}$ is defined in . Then we have $$\begin{aligned}
\Omega_{\mathcal{M}}^{(n)}
&= \iota_{n}^{*}\Omega \\
&= -\iota_{n}^{*}\d\Theta \\
&= -\d\iota_{n}^{*}\Theta \\
&= -\d\Theta_{\mathcal{M}}^{(n)},
\end{aligned}$$ and the expression for $\Omega_{\mathcal{M}}^{(n)}$ follows from tedious but straightforward calculations; note that $$\begin{aligned}
\d\mathcal{N}_{\hbar}(\mathcal{B},\delta)
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
-\frac{1}{2}\tr\parentheses{ \mathcal{B}^{-1}\d\mathcal{B} }
-\frac{2}{\hbar}\d\delta
} \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
-\frac{1}{2}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1}\d\mathcal{B}^{(n)} }
-\frac{2}{\hbar}\d\delta
}.
\end{aligned}$$ So it remains to prove the recurrence relation . Using , we have $$\begin{aligned}
\Theta_{\mathcal{M}}^{(n+e_{j})}
&= \iota_{n+e_{j}}^{*}\Theta \\
&= -\hbar {\operatorname{Im}}\ip{ \chi_{n+e_{j}} }{ D_{y}\chi_{n+e_{j}} } \cdot \d{y} \\
&= -\frac{\hbar}{n_{j}+1} {\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{y}\parentheses{ \mathscr{L}^{*}_{j} \chi_{n} } } \cdot \d{y} \\
&= -\frac{\hbar}{n_{j}+1} \Bigl(
{\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{y}\mathscr{L}^{*}_{j} \chi_{n} }
+ {\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{j} D_{y}\chi_{n} }
\Bigr) \cdot \d{y},
\end{aligned}$$ where again no summation is assumed on $j$. Using the properties and of the ladder operators, we have $$\begin{aligned}
{\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{j} D_{y}\chi_{n} } \cdot \d{y}
&= {\operatorname{Im}}\ip{ \mathscr{L}_{j} \mathscr{L}^{*}_{j} \chi_{n} }{ D_{y}\chi_{n} } \cdot \d{y} \\
&= (n_{j}+1) {\operatorname{Im}}\ip{ \chi_{n} }{ D_{y}\chi_{n} } \cdot \d{y} \\
&= -\frac{n_{j}+1}{\hbar} \Theta_{\mathcal{M}}^{(n)}.
\end{aligned}$$ Therefore we obtain the recurrence relation $$\label{eq:rec_rel-Theta-pre}
\Theta_{\mathcal{M}}^{(n+e_{j})}
= \Theta_{\mathcal{M}}^{(n)}
- \frac{\hbar}{n_{j}+1}
{\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{y}\mathscr{L}^{*}_{j} \chi_{n} } \cdot \d{y}.$$ Let us evaluate the second term on the right-hand side. Taking the derivatives of with respect to $(q,p)$, we have $$D_{q_{l}}\mathscr{L}^{*}_{j} = -\frac{\rmi}{\sqrt{2\hbar}}\,\mathcal{B}^{-1/2}_{jk}(\mathcal{A} - \rmi\mathcal{B})_{kl},
\qquad
D_{p_{l}}\mathscr{L}^{*}_{j} = \frac{\rmi}{\sqrt{2\hbar}}\,\mathcal{B}^{-1/2}_{jl},$$ and hence $$\begin{aligned}
\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{q_{l}}\mathscr{L}^{*}_{j} \chi_{n} }
&= -\frac{\rmi}{\sqrt{2\hbar}}\,\mathcal{B}^{-1/2}_{jk}(\mathcal{A} - \rmi\mathcal{B})_{kl} \ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \chi_{n} }, \\
\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{p_{l}}\mathscr{L}^{*}_{j} \chi_{n} }
&= \frac{\rmi}{\sqrt{2\hbar}}\,\mathcal{B}^{-1/2}_{jl} \ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \chi_{n} }.
\end{aligned}$$ However, they both vanish due to the orthogonality of the basis $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$: $$\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \chi_{n} }
= \sqrt{n_{j}+1}\,\tip{ \chi_{n+{e_{j}}} }{ \chi_{n} } = 0.$$ On the other hand, taking the derivatives of with respect to $\mathcal{A}$ and $\mathcal{B}$, we have $$\begin{aligned}
D_{\mathcal{A}_{lr}}\mathscr{L}^{*}_{j}
&= \frac{\rmi}{2\sqrt{2\hbar}}\,\parentheses{
\mathcal{B}^{-1/2}_{jl}(\hat{x} - q)_{r} + \mathcal{B}^{-1/2}_{jr}(\hat{x} - q)_{l}
} \\
&= \frac{\rmi}{4}\, \parentheses{ \mathcal{B}^{-1/2}_{jl} \mathcal{B}^{-1/2}_{rk} + \mathcal{B}^{-1/2}_{jr} \mathcal{B}^{-1/2}_{lk} } ( \mathscr{L}_{k} + \mathscr{L}^{*}_{k} )
\end{aligned}$$ and $$\begin{aligned}
D_{\mathcal{B}_{lr}}\mathscr{L}^{*}_{j}
&= \frac{\rmi}{\sqrt{2\hbar}}\, D_{\mathcal{B}_{lr}}\mathcal{B}^{-1/2}_{js} \bigl(
(\mathcal{A} - \rmi\mathcal{B}) (\hat{x} - q) - (\hat{p} - p)
\bigr)_{s}
+ \frac{1}{2\sqrt{2\hbar}}\, \parentheses{ \mathcal{B}^{-1/2}_{jl} (\hat{x} - q)_{r} + \mathcal{B}^{-1/2}_{jr} (\hat{x} - q)_{l} }\\
&= D_{\mathcal{B}_{lr}}\mathcal{B}^{-1/2}_{js} \mathcal{B}^{1/2}_{su} \mathscr{L}^{*}_{u}
+ \frac{1}{4} \parentheses{ \mathcal{B}^{-1/2}_{jl} \mathcal{B}^{-1/2}_{ru} + \mathcal{B}^{-1/2}_{jr} \mathcal{B}^{-1/2}_{lu} } ( \mathscr{L}_{u} + \mathscr{L}^{*}_{u} ),
\end{aligned}$$ where we used as well as the following identity that follows from : $$\label{eq:x-q_in_As}
\hat{x} - q = \sqrt{\frac{\hbar}{2}}\,\mathcal{B}^{-1/2} ( \mathscr{L} + \mathscr{L}^{*} ).$$ So we have $$\begin{aligned}
\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{\mathcal{A}_{lr}}\mathscr{L}^{*}_{j} \chi_{n} }
&= \frac{\rmi}{4}\,\parentheses{ \mathcal{B}^{-1/2}_{jl} \mathcal{B}^{-1/2}_{rk} + \mathcal{B}^{-1/2}_{jr} \mathcal{B}^{-1/2}_{lk} } \parentheses{
\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}_{k} \chi_{n} }
+ \ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{k} \chi_{n} }
} \\
&= (n_{j}+1)\frac{\rmi}{4}\,\mathcal{N}_{\hbar}(\mathcal{B},\delta)\,\parentheses{ \mathcal{B}^{-1/2}_{jl} \mathcal{B}^{-1/2}_{rj} + \mathcal{B}^{-1/2}_{jr} \mathcal{B}^{-1/2}_{lj} }
\end{aligned}$$ with no summation on the index $j$, since $$\label{eq:inner_products}
\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}_{k} \chi_{n} } = 0,
\qquad
\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{k} \chi_{n} } = \delta_{jk}\, (n_{j}+1) \mathcal{N}_{\hbar}(\mathcal{B},\delta)$$ due to and . On the other hand, we see that the term $\tip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{\mathcal{B}_{lr}}\mathscr{L}^{*}_{j} \chi_{n} }$ is real and hence does not contribute to . As a result, we obtain $$\begin{aligned}
{\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{y}\mathscr{L}^{*}_{j} \chi_{n} } \cdot \d{y}
&= {\operatorname{Im}}\ip{ \mathscr{L}^{*}_{j} \chi_{n} }{ D_{\mathcal{A}_{lr}}\mathscr{L}^{*}_{j} \chi_{n} } \d\mathcal{A}_{lr} \\
&= \frac{n_{j}+1}{2}\,\mathcal{N}_{\hbar}(\mathcal{B},\delta) (\mathcal{B}^{-1/2}\,\d\mathcal{A}\,\mathcal{B}^{-1/2})_{jj},
\end{aligned}$$ and hence substituting this into yields the recurrence relation .
Hamiltonian Dynamics of Semiclassical Wave Packets {#sec:Hamiltonian_Dynamics_of_SWP}
==================================================
Now that we have the symplectic forms $\{ \Omega_{\mathcal{M}}^{(n)} \}_{n\in\N_{0}^{d}}$, it remains to find the Hamiltonians $\{ H^{(n)} \}_{n\in\N_{0}^{d}}$ that correspond to the semiclassical wave packets $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$ in order to formulate Hamiltonian dynamics for them.
In our previous work [@OhLe2013], we found the Hamiltonian $H^{(0)}$ corresponding to the Gaussian $\chi_{0}$ via an asymptotic expansion of the pull-back of the expectation value of the Hamiltonian operator $\hat{H}$, i.e., $$\label{eq:H^0}
\texval{\hat{H}}^{(0)} \defeq \iota_{0}^{*}\texval{\hat{H}}
= \bigl\langle \chi_{0}, \hat{H} \chi_{0} \bigr\rangle \\
= H^{(0)} + O(\hbar^{2}).$$ Then the Hamiltonian system $\ins{X_{H^{(0)}}}{\Omega^{(0)}} = \d{H^{(0)}}$ yields with the second equation replaced by . In this section, we would like to generalize this result to $\chi_{n}$ with an arbitrary $n\in\N_{0}^{d}$.
Semiclassical Hamiltonians
--------------------------
Let us find an asymptotic expansion for the expectation value $$\texval{\hat{H}}^{(n)} \defeq \iota_{n}^{*}\texval{\hat{H}}
= \bigl\langle \chi_{n}, \hat{H} \chi_{n} \bigr\rangle$$ of the Schrödinger operator $\hat{H}$ in with respect to the semiclassical wave packet $\chi_{n}$. We will evaluate the kinetic and potential parts of $\{ H^{(n)} \}_{n\in\N_{0}^{d}}$ separately: It turns out that the kinetic part can be found again via a recurrence relation by induction on $n$, whereas the potential part can be evaluated directly as an asymptotic expansion in $\hbar$ for any $n \in \N_{0}^{d}$ under a reasonable technical assumption on the potential $V$.
\[prop:H\_M\] Suppose that the potential $V$ is in $C^{3}(\R^{d})$ and that there exist $C_{1}, C_{2}, M_{1} \in \R$ such that $C_{1} \le V(x)$ and for any $\alpha \in \N_{0}^{d}$ with $|\alpha| = 3$, $$\label{eq:assumption_on_D3V}
|D^{\alpha}V(x)| \le C_{2} \exp(M_{1}|x|^{2}).$$ Then the expectation value $\texval{\hat{H}}^{(n)}$ for each $n\in\N_{0}^{d}$ has the asymptotic expansion $$\label{eq:exvalH-asymptotic}
\texval{\hat{H}}^{(n)} = H^{(n)} + \mathcal{N}_{\hbar}(\mathcal{B},\delta)\,O(\hbar^{3/2}),$$ where $H^{(n)}\colon \mathcal{M} \to \R$ is defined as $$\label{eq:H_M}
H^{(n)} \defeq \mathcal{N}_{\hbar}(\mathcal{B},\delta) \Biggl\{
\frac{p^{2}}{2m} + \frac{\hbar}{4m} \tr\!\parentheses{ (\mathcal{B}^{(n)})^{-1}(\mathcal{A}^{2} + \mathcal{B}^{2}) } \\
+ V(q) + \frac{\hbar}{4} \tr\!\parentheses{ (\mathcal{B}^{(n)})^{-1} D^{2}V(q) }
\Biggr\},$$ with $\mathcal{B}^{(n)}$ defined in .
\[rem:N\] The quantity $\mathcal{N}_{\hbar}(\mathcal{B},\delta) = \norm{ \chi_{n}(y;\,\cdot\,) }^{2}$ depends on $\hbar$ as shown in . However, as we shall see later in Proposition \[prop:Omega\_M-reduced\], $\mathcal{N}_{\hbar}(\mathcal{B},\delta)$ is conserved along the Hamiltonian dynamics of the parameters $y = (q,p,\mathcal{A},\mathcal{B},\phi,\delta)$ that we derive later. Therefore, upon normalizing the wave packet $\chi_{n}(y;\,\cdot\,)$ in the initial condition by setting $\mathcal{N}_{\hbar}(\mathcal{B},\delta) = 1$, it stays so all time; hence we may assume $\mathcal{N}_{\hbar}(\mathcal{B},\delta) = O(1)$.
The error term becomes $O(\hbar^{2})$ if $V$ is $C^{4}(\R^{d})$ and assuming for $|\alpha| = 4$. In fact, the asymptotic expansion in assumes that $V$ is smooth and satisfies a condition similar to ; see [@OhLe2013 Proposition 7.1].
Note first that the assumption that $V$ is bounded from below guarantees that the Schrödinger operator is essentially self-adjoint. Let us split the expectation value of the Hamiltonian into the kinetic and potential parts, i.e., $$\texval{\hat{H}}^{(n)} = \texval{\hat{T}}^{(n)} + \exval{V}^{(n)}$$ with $\hat{T} \defeq \hat{p}^{2}/(2m)$, and first evaluate the kinetic part. We see that $$\begin{aligned}
\texval{\hat{T}}^{(n+e_{j})}
&= \bigl\langle \chi_{n+e_{j}}, \hat{T} \chi_{n+e_{j}} \bigr\rangle \\
&= \frac{1}{n_{j}+1} \bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \hat{T} \mathscr{L}^{*}_{j} \chi_{n} \bigr\rangle \\
&= \frac{1}{n_{j}+1} \parentheses{
\bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \bigl[\hat{T}, \mathscr{L}^{*}_{j}\bigr] \chi_{n} \bigr\rangle
+ \bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \mathscr{L}^{*}_{j} \hat{T} \chi_{n} \bigr\rangle
},
\end{aligned}$$ but then $$\begin{aligned}
\bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \mathscr{L}^{*}_{j} \hat{T} \chi_{n} \bigr\rangle
&= \bigl\langle \mathscr{L}_{j} \mathscr{L}^{*}_{j} \chi_{n}, \hat{T} \chi_{n} \bigr\rangle \\
&= (n_{j}+1) \bigl\langle \chi_{n}, \hat{T} \chi_{n} \bigr\rangle \\
&= (n_{j}+1) \texval{\hat{T}}^{(n)},
\end{aligned}$$ and hence we have the recurrence relation $$\texval{\hat{T}}^{(n+e_{j})} = \texval{\hat{T}}^{(n)} + \frac{1}{n_{j}+1} \bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \bigl[\hat{T}, \mathscr{L}^{*}_{j}\bigr] \chi_{n} \bigr\rangle.$$
Let us evaluate the second term on the right-hand side. It is straightforward to see that, using , $$\bigl[ \hat{T}, \mathscr{L}^{*} \bigr]
= \frac{1}{m}\sqrt{\frac{\hbar}{2}}\, \mathcal{B}^{-1/2} (\mathcal{A} - \rmi\mathcal{B}) \hat{p},$$ and hence we have $$\begin{aligned}
\bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \bigl[\hat{T}, \mathscr{L}^{*}_{j}\bigr] \chi_{n} \bigr\rangle
&= \frac{1}{m}\sqrt{\frac{\hbar}{2}}\, \mathcal{B}^{-1/2}_{jk}
(\mathcal{A} - \rmi\mathcal{B})_{kl} \bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \hat{p}_{l} \chi_{n} }.
\end{aligned}$$ It is easy to see from the definition of the ladder operators (see also ) that $$\hat{p} - p = \sqrt{\frac{\hbar}{2}}\,\bigl(
(\mathcal{A} - \rmi\mathcal{B}) \mathcal{B}^{-1/2} \mathscr{L}
+ (\mathcal{A} + \rmi\mathcal{B}) \mathcal{B}^{-1/2} \mathscr{L}^{*}
\bigr),$$ and so $$\begin{aligned}
\bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \hat{p}_{l} \chi_{n} }
&= \bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \chi_{n} } p_{l} \\
&\quad + \sqrt{\frac{\hbar}{2}}\,\bigl(
(\mathcal{A} - \rmi\mathcal{B})_{lr} \mathcal{B}^{-1/2}_{rs} \bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}_{s} \chi_{n} }
+ (\mathcal{A} + \rmi\mathcal{B})_{lr} \mathcal{B}^{-1/2}_{rs} \bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{s} \chi_{n} }
\bigr) \\
&= (n_{j}+1)\sqrt{\frac{\hbar}{2}}\,\mathcal{N}_{\hbar}(\mathcal{B},\delta)\,(\mathcal{A} + \rmi\mathcal{B})_{lr} \mathcal{B}^{-1/2}_{rj}
\end{aligned}$$ because, due to the properties and of the ladder operators and the orthogonality of $\{ \chi_{n} \}_{n\in\N_{0}^{d}}$, $$\bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \chi_{n} } = 0,
\qquad
\bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}_{s} \chi_{n} } = 0,
\qquad
\bigip{ \mathscr{L}^{*}_{j} \chi_{n} }{ \mathscr{L}^{*}_{s} \chi_{n} }
= \delta_{js}\,(n_{j}+1)\,\mathcal{N}_{\hbar}(\mathcal{B},\delta).$$ Therefore, $$\begin{aligned}
\bigl\langle \mathscr{L}^{*}_{j} \chi_{n}, \bigl[\hat{T}, \mathscr{L}^{*}_{j}\bigr] \chi_{n} \bigr\rangle
&= (n_{j}+1) \frac{\hbar}{2m}\, \mathcal{N}_{\hbar}(\mathcal{B},\delta)\,\mathcal{B}^{-1/2}_{jk}
(\mathcal{A} - \rmi\mathcal{B})_{kl} (\mathcal{A} + \rmi\mathcal{B})_{lr} \mathcal{B}^{-1/2}_{rj} \\
&= (n_{j}+1) \frac{\hbar}{2m}\, \mathcal{N}_{\hbar}(\mathcal{B},\delta)
\parentheses{ \mathcal{B}^{-1/2}(\mathcal{A}^{2} + \mathcal{B}^{2})\mathcal{B}^{-1/2} }_{jj}
\end{aligned}$$ since both $\mathcal{A}$ and $\mathcal{B}$ are symmetric. As a result, we obtain the recurrence relation $$\texval{\hat{T}}^{(n+e_{j})} = \texval{\hat{T}}^{(n)}
+ \frac{\hbar}{2m}\, \mathcal{N}_{\hbar}(\mathcal{B},\delta)
\parentheses{ \mathcal{B}^{-1/2}(\mathcal{A}^{2} + \mathcal{B}^{2})\mathcal{B}^{-1/2} }_{jj}.$$ It is easy to see by direct calculations that, as in [@OhLe2013], $$\texval{\hat{T}}^{(0)} = \mathcal{N}_{\hbar}(\mathcal{B},\delta) \braces{
\frac{p^{2}}{2m} + \frac{\hbar}{4m} \tr\parentheses{ \mathcal{B}^{-1/2}(\mathcal{A}^{2} + \mathcal{B}^{2})\mathcal{B}^{-1/2} }
}.$$ Hence the recurrence relation yields $$\begin{aligned}
\texval{\hat{T}}^{(n)}
&= \texval{\hat{T}}^{(0)}
+ \frac{\hbar}{2m}\, \mathcal{N}_{\hbar}(\mathcal{B},\delta) \sum_{j=1}^{d} n_{j} \parentheses{ \mathcal{B}^{-1/2}(\mathcal{A}^{2} + \mathcal{B}^{2})\mathcal{B}^{-1/2} }_{jj} \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \braces{
\frac{p^{2}}{2m}
+ \frac{\hbar}{4m} \tr\parentheses{ \mathcal{B}^{-1/2} \Lambda^{(n)} \mathcal{B}^{-1/2}(\mathcal{A}^{2} + \mathcal{B}^{2}) }
} \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \braces{
\frac{p^{2}}{2m}
+ \frac{\hbar}{4m} \tr\parentheses{ (\mathcal{B}^{(n)})^{-1}(\mathcal{A}^{2} + \mathcal{B}^{2}) }
},
\end{aligned}$$ where $\Lambda^{(n)}$ and $\mathcal{B}^{(n)}$ are defined in and .
Next, let us find an asymptotic expansion of the potential term $\exval{V}^{(n)}$. We mimic the technique employed in the proof of Theorem 2.9 in @Ha1998. First, since $V$ is assumed to be $C^{3}$, we have, for any $x \in \R^{d}$, $$V(x) = V(q) + D_{k}V(q) (x - q)_{k} + \frac{1}{2} D^{2}_{kl}V(q) (x - q)^{\otimes^{2}}_{kl}
+ \sum_{|\alpha|=3} \frac{1}{\alpha!} D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha}$$ for some point $\sigma(q,x)$ in the closed ball $\bar{\mathbb{B}}_{|x-q|}(q) \subset \R^{d}$ with radius $|x-q|$ centered at $q$, where we used the shorthands $(x - q)^{\otimes^{2}}_{kl} = (x - q)_{k} (x - q)_{l}$ and $(x - q)^{\alpha} = \prod_{j=1}^{d} (x - q)_{j}^{\alpha_{j}}$. Therefore, $$\begin{aligned}
\exval{V}^{(n)}
&= \ip{ \chi_{n} }{ V \chi_{n} } \\
&= \norm{ \chi_{n} }^{2}\, V(q)
+ \ip{ \chi_{n} }{ (x - q)_{k} \chi_{n} } D_{k}V(q)
+ \frac{1}{2}\ip{ \chi_{n} }{ (x - q)^{\otimes^{2}}_{kl} \chi_{n} } D^{2}_{kl}V(q) \\
&\quad + \sum_{|\alpha|=3} \frac{1}{\alpha!} \ip{ \chi_{n} }{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n} }.
\end{aligned}$$ But then the second term vanishes because, using and in view of and , $$\begin{aligned}
\ip{ \chi_{n} }{ (x - q)_{k} \chi_{n} }
&= \sqrt{\frac{\hbar}{2}}\,\mathcal{B}^{-1/2}_{kl} \ip{ \chi_{n} }{ ( \mathscr{L}_{l} + \mathscr{L}^{*}_{l} ) \chi_{n} } \\
&= \sqrt{\frac{\hbar}{2}}\,\mathcal{B}^{-1/2}_{kl}\parentheses{
\sqrt{n_{l}} \ip{ \chi_{n} }{ \chi_{n-e_{l}} }
+ \sqrt{n_{l}+1} \ip{ \chi_{n} }{ \chi_{n+e_{l}} }
} \\
&= 0.
\end{aligned}$$ On the other hand, using and as well as $\ip{ \mathscr{L}_{j} \chi_{n} }{ \mathscr{L}_{k} \chi_{n} } = \delta_{jk}\, n_{j} \mathcal{N}_{\hbar}(\mathcal{B},\delta)$, we can evaluate the third term as follows: $$\begin{aligned}
\ip{ \chi_{n} }{ (x - q)^{\otimes^{2}}_{kl} \chi_{n} }
&= \ip{ (x - q)_{k} \chi_{n} }{ (x - q)_{l} \chi_{n} } \\
&= \frac{\hbar}{2}\,\mathcal{B}^{-1/2}_{kr}\mathcal{B}^{-1/2}_{ls}
\bigip{ (\mathscr{L}_{r} + \mathscr{L}_{r}^{*})\chi_{n} }{ (\mathscr{L}_{s} + \mathscr{L}_{s}^{*}) \chi_{n} } \\
&= \frac{\hbar}{2}\,\mathcal{B}^{-1/2}_{kr}\mathcal{B}^{-1/2}_{ls} \parentheses{
\bigip{ \mathscr{L}_{r} \chi_{n} }{ \mathscr{L}_{s} \chi_{n} }
+ \bigip{ \mathscr{L}^{*}_{r} \chi_{n} }{ \mathscr{L}^{*}_{s} \chi_{n} }
} \\
&= \frac{\hbar}{2}\,\mathcal{N}_{\hbar}(\mathcal{B},\delta)\,\mathcal{B}^{-1/2}_{kr}(2n_{r}+1)\delta_{rs}\mathcal{B}^{-1/2}_{ls} \\
&= \frac{\hbar}{2}\,\mathcal{N}_{\hbar}(\mathcal{B},\delta)\,\parentheses{ \mathcal{B}^{-1/2}\Lambda^{(n)}\mathcal{B}^{-1/2} }_{kl} \\
&= \frac{\hbar}{2}\,\mathcal{N}_{\hbar}(\mathcal{B},\delta)\, (\mathcal{B}^{(n)})^{-1}_{kl},
\end{aligned}$$ where $\Lambda^{(n)}$ and $\mathcal{B}^{(n)}$ are defined in and . So it remains to show that the last term is $O(\hbar^{3/2})$. Let $R > 0$ and set $$C_{3} \defeq \max_{|\alpha|=3} \max_{x \in \bar{\mathbb{B}}_{R}(q)} \abs{ D^{\alpha}V(x) }.$$ If $x \in \bar{\mathbb{B}}_{R}(q)$ then $\sigma(q,x) \in \bar{\mathbb{B}}_{R}(q)$ as well and hence, for any $\alpha \in \N_{0}^{d}$ with $|\alpha| = 3$, $$\begin{aligned}
\abs{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n}(y;x) }
\le C_{3}\,\hbar^{3/2}\, \abs{ \parentheses{ \frac{x - q}{\sqrt{\hbar}} }^{\alpha} \chi_{n}(y;x) },
\end{aligned}$$ whereas if $x \in \bar{\mathbb{B}}_{R}(q)^{\rm c}$ then, due to the assumption on the potential $V$, there exists $C_{4} > 0$ such that $$\begin{aligned}
\abs{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n}(y;x) }
&\le C_{4} \exp(M_{1}|x|^{2}) \abs{ \chi_{n}(y;x) }.
\end{aligned}$$ Let $\mathbf{1}_{S}$ be the indicator function of an arbitrary subset $S \subset \R^{d}$ and also define the normalized wave packet $$\varphi_{n}(y;x) \defeq \frac{\chi_{n}(y;x)}{\norm{\chi_{n}(y;\,\cdot\,)}}.$$ Then we have, for any $\alpha \in \N_{0}^{d}$ with $|\alpha| = 3$ and any $x \in \R^{d}$, $$\begin{aligned}
\abs{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n}(y;x) }
&= \mathbf{1}_{\bar{\mathbb{B}}_{R}(q)}(x) \abs{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n}(y;x) } \\
&\qquad + \mathbf{1}_{\bar{\mathbb{B}}_{R}(q)^{\rm c}}(x) \abs{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n}(y;x) } \\
&\le C_{3}\, \hbar^{3/2}\, \norm{\chi_{n}(y;\,\cdot\,)}\, \mathbf{1}_{\bar{\mathbb{B}}_{R}(q)}(x)\,\abs{ \parentheses{ \frac{x - q}{\sqrt{\hbar}} }^{\alpha} \varphi_{n}(y;x) } \\
&\quad+ C_{4} \norm{\chi_{n}(y;\,\cdot\,)}\, \mathbf{1}_{\bar{\mathbb{B}}_{R}(q)^{\rm c}}(x)\, \exp(M_{1}|x|^{2}) \abs{ \varphi_{n}(y;x) }.
\end{aligned}$$ However, the norm of the first term is $O(\hbar^{3/2})$ because, as shown in @Ha1998 [Eq. (3.30)], $$\norm{ \parentheses{ \frac{x - q}{\sqrt{\hbar}} }^{\alpha} \varphi_{n}(y;\,\cdot\,) } = O(1),$$ whereas $$\norm{ \exp(M_{1}|x|^{2}) \varphi_{n}(y;\,\cdot\,) } = o(\hbar^{\gamma})$$ for any real number $\gamma$. Hence $$\norm{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n} } = \norm{\chi_{n}(y;\,\cdot\,)}\,O(\hbar^{3/2}),$$ and thus by the Cauchy–Schwarz inequality, $$\begin{aligned}
\sum_{|\alpha|=3} \frac{1}{\alpha!} \abs{ \ip{\chi_{n}}{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n} } }
&\le \norm{\chi_{n}} \sum_{|\alpha|=3} \frac{1}{\alpha!} \norm{ D^{\alpha}V(\sigma(q,x)) (x - q)^{\alpha} \chi_{n} } \\
&\le \norm{\chi_{n}}^{2}\,O(\hbar^{3/2}) \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta)\,O(\hbar^{3/2}).
\end{aligned}$$ As a result, we obtain $$\begin{aligned}
\exval{V}^{(n)}
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
V(q) + \frac{\hbar}{4}\,(\mathcal{B}^{(n)})^{-1}_{kl} D^{2}_{kl}V(q)
+ O(\hbar^{3/2})
} \\
&= \mathcal{N}_{\hbar}(\mathcal{B},\delta) \parentheses{
V(q) + \frac{\hbar}{4}\tr\!\parentheses{ (\mathcal{B}^{(n)})^{-1} D^{2}V(q) }
+ O(\hbar^{3/2})
}.
\end{aligned}$$ Hence we have the asymptotic expansion along with .
Hamiltonian Dynamics of Semiclassical Wave Packets {#hamiltonian-dynamics-of-semiclassical-wave-packets}
--------------------------------------------------
Now that we have both the symplectic forms $\{ \Omega_{\mathcal{M}}^{(n)} \}_{n\in\N_{0}^{d}}$ and the Hamiltonians $\{ H^{(n)} \}_{n\in\N_{0}^{d}}$ associated with the semiclassical wave packets $\{ \chi_{n}(y;\,\cdot\,) \}_{n\in\N_{0}^{d}}$, we may formulate Hamiltonian dynamics for each of them:
\[thm:Hamiltonian\_system\] Suppose that the potential $V$ satisfies the conditions stated in Proposition \[prop:H\_M\]. Then, for any $n \in \N_{0}^{d}$, the Hamiltonian vector field $X_{H^{(n)}} \in \mathfrak{X}(\mathcal{M})$ associated with the semiclassical wave packet $\chi_{n}(y;\,\cdot\,)$ is defined by ${\bf i}_{X_{H^{(n)}}} \Omega_{\mathcal{M}}^{(n)} = \mathbf{d}H^{(n)}$ or $$\label{eq:SemiclassicalSystem}
\begin{array}{c}
\DS
\dot{q} = \frac{p}{m},
\qquad
\dot{p} = -D_{q}V^{(n)}_{\hbar}(q,\mathcal{B}),
\medskip\\
\DS
\dot{\mathcal{A}} = -\frac{1}{m}\parentheses{ \mathcal{A}^{2} - \frac{1}{2}(\mathcal{B}^{(n)}\Lambda^{(n)}\mathcal{B} + \mathcal{B}\Lambda^{(n)}\mathcal{B}^{(n)}) } - D^{2}V(q),
\qquad
\dot{\mathcal{B}}^{(n)} = -\frac{1}{m}(\mathcal{A}\mathcal{B}^{(n)} + \mathcal{B}^{(n)}\mathcal{A}),
\medskip\\
\DS
\dot{\phi} = \frac{p^{2}}{2m} - V(q) - \frac{\hbar}{2m} \tr(\Lambda^{(n)}\mathcal{B}),
\qquad
\dot{\delta} = \frac{\hbar}{2m} \tr\mathcal{A},
\end{array}$$ where the corrected potential $V^{(n)}_{\hbar}$ is defined as $$\label{eq:V^n}
V^{(n)}_{\hbar}(q,\mathcal{B})
\defeq
V(q) + \frac{\hbar}{4}\tr\!\parentheses{ (\mathcal{B}^{(n)})^{-1} D^{2}V(q) }.$$
For the special case with $n = (\bar{n}, \dots, \bar{n}) \in \N_{0}^{d}$ with $\bar{n} \in \N_{0}$, we have $\Lambda^{(n)} = (2\bar{n}+1)\,I_{d}$ and $\mathcal{B}^{(n)} = (2\bar{n}+1)^{-1} \mathcal{B}$. Hence the equations for $\mathcal{A}$ and $\mathcal{B}$ simplify to those in .
The assertion follows from tedious but straightforward calculations using the formulas for the symplectic forms $\{ \Omega_{\mathcal{M}}^{(n)} \}_{n\in\N_{0}^{d}}$ and the Hamiltonians $\{ H^{(n)} \}_{n\in\N_{0}^{d}}$ from Propositions \[prop:Omega\_M\] and \[prop:H\_M\], respectively.
Symplectic Reduction by Phase Symmetry {#sec:Symplectic_Reduction}
======================================
The Hamiltonian $H^{(n)}$ found in does not depend on the phase variable $\phi$ and hence is invariant under the $\mathbb{S}^{1}$ phase shift action. Therefore, we can reduce the Hamiltonian dynamics to a lower-dimensional one by the symplectic (Marsden–Weinstein) reduction [@MaWe1974] (see also @MaMiOrPeRa2007 [Sections 1.1 and 1.2]) as is done for the Gaussian case in our earlier work [@OhLe2013]. The resulting [*reduced*]{} symplectic structure is much simpler than $\Omega_{\mathcal{M}}^{(n)}$ from Proposition \[prop:H\_M\] and moreover takes an appealing form: It is given by the classical symplectic form plus an $O(\hbar)$ “correction” term for any $n \in \N_{0}^{d}$ and similarly for the Hamiltonian as well, hence generalizing the results for the Gaussian case ($n = 0$) from our earlier work [@OhLe2013 Theorem 4.1].
Reduced Symplectic Structures
-----------------------------
\[prop:Omega\_M-reduced\] Let $\Phi\colon \mathbb{S}^{1} \times \mathcal{M} \to \mathcal{M}$ be the $\mathbb{S}^{1}$-action on $\mathcal{M}$ defined for any $\theta \in \mathbb{S}^{1}$ as $$\label{eq:Phi}
\Phi_{\theta}: \mathcal{M} \to \mathcal{M};
\quad
(q, p, \mathcal{A}, \mathcal{B}, \phi, \delta) \mapsto (q, p, \mathcal{A}, \mathcal{B}, \phi + \hbar\,\theta, \delta).$$ Then the corresponding momentum map $\mathbf{J}_{\!\mathcal{M}}^{(n)}\colon \mathcal{M} \to \mathfrak{so}(2)^{*} \cong \R$ is given by $$\label{eq:mathbfJ}
\mathbf{J}_{\!\mathcal{M}}^{(n)}(y) = -\hbar\,\mathcal{N}_{\hbar}(\mathcal{B},\delta),$$ and the Marsden–Weinstein quotient $$\overline{\mathcal{M}}_{\hbar}^{(n)} \defeq (\mathbf{J}_{\!\mathcal{M}}^{(n)})^{-1}(-\hbar)/\mathbb{S}^{1} = T^{*}\R^{d} \times \Sigma_{d}$$ is equipped with the reduced symplectic form $$\label{eq:Omega-reduced}
\begin{split}
\overline{\Omega}_{\hbar}^{(n)}
&= \mathbf{d}q_{i} \wedge \mathbf{d}p_{i} + \frac{\hbar}{4} (\mathcal{B}^{(n)})^{-1}_{jr} (\mathcal{B}^{(n)})^{-1}_{sk}\,\d\mathcal{A}_{jk} \wedge \d\mathcal{B}^{(n)}_{rs}
\\
&= \mathbf{d}q_{i} \wedge \mathbf{d}p_{i} + \frac{\hbar}{4} \d(\mathcal{B}^{(n)})^{-1}_{jk} \wedge \d\mathcal{A}_{jk},
\end{split}$$ where $\mathcal{B}^{(n)}$ is defined in .
Let us first find the momentum map corresponding to the above action. It is easy to see that the action $\Phi$ leaves the one-form $\Theta_{\mathcal{M}}^{(n)}$ invariant, i.e., $\Phi_{\theta}^{*}\Theta_{\mathcal{M}}^{(n)} = \Theta_{\mathcal{M}}^{(n)}$ for any $\theta \in \mathbb{S}^{1}$, and hence $\Phi$ is symplectic with respect to $\Omega_{\mathcal{M}}^{(n)}$, i.e., $\Phi_{\theta}^{*}\Omega_{\mathcal{M}}^{(n)} = \Omega_{\mathcal{M}}^{(n)}$ for any $\theta \in \mathbb{S}^{1}$. The infinitesimal generator of the above action corresponding to an arbitrary element $\xi$ in the Lie algebra $\mathfrak{so}(2) \cong \R$ is $$\xi_{\mathcal{M}}(y) \defeq \left. \od{}{\eps} \Phi_{\eps\xi}(y) \right|_{\eps=0}
= \hbar\,\xi\,\pd{}{\phi}.$$ Since $\Phi$ leaves $\Theta_{\mathcal{M}}^{(n)}$ invariant for any $n \in \N_{0}^{d}$, the momentum map $\mathbf{J}_{\!\mathcal{M}}^{(n)}$ with respect to the symplectic structure $\Omega_{\mathcal{M}}^{(n)}$ for any $n \in \N_{0}^{d}$ is defined as $$\ip{ \mathbf{J}_{\!\mathcal{M}}^{(n)}(y) }{\xi} = \ip{ \Theta_{\mathcal{M}}^{(n)}(y) }{ \xi_{\mathcal{M}}(y) } = -\hbar\,\mathcal{N}_{\hbar}(\mathcal{B},\delta)\,\xi$$ for any $\xi \in \mathfrak{so}(2)$. Hence we obtain . So the level set $(\mathbf{J}_{\!\mathcal{M}}^{(n)})^{-1}(-\hbar)$ is given by the set of those parameters that normalize the wave packet $\chi_{n}(y;\,\cdot\,)$, i.e., $$\begin{aligned}
(\mathbf{J}_{\!\mathcal{M}}^{(n)})^{-1}(-\hbar)
&= \setdef{ (q,p,\mathcal{A},\mathcal{B},\phi,\delta) \in \mathcal{M} }{ \mathcal{N}_{\hbar}(\mathcal{B},\delta) = 1 } \\
&= \setdef{ y = (q,p,\mathcal{A},\mathcal{B},\phi,\delta) \in \mathcal{M} }{ \bigl\| \chi_{n}(y;\,\cdot\,) \bigr\| = 1 }.
\end{aligned}$$ Therefore, one may solve $\mathcal{N}_{\hbar}(\mathcal{B},\delta) = 1$ for $\delta$ (see ) to have the inclusion $i_{\hbar}\colon (\mathbf{J}_{\!\mathcal{M}}^{(n)})^{-1}(-\hbar) \to \mathcal{M}$ defined as $$i_{\hbar}\colon (q,p,\mathcal{A},\mathcal{B},\phi) \mapsto \parentheses{ q,p,\mathcal{A},\mathcal{B},\phi,\frac{\hbar}{4}\ln\parentheses{\frac{(\pi\hbar)^{d}}{\det\mathcal{B}}} },$$ and hence we have $$i_{\hbar}^{*}\Theta_{\mathcal{M}}^{(n)}
= p_{i}\,\mathbf{d}q_{i} - \frac{\hbar}{4}\tr\parentheses{ (\mathcal{B}^{(n)})^{-1} \d\mathcal{A} } - \mathbf{d}\phi.$$ Therefore, $$\begin{aligned}
i_{\hbar}^{*}\Omega_{\mathcal{M}}^{(n)}
&= -i_{\hbar}^{*}\mathbf{d}\Theta_{\mathcal{M}}^{(n)} \\
&= -\mathbf{d}\bigl( i_{\hbar}^{*}\Theta_{\mathcal{M}}^{(n)} \bigr) \\
&= \mathbf{d}q_{i} \wedge \mathbf{d}p_{i} + \frac{\hbar}{4}\,\d(\mathcal{B}^{(n)})^{-1}_{jk} \wedge \d\mathcal{A}_{jk}.
\end{aligned}$$ However, we have the quotient map $$\pi_{\hbar}\colon (\mathbf{J}_{\!\mathcal{M}}^{(n)})^{-1}(-\hbar) \to \overline{\mathcal{M}}_{\hbar}^{(n)} \defeq (\mathbf{J}_{\!\mathcal{M}}^{(n)})^{-1}(-\hbar)/\mathbb{S}^{1};
\quad
(q,p,\mathcal{A},\mathcal{B},\phi) \mapsto (q,p,\mathcal{A},\mathcal{B}),$$ and see that $\overline{\Omega}_{\hbar}^{(n)}$ shown in satisfies $\pi_{\hbar}^{*}\overline{\Omega}_{\hbar}^{(n)} = i_{\hbar}^{*}\Omega_{\mathcal{M}}^{(n)}$ and hence defines the reduced symplectic form on $\overline{\mathcal{M}}_{\hbar}^{(n)}$; note that $\pi_{\hbar}^{*}$ is injective because $\pi_{\hbar}$ is a surjective submersion.
Reduced Hamiltonian Dynamics
----------------------------
Since the Hamiltonian $H^{(n)}$ does not depend on the phase variable $\phi$, it has the $\mathbb{S}^{1}$-symmetry under the action $\Phi$ defined in . Therefore we can reduce the Hamiltonian dynamics to the reduced symplectic manifold $\overline{\mathcal{M}}_{\hbar}$:
\[thm:ReducedSemiclassicalSystem\] Suppose that the potential $V$ satisfies the conditions stated in Proposition \[prop:H\_M\]. Then the Hamiltonian system on $\mathcal{M}$ for the semiclassical wave packet $\chi_{n}(y;\,\cdot\,)$ is reduced by the above $\mathbb{S}^{1}$-symmetry to the Hamiltonian system $$\label{eq:ReducedHamiltonianSystem}
{\bf i}_{X_{\overline{H}^{(n)}_{\hbar}}} \overline{\Omega}^{(n)}_{\hbar} = \mathbf{d}\overline{H}^{(n)}_{\hbar}$$ defined on $\overline{\mathcal{M}}_{\hbar}^{(n)} = T^{*}\R^{d} \times \Sigma_{d}$ with the reduced symplectic form and the reduced Hamiltonian $\overline{H}^{(n)}\colon \overline{\mathcal{M}}_{\hbar}^{(n)} \to \R$ given by $$\label{eq:H-reduced}
\begin{split}
\overline{H}^{(n)}_{\hbar}
&= \frac{p^{2}}{2m} + V(q)
+ \frac{\hbar}{4} \parentheses{
\frac{1}{m} \tr\!\parentheses{ (\mathcal{B}^{(n)})^{-1}(\mathcal{A}^{2} + \mathcal{B}^{2}) }
+ \tr\parentheses{ (\mathcal{B}^{(n)})^{-1} D^{2}V(q) }
} \\
&= \frac{p^{2}}{2m} + \frac{\hbar}{4m} \tr\!\parentheses{ (\mathcal{B}^{(n)})^{-1}(\mathcal{A}^{2} + \mathcal{B}^{2}) } + V^{(n)}_{\hbar}(q,\mathcal{B}),
\end{split}$$ where the corrected potential $V^{(n)}_{\hbar}$ is defined in . Specifically, gives the reduced set of the semiclassical equations: $$\label{eq:ReducedSemiclassicalSystem}
\begin{array}{c}
\DS
\dot{q} = \frac{p}{m},
\qquad
\dot{p} = -D_{q}V^{(n)}_{\hbar}(q,\mathcal{B}),
\medskip\\
\DS
\dot{\mathcal{A}} = -\frac{1}{m}\parentheses{ \mathcal{A}^{2} - \frac{1}{2}(\mathcal{B}^{(n)}\Lambda^{(n)}\mathcal{B} + \mathcal{B}\Lambda^{(n)}\mathcal{B}^{(n)}) } - D^{2}V(q),
\qquad
\dot{\mathcal{B}}^{(n)} = -\frac{1}{m}(\mathcal{A}\mathcal{B}^{(n)} + \mathcal{B}^{(n)}\mathcal{A}).
\end{array}$$
Clearly the Hamiltonian $H^{(n)}$ has the $\mathbb{S}^{1}$-symmetry, i.e., $H^{(n)} \circ \Phi_{\theta} = H^{(n)}$ for any $\theta \in \mathbb{S}^{1}$ and so, by the Marsden–Weinstein reduction [@MaWe1974] (see also @MaMiOrPeRa2007 [Sections 1.1 and 1.2]), the Hamiltonian system reduces to the reduced one with the reduced Hamiltonian $\overline{H}^{(n)}_{\hbar}$ defined as $\overline{H}^{(n)}_{\hbar} \circ \pi_{\hbar} = H^{(n)} \circ i_{\hbar}$, which yields . It is a straightforward calculation to see that yields .
Numerical Results {#sec:Numerical_Results}
=================
Problem Setting: Escape from Cubic Potential Well
-------------------------------------------------
We performed numerical experiments with the simple one-dimensional potential (i.e., $d = 1$) $$\label{eq:V}
V(x) = 2x^{2} + x^{3} + 0.1x^{4}$$ and $m = 1$, and different values of index $n$ and parameter $\hbar$. This example is a slightly modified version of an example from @KeLaOh2016 [Section 6.4], which in turn is a rescaled version of the cubic potential example from @PrPe2000 with an additional quartic confinement term to make sure that the potential is bounded from below; see the assumptions in Proposition \[prop:H\_M\].
The initial position of the particle in the phase space is $(q(0),p(0)) = (0.25,1)$; this gives the classical total energy $H_{\text{cl}} \simeq 0.641$. This is below the local maximum $V_{1} \simeq 1.703$ of the potential (located at $x \simeq 1.73$) and hence the solution $(q(t),p(t))$ to the classical Hamiltonian system in gives a periodic orbit confined in the potential well; see Fig. \[fig:V\].
(-3,-0.35) rectangle (1,2); (-3,0) – (0.75,0) node\[below\] [$x$]{}; (0,-1) – (0,2.5); plot (,[2\*pow(,2) + pow(,3) + 0.1\*pow(,4)]{}); at (0.8,2) [$V(x)$]{}; (0.25,0.075) – (0.25,-0.075) node\[below\] [$-0.25$]{}; (0.25,0) circle (1pt); (0.25,0) – (-0.15,0); at (-10/3,0) [$-10/3$]{}; (-10/3,0) – (-10/3,1.85); (-0.686347,0.641016) – (0.503472,0.641016); at (0,0.641016) [$H_{\text{cl}}$]{}; (-1.73444,1.70386) – (0,1.70386); at (0,1.70386) [$V_{1} \simeq 1.703$]{}; (-1.73444,1.70386) – (-1.73444,0) node\[below\] [$-1.73$]{}; (-2.5,0.075) – (-2.5,-0.075) node\[below\] [$-2.5$]{};
However, the semiclassical Hamiltonian dynamics with the same initial condition $(q(0),p(0))$ may not be confined in the potential well because it is a Hamiltonian system in the higher-dimensional space $T^{*}\R^{d} \times \Sigma_{d}$. We set the initial condition for $(\mathcal{A},\mathcal{B})$ as $(\mathcal{A}(0),\mathcal{B}(0)) = (0,1)$; the phase is set as $\phi(0) = 0$ because it is irrelevant to the dynamics of observables; $\delta(0)$ is set so that the initial wave function is normalized. This means that the initial wave function $$\label{eq:psi-IC}
\psi(0,x) = \chi_{n}(q(0),p(0),\mathcal{A}(0),\mathcal{B}(0),\phi(0),\delta(0);x)$$ is the Hermite function with index $n\in\N_{0}$ because the ground state $\chi_{0}$ and the ladder operators become those of the harmonic oscillator; see and .
Results
-------
We computed the solutions of the classical system (the first two equations of ) and the Hamiltonian system for the semiclassical wave packet $\chi_{n}$. Also, for a reference solution $(\exval{\hat{x}}(t),\exval{\hat{p}}(t))$ of the expectation value dynamics, we used a method based on Egorov’s Theorem with the phase space density developed in [@KeLaOh2016] (essentially equivalent to the Initial Value Representation (IVR) method [@Mi1970; @Mi1974b; @WaSuMi1998; @Mi2001] often used by chemical physicists). It is known that such Egorov-type/IVR algorithms give an $O(\hbar^{2})$ approximation to the expectation value dynamics of the Schrödinger equation , and hence provide a very good alternative to the exact solution in the semiclassical regime $\hbar \ll 1$.
We used the Störmer–Verlet method [@Ve1967] to solve the classical Hamiltonian system and the variational splitting integrator of @FaLu2006 (see also @Lu2008 [Section IV.4]) for the semiclassical solution; the time step is $0.01$ in all the cases. It is easy to show that the variational splitting integrator preserves the symplectic structure , and its limit as $\hbar \to 0$ gives the Störmer–Verlet method [@Ve1967]. The Egorov-type algorithm involves averaging of solutions of the classical Hamiltonian system with numerous initial conditions sampled with respect to the initial phase space densities corresponding to the initial wave function . Again the classical Hamiltonian system is solved using the Störmer–Verlet method and $100,000$ initial conditions are sampled to ensure accuracy.
The phase space plots of the results with $n = 1,3,5,10$ are shown in Figs. \[fig:0.05\]–\[fig:0.01\] for $\hbar = 0.05, 0.025, 0.01$. Fig. \[fig:0.025\_t-H\] shows the time evolution of the classical energy $H_{\text{cl}} = p^{2}/(2m) + V$ along the classical solution, the semiclassical energy $\overline{H}^{(n)}_{\hbar}$ along the semiclassical solution, and the expectation value $\texval{\hat{H}}$ along the solutions of the Egorov-type algorithm. They are shown for $0 \le t \le T$ where $T \simeq 3.39$ is the period of the classical solution. The classical solution is trapped inside the potential well as explained earlier. On the other hand, the semiclassical energy or Hamiltonian $\overline{H}^{(n)}_{\hbar}$ in becomes larger for larger values of $\hbar$ and $n$, and significantly deviates from the classical energy $H_{\text{cl}}$; see Fig. \[fig:0.025\_t-H\] to see how $\overline{H}^{(n)}_{\hbar}$ changes as $\hbar$ becomes larger. As a result, the solutions escape from the potential well for relatively large values of $\hbar$ and $n$ whereas it is trapped inside the well for some small values of $\hbar$ and $n$. However, note that, unlike the classical case, $\overline{H}^{(n)}_{\hbar} < V_{1}$ does not necessarily imply that the trajectory is trapped inside the well because, as mentioned above, the semiclassical dynamics is a Hamiltonian system on $T^{*}\R^{d} \times \Sigma_{d}$ and so the level set of the Hamiltonian does not necessarily define a closed curve in $T^{*}\R^{d}$ even with $\overline{H}^{(n)}_{\hbar} < V_{1}$. This is in fact the case for, e.g., $\hbar = 0.025$ and $n = 5, 10$; see Figs. \[fig:0.025\] and \[fig:0.025\_t-H\].
More importantly, [*the semiclassical solutions show a very good agreement with the reference solutions computed by the Egorov-type algorithm*]{}. Note that these solutions are computed using completely different methods: one from a single semiclassical Hamiltonian system whereas the other by sampling numerous solutions of the classical Hamiltonian system.
However, there is an issue with the semiclassical solutions as well. The solutions of the semiclassical system deviate from the reference solutions after a while as we can see in some of the solutions in the figures. In fact, it is known that approximation methods using semiclassical wave packets are usually valid only in the Ehrenfest time scale, i.e., $t \sim \ln(1/\hbar)$; see, e.g., @HaJo2000, @CoRo2012, and @ScVaTo2012. It is because the wave packet becomes very widespread, i.e., the parameter $\mathcal{B}$—which controls the width of the wave packet—becomes significantly small in the Ehrenfest time scale. The Ehrenfest time scales in our settings are roughly the same as the period $T \simeq 3.39$ of the classical solution. This issue seems to exacerbate the errors in the numerical solution; see, for example the behavior of the Hamiltonian for $\hbar = 0.025$ and $n = 10$ in Fig. \[fig:0.025\_t-H\].
[^1]: @Ha1998 uses parameters $A, B \in \Mat_{d}(\C)$, which are related to $Q$ and $P$ as $A = Q$ and $B = -\rmi P$.
|
---
abstract: 'The Online Encyclopedia of Integer Sequences (OEIS) is a catalog of integer sequences. We are particularly interested in the number of occurrences of $N(n)$ of an integer $n$ in the database. This number $N(n)$ marks the importance of $n$ and it varies noticeably from one number to another, and from one number to the next in a series. “Importance" can be mathematically objective ($2^{10}$ is an example of an “important" number in this sense) or as the result of a shared mathematical culture ($10^9$ is more important than $9^{10}$ because we use a decimal notation). The concept of algorithmic complexity [@kolmo; @chaitin; @levin] (also known as Kolmogorov or Kolmogorov-Chaitin complexity) will be used to explain the curve shape as an “objective" measure. However, the observed curve is not conform to the curve predicted by an analysis based on algorithmic complexity because of a clear gap separating the distribution into two clouds of points. A clear zone in the value of $N(n)$ first noticed by Philippe Guglielmetti[^1]. We shall call this gap “Sloane’s gap".'
author:
- |
Nicolas Gauvrit$^1$, Jean-Paul Delahaye$^2$ and Hector Zenil$^2$\
$^{1}$LADR, EA 1547, Centre Chevaleret, Université de Paris VII\
[email protected]\
$^{2}$LIFL, Laboratoire d’Informatique Fondamentale de Lille\
UMR CNRS 8022, Université de Lille I\
{hector.zenil,delahaye}@lifl.fr
title: 'Sloane’s Gap: Do Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS?'
---
Introduction
============
The Sloane encyclopedia of integer sequences[@sloane3][^2] (OEIS) is a remarkable database of sequences of integer numbers, carried out methodically and with determination over forty years[@cipra]. As for May 27, 2011, the OEIS contained 189,701 integer sequences. Its compilation has involved hundreds of mathematicians, which confers it an air of homogeneity and apparently some general mathematical objectivity–something we will discuss later on.
When plotting $N(n)$ (the number of occurrences of an integer in the OEIS) two main features are evident:\
(a) Statistical regression shows that the points $N(n)$ cluster around $k/n^{1.33}$, where $k = 2.53 \times 10^8$.\
(b) Visual inspection of the graph shows that actually there are two distinct sub-clusters (the upper one and the lower one) and there is a visible gap between them. We introduce and explain the phenomenon of “Sloane’s gap.”
The paper and rationale of our explanation proceeds as follows:\
We explain that (a) can be understood using algorithmic information theory. If $U$ is a universal Turing machine, and we denote $m(x)$ the probability that $U$ produces a string $x$, then $m(x) = k 2^{-K(x)+O(1)}$, for some constant $k$, where $K(x)$ is the length of the shortest description of $x$ via $U$. $m(.)$ is usually refered to as the Levin’s universal distribution or the Solomonoff-Levin measure [@levin]. For a number $n$, viewed as a binary string via its binary representation, $K(n) \leq \log_{2} n + 2 \log_{2} \log_{2} n + O(1)$ and, for most $n$, $K(n) \geq \log_{2} n$. Therefore for most $n$, $m(n)$ lies between $k/( n (\log_{2} n)^2)$ and $k/n$. Thus, if we view OEIS in some sense as a universal Turing machine, algorithmic probability explains (a).
Fact (b), however, is not predicted by algorithmic complexity and is not produced when a database is populated with automatically generated sequences. This gap is unexpected and requires an explanation. We speculate that OEIS is biased towards social preferences of mathematicians and their strong interest in certain sequences of integers (even numbers, primes, and so on). We quantified such a bias and provided statistical facts about it.
Presentation of the database
============================
The encyclopedia is represented as a catalogue of sequences of whole numbers and not as a list of numbers. However, the underlying vision of the work as well as its arrangement make it effectively a dictionary of numbers, with the capacity to determine the particular properties of a given integer as well as how many known properties a given integer possesses.
A common use of the Sloane encyclopedia is in determining the logic of a sequence of integers. If, for example, you submit to it the sequence 3, 4, 6, 8, 12, 14, 18, 20..., you will instantly find that it has to do with the sequence of prime augmented numbers, as follows: 2+1, 3+1, 5+1, 7+1, 11+1, 13+1, 17+1, 19+1...
Even more interesting, perhaps, is the program’s capacity to query the database about an isolated number. Let us take as an example the Hardy-Ramanujan number, 1729 (the smallest integer being the sum of two cubes of two different shapes). The program indicates that it knows of more than 350 sequences to which 1729 belongs. Each one identifies a property of 1729 that it is possible to examine. The responses are classified in order of importance, an order based on the citations of sequences in mathematical commentaries and the encyclopedia’s own cross-references. Its foremost property is that it is the third Carmichael number (number $n$ not prime for which $\forall a\in \mathbb{N}^{\ast },$ $n|a^{n}-a)$). Next in importance is that 1729 is the sixth pseudo prime in base 2 (number $n$ not prime such that $n|2^{n-1}-1$). Its third property is that it belongs among the terms of a simple generative series. The property expounded by Ramanujan from his hospital bed appears as the fourth principle. In reviewing the responses from the encyclopedia, one finds further that:
- 1729 is the thirteenth number of the form $n^3+1$;
- 1729 is the fourth “factorial sextuple", that is to say, a product of successive terms of the form $6n +1$: $1729=1\times 7\times 13\times 19$;
- 1729 is the ninth number of the form $n^{3}+(n+1)^{3}$;
- 1729 is the sum of the factors of a perfect square ($33^2$);
- 1729 is a number whose digits, when added together yield its largest factor ($1+7+2+9=19$ and $1729 =7\times 13\times 19$);
- 1729 is the product of 19 a prime number, multiplied by 91, its inverse;
- 1729 is the total number of ways to express 33 as the sum of 6 integers.
The sequence encyclopedia of Neil Sloane comprises more than 150000 sequences. A partial version retaining only the most important sequences of the database was published by Neil Sloane and Simon Plouffe[@sloane2] in 1995. It records a selection of 5487 sequences[@sloane2] and echoes an earlier publication by Sloane [@sloane].
Approximately forty mathematicians constitute the “editorial committee" of the database, but any user may propose sequences. If approved, they are added to the database according to criteria of mathematical interest. Neil Sloane’s flexibility is apparent in the ease with which he adds new sequences as they are proposed. A degree of filtering is inevitable to maintain the quality of the database. Further, there exist a large number of infinite families of sequences (all the sequences of the form ($kn$), all the sequences of the form ($k^n$), etc.), of which it is understood that only the first numbers are recorded in the encyclopedia. A program is also used in the event of a failure of a direct query which allows sequences of families that are not explicitly recorded in the encyclopedia to be recognized.
Each sequence recorded in the database appears in the form of its first terms. The size of first terms associated with each sequence is limited to approximately 180 digits. As a result, even if the sequence is easy to calculate, only its first terms will be expressed. Next to the first terms and extending from the beginning of the sequence, the encyclopedia proposes all sorts of other data about the sequence, e.g., the definitions of it and bibliographical references.
Sloane’s integer encyclopedia is available in the form of a data file that is easy to read, and that contains only the terms retained for each sequence. One can download the data file free of charge and use it–with mathematical software, for example–to study the expressed numbers and conduct statistical research about the givens it contains.
One can, for example, ask the question: “Which numbers do not appear in Sloane’s encyclopedia?" At the time of an initial calculation conducted in August 2008 by Philippe Guglielmetti, the smallest absent number tracked down was 8795, followed in order by 9935, 11147, 11446, 11612, 11630,... When the same calculation was made again in February 2009, the encyclopedia having been augmented by the addition of several hundreds of new sequences, the series of absent numbers was found to comprise 11630, 12067, 12407, 12887, 13258...
The instability over time of the sequence of missing numbers in the OEIS suggests the need for a study of the distribution of numbers rather than of their mere presence or absence. Let us consider the number of properties of an integer, $N(n)$, while measuring it by the number of times $n$ appears in the number file of the Sloane encyclopedia. The sequence $N(n)$ is certainly unstable over time, but it varies slowly, and certain ideas that one can derive from the values of $N(n)$ are nevertheless quite stable. The values of $N(n)$ are represented in Figure 1. In this logarithmic scale graph a cloud formation with regular decline curve is shown.
Let us give a few examples: the value of $N(1729)$ is 380 (February 2009), which is fairly high for a number of this order of magnitude. For its predecessor, one nevertheless calculates $N(1728)=622$, which is better still. The number 1728 would thus have been easier for Ramanujan! Conversely, $N(1730)=106$ and thus 1730 would have required a more elaborate answer than 1729.
The sequence $\left( N(n)\right) _{n\in \mathbb{N}^{\ast }}$ is generally characterized by a decreasing curve. However, certain numbers $n$ contradict this rule and possess more properties than their predecessors: $N(n)>N(n-1)$.
We can designate such numbers as “interesting". The first interesting number according to this definition is 15, because $N(15)=34\,183$ and $N(14)=32\,487$. Appearing next in order are 16, 23, 24, 27, 28, 29, 30, 35, 36, 40, 42, 45, 47, 48, 52, 53, etc.
We insist on the fact that, although unquestionably dependent on certain individual decisions made by those who participate in building the sequence database, the database is not in itself arbitrary. The number of contributors is very large, and the idea that the database represents an objective view (or at least an intersubjective view) of the numeric world could be defended on the grounds that it comprises the independent view of each person who contributes to it and reflects a stable mathematical (or cultural) reality.
Indirect support for the idea that the encyclopedia is not arbitrary, based as it is on the cumulative work of the mathematical community, is the general cloud-shaped formation of points determined by $N(n)$, which aggregates along a regular curve (see below).
Philippe Guglielmetti has observed that this cloud possesses a remarkable characteristic[^3]: it is divided into two parts separated by a clear zone, as if the numbers sorted themselves into two categories, the more interesting above the clear zone, and the less interesting below the clear zone. We have given the name “Sloane’s Gap" to the clear zone that divides in two the cloud representing the graph of the function $n\longmapsto N(n)$. Our goal in this paper is to describe the form of the cloud, and then to formulate an explanatory hypothesis for it.
Description of the cloud
========================
Having briefly described the general form of the cloud, we shall direct ourselves more particularly to the gap, and we will investigate what characterizes the points that are situated above it.
General shape
-------------
The number of occurrences $N$ is close to a grossly decreasing convex function of $n$, as one can see from Figure 1.
A logarithmic regression provides a more precise idea of the form of the cloud for $n$ varying from 1 to 10000. In this interval, the coefficient of determination of the logarithmic regression of $\ln \left( N\left( n\right) \right)$ in $n$ is of $r^2=.81$, and the equation of regression gives the estimation: $$\ln \left( N\left( n\right) \right) \simeq -1.33\ln (n)+14.76$$ or $$\hat{N}\left( n\right) =\frac{k}{n^{1.33}},$$ where $k$ is a constant having the approximate value $2.57\times 10^8$, and $\hat{N}$ is the estimated value for $N$.
Thus the form of the function $N$ is determined by the equation above. Is the existence of Sloane’s gap natural then, or does it demand a specific explanation? We note that to our knowledge, only one publication mentions the existence of this split [@delahaye].
Defining the gap
----------------
In order to study the gap, the first step is to determine a criterion for classification of the points. Given that the “gap" is not clearly visible for the first values of $n$, we exclude from our study numbers less than 300.
One empirical method of determining the boundary of the gap is the following: for the values ranging from 301 to 499, we use a straight line adjusted “by sight", starting from the representation of $\ln(N)$ in functions of $n$. For subsequent values, we take as limit value of $n$ the 82nd percentile of the interval $[n-c,n+c]$. $c$ is fixed at 100 up to $n=1000$, then to 350. It is clearly a matter of a purely empirical choice that does not require the force of a demonstration. The result corresponds roughly to what we perceive as the gap, with the understanding that a zone of uncertainty will always exist, since the gap is not entirely devoid of points. Figure 2 shows the resulting image.
Characteristics of numbers “above"
----------------------------------
We will henceforth designate as $A$ the set of abscissae of points classified “above" the gap by the method that we have used. Of the numbers between 301 and 10000, 18.2% are found in A– 1767 values.
In this section, we are looking for the properties of these numbers. Philippe Guglielmetti has already remarked that the prime numbers and the powers of two seem to situate themselves more frequently above the gap. The idea is that certain classes of numbers that are particularly simple or of particular interest to the mathematician are over-represented.
### Squares
83 square numbers are found between 301 and 10000. Among these, 79 are located above the gap, and 4 below the gap, namely, numbers 361, 484, 529, and 676. Although they may not be elements of $A$, these numbers are close to the boundary. One can verify that they collectively realize the local maximums for $\ln (N)$ in the set of numbers classified under the cloud. One has, for example, $N(361)=1376$, which is the local maximum of $\left\{ N\left( n\right), n\in \left[ 325,10~000\right] \backslash A\right\} $. For each of these four numbers, Table 1 gives the number of occurrences $N$ in Sloane’s list, as well as the value limit that they would have to attain to belong to $A$.
95.2% of squares are found in $A$, as opposed to 17.6% of non-squares. The probability that a square number will be in $A$ is thus 5.4 times greater than that for the other numbers.
$$\begin{tabular}{c|c|c}
\hline
$n$ & $N\left( n\right) $ & value limit \\
\hline
361 & 1376 & 1481 \\
484 & 976 & 1225 \\
529 & 962 & 1065 \\
676 & 706 & 855\\
\hline
\end{tabular}$$
Table 1–List of the square numbers $n$ found between 301 and 10000 not belonging to $A$, together with their frequency of occurrence and the value of $N(n)$ needed for $n$ to be classified in $A$.
### Prime numbers
The interval under consideration contains 1167 prime numbers. Among them, 3 are not in $A$: the numbers 947, 8963, and 9623. These three numbers are very close to the boundary. 947 appears 583 times, while the limit of $A$ is 584. Numbers 8963 and 6923 appear 27 times each, and the common limit is 28.
99.7% of prime numbers belong to $A$, and 92.9% of non-prime numbers belong to the complement of $A$. The probability that a prime number will belong to $A$ is thus 14 times greater than the same probability for a non-prime number.
### A multitude of factors
Another class of numbers that is seemingly over-represented in set $A$ is the set of integers that have “a multitude of factors". This is based on the observation that the probability of belonging to $A$ increases with the number of prime factors (counted with their multiples), as can be seen in Figure 3. To refine this idea we have selected the numbers $n$ of which the number of prime factors (with their multiplicty) exceeds the 95th percentile, corresponding to the interval $[n-100,n+100]$.
811 numbers meet this criterion. Of these, 39% are found in $A$, as opposed to 16.3% for the other numbers. The probability that a number that has a multitude of prime factors will belong to $A$ is thus 2.4 times greater than the same probability for a number that has a smaller number of factors. Table 2 shows the composition of $A$ as a function of the classes that we have considered.
$$\begin{tabular}{c|c|c|c}
\hline
class & number in $A$ & \% of $A$ & \% (cumulated) \\
\hline
primes & 1164 & 65.9 & 65.9 \\
squares & 79 & 4.5 & 70.4 \\
many factors & 316 & 17.9 & 87.9\\
\hline
\end{tabular}$$
Table 2–For each class of numbers discussed above, they give the number of occurrences in $A$, the corresponding percentage and the cumulative\
percentage in $A$.
### Other cases
The set $A$ thus contains almost all prime numbers, 95% of squares, and a significant percentage of numbers that have a multitude of factors and all the numbers possessing at least ten prime factors (counted with their multiplicity).
These different classes of numbers by themselves represent 87.9% of $A$. Among the remaining numbers, some evince outstanding properties, for example, linked to decimal notation, as in: 1111, 2222, 3333$\ldots$. Others have a simple form, such as 1023, 1025, 2047, 2049... that are written $2^n+1$ or $2^n-1$.
When these cases that for one reason or another possess an evident “simplicity" are eliminated, there remains a proportion of less than 10% of numbers in $A$ for which one cannot immediately discern any particular property.
Explanation of the cloud-shape formation
========================================
Overview of the theory of algorithmic complexity
------------------------------------------------
Save in a few exceptional cases, for a number to possess a multitude of properties implies that the said properties are simple, where simple is taken to mean “what may be expressed in a few words". Conversely, if a number possesses a simple property, then it will possess many properties. For example, if $n$ is a multiple of 3, then $n$ will be a even multiple of 3 or a odd multiple of 3. Being a “even multiple of 3" or “odd multiple of 3" is a little more complex than just being a “multiple of 3", but it is still simple enough, and one may further propose that many sequences in Sloane’s database are actually sub-sequences of other, simpler ones. In specifying a simple property, its definition becomes more complex (by generating a sub-sequence of itself), but since there are many ways to specify a simple property, any number that possesses a simple property necessarily possesses numerous properties that are also simple.
The property of $n$ corresponding to a high value of $N(n)$ thus seems related to the property of admitting a “simple" description. The value $N(n)$ appears in this context as an indirect measure of the simplicity of $n$, if one designates as “simple" the numbers that have properties expressible in a few words.
Algorithmic complexity theory[@kolmo; @chaitin; @levin] assigns a specific mathematical sense to the notion of simplicity, as the objects that “can be described with a short definition". Its modern formulation can be found in the work of Li and Vitanyi[@li], and Calude[@calude].
Briefly, this theory proposes to measure the complexity of a finite object in binary code (for example, a number written in binary notation) by the length of the shortest program that generates a representation of it. The reference to a universal programming language (insofar as all computable functions can possess a program) leads to a theorem of invariance that warrants a certain independence of the programming language.
More precisely, if $L_{1}$ and $L_{2}$ are two universal languages, and if one notes $K_{L_{1}}$ (resp. $K_{L_{2}}$) algorithmic complexity defined with reference to $L_{1}$ (resp. to $L_{2}$), then there exists a constant $c$ such that $|K_{L_1}(s)-K_{L_2}(s)| < c $ for all finite binary sequences $s$.
A theorem (see for example \[theorem 4.3.3. page 253 in [@li]\]) links the probability of obtaining an object $s$ (by activating a certain type of universal TM–called optimal–running on binary input where the bits are chosen uniformly random) and its complexity $K(s)$. The rationale of this theorem is that if a number has many properties then it also has a simple property.
The translation of this theorem for $N(n)$ is that if one established a universal language $L$, and established a complexity limit $M$ (only admitting descriptions of numbers capable of expression in fewer than $M$ symbols), and counted the number of descriptions of each integer, one would find that $\frac{N(n)}{M}$ (where $M=\sum_{i\in \mathbb{N}}N(i)$) is approximately proportional to: $\frac{1}{2^{K\left( n\right) }}$: $$\frac{N(n)}{M}=\frac{1}{2^{K\left( n\right) +O(\ln (\ln (n)) )}}.$$
Given that $K(n)$ is non computable because of the undecidability of the halting problem and the role of the additive constants involved, a precise calculation of the expected value of $N(n)$ is impossible. By contrast, the strong analogy between the theoretical situation envisaged by algorithmic complexity and the situation one finds when one examines $N(n)$ inferred from Sloane’s database, leads one to think that $N(n)$ should be asymptotically dependent on $\frac{1}{2^{K\left( n\right) }}$. Certain properties of $K(n)$ are obliquely independent of the reference language chosen to define $K$. The most important of these are:
- $K(n)<\log _{2}(n)+2\log _{2}(\log_{2}(n))+c^{\prime }$ ($c^{\prime }$ a constant)
- the proportion of $n$ of a given length (when written in binary) for which $K(n)$ recedes from $\log_2(n)$ decreases exponentially (precisely speaking, less than an integer among $2^q$ of length $k$, has an algorithmic complexity $K(n)\leq k-q$).
In graphic terms, these properties indicate that the cloud of points obtained from writing the following $\frac{1}{2^{K\left( n\right) }}$ would be situated above a curve defined by $$f(n)\approx \frac{h}{2^{\log _{2}\left( n\right) }}=\frac{h}{n}$$ ($h$ being a constant), and that all the points cluster on the curve, with the density of the points deviating from the curve decreasing rapidly.
This is indeed the situation we observe in examining the curve giving $N(n)$. The theory of algorithmic information thus provides a good description of what is observable from the curve $N(n)$. That justifies an a posteriori recourse to the theoretical concepts of algorithmic complexity in order to understand the form of the curve $N(n)$. By contrast, nothing in the theory leads one to expect a gap like the one actually observed. To the contrary, continuity of form is expected from the fact that $n+1$ is never much more complex than $n$.
To summarize, if $N(n)$ represented an objective measure of the complexity of numbers (the larger $N(n)$ is, the simpler $n$ ), these values would then be comparable to those that yield $\frac{1}{2^{K\left( n\right) }}$. One should thus observe a rapid decrease in size, and a clustering of values near the base against an oblique curve, but one should not observe a gap, which presents itself here as an anomaly.
To confirm the conclusion that the presence of the gap results from special factors and render it more convincing, we have conducted a numerical experiment.
We define random functions $f$ in the following manner (thanks to the algebraic system $Mathematica$):
1. Choose at random a number $i$ between 1 and 5 (bearing in mind in the selection the proportions of functions for which $i=1$, $i=2$, $\ldots$, $i =5$ among all those definable in this way).
2. If $i=1$, $f$ is defined by choosing uniformly at random a constant $k\in \left\{ 1,...,9\right\}$, a binary operator $\varphi $ from among the following list: $+$, $\times$, and subtraction sign, in a uniform manner, and a unary operand $g$ that is identity with probability .8, and the function squared with probability .2 (to reproduce the proportions observed in Sloane’s database). One therefore posits $f_i(n)=\varphi (g(n),k).$
3. If $i\geq 2$, $f_i$ is defined by $f_i(n) =\varphi (g(f_{i-1}(n)),k), $ where $k$ is a random integer found between 1 and 9, $g$ and $\varphi $ are selected as described in the point 2 (above), and $f_{i-1}$ is a random function selected in the same manner as in 2.
For each function $f$ that is generated in this way, one calculates $f(n)$ for $n=1, $…$, 20.$ These terms are regrouped and counted as for $N(n)$. The results appear in Figure 4. The result confirms what the relationship with algorithmic complexity would lead us to expect. There is a decreasing oblique curve with a mean near 0, with clustering of the points near the base, but no gap.
The gap: A social effect?
-------------------------
This anomaly with respect to the theoretical implications and modeling is undoubtedly a sign that what one sees in Sloane’s database is not a simple objective measure of complexity (or of intrinsic mathematical interest), but rather a trait of psychological or social origin that mars its pure expression. That is the hypothesis that we propose here. Under all circumstances, a purely mathematical vision based on algorithmic complexity would encounter an obstacle here, and the social hypothesis is both simple and natural owing to the fact that Sloane’s database, while it is entirely “objective", is also a social construct.
Figure 5 illustrates and specifies our hypothesis that the mathematical community is particularly interested in certain numbers of moderate or weak complexity (in the central zone or on the right side of the distribution), and this interest creates a shift toward the right-hand side of one part of the distribution (schematized here by the grey arrow). The new distribution that develops out of it (represented in the bottom figure) presents a gap.
We suppose that the distribution anticipated by considerations of complexity is deformed by the social effect concomitant with it: mathematicians are more interested in certain numbers that are linked to selected properties by the scientific community. This interest can have cultural reasons or mathematical reasons (as per results already obtained), but in either case it brings with it an over-investment on the part of the mathematical community. The numbers that receive this specific over-investment are not in general complex, since interest is directed toward them because certain regularities have been discovered in them. Rather, these numbers are situated near the pinnacle of a theoretical asymmetrical distribution. Owing to the community’s over-investment, they are found to have shifted towards the right-hand side of the distribution, thus explaining Sloane’s gap.
It is, for example, what is generated by numbers of the form $2^n +1$, all in A, because arithmetical results can be obtained from this type of number that are useful to prime numbers. Following some interesting preliminary discoveries, scientific investment in this class of integers has become intense, and they appear in numerous sequences. Certainly, $2^n+1$ is objectively a simple number, and thus it is normal that it falls above the gap. Nevertheless, the difference in complexity between $2^n+1$ and $2^n+2$ is weak. We suppose that the observed difference also reflects a social dynamic which tends to augment $N(2^n+1)$ for reasons that complexity alone would not entirely explain.
Conclusion
==========
The cloud of points representing the function $N$ presents a general form evoking an underlying function characterized by rapid decrease and “clustering near the base" (local asymmetrical distribution). This form is explained, at least qualitatively, by the theory of algorithmic information.
If the general cloud formation was anticipated, the presence of Sloane’s gap has, by contrast, proved more challenging to its observers. This gap has not, to our knowledge, been successfully explained on the basis of uniquely numerical considerations that are independent of human nature as it impinges on the work of mathematics. Algorithmic complexity anticipates a certain “continuity" of $N$, since the complexity of $n+1$ is always close to that of $n$. The discontinuity that is manifest in Sloane’s gap is thus difficult to attribute to purely mathematical properties independent of social contingencies.
By contrast, as we have seen, it is explained very well by the conduct of research that entails the over-representation of certain numbers of weak or medium complexity. Thus the cloud of points representing the function $N$ shows features that can be understood as being the result of at the same time human and purely mathematical factors.
[99]{}
CALUDE, C.S. *Information and Randomness: An Algorithmic Perspective.* (Texts in Theoretical Computer Science. An EATCS Series), Springer; 2nd. edition, 2002. CHAITIN, G.J. *Algorithmic Information Theory,* Cambridge University Press, 1987. CIPRA, B. *Mathematicians get an on-line fingerprint file,* Science, 205, (1994), p. 473. DELAHAYE, J.-P. *Mille collections de nombres,* Pour La Science, 379, (2009), p. 88-93. DELAHAYE, J.-P., ZENIL, H. “On the Kolmogorov-Chaitin complexity for short sequences", in CALUDE, C.S. (ed.) *Randomness and Complexity: from Chaitin to Leibniz*, World Scientific, p. 343-358, 2007. KOLMOGOROV, A.N. *Three approaches to the quantitative definition of information*. Problems of Information and Transmission, 1(1): 1–7, 1965. LEVIN, L. *Universal Search Problems.* 9(3): 265-266, 1973 (c). (submitted: 1972, reported in talks: 1971). English translation in: B.A.Trakhtenbrot. *A Survey of Russian Approaches to Perebor (Brute-force Search) Algorithms.* Annals of the History of Computing 6(4): 384-400, 1984. LI, M., VITANYI, P. *An introduction to Kolmogorov complexity and its applications,* Springer, 1997. SLOANE, N.J.A. *A Handbook of Integer Sequences,* Academic Press, 1973. SLOANE, N.J.A. *The on-line encyclopedia of integer sequences,* Notices of the American Mathematical Society, 8, (2003), p. 912-915. SLOANE, N.J.A. PLOUFFE, S. *The Encyclopedia of Integer Sequences,* Academic Press, 1995.
[^1]: On his site <http://drgoulu.com/2009/04/18/nombres-mineralises/> last consulted 1 June, 2011.
[^2]: The encyclopedia is available at: http://oeis.org/, last consulted 26 may, 2011.
[^3]: Personal communication with one of the authors, 16th of February, 2009.
|
---
abstract: |
Near-infrared (NIR) spectra of the subluminous Type Ia supernova SN 1999by are presented which cover the time evolution from about 4 days before to 2 weeks after maximum light. Analysis of these data was accomplished through the construction of an extended set of delayed detonation (DD) models covering the entire range of normal to subluminous SNe Ia. The explosion, light curves (LC), and the time evolution of the synthetic spectra were calculated self-consistently for each model with the only free parameters being the initial structure of the white dwarf (WD) and the description of the nuclear burning front during the explosion. From these, one model was selected for SN 1999by by matching the synthetic and observed optical light curves, principly the rapid brightness decline. DD models require a minimum amount of burning during the deflagration phase which implies a lower limit for the $^{56}Ni$ mass of about $0.1 M_\odot$ and consequently a lower limit for the SN brightness. The models which best match the optical light curve of SN 1999by were those with a $^{56}Ni$ production close to this theoretical minimum. The data are consistent with little or no interstellar reddening ($E(B-V) \leq
0.12^m$) and we derive a distance or $11 \pm 2.5$ Mpc for SN 1999by in agreement with other estimates.
Without any modification, the synthetic spectra from this subluminous model match reasonably well the observed IR spectra taken on May 6, May 10, May 16 and May 24, 1999. These dates correspond roughly to $-4$ d, 0 d, and 6 d and 14 d after maximum light. Prior to maximum, the NIR spectra of SN 1999by are dominated by products of explosive carbon burning (O, Mg), and Si. Spectra taken after maximum light are dominated by products of incomplete Si burning. Unlike the behavior of normal Type Ia SNe, lines from iron-group elements only begin to show up in our last spectrum taken about two weeks after maximum light. The implied distribution of elements in velocity space agrees well with the DD model predictions for a subluminous SN Ia. Regardless of the explosion model, the long duration of the phases dominated by layers of explosive carbon and oxygen burning argues that SN 1999by was the explosion of a white dwarf at or near the Chandrasekhar mass.
The good agreement between the observations and the models without fine-tuning a large number of free parameters suggests that DD models are a good description of at least subluminous Type Ia SNe. Pure deflagration scenarios or mergers are unlikely and helium-triggered explosions can be ruled out. However, problems for DD models still remain, as the data seem to be at odds with recent 3-D models of the deflagration phase which predict significant mixing of the inner layers of the white dwarf prior to detonation. Possible solutions include the effects of rapid rotation on the propagation of nuclear flames during the explosive phase of burning, or extensive burning of carbon just prior to the runaway.
author:
- Peter Höflich
- 'Christopher L. Gerardy & Robert A. Fesen'
- Shoko Sakai
title: Infrared Spectra of the Subluminous Type Ia Supernova 1999by
---
Introduction
============
On April 30, SN 1999by was independently discovered at about $15^m$ by @arbor99, and @papenkova99. Images with the Katzman Automatic Imaging Telescope [@treffers97; @li99] of the same field provide an upper limit of $19.3^m$ on April 25.2 UT. The supernova (SN) was found in the Sb galaxy NGC 2841 which has been the host of three previous supernovae: SN 1912A, SN 1957A, and SN 1972R [@papenkova99]. Based on optical spectra, SN 1999by was identified as a Type Ia SN [@gerardy99]. Shortly thereafter, @garnavich99 reported that SN 1999by showed stronger than normal 5800 Å absorption and depressed flux near 4000 Å, suggesting that SN 1999by would be a significantly subluminous Type Ia event.
According to @bonanos99, SN 1999by reached a maximum light of $m_B=13.80^m \pm 0.02 ^m$ on UT 1999 May 10.5, and a maximum in the $V$ band of $m_V=13.36^m \pm 0.02 ^m$. The Lyon/Meudon Extragalactic Database (LEDA)[^1], gives the heliocentric radial velocity of NGC 2841, corrected for Virgo infall, as $811.545$ km s$^{-1}$. Using $H_0=65$ km s$^{-1}$ Mpc$^{-1}$ puts the distance to NGC 2841 at 12.5 Mpc with a distance modulus of $30.5^m$. Using this distance modulus and the photometry of @bonanos99, the absolute peak magnitudes of SN 1999by are $M_B=-16.68^m$ and $M_V=-17.12^m$. Thus, SN 1999by is underluminous by roughly 2.5 magnitudes as compared to a typical Type Ia supernova (SN Ia).
Detailed analysis of the optical light curves (LCs) by @toth00 confirm these basic results. The light curve of SN 1999by shows a very steep post-maximum brightness decline $M_V(\Delta M_{\Delta t=15d})$ of $1.35^m$ to $1.45^m$. Based on detailed fits of the LCs, @toth00 find the interstellar reddening to be $E(B-V) \leq 0.1^m$ which is in agreement with the values for galactic reddening given by @schlegel98 [$E(B-V)=0.015^m$], and @burstein82 [$E(B-V) \approx 0^m$]. Recently, @garnavich provided detailed optical LCs and spectra and found values consistent with previous data.
These measurements imply that SN 1999by was as underluminous as SN 1991bg, the prototypical example of the subluminous Type Ia subclass [@filippenko92; @leibundgut93]. Other members include SN 1992K [@hamuy96a; @hamuy96b], SN 1997cn [@turatto98], and SN 1998de [@modjaz00]. Some defining characteristics of the subclass are rapidly declining light curves ($M_B(\Delta M_{\Delta t=15d}) \simeq 1.9^m$), peak magnitudes fainter than normal by 2–3 mag, and redder colors at maximum light ($(B-V)_{\rm max}$ $\simeq$ 0.4–0.5$^m$). One of the currently active areas in SNe Ia research concerns the nature of these subluminous events. Theoretical interpretations of subluminous SN Ia include all three types of explosion mechanism: the centrally triggered detonation of a sub-Chandrasekhar mass WD, deflagration and delayed detonations of massive WD and two merging WDs (see sect. 3 & Woosley & Weaver, 1994 ; Höflich et al., 1995; Nugent et al., 1995, Milne et al. 1999). The possibility has also been raised that SN1991bg-like SNe Ia should not be classified with other SNe Ia at all as they may arise from different progenitors. SN 1999by is one of the best observed SNe Ia with data superior to that of SN 1991bg. In addition to the studies of optical light curves and spectra mentioned above, detailed polarization spectra of SN 1999by have been obtained and analyzed [@howell01]. Whereas ‘normal’ SNe Ia tend to show little or no polarization [e.g. @wang97], this supernova was significantly polarized, up to 0.7%, indicating an overall asphericity of $\approx 20\%$. This result suggests that there may be a connection between the observed asphericity and the subluminosity in SNe Ia.
In recent years it has become apparent that infrared spectroscopic observations can be used as valuable tools to determine the chemical structure of SN Ia envelopes. For instance, near-infrared spectra can be used to locate the boundaries between explosive carbon and oxygen burning, or between complete and incomplete silicon burning by measuring , and iron-group lines [@wheeler98]. However, the difficulties involved in obtaining high quality infrared spectra of supernovae has limited IR studies to a small handful of objects. The study of SN 1994D by @meikle96 is the only one to include both pre- and post-maximum spectra. Since most of their spectra was of the 1–1.3 region, @meikle96 concentrated on determining the origin of a feature at 1.05 . They concluded that this feature might be due to either 1.083 or 1.0926 , but found difficulties with both identifications. Detailed modeling of SN 1994D and SN 1986G was able to reproduce the basic NIR spectral features and their evolution [@hoflich97; @wheeler98]. In particular, the identification of the 1.05 feature in SN 1994D as due to could be established (recently confirmed by @lentz01), and a broad feature, appearing between $\approx 1.5$ and 1.9 was identified as a blend of iron-group elements.
We note that the nature of subluminous SNe Ia is important for the use of SNe Ia as distance indicators. In particular, the calibration of the brightness decline relation depends critically on subluminous SNe Ia because they significantly increase the required baseline observations. From this perspective, it may turn out to be critical to determine whether all SNe Ia form a homogeneous class of objects or not. However, current limitations of model calculations will not allow us to improve the accuracy of the current estimates for the absolute distance to SN 1999by, and the possible implications of our results on the use of SNe Ia for cosmology is not the subject of this paper. In this paper, we present near-infrared spectra of SN 1999by covering the time evolution of the supernova from about four days before to two weeks after maximum light. Detailed models of the SN explosion based on a delayed detonation (DD) scenario are used to analyze the data. The explosion models, light curves, and synthetic spectra are calculated in a self-consistent manner. Given the initial structure of the progenitor and description of the nuclear burning front, the light curves and spectra are calculated from the explosion model without any further tuning. The observations and data reduction are discussed in §2. In §3, general properties of the explosion models, light curves and spectra are discussed for normal and subluminous SNe Ia within the DD scenario. In §4, a specific model for this SN is chosen by matching the properties of the predicted light curve to those observed in optical studies of SN 1999by. Synthetic optical and IR spectra for this model are then presented and the latter are compared to the observed IR spectra. Detailed line identifications for the infrared features are provided. The effects on the predicted spectra of large scale mixing during the deflagration phase also examined. In §5, we discuss the implications for progenitors and alternative explosion mechanisms. We close in §6 with a final discussion of the results for SN 1999by, and put our findings in context with other SNe Ia and different explosion scenarios.
Observations
============
Low-dispersion (R $\approx$ 700), near-infrared spectra of SN 1999by from 1.0–2.4 were obtained with the 2.4m Hiltner Telescope at MDM Observatory during the nights of 6–10 May 1999. The data were collected using TIFKAM (a.k.a. ONIS), a high-throughput infrared imager and spectrograph with an ALLADIN 512 $\times$ 1024 InSb detector. This instrument can be operated with standard *J*, *H*, and *K* filters for broadband imaging, or with a variety of grisms, blocking filters, and an east-west oriented $0\farcs 6$ slit for low ($R \approx 700$) and moderate ($R \approx 1400$) resolution spectroscopy.
SN 1999by was observed with three different spectroscopic setups, which covered the 0.96–1.80 , 1.2–2.2 , and 1.95–2.4 wavelength regions. The observations were broken into multiple 600 s exposures. Between each exposure, the supernova light was dithered along the slit to minimize the effect of detector defects and provide first-order background subtraction. Total exposure times vary from spectrum to spectrum and are listed in the log of NIR observations in Table \[nirlog\].
Wavelength calibration of the spectra was achieved by observing neon, argon, and xenon arc lamps. The spectra were corrected for telluric absorption by observing A stars and early G dwarfs at similar airmass, chosen from the Bright Star Catalog [@BSC]. Applying the procedure described by @hanson98, stellar features were removed from the G dwarf spectra by dividing by a normalized solar spectrum [@solar1; @solar2][^2]. The results were used to correct for telluric absorption in the A stars. Hydrogen features in the corrected A star spectra were removed from the raw A star spectra and the results were used to correct the target data for telluric absorption. \[For further discussion of this procedure see @hanson98 [@HCR96], and references therein.\] The instrumental response was calibrated by matching the continuum of the A star telluric standards to the stellar atmosphere models of @kurucz93.
The first three nights (6-8 May) were photometric, and *J*, *H*, and *K* broadband images of SN 1999by and @persson photometric standards. The resulting photometry is listed in Table \[nirlog\] and was used to set the flux levels of the corresponding spectra. A square bandpass was assumed and the flux levels thus attained are believed accurate to $\sim$ 20–30%. For May 9 and 10 no flux information was available, but the observed spectral bands had large overlaps, and the relative flux levels were set by matching the data in the overlapping regions.
Near-infrared spectra of a star $29\farcs 5$ north and $11\farcs 7$ west of SN 1999by [@papenkova99] were also obtained and reduced in the same manner. This star was then used as a local relative spectrophotometric standard to reduce spectroscopic data from later observing runs. Spectra of SN 1999by covering 1.0–1.8 were collected on five more nights during the following two weeks (16 May, 18 May, 20 May, 21 May and 24 May). The first three were obtained by P. Martini and A. Steed using TIFKAM on the 2.4m telescope at MDM, and the last two by S. Sakai using TIFKAM on the 2.1m telescope at the Kitt Peak National Observatory.
Figure \[nirspec\] shows a plot of all the NIR spectra. Since absolute fluxes were only available for the first three spectra, the data are presented in arbitrary flux units, and they have been shifted vertically for clarity. Regions of very low S/N due to strong telluric absorption have been omitted. The epochs listed are relative to 10 May 1999, the date of $V_{Max}$ given by @bonanos99. Like all the subsequent plots, the data have been corrected to the 638 km s$^{-1}$ redshift of NGC 2841 [@devaucouleurs91].
Models for the Explosion, Light Curves, and Spectra
===================================================
There is general agreement that SNe Ia result from some process involving the combustion of a degenerate white dwarf [@hoyle60]. Within this general picture, three classes of models have been considered: (1) An explosion of a CO-WD, with mass close to the Chandrasekhar limit, which accretes mass through Roche-lobe overflow from an evolved companion star [@whelan73]. The explosion is triggered by compressional heating near the WD center. (2) An explosion of a rotating configuration formed from the merging of two low-mass WDs, caused by the loss of angular momentum through gravitational radiation [@webbink84; @iben84; @paczynski85]. (3) Explosion of a low mass CO-WD triggered by the detonation of a helium layer [@nomoto80; @woosley80; @woosley86].
Only the first two models appear to be viable for normal Type Ia SNe as the third, the sub-Chandrasekhar WD model, has been ruled out on the basis of predicted light curves and spectra [@hoflich96b; @nugent97]. Still, theoretical interpretations of subluminous SN Ia include all three types of explosion mechanism [@woosley94; @hkw95; @nugent95; @milne99]. The possibility has also been raised that the subluminous SN 1991bg-like supernovae may arise from different progenitors and should not be classified with other SNe Ia at all.
Within $M_{Ch}$ models, it is believed that the burning front starts as a subsonic deflagration. However, the time evolution of the burning front is still an open question. That is, whether the deflagration front burns through the entire WD [@nomoto84], or alternatively, transitions into a supersonic detonation mode as suggested in the delayed detonation (DD) model [@khokhlov91; @woosley94; @yamaoka92]. DD models have been found to reproduce the optical and infrared light curves and spectra of ‘typical’ SNe Ia reasonably well (@hoflich95, H95 hereafter; @hk96 [@hoflich96b; @fisher98; @nugent97; @wheeler98; @lentz01]).
In addition, DD models provide a natural explanation for the brightness decline relation $M (\Delta M_{\Delta t=15d})$ observed in the light curves of SNe Ia [@phillips87; @hamuy95; @hamuy96b; @suntzeff99]. This is a consequence of the temperature dependence of the opacity and the lack of a dependence of the explosion energy on the amount of $^{56}Ni$ produced (@hmk93 [@khokhlov93; @hoflich96b; @mazzali01]). Thus, within the delayed detonation scenario, both normal and subluminous SNe Ia can be explained as variants of a single phenomenon (H95, @hkw95, @umeda99). This makes the DD model an attractive scenario and it is used as the basis for our present analysis.
In our models, the explosions LCs and spectra are all calculated self-consistently. Given the initial structure of the progenitor and a description of the nuclear burning front, the light curves and spectra are calculated directly from the explosion model without any additional parameters. Since the predicted observables follow directly from the supernova model, this approach provides a direct link between the observations and the explosion physics and progenitor properties.
Finally, we note that detailed analyses of observed SN Ia spectra and light curves indicate that mergers [@benz90] and pure deflagration SNe (such as the W7 model) could contribute to the population of bright SNe Ia (@hk96, hereafter HK96; @hatano00). On the other hand, it is not obvious that subluminous SNe Ia and the brightness decline relation can be understood within pure deflagration models. In particular, the amount of burning, and consequently the explosion energy, will be strongly correlated with $^{56}Ni$ production. Since no detailed studies for deflagration models have been performed, these models cannot be ruled out but, as we discuss in §5, the IR spectra of SN 1999by are hard to reconcile with a pure deflagration.
Numerical Methods
-----------------
### The Explosion
Despite recent progress in our understanding of nuclear burning fronts, current 3-D models are not sufficiently evolved to allow for a consistent treatment of the burning front throughout all phases [@khokhlov01 see also §4.3 & §5]. Thus detailed models rely on parameterized descriptions guided by 3-D results. Within spherical models, the transition from deflagration to detonation can be conveniently parameterized by a density $\rho_{tr}$ which has been found to be the dominating factor for the determining chemical structure, light curves and spectra.
We have calculated explosion models using a one-dimensional radiation-hydro code (HK96) that solves the hydrodynamical equations explicitly by the piecewise parabolic method [@collela84]. Nuclear burning is taken into account using an extended network of 606 isotopes from n,p to $^{74}Kr$ [@thieleman96 and references therein]. The propagation of the nuclear burning front is given by the velocity of sound behind the burning front in the case of a detonation wave, and in a parameterized form during the deflagration phase calibrated by detailed 3-D calculations [e.g. @khokhlov01]. We use the parameterization as described in @dominguez00. For a deflagration front at distance $r_{burn}$ from the center, we assume that the burning velocity is given by $v_{\rm burn}=max(v_{t}, v_{l})$, where $v_{l}$ and $v_{t}$ are the laminar and turbulent velocities with $$v_{t}= 0.2 ~\sqrt{\alpha_{T} ~ g~L_f},~
\eqno{[1]}$$ with $$\alpha_T ={(\alpha-1)/( \alpha +1})$$ and $$\alpha ={\rho^+(r_{\rm burn})/ \rho^-(r_{\rm burn})}.$$ Here $\alpha _T$ is the Atwood number, $L_f$ is the characteristic length scale, and $\rho^+$ and $\rho^-$ are the densities in front of and behind the burning front, respectively. The quantities $\alpha$ and $L_f$ are directly taken from the hydrodynamical model at the location of the burning front and we take $L_f=r_{\rm burn(t)}$. The transition density is treated as a free parameter.
### The Light Curves
From these explosion models, the subsequent expansion, bolometric and broad band light curves (LC) are calculated following the method described by @hoflich98, and references therein. The LC-code is the same used for the explosion except that $\gamma$-ray transport is included via a Monte Carlo scheme and nuclear burning is neglected. In order to allow a more consistent treatment of the expansion, we solve the time-dependent, frequency-averaged radiation moment equations. The frequency-averaged variable Eddington factors and mean opacities are calculated by solving the frequency-dependent transport equations. About one thousand frequencies (in one hundred frequency groups) and about nine hundred depth points are used. At each time step, we use T(r) to determine the Eddington factors and mean opacities by solving the frequency-dependent radiation transport equation in a co-moving frame and integrate to obtain the frequency-averaged quantities. The averaged opacities have been calculated assuming local thermodynamic equilibrium (LTE). Both the monochromatic and mean opacities are calculated in the narrow line limit. Scattering, photon redistribution, and thermalization terms used in the light curve opacity calculations are taken into account.
In previous works (e.g. @hk96), the photon redistribution and thermalization terms have been calibrated for a sample of spectra using the formalism of the equivalent two level approach (H95). For increased consistency, we use the same equations and atomic models but solve the rate equations simultaneously with the light curves calculation at about every 100$^{th}$ time step, at the expense of some simplifications in the NLTE-part compared to H95. For the opacities we use the narrow line limit, and for the radiation fields we use the solution of the monochromatic radiation transport using $\approx 1000$ frequency groups.
### Spectral Calculations
Our non-LTE code (H95, and references therein) solves the relativistic radiation transport equations in a co-moving frame. The spectra are computed for various epochs using the chemical, density, and luminosity structure and $\gamma$-ray deposition resulting from the light curve coder. This provides a tight coupling between the explosion model and the radiative transfer. The effects of instantaneous energy deposition by $\gamma$-rays, the stored energy (in the thermal bath and in ionization) and the energy loss due to the adiabatic expansion are taken into account. Bound-bound, bound-free and free-free opacities are included in the radiation transport which has been discretized by about $2 \times 10^4$ frequencies and 97 radial points.
The radiation transport equations are solved consistently with the statistical equations and ionization due to $\gamma$-radiation for the most important elements and ions. Typically, between 27 to 137 bound levels are taken into account. We use (27/123/242), (43/129/431), (20/60/153), (35/212/506), (41/195/742), (62/75/592), (137/3120/7293), (84/1355/5396), (71/865/3064) where the first, second and third numbers in brackets denote the number of levels, bound-bound transitions, and number of discrete lines for the radiation transport. The latter number is larger because nearby levels within multiplets have been merged for the rates.
The neighboring ionization stages have been approximated by simplified atomic models restricted to a few NLTE levels + LTE levels. The energy levels and cross sections of bound-bound transitions are taken from @kurucz93 starting at the ground state. The bound-free cross sections are taken from TOPBASE 0.5 as implemented in Straßburg [@mendoza93]. Collisional transitions are treated in the ‘classical’ hydrogen-like approximation [@mihalas78] that relates the radiative to the collisional gf-values. All form factors are set to 1.
About 10$^{6}$ additional lines are included (out of a line list of $4 \times 10^7$) assuming LTE-level populations. The scattering, photon redistribution, and thermalization terms are computed with an equivalent-two-level formalism that is calibrated using NLTE models.
Modeling Results
----------------
Spherical dynamical explosions, light curves, and spectra are calculated for both normal bright and subluminous SNe Ia. Due to the one-dimensional nature of the models, the moment of the transition to a detonation is a free parameter. The moment of deflagration-to-detonation transition (DDT) is conveniently parameterized by introducing the transition density, $\rho_{\rm tr}$, at which DDT happens.
Within the DD scenario, the free model parameters are: 1) The chemical structure of the exploding WD, 2) Its central density $\rho_c$ at the time of the explosion, 3) The description of the deflagration front, and 4) The density $\rho_{tr}$ at which the transition from deflagration to detonation occurs. Note that the density structure of a WD only weakly depends on the temperature because it is highly degenerate. The central density $\rho_c$ depends mainly on the history of accretion. Model parameters have been chosen for the progenitor and $\rho_c$ which allow us to reproduce light curves and spectra of ‘typical’ Type Ia supernovae.
In all models, the structure of the exploding C/O white dwarf is based on a model star with 5 M$_{\sun}$ at the main sequence and solar metallicity. Through accretion, this core has been grown close to the Chandrasekhar limit (see Model 5p0y23z22 in @dominguez01). At the time of the explosion of the WD, its central density is 2.0$\times 10^9$ g cm$^{-3}$ and its mass is close to 1.37$M_\odot$. The transition density $\rho_{tr}$ has been identified as the main factor which determines the $^{56}Ni$ production and, thus the brightness of a SNe Ia (H95; @hkw95; @iwamoto99). The transition density $\rho_{tr}$ from deflagration to detonation is varied between 8 and 27 $\times 10^6$ g cm$^{-3}$ to span the entire range of brightness in SNe Ia. The models are identified by [*5p0z22.ext*]{} where [*ext*]{} is the value of $\rho_{tr}$ in $10^6$ g cm$^{-3}$. Some of the basic quantities are given in Table \[modelprop\].
### Explosion Models
Here we will restrict our discussion to the basic features of DD models which are relevant for the spectral analysis of SN 1999by. For a more detailed discussion of the hydrodynamical evolution of DD-models, see @khokhlov91, H95, and references therein.
During the deflagration phase, the WD is lifted, starts to expand, and after burning about 0.22 to 0.28 $M_\odot$, the transition to a detonation is triggered. Figures \[density\] and \[abundance\] show the density, velocity and chemical structures for representative models after a homologous expansion has been achieved. The final velocity and density structures are rather similar. The expansion velocities decrease slightly with $\rho_{tr}$ because a significant amount of oxygen remains unburned and did not contribute to the energy production. In 5p0z22.8, oxygen remained unburned in the outer $\approx 0.4 M_\sun $. Overall, the density is decreasing with radius because burning took place throughout the entire WD except in the very outer layers. In contrast, pure deflagration model such as W7 [@nomoto84], pulsating delayed detonation [@hkw95], and merger models [@khokhlov93] show a significant density bump at the boundary between burned and unburned layers.
Iron group elements are produced in the inner layers where density and temperature stay high for a sufficient time during the explosion, whereas intermediate mass elements are produced in the layers above where nuclear statistical equilibrium (NSE) has no time to set in. In the outer zones, products of explosive oxygen (e.g. Si, S) and explosive carbon burning (O, Mg, Ne) and some Si are seen. Only a very thin layer of unburned C/O remains.
For layers with final velocities $v\leq 3000$ km s$^{-1}$, the densities in the early stages of the explosion are sufficiently high for electron capture. This results in the production of $^{54,56}$Fe & $^{58}$Ni whereas in the outer layers only $^{56}$Ni is produced. We employ the same description for the deflagration front in all models and, consequently, the very inner structures are almost identical in all models. A small dip in the Ni abundance is present at the DDT because the time scales for burning to NSE are comparable to the expansion time scales at densities of $\approx 10^7$ g cm$^{-3}$. This feature can be expected to be smeared out in multi-dimensional calculations [@livne99].
After the DDT, a detonation front develops and the burning of fuel is triggered by compression of unburned material. Consequently, burning takes place under higher densities compared to a deflagration. In general, additional material is burned up to NSE and this extends the layers of $^{56}$Ni for all $\rho_{tr} \ge 1.2 \times 10^7$ g cm$^{-3}$. For models with smaller $\rho_{tr}$, burning only proceeds up to Si despite the compression. This increases the production of intermediate mass elements (e.g. Si, S) at the expense of $^{56}$Ni. In our models, the efficiency for $^{56}$Ni production drops rapidly as a function of decreasing $\rho_{tr}$ (see Table \[modelprop\]). We note that for $\rho_{tr} \leq 6 \times 10^{6}$ g cm$^{-3}$, nuclear burning stops all together for $\rho < \rho_{th}$ and, unlike any observed SN, one would see a very slowly expanding C/O envelope heated by little $^{56}$Ni.
There exists a rather strict lower limit for the absolute brightness of SNe Ia within the Chandrasekhar mass models. This can be understood as follows. The absolute brightness of a SNe Ia is mainly determined by the $^{56}Ni$. In general, $^{56}Ni$ is produced both during the deflagration and the detonation phase. For the most subluminous SNe Ia however, no $^{56}Ni$ is produced during the detonation. Within the $M_{Ch}$ scenario and DD-models in particular, we need to burn a minimum of $\approx 0.2 M_\odot$ during the deflagration to achieve the required ‘lift/pre-expansion’ of the WD. During the deflagration phase, iron group elements are produced. The fraction of $^{56}Ni$ depends on the amount of material which undergoes electron capture, i.e. on the central density and, to a smaller extent, on the description of the burning front (@brachwitz01; @dominguez01). For a model with $\rho_c = 2 \times 10^9$ g cm$^{-3}$, we find a lower limit of $M(^{56}Ni)\approx 0.09 M_\odot$. Recently, @brachwitz01 studied the influence of $\rho_{c}$ between 1.5 to $8 \times 10^9$ g cm$^{-3}$ on the nuclear burning for similar DD-models with a high transition density which may describe ‘normal’ SN Ia. In those models, the $^{56}Ni$ produced during the deflagration phase varies by $\leq 25 \%$ which provides an estimate for the possible variation in the minimum $^{56}Ni $ production. We conclude that there exists a lower limit of $M(^{56}Ni) \approx 0.1 M_{\sun} \pm 25 \%$ within DD-models.
In principle, 3-D effects may change the lower limit for the $^{56}Ni$ production. However, all current 3-D calculations show that rising blobs are formed at high densities which undergo complete Si-burning because the burning time scales to Ni shorter than the hydro time scales. In the case of a DDT or a transition to a phase of very fast deflagration, the amount of burning will likely be similar to the 1-D case. Thus, we do not expect a significant change in the minimum ${56}Ni$ production.
### General properties of theoretical light curves
Optical light curves and spectra of SN Ia are not the subject of this paper but we used their general properties for selecting a model for SN1999by.
Within the DD scenario, both normal and subluminous SNe Ia can be produced (H95, @hkw95, HKW95 hereafter; @umeda99). The amount of $^{56}$Ni ($M_{^{56}Ni}$) depends primarily on $\rho_{tr}$ (H95; HKW95; @umeda99) and to a much lesser extent on the assumed value of the deflagration speed, initial central density of the WD, and initial chemical composition (ratio of carbon to oxygen). Models with smaller transition density give less nickel and hence both lower peak luminosity and lower temperatures (HKW95, @umeda99). In DDs, almost the entire WD is burned and the total production of nuclear energy is almost constant. This and the dominance of $\rho_{tr}$ for determining the $^{56}Ni$ production are the basis of why, to first approximation, SNe Ia appear to be a one-parameter family.
The observed $M_V(\Delta M_{\Delta t=15d})$ relation can be understood as an opacity effect [@hoflich96b], namely, the consequence of rapidly dropping opacities at low temperatures [@hmk93; @khokhlov93; @mazzali01]. Less Ni means lower temperature and consequently, reduced mean opacities because the emissivity is shifted from the UV toward longer wavelengths with less line blocking. A more rapidly decreasing photosphere causes a faster release of the stored energy and, as a consequence, steeper declining LCs with decreasing brightness. The DD models give a natural and physically well-motivated origin of the $M_V(\Delta M_{\Delta t=15d})$ relation of SNe Ia (@hamuy95 [@hamuy96b; @phillips87; @suntzeff99]).
The broad band *B* and *V* light curves are shown in Figure \[lcs\]. Some general properties are given in Table \[modelprop\] and Figure \[dm15\]. We note that the decline of the *B* and *V* LC during the early $^{56}Co$ tail is slightly smaller for the most subluminous models (e.g. 5p0z22.08) compared to the intermediate models (e.g. 5p0z22.16) because, at day 40, the $\gamma $-rays are almost fully trapped in the former whereas, for the latter, the escape probability of $\gamma $-rays is already increasing (HKW95). With varying $\rho_{tr}$, we find rise times between 13.9 to 19.0 days, and the maximum brightness $M_V$ spans a range $-17.21^m$ and $-19.35^m$ with $B-V$ between $+0.47^m$ to $-0.03^m$. The value of $M_V$ is primarily determined by the transition density. Variations in the progenitors (main sequence mass and metallicity) or the central density at the point of ignition causes an spread around $M_V$ of $\approx0.2~...~0.3^m$ [@hoflich98; @dominguez01]. As discussed above, the brightness decline relation $M_V(\Delta M_{\Delta t=15d})$ can be understood as an opacity effect or, more precisely, it is due to the drop of the opacity at low temperatures. Therefore, subluminous models are redder and the decline is steeper. In our series of models, $M_V(\Delta M_{\Delta t=15d})$ ranges between $1.45^m$ and $0.91^m$. However, $M_V(\Delta M_{\Delta t=15d})$ is not a strictly linear relation (Fig. \[dm15\]). For bright SNe Ia ($-19.35^m \leq M_V \leq -18.72^m$), called [*Branch-normals*]{} ([@branch96; @branch99]), the linear decline is relatively flat, followed by a steeper decline toward the subluminous models. For normal-luminosity models, the photospheric temperature is well above the critical value at which the opacity drops ($\approx 7000 ~... 8000 K$; @hmk93) whereas the opacity drops rapidly in subluminous models soon after maximum light.
Analysis of SN1999by
====================
Selection of the SN1999by Model
-------------------------------
Detailed B and V light curves of SN 1999by have been collected and published by @toth00 and optical spectra around maximum light are available from observations at McDonald Observatory [@howell01]. Recently, SN 1999by spectra and LCs have been provided by @garnavich. Within our series of models, we selected the most suitable model based on the optical photometry. Unfortunately, there is a gap in the observed data starting about one week after maximum light, a phase which is most suitable for selecting the appropriate models. Thus, as the main discriminator, we use the complete LCs as reconstructed by @toth00.
SN 1999by showed a decline rate $M_V(\Delta M_{\Delta t=15d})$ between $ 1.35^m$ and $1.45^m$ (@toth00) clearly ruling out all but the most subluminous models with transition densities of $8$ and $10 \times 10^6$ g cm$^{-3}$ and maximum brightness of $-17.21^m$ and $-17.35^m$, respectively (Table \[modelprop\]). Both models are consistent with the early light curves given in @toth00 i and @garnavich (Fig.\[lc.obs\]). From the models and the early LC, the time of the explosion can be placed around April 27 $\pm$ 1 d, 1999. The model post-maximum LC in B is less reliable because it depends very sensitively on the size and time evolution of line blanketing and the thermalization. $M_B(\Delta M_{\Delta t=15d})$ is $1.64^m$ and $1.73^m$ for 5p0z22.10 and 5p0z22.8, respectively. This compares reasonably well with $M_B(\Delta M_{\Delta t=15d}) = 1.87^m$ observed for SN1999by [@howell01]. In the LCs (Fig. \[lc.obs\]), discrepancies of about $0.3 ^m$ show up after the time gap in the observations. At these times, the envelope of a subluminous SN Ia becomes transparent, and strong emission features emerge (see below). Thus these discrepancies are likely a consequence of discretization errors due to the use use of only 1000 frequency for the LC calculations.
According to @bonanos99, $B-V = 0.44 \pm 0.04^m$ near maximum light on May 10, 1999. @garnavich give $B-V = 0.50 \pm 0.03^m$. We find $B-V = 0.47^m$ and $0.42^m$ for 5p0z22.8 and 5p0z22.10, respectively. In previous studies, we find intrinsic uncertainties in B-V to be $\approx 0.05^m$ for our models at maximum light [@hoflich95]. Thus, the interstellar reddening can be expected to be $E_{B-V} \leq 0.11^m$ consistent with the values for our galaxy given by @burstein82 and @schlegel98 and the estimates for the total extinction derived by @toth00. For our models, if we assume $m_V = 13.10 \pm 0.05^m$, we then derive a distance modulus of $M_V-m_V = 30.39^m \pm 0.12^m$ ($-0$ to 0.35$^m$). The first and second error-terms originate from the observational uncertainties plus the brightness range of the models and the reddening correction, respectively. >From our models, we can derive a distance to NGC 2841 of $ 11 \pm 2.5$ Mpc and $12 \pm 1$ Mpc if we include and neglect interstellar reddening, respectively. Both from observations [@schlegel98] and our models, very small reddening is preferred. Following @schlegel98, we adopt $E(B-V)=0.015^m$ for all comparisons with the observations.
Based on the density, temperature and chemical structure predicted from the explosion model, $\gamma-ray$ transport and light curve calculations, we have constructed detailed NLTE-spectra for 11 d, 15 d, 22 d and 29 d after the explosion (days $-4$, 0, +7 and +14 after maximum light). For the subluminous model with the transition density $\rho_{tr}=8\times 10^6$ g cm$^{-3}$ (Fig. \[density\]), the evolution of the temperature, energy deposition by $\gamma$-rays, and Rosseland optical depth is given as a function of radius (Fig. \[structures\]). Obviously, the total optical depth drops rapidly with time from 98, 40, 15 to 6 due to the cooling; much faster than quadratic as would be expected from the geometric dilution alone (98, 52, 25 and 12), and thus more rapidly than in normal-bright SNe Ia [@hoflich93; @khokhlov93]). As mentioned above, the reason is the rapid drop in the mean opacity for $T \leq 6000 ... 8000 K$ which causes the outer layers to become almost transparent. Consequently, the photosphere recedes quickly in mass and even starts to shrink in radius just a few days after the explosion. The corresponding expansion velocities at the photosphere are approximately 14000, 10500, 6500 and 4000 km s$^{-1}$ at the four epochs computed.
The low photospheric temperatures are due to the small amount of $^{56}Ni$ and its low expansion velocity which also causes the local trapping of $\gamma$-rays (Fig. \[structures\]). In contrast to normal SNe Ia, most of the decay energy is deposited well below the photosphere up to about 2 weeks after maximum light. This explains, in part, the low ionization stages seen in the IR-spectra (see below).
The location (and thus the velocity) of the photosphere determines which layers in the supernova envelope will form the spectral features. Before maximum light, the spectra sample the layers of explosive carbon burning which are O, Mg, Ne and Si-rich (Fig. \[abundance\]). Up to 2 weeks after maximum light, spectra are formed within layers of incomplete silicon burning which are Si, S and Ca-rich. Only thereafter are the layers which are dominated by iron group elements finally exposed. This is very different from models for normal luminosity SNe Ia for which the iron rich layers are already exposed near maximum light. These profound differences are key for our understanding of the optical and IR spectra of subluminous SNe Ia and the differences compared to normal SNe Ia.
Optical spectra are not the main subject of this paper, and detailed spectra have not been available for this analysis except for maximum light (Fig. \[mod.opt2\], @howell01). However, for completeness, the time sequence of optical spectra is given in Figure \[mod.opt\]. Overall, the spectra show an evolution typical for SN Ia, (e.g. SN 1994D, H95). They can be understood as a consequence of the declining temperature and total optical depth with time, and the rapidly dropping temperature. The Doppler shift of the absorption component decreases rapidly because of the receding photosphere. For subluminous models, the entire envelope becomes almost transparent and the spectra start to enter the nebular phase at about two weeks after explosion. This marks the end of the applicability of our atomic models. Some features (see Fig. \[mod.opt2\]) are typical of subluminous SNe Ia and are a consequence of the low luminosity and photospheric temperature around maximum light. The line at 5800 Å is strong relative to the line at 6150 Å, and the flux near 4000 Å was depressed as in other subluminous SNe, and features due to were strong as was the line at 7500 Å[@garnavich99]. As already suggested by @nugent97, the strong absorption and the relative strength of the Si lines can be understood as a temperature effect. Within DD models with $ M_{Ch}$ progenitors, the strong line is a direct consequence of the fact that the spectra at maximum light are formed in massive layers of explosive carbon burning.
While the structures and synthetic spectra of the models 5p0z22.8 and 5p0z22.10 are very similar, 5p0z22.8 is slightly preferred from a consideration of the expansion velocities at the photosphere which are 10500 and 11500 km s$^{-1}$, respectively. Therefore, we will will adopt 5p0z22.8 in the following. Based on parameterized LTE model atmospheres (SYNOW, @fisher99), a detailed analysis of the Doppler shifts of lines has been provided by @garnavich. They found in the entire range between 11,300 down to 6500 km s$^{-1}$ which is consistent with our hydrodynamical models (see Fig. \[structures\]).
Analysis of the IR-spectra of SN 1999by
---------------------------------------
Near IR-spectra (0.9 to 2.5 $\mu$m) of 5p0z22.8 at day 11, 15, 22 and 29 after explosion are compared to those of SN1999by at May 6, 10, 16, and 24 (Figs. ref[ir.may6]{}–\[ir.may24\]). The spectra are characterized by overlapping P-Cygni and absorption lines, and emission features. Unless noted otherwise, the observed wavelengths quoted the emission component. The overall energy distribution in the continuum is determined by bound-free and Thomson opacities and the temperature evolution. In general, the spectra and their time evolution agree reasonably well with the model predictions and all the major features can be identified.
Few detailed analyses and line identifications of IR spectra have been performed [@meikle96; @bowers97; @hoflich97; @wheeler98; @hernandez00] and these were limited to spectra of SN 1994D, SN 1998bu and post-maximum spectra of SN1986G. To put our analysis given below in context with normal-luminosity SNe Ia, we will refer to those works as reference points.
### Early time IR-spectra
The observed NIR spectra from May 6 and 10 (Figs. \[ir.may6\] & \[ir.may10\]) are quite similar, and comparison with the model spectra show that both of these spectra are formed in layers of explosive carbon burning. Spectral features can be attributed to , , and . Typically, several thousand overlapping lines contribute to the overall opacity but few strong features leave their mark on the spectrum.
Some weak features at about 2.1 and 2.38 are due to (21369, 21432, 24041, 24044, 24125 Å) and (21920, 21990 Å). The strong, broad feature between 1.62 and 1.75 is a blend produced by lines (16760, 16799, 174119, 17717 Å) and (16907, 16977, 17183, 17184 Å). Its blue edge is determined by . A similar feature is also present in normal luminosity SNe Ia but does not contribute because the corresponding spectra are formed in layers of explosive oxygen burning.
The strong feature that peaks near 1.35 is due to (13650 Å), and weaker features with peaks near 1.23 and 1.43 are a multiplet (13692 to 13696 Å) and a blend of (14542Å) and (14454 Å), respectively. A very weak feature in the synthetic spectrum near 1.23 can be attributed to (13164 Å), but is well below the noise level of the observed spectra.
Features with P-Cygni absorption minima at 1.12, 1.06, 1.03 and 1.0 are produced by (11620, 11600 Å), (11302, 11286 Å), (10914, 10915 & 10950 Å), (10683 & 10691 Å), and (10092 Å), respectively. The multiplet of 1.09 is also prominent in normal luminosity SNe Ia but the other features are weak and blended with , and .
The observed spectra go down to about 9700 Å. However, there are some interesting features at shorter wavelengths which are worth mentioning. Foremost, the model predicts a very strong feature at about 9100 Å due to (9405 Å). Its velocity provides an important constraint on the minimum velocity for the unburned region, and possible mixing. Other features are produced by (9344 Å), (IR-triplet), and (8446 Å) and (8246 Å).
Despite the overall agreement, some discrepancies remain. First, the line at 10691Å is somewhat too strong, and its Doppler shift on May 6 is too large by about 1500 km s$^{-1}$. This feature is formed well above the photosphere (13000 km s$^{-1}$). This might indicate that SN 1999by has a slightly lower temperature in the outer layers compared to the model, which would cause less excitation of carbon and, might cut off the high velocity contribution to the absorption. Secondly, in the model spectrum there seems to be a lack of emission produced by near 1.18 . We note that similar problems with Si are also commonly present in the optical line at about 6380 Å for normal SNe Ia [@hoflich95; @lentz00]. They may be related to the excitation process of Si but their origin is not well understood.
### IR-spectra after maximum light
The spectrum from May 16 (Fig. \[ir.may16\]) is formed in layers of explosive oxygen burning. The lines can be attributed to , , and . All features due to and seen in the pre-maximum spectra can still be identified. No lines due to and are present, but strong lines of and appear. Strong features at about 1.14 and 1.1 are due to lines (11784, 11839 & 11849 Å) and (10821 Å) & (11800 Å), and the broad feature at 0.91 (not covered in this spectrum, but included for completeness) is a blend of (9890 Å) and (10009Å). Weaker lines (, 9997Å; , 10455 Å) contribute to blends around 1.03 .
Some weak lines due to and lines occur throughout the spectrum but the most characteristic feature in post maximum IR-spectra of normal SNe Ia is missing. Starting a few days after maximum, normal SNe Ia show a wide emission feature between 1.5 and 1.8 produced by thousands of , & lines. Typically, their emission flux is about twice as large as the adjoining continuum. This feature marks the lower velocity end of the region of complete silicon burning, and it seems to be common in all normal luminosity SNe Ia such as SN 1986G, SN 2000br and SN 2000cx (@wheeler98; Marion 2001, priv. comm.; @rudy01). In 5p0z22.8, this broad emission feature is not seen because the photosphere has yet to recede to the layers of complete silicon burning. On May 24th (Fig. \[ir.may24\]) this feature finally appears but is very weak. Overall, two weeks after maximum light, the spectrum in dominated by a large number of and lines. Strong features of are around 9600 Å 1.2 $\mu$m and 1.4 $\mu$m with strong lines from 9500 to 12443 Å, and 14000Å – 16700 Å.
In our synthetic spectrum the , & feature 1.6 to 1.8 $\mu$m does not show a local minimum at $\approx 1.7 \mu m$ as observed in SN1999by, and as also predicted by normal bright models for SN1986G [@hoflich97; @wheeler98]. In our subluminous model, the emission near 1.7 comes from the central Ni-rich region. The absence of such emission in the observed spectrum may indicate that the very central region ($M(r)\leq 0.2 M_\odot$) of the model is too transparent compared SN1999by. This discrepancy is not critical for measuring the chemical structure of the envelope because this is determined by the blue edge. The difference in opacity may be caused by missing iron lines in the line list (§3.1.3), or an underestimation of the excitation by non-thermal electrons because we assume local deposition for the electrons from the $\beta^+$ decay of $^{56}Co$.
Mixing processes
----------------
As we have seen, model 5p0z22.8 reproduces the basic features of both optical and IR spectra, including the time evolution. This suggests that DD models are a reasonably good description for subluminous SNe Ia.
Unfortunately, we are still missing a handle on an important piece of physics. The propagation of a detonation front is well understood but the description of the deflagration front and the deflagration to detonation transition (DDT) pose problems. Currently, state of the art allows us to follow the front only through the phase of linear instabilities, i.e. only the early part of the deflagration phase. The resulting structures of these calculations cannot account for the observations of typical SNe Ia (Khokhlov 2001), because a significant fraction of the WD ($\approx 0.5 M_\odot$) remains unburned.
Nevertheless, these calculations point toward a more general problem, or toward a better understanding of the nature of subluminous SNe Ia. On a microscopic scale, a deflagration propagates due to heat conduction by electrons. Though the laminar flame speed in SNe Ia is well known, the front has been found to be Rayleigh-Taylor (RT) unstable, increasing the effective speed of the burning front [@nomoto76]. More recently, significant progress has been made toward a better understanding of the physics of flames. Starting from static WDs, hydrodynamic calculations of the deflagration fronts have been performed in 2-D [@niemeyer95; @lisewski00], and full 3-D [@khokhlov95; @khokhlov01].
The calculations by @khokhlov01 demonstrated a complicated morphology for the burning front. Plumes of burned material will fill a significant fraction of the WD, and unburned or partially-burned material can be seen near the center. Thus iron-rich elements will not be confined to the central region as in 1-D models. @khokhlov01 finds that while the expansion of the envelope becomes almost spherical, the inhomogeneous chemical structure will fill about 50 to 70% of the star (in mass). If a DDT occurs at densities needed to reproduce normal-bright SNe Ia, most of the unburned fuel in these regions will be burned to iron-group elements during the detonation phase. This will eliminate the chemical inhomogeneities. However, in our subluminous model, the chemical structure imposed during the deflagration phase must be expected to survive because no or very little $^{56}Ni$ is produced during the detonation.
In order to test for this effect, we have mixed the inner layers of model 5p0z22.8 up to an expansion velocity of 8000 km s$^{-1}$. This brings iron-group elements into the outer regions and mixes intermediate mass elements (Si, S, Ca and some O) into the inner, hotter layers. The layers with expansion velocities less than 8000 km s$^{-1}$ become visible a few days after maximum light (see above). In Figure \[mix.may16\], a comparison is shown between the spectrum of SN1999by on May 16th, 1999, and the synthetic spectrum about one week after maximum light, and problems with the spectral fit are apparent. As can be expected from the discussion above, the effective extension of the iron-rich layers to 8000 km s$^{-1}$ causes the 1.5 to 1.8 blend to become rather strong, and line blanketing due to iron-group elements ( & ) become too strong in the region around 1 to 1.2 . Moreover, at about 0.99, 1.02, 1.05 & 1.16 , (absorption components) strong blends can be seen due to (11302, 11287 & 11297Å), (11715, 11745, 13650 & 13650 Å), (9890, 9931, 9997, 11839 & 11745 Å) and (10455 & 108212 Å).
We have to take into account that our approach assumes microscopic mixing whereas RT instabilities provide large scale inhomogeneities. If $^{56}Ni$ and the intermediate mass elements are separated, direct excitation of intermediate mass elements may be reduced. With respect to the radiation transport, the main difference is the covering factor. This will effect line blanketing and thus the line blanketing between 1 to 1.2 $\mu$m may not pose a problem. However, the emission features between 1.5 to 1.8 $\mu$m will not go away. The reduced excitation by non-thermal electrons and $\gamma$’s in case of macroscopic mixing are important at low optical depths. Less excitation may reduce the problem with the O, Si, S & Ca lines but the problem will no disappear because the corresponding features are formed close to the photosphere where the excitation is thermal in nature.
These problems strongly suggest that large scale mixing did not occur in SN 1999by. This may provide a key for our understanding of the nature of these objects.
Implications for the SN Ia progenitors, and alternative scenarios
=================================================================
As previously discussed, the pre-conditioning of the WD before the explosion may provide a key for understanding the nature of subluminous SNe Ia. Such pre-conditioning may include the main sequence mass of the progenitor mass and its metallicity [@dominguez01; @hoflich98; @iwamoto99], the accretion history [@langer00], or large-scale velocity fields such as turbulence prior to the runaway (@hs01). Up to now, none of these effects have been included in detailed 3-D calculations. Therefore, the suggestions (below) require further investigation before taken as explanation.
Beyond its subluminous nature, SN 1999by showed another peculiarity with respect to normal-bright SNe Ia. The observed polarization of SN 1999by was rather high ($P \approx 0.7 \%$) with a well-defined axis of symmetry, whereas normal SNe Ia show little or no polarization ($P \leq 0.2 \%$; @wang01). >From a detailed analysis of the polarization spectra, @howell01 concluded that the overall geometry of SN 1999by showed a large scale, probably rotational asymmetry of about 20%. The authors suggested several explanations. Either, SN 1999by may be the result of the explosion of a rapidly rotating WD close to the break-up, or a result of a merger scenario. In a rapidly rotating WD, large-scale circulations [@erigichi85] might influence the deflagration phase of explosive burning, perhaps by breaking up large eddies and preventing their rise.
Alternatively, it should be remembered that the chemical structure of the exploding WD depends on the pre-expansion of the WD prior to the detonation phase. More precisely, we need to burn about $0.2 M_\odot$ to lift the WD in its gravitational potential. This mass is significantly more than is burned during the smoldering phase (the phase of slow convective burning just prior to the explosion). Even in the presence of strong turbulence, the amount of burned material is much less ($\leq 10^{-2} M_\odot$, @hs01). In all models for the ignition (e.g. @nomoto82 [@garcia95]), the central temperature rises by non-explosive carbon burning during the last few minutes to hours before the runaway because the heat cannot be dispersed. It may be feasible that large scale velocity fields due to rotation may extend this phase, and significantly increase the amount of burning prior to the explosion. Thus, the pre-expansion in subluminous SNe Ia may occur directly during the smoldering phase, and the deflagration phase may be even skipped, i.e. a detonation occurs promptly, and we may have a smoldering detonation model (SD). In SD-models, no $^{56}Ni$ is produced during the phase of pre-expansion and, consequently, this mechanism does not predict a lower limit for $M_{Ni}$ and, thus, $M_V$. In principle, SD-models may be an attractive alternative also for normal-bright SNe because they omit the need for a DDT but, from the present understanding of the runaway, in general, extensive burning prior to the runaway cannot be expected (see above). From the point of the spectral analysis, this option is intriguing because burning during the smoldering phase precedes only to O and the time scales of burning are sufficient to completely homogenize the chemical structure of the WD.
Finally, there is the question of alternative scenarios for SNe Ia in the context of subluminous SNe Ia. Pure deflagrations of $M_{Ch}$-WDs without any DDT may produce little $ ^{56}Ni$ but, according to current simulations, a large fraction of the WD would remain unburned. Even if this problem were solved in the future, the problem of inhomogeneous chemistry and Si-rich layers close to the center would remain. Helium triggered detonations of sub-Chandrasekhar mass WDs have been suggested in the past. However, their distinct feature is an outer layer of $^{56}Ni$ above a low mass layer of explosive oxygen burning, a layer of incomplete Si burning, and some $^{56}Ni$ close to the center. This chemical structure is in strong contradiction to the optical spectra and LCs of SN 1999by, and their evolution with time.
Alternatively, setting aside the problem of triggering the explosion, central detonations in low-mass WDs have been suggested in the past (e.g., @ruizlapuente93). To produce little $^{56}Ni$, the explosion must occur in WDs with central densities $\leq 1 \times 10^7$ g cm$^{-3}$ which corresponds to $M(WD) \leq 0.7 M_\odot$. In contrast to the DD-models, these models would not show a lower limit for the $^{56}Ni$ production and $M_V$, and they would avoid the problem of mixing during a deflagration phase. Nonetheless, we regard these models as unlikely candidates for SN 1999by because there are a couple of severe problems beside the lack of a triggering mechanism for the explosion. The total mass of the ejecta is lower (by a factor of 2) than is required to keep the line forming region in the layers of explosive carbon and incomplete Si-burning up to two and four weeks after the explosion, respectively. The specific production of nuclear energy does not depend on $M_{WD}$. In this case, the diffusion time scales and, thus, the rise times to maximum light are $ \propto M_{WD}$ [e.g. @pinto00]. Compared to the LCs for a WD with $M_{Ch}$, we expect rise times to be shorter by $\approx 2$ which is incompatible with SN 1999by observations.
This leaves the merging scenario as an alternative as we may expect a significant asphericity in the explosion if the merging occurs on hydrodynamical time scales. Nevertheless, this alternative is not a very attractive one. Based on 1-D parameterized models, @khokhlov93 provided chemical structures for explosions which may resemble merger models. As a general feature, these models show a shell-like structure with a massive outer layer of unburned C/O which encloses a thin, low mass layer of explosive carbon burning. This pattern is not consistent with the IR-spectra for SNe 1999by up to maximum light. However, it should be kept in mind that, besides the limitations of 1-D models for mergers, no chemical structures are available for merging subluminous models. A more severe problem for mergers may be that merger models tend to ignite prematurely and produce a detonation wave that burns the C/O WD to a O/Ne/Mg WD prior to the completion of the merger. Thus, merging may result in an accretion induced collapse rather than a SNe Ia [@saio98].
Summary
=======
IR-spectra of the subluminous SN1999by have been presented which cover the time evolution from about 4 days before to 2 weeks before maximum light. This is the first subluminous SN Ia (and arguably the first SN Ia) for which IR spectra with good time coverage are available. These observations allowed us to determine the chemical structure of the SN envelope.
Based on a delayed detonation model, a self-consistent set of hydrodynamic explosions, light curves, and synthetic spectra have been calculated. This analysis has only two free parameters: the initial structure of the progenitor and the description of the nuclear burning front. The light curves and spectral evolution follow directly from the explosion model without any further tuning, thus providing a tight link between the model physics and the predicted observables.
By varying a single parameter, the transition density at which detonation occurs, a set of models has been constructed which spans the observed brightness variation of Type Ia supernovae. The absolute maximum brightness depends primarily on the $^{56}Ni$ production which, for DD-models, depends mainly on the transition density $\rho_{tr}$. The brightness-decline relation $M_V (\Delta M_{\Delta t =15d})$ observed in SNe Ia is also reproduced in these models. In the DD scenario this relation is a result of the temperature dependence of the opacity, or more precisely, as a consequence of the rapid drop in the opacities for temperatures less than about 7000 to 8000 K.
Within $M_{Ch}$ WD models, a certain amount of burning during the deflagration phase is needed to pre-expand the WD and avoid burning the entire star to $^{56}Ni$. This implies a lower limit for the $^{56}Ni$ mass of about $0.1
M_\odot$ $\pm 25 \%$ and, consequently, implies a minimum brightness for SNe Ia within this scenario. The best model for SN 1999by ($\rho_{tr} = 8 \times 10^7$ g cm$^{-3}$, selected by matching to the predicted and observed optical light curves) is close to this minimum Ni yield. The data are consistent with little or no interstellar reddening ($E(B-V) \leq 0.12^m$), and the derived distance is $11 \pm 2.5$ Mpc or $12 \pm 1$ Mpc if we take the limit for $E(B-V)$ from the models or assume $E(B-V)=0.015^m$ according @schlegel98 for the galaxy.
Without any further modification, this subluminous model has been used to analyse the IR-spectra from May 6, May 10, May 16 and May 24, 1999, which correspond to $-4$ d, 0 d, +7 d and +14 d after maximum light. The photosphere ($\tau_{Thomson} = 1$) recedes from 14000, to 10500, 6500 and 4000 km s$^{-1}$. The observed and theoretical spectra agree reasonably well with respect to the Doppler shift of lines and all strong features could be identified.
Before maximum light, the spectra are dominated by products of about explosive carbon burning (O, Mg), and Si. Spectra taken at +7 d and +14 d after maximum are dominated by products of incomplete Si burning. At about 2 weeks after maximum, the iron-group elements begin to show up. The long duration of the phases dominated by layers of explosive carbon burning and incomplete Si burning implies massive layers of these burning stages that are comparable with our model results. ($\approx 0.45 $ and $ 0.65 M_\odot $ for explosive carbon and incomplete silicon burning, respectively.) This, together with the $^{56}Ni$ mass, argues that SN 1999by was the explosion of a WD at or near the Chandrasekhar mass.
Finally, we note that the observed IR spectra are at odds with recent 3-D calculations [@khokhlov01] which predict that large scale chemical inhomogeneities filling 50 – 70% (in mass) of the WD will be formed during the deflagration phase. When the effect of such inhomogeneous mixing is tested by mixing the inner layers of our SN 1999by model, significant differences appear between the model spectra and the observed data. This suggests that no significant large-scale mixing took place in SN 1999by. The lack of observed mixing and the asphericity seen in SN 1999by may be important clues into the nature of subluminous SNe Ia, and may be related to the reason for a low DDT transition density (and hence the low luminosity). Alternatively, we may have an extended smoldering phase of the WD prior to the explosion which skips the deflagration phase altogether (see §5). In either case, the pre-conditioning of the WD prior to the explosion seems to be a key for understanding SNe Ia.
We would like to thank Paul Martini and Adam Steed for aiding us with data acquisition and J. C. Wheeler for carefully reading the manuscript and providing helpful suggestions. P. H. would like to thank NASA for its support by NASA grant NAG5-7937. C. L. G. and R. A. F.’s research is supported by NSF grant 98-76703. The calculations for the explosion, light curves and spectra were performed on a Beowulf-cluster financed by the John W. Cox – Fund of the Department of Astronomy at the University of Texas.
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[ccccc]{} 6 May & -4 d & 0.96–1.8 & 4$\times$600 s & Gerardy\
” & ” & 1.95–2.4 & 3$\times$600 s & ”\
7 May & -3 d & 0.96–1.8 & 3$\times$600 s & ”\
” & ” & 1.95–2.4 & 3$\times$600 s & ”\
8 May & -2 d & 0.96–1.8 & 3$\times$600 s & ”\
9 May & -1 d & 0.96–1.8 & 3$\times$600 s & ”\
” & ” & 1.2–2.2 & 3$\times$600 s & ”\
10 May & 0 d & 0.96–1.8 & 3$\times$600 s & ”\
” & ” & 1.2–2.2 & 3$\times$600 s & ”\
” & ” & 1.95–2.4 & 3$\times$600 s & ”\
16 May & 6 d & 0.96–1.8 & 3$\times$600 s & Martini & Steed\
18 May & 8 d & 0.96–1.8 & 3$\times$600 s & ”\
20 May & 10 d & 0.96–1.8 & 3$\times$600 s & ”\
21 May & 11 d & 0.96–1.8 & 3$\times$600 s & Sakai\
24 May & 14 d & 0.96–1.8 & 3$\times$600 s & ”\
Date & Epoch & J & H & K\
\
6 May & -4 d & $13.28 \pm 0.09$ & $13.23 \pm 0.05$ & $13.19 \pm 0.05$\
7 May & -3 d & $13.28 \pm 0.09$ & $13.21 \pm 0.05$ & $13.14 \pm 0.05$\
8 May & -2 d & $13.07 \pm 0.09$ & $13.04 \pm 0.05$ &
[lccccccc]{} 5p0z22.8 & 8. & 0.095 & -17.21 & 13.9 & 0.47 & 1.45\
5p0z22.10 & 10. & 0.107 & -17.35 & 14.1 & 0.42 & 1.37\
5p0z22.12 & 12. & 0.137 & -17.63 & 14.7 & 0.38 & 1.32\
5p0z22.14 & 14. & 0.153 & -17.72 & 14.9 & 0.22 & 1.30\
5p0z22.16 & 16. & 0.268 & -18.72 & 15.8 & 0.14 & 1.26\
5p0z22.18 & 18. & 0.365 & -18.82 & 16.6 & 0.08 & 1.21\
5p0z22.20 & 20. & 0.454 & -18.96 & 17.0 & 0.02 & 1.19\
5p0z22.23 & 23. & 0.561 & -19.21 & 18.2 & -0.02 & 1.05\
5p0z22.25 & 25. & 0.602 & -19.29 & 18.6 & -0.02 & 1.00\
5p0z22.27 & 27. & 0.629 & -19.35 & 19.0 & -0.03 & 0.91\
![Near-infrared spectral evolution of SN1999by. Epochs are relative to the date of *V*$_{\rm Max}$, May 10, 1999. For clairity, the spectra have been shifted vertically, and regions of very low S/N due to telluric absorption have been omitted.[]{data-label="nirspec"}](f1.eps){width="15.2cm"}
![Density (blue, dotted) and velocity (red, solid) as a function of the mass for models with $\rho_{tr}=$ 8, 16 and 25 $\times 10^6 g/cm^{-3}$ (from top to bottom).[]{data-label="density"}](f2.eps){width="12.cm"}
![Abundances of stable isotopes as a function of the expansion velocity for models with $\rho_{tr}=$ 8, 16 and 25 $\times 10^6 g/cm^{-3}$ (from top to bottom). In addition, $^{56}Ni$ is given. The curves with the highest abundance close to the center correspond to $^{54}Fe$, $^{58}Ni$ and $^{56}Fe$.[]{data-label="abundance"}](f3.eps){width="10.2cm"}
![*B* (left) and *V* (right) light curves for models with $ \rho_{tr} $ = 8, 12, 16, 20 and 25 $\times 10^{6}g/cm^3$ (from bottom to top).[]{data-label="lcs"}](f4.eps){width="10.2cm"}
![Maximum brightness $M_V$ as a function of the $^{56}Ni$ mass (upper left), $\rho_{tr}$ (lower left), and $M_V(\Delta M_{\Delta t=15d})$ (lower right), and $(B-V)$ as a function of $M_V$ (upper right). For all panels except the lower right, the points correspond to models with $\rho_{tr}$ of 8, 10, 12, 14, 16, 18, 20, 23, 25 and 27 $\times 10^{6}g/cm^3$ from left to right. In the lower right panel, this order is reversed.[]{data-label="dm15"}](f6.eps){width="8.2cm"}
![Temperature (left), Rosseland optical depth (center), and energy deposition by $\gamma$-rays (right), as functions of radius at day 10, 14, 21 and 28 (from top to bottom).[]{data-label="structures"}](f5.eps){width="13.2cm"}
![ Evolution of the synthetic UV and optical spectra for the subluminous model 5p0z22.8 at 10.6, 15, 22 and 29 days after the explosion. The flux has been normalized at 7000 Å. []{data-label="mod.opt"}](f7.eps){width="10.2cm"}
![ Comparison of the observed *B* (left) and *V* (right) light curves of SN1999by with the predicted light curves of model 5p0z22.08.[]{data-label="lc.obs"}](f8.eps){width="9.2cm"}
![Comparison of the optical spectrum of SN1999by on May 10, 1999, (blue) with the theoretical spectrum of 5p0z22.8 at maximum light (red, 15 days after the explosion).[]{data-label="mod.opt2"}](f9.eps){width="10.2cm"}
![Comparison of the observed NIR spectrum of SN1999by on May 6, 1999, with the theoretical IR-spectrum of 5p0z22.8 $\approx 4$ days before maximum light (10 days after the explosion). Between 1.35 and 1.45 , and longward of 2.45 , the S/N in the observed spectrum is very low due to strong telluric absorption.[]{data-label="ir.may6"}](f10a.eps "fig:"){width="10.2cm"} ![Comparison of the observed NIR spectrum of SN1999by on May 6, 1999, with the theoretical IR-spectrum of 5p0z22.8 $\approx 4$ days before maximum light (10 days after the explosion). Between 1.35 and 1.45 , and longward of 2.45 , the S/N in the observed spectrum is very low due to strong telluric absorption.[]{data-label="ir.may6"}](f10b.eps "fig:"){width="10.2cm"}
![Comparison of the observed NIR spectrum of SN1999by on May 10, 1999, with the theoretical IR-spectrum of 5p0z22.8 at maximum light (15 days after the explosion). Between 1.35 and 1.45 , between 1.8 and 1.9 , and longward of 2.45 , the S/N in the observed spectrum is very low due to strong telluric absorption.[]{data-label="ir.may10"}](f11a.eps "fig:"){width="10.2cm"} ![Comparison of the observed NIR spectrum of SN1999by on May 10, 1999, with the theoretical IR-spectrum of 5p0z22.8 at maximum light (15 days after the explosion). Between 1.35 and 1.45 , between 1.8 and 1.9 , and longward of 2.45 , the S/N in the observed spectrum is very low due to strong telluric absorption.[]{data-label="ir.may10"}](f11b.eps "fig:"){width="10.2cm"}
![Comparison of the observed NIR spectrum of SN1999by on May 16, 1999, with the theoretical IR-spectrum of 5p0z22.8 seven days after maximum light (22 days after the explosion). Between 1.35 and 1.45 the S/N in the observed spectrum is very low due to strong telluric absorption.[]{data-label="ir.may16"}](f12.eps){width="10.2cm"}
![Comparison of the observed NIR spectrum of SN1999by on May 24, 1999, with the theoretical IR-spectrum of 5p0z22.8 14 days after maximum light (29 days after the explosion). Between 1.35 and 1.45 the S/N in the observed spectrum is very low due to strong telluric absorption.[]{data-label="ir.may24"}](f13.eps){width="10.2cm"}
![Comparison of the observed NIR spectrum of SN1999by on May 16, 1999, with the theoretical IR-spectrum of 5p0z22.8 seven days after maximum light if we impose mixing in the inner 0.7 $M_\odot$. Line blanketing due to a large number of FeII and CoII lines becomes strong as on May 24, 1999 in the unmixed model (see Fig. \[ir.may16\]).[]{data-label="mix.may16"}](f14.eps){width="10.2cm"}
[^1]: http://www-obs.univ-lyon1.fr/leda/home\_leda.html
[^2]: NSO/Kitt Peak FTS data used here were produced by NSF/NOAO.
|
---
abstract: |
The paper uses a feature of calculating the Riemann Zeta function in the critical strip, where its approximate value is determined by partial sums of the Dirichlet series, which it is given.
These expressions are called the first and second approximate equation of the Riemann Zeta function.
The representation of the terms of the Dirichlet series by vectors allows us when analyzing the polyline formed by these vectors:
1\) explain the geometric meaning of the generalized summation of the Dirichlet series in the critical strip;
2\) obtain formula for calculating the Riemann Zeta function;
3\) obtain the functional equation of the Riemann Zeta function based on the geometric properties of vectors forming the polyline;
4\) explain the geometric meaning of the second approximate equation of the Riemann Zeta function;
5\) obtain the vector equation of non-trivial zeros of the Riemann Zeta function;
6\) determine why the Riemann Zeta function has non-trivial zeros on the critical line;
7\) understand why the Riemann Zeta function cannot have non-trivial zeros in the critical strip other than the critical line.
The main result of the paper is a definition of possible ways of the confirmation of the Riemann hypothesis based on the properties of the vector system of the second approximate equation of the Riemann Zeta function.
author:
- |
Kirill V. Kapitonets\
BAUMAN MSTU, GRADUATE 1990\
MCC EuroChem\
Moscow\
Russian Federation\
`[email protected]`\
title: ANALYSIS OF THE RIEMANN ZETA FUNCTION
---
Introduction
============
A more detailed title of the paper can be formulated as follows:
*Geometric analysis of expressions that determine a value of the Riemann zeta function in the critical strip.*
However, this name will not fully reflect the content of the paper. It is necessary to refer to the sequence of problems. The starting point of the analysis of the Riemann zeta function was the question:
*Why the Riemann zeta function has non-trivial zeros.*
A superficial study of the question led to the idea of geometric analysis of the Dirichlet series.
Preliminary geometric analysis allowed to obtain the following results:
1\) definition of an expression for calculating a value of the Riemann zeta function in the critical strip;
2\) derivation of the functional equation of the Riemann zeta function based on the geometric properties of vectors;
3\) explanation of the geometric meaning of the second approximate equation of the Riemann zeta function.
The most important result of the preliminary geometric analysis was the question why the Riemann zeta function cannot have non-trivial zeros in the critical strip except the critical line.
Therefore, the extended title of the paper is follows:
*Geometric analysis of the expressions defining a value of the Riemann zeta function in the critical strip with the aim of answering the question of why the Riemann zeta function cannot have non-trivial zeros in the critical strip, except for the critical line.*
At once, it is necessary to define the upper limit of the paper. Although that the paper explores the main question of the Riemann hypothesis, we are not talking about its proof.
The results of the geometric analysis of the expressions defining a value of the Riemann zeta function in the critical strip is the definition of the possible ways of the confirmation of the Riemann hypothesis based on the results of this analysis.
In addition, if we are talking about the boundaries, then we will mark the lower limit of the paper. We will not use the latest advances in the theory of the Riemann zeta function and improve anyone’s result (we will state our position on this later).
We turn to the origins of the theory of the Riemann zeta function and base on the very first results obtained by Riemann, Hardy, Littlewood and Titchmarsh, as well as by Adamar and Valle Poussin.
In addition, of course, we cannot do without the official definition of the Riemann hypothesis:
*The Riemann zeta function is the function of the complex variable s, defined in the half-plane Re(s) > 1 by the absolutely convergent series:* $$\label{zeta_dirichlet}\zeta(s) = \sum\limits_{n=1}^{\infty}\frac{1}{n^s};$$ *and in the whole complex plane C by analytic continuation. As shown by Riemann, $\zeta(s)$ extends to C as a meromorphic function with only a simple pole at s =1, with residue 1, and satisfies the functional equation:* $$\label{zeta_func_eq}\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)=\pi^{-(1-s)/2}\Gamma(\frac{1-s}{2})\zeta(1-s);$$ *Thus, in terms of the function $\zeta(s)$, we can state Riemann hypothesis: The non-trivial zeros of $\zeta(s)$ have real part equal to 1/2.*
We define two important concepts of the Riemann zeta function theory: the critical strip and the critical line.
Both concepts relate to the region where non-trivial zeros of the Riemann zeta function are located.
Adamar and Valle-Poussin independently in the course of the proof of the theorem about the distribution of prime numbers, which is based on the behavior of non-trivial zeros of the Riemann zeta function , showed that all non-trivial zeros Riemann zeta function is located in a narrow strip $0<Re(s)<1$. In the theory of the Riemann zeta function, this strip is called *critical strip*.
The line $Re(s)=1/2$ referred to in the Riemann hypothesis is called *critical line*.
It is known that the Dirichlet series (\[zeta\_dirichlet\]), which defines the Riemann zeta function, diverges in the critical strip, therefore, to perform an analytical continuation of the Riemann zeta function, *the method of generalized summation* of divergent series, in particular the generalized Euler-Maclaurin summation formula is used.
We should note that the generalized Euler-Maclaurin summation formula is used in the case where the partial sums of a divergent series are suitable for computing the generalized sum of this divergent series.
From the theory of generalized summation of divergent series [@HA3], it is known that any method of generalized summation, if it yields any result, yields the same result with other methods of generalized summation.
On the one hand, this fact confirms the uniqueness of the analytical continuation of the function of a complex variable, and on the other hand, apparently, the principle of analytical continuation extends in the form of a generalized summation to the functions of a real variable, which are defined in divergent series.
Now we can formulate the principles of geometric analysis of expressions that define a value of the Riemann zeta function in the critical strip:
1\) a complex number can be represented geometrically using points or vectors on the plane;
2\) the terms of the Dirichlet series, which defines the Riemann zeta function, are complex numbers;
3\) a value of the Riemann zeta function in the region of analytic continuation can be represented by any method of generalized summation of divergent series.
What exactly is the geometric analysis of values of the Riemann zeta function, we will see when we construct a polyline, which form a vector corresponding to the terms of the Dirichlet series, which defines the Riemann zeta function.
All results in the paper are obtained empirically by calculations with a given precision (15 significant digits).
Representation of the Riemann zeta function value by a vector system
====================================================================
This section presents detailed results of geometric analysis of expressions that determine a value of the Riemann zeta function in the critical strip.
The possibility of such an analysis is due to two important facts:
1\) representation of complex numbers by vectors;
2\) using of methods of generalized summation of divergent series.
Representation of the partial sum of the Dirichlet series by a vector system
----------------------------------------------------------------------------
Each term of the Dirichlet series (\[zeta\_dirichlet\]), which defines the Riemann zeta function, is a complex number $x_n+iy_n$, hence it can be represented by the vector $X_n=(x_n, y_n)$.
To obtain the coordinates of the vector $X_n(s), s=\sigma+it$, we first present the expression $n^{-s}$ in the exponential form of a complex number, and then, using the Euler formula, in the trigonometric form of a complex number: $$\label{x_vect_direchlet}X_n(s)=\frac{1}{n^{s}}=\frac{1}{n^{\sigma}}e^{-it\log(n)}=\frac{1}{n^{\sigma}}(\cos(t\log(n))-i\sin(t\log(n)));$$
Then the coordinates of the vector $X_n(s)$ can be calculated by formulas: $$\label{x_n_y_n}x_n(s)=\frac{1}{n^{\sigma}}\cos(t\log(n)); y_n(s)=-\frac{1}{n^{\sigma}}\sin(t\log(n));$$
*Using the rules of analytical geometry, we can obtain the coordinates corresponding to the partial sum $s_m(s)$ of the Dirichlet series (\[zeta\_dirichlet\]):* $$\label{s_m_x_s_m_y}s_m(s)_x=\sum_{n=1}^{m}{x_n(s)}; s_m(s)_y=\sum_{n=1}^{m}{y_n(s)};$$
![Polyline, $s=1.25+279.229250928i$[]{data-label="fig:s1_1"}](s1_1.jpg)
We construct a polyline corresponding to partial sums of $s_m(s)$. Select a value *in the convergence region* of the Dirichlet series (\[zeta\_dirichlet\]), for example, $s=1.25+279.229250928 i$.
We also display the vector $(0.69444570272324, 0.61658346971775)$, which corresponds to a value of the Riemann zeta function at $s=1.25+279.229250928 i$.
We will display the first $m=90$ vectors (later we will explain why we chose such a number) so that the vectors follow in ascending order of their numbers (fig. \[fig:s1\_1\]).
Now we change a value of the real part $s=0.75+279.229250928 i$ and move to the region where the Dirichlet series (\[zeta\_dirichlet\]) *diverges.*
![Polyline, $s=0.75+279.229250928i$[]{data-label="fig:s3_1"}](s3_1.jpg)
We also display the vector $(0.22903651233853, 0.51572970834588)$, which corresponds to a value of the Riemann zeta function at $s=0.75+279.229250928 i$.
We observe (fig. \[fig:s3\_1\]) an increase of size of the polyline, but the qualitative behavior of the graph does not change. *The polyline twists around a point corresponding to a value of the Riemann zeta function.*
To see what exactly is the difference, we need to consider the behavior of vectors with smaller numbers, for example, in the range $m=(300, 310)$ with relation to vectors with large numbers, for example, in the range $m=(500, 510)$.
![Part of polyline, $s=1.25+279.229250928i$[]{data-label="fig:s1_2_1"}](s1_2_1.jpg)
We see (fig. \[fig:s1\_2\_1\]) that when $\sigma=1.25$ the polyline is *a converging* spiral, as the radius of the spiral in the range of $m=(300, 310)$ is greater than the radius of the spiral in the range of $m=(500, 510)$.
![Part of polyline, $s=0.75+279.229250928i$[]{data-label="fig:s3_2_1"}](s3_2_1.jpg)
Unlike the first case, when $\sigma=0.75$ we see (fig. \[fig:s3\_2\_1\]) that a polyline is *a divergent* spiral, since the radius of the spiral in the range $m=(300, 310)$ is less than the radius of the spiral in the range $m=(500, 510)$.
But regardless of whether the spiral converges or diverges, in both cases we see that the center of the spiral corresponds to the point corresponding to a value of the Riemann zeta function (we will later show that it is indeed).
This fact can be considered a geometric explanation of the method of generalized summation:
*The coordinates of the partial sums $s_m(s)_x$ and $s_m(s)_y$ of the Dirichlet series (\[zeta\_dirichlet\]) vary with relation to some middle values of $x$ and $y$, which we take as a value of the Riemann zeta function, only in one case the coordinates of the partial sums converge infinitely to these values, and in the other case they diverge infinitely with relation to these values.*
Properties of vector system of the partial sums of the Dirichlet series – the Riemann spiral
--------------------------------------------------------------------------------------------
In order not to use in the further analysis of the long definitions of the vector system, which we are going to study in detail, we introduce the following definition:
*The Riemann spiral is a polyline formed by vectors corresponding to the terms of the Dirichlet series (\[zeta\_dirichlet\]) defining the Riemann zeta function, in ascending order of their numbers.*
A preliminary analysis of *Riemann spiral* showed that vectors arranged in ascending order of their numbers form a converging or divergent spiral with a center at a point corresponding to a value of the Riemann zeta function.
We also see (fig. \[fig:s3\_1\]) that the Riemann spiral has several more centers where the vectors first form a converging spiral and then a diverging spiral.
Comparing (fig. \[fig:s1\_1\]) and (fig. \[fig:s3\_1\]) we see that the number of such centers does not depend on the real part of a complex number.
While comparing (fig. \[fig:s3\_1\]) and (fig. \[fig:s7\_1\]), we see that the number of such centers increases when the imaginary part of a complex number increases.
![The Riemann spiral, $s=0.75+959.459168807i$[]{data-label="fig:s7_1"}](s7_1.jpg)
We can explain this behavior of Riemann spiral vectors if we return to the exponential form (\[x\_vect\_direchlet\])of the record of terms of the Dirichlet series (\[zeta\_dirichlet\]).
The paradox of the Riemann spiral is that with unlimited growth of the function $t\log(n)$, the absolute angles $\varphi_n(t)$ of its vectors behave in a pseudo-random way (fig. \[fig:absolute\_angles\]), because we can recognize angles only in the range $[0, 2\pi]$. $$\label{phi_n}\varphi_n(t)=t\log(n)mod\ 2\pi;$$
![Absolute angles $\varphi_n(t)$ of vectors of the Riemann spiral, rad, $s=0.75+959.459168807i$[]{data-label="fig:absolute_angles"}](absolute_angles.jpg)
The angles between the vectors $\Delta\varphi_n(t)$, if they are measured not as visible angles between segments, but as angles between directions, permanently grow (fig. \[fig:relative\_angles\]) to a value $2\pi$, then they sharply decrease to a value $0$ and again grow to a value $2\pi$. $$\label{delta_phi_n}\Delta\varphi_n(t)=\varphi_n(t)-\varphi_{n-1}(t);$$
On the last part this growth is *asymptotic,* i.e. no matter how large the vector number $n$ is, a value $\log(n)$ will never be equal to a value $\log(n+1)$.
![Relative angles $\Delta\varphi_n(t)$ of vectors of the Riemann spiral, rad, $s=0.75+959.459168807i$[]{data-label="fig:relative_angles"}](relative_angles.jpg)
In consequence of the revealed properties of the Riemann spiral vectors, we observe two types of singular points:
1\) *reverse points* where the visible twisting of the vectors is replaced by untwisting, these points are multiples of $(2k-1)\pi$;
2\) *inflection points*, in which the visible untwisting of the vectors is replaced by twisting, these points are multiples of $2k\pi$;
Now we can explain why for a value of $s=1.25+279.229250928 i$ we took 90 vectors to construct the Riemann spiral:
$$\label{m1}279.229250928/\pi=88.881431082;$$
Therefore, the first reverse point is between the 88th and 89th vectors, we just rounded the number of vectors to a multiple.
*This is the number of vectors we must use to build the Riemann spiral in order to vectors fully twisted around the point corresponding to a value of the Riemann zeta function.*
In addition, we can determine the number of reverse points $m$, as we remember, this number determines the range in which (fig. \[fig:relative\_angles\]) the periodic monotonous increase of angles between the Riemann spiral vectors is observed, moreover this number plays an important role (as we will see later) in the representation of the Riemann zeta function values by the vector system.
We can determine the number of reverse points $m$ from the condition that between two reverse points there is at least one vector:
$$\label{m2_eq}\frac{t}{(2m-1)\pi}-\frac{t}{(2m+1)\pi}=1;$$
From this equation we find:
$$\label{m2}m=\sqrt{\frac{t}{2\pi}+\frac{1}{4}};$$
At the end of the consideration of the static parameters of the Riemann spiral, we perform an analysis of its radius of curvature:
$$\label{curvature_radius_eq}r_n=\frac{|X_n|\cos(\Delta\varphi_n)}{\sqrt{1-\cos(\Delta\varphi_n)^2}};$$
where $$\label{cos_delta_varphi_n}\cos(\Delta\varphi_n)=\frac{(X_n,X_{n-1})}{|X_n||X_{n-1}|};$$ $$\label{module}|X_n|=\sqrt{x_n^2+y_n^2};$$ $$\label{scalar_product}(X_n,X_{n-1})=x_nx_{n-1}+y_ny_{n-1};$$
![Curvature radius of the Riemann spiral, $s=0.75+959.459168807i$[]{data-label="fig:curvature_radius"}](curvature_radius.jpg)
We see (fig. \[fig:curvature\_radius\]) that the Riemann spiral has *an alternating sign* of radius of curvature. This is the only spiral that has this property.
The maximum negative value of the radius of curvature takes at the reverse point, and the maximum positive is at the inflection point.
Derivation of an empirical expression for the Riemann zeta function
-------------------------------------------------------------------
We study in detail the behavior of Riemann spiral vectors after the first reverse point: $$m=\frac{t}{\pi};$$
As we know, the Riemann spiral vectors in the critical strip form a divergent spiral (fig. \[fig:s3\_2\_1\]).
Consider the Riemann spiral after the first reverse point on the left boundary of the critical strip (Fig. \[fig:s8\_2\_1\]) i.e. when $\sigma=0$.
![Part of polyline, $s=0+279.229250928i$[]{data-label="fig:s8_2_1"}](s8_2_1.jpg)
We see that the size of the polyline has increased in comparison with (fig. \[fig:s3\_2\_1\]), but the behavior of the vectors has not changed, as their numbers increase, the vectors tend to form a regular polygon and the quantity of edges grows with the growth of vector numbers.
And as we understand, as a result of the vectors tend to form a circle of the greater radius, than the greater the number of vectors, and the center of the circle tends to the center of the spiral, consequently, to the point, which corresponds to a value of the Riemann zeta function.
In analytic number theory, this fact[^1] is known as the first approximate equation of the Riemann zeta function: $$\label{zeta_eq_1}\zeta(s)=\sum_{n\le x}{\frac{1}{n^s}}-\frac{x^{1-s}}{1-s}+\mathcal{O}(x^{-\sigma});
\sigma>0; |t|< 2\pi x/C; C > 1$$
As we know, Hardy and Littlewood obtained this approximate equation based on the generalized Euler-Maclaurin summation method.
It is known from the theory of generalized summation of divergent series [@HA3] that we can apply *any other method* of generalized summation (if it gives any value) and get the same result.
We use the Riemann spiral to determine another method of generalized summation.
Consider the first 30 vectors of the Riemann spiral after the first reverse point (fig. \[fig:q\_1\]), here the vectors form a star-shaped polygon, with the point that corresponds to a value of the Riemann zeta function at the center of this polygon, as in the case of a divergent spiral (fig. \[fig:s3\_2\_1\]).
![First 30 vectors after the first reverse point, $s=0.75+279.229250928i$[]{data-label="fig:q_1"}](q_1.jpg)
Connect the middle of the segments formed by vectors and get 29 segments (fig. \[fig:q\_2\]).
![The first step of generalized summation, $s=0.75+279.229250928i$[]{data-label="fig:q_2"}](q_2.jpg)
We see that the star-shaped polygon has decreased in size, and the point that corresponds to a value of the Riemann zeta function is again at the center of this polygon.
![The twentieth step of generalized summation, $s=0.75+279.229250928i$[]{data-label="fig:q_20"}](q_20.jpg)
We will repeat the operation of reducing the polygon (fig. \[fig:q\_20\]) until there is one segment left.
*Calculations show that when calculating the coordinates of vectors with an accuracy of 15 characters, the coordinates of the middle of this segment with an accuracy of not less than 13 digits match a value of the zeta function of Riemann, for example, at point $0.75+279.229250928i$ the exact value of [@ZF] equals $0.22903651233853+0.51572970834588i$, and in the calculation of midpoints of the segments we get a value $0.22903651233856+0.51572970834589i$.*
To calculate values of the Riemann zeta function, a reduced formula can be obtained by the described method. Write sequentially the expressions to calculate the midpoints of the segments and substitute successively the obtained formulas one another: $$\label{a}a_i = \frac{x_i+x_{i+1}}{2};$$ $$\label{b}b_i =\frac{a_i+a_{i+1}}{2}=\frac{x_i+2x_{i+1}+x_{i+2}}{4};$$ $$\label{c}c_i =\frac{b_i+b_{i+1}}{2}= \frac{x_i+3x_{i+1}+3x_{i+2}+x_{i+3}}{8};$$ $$\label{d}d_i =\frac{c_i+c_{i+1}}{2}=\frac{x_i+4x_{i+1}+6x_{i+2}+4x_{i+3}+x_{i+4}}{16};$$
We write the same formulas for the coordinates $y_k$.
In the numerator of each formula (\[a\]-\[d\]) we see the sum of vectors multiplied by binomial coefficients, which correspond to the degree of Newton’s binomial, equal to the number of vectors minus one, and in the denominator the degree of two, equal to the number of vectors minus one.
Now we can write down the abbreviated formula:
$$\label{s_x_s_y}s_x=\frac{1}{2^m}\sum_{k=0}^{m}\Big(_m^k\Big)x_k; s_y=\frac{1}{2^m}\sum_{k=0}^{m}\Big(_m^k\Big)y_k;$$
where $x_k$ and $y_k$ are coordinates of partial sums (\[s\_m\_x\_s\_m\_y\]) of Dirichlet series.
*We obtained the formula of the generalized Cesaro summation method $(C, k)$ [@HA3].*
It should be noted that to calculate the coordinates of the center of the star-shaped polygon when a value of the imaginary part of the complex number is equal to 279.229250928, we use 30 vectors after the first reverse point, while starting with a value of the imaginary part of a complex number equal to 1000, it is enough to use only 10 vectors after the first reverse point.
We obtained a result that applies not only to the Riemann zeta function, but also to all functions of a complex variable that have an analytic continuation.
A result that relates not only to the analytic continuation of the Riemann zeta function, but to the analytic continuation as the essence of any function of a complex variable.
A result that refers to any function and any physical process where such functions are applied, and hence a result that defines the essence of that physical process.
Euler’s intuitive belief[^2] that a divergent series can be matched with a certain value [@EL], has evolved into the fundamental theory[^3] of generalized summation of divergent series, which Hardy systematically laid out in his book [@HA3].
The essence of our conclusions is as follows:
1\) When it comes to analytical continuation of some function of a complex variable, it automatically means that there are at least two regions of definition of this function:
a\) the region where the series by which this function is defined converges;
b\) the region where the same series diverges.
*Actually, the question of analytical continuation arises because there is an region where the series that defines the function of a complex variable diverges.*
Thus, the analytical continuation of any function of a complex variable is inseparably linked to the fact that there is a divergent series.
We can say that this is the very essence of the analytical continuation.
2\) Analytical continuation is possible only if there is some method of generalized summation that will give a result, i.e. some value other than infinity.
This value will be a value of the function, in the region where the series by which this function is defined diverges.
3\) The most important thing in this question is that the series by which the function is defined must behave asymptotically in the region where this series infinitely converges and in the region where it infinitely diverges.
And then we come to an important point:
*The function of a complex variable, if it has an analytical continuation (and therefore is given by a series that converges in one region and diverges in another, i.e. has no limit of partial sums of this series) is determined by the asymptotic law of behavior of the series by which this function is given and it is no matter whether this series converges or diverges, a value of this function will be the asymptotic value with relation to which this series converges or diverges.*
Analytical continuation is possible only if the series with which the function is given has asymptotic behavior, i.e. its values oscillate with relation to the asymptote, which is a value of the function.
In the case of a function of a complex variable, there are two such asymptotes, and in the case of a function of a real variable, such an asymptote is one.
If the Riemann zeta function is given asymptotic values, then it also has *asymptotic value of zero,* hence it may seem that the Riemann hypothesis cannot be proved.
As we will show later, a value of the Riemann zeta function can be given by a finite vector system, the sum of which gives *the exact value of zero* if these vectors form a polygon.
*In this regard, we can conclude that the Riemann hypothesis can be confirmed only if such a finite vector system exists and only using the properties of the vectors of this system.*
One can disagree with the conclusion that it is possible to confirm the Riemann hypothesis using a finite vector system, but we will go this way, because the chosen method of geometric analysis of the Riemann zeta function allows us to penetrate into the essence of the phenomenon.
Derivation of an empirical expression for the functional equation of the Riemann zeta function
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We already know one dynamic property of the Riemann spiral, which is that the number of reverse points increases with the growth of the imaginary part of a complex number.
We will study *the reverse points* and *inflection points,* which, as we will see later, are not just special points of the Riemann spiral, they are its essential points that define the essence of the Riemann spiral and the Riemann zeta function.
We will use our empirical formula for calculating a value of the Riemann zeta function, which corresponds to the Cesaro generalized summation method.
As we saw (fig. \[fig:s7\_1\]), the Riemann spiral vectors at any reverse point where the Riemann spiral has a negative radius of curvature, up to the reverse point form a converging spiral, and after the reverse point form a divergent spiral.
This fact allows us to apply the formula for calculating a value of the Riemann zeta function at any reverse point, because this formula allows us to calculate the coordinates of any center of the divergent spiral, which form the Riemann spiral vectors.
For the convenience of further presentation, we introduce the following definition:
*Middle vector of the Riemann spiral is a directed segment connecting two adjacent centers of the Riemann spiral, drawn in the direction from the center with a smaller number to the center with a larger number (fig. \[fig:average\_vectors\]) when numbering from the center corresponding to value of the Riemann zeta function.*
![The middle vectors of the Riemann spiral , $s=0.25+5002.981i$[]{data-label="fig:average_vectors"}](average_vectors.jpg)
To identify modulus of middle vectors of the Riemann spiral first using the formula (\[s\_x\_s\_y\]) we define the coordinates of the centers of the Riemann spiral.
To calculate the coordinates of the first center of the Riemann spiral (a value of the Riemann zeta function) we will use 30 vectors, to calculate the coordinates of the second center - 20 vectors.
To increase the accuracy, we can choose a different number of vectors for each center of the Riemann spiral, but since we choose a sufficiently large value of the imaginary part of a complex number and a sufficiently small number of vectors, it will be enough to use 5 vectors to calculate the coordinates starting from the third center of the Riemann spiral.
Then, using the formula to determine the distance between two points, we find the modulus of the middle vectors: $$\label{dist}|Y_n|=\sqrt{(x_{n+1}-x_n)^2+(y_{n+1}-y_n)^2};$$
We calculate the modulus of the first six middle vectors of the Riemann spiral for values of the real part in the range from 0 to 1 in increments of 0.1 and a fixed value of the imaginary part of a complex number - 5000 and then we approximate the dependence of the modulus of the middle vector of the Riemann spiral on its sequence number (fig. \[fig:v1\_5000\_6\]).
![The dependence of the modulus of the middle vector from the sequence number, $Re(s)=(0.0, 0.1, 0.2, 0.3)$, $Im(s)=5000$[]{data-label="fig:v1_5000_6"}](v1_5000_6.jpg)
The best method of approximation of dependence of the modulus of the middle vector from the sequence number (fig. \[fig:v1\_5000\_6\], where $x=n$) for different values of the real part of a complex number (and the fixed imaginary part of a complex number) is a power function $$\label{depence_1}|Y_n|=An^B;$$
It should be noted that the accuracy of the approximation increases or with a decrease in the number of vectors (fig. \[fig:v0\_5000\_5\]), or by increasing a value of the imaginary part of a complex number (fig. \[fig:v0\_8000\_6\]), this fact indicates the *asymptotic* dependence of the obtained expressions.
![The dependence of the modulus of the middle vector for the five vectors , $s=0+5000i$[]{data-label="fig:v0_5000_5"}](v0_5000_5.jpg)
![Dependence of the middle vector modulus for a larger value of the imaginary part of a complex number , $s=0+8000i$[]{data-label="fig:v0_8000_6"}](v0_8000_6.jpg)
Now approximate the dependence of the coefficients $A$ and $B$ on a value of the real part of a complex number.
We start with the coefficient $B$, because at this stage we will complete its analysis, and for the analysis of the coefficient $A$ additional calculations will be required.
![Dependence of the coefficient $B$ on the real part of a complex number, $Im(s)=5000$[]{data-label="fig:factor2_5000"}](factor2_5000.jpg)
The coefficient $B$ has a linear dependence (fig. \[fig:factor2\_5000\], where $x=\sigma$) from a value of the real part of a complex number. Therefore, taking into account the identified asymptotic dependence, we can rewrite the expression (\[depence\_1\]) for the modulus of the middle vectors of the Riemann spiral: $$\label{depence_2}|Y_n|=A\frac{1}{n^{1-\sigma}};$$
The best way to approximate the dependence of the coefficient $A$ (fig. \[fig:factor1\_5000\], where $x=\sigma$) from a value of the real part of a complex number is the exponent: $$\label{depence_3}A=Ce^{D\sigma};$$
![Dependence of the coefficient $A$ on the real part of a complex number, $Im(s)=5000$[]{data-label="fig:factor1_5000"}](factor1_5000.jpg)
We first find the ratio of the coefficients $C$ and $D$, for this we calculate $\log(C)$: $$\label{depence_4}2\log(C)=6.67772573;$$
Taking into account the revealed asymptotic dependence $2\log(C)=D$, we can rewrite the expression (\[depence\_3\]) for the coefficient $A$: $$\label{depence_5}A=Ce^{-2\log(C)\sigma}=e^{log(C)-2\log(C)\sigma}=e^{2log(C)(\frac{1}{2}-\sigma)}=(C^2)^{\frac{1}{2}-\sigma};$$
Now, taking into account the identified asymptotic dependence $|Y_n|=A=C$ when $\sigma=0$, we calculate the modulus of the first middle vector when $\sigma=0$ for different values of the imaginary part of a complex number in the range from 1000 to 9000 in increments of 1000 and then approximate the dependence of the coefficients $C$ on a value of the imaginary part of a complex number.
![Dependence of the coefficient $C$ on the imaginary part of a complex number, $Re(s)=0$[]{data-label="fig:factor1"}](factor1.jpg)
The best way to approximate the dependence of the coefficient $C$ (Fig. \[fig:factor1\], where $x=t$) from a value of the imaginary part of a complex number is a power function: $$\label{depence_6}C=Et^F;$$
Taking into account the revealed asymptotic dependence consider $$F=\frac{1}{2};$$
then we can write the final expression for the modulus of the middle vectors of the Riemann spiral: $$\label{depence_8}|Y_n|=(E^2t)^{\frac{1}{2}-\sigma}\frac{1}{n^{1-\sigma}};$$
where $E^2=0.159154719364$ some constant, the meaning of which we learn later.
We obtained an asymptotic expression (\[depence\_8\]) for the modulus of the middle vectors of the Riemann spiral, which becomes, as we found out, more precisely when the imaginary part of a complex number increases.
*As a consequence of the asymptotic form of the resulting expression, we can apply it to any middle vector, even if we can no longer calculate its coordinates or the calculated coordinates give an inaccurate value of the middle vector modulus.*
Therefore, we can obtain any quantity of middle vectors necessary to construct *the inverse Riemann spiral.*
So we can get an infinite series, which is given by the middle vectors of the Riemann spiral.
By comparing the expression for the modulus of vectors (\[x\_n\_y\_n\]) of the Riemann spiral and the expression for the modulus of its middle vectors (\[depence\_8\]), we can assume that the infinite series formed by the middle vectors of the Riemann spiral sets a value of the Riemann zeta function $\zeta(1-s)$.
*So we can assume that values of the Riemann zeta function $\zeta(s)$ and $\zeta(1-s)$ are related through an expression for the middle vectors of the Riemann spiral.*
To complete the derivation of the dependence of the Riemann zeta function $\zeta(s)$ and $\zeta(1-s)$, it is necessary to determine the dependence of the angles between the middle vectors of the Riemann spiral and construct an inverse Riemann spiral whose vectors, as we show further, asymptotically twist around the zero of of the complex plane.
If to determine the coordinates of the centers of Riemann spiral, we used the *reverse points,* to determine the angles between middle vectors of the Riemann spiral, we will use *inflection points.* We see (fig. \[fig:average\_vectors\]) that the inflection points are not only the points at which the visible untwisting of the vectors is replaced by the twisting, but also the points at which the middle vectors intersect the Riemann spiral.
One can show by computing that the angles between the middle vectors and the Riemann spiral at the intersection points are asymptotically equal to $\pi/4$, then the angles between the middle vectors can be equated to the angles between the Riemann spiral vectors at the inflection points.
As we remember, the inflection points are multiples of $2k\pi$, then using the properties of the logarithm, we can find the angle between the first and any other middle vector of the Riemann spiral: $$\label{depence_9}\beta_k=\alpha_k-\alpha_1=t\log\Big(\frac{t}{2\pi}\Big)-t\log(k)-t\log\Big(\frac{t}{2\pi}\Big)=-t\log(k);$$
We have obtained an expression that shows that the angles between the first middle vector of the Riemann spiral and any other middle vector are equal in modulus and opposite in sign to the angles between the first vector and corresponding vector of the Riemann spiral, the negative sign shows that the middle vectors have a special kind of symmetry (which we will consider later) with relation to the vectors of value of the Riemann spiral.
Knowing the coordinates of the first middle vector, which we calculated with sufficient accuracy, we can now find the angles and modulus of the remaining middle vectors using the obtained asymptotic expressions and construct the inverse Riemann spiral (Fig. \[fig:reverse\_spiral\_approx\]).
![Inverse Riemann spiral, $s=0.25+5002.981i$[]{data-label="fig:reverse_spiral_approx"}](reverse_spiral_approx.jpg)
Now at $n=k$ we can write the final formula to calculate the coordinates of the middle vectors $Y_n$: $$\label{h_x_n_h_y_n}\tilde{x}_n(s)=(E^2t)^{\frac{1}{2}-\sigma}\frac{1}{n^{1-\sigma}}\cos(\alpha_1-t\log(n)); \tilde{y}_n(s)=(E^2t)^{\frac{1}{2}-\sigma}\frac{1}{n^{1-\sigma}}\sin(\alpha_1-t\log(n));$$
Using Euler’s formula for complex numbers we write the expression for the middle vectors of the Riemann spiral in exponential form: $$\label{y_n_exp}Y_n(s)=(E^2t)^{\frac{1}{2}-\sigma}\frac{1}{n^{1-\sigma}}e^{-i(\alpha_1-t\log(n))};$$
Using the rules of analytical geometry, we can obtain the coordinates of the vector corresponding to the partial sum of $\hat{s}_m(s)$ inverse Riemann spiral: $$\label{h_s_m_x_h_s_m_y}\tilde{s}_m(s)_x=\zeta(s)_x-\sum_{n=1}^{m}{\tilde{x}_n(s)}; \tilde{s}_m(s)_y=\zeta(s)_y-\sum_{n=1}^{m}{\tilde{y}_n(s)};$$
To verify that the obtained expression for the middle vectors of the Riemann spiral defines the relation values of the Riemann zeta function $\zeta(s)$ and $\zeta(1-s)$, consider the middle vectors of the Riemann spiral near the point of zero (fig. \[fig:reverse\_spiral\_zero\]).
![Middle vectors of the inverse Riemann spiral near the point of zero, $s=0.25+5002.981i$[]{data-label="fig:reverse_spiral_zero"}](reverse_spiral_zero.jpg)
We see that the middle of the vectors twist around the zero point, in the same way as the vectors of the Riemann spiral twist around the point with the coordinates of a value of the Riemann zeta function[^4].
Now we can write the final equation that relates values of the Riemann zeta function $\zeta(s)$ and $\zeta(1-s)$ *taking into account the rules of generalized summation of divergent series:* $$\label{zeta_func_approx}\sum_{n=1}^{\infty}{\frac{1}{n^s}}-(E^2t)^{\frac{1}{2}-\sigma}e^{-i\alpha_1}\sum_{n=1}^{\infty}{\frac{1}{n^{1-s}}}=0;$$
where $\alpha_1$ is the angle of the first middle vector of the Riemann spiral.
*This fact shows the asymptotic form of functional equation and the geometric nature of the Riemann zeta function, based on the significant role of turning points and inflection points of the Riemann spiral, which determine the middle vectors and the inverse Riemann spiral.*
We can try to find an asymptotic expression for the angle $\alpha_1$ of the first middle vector of the Riemann spiral in a geometric way, but this will not significantly improve the result.
We use the results of the analytical theory of numbers to an arithmetic way to find the exact expression for the angle $\alpha_1$ of the first middle vector of the Riemann spiral and to determine the meaning of the constant $E^2=0.159154719364$.
Derivation of empirical expression for CHI function
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The functional equation of the Riemann zeta function [@TI] has several equivalent entries: $$\label{zeta_func_eq2}\zeta(s)=\chi(s)\zeta(1-s);$$
where $$\label{chi_eq}\chi(s)=\frac{(2\pi)^s}{2\Gamma(s)\cos(\large\frac{\pi s}{2})}=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s)=\pi^{s-\frac{1}{2}}\frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})};$$
Comparing (\[zeta\_func\_approx\]) and (\[zeta\_func\_eq2\]), we get: $$\label{chi_eq_app}\chi(s)=(E^2t)^{\frac{1}{2}-\sigma}e^{-i\alpha_1};$$
We find the same expression in Titchmarsh [@TI]:
in any fixed strip $\alpha\le\sigma\le\beta$, when $t\to \infty$: $$\label{chi_eq_app2}\chi(s)= \Big(\frac{2\pi}{t}\Big)^{(\sigma+it-\frac{1}{2})}e^{i(t+\frac{\pi}{4})}\Big\{1+\mathcal{O}\Big(\frac{1}{t}\Big)\Big\};$$
We write the expression (\[chi\_eq\_app2\]) in the exponential form of a complex number: $$\label{chi_eq_ex}\chi(s)= \Big(\frac{t}{2\pi}\Big)^{(\frac{1}{2}-\sigma)}e^{-i(t(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{4}+\tau(s))};$$
where $$\label{tau}\tau(s)= \mathcal{O}\Big(\frac{1}{t}\Big);$$
By matching (\[chi\_eq\_app\]) and (\[chi\_eq\_ex\]), we obtain the angle of the first middle vector of the Riemann spiral: $$\label{alpha1}\alpha_1=t(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{4}+\tau(s);$$
And define the meaning of the constant $E^2=0.159154719364$: $$\label{e_2}E^2=\frac{1}{2\pi}=0.159154943091;$$
Now we can write the asymptotic equation (\[zeta\_func\_approx\]) in its final form: $$\label{zeta_func_approx2}\sum_{n=1}^{\infty}{\frac{1}{n^s}}-\Big(\frac{t}{2\pi}\Big)^{\frac{1}{2}-\sigma}e^{-i\alpha_1}\sum_{n=1}^{\infty}{\frac{1}{n^{1-s}}}=0;$$
We will find the empirical expression for remainder term $\tau(s)$ of the CHI function, which defines the ratio of the modulus and the argument of the exact $\chi(s)$ and the approximate $\tilde\chi(s)$ value of the CHI function: $$\label{chi_eq_rem}\tau(s)=\Delta\varphi_{\chi}+\log\Big(\frac{|\chi(s)|}{|\tilde\chi(s)|}\Big)i;$$
where $$\label{delta_varphi_chi}\Delta\varphi_{\chi}=Arg(\chi(s))-Arg(\tilde\chi(s));$$
The exact values[^5] CHI functions we find from the functional equation of the Riemann zeta function, substituting the exact values of the Riemann zeta function: $$\label{chi_eq_ex2}\chi(s)=\frac{\zeta(s)}{\zeta(1-s)};$$
Approximate values of the CHI function we find from the expression (\[chi\_eq\_ex\]), dropping the function $\tau(s)$. $$\label{chi_eq_app3}\tilde\chi(s)= \Big(\frac{t}{2\pi}\Big)^{(\frac{1}{2}-\sigma)}e^{-i(t(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{4})};$$
The calculation of values of the Riemann zeta function is currently available in different mathematical packages. To calculate the exact values of the Riemann zeta function we will use the Internet service [@ZF].
We will use 15 significant digits, because this accuracy is enough to analyze the CHI function.
We will calculate values of the Riemann zeta function in the numerator (\[chi\_eq\_ex2\]) for values of the real part of a complex number in the range from 0 to 1 in increments of 0.1 and values of the imaginary part of a complex number in the range from 1000 to 9000 in increments of 1000. Note that the range of values of the real part of the a complex number contains 2m+1 value, which are related by the following relation: $$\label{sigma} 1-\sigma_{k+1}=\sigma_{2m-k+1};$$
where k varies between 0 and 2m, hence, for k=m: $$\label{sigma_m}\sigma_{m+1} = 0.5;$$ We use, on the one hand, the property of the Riemann zeta function as a function of a complex variable: $$\label{zeta_conj}\overline{\zeta(1-\sigma+it)}=\zeta(\overline{1-\sigma+it})=\zeta(1-\sigma-it)=\zeta(1-s);$$
From other side: $$\label{zeta_conj2}\overline{\zeta(1-\sigma+it)}=Re(\zeta(1-\sigma+it))-Im(\zeta(1-\sigma+it))i;$$
Equate (\[zeta\_conj\]) and (\[zeta\_conj2\]): $$\label{zeta_right}\zeta(1-s)=\zeta(1-\sigma-it)=Re(\zeta(1-\sigma+it))-Im(\zeta(1-\sigma+it))i;$$
Now we use the relation (\[sigma\]): $$\label{zeta_right_req}\zeta(1-s_{k+1})=Re(\zeta(\sigma_{2m-k+1}+it))-Im(\zeta(\sigma_{2m-k+1}+it))i=Re(\zeta(s_{2m-k+1})-Im(s_{2m-k+1})i;$$
We use the resulting formula to compute the Riemann zeta function in the denominator (\[chi\_eq\_ex2\]) based on values computed for the numerator. Then we get an expression for CHI function: $$\label{chi_req}\chi(s_{k+1})=\frac{(Re(\zeta(s_{k+1}))+Im(\zeta(s_{k+1}))i)(Re(\zeta(s_{2m-k+1}))+Im(\zeta(s_{2m-k+1}))i)}{Re(\zeta(s_{2m-k+1}))^2+Im(\zeta(s_{2m-k+1}))^2};$$
We will open brackets and write separate expressions for the real part of the CHI function: $$\label{chi_req_re}Re(\chi(s_{k+1}))=\frac{Re(\zeta(s_{k+1}))Re(\zeta(s_{2m-k+1}))-Im(\zeta(s_{k+1}))Im(\zeta(s_{2m-k+1}))}{Re(\zeta(s_{2m-k+1}))^2+Im(\zeta(s_{2m-k+1}))^2};$$
and for imaginary: $$\label{chi_req_im}Im(\chi(s_{k+1}))=\frac{Re(\zeta(s_{k+1}))Im(\zeta(s_{2m-k+1}))+Im(\zeta(s_{k+1}))Re(\zeta(s_{2m-k+1}))}{Re(\zeta(s_{2m-k+1}))^2+Im(\zeta(s_{2m-k+1}))^2};$$
We will consider this to be the exact value of CHI function, because we can calculate it with a given accuracy.
Now write the expression for the real part of the approximate value of the CHI function: $$\label{chi_req_re_app}Re(\tilde\chi(s_{k+1}))=(\frac{2\pi}{t})^{(\sigma-\frac{1}{2})}\cos(t(\log(\frac{2\pi}{t})+1)+\frac{\pi}{4});$$
and for imaginary: $$\label{chi_req_im_app}Im(\tilde\chi(s_{k+1}))=(\frac{2\pi}{t})^{(\sigma-\frac{1}{2})}\sin(t(\log(\frac{2\pi}{t})+1)+\frac{\pi}{4});$$
![The ratio of the exact $|\chi(s)|$ and approximate $|\tilde\chi(s)|$ module of the CHI function[]{data-label="fig:ratio_chi"}](ratio_chi.jpg)
Calculate the modulus for the exact $|\chi(s)|$ value of CHI function: $$\label{chi_abs}|\chi(s)|=\sqrt{Re(\chi(s))^2+Im(\chi(s))^2};$$
and for the approximate $|\tilde\chi(s)|$ value of CHI function:
$$\label{chi_abs_app}|\tilde\chi(s)|=\sqrt{Re(\tilde\chi(s))^2+Im(\tilde\chi(s))^2};$$
and the angle between the exact $\chi(s)$ and approximate $\tilde\chi(s)$ value of CHI function: $$\label{chi_angle}\Delta\varphi_{\chi}=Arg(\chi(s))-Arg(\tilde\chi(s))= \arccos(\frac{Re(\chi(s))}{|\chi(s)|})-\arccos(\frac{Re(\tilde\chi(s))}{|\tilde\chi(s)|});$$
We construct graphs of the ratio of the modulus of the exact $|\chi(s)|$ and approximate $|\tilde\chi(s)|$ values of the CHI function of the real part of a complex number of numbers (fig. \[fig:ratio\_chi\]).
*The graph shows (fig. \[fig:ratio\_chi\]) that the ratio of the modulus of the exact $|\chi(s)|$ and approximate $|\tilde\chi(s)|$ values CHI functions can be taken as 1, therefore, we can say that:* $$\label{tau2}\tau(s)=\Delta\varphi_{\chi};$$
We construct graphs of the dependence of $\Delta\varphi_{\chi}$ on the real part of a complex number (fig. \[fig:angle\_chi\_real\]) and from the imaginary part of a complex number (fig. \[fig:angle\_chi\_complex\]).
![The angle $\Delta\varphi_{\chi}$ between the exact $\chi(s)$ and approximate $\tilde\chi(s)$ value of the CHI function (on the real part)[]{data-label="fig:angle_chi_real"}](angle_chi_real.jpg)
![The angle $\Delta\varphi_{\chi}$ between the exact $\chi(s)$ and approximate $\tilde\chi(s)$ value of the CHI function (on the imaginary part)[]{data-label="fig:angle_chi_complex"}](angle_chi_complex.jpg)
$\tau(s)$ provided (\[tau2\]) is the argument of remainder term of the CHI function, hence it is a combination of the arguments of remainder terms of the product $\Gamma(s)\cos(\pi s/2)$ in (\[chi\_eq\]).
The argument of the remainder term $\mu(s)$ of the gamma function can be obtained from an expression we can find in Titchmarsh [@TI]: $$\label{gamma_app}\log(\Gamma(\sigma+it))=(\sigma+it-\frac{1}{2})\log{it}-it+\frac{1}{2}\log{2\pi}+\mu(s);$$
The most significant researches of remainder term of the gamma function can be found in the paper of Riemann [@SI] and Gabcke [@GA], they independently and in different ways obtain an expression for the argument of the remainder term of the gamma function when $\sigma=1/2$.
We use the expression explicitly written by Gabcke [@GA]: $$\label{mu}\mu(t)=\frac{1}{48t}+\frac{1}{5760t^3}+\frac{1}{80640t^5}+\mathcal{O}(t^{-7});$$
We construct a graph of the dependence of the argument of the remainder term $\mu(t)$ of the gamma function from the imaginary part of a complex number when $\sigma=1/2$ (fig. \[fig:remainder\_gamma\]).
![The remainder term of the gamma function, $\sigma=1/2$[]{data-label="fig:remainder_gamma"}](remainder_gamma.jpg)
We compare the obtained graph with the graph of dependence $\Delta\varphi_{\chi}$ on the imaginary part of a complex number when $\sigma=1/2$ (fig. \[fig:angle\_chi\_complex\_0\_5\]).
![The angle $\Delta\varphi_{\chi}$ between the exact $\tilde(s)$ and approximate $\hat\chi(s)$ value of the CHI function, $\sigma=1/2$[]{data-label="fig:angle_chi_complex_0_5"}](angle_chi_complex_0_5.jpg)
These graphs correspond to each other up to sign and constant value, i.e. the absolute value of the angle between the exact $\chi(s)$ and approximate $\tilde\chi(s)$ value of the CHI function is exactly two times greater than a value of the argument of the remainder term of the gamma function when $\sigma=1/2$.
*But a more significant result is obtained by dividing the angle values $\Delta\varphi_{\chi}$ between the exact $\chi(s)$ and approximate $\tilde\chi(s)$ value of the CHI function by a value of the argument of remainder term $\mu(t)$ of the gamma function when $\sigma=1/2$.*
The result of this operation we get the functional dependence $\lambda(\sigma)$ values of the argument of the remainder term $\tau(s)$ of CHI function (with different values of the real part of a complex number) from a value of the argument of the remainder term $\mu(t)$ of the gamma function when $\sigma=1/2$.
![Factor $\lambda(\sigma)$ (from the real part of a complex number)[]{data-label="fig:lambda_func"}](lambda_func.jpg)
$$\label{tau3}\tau(s)=\Delta\varphi_{\chi}=\lambda(\sigma)\mu(t);$$
The coefficient of $\lambda(\sigma)$ shows which number to multiply a value of the argument of the remainder term $\mu(t)$ of the gamma function in the form of the Riemann-Gabcke to get a value of the argument of the remainder term $\tau(s)$ of CHI function.
We will find an explanation for this paradoxical identity in the further study of expressions that determine a value of the Riemann zeta function.
We obtained an exact expression for the angle $\alpha_1$ of the first middle vector of the Riemann spiral:
$$\label{alpha1_ex}\alpha_1=t(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{4}+\lambda(\sigma)\mu(t);$$
As well as the exact expression for the CHI function: $$\label{chi_eq_ex3}\chi(s)= \Big(\frac{t}{2\pi}\Big)^{(\frac{1}{2}-\sigma)}e^{-i(t(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{4}+\lambda(\sigma)\mu(t))};$$
Now we can perform the exact construction of the inverse Riemann spiral.
Representation of the second approximate equation of the Riemann zeta function by a vector system
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We write the second approximate equation of the Riemann zeta function [@HA1] in vector form. $$\label{zeta_eq_2}\zeta(s)=\sum_{n\le x}{\frac{1}{n^s}}+\chi(s)\sum_{n\le y}{\frac{1}{n^{1-s}}}+\mathcal{O}(x^{-\sigma})+\mathcal{O}(|t|^{1/2-\sigma}y^{\sigma-1});$$ $$\label{zeta_eq_2_cond}0<\sigma <1; 2\pi xy=|t|;$$ $$\label{chi_eq2}\chi(s)=\frac{(2\pi)^s}{2\Gamma(s)\cos(\large\frac{\pi s}{2})}=2^s\pi^{s-1}\sin(\frac{\pi s}{2})\Gamma(1-s);$$
We use the exponential form of a complex number, then , using Euler’s formula, go to the trigonometric form of a complex number and get the coordinates of the vectors. Put $x=y$, then for $m=\Big[\sqrt{\frac{t}{2\pi}}\Big]$ we get: $$\label{zeta_eq_2_vect}\zeta(s)=\sum_{n=1}^{m}{X_n(s)}+\sum_{n=1}^{m}{Y_n(s)}+R(s);$$
where $$\label{x_vect}X_n(s)=\frac{1}{n^{s}}=\frac{1}{n^{\sigma}}e^{-it\log(n)}=\frac{1}{n^{\sigma}}(\cos(t\log(n))-i\sin(t\log(n)));$$ $$\label{y_vect}Y_n(s)=\chi(s)\frac{1}{n^{1-s}}=\chi(s)\frac{1}{n^{1-\sigma}}e^{it\log(n)}
=\chi(s)\frac{1}{n^{1-\sigma}}(\cos(t\log(n))+i\sin(t\log(n));$$
$R(s)$ - some function of the complex variable, which we will estimate later using the exact values of $\zeta(s)$ and $\chi(s)$.
The vector system (\[zeta\_eq\_2\_vect\]) determines a value of $\zeta(s)$ at each interval: $$\label{m_interval}t\in[2\pi m^2,2\pi (m+1)^2);
m=1,2,3...$$
![Finite vector system, $s=0.25+5002.981i$[]{data-label="fig:finite_vector_system"}](finite_vector_system.jpg)
We see that the first sum (\[zeta\_eq\_2\_vect\]) corresponds to the vectors of the Riemann spiral, and the second is the middle vector of the Riemann spiral.
Thus, we can explain the geometric meaning of the second approximate equation of the Riemann zeta function.
![Gap of vector system, $s=0.25+5002.981i$[]{data-label="fig:remainder_gap"}](remainder_gap.jpg)
In the analysis of Riemann spiral we received the quantity of reverse points: $$m=\sqrt{\frac{t}{2\pi}+\frac{1}{4}}$$ of the conditions that between two reverse points is at least one vector of the Riemann spiral.
If we look at the inverse Riemann spiral (fig. \[fig:reverse\_spiral\_approx\]), we note that the number of reverse points, which corresponds to one side of the middle vectors of the Riemann spiral, and on the other hand, the vectors of value of the Riemann spiral, which had not yet twist in a convergent and then a divergent spiral.
*If we remove the vectors that twist into spirals, we get the vector system of (fig. \[fig:finite\_vector\_system\]), which corresponds to the second approximate equation of the Riemann zeta function.*
We can also explain the geometric meaning of the remainder term of the second approximate equation of the Riemann zeta function.
As the scale of the picture of the vector system increases (fig. \[fig:finite\_vector\_system\]), we see a gap between the vectors and the middle vectors of the Riemann spiral (fig. \[fig:remainder\_gap\]), this gap is the remainder term of the second approximate equation of the Riemann zeta function.
The axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function - conformal symmetry
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In the process of geometric derivation of the functional equation of the Riemann zeta function we found that the angles (\[depence\_9\]) between the first middle vector of the Riemann spiral and any other middle vector are equal in modulus and opposite in sign to the angles between the first vector and appropriate vector of the Riemann spiral.
The same result (\[x\_vect\]) and (\[y\_vect\]) we obtained when writing the second approximate equation of the Riemann zeta function in vector form.
We show 1) that the angles between any two middle vectors $Y_i$ and $Y_j$ of the Riemann spiral are equal in modulus and opposite in sign, if we measure the angle from the respective vectors, to the angles between the corresponding vectors $X_i$ and $X_j$ of the Riemann spiral and 2) there is a line that has angles equaled modulus and opposite in sign, if we measure the angle from this line, with any pair of corresponding vectors $Y_i$ and $X_i$ of the Riemann spiral (Lemma 1).
Put in accordance with the first middle vector $Y_1$, the two random middle vectors $Y_i$ and $Y_j$ of the Riemann spiral, the vector $X_1$ and two relevant vectors $X_i$ and $X_j$ of the Riemann spiral the segments $A_1A_2$, $A_2A_3$, $A_3A_4$, $A_1'A_2'$ , $A_2'A_3'$ and $A_3'A_4'$ respectively.
Consider (fig. \[fig:conformal\_symmetry\_lemma\]) two polyline formed by vertices $A_1A_2A_3A_4$ and $A_1'A_2'A_3'A_4 '$ respectively, and oriented arbitrarily. Then edges $A_2A_3$ and $A_3A_4$ have angles with the edge $A_1A_2$ is equal in modulus and opposite in sign to the angles, which have edge $A_2'A_3'$ and $A_3'A_4'$ respectively, with the edge $A_1'A_2'$, if we measure angles from the edges $A_1A_2$ and $A_1'A_2'$ respectively.
1\) We show that the angle between edges $A_2A_3$ and $A_3A_4$ is equal in modulus and opposite in sign to the angle between the edges $A_2'A_3'$ and $A_3'A_4 '$, if we measure the angle from the corresponding edges, for example, from $A_3A_4$ and $A_3'A_4'$.
![Polylines with equal angles[]{data-label="fig:conformal_symmetry_lemma"}](conformal_symmetry_lemma.jpg)
A\) Consider first (fig. \[fig:conformal\_symmetry\_lemma\]) case when edges $A_1A_2$, $A_3A_4$ and edges $A_1'A_2'$, $A_3'A_4'$ respectively are not parallel to each other.
Continue edges $A_1A_2$, $A_3A_4$ and edges $A_1'A_2'$, $A_3'A_4'$ to the intersection, we obtain the triangles $A_2B_2A_3$ and $A_2'B_2'A_3'$ respectively (fig. \[fig:conformal\_symmetry\_lemma\]).
These triangles are congruent by two angles, because the angles $A_1B_2A_4$ and $A_1'B_2'A_4'$ are equal in modulus, as the angle of the edges $A_3A_4$ and $A_3'A_4'$ with edges $A_1A_2$ and $A_1'A_2'$ respectively, and the angles $B_2A_2A_3$ and $B_2'A_2'A_3'$ are adjacent angles $A_1A_2A_3$ and $A_1'A_2'A_3'$ respectively which are also equal in modulus, as the angle of the edges $A_2A_3$ and $A_2'A_3'$ with edges $A_1A_2$ and $A_1'A_2'$ respectively.
Hence, the angles $A_2A_3A_4$ and $A_2'A_3'A_4'$ are equal in modulus as the angles adjacent to the angles $A_2A_3B_2$ and $A_2'A_3'B_2'$ respectively, which are equal as corresponding angles of congruent triangles.
B\) Now consider (fig. \[fig:conformal\_symmetry\_2\_lemma\]) case when edges $A_1A_2$, $A_3A_4$ and edges $A_1'A_2'$, $A_3'A_4'$ respectively parallel to each other.
![Polylines with equal angles, parallel edges[]{data-label="fig:conformal_symmetry_2_lemma"}](conformal_symmetry_2_lemma.jpg)
The continue of edges $A_1A_2$, $A_3A_4$ and edges $A_1'A_2'$, $A_3'A_4'$ respectively do not intersect, because they form parallel lines $A_1B_2$, $A_4B_3$ and $A_1'B_2'$, $A_4'B_3'$ respectively.
Thus edges $A_2A_3$ and $A_2'A_3'$ are intersecting lines of those parallel lines, respectively.
Hence, the angles $A_2A_3A_4$ and $A_2'A_3'A_4'$ are equal in modulus as corresponding angles at intersecting lines of two parallel lines because the angles $A_1A_2A_3$ and $A_1'A_2'A_3'$ are equal in modulus, as the angle of the edges $A_2A_3$ and $A_2'A_3'$ with edges $A_1A_2$ and $A_1'A_2'$ respectively.
If we measure the angle $A_2A_3A_4$ from edge $A_3A_4$, it is necessary to count its anti-clockwise, then it has a positive sign.
If we measure the angle $A_2'A_3'A_4'$ from edge $A_3'A_4'$, it is necessary to count its clockwise, then it has a negative sign.
Therefore, the angle between the edges $A_2A_3$ and $A_3A_4$ equal in modulus and opposite in sign to the angle between the edges $A_2'A_3'$ and $A_3'A_4'$, if we measure the angle from the edges $A_3A_4$ and $A_3'A_4'$ respectively.
2\) Now we continue edges $A_1A_2$ and $A_1'A_2'$ to the intersection and divide the angle $A_2O_1A_2'$ into two equal angles, we will get a line $O_1O_3$, which has equal in modulus and opposite in sign angles, if we measure the angle from this line, to edges $A_1A_2$ and $A_1'A_2'$ (fig. \[fig:conformal\_symmetry\_lemma\]).
We show that line $O_1O_3$ also is equal in modulus and opposite in sign to angles, if we measure the angle from this line, with edges $A_2A_3$, $A_3A_4$ and $A_2'A_3'$, $A_3'A_4'$ respectively.
A\) Consider first (fig. \[fig:conformal\_symmetry\_lemma\]) case when edges $A_2A_3$ and $A_2'A_3'$ are not parallel to each other (sum of angles$A_2O_1O_3$, $A_1A_2A_3$ and $A_2'O_1O_3$, $A_1'A_2'A_3'$ is not equal to $\pi$).
Continue edges $A_2 A_3$, $A_2'A_3'$ to the intersection with the line $O_1 O_3$, we get the triangles $O_1A_2 O_2$ and $O_1A_2'O_2'$ respectively.
These triangles are congruent by two angles, because the angles $A_2O_1O_3$ and $A_2'O_1O_3$ are equal in modulus by build, and the angles $A_1A_2A_3$ and $A_1'A_2'A_3'$ are equal in modulus, as the angle of the edges $A_1A_2$ and $A_1'A_2'$ with edges $A_2A_3$ and $A_2'A_3'$ respectively.
Hence, angles $A_2O_2O_1$ and $A_2'O_2'O_1$ are equal in modulus as the corresponding angles of the congruent triangles.
B\) Now consider (fig. \[fig:conformal\_symmetry\_3\_lemma\]) case when edges $A_2A_3$ and $A_2'A_3'$ are parallel to each other (sum of angles$A_2O_1O_3$, $A_1A_2A_3$ and $A_2'O_1O_3$, $A_1'A_2'A_3'$ is equal to $\pi$).
![Polylines with equal angles, sum of angles $\pi$[]{data-label="fig:conformal_symmetry_3_lemma"}](conformal_symmetry_3_lemma.jpg)
In this case, lines $O_1A_2$ and $O_1A_2'$ are intersecting two parallel lines $O_1O_3$, $A_2O_2$ and $O_1O_3$, $A_2'O_2'$ respectively, since the angles $A_2O_1O_3$ and $A_2'O_1O_3$ are equal in modulus by build, and the angles $A_1A_2A_3$ and $A_1'A_2'A_3'$ are equal in modulus as the angle of the edges $A_1A_2$ and $A_1'A_2'$ with edges $A_2A_3$ and $A_2'A_3'$ respectively, and the sum of the respective angles at intersecting lines is equal to $\pi$.
Therefore, the angles between the line segments $A_2A_3$ and $A_2'A_3'$ and line $O_1O_3$ is equal to zero because $A_2A_3$ and $A_2'A_3'$ are parallel to line $O_1O_3$.
Continue edges $A_3A_4$, $A_3'A_4'$ to the intersection with the straight line $O_1O_3$, get the triangles $O1B_2O_3$ and $O_1B_2'O_3'$ respectively (fig. \[fig:conformal\_symmetry\_lemma\]).
These triangles are congruent by two angles, because the angles $A_2O_1O_3$ and $A_2'O_1O_3$ are equal in modulus by build, and the angles $A_1B_2A_4$ and $A_1'B_2'A_4'$ are equal in modulus, as the angle of the edges $A_1A_2$ and $A_1'A_2'$ with edges $A_3A_4$ and $A_3'A_4'$ respectively.
Hence, angles $B_2O_3O_1$ and $B_2'O_3'O_1$ are equal in modulus as the corresponding angles of the congruent triangles.
If we measure the angles $A_2O_2O_1$ and $B_2O_3O_1$ from a line $O_1O_3$, they need to count anti-clockwise, then either have a positive sign.
If we measure the angles $A_2'O_2'O_1$ and $B_2'O_3'O_1$ from a line $O_1O_3$, they need to count clockwise, then either have a negative sign.
Therefore, line $O_1O_3$ has equal in modulus and opposite in sign angles, if we measure the angle from this line, with edges $A_1A_2$, $A_2A_3$, $A_3A_4$ and $A_1'A_2'$, $A_2'A_3'$, $A_3'A_4'$ respectively. $\square$
According to the Lemma 1, the vector system of the second approximate equation of the Riemann zeta function has *a special kind of symmetry* when there is a line that has angles equal in modulus and opposite in sign, if we measure its from this line, with any pair of corresponding vectors $Y_i$ and $X_i$ of the Riemann spiral.
It should be noted that this symmetry of angles is kept when $\sigma=1/2$ when, as we will show later, the vector system of the second approximate equation of the Riemann zeta function has *mirror symmetry.*
To distinguish these two types of symmetry, we give a name to a special kind of symmetry of the vector system of the second approximate equation of the Riemann zeta function by analogy with the conformal transformation in which the angles are kept.
*Conformal symmetry - a special kind of symmetry in which there is a line that has equal modulus and opposite sign angles, if we measure the angle from this line, with any pair of corresponding segments.*
The angle $\hat\varphi_M$ of the axis of mirror symmetry is equal to the angle $\varphi_M$ of the axis of conformal symmetry $$\label{phi_m}\hat\varphi_M=\varphi_M=\frac{Arg(\chi(s))}{2}+\frac{\pi}{2};$$
In other words, it is the same line if we draw the axis of conformal symmetry at the same distance from the end of the first middle vector $Y_1$ and from the end of the first vector $X_1$ of the Riemann spiral.
Mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function
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In 1932 Siegel published notes of Riemann [@SI] in which Riemann, unlike Hardy and Littlewood, represented the remainder term of the second approximate equation of the Riemann zeta function explicitly: $$\label{zeta_eq_2_zi}\zeta(s)=\sum_{l=1}^{m}{l^{-s}}+\frac{(2\pi)^s}{2\Gamma(s)\cos(\frac{\pi s}{2})}\sum_{l=1}^{m}{l^{s-1}}+(-1)^{m-1}\frac{(2\pi) ^{\frac{s+1}{2}}}{\Gamma(s)}t^{\frac{s-1}{2}}e^{ \frac{\pi is}{2}- \frac{ti}{2}- \frac{\pi i}{8}}\mathcal{S};$$ $$\label{rem_sum}\mathcal{S}=\sum_{0\le 2r\le k\le n-1}{\frac{2^{-k}i^{r-k}k!}{r!(k-2r)!}a_kF^{(k-2r)}(\delta)}+\mathcal{O}\Big(\big(\frac{3n}{t}\big)^{\frac{n}{6}}\Big);$$ $$\label{rem_param}n\le 2\cdot 10^{-8}t;
m=\Big[\sqrt{\frac{t}{2\pi}}\Big]; \\
\delta=\sqrt{t}-(m+\frac{1}{2})\sqrt{2\pi};$$ $$\label{rem_func}F(u) =\frac{\cos{(u^2+\frac{3\pi}{8})}}{\cos{(\sqrt{2\pi}u)}};$$
We use an approximate expression (\[gamma\_app\]) for the gamma function to write the expression for the remainder term of the second approximate equation of the Riemann zeta function in exponential form. $$\label{rem_exp}R(s) = (-1)^{m-1}\Big(\frac{t}{2\pi}\Big)^{-\frac{\sigma}{2}}e^{-i[\frac{t}{2}(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{8}+\mu(s)]}\mathcal{S},$$
Given an estimate for the sum of $\mathcal{S}$, which can be found, for example, Titchmarsh [@TI]: $$\label{sum_app}\mathcal{S}=\frac{\cos{(\delta^2+\frac{3\pi}{8})}}{\cos{(\sqrt{2\pi}\delta)}}+\mathcal{O}(t^{-\frac{1}{2}});$$ we can identify the expression for the argument of the remainder term of the second approximate equation of the Riemann zeta function: $$\label{arg_rem}Arg(R(s))=-(\frac{t}{2}(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{8}+\mu(s));$$
Taking into account the expression (\[mu\]) for the argument of the remainder term of the gamma function when $\sigma=1/2$, we obtain an expression for the argument of the remainder term of the second approximate equation of the Riemann zeta function on the critical line. $$\label{arg_rem_2}Arg(\hat R(s))=-(\frac{t}{2}(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{8}+\mu(t));$$
In the derivation of exact expressions for the CHI function, we found the expression (\[tau3\]), which shows that when $\sigma=1/2$ value of the argument of remainder term $\tau(s)$ of CHI function is exactly twice the argument of the remainder term $\mu(t)$ of gamma functions (\[mu\]) in the form of the Riemann-Gabcke. $$\label{arg_chi}Arg(\hat\chi(s))=-(t(\log{\frac{t}{2\pi}}-1)-\frac{\pi}{4}+2\mu(t));$$
Comparing the expression (\[arg\_rem\_2\]) for the argument, the remainder term of the second approximate equation of the Riemann zeta function of and the expression (\[arg\_chi\]) for the argument CHI-function on the critical line, we find a fundamental property of the vector of the remainder term of the second approximate equation of the Riemann zeta function: $$\label{arg_rem_3}Arg(\hat R(s))=\frac{Agr(\hat\chi(s))}{2};$$
*On the critical line, the argument of the remainder term of the second approximate equation of the Riemann zeta function is exactly half the argument of the CHI function.*
Comparing (\[phi\_m\]) and (\[arg\_rem\_3\]) we see that when $\sigma=1/2$, the vector of the remainder term $\hat R(s)$ is perpendicular to the axis of symmetry of the vector system of the second approximate equation of the zeta function of Riemann: $$\label{arg_rem_4}Arg(\hat R(s))=\varphi_L;$$
As we will show later, this fact is fundamental in the existence of non-trivial zeros of the Riemann zeta function on the critical line.
When changing the imaginary part of a complex number, the vectors of the vector system of the second approximate equation of the Riemann zeta function can occupy an any position in the entire range of angles $[0, 2\pi]$, in consequence of which they form a polyline with self-intersections (fig. \[fig:finite\_vector\_system\]), which complicates the analysis of this vector system.
To obtain the polyline formed by the vectors $X_n$ and the middle vectors of the Riemann spiral $Y_n$, without self-intersections, the vectors can be ordered by a value of the angle, then they will form a polyline, which has no intersections (fig. \[fig:finite\_vector\_system\_perm\]).
![Permutation of vectors of the Riemann spiral vector system, $s=0.25+5002.981i$[]{data-label="fig:finite_vector_system_perm"}](finite_vector_system_perm.jpg)
We determine the properties of the vector system of the second approximate equation of the Riemann zeta function in the permutation of vectors.
As it is known from analytical geometry, the sum of vectors conforms the permutation law, i.e. it does not change when the vectors are permuted.
Thus, the permutation of the vectors of the vector system of the second approximate equation of the Riemann zeta function does not affect a value of the Riemann zeta function.
Since conformal symmetry, by definition, depends only on angles and does not depend on the actual position of the segments, then the permutation of the vectors of the vector system of the second approximate equation of the Riemann zeta function, conformal symmetry is kept because the angles between the lines formed by the vectors and axis of symmetry do not change.
While, for mirror symmetry, the permutation of vectors forms a new pair of vertices and it is necessary to determine that they are on the same line perpendicular to the axis of symmetry and at the same distance from the axis of symmetry.
![Permutation vectors, new pair of vertices[]{data-label="fig:vector_permutation"}](vector_permutation.jpg)
Consider (fig. \[fig:vector\_permutation\]) symmetrical polygon $S_1$ formed by the vertices $A_1A_2A_3$ $A_3'A_2'A_1'$.
The axis of symmetry $O_1O_3$ divides segments $A_1'A_1$, $A_2'A_2$ and $A_3'A_3$ into equal segments, because vertices $A_1'$ and $A_1$, $A_2'$ and $A_2$, $A_3'$ and $A_3$ is mirror symmetrical relative to the axis of symmetry $O_1O_3$.
Change the edges $A_1'A_2'$, $A_2'A_3'$ and $A_1A_2$, $A_2A_3$ of their places, we get two new vertices $\hat A_2'$ and $\hat A_2$.
We connect vertices $\hat A_2'$, $O_3$ and $\hat A_2$, $O_3$, we get the triangles $T_1'$ and $T_1$ are formed respectively the vertices of $\hat A_2'A_3'O_3$ and $\hat A_2A_3O_3$.
We show that the triangles $T_1'$ and $T_1$ are equal.
Parallelograms $A_1'A_2'A_3'\hat A_2'$ and $A_1A_2A_3\hat A_2$ are equal by build, therefore, angles $A_2'A_3'\hat A_2'$ and $A_2A_3\hat A_2$ are equal;
In the source polygon $S_1$, the angles $A_2'A_3'O_3$ and $A_2A_3O_3$ are equal, hence angles $\hat A_2'A_3'O_3$ and $\hat A_2A_3O_3$ are equal as the difference of equal angles;
Then the triangles $T_1'$ and $T_2$ are equal by the equality of the two edges $\hat A_2'A_3'$, $\hat A_2A_3$ and $A_3'O_3$, $A_3O_3$ and the angle between them $\hat A_2'A_3'O_3$ and $\hat A_2A_3O_3$;
Connect vertices $\hat A_2'$ and $\hat A_2$, we get the intersection of $\hat O_2$ segments $\hat A_2'\hat A_2$ and the axis of symmetry $O_1O_3$.
The angles $\hat A_2'O_3A_3'$ and $\hat A_2O_3A_3$ are equal as the corresponding angles of equal triangles $T_1'$ and $T_1$, hence angles $\hat A_2'O_3\hat O_2$ and $\hat A_2O_3\hat O_2$ are equal;
Then the triangle $\hat A_2'O_3\hat A_2$ is isosceles and the segment $\hat O_2O_3$ is its height, because the bisector in an equilateral triangle is its height.
Therefore, the vertices $\hat A_2'$ and $\hat A_2$ lie on the line perpendicular to the axis of symmetry $O_1O_3$ and they have the same distance from the axis of symmetry $O_1O_3$.
Thus, when the corresponding edges of the symmetric polygon are permuted, the mirror symmetry is kept (Lemma 2). $\square$
Now we define the property of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$.
Consider (fig. \[fig:conformal\_vectors\]) polygon, formed by the vertices $A_1A_2A_3A_3'A_2'A_1' $, whose edges $A_1'A_2'$, $A_2'A_3'$ and $A_1A_2$, $A_2A_3$ are equal, have conformal symmetry with relation to the axis of symmetry $O_1O_3$ and vertices $A_1'$ and $A_1$ is mirror symmetrical relative to the axis of symmetry $O_1O_3$.
![Equal vectors possessing conformal symmetry[]{data-label="fig:conformal_vectors"}](conformal_vectors.jpg)
We show that the vertices $A_2'$, $A_3'$ and $A_2$, $A_3$ respectively are mirror symmetric about the axis of symmetry $O_1O_3$ (Lemma 3).
The axis of symmetry $O_1O_3$ divides the segment $A_1'A_1$ into equal segments and segment $A_1'A_1$ is perpendicular to the axis of symmetry $O_1O_3$.
The angles $A_1A_2A_3$ and $A_1'A_2'A_3'$ are equal by Lemma 1, since edges $A_1A_2$, $A_2A_3$ and $A_1'A_2'$, $A_2'A_3'$ respectively have conformal symmetry.
Construct segments $O_1A_2'$, $O_1A_3'$ and $O_1A_2$, $O_1A_3$.
Triangles $O_1A_1'A_2'$ and $O_1A_1A_2$ are equal by the equality of the two sides $O_1A_1'$, $A_1'A_2'$ and $O_1A_1$, $A_1A_2$ respectively and the angle between them $O_1A_1'A_2'$ and $O_1A_1A_2$.
The angles $O_1A_2'A_3'$ and $O_1A_2A_3$ are equal as parts of the equal angles $A_1'A_2'A_3'$ and $A_1A_2A_3$ since angles $A_1'A_2'O_1$ and $A_1A_2O_1$ are equal as the corresponding angles of equal triangles.
Triangles $O_1A_2'A_3'$ and $O_1A_2A_3$ are equal by the equality of the two sides $O_1A_2'$, $A_2'A_3'$ and $O_1A_2$, $A_2A_3$ respectively and the angle between them $O_1A_2'A_3'$ and $O_1A_2A_3$.
The angles $O_3O_1A_2'$ and $O_3O_1A_2$ are equal as parts os the equal angles $A_1'O_1O_3$ and $A_1O_1O_3$ since angles $A_1'O_1A_2'$ and $A_1O_1A_2$ are equal as the corresponding angles of equal triangles.
The angles $O_3O_1A_3'$ and $O_3O_1A_3$ are equal as parts of the equal angles $A_1'O_1O_3$ and $A_1O_1O_3$ since angles $A_1'O_1A_2'$, $A_1O_1A_2$ and $A_2'O_1A_3'$, $A_2O_1A_3$ respectively are equal as the corresponding angles of equal triangles.
Triangles $A_3'O_1A_3$ and $A_2'O_1A_2$ are isosceles and segments $O_1O_2$ and $O_1O_3$ are respectively their height, hence the segments $O_1O_2$ and $O_1O_3$ divide segments $A_2'A_2$ and $A_3 A_3'$ into equal parts and segments $A_2'A_2$ and $A_3 A_3'$ are perpendilular axis of symmetry $O_1O_3$. $\square$
We construct a polyline formed by the vectors of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ (fig. \[fig:mirror\_symmetry\]).
In accordance with (\[x\_vect\]) and (\[y\_vect\]) for $X_n$ and $Y_n$ respectively all segments $A_1'A_2', A_2'A_3', ... \\A_{m-2}'A_{m-1}', A_{m-1}'A_m'$ and $A_1A_2, A_2A_3, ... A_{m-2}A_{m-1}, A_{m-1}A_m$ when $\sigma=1/2$ are equal.
We draw the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function through the middle of the segment formed by the vector $\hat R(s)$ of the remainder term when $\sigma=1/2$.
We obtain two mirror-symmetric vertices $A_m ' $ and $A_m$ besause when $\sigma=1/2$ the vector $\hat R(s)$ of the remainder term is perpendicular to the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function.
![Mirror symmetry of the Riemann spiral vector system, $s=0.5+5002.981i$[]{data-label="fig:mirror_symmetry"}](mirror_symmetry.jpg)
*According to Lemma 3, the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ has mirror symmetry* [^6].
Corollary 1. The vector $A_1'A_1$ corresponds to the vector of a value of the Riemann zeta function when $\sigma=1/2$, therefore, when $\sigma=1/2$, the argument of the Riemann zeta function up to the sign corresponds to the direction of the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function.
Corollary 2. Consider the projection of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ on the axis of symmetry $M$ of this vector system (fig. \[fig:mirror\_symmetry\]).
Segments $A_1'A_1$ and $A_m'A_m$ is perpendicular to the axis of symmetry of $M$, hence, their projection on axis of symmetry $M$ equal to zero.
The projections of the vectors $X_n$ and $Y_n$ on the axis of symmetry $M$ are equal in modulus and opposite sign, hence $$\label{x_n_y_n_m}(\sum_{n=1}^{m}{X_n})_M+(\sum_{n=1}^{m}{Y_n})_M=0;$$
Therefore $$\label{zeta_app_m}(\sum_{n=1}^{m}{X_n})_M+(\sum_{n=1}^{m}{Y_n})_M+\hat R(s)_M=0;$$
and $$\label{zeta_m}\hat \zeta(s)_M=0;$$
Corollary 3. Consider the projection of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ on the normal $L$ to the axis of symmetry $M$ of this vector system (fig. \[fig:mirror\_symmetry\]).
According to the rules of vector summation $$\label{zeta_app_l}\hat \zeta(s)=\hat \zeta(s)_L=(\sum_{n=1}^{m}{X_n})_L+(\sum_{n=1}^{m}{Y_n})_L+\hat R(s)_L;$$
*This expression, as we will show later, called the Riemann-Siegel formula, is used to find the non-trivial zeros of the Riemann zeta function on the critical line.*
It should be noted that, while, vector $X_1$ remains fixed relative to the axes $x=Re(s)$ and $y=Im(s)$, relative to the normal of $L$ to the axis of symmetry and the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function vectors $X_n$ and $Y_n$ are rotated, in accordance with (\[x\_vect\]) and (\[y\_vect\]) towards each other with equal speeds and the angles of the vector of the remainder term $R(s)$ remains fixed when $\sigma=1/2$.
This rotation of the vectors $X_n$ and $Y_n$ leads to the cyclic behavior of the projection $\hat\zeta(s)_L$ of the Riemann zeta function on the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function on *the critical line,* i.e. $\hat\zeta(s)_L=\zeta(1/2+it)_L$ alternately takes the maximum positive and maximum negative value, therefore, $\hat\zeta(s)_L$ cyclically takes a value of zero.
Projections $\zeta(s)_L$ and $\zeta(s)_M$ of the Riemann zeta function respectively on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function in the critical strip when $\sigma\ne 1/2$ have the same cyclic behavior, i.e. $\zeta(s)_L$ and $\zeta(s)_M$ alternately take the maximum positive and maximum negative value, hence $\zeta(s)_L$ and $\zeta(s)_M$ is cyclically takes a value of zero, and, as we will show later, if $\zeta(s)_L$ or $\zeta(s)_M$ are set to zero for any value of $\sigma+it$, then they take a value of zero for value $1-\sigma+it$ also.
Non-trivial zeros of the Riemann zeta function
----------------------------------------------
Using the vector equation of the Riemann zeta function (\[zeta\_eq\_2\_vect\]), we can obtain the vector equation of the non-trivial zeros of the Riemann zeta function. $$\label{zeta_eq_2_zero}\sum_{n=1}^{m}{X_n(s)}+\sum_{n=1}^{m}{Y_n(s)}+R(s)=0;$$
Denote the sums of vectors $X_n$ and $Y_n$: $$\label{l_1_l_2}L_1=\sum_{n=1}^{m}{X_n(s)}; L_2=\sum_{n=1}^{m}{Y_n(s)};$$
The vectors $L_1$ and $L_2$ are invariants of the vector system of the second approximate equation of the Riemann zeta function, since they do not depend on the order of the vectors $X_n$ and $Y_n$, nor on their quantity.
We can now determine the geometric condition of the non-trivial zeros of the Riemann zeta function: $$\label{zeta_zero_vect}L_1+L_2+R=0;$$
This condition means that when the Riemann zeta function takes a value of non-trivial zero when $\sigma=1/2$, the vectors $L_1$, $L_2$ and $R$ form *an isosceles* triangle, because when $\sigma=1/2$ $|L_1|=|L_2|$, and when $\sigma\ne 1/2$, if the Riemann zeta function takes a value of non-trivial zero, these vectors must form a triangle of *general form,* because when $\sigma\ne 1/2$ $|L_1|\ne |L_2|$.
According to Hardy’s theorem [@HA2], the Riemann zeta function has an infinite number of zeros when $\sigma=1/2$.
*This fact is confirmed by the projection values $\hat\zeta(s)_L$ and $\hat\zeta(s)_M$ of the Riemann zeta function respectively on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function on the critical line.*
The Riemann zeta function takes a value of non-trivial zero when $\sigma=1/2$ every time when the projection $\hat\zeta(s)_L$ takes a value of zero, because the projection $\hat\zeta(s)_M$ when $\sigma=1/2$, according to the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function on the critical line are equel to zero identicaly.
*Geometric meaning of non-trivial zeros of the Riemann zeta function means that when the Riemann zeta function takes a value of non-trivial zero when $\sigma=1/2$, the vector system of the second approximate equation of the Riemann zeta function, as a consequence of the mirror symmetry of this vector system on the critical line, in accordance with Lemma 3, forms a symmetric polygon, since the sum of the vectors forming the polygon is equal to zero.*
Using the vector system of the second approximate equation of the Riemann zeta function, we can also explain the geometric meaning of the Riemann-Siegel function [@SI; @GA; @TI], which is used to compute the non-trivial zeros of the Riemann zeta function on the critical line: $$\label{zeta_ri_si}Z(t)=e^{\theta i}\zeta(\frac{1}{2}+it);$$
where $$\label{teta}e^{\theta i}=\Big(\chi(\frac{1}{2}+it)\Big)^{\frac{1}{2}};$$
According to the rules of multiplication of complex numbers, the Riemann-Siegel function determines the projection $\hat\zeta(s)_L$ of the Riemann zeta function on the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function, since $$\label{teta2}\theta=\frac{Arg(\chi(\frac{1}{2}+it))}{2}=\varphi_L;$$
Which corresponds to the results of our research, i.e. $\hat \zeta(s)=\hat \zeta(s)_L$, since when $\sigma=1/2$ in accordance with the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function on the critical line $\hat \zeta(s)_M=0$.
In conclusion of the research of the vector system of the second approximate equation of the Riemann zeta function, we establish another fundamental fact.
The first middle vector $Y_1$ of the Riemann spiral rotates relative to the axes $x=Re(s)$ and $y=Im(s)$ around a fixed first vector $X_1$ of the Riemann spiral, and in accordance with the argument of the CHI function (\[chi\_eq\_app\]) does $N(t)$ complete rotations: $$\label{x_1_num}N(t)=\frac{|Arg(\chi(s))|}{2\pi};$$
Later we show that when $\sigma=1/2$ the first middle vector $Y_1$ of the Riemann spiral passes through the zero of the complex plane average once for each complete rotation [^7] since, in accordance with the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function on the critical line, the end of first middle vector $Y_1$ makes reciprocating motions along the normal $L$ to the axis of symmetry of this vector system, since this normal passes through the end of the first vector $X_1$ of the Riemann spiral, which is at the zero of the complex plane.
*Thus, we can determine the number of non-trivial zeros of the Riemann zeta function on the critical line (as opposed to the Riemann-von Mangoldt formula, which determines the number of non-trivial zeros of the Riemann zeta function in the critical strip) via the number of complete rotations of the first middle vector $Y_1$ of the Riemann spiral:* $$\label{zeta_zero_num}N_0(t)=\Bigg[\frac{|Arg(\chi(s))-\alpha_2|}{2\pi}\Bigg]+2;$$
where $\alpha_2$ argument to the second base point [^8] of the first middle vector $Y_1$ of the Riemann spiral.
Variants of confirmation of the Riemann hypothesis
==================================================
Before proceeding to the variants of confirmation of the Riemann hypothesis based on the analysis of the vector system of the second approximate equation of the Riemann zeta function, we consider two traditional approaches.
The authors of the first approach estimate the proportion of non-trivial zeros $k$ and the proportion of simple zeros $k^*$ on the critical line compared to $N(T)$ the number of non-trivial zeros of the Riemann zeta function in the critical strip. $$\label{k}k=\lim_{T\to\infty}\inf\frac{N_0(T)}{N(T)};$$ $$\label{k_s}k^*=\lim_{T\to\infty}\inf\frac{N_{0s}(T)}{N(T)};$$
In the critical strip $N(T)$, the number of non-trivial zeros of the Riemann zeta function is determined by the Riemann-von Mangoldt formula [@TI]: $$\label{n_mn}N(T)=\frac{T}{2\pi}(\log{\frac{T}{2\pi}}-1)+\frac{7}{8}+S(T)+\mathcal{O}(\frac{1}{T});$$ $$\label{s_mn}S(T)=\frac{1}{\pi}Arg(\zeta(\frac{1}{2}+iT))=\mathcal{O}(\log T), T\to\infty;$$
This expression [@BU] is used to determine the proportion of non-trivial zeros $k$ and the proportion of simple zeros $k^*$ on the critical line: $$\label{k_2}k\ge 1-\frac{1}{R}\log\Big(\frac{1}{T}\int_{1}^{T}|V\psi(\sigma_0+it)|^2dt\Big)+o(1);$$ $$\label{k_s_2}k^*\ge 1-\frac{1}{R}\log\Big(\frac{1}{T}\int_{2}^{T}|V\psi(\sigma_0+it)|^2dt\Big)+o(1);$$
where $V(s)$ is some function whose number of zeros is the same as the number of zeros $\zeta(s)$ in a contour bounded by a rectangle (i. e. not on the critical line): $$\label{rect}\frac{1}{2}<\sigma<1; 0<t<T;$$
$\psi(s)$ some mollifier function that has no zeros and compensates for the change of $|V(s)|$.
R is some positive real number
and $$\label{sigma_0}\sigma_0=\frac{1}{2}-\frac{R}{\log T};$$
In recent papers [@BU; @FE] the function $V(s)$ is used in the form of Levenson [@LE]. $$\label{v}V(s)=Q(-\frac{1}{\log T}\frac{d}{dt})\zeta(s);$$
where $Q(x)$ is a real polynomial with $Q(0)=1$ and $Q'(x)=Q'(1-x)$.
In this approach, the authors use different kinds of polynomials $Q(x)$, mollifier functions $\psi (s)$, as well as different methods of approximation and evaluation of the integral $$\label{int}\int_{1}^{T}|V\psi(\sigma_0+it)|^2dt;$$
In the paper [@BU] in 2011 it is proved that $$\label{k_bu}k\ge .4105; k^*\ge .4058$$
In the parallel paper [@FE] also in 2011 it is proved that $$\label{k_fe}k\ge .4128;$$
To understand the dynamics of the results in this direction, we compare them with the paper [@CO] in which in 1989. $$\label{k_co}k\ge .4088; k^*\ge .4013$$
*The complexity of this approach lies in the fact that different methods of approximation of the integral (\[int\]) allow to obtain an insufficiently accurate result.*
We hope, the authors of [@PR] in 2019 found a way to accurately determine the number of non-trivial zeros in the critical strip and on the critical line, because they claim to have obtained the result $k=1$.
The second traditional direction of confirmation of the Riemann hypothesis is connected with direct verification of non-trivial zeros of the Riemann zeta function.
As we have already mentioned, the Riemann-Siegel formula (\[zeta\_ri\_si\]) is used for this.
One of the recent papers [@GO], which besides computing $10^{13}$ of the first non-trivial zeros of the Riemann zeta function on the critical line offers a statistical analysis of these zeros and an improved approximation method of the Riemann-Siegel formula, was published in 2004.
*The complexity of this approach lies in the fact that it is impossible to calculate all the non-trivial zeros of the Riemann zeta function, therefore, by this method of confirmation the Riemann hypothesis it can be refuted rather than confirm.*
It should be noted that there is *a contradiction* between the results of the first and second method of confirmation of the Riemann hypothesis.
The determination of the proportion of non-trivial zeros on the critical line is in no way related to *the specific interval* of the imaginary part of a complex number, the authors of the first approach do not try to show that there is a sufficiently large interval where the Riemann hypothesis is true in accordance with the second method of confirmation the Riemann Hypothesis.
In other words, by determining the proportion of non-trivial zeros of the Riemann zeta function on the critical line, the authors of the first direction of confirmation of the Riemann hypothesis *indirectly indicate that part of the non-trivial zeros of the Riemann zeta function lies not on the critical line at any interval*, i.e. even where these zeros are already verified by the second method.
The site [@WA] collects some unsuccessful attempts to prove the Riemann hypothesis, some of them are given with a detailed error analysis.
In his presentation, Peter Sarnak [@SA], in the course of analyzing the different approaches to proving the Riemann hypothesis, mentions that about three papers a week are submitted for consideration annually.
However, the situation with the proof of the Riemann hypothesis remains uncertain and we tend to agree with Pete Clark [@KL]:
*So far as I know, there is no approach to the Riemann Hypothesis which has been fleshed out far enough to get an even moderately skeptical expert to back it, with any odds whatsoever. I think this situation should be contrasted with that of Fermat’s Last Theorem \[FLT\]: a lot of number theorists, had they known in say 1990 that Wiles was working on FLT via Taniyama-Shimura, would have found that plausible and encouraging.*
Based on the research results described in the second section of our paper, we propose several new approaches to confirm the Riemann hypothesis based on the properties of the vector system of the second approximate equation of the Riemann zeta function:
1\) the first method is based on determining the exact number (\[zeta\_zero\_num\]) of non-trivial zeros of the Riemann zeta function on the critical line;
2\) the second method of confirmation of the Riemann hypothesis is based on the analysis of projections $\zeta(s)_L$ and $\zeta(s)_M$ of the Riemann zeta function respectively on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function in the critical strip;
3\) the third method is based on the vector condition (\[zeta\_zero\_vect\]) of non-trivial zeros of the Riemann zeta function.
Determining the exact number (\[zeta\_zero\_num\]) of non-trivial zeros of the Riemann zeta function on the critical line is based on several facts:
1\) the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ has mirror symmetry (\[zeta\_m\]);
2\) the vector system of the second approximate equation of the Riemann zeta function rotates around the end of the first vector $X_1$ of the Riemann spiral (\[phi\_m\]) in a fixed coordinate system of the complex plane;
3\) vectors $X_n$ and the middle vectors $Y_n$ of the Riemann spirals rotate in opposite directions (\[x\_vect\]) and (\[y\_vect\]) in the moving coordinate system formed by a normal $L$ to the axis of symmetry and axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function.
The first middle vector $Y_1$ of the Riemann spiral rotates with the vector system of the second approximate equation of the Riemann zeta function around the end of the first vector $X_1$ of the Riemann spiral (\[phi\_m\]) in the fixed coordinate system of the complex plane, so the first middle vector $Y_1$ of the Riemann spiral, in accordance with (\[chi\_eq\_app\]), periodically passes *base point* where it takes a position opposite to the first vector $X_1$ of the Riemann spiral: $$\label{base_point}Arg(\chi(\frac{1}{2}+it_k))=(2k-1)\pi;$$
The argument of *the base points* of the first middle vector $Y_1$ of the Riemann spiral differs from the argument of the Gram points [@GR] by $\pi/2$: $$\label{gram_point}\theta=\frac{Arg(\chi(\frac{1}{2}+it_n))}{2}=(n-1)\pi;$$
*We consider that the complete rotation of the first middle vector $Y_1$ of the Riemann spiral is the rotation from any base point to the next base point.*
![Base point \#4520, the first middle vector is above the real axis, $s=0.5+5001.099505i$[]{data-label="fig:gram_point_4520"}](gram_point_4520.jpg)
In accordance with the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$, at the base points the end of the first middle vector $Y_1$ of the Riemann spiral is on the imaginary axis of the complex plane, so the vector can occupy one of two positions above (fig. \[fig:gram\_point\_4520\]) or bottom (fig. \[fig:gram\_point\_4525\]) of the real axis of the complex plane.
![Base point \#4525, the first middle vector is below the real axis, $s=0.5+5005.8024855i$[]{data-label="fig:gram_point_4525"}](gram_point_4525.jpg)
Since the vectors $X_n$ and the middle vectors $Y_n$ of the Riemann spiral rotate in opposite directions (\[x\_vect\], \[y\_vect\]) in the moving coordinate system formed by the normal $L$ to the axis of symmetry and the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function, in the fixed coordinate system of the complex plane, the first middle vector $Y_1$ of the Riemann spiral, when it is below the real axis of the complex plane, rotates towards the first vector $X_1$ Riemann Spirals and opposite when the first middle vector $Y_1$ of the Riemann spiral is on top of the real axis of the complex plane, it rotates away from the first vector $X_1$ of the Riemann spiral.
If the first middle vector $Y_1$ of the Riemann spiral at the base point rotates towards the first vector $X_1$ of the Riemann spiral, then, in accordance with the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$, until the rotation of the first middle vector $Y_1$ of the Riemann spiral is completed, the axis of symmetry $M$ of this vector system crosses the zero of the complex plane (fig. \[fig:root\_4525\]).
Conversely, if the first middle vector $Y_1$ of the Riemann spiral at the base point rotates away from the first vector $X_1$ of the Riemann spiral, then from the beginning of the rotation of the first middle vector $Y_1$ of the Riemann spiral, the axis of symmetry $M$ of this vector system has already crossed the zero of the complex plane (fig. \[fig:root\_4520\]).
![Non-trivial zero of the Riemann zeta function \#4525, after the base point, $s=0.5+5006.208381106i$[]{data-label="fig:root_4525"}](root_4525.jpg)
![Non-trivial zero of the Riemann zeta function \#4520, up to the base point, $s=0.5+5000.834381i$[]{data-label="fig:root_4520"}](root_4520.jpg)
In the result of the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function, when the axis of symmetry $M$ of this vector system crosses the zero of the complex plane, the end of the first middle vector $Y_1$ of the Riemann spiral also crosses the zero of the complex plane, so the vector system of the second approximate equation of the Riemann zeta function at this point forms a closed polyline.
*It is known from analytical geometry that the sum of vectors forming a closed polyline is equal to zero.*
Therefore, when the first middle vector $Y_1$ of the Riemann spiral is at the base point, we can be sure that either before that point or after that point the Riemann zeta function takes a value of non-trivial zero.
*This fact allows us to conclude that one non-trivial zero of the Riemann zeta function corresponds to one complete rotation of the first middle $Y_1$ vector of the Riemann spiral.*
In accordance with of different combinations of the location of the first middle vector $Y_1$ of the Riemann spiral relative to the real axis of the complex plane at adjacent base points, there may be a different number of non-trivial zeros of the Riemann zeta function in the intervals between the base points:
a\) one non-trivial zero if the first middle vector $Y_1$ of the Riemann spiral occupies the same positions at two serial base points;
b\) two zeros if at the first base point the first middle vector $Y_1$ of the Riemann spiral is below and at the next base point is above the real axis of the complex plane;
c\) no zero, if, on the contrary, at the first base point the first middle vector $Y_1$ of the Riemann spiral is above, and at the next base point, is below the real axis of the complex plane.
Denote the types of base points:
$a_1$ - if the first middle vector $Y_1$ of the Riemann spiral occupies a position above[^9] the real axis of the complex plane;
$a_2$ - if the first middle vector $Y_1$ of the Riemann spiral occupies the position below the real axis of the complex plane.
Then we can define a sequence of base points of the same types, which correspond to the first type of interval, which has one non-trivial zero: $$A_1=a_1a_1;$$ $$A_2=a_2a_2;$$
and sequence base points with different types, which correspond to the second and third interval type, respectively: $$B=a_2a_1;$$ $$C=a_1a_2;$$
It is obvious that the sequence $A_1$ and $A_2$ can’t follow each other, because $$a_1a_1a_2a_2=A_1 CA_2$$
or $$a_2 a_2a_1 a_1=A_2 BA_1$$
It is also clear that each other can not follow the sequence $B$, since $$a_2a_1 a_2 a_1=BC$$
and each other can not follow sequence $C$, since $$a_1a_2a_1a_2=CBC$$
Therefore, if at some interval between two base points of the Zeta-function of Riemann has no non-trivial zeros, i.e. is the interval of type $C$, then at another interval the Zeta-function of Riemann will have two non-trivial zero, it is the interval of type $B$, since these intervals appear every time when the type of base point changed, such as $$a_1a_2a_1=CB$$ or more long chain $$a_1a_2a_2a_2a_1a_1a_1a_1a_1a_2a_2a_1=CA_2A_2BA_1A_1A_1A_1CA_2B$$
*Thus, the total number of non-trivial zeros of the Riemann zeta function at critical line always corresponds to the number of base points or the number of complete rotations of the first middle $Y_1$ vector of the Riemann spiral around the end of the first vector $X_1$ of the Riemann spiral in the fixed coordinate system of the complex plane.*
We used this property to obtain the expression (\[zeta\_zero\_num\]) the number of non-trivial zeros of the Riemann zeta function on the critical line through the angle of the first middle vector $Y_1$ of the Riemann spiral.
We substitute the exact expression (\[arg\_chi\]) argument CHI functions in (\[zeta\_zero\_num\]): $$\label{zeta_zero_num_0}N_0(T)=\Bigg[\Big|\frac{T}{2\pi}(\log{\frac{T}{2\pi}}-1)-\frac{1}{8}+\frac{2\mu(T)-\alpha_2}{2\pi}\Big|\Bigg]+2;$$
where $\mu(T)$ is the remainder term of the gamma function (\[mu\]) when $\sigma=1/2$;
$\alpha_2$ argument of the CHI function at the second base point.
Comparing the expression for the number of non-trivial zeros in the critical strip (\[n\_mn\]) and the expression for the number of non-trivial zeros on the critical line (\[zeta\_zero\_num\_0\]), we see that these values match.
We are in a paradoxical situation where we know the exact number of zeros on the critical line, because $\mu(T)\to 0$ at $T\to \infty$ and do not know the exact number of zeros in the critical strip, because in the most optimistic estimate [@TR]: $$\label{s_tr}|S(T)|<1.998+0.17\log(T); T>e;$$
*Thus, based on the last estimate (\[s\_tr\]) of the remainder term of the Riemann-von Mangoldt formula, we can say that almost all non-trivial zeros of the Riemann zeta function lie on the critical line.*
It should be noted that for the final solution of the problem by this method it is not enough to show that $$\label{s_li}|S(T)|<\mathcal{O}(\frac{\log(T)}{\log\log(T)}); T\to \infty;$$
Although that this result was obtained by Littlewood [@LI] provided that the Riemann hypothesis is true, since the expression $(2\mu(T)-\alpha_2)/2\pi$ has a limit at $T\to \infty$, and the expression $\log(T)/\log\log (T)$ has no such limit, although it grows very slowly.
*In other words, by comparing the number of non-trivial zeros of the Riemann zeta function in the critical strip and on the critical line, it is almost impossible to confirm the Riemann hypothesis.*
Using the method of analyzing the vector system of the second approximate equation of the Riemann zeta function, we can offer a confirmation of the Riemann hypothesis *from the contrary.*
This approach is to show that Riemann zeta functions *cannot have non-trivial zeros* when $\sigma\ne 1/2$.
This problem is solved in different ways by the second and third methods of confirmation of the Riemann hypothesis, based on the properties of the vector system of the second approximate equation of the Riemann zeta function.
We have already considered the dynamics of the first middle vector $Y_1$ of the Riemann spiral at the base points when $\sigma=1/2$, now consider the dynamics of the first middle vector $Y_1$ of the Riemann spiral at the base points when $\sigma\ne 1/2$.
In accordance with (\[alpha1\_ex\]) base point when $\sigma\ne 1/2$ are in the neighborhood $\epsilon=\mathcal{O}(t^{-1})$ points when $\sigma=1/2$, i.e. they have practically the same value at $t\to \infty$.
Therefore, at the base points when $\sigma\ne 1/2$ the first middle vector $Y_1$ of the Riemann spiral occupies a position opposite to the first vector $X_1$ of the Riemann spiral, while, in accordance with the violation of the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function, the end of the first middle vector $Y_1$ of the Riemann spiral *cannot be* on the imaginary axis of the complex plane.
Therefore, when $\sigma<1/2$, the end of the first middle vector $Y_1$ of the Riemann spiral is on the left (fig. \[fig:gram\_point\_4525\_left\]), when $\sigma>1/2$ it is on the right (fig. \[fig:gram\_point\_4525\_right\]) from the imaginary axis of the complex plane, regardless of the position relative to the real axis of the complex plane.
![Base point \#4525, the first middle vector of the Riemann spiral is on the left, $s=0.35+5005.8024855i$[]{data-label="fig:gram_point_4525_left"}](gram_point_4525_left.jpg)
![Base point \#4525, the first middle vector of the Riemann spiral is on the right, $s=0.65+5005.8024855i$[]{data-label="fig:gram_point_4525_right"}](gram_point_4525_right.jpg)
It is easy to notice that at the base point of the vector values of the Riemann zeta function at values $\sigma+it$ and $1 - \sigma+it$ deviate from the normal $L$ to the axis of symmetry of the vector system of the second approximate equation in different directions at *the same angle* (fig. \[fig:gram\_point\_4525\_two\_angles\]).
![The deviation of the vector of values of the Riemann zeta function in the base point \#4525, when $\sigma=0.35$ and $\sigma=0.65$[]{data-label="fig:gram_point_4525_two_angles"}](gram_point_4525_two_angles.jpg)
This behavior of vectors of values of the Riemann zeta function for values of the complex variable symmetric about the critical line can be easily explained by the arithmetic of arguments of a complex numbers at the base point. $$\label{arg1}Arg(\zeta(s))_B=Arg(\chi(s))_B+Arg(\zeta(1-s))_B$$
At the base point $$\label{arg2}Arg(\chi(s))_B=\pi;$$
since $$\label{zeta_conj3}\zeta(1-\sigma+it)=\zeta(\overline{1-\sigma-it})=\zeta(\overline{1-s})=\overline{\zeta(1-s)};$$ $$\label{arg3}Arg(\zeta(1-s))=-Arg(\overline{\zeta(1-s)})=-Arg(\zeta(1-\sigma+it));$$
then at the base point $$\label{arg4}Arg(\zeta(\sigma+it))_B=\pi-Arg(\zeta(1-\sigma+it))_B;$$
This ratio of arguments is kept in the moving coordinate system formed by the normal $L$ to the axis of symmetry and the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function *for any values* of the complex variable symmetric about the critical line. $$\label{arg5}Arg(\zeta(\sigma+it))-\frac{Arg(\chi(\sigma+it))}{2}=-(Arg(\zeta(1-\sigma+it))-\frac{Arg(\chi(\sigma+it))}{2});$$
The vector system of the second approximate equation of the Riemann zeta function rotates relative to the end of the first vector $X_1$ of the Riemann spiral (\[phi\_m\]), while, in accordance with (\[x\_vect\], \[y\_vect\]), each next vector of this vector system rotates relative to the previous vector in the same direction as the entire vector system, therefore, the angle of twist of the vectors *grows monotonically.*
Thus, in accordance with (\[x\_vect\], \[y\_vect\]) when $\sigma<1/2$, the angle of twist of vectors *grows faster* than when $\sigma=1/2$, because the angles between the vectors are equal, but when $\sigma<1/2$, the modulus of each vector is greater than the modulus of the corresponding vector when $\sigma=1/2$, so the vector system when $\sigma<1/2$ is twisted *at a greater angle* than in the case $\sigma=1/2$.
While, when $\sigma>1/2$, the angle of twist of vectors *grows slower* than when $\sigma=1/2$, because the angles between the vectors are equal to, but $\sigma>1/2$, the modulus of each vector is less than the modulus of the corresponding vector when $\sigma=1/2$, so the vector system when $\sigma>1/2$ is twisted *at a smaller angle* than in the case $\sigma=1/2$.
![Base point \#4525, the vector of values of the Riemann zeta function parallel to the axis of symmetry, $s=0.35+5006.186i$[]{data-label="fig:gram_point_4525_symmetry1"}](gram_point_4525_symmetry1.jpg)
In accordance with the identified ratio of the arguments (\[arg5\]) and a monotonic increase of the angle of twist of vectors, we can conclude that the vectors of value of the Riemann zeta function, when values of the complex variable are symmetric about the critical line, rotate in the moving coordinate system formed by a normal of $L$ to the axis of symmetry and axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function in opposite directions with the same speed and therefore at different values of the imaginary part of a complex number has a special positions:
a\) directed in different sites, along the axis of symmetry $M$ (fig. \[fig:gram\_point\_4525\_symmetry1\], \[fig:gram\_point\_4525\_symmetry2\]);
b\) directed in the same site, along the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function (fig. \[fig:gram\_point\_4525\_normal1\], \[fig:gram\_point\_4525\_normal2\]).
![Base point \#4525, the vector of values of the Riemann zeta function parallel to the axis of symmetry, $s=0.65+5006.186i$[]{data-label="fig:gram_point_4525_symmetry2"}](gram_point_4525_symmetry2.jpg)
![Base point \#4525, the vector of values of the Riemann zeta function parallel to the normal to the axis of symmetry, $s=0.35+5006.484i$[]{data-label="fig:gram_point_4525_normal1"}](gram_point_4525_normal1.jpg)
![Base point \#4525, the vector of values of the Riemann zeta function parallel to the normal to the axis of symmetry, $s=0.65+5006.484i$[]{data-label="fig:gram_point_4525_normal2"}](gram_point_4525_normal2.jpg)
While, when $\sigma=1/2$, in accordance with the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$, the vector of value of the Riemann zeta function rotates so that it is always located along the normal $L$ to the axis of symmetry of this coordinate system (fig. \[fig:gram\_point\_4525\]).
According to the position relative to the real axis of the complex plane at the base point when $\sigma\ne 1/2$, the first middle vector $Y_1$ of the Riemann spiral, when it is below the real axis of the complex plane, rotates towards the first vector $X_1$ of the Riemann spiral and, conversely, when the first middle vector $Y_1$ of the Riemann spiral is above the real axis of the complex plane, it rotates away from the first vector $X_1$ of the Riemann spiral.
If the first middle vector $Y_1$ of the Riemann spiral at the base point when $\sigma\ne 1/2$ rotates towards the first vector $X_1$ of the Riemann spiral, then, in accordance with the conformal symmetry of the vector system of the second approximate equation of the Riemann zeta function when $\sigma\ne 1/2$, until the rotation of the first middle vector $Y_1$ of the Riemann spiral is completed, it will take a special position when the axis of symmetry $M$ of this vector system passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral.
Conversely, if the first middle vector $Y_1$ of the Riemann spiral at the base point when $\sigma\ne 1/2$ rotates away from the first vector $X_1$ of the Riemann spiral, from the beginning of the rotation of the first middle vector $Y_1$ of the Riemann spiral, it already occupied a special position when the axis of symmetry $M$ of this vector system is passed through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral.
This special position of the first middle vector $Y_1$ of the Riemann spiral corresponds to the special position of the vector of value of the Riemann zeta function when it locates along to the axis of symmetry $M$ (fig. \[fig:gram\_point\_4525\_symmetry1\], \[fig:gram\_point\_4525\_symmetry2\]) of the vector system of the second approximate equation of the Riemann zeta function.
According to the rules of summation of vectors in this special position of the first middle vector $Y_1$ of the Riemann spiral, the projection of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry of this vector system is equal to zero, hence $$\label{zeta_l_y1}\zeta(s)_L=0;$$
By increasing the imaginary part of a complex number, the first middle vector $Y_1$ of the Riemann spiral will move to another special position when the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral at that moment it rotated by an angle $\pi/2$ from the first special position in the moving coordinate system formed by the normal $L$ to the axis of symmetry and the axis of symmetry $M$ of this vector system.
Second special position of the first middle vector $Y_1$ of the Riemann spiral corresponds to the special position of the vector of value of the Riemann zeta function when it locates along the normal $L$ to the axis of symmetry (fig. \[fig:gram\_point\_4525\_normal1\], \[fig:gram\_point\_4525\_normal2\]) of the vector system of the second approximate equation of the Riemann zeta function.
According to the rules of summation of vectors in this special position of the first middle vector $Y_1$ of the Riemann spiral, the projection of the vector system of the second approximate equation of the Riemann zeta function on the axis of symmetry $M$ of this vector system is equal to zero, hence $$\label{zeta_m_y1}\zeta(s)_M=0;$$
Thus, when we performed an additional analysis of the vector system of the second approximate equation of the Riemann zeta function in accordance with the identified ratio of the arguments (\[arg5\]) and monotonic increase of the angle of twist of vectors we found that each base point corresponds to *two special positions* of the first middle vector $Y_1$ of the Riemann spiral, in which when $\sigma\ne 1/2$ the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system *alternately* take a value of zero and when $\sigma=1/2$ one of which corresponds to the non-trivial zero of the Riemann zeta function.
Now we need to make sure that the modulus of the Riemann zeta function cannot take a value of zero except for the special position of the first middle vector $Y_1$ of the Riemann spiral when $\sigma=1/2$.
We will analyzing the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system starting from the boundary of the critical strip, where the Riemann zeta function has no non-trivial zeros, i.e. for values $\sigma=0$ and $\sigma=1$.
Construct the graphs of the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system when $\zeta(1+it)_L=0$, in accordance with the ratio of the arguments (\[arg5\]) of the vectors of values of the Riemann zeta function, when values of the complex variable symmetric about the critical line, we obtain another equality $\zeta(0+it)_L=0$ (fig. \[fig:gram\_point\_4525\_zeta\_1\_l\]).
![Base point \#4525, projection of the vector system on the normal $L$ to the axis of symmetry when $\zeta(1+5006,09072i)_L=0$[]{data-label="fig:gram_point_4525_zeta_1_l"}](gram_point_4525_zeta_1_l.jpg)
![Base point \#4525, projection of the vector system on the axis of symmetry $M$ when $\zeta(1+5006,09072i)_L=0$[]{data-label="fig:gram_point_4525_zeta_1_m"}](gram_point_4525_zeta_1_m.jpg)
Analysis of the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system shows that in the interval $A_k$ of the imaginary part of a complex number from $t_1$: $\zeta(1+it_1)_L=0$ to $t_2$: $\zeta(1/2+it_2)_L=0$, any value of $t$ corresponds to a value $0<\sigma<1$: $\zeta(\sigma+it)_L=0$.
In other words, in the interval $A_k$ for each value of the imaginary part of a complex number, the graph of the function $\zeta(\sigma+it)_L=0$ crosses the abscissa axis twice for the symmetric values $\sigma+it$ and $1-\sigma+it$ until when $\sigma=1/2$ it reaches a value of the non-trivial zero of the Riemann zeta function.
While the graph of the function $\zeta(\sigma+it) _M=0$ at any value $t$ of the imaginary part of a complex number from the interval $A_k$ crosses the abscissa axis only once, at the point $\sigma=1/2$ (fig. \[fig:gram\_point\_4525\_zeta\_1\_m\]).
Now we construct the graphs of the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system when $\zeta(1+it)_M=0$, in accordance with the ratio of the arguments (\[arg5\]) of the vectors of value of the Riemann zeta function, when values of the complex variable symmetric about the critical line, we obtain another equality $\zeta(0+it)_M=0$ (fig. \[fig:gram\_point\_4525\_zeta\_2\_m\]).
![Base point \#4525, projection of the vector system on the normal $L$ to the axis of symmetry when $\zeta(1+5006,4559i)_M=0$[]{data-label="fig:gram_point_4525_zeta_2_l"}](gram_point_4525_zeta_2_l.jpg)
![Base point \#4525, projection of the vector system on the axis of symmetry $M$ when $\zeta(1+5006,4559i)_M=0$[]{data-label="fig:gram_point_4525_zeta_2_m"}](gram_point_4525_zeta_2_m.jpg)
Analysis of the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system shows that in the interval $C_k$ of the imaginary part of a complex number from $t'_1$: $\zeta(1+it'_1)_M=0$ to $t'_2$: $\zeta(1/2+it'_2)_M=0$, any value of $t'$ corresponds to a value $0<\sigma<1$: $\zeta(\sigma+it')_M=0$.
In other words, in the interval $C_k$ for each value of the imaginary part of a complex number, the graph of the function $\zeta(\sigma+it')_M=0$ crosses the abscissa axis three times for the symmetric values $\sigma+it'$, $1-\sigma+it'$ and at the point $\sigma=1/2$.
While the graph of the function $\zeta(\sigma+it') _L=0$ at any value $t'$ of the imaginary part of a complex number from the interval $C_k$ never crosses the abscissa axis (fig. \[fig:gram\_point\_4525\_zeta\_2\_l\]).
Therefore, in the interval $C_k$ for any value of the imaginary part of a complex number and any value of the real part of a complex number $0<\sigma<1$, the Riemann zeta function has no non-trivial zeros.
![Base point \#4525, projection of the vector system on the normal $L$ to the axis of symmetry when $\zeta(1/2+5006,208381106i)_L=0$[]{data-label="fig:gram_point_4525_zeta_3_l"}](gram_point_4525_zeta_3_l.jpg)
![Base point \#4525, projection of the vector system on the axis of symmetry $M$ when $\zeta(1/2+5006,208381106i)_L=0$[]{data-label="fig:gram_point_4525_zeta_3_m"}](gram_point_4525_zeta_3_m.jpg)
Analysis of the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system in the interval of $B_k$ between intervals $A_k$ and $C_k$ and in the interval $D_k$ between intervals $C_k$ and $A_{k+1}$ shows that the projection at any value of the imaginary part of a complex number and any value of the real part of a complex number when $0<\sigma<1$ is not equal to zero, therefore, in the interval of $B_k$ and $D_k$ the Riemann zeta function has no non-trivial zeros.
It should be noted that the sign of the projection of the vector system on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function changes at each interval $C_k$, i.e. it depends on the number of the base point (fig. \[fig:graphics\_projections\]).
While the sign of the projection of the vector system on the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function depends on the location of the first middle vector $Y_1$ of the Riemann spiral at the base point (fig. \[fig:graphics\_projections\]), i.e. it changes at each interval $A_k$.
Thus, when we performed an additional analysis of the vector system of the second approximate equation of the Riemann zeta function we found that each base point corresponds to *four intervals* in which the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system take certain values and only in one interval when $\sigma=1/2$ they can be zero at the same time (fig. \[fig:gram\_point\_4525\_zeta\_3\_l\] and \[fig:gram\_point\_4525\_zeta\_3\_m\]), in this moment the Riemann zeta function takes a value of non-trivial zero.
Now that we know all possible variants of the ratio of the projection of vectors system of the second approximate equation of the Riemann zeta function on normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system we found out *what and why* can be these projections for any value of the real part of a complex number in the critical strip, and on the boundary of the critical strip, i.e. for values $\sigma=0$ and $\sigma=1$, where the Riemann zeta function has no non-trivial zeros, as it proved Adamar and vallée Poussin.
We have found out how these projections change when the imaginary part of the complex number changes and how these changes are related to the number of the base point and the position of the first middle vector $Y_1$ of the Riemann spiral at the base point, we have to answer two questions:
1\) Why should these projections have such ratios for any base point?
2\) Why can’t there be any other reason for the modulus of the Riemann zeta function to go to zero when $\sigma\ne 1/2$?
The first question has already been answered in the analysis of the projections of the vector system on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function:
1\) The vector system of the second approximate equation of the Riemann zeta function rotates when the imaginary part of the complex number changes, which we can confirm this by the equation of the axis of symmetry (\[phi\_m\]) of this vector system;
2\) The vector system periodically passes the base points, where it is convenient to fix its special properties;
3\) The special properties of the vector system are determined by the position of the first middle vector $Y_1$ of the Riemann spiral at the base point relative to the real axis of the complex plane and the axes of the moving coordinate system formed by the normal $L$ to the axis of symmetry and the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function;
4\) The arguments of $\zeta(s)$ and $\overline{\zeta(1-s)}$ have axial symmetry around the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function;
5\) The vectors of value of $\zeta(s)$ and $\overline{\zeta(1-s)}$ rotate in different directions at the same speed in the moving coordinate system formed by the normal $L$ to the axis of symmetry and the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function;
6\) Projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system are determined by the vectors of value of $\zeta(s)$ and $\overline{\zeta(1-s)}$.
*Thus, the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this vector system have periodic properties (which we described earlier) with relation to the base points, so these properties of the projections of the vector system must be observed at any base point, and therefore for all values of the complex variable, where this vector system determines a value of the Riemann zeta function, i.e., on the entire complex plane except the real axis, where, as is known, the Riemann zeta function has only trivial zeros.*
To answer the second question, why there can be no other reason for the modulus of the Riemann zeta function to zero when $\sigma\ne 1/2$, it is necessary to consider possible variants of such a zero transformation, for example:
1\) The interval $A_k$, where $\zeta(s)_L=0$, intersects with the interval $C_k$, where $\zeta(s)_M=0$, for any value of a complex variable when $\sigma\ne 1/2$;
2\) The condition $\zeta(s)_L=0$ and $\zeta(s)_M=0$ is satisfied in the interval $B_k$ or $D_k$ , where $\zeta(s)_L\ne 0$ and $\zeta(s)_M\ne 0$, for any value of the complex variable in the critical strip.
*It is obvious that the module $\zeta(s)$ can not arbitrarily be reduced to zero because this would require that the zero was reduced module of all vectors $X_n$ and $Y_n$ of the Riemann spiral, which contradicts (\[x\_vect\], \[y\_vect\]).*
Somebody can think of other reasons why the modulus of the Riemann zeta function can transform to zero when $\sigma\ne 1/2$, we hope that upon careful examination they will all be refuted, because the identified properties of the projections of the vector system of the second approximate equation of the Riemann zeta function on the normal $L$ to the axis of symmetry and on the axis of symmetry $M$ of this system show the direction of confirmation why the Riemann zeta function may not have non-trivial zeros when $\sigma\ne 1/2$.
Now we consider another method of confirmation the Riemann hypothesis, based on the properties of the vector system of the second approximate equation of the Riemann zeta function.
We need to find out in which cases the invariants $L_1$ and $L_2$ (\[l\_1\_l\_2\]) of this vector system and the vector of the remainder term $R$ of the second approximate equation of the Riemann zeta function can form a triangle (\[zeta\_zero\_vect\]).
Consider the invariants $L_1$ and $L_2$ at the special points of the first middle vector $Y_1$ of the Riemann spiral, when the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral (fig. \[fig:gram\_point\_4525\_symmetry\_inv\]), and when the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral (fig. \[fig:gram\_point\_4525\_normal\_inv\]).
![Base point \#4525, the vector of values of the Riemann zeta function parallel to the axis of symmetry, $s=0.35+5006.186i$ and $s=0.65+5006.186i$[]{data-label="fig:gram_point_4525_symmetry_inv"}](gram_point_4525_symmetry_inv.jpg)
![Base point \#4525, the vector of values of the Riemann zeta function parallel to the normal to the axis of symmetry, $s=0.35+5006.484i$ and $s=0.65+5006.484i$[]{data-label="fig:gram_point_4525_normal_inv"}](gram_point_4525_normal_inv.jpg)
In all other cases, in accordance with the continuity of values of the Riemann zeta function, the invariants $L_1$ and $L_2$ will occupy different intermediate positions (fig. \[fig:gram\_point\_4525\_intermediate\_inv\]).
![Base point \#4525, intermediate position of the vector of values of the Riemann zeta function, $s=0.35+5006.186i$ and $s=0.65+5006.186i$[]{data-label="fig:gram_point_4525_intermediate_inv"}](gram_point_4525_intermediate_inv.jpg)
When the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral (fig. \[fig:gram\_point\_4520\_normal\_inv\_opposite\]) invariants $L_1$ and $L_2$ can occupy a position close to the trapezoid, but in any case, obviously, can not form a triangle.
![Base point \#4520, the vector of values of the Riemann zeta function parallel to the normal to the axis of symmetry, $s=0.35+5001.415i$ and $s=0.65+5001.415i$[]{data-label="fig:gram_point_4520_normal_inv_opposite"}](gram_point_4520_normal_inv_opposite.jpg)
*It is obvious that the invariants $L_1$ and $L_2$ can occupy the position closest to the triangle only when the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral (fig. \[fig:gram\_point\_4525\_symmetry\_inv\]), so we will perform further analysis of the invariants $L_1$ and $L_2$ in this special position of the first middle vector $Y_1$ of the Riemann spiral.*
It should be noted that in this position of the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ in accordance with the mirror symmetry, this vector system forms a closed polyline, and therefore the Riemann zeta function takes a value of non-trivial zero (fig. \[fig:gram\_point\_4525\_symmetry\_inv\_zero\]).
![Base point \#4525, non-trivial zero of the Riemann zeta function, $s=0.5+5006.208381106i$[]{data-label="fig:gram_point_4525_symmetry_inv_zero"}](gram_point_4525_symmetry_inv_zero.jpg)
In the ordinate of the non-trivial zero of the Riemann zeta function, the projection modulus of the invariant $L_1$ on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function increases when $\sigma$ decreases and, conversely, it decreases when $\sigma$ increases.
The projection of the invariant $L_2$ on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function changes according to the sign of the projection of its gradient on the axis of symmetry $M$: $$\label{grad_l_2}grad_M L_2=\sum_{n=1}^{m}\Big(\frac{\partial Y_n}{\partial\sigma}\Big)_M;$$
If the sign of the projection of the invariant $L_2$ on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function is equal to the sign of the projection of its gradient on the axis of symmetry $M$, then the projection of the invariant $L_2$ on the axis of symmetry $M$ increases when $\sigma$ increases (fig. \[fig:gram\_point\_4525\_symmetry\_3\_inv\]) respectively while $\sigma$ decreases it decreases also (fig. \[fig:gram\_point\_4525\_symmetry\_2\_inv\]) and, conversely, if the sign of the projection of the invariant $L_2$ on the axis of symmetry $M$ is not equal to the sign of the projection of its gradient on the axis of symmetry $M$, then when $\sigma$ increases the projection of the invariant $L_2$ on the axis of symmetry $M$ decreases (fig. \[fig:gram\_point\_4525\_symmetry\_5\_inv\]) respectively while $\sigma$ decreases it increases (fig. \[fig:gram\_point\_4525\_symmetry\_4\_inv\]).
![Base point \#4525, the ordinate of the non-trivial zero of the Riemann zeta function, positive gradient, $s=0.65+5006.208381106i$[]{data-label="fig:gram_point_4525_symmetry_3_inv"}](gram_point_4525_symmetry_3_inv.jpg)
![Base point \#4525, the ordinate of the non-trivial zero of the Riemann zeta function, positive gradient, $s=0.35+5006.208381106i$[]{data-label="fig:gram_point_4525_symmetry_2_inv"}](gram_point_4525_symmetry_2_inv.jpg)
![Base point \#4521, the ordinate of the non-trivial zero of the Riemann zeta function, negative gradient, $s=0.65+5001.889773627i$[]{data-label="fig:gram_point_4525_symmetry_5_inv"}](gram_point_4525_symmetry_5_inv.jpg)
![Base point \#4521, the ordinate of the non-trivial zero of the Riemann zeta function, negative gradient, $s=0.35+5001.889773627i$[]{data-label="fig:gram_point_4525_symmetry_4_inv"}](gram_point_4525_symmetry_4_inv.jpg)
The sign of the projection of the gradient of the invariant $L_2$ on the axis of symmetry $M$ depends on the distribution of angles and modulus of the middle vectors, which becomes obvious if we arrange the middle vectors in increasing order of their angles (fig. \[fig:gram\_point\_4525\_symmetry\_inv\_zero\]).
Researches show that when $\sigma\ne 1/2$ at the point when $\zeta(s)_M=0$ the sign of the projection of the gradient of the invariant $L_2$ on the axis of symmetry $M$ is kept.
Now, when we consider the invariants $L_1$ and $L_2$ at the point where the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral (fig. \[fig:gram\_point\_4525\_symmetry\_inv\]), it is sufficient to consider the sum of projections of vectors $L_1$, $L_2$ and R on the axis of symmetry $M$, because at this point the sum of projections of vectors $L_1$, $L_2$ and R on the normal $L$ to the axis of symmetry is equal to zero.
Consider separately the sum of projections of invariants $L_1$ and $L_2$ on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function: $$\label{delta_l}\Delta L=\sum_{n=1}^{m}{(X_n(s))_M}+\sum_{n=1}^{m}{(Y_n(s))_M};$$
and the projection of vector R of the remainder term of the second approximate equation of the Riemann zeta function: $$\label{delta_r}\Delta R=R\sin(\Delta\varphi_R);$$
where $\Delta\varphi_R$ is the deviation of the vector of the remainder term R of the second approximate equation of the Riemann zeta function from the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function.
Then the vector condition (\[zeta\_zero\_vect\]) of the non-trivial zero of the Riemann zeta function can be rewritten as follows: $$\label{zeta_zero_module}|\Delta L|=|\Delta R|;$$
We already know that in accordance with the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$ at the point when the axis of symmetry $M$ of this vector system passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral (fig. \[fig:gram\_point\_4525\_symmetry\_inv\_zero\]), the Riemann zeta function takes a value of non-trivial zero.
Thus $\Delta L=0$ and $\Delta R=0$.
When $\sigma\ne 1/2$, the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function is broken, in other words, $\Delta L\ne 0$ and $\Delta R\ne 0$, hence the change in $\Delta L$, if the Riemann zeta function can take a value of non-trivial zero, must be compensated by the change in $\Delta R$.
Consider the dependence of the sum of projections of the invariants $L_1$ and $L_2$ (fig. \[fig:projections\_sum\_l1\_l2\]) and the projection of the vector $R$ of the remainder term (fig. \[fig:projection\_r\]) on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function from the real part of a complex number at the point when the axis of symmetry $M$ of this vector system passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral.
![Base point \#4525, sum of projections of invariants $L_1$ and $L_2$, ordinate $\zeta(0.35+5006.186i)_L=0$[]{data-label="fig:projections_sum_l1_l2"}](projections_sum_l1_l2.jpg)
![Base point \#4525, projection of the remainder term $R$, ordinate $\zeta(0.35+5006.186i)_L=0$[]{data-label="fig:projection_r"}](projection_r.jpg)
It is obvious that these functions are equal only at the point $\sigma=1/2$, i.e. on the critical line.
Consider the *boundary function* that separates values $\Delta L$ and $\Delta R$ (fig. \[fig:boundary\_function\]), at the point where the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral: $$\label{boundary_function}F(s)=A \Big(\frac{\sum_{n=1}^{m}{|Y_n(s)|}}{\sum_{n=1}^{m}{|X_n(s)|}}-1\Big);$$
where $A$ is some constant greater than zero.
![Base point \#4525, boundary function $F(s)$, ordinate $\zeta(0.35+5006.186i)_L=0$, $A=0.5$[]{data-label="fig:boundary_function"}](boundary_function.jpg)
![Graphics of projections of the Riemann zeta function[]{data-label="fig:graphics_projections"}](graphics_projections.jpg)
It is obvious that the sum of projections of invariants $L_1$ and $L_2$ (fig. \[fig:projections\_sum\_l1\_l2\]) on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=0$ is determined by a value of the projection $\zeta(0+it)_M$ (fig. \[fig:graphics\_projections\]), because in this case, $|\Delta L|\gg|\Delta R|$ (fig. \[fig:projection\_r\]), therefore, the sum of projections of invariants $L_1$ and $L_2$ on the axis of symmetry $M$ at all points except $\sigma=1/2$ modulo more than values of the boundary function, while values the projection of the vector $R$ of the remainder term on the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function in all points except $\sigma=1/2$ modulo smaller than values of the boundary functions.
Now we need to find out the dependence of the angle of deviation $\Delta\varphi_R$ of vector remainder member of the second approximate equation of the Riemann zeta function from normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function, because a value of this angle determines a value of $\Delta R$.
Although that Riemann recorded the remainder term (\[zeta\_eq\_2\_zi\]) of the second approximate equation of the Riemann zeta function explicitly, Riemann and other authors use the argument of the remainder term only for the particular case of (\[arg\_rem\_2\]) when $\sigma=1/2$, so we use the explicit expression for the CHI functions (\[chi\_eq\_ex3\]) and will calculate the argument of the remainder term using the exact values of the Riemann zeta function [@ZF]: $$\label{delta_varphi_r}\Delta\varphi_R=\frac{1}{2}Arg(\chi(s))-Arg(R(s))=\frac{1}{2}\arccos(\frac{Re(\chi(s))}{|\chi(s)|})-\arccos(\frac{Re(R(s))}{|R(s)|});$$
where $$\label{r_vect}R(s)=\zeta(s)-\sum_{n=1}^{m}{X_n(s)}-\sum_{n=1}^{m}{Y_n(s)};$$ $$\label{r_module}|R(s)|=\sqrt{Re(R(s))^2+Im(R(s))^2};$$
The calculations show a linear relationship (fig. \[fig:delta\_varphi\_r\_real\]) of the angle of deviation $\Delta\varphi_R$ of the vector of the remainder term of the second approximate equation of the Riemann zeta function from normal $L$ to the axis of symmetry of this vector system from the real part of a complex number.
![Base point \#4525, the deviation angle of vector $\Delta\varphi_R$ of remainder term, rad, ordinate $\zeta(0.35+5006.186i)_L=0$[]{data-label="fig:delta_varphi_r_real"}](delta_varphi_r_real.jpg)
![The deviation angle of vector $\Delta\varphi_R$ of remainder term, rad (from the imaginary part of a complex number)[]{data-label="fig:delta_varphi_r_complex"}](delta_varphi_r_complex.jpg)
![Fractional part of $\sqrt{t/2\pi}$ (from imaginary part of complex number)[]{data-label="fig:frac_part_m_complex"}](frac_part_m_complex.jpg)
![The maximum derivation angle of the vector $\Delta\varphi_R$ of the remainder term, rad (from the imaginary part of a complex number)[]{data-label="fig:delta_varphi_r_max_complex"}](delta_varphi_r_max_complex.jpg)
*This dependence is confirmed for any values of the imaginary part of a complex number, since it is determined by the form of the polyline formed by the vector system of the second approximate equation of the Riemann zeta function, for symmetric values $\sigma+it$ (fig. \[fig:gram\_point\_4525\_left\]) and $1-\sigma+it$ (fig. \[fig:gram\_point\_4525\_right\]).*
It can be shown that these polylines are mirror congruent, hence all the angles of these polylines are mirror congruent, including the arguments of the remainder terms.
The angle of deviation $\Delta\varphi_R$ of the vector of the remainder term of the second approximate equation of the Riemann zeta function from normal $L$ to the axis of symmetry of the vector system of the second approximate equation dzeta-functions of Riemann has a periodic dependence (fig. \[fig:delta\_varphi\_r\_complex\]) from the imaginary part of a complex number, with a period equal to an interval (\[m\_interval\]), where fractional part of the expression $\sqrt{t/2\pi}$, which varies from 0 to 1 (fig. \[fig:frac\_part\_m\_complex\]), has the periodic dependence also.
The angle of deviation $\Delta\varphi_R$ of the vector of the remainder term of the second approximate equation of the Riemann zeta function from normal $L$ to the axis of symmetry of this vector system has the maximum value at the boundaries of the intervals (\[m\_interval\]), and the absolute value of the maximum variance asymptotically decreases (fig. \[fig:delta\_varphi\_r\_max\_complex\]) with the growth of the imaginary part of a complex number, which corresponds to the evaluation of the remainder term of the gamma function: $$\label{mu_limit}\mu(s)\to 0, t\to\infty;$$
*Thus, at the point where the axis of symmetry $M$ of the vector system of the second approximate equation of the Riemann zeta function passes through both the zero of the complex plane and the end of the first middle vector $Y_1$ of the Riemann spiral, the condition (\[zeta\_zero\_module\]) can be true only when $\sigma=1/2$, since $\sigma\ne 1/2$ $|\Delta L|>|\Delta R|$ on any interval (\[m\_interval\]).*
Somebody can think of different options, why expression (\[boundary\_function\]) cannot be *bounding function*, but we hope that they will all be refuted.
Summary
=======
Was Riemann going to speak at the Berlin Academy of Sciences on the occasion of his election as a corresponding member to present a proof of the asymptotic law of the distribution of prime numbers?
Only Riemann himself could have answered this question, but we are inclined to assume that he did not intend to.
Neither before nor after (Riemann died seven years later) he did not return to the subject publicly.
Riemann did not publish any paper in progress, so the paper on the analysis of the formula, which is now called the Riemann-Siegel formula was published by Siegel based on Riemann’s notes, but this paper is rather a development of the analytical analysis of the Riemann zeta function, rather than a proof of the asymptotic distribution law of prime numbers.
The speech was rather about the analytical function of the complex variable.
Riemann showed how bypassing the difficult arguments about the convergence of the series, which defines the function, we can get its analytical continuation using the residue theorem.
Riemann also used a feature of a function of a complex variable for which zeros are its singular points, which carry basic information about the function.
It was in connection with this feature of the complex variable function that the Riemann hypothesis appeared - the hypothesis of the distribution of zeros of the Riemann zeta function.
Strange is also the absence of any mention of the representation of complex numbers by points on the plane, although Riemann in his report uses the rotation of the zeta function by the angle $Arg(\chi(s))/2$, which is certainly an operation on complex numbers as points on the plane.
Such relation to complex numbers seems even more strange in light of the fact that Riemann was a student of Gauss, who was one of the first to introduce the representation of complex numbers by points on the plane.
*In other words, we tend to assume that the zeta function was chosen to show how we can solve the problem by methods of functions of a complex variable, Riemann is not important the problem itself, it is important approach and methods that gives the theory of functions of a complex variable as the apotheosis of the theory of analytic functions.*
Although that a theorem of the distribution of prime numbers is proved analytically, i.e. using the analytical Riemann zeta function, later it was found a proof that uses the functions of a real variable, i.e. an elementary proof.
Thus, the role of the Riemann zeta function has shifted towards regularization, the so-called methods of generalized summation of divergent series.
The Riemann hypothesis seems to belong to such problems, since the generalized Riemann hypothesis deals with an entire class of Dirichlet L-functions.
Zeta function regularization is also used in physics, particularly in quantum field theory.
We prefer to conclude that the main role of the Riemann zeta function is in the understanding of generalized methods for summing asymptotic divergent series and as a consequence of constructing an analytic continuation of the functions of a complex variable.
Based on that on numerous forums seriously is discussed infinite sum of natural numbers: $$\label{123}1+2+3+ = -\frac{1}{12};$$
only a few understand the essence of the generalized summation or regularization of divergent series.
Here mathematics encounters philosophy, namely with *the law of unity of opposites.*
The essence of generalized summation is *regularization* (that is why the second name of the method is regularization) - it means that a divergent series cannot exist without its *opposite* - convergent series.
In other words, there is only *one series*, but it converges in one region and diverges in another.
The most natural such series is the Dirichlet series: $$\label{dirichlet}\sum_{n=1}^{\infty}\frac{1}{n^s};$$
which in the real form was researched by Euler (he first raised the question of the need for the concept of generalized summation), and in the complex form it was considered by Riemann.
Riemaann left the question of generalized summation, skillfully replacing the divergent Dirichlet series by already regularized integral of gamma functions of a complex variable: $$\label{gamma_int}\Gamma (z)={\frac {1}{e^{i2\pi z}-1}}\int \limits _{L}\!t^{\,z-1}e^{-t}\,dt;$$
The essence of unity summation of an infinite series is in the definition of sum this series in the region where this series converges and in the region where *the same* series diverges.
The misconception begins in the definition of *infinite sum.* In the case of a convergent series, it only seems to us that we can find the sum of this series. In fact, we find *limit of partial sums* because in accordance with the convergence of the series, such a limit exists (by definition).
Euler first formulated the need for a different concept of the word sum in the application to the divergent series (or rather to the series in the region where it diverges), he explained this by the practical need to attach some value divergent series.
We now know that this value is found as *limit of generalized partial sums* of an infinite series.
And the main condition that is imposed on the method of obtaining generalized partial sums (except that there must be a limit) is *regularity*, i.e. the limit of generalized partial sums in the region where the series converges must be equal to the limit of partial sums of this series.
This understanding, as Hardy observed, came only with the development of the theory of the function of a complex variable, namely the notion of *analytic continuation*, which is closely linked to the infinite series that defines the analytic function, and is also closely linked to the fact that this infinite series converges in one region and diverges in another.
An analytic continuation of a function of a complex variable, if it is possible, is unique and this fact (which has a regorous proof) is possible only if there exists a limit of generalized partial sums of an infinite series by which the analytic function is defined, in the region where this series diverges.
In the theory of generalized summation of divergent series, it is also rigorously proved that if the limit of generalized partial sums exists for two different regularization methods (obtaining generalized partial sums), then it has the same value.
*It is this correspondence of different methods of obtaining generalized partial sums and analytic continuation of the function of a complex variable that Hardy had in mind.*
The Dirichlet series $\sum\limits_{n=1}^{\infty}\frac{1}{n^s}$ in complex form defines the Riemann zeta function.
As is known, this series diverges in the critical strip, where the Riemann zeta function has non-trivial zeros.
Hence all non-trivial zeros of the Riemann zeta function are *limit* of generalized partial sums of the Dirichlet series, while the trivial zeros of the Riemann zeta function define odd Bernoulli numbers, which are all zero.
In the Riemann zeta function theory to regularize the Dirichlet series the Euler-MacLaren formula [^10] is traditionally used, as we mentioned earlier, this formula is used if the partial sums of a divergent series are suitable for calculating the generalized sum of that divergent series.
The Euler-MacLaren generalized summation formula allows us to move from an infinite sum to an improper integral, i.e. to the limit of generalized partial sums of the Dirichlet series.
The geometric analysis of partial sums of the Dirichlet series, which defines the Riemann zeta function, allowed us to make an important conclusion that a generalized summation of infinite series is possible if this series diverges asymptotically.
And then we get into the essence of the definition of analytic functions through *asymptotic* infinite series, and a value of the function equal to a value of the asymptote in any case or when the series asymptotically converge or when a series diverges asymptotically, while in order to find the limit of this asymptote when the series converges, use partial sums of this series and when the series diverges, to find the limit of the asymptote, we must use the regular generalized partial sum.
And then we can obtain an analytic continuation of the function given by the *asymptotic infinite series.*
Therefore, the result obtained at the very beginning of the research, namely the use of an alternative method of generalized summation of Cesaro, which was obtained geometrically, is as important as the description of the various options to confirm the Riemann hypothesis.
Conclusion
==========
We believe that the method of geometric analysis of Dirichlet series, based on the representation of complex numbers by points on the plane, will complement the set of tools of analytical number theory.
The method described in this paper allows us to get into the essence of the function of a complex variable, to identify regularities that explain any value of this function (including in the region where the series that difine this function diverges) and most importantly it gives *an idea of the exact value of zero* as the sum of vectors that form a closed polyline.
After going through the analysis of the vector system of the second approximate equation of the Riemann zeta function, we can formulate the results of the second method of confirmation the Riemann hypothesis without using this vector system, since we only needed it to find the key points that indicate that the Riemann zeta function *cannot have non-trivial zeros in the critical strip, except for the critical line.*
Obviously we can move from the fixed coordinate system formed by the axes $x=Re(s)$ and $y=Im(s)$, to the moving coordinate system formed by axes $L$ with angle $\varphi_L=Arg(\chi(s))/2$ and by axes $M$ with angle $\varphi_M=(Arg(\chi(s))+\pi)/2$ passing through the zero of the complex plane.
Then from the functional equation of the Riemann zeta function: $$\label{zeta_func_eq3}\zeta(s)=\chi(s)\zeta(1-s);$$
and equality arguments of functions: $$\label{arg6}Arg(\zeta(1-s))=-Arg(\zeta(\overline{1-s}));$$
using the arithmetic of the arguments of complex numbers, when we rotate the vector of a value of Riemann zeta function by the angle $Arg(\chi(s))/2$ in the negative direction, we obtain: $$\label{zeta_arg}Arg(\zeta(s))-\frac{Arg(\chi(s))}{2}=-(Arg(\zeta(\overline{1-s}))-\frac{Arg(\chi(s))}{2});$$
Consequently, in the moving coordinate system formed by the axes $L$ and $M$, the angles of the vectors of value of $\zeta(s)$ and $\zeta(\overline{1-s})$ are symmetric about the axis $L$, thus, in accordance with the symmetry of the angles, the vector of value of $\hat\zeta(s)=\zeta (1/2+it)$ is always directed along the axis $L$.
In other words, in the moving coordinate system formed by the axes $L$ and $M$, the vector of value of $ \hat\zeta(s)=\zeta(1/2+it)$ remains fixed and only changes its modulus and sign, while the vectors of value of $\zeta(s)$ and $\zeta(\overline{1-s})$ when $\sigma\ne 1/2$ rotate in this moving coordinate system in different directions with the same speed.
Therefore:
a\) the projection $\hat\zeta(s) _M=\zeta(1/2+it)_M$ is always zero;
b\) the projections $\zeta(s)_L$ and $\zeta(\overline{1-s})_L$ are periodically equal to zero when the vectors of value of $\zeta(s)$ and $\zeta(\overline{1-s})$ both locate along the axis $M$ they have opposite directions and at this point the projections $\zeta(s)_M$ and $\zeta(\overline{1-s})_M$ are not equal to zero, because $\zeta(s)_L$ is an odd harmonic function, and $\zeta(s)_M$ is an even harmonic functionthat conjugate to $\zeta(s)_L$;
c\) the projections $\zeta(s)_M$ and $\zeta(\overline{1-s})_M$ are periodically equal to zero when the vectors of value of $\zeta(s)$ and $\zeta(\overline{1-s})$ both locate along the axis $L$ while they have the same direction and at this point the projections $\zeta(s)_L$ and $\zeta(\overline{1-s})_L$ are not equal to zero, since $\zeta(s)_L$ is an odd harmonic function, and $\zeta(s)_M$ is an even harmonic functionthat conjugate to $\zeta(s)_L$;
So only $\hat\zeta(s)_M$ and $\hat\zeta(s)_L$ can be equal to zero at the same time, because $\hat\zeta(s)_L=\zeta(1/2+it)_L$ is an odd harmonic function, and $\hat\zeta(s)_M=\zeta(1/2+it)_M$ is a harmonic function *identically equal to zero* (fig. \[fig:graphics\_projections\_2\]).
*Therefore, when $\sigma\ne 1/2$, the Riemann zeta function cannot be zero, because when $\sigma\ne 1/2$, the projections $\zeta(s)_L$ and $\zeta(s)_M$ cannot be equal to zero at the same time* (fig. \[fig:graphics\_projections\_3\]).
![Graphics of projections of the Riemann zeta function, $\sigma=1/2$[]{data-label="fig:graphics_projections_2"}](graphics_projections_2.jpg)
![Graphics of projections of the Riemann zeta function, $\sigma=0$[]{data-label="fig:graphics_projections_3"}](graphics_projections_3.jpg)
There may be a Lemma on conjugate harmonic functions identically nonzero, but we have not found it, just like the Lemma on the symmetric polygon we proved earlier (Lemma 3).
Therefore, to confirm our conclusions, we are forced to return to the vector system of the second approximate equation of the Riemann zeta function.
Vertical marks on the graphs (fig. \[fig:graphics\_projections\_2\] and \[fig:graphics\_projections\_3\]) is *the base point*, corresponding to solution of the equation: $$\label{}Arg(\chi(\frac{1}{2}+it_k))=(2k-1)\pi;$$
Using the mirror symmetry property of the vector system of the second approximate equation of the Riemann zeta function when $\sigma=1/2$, we previously defined two types of base points:
$a_1$ - if the first middle vector of the Riemann spiral at the base point is above or along the real axis of the complex plane (the second position corresponds to the non-trivial zero of the Riemann zeta function at the base point is a likely event), in this case the non-trivial zero of the Riemann zeta function that corresponds to this base point is located between this and the previous base point;
$a_2$ - if the first middle vector of the Riemann spiral at the base point is below the real axis of the complex plane, then the non-trivial zero of the Riemann zeta function that corresponds to this base point is located between this and the next base point;
as well as four types of intervals between base points of different types:
$A_1=a_1a_1$ and $A_2=a_2a_2$ - intervals of this kind contain one non-trivial zero of the Riemann zeta function;
$B=a_2a_1$ - interval of this kind contains two non-trivial zeros of the zeta function of Riemann;
$C=a_1a_2$ - interval of this kind does not contain any non-trivial zero of the Riemann zeta function.
We also found that there cannot be the following combinations of intervals:
$A_1A_2$, $A_2A_1$, $BB$ and $CC$;
Hence we have *a fixed set* of combinations of intervals:
$A_1A_1=a_1a_1a_1$, $A_1C=a_1a_1a_2$, $CB=a_1a_2a_1$, $CA_2=a_1a_2a_2$, $BA_1=a_2a_1a_1$, $BC=a_2a_1a_2$, $A_2B=a_2a_2a_1$, $A_2A_2=a_2a_2a_2$;
It is obvious that the sequence of base points which are represented in the graphs (fig. \[fig:graphics\_projections\_2\] and \[fig:graphics\_projections\_3\]), contains all the possible combinations of intervals:
$a_2a_2a_2a_1a_1a_1a_2a_1a_2a_2=A_2A_2BA_1A_1BCBA_2$;
In accordance with the properties of the vector system of the second approximate equation of the Riemann zeta function, we can determine the sign of the function $\zeta(\sigma+it)_L$ and the sign of its first derivative at each base point.
This requires:
1\) determine the type of base point by the position of the first middle vector Riemann spiral relative to the real axis of the complex plane at the base point;
2\) determine the direction of the normal $L$ to the axis of symmetry of the vector system of the second approximate equation of the Riemann zeta function at the base point;
Then
A\) If the first middle vector Riemann sprial is below the real axis of the complex plane, the function $\zeta (\sigma+it)_L$ will have the sign opposite to the direction of the normal $L$ to the axis of symmetry;
B\) If the first middle vector of the Riemann sprial is above the real axis of the complex plane, the function $\zeta (\sigma+it)_L$ will have a sign corresponding to the direction of the normal $L$ to the axis of symmetry;
C\) The sign of the first derivative of the function $\zeta(\sigma+it)_L$ always has a sign corresponding to the direction of the normal $L$ to the axis of symmetry;
These rules are executed for any combination of intervals from *a fixed set,* so they are executed for any combination of intervals that can occur.
In the moving coordinate system formed by axes $L$ and $M$ when $\sigma\ne 1/2$, the vector of values of the Riemann zeta function must rotate on the angle $\pi/2$ from the position corresponding to $\zeta(\sigma+it)_L=0$ to the position corresponding to $\zeta(\sigma+it)_M=0$ as well as from the position corresponding to $\zeta(\sigma+it)_M=0$ to position corresponding to $\zeta(\sigma+it)_L=0$, hence *all the zeros of the function $\zeta(\sigma+it)_M$ when $\sigma\ne 1/2$, lie between the zeros of the function $\zeta(\sigma+it)_L$.*
Corollary 1. The function $\zeta(1/2+it)_L$ has an infinite number of zeros (this statement corresponds to Hardy’s theorem [@HA2] on an infinite number of non-trivial zeros of the Riemann zeta function on the critical line).
Corollary 2. The number of non-trivial zeros of the Riemann zeta function on the critical line corresponds to the number of base points: $$\label{}N_0(T)=\Bigg[\Big|\frac{T}{2\pi}(\log{\frac{T}{2\pi}}-1)-\frac{1}{8}+\frac{2\mu(T)-\alpha_2}{2\pi}\Big|\Bigg]+2;$$
where $\mu(T)$ is the remainder term of the gamma function (\[mu\]) when $\sigma=1/2$;
$\alpha_2$ argument of the CHI function at the second base point.
Corollary 3. The function $\zeta(\sigma+it)_L$ when $\sigma\ne 1/2$ has an infinite number of zeros.
Corollary 4. The function $\zeta(\sigma+it)_M$ when $\sigma\ne 1/2$ has an infinite number of zeros.
Corollary 5. Zeros of function $\zeta(\sigma+it)_L$ and functions $\zeta(\sigma+it)_M$ when $\sigma\ne 1/2$ are not the same, because all zeros of function $\zeta(\sigma+it)_M$ when $\sigma\ne 1/2$ lie between zeros of function $\zeta(\sigma+it)_L$.
Based on the obtained results, we believe that the methods of confirmation of the Riemann hypothesis based on the properties of the vector system of the second approximate equation of the Riemann zeta function will soon lead to its proof.
Gratitudes
==========
Special thanks to Professor July Dubensky, who allowed to speak at the seminar and instilled confidence in the continuation of the research. Colleagues and friends who listened to the first results and supported throughout the research. My wife, who helped to prepare the presentation at the seminar, supported and believed in success.
The organizers of the conference of Chebyshev collection in Tula, inclusion in the list of participants of this conference allowed to prepare the first version of the paper, but a exclusion from the list of participants mobilized and allowed to obtain new important results of the research.
[1]{}
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S. Feng, Zeros of the Riemann zeta function on the critical line, http://arxiv.org /abs/1003.0059. T. S. Trudgian, A modest improvement on the function S(T), 2012, Mathematics of Computation, vol. 81, no. 278, pp. 1053-1061, Available at: https://arxiv.org/abs/1010.4596 Raymond Manzoni, Simpler zeta zeros Available at: https://math.stackexchange.com/questions/805396/simpler-zeta-zeros/815480\#815480 https://aimath.org/wp-content/uploads/bristol-2018-slides/Sarnak-talk.pdf N. Preobrazhenskay, S. Preobpagenskii, 100% of the zeros of the Riemann zeta-function are on the critical line, https://arxiv.org/abs/1805.07741 http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/RHproofs.htm http://upbyte.net/news/vychislenie\_dzeta\_funkcii\_rimana/2015-07-20-703
[^1]: The first is the correspondence to the graph of partial sums of the Dirichlet series considered by Erickson in his paper [@ER]
[^2]: These inconveniences and apparent contradictions can be avoided if we give the word sum a meaning different from the usual. Let us say that the sum of any infinite series is a finite expression from which the series can be derived.
[^3]: It is impossible to state Euler’s principle accurately without clear ideas about functions of a complex variable and analytic continuation.
[^4]: This fact was already considered by the forum Stack Exchange [@MA], but nobody try to compute the Riemann zeta function with geometric method, which corresponds to the method of generalized summation Cesaro.
[^5]: The exact value will be understood as a value obtained with a given accuracy.
[^6]: We later show that the mirror symmetry of the vector system of the second approximate equation of the Riemann zeta function is also determined by the argument of the Riemann zeta function when $\sigma=1/2$.
[^7]: The research of the vector system of the second approximate equation of the Riemann zeta function on the critical line shows that if the first middle vector $Y_1$ of the Riemann spiral for any complete rotation around the fixed first vector $X_1$ of the Riemann spiral never passes through the zero of the complex plane, then for another complete rotation it passes through the zero value of the complex plane twice.
[^8]: base point is a value of a complex variable in which the first middle vector $Y_1$ occupies the position opposite to the first vector $X_1$ of the Riemann spiral.
[^9]: If the Riemann zeta function can take a value of non-trivial zero at the base point, then the first middle vector $Y_1$ will occupy the position of the first vector $X_1$ of the Riemann spiral (we will not consider this state in detail), we assume that this position belongs to the base point type $a_1$.
[^10]: unfortunately, the fact that the Euler-McLaren summation formula is used for the Riemann zeta function as a method of generalized summation of divergent series is not mentioned in all textbooks.
|
---
author:
- 'Bum-Suk Yeom'
- Young Sun Lee
- 'Jae-Rim Koo'
- 'Timothy C. Beers'
- Young Kwang Kim
date: 'Received ; accepted '
title: ' Origin and Status of Low-Mass Candidate Hypervelocity Stars '
---
Introduction {#intro}
============
Hypervelocity stars (HVSs) are unbound and rare fast-moving objects in the Galactic halo, possessing space velocities that exceed the Galactic escape speed. The first HVS was discovered from a radial velocity survey of faint blue horizontal branch stars [@brown2005]. It is a 3 $M_\odot$ main-sequence B-type star moving with a Galactic rest-frame velocity of about 700 [km s$^{-1}$]{} at a distance of about 100 kpc. Since then, about 20 B-type HVS candidates have been discovered in the Galactic halo [@brown2015].
These intriguing objects are believed to originate from the so-called “Hills mechanism”, which is associated with the supermassive black hole (SMBH) at the Galactic Center. This theory suggests that, in the case of a binary system interacting with the SMBH, the SMBH can destroy the binary system and eject one of its stars, attaining speeds up to $\sim$ 1000 [km s$^{-1}$]{} [@hills1988; @yu2003]. It is known that this mechanism can also produce HVSs bound to the Milky Way (MW) [@bromley2009; @brown2014].
In addition to the HVSs, there are other types of fast-moving stars, referred to as “runaway stars” among O- and B-type stars, which have peculiar velocities larger than 40 [km s$^{-1}$]{} [e.g., @gies1987; @stone1991; @tetzlaff2011]. Several scenarios have been proposed to explain the runaway stars. The binary ejection mechanism [@blaauw1961; @tauris1998; @tauris2015] postulates that these objects could be formed in a binary system and “released” out of their system by the explosive death of their companion in the Galactic disk. The dynamical ejection mechanism proposed by @poveda1967 assumes that a star can be ejected by multi-body interactions in a high-density environment such as star clusters. Another explanation to account for these stars is the ejection from a star-forming galaxy such as the Large Magellanic Cloud (LMC) [@boubert2016]. An alternative theory suggests that these objects are the members of a tidally disrupted dwarf galaxy [@abadi2009]. Even though there exist many scenarios to explain the HVSs and runaway stars, full understanding of their origin remains elusive.
In spite of the ambiguity of their origin, high-velocity stars have received attention because they can provide a means for measuring the local escape velocity at a given distance from the Galactic Center, which can in turn constrain the total mass of the MW, still uncertain by more than a factor of two [@xue2008; @watkins2010]. Even though the origin of these objects is uncertain, we can utilize their general characteristics to infer where they originated. What HVSs have in common is that they are young, massive main-sequence stars typically found at present distances beyond 50 kpc from the Sun [See @brown2015AR and references therein]. Therefore, we can hypothesize that the high-velocity stars may originate from star-forming regions in the disk, bulge, or dwarf satellites of the MW.
Although most of the currently known HVSs and runaway stars are early type (high-mass) main-sequence stars, one might expect that the proposed ejection mechanisms could work for any stellar type, leading to the prospect of identifying low-mass HVSs [@kollmeier2007]. For this reason, various efforts to search for such stars have been carried out from the stellar database constructed by large spectroscopic surveys such as Sloan Digital Sky Survey [SDSS; @york2000] and Large Sky Area Multi-Object Fibre Spectroscopic Telescope [LAMOST; @cui2012].
Indeed, @kollmeier2009 and @li2012 identified 6 and 13 HVS candidates, respectively, from SDSS. In addition, @zhong2014 reported 28 HVS candidates, 17 of which are F-, G-, and K-type dwarf stars. @li2015 reported another 19 low-mass HVS candidates from LAMOST. @palladino2014 [hereafter Pal14] also reported the discovery of 20 low-mass G-, and K-type HVS candidates from Sloan Extension for Galactic Understanding and Exploration [SEGUE; @yanny2009]. One interesting aspect of many of these low-mass candidates is that they appeared to be associated with birth in the Galactic disk, rather than the Galactic Center, as is the case for the high-mass HVSs.
Previous studies of low-mass HVSs or runaway star candidates were carried out exclusively on the basis of derived stellar kinematics, since proper motion information, when combined with an observed radial velocity and distance estimate, provides the full space velocity of a star. In keeping with this, it is interesting to note that HVSs thought to have a Galactic Center origin (e.g., the high-mass HVSs) are often well-separated in their proper motions from likely disk origins, which is not the case for the low-mass runaway-star candidates. For this reason, one might consider use of their detailed chemical-abundance patterns in order to identify the possible birthplaces of the runaway-star candidates – bulge, disk, or halo, and thereby constrain the possible ejection mechanisms of such stars.
We note here that a number of recent studies claim that most of the low-mass HVS candidates and runaway stars identified thus far are bound to the MW. For example, @ziegerer2015 have re-calculated the proper motions for the 14 HVS candidates that Pal14 reported, using images from SDSS[^1], Digitized Sky Survey[^2] (DSS), and UKIDSS[^3] [@lawrence2007]. They found that the newly measured proper motions are much smaller than the ones used by Pal14, and, as a result, all of their HVS candidates are bound to the MW under three different potentials for the MW. More recently, @boubert2018 reported using [$Gaia$]{} Data Release 2 [DR2; @gaia2018] proper motions and radial velocities that all late-type stars, which have been claimed to be HVSs previously are likely to be bound to the MW, except one object (LAMOST J115209.12$+$120258.0).
Prior to clarification on the proper motions for low-mass HVS candidates in the Pal14 sample, we carried out follow-up spectroscopic observations and obtained medium-resolution ($R$ = 6000) spectra for six of them, in order to study their chemical abundance patterns. As we report in this paper, we make use of [*chemical tagging*]{} [@freeman2002], in an attempt to understand their characteristics and likely parent populations of their birthplaces. This approach has already proven to be useful to constrain the origin of HVSs or runaway stars [@hawkins2018]. In addition, we make use of the greatly improved proper motion information from [*Gaia*]{} DR2 to carry out a kinematics analysis of our program objects. Although we confirm that none of our program stars unbound, and thus are no longer viable HVS candidates, for simplicity we refer to them as HVS candidates through this paper. This paper is organized as follows. The spectroscopic observations and reduction of the six low-mass candidate HVSs are described in Section \[sec2\]. In Section \[sec3\], we determine the stellar parameters and chemical abundances for our program stars. In Section \[sec4\], we present results of the analysis of the chemical and kinematic properties of our objects. Section \[sec5\] discusses the characteristics and possible origin of each HVS candidate. A summary of our results, and brief conclusions are provided in Section \[sec6\].
\[t\] ![Spectra of standard stars in the wavelength range used to derive stellar parameters and chemical abundances.[]{data-label="figure1"}](spectra_std.eps "fig:"){width="\columnwidth"}
\[t\]![Same as in Figure \[figure1\], but for our program stars.[]{data-label="figure2"}](spectra_obj.eps "fig:"){width="\columnwidth"}
Spectroscopic Observations and Reduction {#sec2}
========================================
Among the 20 HVS candidates reported by Pal14, we obtained spectroscopy for six stars during the period between January and May, 2014, including two stars that were not studied by @ziegerer2015. The spectra were obtained with the Dual Imaging Spectrograph (DIS) on the Apache Point Observatory 3.5 m telescope in New Mexico. We selected the grating combination B1200/R1200, which has wavelength coverage of 4200 – 5400[Å]{} and 5600 – 6700[Å]{} in the blue and red channels, respectively, along with a 1.5 arcsec slit, a yielding spectral resolving power of $R$ = 6000, sufficient to perform a chemical abundance analysis for individual $\alpha$-elements (Mg, Ca, Si, and Ti) and iron-group elements (Cr, Fe, and Ni).
In addition, we observed eight well-studied disk stars as comparison stars, to verify the accuracy of our derived stellar parameters ($T_{\rm eff}$, $\log~g$, and \[Fe/H\]) and chemical abundances for our program stars. For the comparison stars, we took several exposures to obtain the spectra at various signal-to-noise (S/N) ratios, enabling an evaluation of the effect of S/N on the estimated stellar parameters and chemical abundances of our target stars. As our targets are mostly faint ($r_{\rm 0} >$ 16.0), we took one or two exposures between 60 and 75 minutes each. Table \[tab1\] lists details of the observations.
We followed the standard spectroscopic reduction steps such as aperture extraction, wavelength calibration, and continuum normalization using IRAF[^4]. Radial velocities were measured using the cross-correlation function by applying *xcsao* task of the *rvsao* package [@kurtz1998] in IRAF. In that process, we used a synthetic model spectrum of $T_{\rm{eff}} = 6000$ K and log $g$ = 4.0 as a template. We also considered the night-sky emission lines to check for any additional instrumental shifts. The spectra of each star were co-added after application of radial velocity corrections, to obtain a final spectrum with higher S/N, typically S/N $\sim$ 30 – 50. The spectra of our standard stars and program stars are shown in Figures \[figure1\] and \[figure2\], respectively.
\[t\] ![Residual plots for the stellar parameters between our estimated values and the reference values for our comparison stars, as a function of S/N. The blue star symbol with an error bar indicates the median value and its median absolute deviation (MAD) calculated in each S/N range. There are four S/N ranges shown ($<$ 50, 50 – 100, 100 – 150, and $>$ 150), separated by the vertical dotted lines.[]{data-label="figure3"}](stellar_param_residual.eps "fig:"){width="\columnwidth"}
\[t\]![Same as in Figure \[figure3\], but for the chemical abundances of Mg, Si, Ca, Ti, Cr, and Ni. Note that the lower number of points for S/N $<$ 50 results from a lower number of abundance measurements, due to too weak absorption features.[]{data-label="figure4"}](stellar_abun_residual.eps "fig:"){width="\columnwidth"}
Star $T_{\rm eff}$ (K) $\log~g$ \[Fe/H\] \[Mg/H\] \[Si/H\] \[Ca/H\] \[Ti/H\] \[Cr/H\] \[Ni/H\] References
----------- ------------------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------------------
HIP 33582 5782 4.30 –0.68 –0.23 –0.36 –0.50 –0.33 –0.64 $\cdots$ 1, 3, 6, 8, 10
HIP 44075 5880 4.10 –0.90 –0.53 –0.55 –0.60 –0.57 –0.89 –0.87 1, 2, 3, 4, 5, 7, 11
HIP 59330 5749 4.02 –0.75 –0.45 –0.42 –0.58 –0.47 –0.74 –0.75 1, 3, 4
HIP 60551 5724 4.38 –0.86 –0.53 –0.52 –0.61 –0.59 –0.85 –0.76 1, 3, 4
HIP 62882 5692 3.81 –1.26 –0.73 –0.79 –0.93 –0.99 –1.30 –1.21 1, 3, 5, 11
HIP 64426 5892 4.19 –0.76 –0.39 –0.50 –0.62 –0.58 –0.81 –0.77 1, 3, 4, 8
HR 2233 6240 3.97 –0.19 –0.06 –0.11 –0.11 0.11 –0.22 –0.22 4, 5, 6, 7, 11
HR 2721 5872 4.30 –0.32 –0.13 –0.21 –0.25 –0.17 –0.35 –0.33 4, 5, 7, 9, 11
Estimation of Stellar Parameters and Chemical Abundances {#sec3}
========================================================
We determined estimates of the stellar parameters ($T_{\rm eff}$, $\log~g$, and \[Fe/H\]), and elemental abundances for Mg, Si, Ca, Ti, Cr, and Ni using the Stellar Parameters And chemical abundances estimator [SP\_Ace; @boeche2016] code. The SP\_Ace code employs similar methodology to that used in the RAdial Velocity Experiment [RAVE; @steinmetz2006] chemical abundance pipeline [@boeche2011], which was developed to derive elemental abundances for the stellar spectra obtained by the RAVE survey. SP\_Ace estimates the stellar parameters and abundances based on a library of the equivalent widths (EWs) for 4643 absorption lines. The EWs are generated from a synthetic grid in the ranges of $3600<T_{\rm{eff}}<7400~\rm{K}$, $0.2<\log~g<5.4$, and $-2.4<\rm{[M/H]}<0.4$. Each spectrum in the synthetic grid was synthesized by MOOG [@sneden1973], after adopting the ALTAS9 model atmospheres [@castelli2003]. Based on input trial values of $T_{\rm{eff}}$, $\log~g$, and \[elements/H\], SP\_Ace calculates by interpolation the expected EWs using the library to generate a normalized model spectrum. Then, using the Levenberg-Marquadt method, it attempts to minimize the $\chi^2$ between the model and the observed spectrum to determine the stellar parameters and chemical abundances. As the lines used to measure the abundances of the individual elements and stellar parameters in SP\_Ace are well described in @boeche2016, we refer the interested reader to their paper.
Even though we obtained spectra of both the blue and red channels with the DIS instrument, we exclusively used the red channel spectra, because the adopted wavelength ranges (5212 – 6860[Å]{} and 8400 – 8920[Å]{}) used by SP\_Ace cover a much larger wavelength range in the red. We applied SP\_Ace to the spectra of the co-added spectra of our program stars as well as to the spectra of individual exposures of the reference stars to derive final estimates of the stellar parameters and chemical abundances.
Before we finalized the stellar parameters and chemical abundances of our program stars for the analysis of the abundance patterns, we first compared our derived values with the values for the standard stars from the various references, as a function of S/N, in order to check for systematic offsets and estimate the precision of the derived stellar parameters and chemical abundances. Table \[tab2\] lists the adopted stellar parameters and chemical abundances of the comparison stars from various references.
Figure \[figure3\] shows residual plots of $T_{\rm eff}$, $\log~g$, and \[Fe/H\] between our values and the reference values, as a function of S/N, for the standard stars. The blue star symbol with an error bar indicates the median value and median absolute deviation (MAD), respectively, in each S/N bin. We considered four bins ($<$ 50, 50 – 100, 100 – 150, and $>$ 150), separated by the vertical dotted lines in the figure. For [$T_{\rm eff}$]{} and [$\log~g$]{}, we do not notice any significant trend with S/N. In the case of the metallicity, our derived value increases with decreasing S/N, as expected. As the typical S/N of the spectra of our program stars is less than 50, we checked the systematic offset of each parameter in the range of S/N $<$ 50. We found a median offset of $T_{\rm eff}$ = 99 $\pm$ 183 K, $\log~g$ = –0.16 $\pm$ 0.34 dex, and \[Fe/H\] = –0.04 $\pm$ 0.09 dex, respectively, in the sense that our values are higher for [$T_{\rm eff}$]{} and lower for [$\log~g$]{} and [\[Fe/H\]]{}. The uncertainty is the MAD. We decided to adjust the parameter scales by these offsets for derivation of the final estimates for our program stars.
We carried out a similar exercise for the chemical abundances (Mg, Si, Ca, Ti, Cr, and Ni); Figure \[figure4\] displays the results. Similar to \[Fe/H\], there is a tendency for larger derived abundances with decreasing S/R. For S/N $<$ 50, we found median offsets with of \[Mg/H\] = –0.16 $\pm$ 0.03 dex, \[Si/H\] = –0.06 $\pm$ 0.14 dex, \[Ca/H\] = –0.10 $\pm$ 0.11 dex, \[Ti/H\] = –0.08 $\pm$ 0.15 dex, \[Cr/H\] = –0.21 $\pm$ 0.17 dex, and \[Ni/H\] = –0.15 $\pm$ 0.15 dex. The uncertainty is the MAD. We applied these offsets to our program stars. Table \[tab3\] lists the offset-corrected stellar parameters and chemical abundances for our program stars. The value of \[$\alpha$/Fe\] is a mean of the four ratios \[Mg/Fe\], \[Si/Fe\], \[Ca/Fe\], and \[Ti/Fe\]. The typical errors on the abundances for our target stars are $\sigma$\[Mg/Fe\] = 0.27 dex, $\sigma$\[Si/Fe\] = 0.17 dex, $\sigma$\[Ca/Fe\] = 0.15 dex, $\sigma$\[Ti/Fe\] = 0.18 dex, $\sigma$\[Cr/Fe\] = 0.20 dex, and $\sigma$\[Ni/Fe\] = 0.18 dex. These uncertainties and the uncertainties on [$T_{\rm eff}$]{}, [$\log~g$]{}, and [\[Fe/H\]]{} are calculated by adding in quadrature the internal error from SP\_Ace and the MAD in the range of S/N $<$ 50 for the standard stars. We use these corrected values for the analysis of chemical properties throughout this paper.
As setting S/N values to different limits in Figures \[figure3\] and \[figure4\] can result in different levels of the offsets in each parameter and abundance, we carried out an exercise to evaluate its effect by applying different limits of S/N, for example bins of S/N $<$ 50, S/N = 50 – 90, S/N = 90 – 150, and S/N $>$ 150. We did not find any significant difference from our original S/N limits, but the offsets all agreed within the errors. In particular, the differences in the chemical abundances from various different S/N limits are less than 0.02 dex, implying that these effect do not affect interpretation of the chemical properties of our program stars. The reason for the small offsets among the different S/N bins is that we take a median value of the several points in each SNR bin.
Also note that the reason for the smaller offset at the low S/N in Figures \[figure3\] and \[figure4\] is that, as the noise in a low S/N spectrum can mimic extra absorption, the estimated abundance tends to be larger. We generally see this tendency in Figures \[figure3\] and \[figure4\] for our standard stars. For this reason, we adjusted the stellar parameters and chemical abundances of our programs by the offsets derived from S/N $<$ 50.
Kinematic and Chemical Properties of Our Program Stars {#sec4}
======================================================
Kinematic Properties {#sec41}
--------------------
The contradicting results on the status of the SEGUE low-mass candidates HVSs between Pal14 and @ziegerer2015 stem from the proper motions that they adopted. Pal14 used the SDSS proper motions [@munn2004; @munn2008], while @ziegerer2015 derived the proper motions with the images from SDSS, DSS, and UKIDSS. Table \[tab4\] summarizes these different sets of proper motions for our target stars, along with those reported in [$Gaia$]{} DR2.
Comparison of the three sets of the proper motions reveals that the proper motions adopted by Pal14 are consistently larger than the other two; the proper motions of the four stars that @ziegerer2015 re-derived are closer to the [$Gaia$]{} proper motions. Munn (priv. communication) notes that the reported proper motions from Pal14, based on SDSS data, are almost certainly the result of mis-identification of the targets with other nearby stars. The SDSS proper motions of the other two stars (Pal14 IDs 7 and 10) that @ziegerer2015 could not measure the proper motions for are also larger than those from [$Gaia$]{} DR2. These smaller proper motions imply that our objects are probably bound to the MW. One program star, Pal14 ID 4, exhibits a larger proper motion than the other program stars, and deserves further investigation.
\[t\] ![Toomre diagram for our sample of stars. The blue symbols indicate the velocities calculated with the SDSS proper motions, while the red symbols are based on the [*Gaia*]{} DR2 proper motions. The black and red circles roughly represent the kinematic boundaries of the thin and thick disks, at a constant velocity of 70 km s$^{-1}$ and 180 km s$^{-1}$ [@venn2004], respectively. The blue line indicates the local Galactic escape speed of $V_{\rm{esc}}$ = 533 km s$^{-1}$ [@piffl2014]. The numbers besides each star are the sample ID used by Pal14.[]{data-label="figure5"}](toomre.eps "fig:"){width="\columnwidth"}
We computed space velocity components and orbital parameters, as listed in Table \[tab5\], using radial velocities measured from the obtained spectra, proper motions from [*Gaia*]{} DR2, and distances estimated by SEGUE Stellar Parameter Pipeline [SSPP; @allende2008; @lee2008a; @lee2008b], based on the methodology of @beers00 [@beers12]. The quoted distance uncertainty is on the order of 15–20%. Because the parallaxes to determine the distance do not exist in [*Gaia*]{} DR2 for two objects of our program stars, we adopted the photometric distances from the SSPP, even though the distance uncertainties of four program stars derived by [*Gaia*]{} DR2 are much smaller (around 10%). We also compared the photometric distances of our four programs with those from the [$Gaia$]{} DR2 parallaxes, and confirmed that our photometric distances agreed with the [$Gaia$]{} distances within the error ranges.
{width="\textwidth"}
The $U$, $V$, and $W$ velocity components were calculated assuming 220 km s$^{-1}$ of the rotation velocity of the local standard of rest (LSR) and ($U_\odot$,$V_\odot$,$W_\odot$) = (–10.1, 4.0, 6.7) [km s$^{-1}$]{} of the solar peculiar motion [@hogg2005]. Each velocity component is positive in the radially outward direction from the Galactic center for $U$, the direction of Solar rotation for $V$, and the direction of the North Galactic Pole for $W$.
The orbital parameters for individual objects were computed under the Galactic gravitational potential *MWPotential2014* [@bovy2015], which is composed of a bulge with a power-law density, a disk parametrized by a Miyamoto-Nagai potential, with mass 6.8 $\times\ 10^{10}\
M_{\odot}$, and a dark-matter Navarro-Frenk-White halo potential. The adopted position of the Sun from the Galactic center is $R_{\odot}$ = 8.0 kpc. Among the orbital quantities, we derived the minimum ($r_{\rm min}$) and maximum ($r_{\rm max}$) distances from the Galactic center, and the maximum distance ($Z_{\rm max}$) from the Galactic plane during the orbit of a given star. Additionally, the eccentricity ($e$) was obtained from $e =$ ($r_{\rm max}-r_{\rm min}$)/($r_{\rm
max}+r_{\rm min}$). Table \[tab5\] lists the calculated velocity components and the derived orbital parameters. The error on each velocity and orbital parameter in the table was estimated from the standard deviation of the distribution of a sample of stars randomly resampled 100 times, assuming a normal error distribution for the distance, radial velocity, and proper motion.
Figure \[figure5\] is a Toomre diagram for our program stars, which can be used to kinematically classify different Galactic stellar components. In the figure, the blue and red squares represent the velocities computed with the proper motions from SDSS and [*Gaia*]{} DR2, respectively. Black and red circles roughly delineate the boundaries of the thin and thick disks, at constant velocities of 70 km s$^{-1}$ and 180 km s$^{-1}$ [@venn2004], respectively. The local Galactic escape speed of $V_{\rm{esc}}$ = 533 [km s$^{-1}$]{} [@piffl2014] is plotted as a blue circle. The figure clearly indicates that our spectroscopically-observed HVS candidates are all bound to the MW when the [*Gaia*]{} DR2 proper motions are used. According to this diagram, kinematically, two of our stars appear to belong to the thick disk, while four of them are likely members of the thin disk.
This component separation is confirmed by the following test. With the space velocity components ($U$, $V$, $W$) of our program stars in hand, we performed a test to compute the likelihood of belonging to the thin disk, thick disk, and the halo, following the methodology of @bensby2003. The basic idea is that by assuming that a stellar population in the thin disk, thick disk, or the halo has Gaussian distributions with different space velocities ($U$, $V$, $W$) and asymmetric drifts, we attempt to separate our HVS candidates into the thin disk, thick disk, or halo component by calculating the probability of belonging to each component. In that process, we adopted the local stellar densities, velocity dispersions in $U$, $V$, and $W$, and the asymmetric drifts listed in Table 1 of @bensby2003.
{width="\textwidth"}
Based on the computed probability of each program star, we derived a relative likelihood of being each component, by comparing the probability of being a member of a given component among the three, and assigned a star into a component with the higher likelihood. For example, if a star has a higher likelihood of being the thick disk relative to being the thin disk, that is Pr(Thick)/Pr(Thin) $>$ 5, this star is assigned the thick disk. On the other hand, the thin disk is assigned when a star has Pr(Thick)/Pr(Thin) $<$ 0.5.
The results of this test for our program stars revealed that only Pal14 IDs 4 and 10 have Pr(Thick)/Pr(Thin) much larger than 5, and the rest of the stars have less than 0.05. The probability of being in the halo population is much less, Pr(Halo)/Pr(Thick) $<$ 0.01. Only one star with ID 4 has Pr(Halo)/Pr(Thick) $\sim$ 0.9, which is still too small to be a halo star. Thus, this exercise proves that our program stars belong to the thin or thick disk.
We also compared the Galactic rest-frame velocity ($V_{\rm GRF}$) for each star with the local escape velocity to check on the probability that it is bound to the MW. A total of one million Monte Carlo realizations were carried out to calculate the Galactic rest-frame velocities after randomly resampling from a normal error distribution of the radial velocities, distances, and proper motions from [*Gaia*]{} DR2. The bound probability is defined by the fraction of the stars that exceed the local escape velocity to the total number of stars in the simulation. We found that all our samples are, as expected, bound to the MW.
To investigate the orbital characteristics of our program stars, we integrated the orbit of each star over 2 Gyr from its current position. Figure \[figure6\] shows the projected orbital trajectories of our program stars into the plane of $Z$ and $R$ (left panels) and $X$ and $Y$ (right panels). $Z$ is the distance from the Galactic plane, and $R$ is the distance from the Galactic center projected onto the Galactic plane. $X$ and $Y$ are based on the Cartesian reference system, in which the center of the Galaxy is at the location at (0, 0) kpc, and the Sun is located at ($X$, $Z$)=(8.0, 0.0) kpc. The filled circle indicates the current location of each star. In the figure, we clearly see that four objects (Pal14 IDs 7, 15, 16, and 17) spend most of their time on orbits outside the Solar circle (upper-left panel) with nearly circular orbits (upper-right panel), with eccentricities less than $e <$ 0.15. Judging from the orbits and $U$, $V$, $W$ velocities, the Pal14 ID 7 and 17 stars are typical thin-disk stars, as they are confined to $|Z| < $ 0.7 kpc, while Pal14 ID 15 and 16 stars appear to belong to the thick disk, as they exhibit excursions above $|Z| >$ 1.5 kpc.
The bottom panels of Figure \[figure6\] indicate that the other two stars in our sample (Pal14 IDs 4 and 10) are mostly inside the Solar radius (lower-left panel), with relatively high eccentricity orbits (lower right); $e$ = 0.83 for Pal14 ID 4 and $e$ = 0.41 for Pal14 ID 10. Even though Pal14 ID 4 exhibits thick-disk kinematics, an external origin from a disrupted dwarf galaxy cannot be ruled out for this object, due to the high eccentricity.
Chemical Properties
-------------------
Even though the kinematics provide valuable information on which Galactic component a given star is likely to be a member, the orbits of disk stars can change over the course of Galactic evolution due to the perturbations by transient spiral patterns or giant molecular clouds. However, since the abundance of a chemical element for dwarf stars is essentially invariant during its main sequence lifetime, this can provide additional information on its likely parent Galactic component, as we explore in this section for our program HVS candidates.
Among the chemical elements, the so-called $\alpha$-elements such as Mg, Si, Ca, and Ti, are good indicators of the star-formation history (duration and intensity) of a stellar population [@tinsley1979]. These elements are produced by successive capture of $\alpha$-particles in massive stars, which explode as core-collapse supernova (CCSN) that enrich the surrounding interstellar medium (ISM) with these elements. At early times, the ISM of a stellar population is enriched by CCSNs, whereas at later times, by Type Ia SNe, which produce more iron-peak elements. Consequently, the large enhancements of the $\alpha$-elements relative to Fe in a stellar population indicates that it experienced rapid star formation, while the lower values of this ratio suggests slower, prolonged star formation.
Figure \[figure7\] exhibits the distributions of Mg, Si, Ca, Ti, Cr, and Ni abundances with respect to Fe, as a function of \[Fe/H\], for four different Galactic stellar populations – thin disk (red circle), thick disk (filled-blue circle), bulge (green triangle), halo (filled-orange square), and LMC stars (black circles). Our program stars are represented by star symbols. The chemical abundances of each star in Galactic stellar populations are adopted from the following references: bulge stars from @alvesbrito2010 [@johnson2014], halo stars from @alvesbrito2010 [@reddy2006], thin and thick disk stars from @alvesbrito2010 [@bensby2003; @reddy2006]; the LMC stars are from @swaelmen2013.
From inspection of Figure \[figure7\], the general trends for each population can be summarized as follows. The thick-disk and halo stars exhibit similar trends, as they are rich in $\alpha$-elements, but relatively lower in Cr and Ni. The thick-disk stars are mostly more metal-rich (\[Fe/H\] $>$ –0.9) than the halo stars (\[Fe/H\] $<$ –0.8). The bulge stars display a wide range of metallicity, with enhanced $\alpha$-elements in the metallicity region overlapping with the thick disk. The level of their $\alpha$-abundances diminishes with increasing metallicity, later joining the thin disk. However, one distinct pattern is that the Ca abundances (somewhat true for Ni as well) of the bulge stars are consistently higher than the thick- and thin-disk populations. Therefore, the Ca abundance plays a key role in distinguishing a bulge star from a disk star. The LMC stars exhibit systematically lower abundances for Mg, Si, Ca, and Ni elements than the other Galactic components, but overlap with other Galactic stars in Ti and Cr. We included the LMC stars in the figure because, as claimed by @boubert2016, there is a possibility that runaway stars may come from the LMC. We also note that there is no clear distinction among the Galactic components in the Fe-peak element Cr. The Ni abundance of bulge stars are relatively higher in the range of –0.5 $<$ \[Fe/H\] $<$ 0.2 than any other Galactic populations. Our program stars, indicated with star symbols, all have metallicity larger than \[Fe/H\] = –0.7; none of them are likely halo stars.
Comparing the chemical characteristics of each Galactic component with those of each star in our program sample, while some of our program stars overlap with one or two Galactic stellar populations, most of our stars exhibit somewhat deviant abundances from comparison stellar populations. The Pal14 ID 16 star, which is the most-Fe rich object in our sample, appears to belong to the thin disk, because the Mg, Si, Ca, and Cr abundances agree with those of the thin-disk stars, although the Ti and Ni abundances are lower than the other thin-disk stars. Its chemistry is also overlapped with that of the bulge population. Pal 14 ID 17 is likely to be associated with the thin disk, as the level of most of elements is similar to that of the thin-disk stars, considering the error bars on Cr and Ni. Pal14 ID 4 is likely to be a bulge star because of the enhancement of all of its $\alpha$-elements. Pal14 ID 15 appears to be a metal-poor thin-disk star, as its $\alpha$-element abundances are lower than the thick disk, although the Cr abundance stands out with respect to the other elements. Pal14 ID 7 may be a thick-disk star, because its Mg, Si, and Ti are enhanced and the Ca abundance does not point to the bulge component, but rather the LMC. Taking into account the large error bar for Mg, Pal14 ID 10 appears to belong to the thick disk, with enhanced Si and Ti.
SDSS ID Pal14 ID Galactic component Possible origin(s)
--------------------- ---------- -------------------- --------------------
J113102.87+665751.1 4 Bulge Accreted or heated
J064337.13+291410.0 7 Thick $\cdots$
J172630.60+075544.0 10 Thick Runaway or heated
J095816.39+005224.4 15 Thin Runaway
J074728.84+185520.4 16 Thin Runaway
J064257.02+371604.2 17 Thin $\cdots$
Discussion {#sec5}
==========
Even though our HVS stars turned out to be disk stars, based on the kinematic probabilistic membership assignment as described in Section \[sec41\], some of them have very distinct dynamical properties compared to the canonical disk stars, which we discuss below. In what follows, we consider a star with high eccentricity to be either an accreted or heated disk star, as stars from a disrupted dwarf galaxy are expected to exhibit high eccentricities, and dynamical heating mechanism can also produce high eccentricity stars. We regard stars with low eccentricity but comparatively large excursions from the Galactic plane to be possible runaway stars.
SDSS J113102.87$+$665751.1 (Pal14 ID 4)
---------------------------------------
As this star has \[Fe/H\] = –0.49, and a high value of \[$\alpha$/Fe\] = 0.38, one might consider this object to be a typical thick-disk star. However, the several individual $\alpha$-elements appear to be more enhanced with respect to the thick-disk population, and relatively closer to the bulge population, especially the Ca abundance, which is the key element to distinguish between the thick-disk and bulge stars. Thus, it is reasonable to infer that it is a bulge star.
The calculated orbital parameters for this star are $Z_{\rm max}$ = 1.04 kpc, $e$ = 0.83, $r_{\rm min}$ = 0.90 kpc, and $r_{\rm max}$ = 8.71 kpc; hence its orbit spans from the near bulge through the Solar radius with a very high eccentricity. Note that our $r_{\rm min}$ value is rather smaller than that of @ziegerer2015 due to the slightly larger proper motions from [*Gaia*]{} DR2. These chemical and dynamical characteristics suggest that this star could be born in the bulge, and expelled to reach the current location by mechanisms such as dynamical ejection in high stellar density environment or binary supernova ejection. However, its Galactic rest-frame velocity of $V_{\rm GRF}$ = 49.1 [km s$^{-1}$]{} is too small to consider that origin likely. Rather, its orbit points to it being a former member of a disrupted dwarf galaxy or a dynamically heated disk star.
SDSS J064337.13$+$291410.0 (Pal14 ID 7)
---------------------------------------
This star has \[Fe/H\] = –0.60 and \[$\alpha$/Fe\] = 0.22, properties associated with a typical thick-disk star. Looking into the individual $\alpha$-elements, the Mg and Ti abundances are relatively higher and the Ca abundance is lower than most stars of the thick-disk population. Only the Si and Ni abundances are close to those of other thick-disk stars.
The kinematic characteristics also suggests this object is a thick-disk star, as the obtained orbital parameters are $Z_{\rm max}$ = 0.65 kpc, $e$ = 0.13, $r_{\rm min}$ = 8.56 kpc, and $r_{\rm max}$ = 11.11 kpc. Combining the chemical and kinematic characteristics, this star may be born in the outer disk, and now located close to the Solar circle.
SDSS J172630.60$+$075544.0 (Pal14 ID 10)
----------------------------------------
Given a metallicity of \[Fe/H\] = –0.69 and \[$\alpha$/Fe\] = 0.15, this object is also regarded as a likely thick-disk star. Yet, the abundances of four $\alpha$-elements exhibit a complex pattern; the Ca abundance is in the thick disk region, whereas the Si and Ti abundances are higher than the rest of the thick-disk population. The Ni abundance is also larger than the thick disk. Taking into account the large uncertainty of the Mg abundance, this object is considered as a thick-disk star.
Kinematically, however, this object has an eccentricity of $e$ = 0.41 and $r_{\rm min}$ = 3.77 kpc, and its vertical height is as high as $Z_{\rm max}$ = 2.53 kpc. Combined with the chemical signatures, this object may be a dynamically heated disk star. It is also possible that this star may be a disk runaway star, originating from near the bulge, or a heated disk star, as it reaches a comparatively large vertical height. The high eccentricity also indicates a possible external origin from a dissolved dwarf galaxy, but the extent of its orbit (only exploring 2 kpc away) appears to be too small to be commensurate with an accreted origin.
SDSS J095816.39$+$005224.4 (Pal14 ID 15)
----------------------------------------
The metallicity of \[Fe/H\] = –0.55 and \[$\alpha$/Fe\] = –0.05 for this star indicate a metal-poor thin-disk star. The chemical abundances of individual elements agree with those of the thin-disk population within the error bars, except the lower Ti and higher Cr abundances than those of any Galactic population.
The derived orbital parameters of $r_{\rm min}$ = 8.36 kpc, $Z_{\rm
max}$ = 1.67 kpc, and $e$ = 0.06 also suggest a thin-disk star. As this star reaches up to $Z$ = 1.67 kpc, this is a good candidate for a disk runaway star ejected by a SNe explosion in a binary system. As @bromley2009 simulated, some runaway stars exhibit similar kinematics to the disk or halo stars.
SDSS J074728.84$+$185520.4 (Pal ID 16)
--------------------------------------
This object has the highest metallicity in our sample, \[Fe/H\] = 0.27, and its \[$\alpha$/Fe\] value is –0.20. All six elements show relatively lower abundance ratios than the Sun, which mimic the patterns of a metal-rich thin-disk star. Hence, one may consider this to belong to the thin disk.
The orbital parameters of $r_{\rm min}$ = 10.77 kpc, $Z_{\rm max}$ = 2.05 kpc, and $e$ = 0.06 also imply a thin-disk star. As can be seen from Figure \[figure6\], this object resides in the outer disk and undergoes large excursions, with a small eccentricity, above the Galactic plane. Thus, it is more plausible to infer that this object is a runaway star from the thin-disk population.
SDSS J064257.02$+$371604.2 (Pal ID 17)
--------------------------------------
Pal14 ID 17 is a metal-rich (\[Fe/H\] = –0.12) star with an essentially Solar alpha ratio (\[$\alpha$/Fe\] = –0.02). The abundances of the individual $\alpha$-elements are overlapped with those of the thin-disk population, whereas the iron-peak elements appear more consistent with the bulge. Its orbital parameters ($r_{\rm min}$ = 8.20 kpc, $Z_{\rm max}$ = 0.56 kpc, and $e$ = 0.10) also point to a typical thin-disk star.
Summary and Conclusions {#sec6}
=======================
We have presented a chemodynamical analysis of six low-mass dwarf stars, alleged to be HVSs by Pal14, in order to determine from which Galactic component they originate. Based on kinematic analysis using accurate [$Gaia$]{} DR2 proper motions, we confirm that all six objects are bound to the MW, as noted previously by @ziegerer2015. Our conclusion is also upheld by the recent study by @boubert2018, who performed a detailed investigation of late-type HVS candidates with proper motions from [$Gaia$]{} DR2, and found that almost all known late-type HVS candidates are bound to the Galaxy. The HVS status for the low-mass stars identified by Pal14 is mainly due to the incorrect assignment of the proper motions.
Nonetheless, we have attempted to characterize the parent Galactic stellar components and origins of our program stars by taking into account a comparison of their abundance patterns with various Galactic stellar components and their orbital properties simultaneously. We note that the kinematic probabilistic assignment of their membership to the Galactic component has revealed that our program stars belong to the thin or thick disk. However, since four of the six stars exhibit distinct dynamical properties and chemical characteristics compared to the canonical disk stars, we cannot rule out exotic origins such as runaway, disk heating, and accretion from dwarf galaxies as discussed in the previous section, and summarize below and in Table \[tab6\].
We identify two typical disk stars (Pal14 IDs 7 and 17); Pal14 ID 7 is a typical thick disk star, while Pal ID 17 is a typical thin-disk star, as these stars have similar abundance patterns to stars of Galactic thick and thin disks, respectively, and have nearly circular orbits.
One star (Pal14 ID 4) may originate from the Galactic bulge, as its orbit passes close to the bulge with a high eccentricity. Moreover, the abundances of the $\alpha$-elements for this star all agree with those of other Galactic bulge stars. One may think that this star may be ejected from the bulge by dynamical ejection or SNe ejection mechanism. However, its $V_{\rm GRF}$ is too small to consider such an origin likely. Rather, its high-eccentricity orbit suggests an accreted origin from a disrupted dwarf galaxy, or dynamical heating.
Pal14 ID 10 may be a runway or heated disk star, as it reaches as high as 2.53 kpc from the Galactic plane during its orbit, and most of the chemical abundances within the derived uncertainties are similar to the thick-disk population.
Pal14 ID 15 appears to be a runaway from the Galactic disk, because it exhibits very large excursion ([$Z_{\rm max}$]{} = 1.67 kpc) from the Galactic plane with small eccentricity ($e$ = 0.06) in an orbit reaching beyond the Solar radius. The similar chemical abundance pattern to other metal-poor thin-disk stars support the idea as well.
Finally, Pal14 ID 16, which has the highest metallicity in our sample, exhibits similar chemical abundances to a very metal-rich thin-disk star, while its orbit exhibits a large excursion ([$Z_{\rm max}$]{} = 2.05 kpc) from the Galactic plane at a location beyond $R$ = 11 kpc. We consider this star as a runaway star from the Galactic disk.
Although all of our spectroscopically observed candidate HVSs turned out to be bound to the Galaxy, and not even particularly fast-moving objects, some of our program stars exhibit exotic orbits. Thus, future higher resolution spectroscopic follow-up observations for these curious stars may be of interest, and provide a better understanding of their origin.
We thank anonymous referees for their careful review of this manuscript, and for pointing out a number of places where we could improve the clarity of the presentation.
This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. This work has made use of data from the European Space Agency (ESA) mission [*Gaia*]{}[^5], processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC) [^6]. Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
Funding for SDSS-III was provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science. The SDSS-III Web site is <http://www.sdss3.org/>.
Y.S.L. acknowledges support from the National Research Foundation (NRF) of Korea grant funded by the Ministry of Science and ICT (No.2017R1A5A1070354 and NRF-2018R1A2B6003961). T.C.B. acknowledges partial support for this work from grant PHY 14-30152; Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-CEE), awarded by the US National Science Foundation.
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[^1]: <http://skyserver.sdss3.org/public/en/tools/chart/navi.aspx>
[^2]: <http://archive.stsci.edu/cgi-bin/dss_plate_finder>
[^3]: <http://www.ukidss.org/>
[^4]: IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.
[^5]: <https://www.cosmos.esa.int/gaia>
[^6]: <https://www.cosmos.esa.int/web/gaia/dpac/consortium>
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abstract: 'A review on the current state of mode physics in classical pulsators is presented. Two, currently in use, time-dependent convection models are compared and their applications on mode stability are discussed with particular emphasis on the location of the Delta Scuti instability strip.'
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Introduction
============
Stars with relatively low surface temperatures show distinctive envelope convection zones which affect mode stability. Among the first problems of this nature was the modelling of the red edge of the classical instability strip (IS) in the Hertzsprung-Russell (H-R) diagram. The first pulsation calculations of classical pulsators without any pulsation-convection modelling predicted red edges which were much too cool and which were at best only neutrally stable. What follows were several attempts to bring the theoretically predicted location of the red edge in better agreement with the observed location by using time-dependent convection models in the pulsation analyses (Dupree 1977; Baker & Gough 1979; Gonzi 1982; Stellingwerf 1984). More recently several authors, e.g. Bono et al. (1995, 1999), Houdek (1997, 2000), Xiong & Deng (2001, 2007), Dupret et al. (2005) were successful to model the red edge of the classical IS.
\[fig:1\]
These authors report, however, that different physical mechanisms are responsible for the return to stability. For example, Bono et al. (1995) and Dupret et al. (2005) report that it is mainly the convective heat flux, Xiong & Deng (2001) the turbulent viscosity, and Baker & Gough (1979) and Houdek (2000) predominantly the momentum flux (turbulent pressure $p_{\rm t}$) that stabilizes the pulsation modes at the red edge.
[ll]{} Balance between buoyancy & &Kinetic theory of accelerating\
turbulent drag (Unno 1967, 1977) &eddies (Gough 1965, 1977a)\
- acceleration terms of convective &- acceleration terms included: $w$,\
fluctuations $w, \Tp$ neglected&$\Tp$ evolve with growth rate $\sigma$\
- nonlinear terms approximated &- nonlinear terms are neglected\
by spatial gradients $\propto1/\ell$ &during eddy growth\
- $\di p^\prime$ neglected in momentum equ.&- $\di p^\prime$ included in Eq. (1)\
- characteristic eddy lifetime: &- $\tau=2/\sigma$ determined stochas-\
$\tau\simeq\ell/2w$ &tically from parametrized shear\
&instability\
- variation $\ell_1=\delta\ell/\ell$ (Unno 1967):&- variation of mixing length\
$\omega\tau<1:\ \ell_1\sim H_1$&according to rapid distortion\
$\omega\tau>1:\ \ell_1\sim r_1$&theory (Townsend 1976), i.e.\
or (Unno 1977): &variation also of eddy shape\
$\ell_1\sim (1+{\rm i}\omega^2\tau^2)^{-1}(H_1-{\rm i}\omega\tau\rho_1/3)$\
($H$ is pressure scale height) &\
- turbulent pressure $p_{\rm t}$ neglected &- $p_{\rm t}=\ob{\rho}$$\ob{ww}$ included in mean\
in hydrostatic support equation &equ. for hydrostatic support\
\[tab:tc\_comp\]
Time-dependent convection models
================================
The authors mentioned in the previous section used different implementations for modelling the interaction of the turbulent velocity field with the pulsation. In the past various time-dependent convection models were proposed, for example, by Schatzman (1956), Gough (1965, 1977a), Unno (1967, 1977), Xiong (1977, 1989), Stellingwerf (1982), Kuhfuß (1986), Canuto (1992), Gabriel (1996), Grigahcène et al. (2005). Here I shall briefly review and compare the basic concepts of two, currently in use, convection models. The first model is that by Gough (1977a,b), which has been used, for example, by Baker & Gough (1979), Balmforth (1992) and by Houdek (2000). The second model is that by Unno (1967, 1977), upon which the generalized models by Gabriel (1996) and Grigahcène et al. (2005) are based, with applications by Dupret et al. (2005).\
Nearly all of the time-dependent convection models assume the Boussinesq approximation to the equations of motion. The Boussinesq approximation relies on the fact that the height of the fluid layer is small compared with the density scale height. It is based on a careful scaling argument and an expansion in small parameters (Spiegel & Veronis 1960; Gough 1969). The fluctuating convection equations for an inviscid Boussinesq fluid in a static plane-parallel atmosphere are $$\begin{aligned}
\vspace{-3pt}
\dt\ui+(\uj\dj\ui-\ob{\uj\dj\ui})&=&-\ob{\rho}^{-1}\di\pp+g\widehat\alpha\Tp\kz\,,\\
\dt\Tp+(\uj\dj\Tp-\ob{\uj\dj\Tp})&=&\beta w-(\ob{\rho}\,\ob{c_{\rm p}})^{-1}\di F_i^\prime\,,
\vspace{-1pt}
\label{eq:bappox}\end{aligned}$$ supplemented by the continuity equation for an incompressible gas, $\dj\uj=0$, where $\bu=(u,v,w)$ is the turbulent velocity field, $\rho$ is density, $p$ is gas pressure, $g$ is the acceleration due to gravity, $T$ is temperature, $c_{\rm p}$ is the specific heat at constant pressure, $\widehat\alpha=-\ob{(\partial\ln\rho/\partial\ln T)_{p}}\,/\,\ob T$, $F_i$ is the radiative heat flux, $\beta$ is the superadiabatic temperature gradient and $\delta_{ij}$ is the Kronecker delta. Primes ($^\prime$) indicate Eulerian fluctuations and overbars horizontal averages. These are the starting equations for the two physical pictures describing the motion of an overturning convective eddy, illustrated in Fig. 1.
In the first physical picture, adopted by Unno (1967), the turbulent element, with a characteristic vertical length $\ell$, evolves out of some chaotic state and achieves steady motion very quickly. The fluid element maintains exact balance between buoyancy force and turbulent drag by continuous exchange of momentum with other elements and its surroundings. Thus the acceleration terms $\dt\ui$ and $\dt\Tp$ are neglected and the nonlinear advection terms provide dissipation (of kinetic energy) that balances the driving terms. The nonlinear advection terms are approximated by $\uj\dj\ui-\ob{\uj\dj\ui}\simeq2w^2/\ell$ and $\uj\dj\Tp-\ob{\uj\dj\Tp}\simeq2wT^\prime/\ell$. This leads to two nonlinear equations which need to be solved numerically together with the mean equations of the stellar structure.
The second physical picture, which was generalized by Gough (1965, 1977a,b) to the time-dependent case, interprets the turbulent flow by indirect analogy with kinetic gas theory. The motion is not steady and one imagines the convective element to accelerate from rest followed by an instantaneous breakup after the element’s lifetime. Thus the nonlinear advection terms are neglected in the convective fluctuation equations (1)-(2) but are taken to be responsible for the creation and destruction of the convective eddies (Gough 1977a,b). By retaining only the acceleration terms the equations become linear with analytical solutions $w\propto\exp(\sigma t)$ and $T^\prime\propto\exp(\sigma t)$ subject to proper periodic spatial boundary conditions, where $t$ is time and $\Re(\sigma)$ is the linear convective growth rate. The mixing length $\ell$ enters in the calculation of the eddy’s survival probability, which is proportional to the eddy’s internal shear (rms vorticity), for determining the convective heat and momentum fluxes. Although the two physical pictures give the same result in a static envelope, the results for the fluctuating turbulent fluxes in a pulsating star are very different (Gough 1977a). The main differences between Unno’s and Gough’s convection model are summarized in Table$\,$1.
Application on mode stability in $\delta$ Scuti stars
=====================================================
Fig. \[fig:2\] displays the mode stability of an evolving 1.7$\,$M$_\odot$ Delta Scuti star crossing the IS. The results were computed with the time-dependent, nonlocal convection model by Gough (1977a,b). As demonstrated in the right panel of Fig. \[fig:2\], the dominating damping term to the work integral $W$ for a star located near the red edge is the contribution from the turbulent pressure fluctuations $W_{\rm t}$. Gabriel (1996) and more recently Grigahcène et al. (2005) generalized Unno’s time-dependent convection model for stability computations of nonradial oscillation modes. They included in their mean thermal energy equation the viscous dissipation of turbulent kinetic energy, $\epsilon$, as an additional heat source. The dissipation of turbulent kinetic energy is introduced in the conservation equation for the turbulent kinetic energy $K:=\ob{\ui\ui}/2$ (e.g. Tennekes & Lumley 1972,$\,\mathsection$3.4; Canuto 1992; : $$\vspace{-3pt}
{\rm D}_tK+\dj(\ob{K\uj}+\ob{\rho}^{-1}\ob{p^\prime\uj})-\nu\partial^2_iK=
-\ob{\ui\uj}\dj U_i+g\widehat\alpha\ob{\uj\Tp}-\epsilon\,,
\label{eq:tke}$$ where ${\rm D}_t$ is the material derivative, $U_i$ is the average (oscillation) velocity, i.e. the total velocity $\tilde\ui=U_i+\ui$, and $\nu$ is the constant kinematic viscosity (in the limit of high Reynolds numbers the molecular transport term can be neglected). The first and second term on the right of Eq. (\[eq:tke\]) are the shear and buoyant productions of turbulent kinetic energy, whereas the last term $\epsilon=\nu\ob{(\dj\ui+\di\uj)^2}/2$ is the viscous dissipation of turbulent kinetic energy into heat. This term is also present in the mean thermal energy equation, but with opposite sign. The linearized perturbed mean thermal energy equation for a star pulsating radially with complex angular frequency $\omega=\omega_{\rm r}+{\rm i}\omega_{\rm i}$ can then be written, in the absence of nuclear reactions, as (‘$\delta$’ denotes a Lagrangian fluctuation and I omit overbars in the mean quantities): $${\rm d}\delta L/{\rm d}m=-{\rm i}\omega c_{\rm p}T({\delta T}/{T}-
\nabla_{\rm ad}{\delta p}/{p})+\delta\epsilon\,,
\label{eq:peq}$$ where $m$ is the radial mass co-ordinate, $\nabla_{\rm ad}=(\partial\ln T/\partial\ln p)_s$ and $L$ is the total (radiative and convective) luminosity. Grigahcène et al. (2005) evaluated $\epsilon$ from a turbulent kinetic energy equation which was derived without the assumption of the Boussinesq approximation. Furthermore it is not obvious whether the dominant buoyancy production term, $g\widehat\alpha\ob{\uj\Tp}$ (see Eq. \[eq:tke\]), was included in their turbulent kinetic energy equation and so Dupret et al. (2005) applied the convection model of Grigahcène et al. (2005) to Delta Scuti and $\gamma$ Doradus stars and reported well defined red edges. The results of their stability analysis for Delta Scuti stars are depicted in Fig. \[fig:3\]. The left panel compares the location of the red edge with results reported by Houdek (2000, see also Fig. \[fig:2\]) and Xiong & Deng (2001). The right panel of Fig. \[fig:3\] displays the individual contributions to the accumulated work integral $W$ for a star located near the red edge of the $n=3$ mode (indicated by the ‘star’ symbol in the left panel). It demonstrates the near cancellation effect between the contributions of the turbulent kinetic energy dissipation , $W_\epsilon$, and turbulent pressure, $W_{\rm t}$, making the contribution from the fluctuating convective heat flux, $W_{\rm c}$, the dominating damping term. The near cancellation effect between $W_\epsilon$ and $W_{\rm t}$ was demonstrated first by Ledoux & Walraven (1958, $\mathsection$65) (see also Gabriel 1996) by writing the sum of both work integrals as: $$\vspace{-2pt}
W_\epsilon+W_{\rm t}=3\pi/2\int_{m_{\rm b}}^M(5/3-\gamma_3)
\Im(\delta p_{\rm t}^*\delta\rho)\rho^{-2}\,{\rm d}m\,,
\vspace{-2pt}
\label{eq:wi}$$ where $M$ is the stellar mass, $m_{\rm b}$ is the enclosed mass at the bottom of the envelope and $\gamma_3\equiv1\!+\!(\partial\ln T/\partial\ln\rho)_s$ ($s$ is specific entropy) is the third adiabatic exponent. Except in ionization zones $\gamma_3\simeq 5/3$ and consequently $W_\epsilon+W_{\rm t}\simeq0$. The convection model by Xiong (1977, 1989) uses transport equations for the second-order moments of the convective fluctuations. In the transport equation for the turbulent kinetic energy Xiong adopts the approximation by Hinze (1975) for the turbulent dissipation rate, i.e. $\epsilon=2\chi k(\ob{\ui\ui}\rho^2/3\ob{\rho^2})^{3/2}$, where $\chi=0.45$ is the Heisenberg eddy coupling coefficient and $k\propto\ell^{-1}$ is the wavenumber of the energy-containing eddies. However, Xiong does not provide a work integral for $\epsilon$ (neither does Unno et al. 1989, $\mathsection$26,30) but includes the viscous damping effect of the small-scale turbulence in his model. The convection models considered here describe only the largest, most energy-containing eddies and ignore the dynamics of the small-scale eddies lying further down the turbulent cascade. Small-scale turbulence does, however, contribute directly to the turbulent fluxes and, under the assumption that they evolve isotropically, they generate an effective viscosity $\nu_{\rm t}$ which is felt by a particular pulsation mode as an additional damping effect. The turbulent viscosity can be estimated as (e.g. Gough 1977b; Unno$\,$et$\,$al.$\,$1989, $\mathsection$20) $\nu_{\rm t}\simeq\lambda(\ob{ww})^{1/2}\ell$, where $\lambda$ is a parameter of order unity. The associated work integral $W_\nu$ can be written in Cartesian co-ordinates as (Ledoux & Walraven 1958, $\mathsection$63) $$\vspace{-2pt}
W_\nu=-2\pi\;\omega_{\rm r}\int_{m_{\rm b}}^M\nu_{\rm t}
\left[e_{ij}e_{ij}-\frac{1}{3}\left(\nabla\cdot\bx\right)^2\right]\,{\rm d}m\,,
\label{eq:Wnu}$$ where $e_{ij}=(\dj\xi_i+\di\xi_j)/2$ and $\bx$ is the displacement eigenfunction. Xiong & Deng (2001, 2007) modelled successfully the IS of Delta Scuti and red giant stars and found the dominating damping effect to be the turbulent viscosity (Eq. \[eq:Wnu\]). This is illustrated in Fig. \[fig:4\] for two Delta Scuti stars: one is located inside the IS (left panel), the other outside the cool edge of the IS (right panel). The contribution from the small-scale turbulence was also the dominant damping effect in the stability calculations by Xiong et al. (2000) of radial p modes in the Sun, although the authors still found unstable modes with orders between $11\le n\le23$. The importance of the turbulent damping was reported first by Goldreich & Keeley (1977) and later by Goldreich & Kumar (1991), who found all solar modes to be stable only if turbulent damping was included in their stability computations. In contrast, Balmforth (1992), who adopted the convection model of Gough (1977a,b), found all solar p modes to be stable due mainly to the damping of the turbulent pressure perturbations, $W_{\rm t}$, and reported that viscous damping, $W_\nu$, is about one order of magnitude smaller than the contribution of $W_{\rm t}$. Turbulent viscosity (Eq. \[eq:Wnu\]) leads always to mode damping, where as the perturbation of the turbulent kinetic energy dissipation, $\delta\epsilon$ (see Eq. \[eq:peq\]), can contribute to both damping and driving of the pulsations (Gabriel 1996). The driving effect of $\delta\epsilon$ was shown by Dupret et al. (2005) for a $\gamma$ Doradus star.
Summary
=======
We discussed three different mode stability calculations of Delta Scuti stars which successfully reproduced the red edge of the IS. Each of these computations adopted a different time-dependent convection description. The results were discussed by comparing work integrals. All convection descriptions include, although in different ways, the perturbations of the turbulent fluxes. Gough (1977a), Xiong (1977, 1989), and Unno$\,$et$\,$al.$\,$(1989) did not include the contribution $W_\epsilon$ to the work integral because in the Boussinesq approximation (Spiegel & Veronis 1960) the viscous dissipation is neglected in the thermal energy equation. In practise, however, this term may be important. Grigahcène et al. (2005) included $W_\epsilon$ but ignored the damping contribution of the small-scale turbulence $W_\nu$, which was found by Xiong & Deng (2001, 2007) to be the dominating damping term. The small-scale damping effect was also ignored in the calculations by Houdek (2000). A more detailed comparison of the convection descriptions has not yet been made but Houdek & Dupret have begun to address this problem.
Discussion
==========
[**-Dalsgaard:**]{}
How does the mixing length affect the red edge of the $\gamma\;$Dor instability strip?
[**:**]{}
The location of the red edge is predominantly determined by radiative damping which gradually dominates over the driving effect of the so-called convective flux blocking mechanism (Dupret et al. 2005). A change in the mixing length will not only affect the depth of the envelope convection zone but also the characteristic time scale of the convection and consequently the stability of g modes with different pulsation periods. A calibration of the mixing length to match the observed location of the $\gamma\;$Dor instability strip will also calibrate the depth of the convection zone at a given surface temperature.
|
---
author:
- 'Wesley C. Fraser [1,2]{}, JJ. Kavelaars [2,1]{}'
bibliography:
- 'AstroElsart\_P32008.bib'
title: 'The Size Distribution of Kuiper belt objects for $D\gtrsim 10$ km'
---
\[1995/12/01\]
**Keywords:** Kuiper Belt
Introduction
============
The Kuiper belt is a population of planetesimals with diameters as large as a few thousand kilometers [@Brown2006] and a mass of approximately a few $0.01 M_\oplus$ [@Gladman2001; @Bernstein2004; @Fuentes2008]. The state of the belt is enigmatic, as such large objects are not likely to form in such a low mass Kuiper belt, over the age of the solar system [@Kenyon2001]. Rather, it is likely that a much more massive initial belt underwent accretion for a time before some large-scale mass-depletion event occurred, such as a stellar passage [@Ida2000; @Levison2004], a sweep through of the mean-motion resonances of Neptune [@Levison2008], or the scattering of KBOs by rogue planets [@Gladman2006]. Whatever the event, it is also responsible, at least in part, for the dynamically excited orbits of KBO populations such as the plutinos or the scattering population [@Gladman2008].
The size distribution of the Kuiper belt contains a fossil record of the end-state of the accretion processes that occurred in that region. Knowledge of the size distribution can constrain disruption strengths of the bodies, formation time-scales in the outer solar system, and the early conditions of the proto-planetary disk. This makes the determination of the size distribution a primary constraint on Kuiper belt formation scenarios.
Because of the large distance to the Kuiper belt, the size distribution is not determined directly, but rather, is inferred from the shape of the observed luminosity function (LF). Early observations determined that the LF for bright objects $(m(R)\lesssim 26)$ was well represented by a power-law with a logarithmic slope $\alpha\sim 0.7$ [@Jewitt1998; @Gladman1998; @Gladman2001; @Allen2002; @Petit2006; @Fraser2008]. This suggests that the size distribution of KBOs is a power-law with logarithmic slope $q\sim4.5$. The deepest survey of KBOs on the ecliptic, presented by @Bernstein2004, found a derth of faint objects demonstrating that the LF must “roll-over” or break to a shallower slope at $m(R)\lesssim27$ [@Bernstein2004]. Interpreting the roll-over in terms of the Kuiper belt size distribution implies that the large-object power-law size distribution breaks at a KBO diameter $D_b\sim 100$ km assuming 6% albedos. Models of accretion in the outer solar system predicted such a break at least an order of magnitude smaller [@Kenyon2001; @Kenyon2002].
The goal of our work was to confirm the results of @Bernstein2004, and accurately measure the shape of the LF at and beyond the roll-over magnitude. From this, the shape of the under-lying KBO size distribution can be inferred. We present here a survey of the ecliptic with limiting magnitude $m(R)\sim 27$, and use this along with previous observations to determine the shape of the LF. In section 2 we present our observations and in sections 3 we present our survey results. In sections 4 and 5 we consider past ecliptic Kuiper belt surveys and consider various LF functional forms. In sections 6 and 7 we present our analysis and a discussion of the results.
Observations, image processing, and characterization
====================================================
New observations were made in service mode with Suprime-Cam on the Subaru 8.2 m telescope [@Miyazaki2002]. Suprime-cam is a 10 chip mosaic CCD camera with a 34’$\times$27’ field-of-view (FOV) and a $0''.202$ pixel scale. Observations were made in the r-band on the nights of April 22nd (night 1), and May 8th (night 2) 2008. The same target field was observed each night, targeting the ecliptic with coordinates $\alpha=\mbox{13}^h\mbox{46}^m\mbox{57}^s\mbox{.7}$, $\delta=-\mbox{10}^o\mbox{44}^{'}\mbox{00}^{''}$. Additionally, each night, a photometric reference image was taken of the Canada-France-Hawaii Supernova Legacy Survey D2 field [@Astier2006]. Details of the observations are presented in Table \[tab:observations\]. Presented in Figure \[fig:seeing\] is a summary of the image quality of both nights; as can be seen, the nights were photometric.
Here we quickly describe our image processing techniques and subsequent moving object search for these data. A more thorough discussion of these techniques can be found in our previous work [@Fraser2008]. We describe in more detail here the additional image processing steps that were necessitated by the use of Suprime-cam.
All images went through the same image pre-processing before the image search step. From the overscan strip, the bias level was measured for each image, and subtracted. It was found that the bias level varied by $\sim100$ ADU (10% of the typical bias value) along the bias strip. Proper removal of this variation required that a fourth order single-dimensional polynomial be fit in a least-squares sense to the overscan region collapsed to a single column, and was done on an image-by-image basis. This polynomial bias-fit was removed from the raw images.
An average flat-field was produced for each chip from the sky-flats provided from the service mode observer (10 from night 1, 6 from night 2) using the *mask\_mkflat\_HA* routine from the sdfred software reductions package. This is the standard image pre-processing package provided and supported by Subaru [@Yagi2002; @Ouchi2004]. This routine detects sources in the sky-flat images. Regions near detected sources are ignored when the average flat is produced.
The Suprime-cam field is vignetted, with $\sim40\%$ throughput-loss near field corners. This vignetting needed to be accounted for when adding artificial sources to the data as the artificial sources must accurately reflect the sensitivity variations across each chip. Thus, the average flat-fields for each chip were separated into a quantum-efficiency variation map, and an illumination pattern. The illumination pattern was created by running a 20 by 20 pixel median filter over the average flat-fields. The quantum-efficiency variation map was set as the normalized difference of the flat-field and the illumination pattern, and was divided out of each of the over-scan subtracted images producing smooth images that revealed the vignetting pattern.
The Suprime-cam FOV is largely distorted such that a Kuiper belt object (KBO), with a rate of motion $\gtrsim 3$”/hr., whose motion would be linear on the sky, would not be linear in the images. Thus, spatial distortions needed to be removed. Image distortions were measured with the *dofit* routine [@Gwyn2008]. This routine compares a reference source list containing true astrometric positions of sources, and positions of those sources within the images; from this, it calculates a 3rd order spatial distortion map. The image of the D2 field used to photometrically calibrate our images also provided a perfect astrometric reference, as sources in the field have been astrometrically calibrated with residuals better than 0.3”. The distortion map was measured from the D2 field image of each night, and then used to resample all images onto a spatially flat image using the *SWarp* package [@Bertin2002]. Flux was conserved during this resampling, and produced images with a 0.1986“ pixel-scale. The spatial rms residuals of the resampling were 0.06”, or $\sim 1/3$ of a pixel as confirmed using the astrometric positions of stars in the USNO catalogs.
The point-source image shape varied significantly across each chip, and warranted the use of a spatially variable point-spread-function (PSF). It was found that a PSF whose shape varied linearly with image position could accurately re-produce the point-source image shape variations. For each chip, a gaussian PSF with look-up table was generated from 15 bright, unsaturated, hand-selected point-sources.
For each chip, a random number of artificial moving sources (between 150 and 250) were generated with random rates and angles of motion consistent with objects on circular Keplerian orbits between 25 and 200 AU, and with random fluxes consistent with that of point sources between 23 and 28.5 mag. These artificial sources were implanted blindly in the data, and the source list was revealed only after the search was complete. Additionally, 10 23rd mag. sources were implanted. These sources had flux sufficient to flag errors in the image combining algorithms.
Artificial sources with sky motions larger than 0.2 pixels were split up into a number of dimmer sources with total flux equal to the original source. The centres of the dimmer sources were shifted to account for the motion of the object in the images. The flux of each artificial source was varied from image to image to match the average brightness variations of 20 reference stars with respect to a reference image (that with the lowest airmass). Additionally, the fluxes of the artificial sources were scaled to account for the vignetting apparent in each chip. This was done by implanting the artificial source into a blank image. The blank image pixel values were then multiplied by the illumination pattern. This was subsequently summed with the vignetted sky images. After artificial object implanting, the illumination pattern was removed from the images. The result was images with spatially smooth backgrounds containing artificial sources with spatially varying sensitivity that matched that of the images.
No attempt was made to subtract a bias pattern from the raw images as no bias frames were provided from the service observations. The sdfred documentation however, claims that this step is unnecessary, as the bias level variations of Suprime-cam are low [@Yagi2002; @Ouchi2004].
To maximize the searchable area of the images, stationary objects were removed by subtracting an image template from the vignetting corrected images. The image template for each chip was created from an average of the search field images using an artificial skepticism routine which places little weight on pixels far away from the pixel average [@Stetson1989]. Every fifth image was used to create the image template, reducing the presence of subtraction residuals created behind all moving sources, while ensuring a high quality template such that all stationary sources were sufficiently removed. The image subtraction was done using *psfmatch3* routine [@Pritchet2005].
Before final image stacks were produced, a high pixel mask was applied to the subtracted images to mask out any spurious hot-pixels and cosmic-ray spikes. The masked-subtracted images were spatially shifted to account for the motions of moving sources. A grid of shift rates ($0.4-4.5$ "/hr.) and angles ($\pm 15^o$ off the ecliptic) were considered which covered the range of motions consistent with bound solar system objects in the Kuiper belt on prograde orbits. In our past searches, we found that the search depth is not very sensitive to the choice in grid spacing between angles, but is very sensitive to the choice in spacing between rates. We chose a rate spacing small enough such that a moving source would never exhibit a trail longer than twice its seeing disk at the rate and angle which best approximated its apparent motion. Our grid consisted of 19 separate rates and 5 separate angles.
The data from each night were photometrically calibrated to the SDSS wavebands by comparing source flux measurements in the D2 field to those quoted in the Mega-pipe project [@Gwyn2008]. A source’s magnitude in the Subaru data is given by $r_{Sub} = -2.5 \log \left(\frac{b}{t}\right) +Z$ where $b$ is the source brightness in ADU, $t$ is the exposure time, and $Z$ is the telescope zeropoint. For each chip, a linear relation of the form $r_{Sub} = C(g'-r')_{Mega} - r'_{Mega}$ was used to transformation the r’-band magnitudes in the Mega-pipe project to Subaru r-band magnitudes. The colour term, $C$, and the zeropoint $Z$ were fit in a least-squares sense to the source-flux measurements in the data, and the Mega-pipe magnitudes using at least 40 sources on each chip. This procedure determined the best fit zeropoint to an accuracy of $\sim0.04$ mag. The results are presented in Table \[tab:calibrations\]. We found that chip 00 exhibits a $~0.4$ magnitude decrease in sensitivity compared to the other 9 chips, which is consistent with other photometrically calibrated Suprime-Cam observations [@Yoshida2007].
The transformations of the r’ and g’-band Mega-pipe magnitudes to the r’, and g’-bands in the SDSS filter set are given by $r'_{Mega}=r'_{SDSS} - 0.024 (g'-r')_{SDSS}$ and $g'_{Mega}=g'_{SDSS}-0.153 (g'-r')_{SDSS}$. Using the colour term $C_{av}=-0.048$ which is the average of colour terms from Table \[tab:calibrations\], we find that the transformation between r-band Subaru and SDSS magnitudes is given by $r'_{SDSS} = r_{Sub} +0.018 (g'-r')_{SDSS}$. For typical KBOs $(g'-r')_{SDSS} \sim 0.7$ mag [@Fraser2008]. Thus, the r-band Subaru and SDSS magnitudes differ by $\sim 0.01$ mag. which is smaller than the uncertainty of the telescope zeropoint. Hence, we use the approximation that for KBOs, $r'_{SDSS}=r_{Sub}$.
Survey and results
==================
From the first night, the grid of image stacks for each chip was manually searched by one operator producing a candidate source list. Moving sources were identified by their appearance in the images; a moving source is round (or nearly) in the image stack that best compensated for the source’s apparent sky motion, and exhibits a characteristic trail in other stacks, helping the operator distinguish moving sources from noise. See @Fraser2008 for a more complete discussion of this technique.
Implanted artificial sources were identified in the candidate source list if the candidate source’s location was within a few pixels of the artificial source’s true centre. The detected artificial sources allowed us to measure our detection efficiency as a function of magnitude, $\eta(r'_{SDSS})$. We represent the detection efficiency of the full field by the functional form
$$\eta(r'_{SDSS})=\frac{\eta_{max}}{2}\left(1-\tanh\frac{r'_{SDSS}-r_{*}}{g}\right)
\label{eq:eff}$$
where $\eta_{max}$ is the maximum efficiency, $r_*$ is the half-maximum detection efficiency magnitude, and $g$ is approximately half the width of the fall from maximum detection efficiency to zero. This function was fit, in a least-squares sense, to the detection efficiency of artificial sources. The best fit parameters were $(\eta_{max},r_{*},g) = (0.932\pm0.008, 26.86\pm0.02, 0.44\pm0.03)$. The best-fit curve represents the measured detection efficiency well (see Figure \[fig:eff\]).
We fit the measured efficiency with $\eta'(r'_{SDSS})=\frac{\eta_{max}}{2}\left(1-\tanh\frac{r'_{SDSS}-r_{*}}{g}\right)\left(1-\tanh\frac{r'_{SDSS}-r_{*}}{g'}\right)$ which has been found to represent the detection efficiency of previous KBO surveys [@Petit2006; @Fuentes2008]. The best-fit curve is presented in Figure \[fig:eff\]. As can be see, there is very little difference between the more complicated efficiency representation and that of Equation \[eq:eff\], and both describe the measurements equally well. Thus, we find that the more complicated function is not warranted or necessary.
Flux measurements were made for all identified moving sources on the shifted image stack which contained the roundest image for that source. Magnitudes were measured in a 3-4.5 pixel radius aperture, and corresponding aperture corrections were determined from the image profiles of ten 23rd mag. artificial sources on each chip. By comparing the flux measurements of the detected artificial sources to their true values, we were able to characterize our flux measurement uncertainties. For background limited sources, the uncertainty in a source’s measured magnitude, $r$, is given by $\Delta r =\gamma 10^\frac{r-Z}{2.5}$ where $Z$ is the telescope zeropoint, and $\gamma$ depends on the telescope and the observations [@Fraser2008]. We found that $\gamma = 1.33$ best described the observed uncertainties.
38 sources were not associated with artificial sources. 36 of these are identified as newly discovered KBOs. Two of these were identified as coincidental overlap of multiple poorly subtracted saturated galaxies. Both of these false detections have fluxes below the 50% detection efficiency. This is consistent with our previous survey - a non-zero false candidate rate occurred for detection efficiencies $\eta<50\%$ [@Fraser2008]. We therefore truncate our detection efficiency at the 50% threshold and ignore the 8 sources faint-ward of this level.
From the positions of the objects on the first night, orbits were calculated using *fit\_radec* [@Bernstein2000] and positions of the objects on night 2 were predicted. While not all of our objects had follow-up detections, we had a 100% follow-up rate for those 7 objects whose predicted positions fell in the FOV of the second night’s observations. This is consistent with the 100% follow-up rate above the 50% detection threshold we report from our past survey [@Fraser2008]. We are thus confident that all detected non-artificial sources with magnitudes above the 50% threshold are real KBOs.
For each detected KBO, three image stacks were created (6 for those with follow-up) of the first half, middle half, and last half of the vignetting removed, non-subtracted images from a night’s observations, using the rate and angle of motion that produced the roundest source image. From these, flux measurements were made. The final source magnitudes we report are the average of all the flux measurements made for that source off the image stacks where the source was at least a few FWHM away from nearby bright stars and galaxies. The results of our survey are presented in Table \[tab:objects\].
Luminosity function
===================
Previous surveys have found that the differential LF (number of KBOs with magnitude $m$ to $m+dm$ per square degree) of bright KBOs ($m(R)\lesssim 26$) mag. is well represented by
$$\Sigma(m)=\ln(10) \alpha 10^{\alpha(m-m_o)}
\label{eq:powerlaw}$$
where $\Sigma(m)$ is the number of objects with magnitude $m$ (usually R-band) per square degree, $\alpha$ is the power-law “slope”, and $m_o$ is the magnitude for which the sky density of objects with magnitudes $\leq m$ is one per square degree (see for example @Trujillo2001b [@Gladman2001; @Petit2006]). In our previous work, we found $\alpha=0.65$ and $m_o(R)=23.43$ for a broad range of surveys [@Fraser2008]. @Bernstein2004 presented the faintest Kuiper belt survey to date. Using a substantial amount of Hubble observations ($\sim 100$ orbits) they achieved a search depth of $m(R)\sim 28.5$ [@Bernstein2005] and discovered three KBOs. The dearth of detections was at least a factor of $\sim10$ below the number expected, requiring that the LF has a steep logarithmic slope for bright objects, that “rolls-over” to shallower slopes for fainter objects.
@Fraser2008 presented a survey in which they searched 3 square degrees of the ecliptic to a depth of $m(R)\sim25.4$ mag. and substantially increased the number of known detections suitable for a determination of the LF faint-ward of $m(R)\sim 23.5$. They found that the LF was well represented by a power-law given by Equation \[eq:powerlaw\]. They concluded that the lack of objects observed by @Bernstein2004 must be caused by a sharp break rather than a broad roll-over, and that this could not occur for $m(R)\leq 24.4$.
Additionally, @Bernstein2004 examined the LFs of the so-called cold and excited KBO populations, $i < 5^o$ and $i>5^o$ respectively. They found that the cold population exhibited a steeper bright object slope than the excited population. The statistical support for such a difference however, was not strong. Simulations that test different Kuiper belt formation scenarios suggest that a difference in the SDs of different KBO populations is caused by different formation conditions/histories for those objects. Thus, a confirmation of the findings by @Bernstein2004 is important, as it provides a very strong constraint on the Kuiper belt formation history.
The survey reported here detected 28 KBOs that can be used to characterize the luminosity function (LF) and SD of KBOs. With these detections, we wish to address three questions:
1. At what magnitude is the large object LF no longer a power-law?
2. Is the observed break consistent with the LF expected if the SD has a sharp break at some object radius? If not, what is the true shape of the LF?
3. Is the SD of the excited population different from that of the cold population?
Luminosity Function forms
-------------------------
In determining the correct LF shape, we consider three functional representations of the differential LF. The first is the power-law given by Equation \[eq:powerlaw\]. The second is the rolling power-law suggested by @Bernstein2004, given by
$$\Sigma(m) = \Sigma_{23} 10^{\alpha(m-23)+\alpha'(m-23)^2}
\label{eq:rolling}$$
where $\alpha$ is the bright object slope, $\alpha'$ is the derivative of the logarithmic slope, and $\Sigma_{23}$ is the number of objects per square degree at 23rd mag.
The third functional form we consider was first presented by @Fraser2008b. They showed that, if the SD is a power-law with slope $q_1$ for large objects that has a sharp break at objects with diameter $D_b$ to slope $q_2$ for small objects, then the *cumulative* LF has the form
$$N\left(<m\right) = A
\begin{cases}
a_1 10^{\alpha_1 m} & \mbox{if $m < m_b$}\\
b_110^{\alpha_2 m} + b_2 10^{\alpha_1 m} + b_3\left(m\right) & \mbox{ if $m_b \leq m \leq m_2$}\\
c_1 +c_2 10^{\alpha_2 m} & \mbox{ if $m > m_2$}.
\end{cases}
\label{eq:Fraser2008}$$
where $\Sigma(m) = \frac{dN(<m)}{dm}$, $\alpha_1=\frac{q_1-1}{5}$ and $\alpha_2=\frac{q_2-1}{5}$. In Equation \[eq:Fraser2008\] $m_b=K+5\log \left(r_1^2 D_{b}^{-1}\right)$ (break magnitude) and $m_2=K+5\log \left(r_2^2 D_{b}^{-1}\right)$ are the magnitudes for which objects smaller than the break diameter are detectable by a given survey for heliocentric distances larger than the Kuiper belt inner edge $r_1$ but smaller than the outer edge $r_2$.
The parameter $K$ is a constant relating the diameter of an object to its apparent magnitude. Choosing a value of $K$ is equivalent to choosing a constant albedo for KBOs. It is apparent that the largest objects have albedos as high as $\sim90$ %, with smaller objects exhibiting much lower values [@Stansberry2007]. Clearly the assumption that albedos are constant is incorrect, at least when considering the largest KBOs. Insufficient knowledge of the albedos for smaller $D\lesssim 500$ km KBOs - the size range which we probe in this work - exists for this assumption to be tested. @Fraser2008 discuss the effects of a varying albedo when a constant is assumed, and demonstrate that the inferred size distribution slope is incorrect. This effect however, is small compared to the uncertainties typical in the LF determination, given the available data.
It is apparent, at least for the observed sources, that an albedo of 6% is a typical value for $D\lesssim 500$ km KBOs [@Stansberry2007], and is the value we assume here. This sets $K\sim18.4$ as the R-band magnitude of a $D\sim1600$ km object at 40 AU with a 6% albedo. This apparently typical albedo might very well be an observational bias. Until more data are available, it will not be known whether 6%, or a much larger value similar to large objects, is a more appropriate choice. Thus, any conclusions about the inferred break size come with this caveat; the real break diameter could be quite different dependening on the true albedos.
This functional form has the advantage that the LF can be interpreted in terms of LF parameters $(\alpha_1,\alpha_2,m_b,\log A)$ or SD parameters $(q_1,q_2,D_b,\log A)$. Both provide equivalent representations of the LF shape, and, assuming an albedo, the SD is directly inferred from the best-fit LF.
The coefficients $a_i$, $b_i$, and $c_i$ of Equation \[eq:Fraser2008\] are functions of the KBO radial distribution. @Fraser2008b have shown that the LF shape for $m_b\leq m \leq m_2$ is strongly dependent on the assumed KBO radial distribution, but the SD slopes $q_1$ and $q_2$, and the break diameter $D_b$ however, can be accurately inferred if data exists sufficiently far away from the break magnitude. They assume that the radial distribution is given by $N(\Delta) \propto \Delta^{-c}$, and suggest that $c=10$ is a good approximation to the radial fall-off observed by @Trujillo2001. We assume the same radial distribution here.
These three functions provide a good sample of LF forms to consider as they progress in levels of complication by a single degree of freedom - the parameters for the different LF forms are $(\alpha,m_o)$ for Equation \[eq:powerlaw\], $(\alpha,\alpha',\Sigma_{23})$ for Equation \[eq:rolling\], and $(\alpha_1,\alpha_2,m_b,A)$ for Equation \[eq:Fraser2008\]. Thus, it is a straight-forward process to determine the number of parameters necessary to describe the true shape of the LF using simple statistics techniques.
Data-sets
---------
In our previous estimates of the LF, we considered observations from surveys for which the detection efficiency was well characterized as a function of magnitude. The F08 sample @Fraser2008 consists of near ecliptic KBO surveys presented in @Jewitt1998 [@Gladman1998; @Gladman2001; @Allen2002; @Petit2006], and @Fraser2008 along with the data from @Trujillo2001b which we subdivided by detection efficiency and astrometric position. The F08b sample is the F08 sample, plus the inclusion of the survey presented by @Bernstein2004 [@Bernstein2005] and [*the survey presented here*]{}.
To measure any differences in the LF between the dynamically excited and cold populations, we define the F08b$_{i<5}$ and F08b$_{i>5}$ samples as those objects with inclinations smaller than and greater than $5^o$ respectively. For these samples, we only consider surveys in which the majority of objects have been observed on arcs longer than 24 hrs, inclinations measured from the observations spanning only one night are not accurate enough to make an inclination cut reliable, and a large contamination between the two samples would occur. Thus we exclude from the F08b$_{i<5}$ and F08b$_{i>5}$ subsamples the Uranus field from @Petit2006, the N11033, N10032W3, UNE, and UNW fields from @Fraser2008, and the survey presented here.
While an inclination cut does not directly probe the different KBO populations, ie. cold/hot-classical objects, resonators, scattered members, etc. [@Gladman2008], it does provide a crude means of separating those KBOs with ‘highly’ excited orbits from those with less excited orbits. The division of $i=5^o$ is chosen to ease comparison with the LF presented by @Bernstein2004. For the F08b$_{i<5}$ and F08b$_{i>5}$ samples, no heliocentric distance cut was made.
Note: the analysis for this manuscript was already complete before the authors were aware of the survey presented by @Fuentes2008. The moderate increase in the total number of detections, $\sim 20\%$, will not substantially alter the results presented here. Thus, we did not include the survey of @Fuentes2008 in our analysis. Their results however, are discussed below.
Analysis \[sec:analysis\]
-------------------------
In determining the correct LF shape and best-fit parameters of the various functional forms, we adopt the same practices as @Fraser2008 [@Fraser2008b]. These are as follows:
1. We cull from each survey all sources faintward of the 50% detection efficiency of that survey, and set the detection efficiency to zero faintward of that point.
2. We offset all magnitudes to R-band using the average KBO colours presented in @Fraser2008.
3. We fit the differential LF, $\Sigma(m)=\frac{dN(<m)}{dm}$ using a maximum likelihood technique.
4. We adopted the same form of the likelihood equation as presented in @Fraser2008 which treats calibration errors and variable sky densities as separate nuisance parameters. We marginalized the likelihood over these parameters before determining the maximum likelihood.
5. We adopted the same prior ranges of the colour and density parameters as presented in @Fraser2008 ($\pm 0.2$ magnitudes for the colour parameters, and $\pm 0.4$ for the logarithm of the density parameters).
In determining the quality of the fits, we utilize the Anderson-Darling statistic
$$\Delta=\int_0^1 \frac{\left(S(m)-P(m)\right)^2}{P(m)\left(1-P(m)\right)}dP(m)
\label{eq:AD}$$
where $P(m)$ is the cumulative probability of detecting an object with magnitude $\leq m$ and $S(m)$ is the cumulative distribution of detections. We calculate the probability, $P(\Delta>\Delta_{obs})$ of finding a value of the Anderson-Darling statistic, $\Delta$, larger than that of the observations for a given LF function and parameter set by bootstrapping the statistic, is. randomly drawing a subsample of objects from that LF equal to that detected in a particular sample, and fitting the LF to that random sample and computing $\Delta$. In the random sampling process, we include random sky-density and colour offsets which represent those expected from the real observations. Values of $P(\Delta>\Delta_{obs})$ near 0 indicate that the functional form is a poor representation of the data.
To determine if the more complicated LF functional forms are statistically warranted, we utilize the log-likelihood test $\chi^2=-2\log\frac{L'}{L}$. The logarithm of the ratio of maximum likelihoods of the simple and more complicated functions, $L$ and $L'$, are distributed as a chi-squared variable with a number of degrees of freedom equal to the difference in free parameters between the two functional forms. A table of $\chi^2$ values can be used to determine the significance of the improvement in the LF fit between the two functional forms. [@Kotz1983].
Results \[sec:Results\]
=======================
F08b Sample
-----------
Presented in Table \[tab:fits\] are the best-fit parameters for the three different LF forms, and $P(\Delta>\Delta_{obs})$ of each fit. The best-fit power-law slope to the F08b data-set, $\alpha=0.58$, is moderately shallower than, but is consistent with the best-fit $\alpha=0.65$ from @Fraser2008. The fit however, is a poor description of the data as evidenced by the low $P(\Delta>\Delta_{obs})<0.02$.
The best-fit of the rolling power-law to the F08b data-set is $(\alpha,\alpha',\Sigma_{23})=(0.8,-0.06,0.82)$. The maximum likelihood value of this fit is increased by more than two orders of magnitude over the power-law which warrants the inclusion of a third degree of freedom, and the fit is an acceptable description of the F08b sample, with $P(\Delta>\Delta_{obs})= 0.4$.
The best-fit of the LF given by Equation \[eq:Fraser2008\], $(\log A,\alpha_1,\alpha_2,m_b) = (23.56,0.76,0.18,24.9)$, is an acceptable fit to the data with $P(\Delta>\Delta_{obs}) = 0.4$. The maximum likelihood value is increased by more than two orders of magnitude over the power-law with a log-likelihood chi-square, $\chi^2 = 11.7$, and by a factor of 2 over the rolling power-law with $\chi^2=1.4$. We find that the best-fit from Equation \[eq:Fraser2008\] is preferred over the power-law at greater than the 3-sigma level. This fit is preferred over the rolling power-law at $\sim 80\%$ significance. Thus we find that both the rolling power-law and broken power-law of Equation \[eq:Fraser2008\] provide equally adequate descriptions of the F08b sample. Note: the Monte-Carlo simulations we have done to calculate $P(\Delta>\Delta_{obs})$ have also demonstrated that on average, the best-fit LF parameters determined from our maximum likelihood technique reproduce the input parameters of the simulations for all LF functionals forms we have considered here.
When the observations of @Bernstein2004 are excluded from the F08b sample, the best-fit power-law has parameters $(\alpha,m_o)=(0.65,23.42)$, which is nearly identical to that found from by @Fraser2008. The power-law is a moderately adequate description of the observations with $P(\Delta>\Delta_{obs}) =0.1$. This result hints that the Kuiper belt LF exhibits a break within the magnitude range of the F08b sample, but fits to the survey data with $m(R)\lesssim27$ do not explicitly require a broken or rolling power-law description.
Presented in Figures \[fig:a1ap\], \[fig:a1N\], and \[fig:a2Db\] are the credible regions of the fits to the F08b sample. The function given by Equation \[eq:Fraser2008\] is derived from a size-distribution similar to that expected from accretion calculations, and thus provides a proper interpretation of the under-lying size-distribution. We find that the LF is consistent with a broken power-law size-distribution with large and small object slopes $q_1=4.8\pm3$ and $q_2=2\pm2$, and a break diameter, $D_b=62\pm 40$ km assuming 6% albedos.
Presented in Figure \[fig:diff\] is a histogram of the F08b sample, the best-fit LFs for the three functional forms we consider, the best-fit power-law LF from @Fraser2008, and the best-fit LF presented by @Fuentes2008. As can be seen, power-laws do not provide acceptable fits to the data with $m(R)\sim28$, and a more complicated function is required. Equations \[eq:rolling\] and \[eq:Fraser2008\] both provide reasonable descriptions of the observations.
The user is cautioned from drawing any further conclusions from Figure \[fig:diff\]. The F08b sample contains data from many different surveys. Thus the sample includes any calibration errors and sky-density variations between surveys which might lead the reader to a false conclusion about the structure of the LF (see @Fraser2008 for a discussion of the magnitude and significance of these effects).
The best-fit LF presented by @Fuentes2008 is a harmonic mean of two power-laws with bright-end slope $\alpha_1=0.7$ and faint-end slope $\alpha_2=0.3$ with a break magnitude of $R_{eq}\sim 24.3$. These parameters are consistent with the slopes and break magnitude of our best-fit non-power-law LFs. They find however, a factor of 3 higher sky-density which is inconsistent at greater than the 3-sigma level with the range of sky densities we find for similar bright-end slopes (see Figure \[fig:a1N\]). The source of the increased number of detections compared to that expected from fits to the F08b sample is unknown.
F08b$_{i>5}$ Sample \[sec:highi\_fit\]
--------------------------------------
When we consider the F08b$_{i>5}$ sample, we find that the best-fit power-law has slope $\alpha=0.55$ with $P(\Delta>\Delta_{obs})<0.04$, and is a poor description of the data. The sample has no objects fainter than $m(R)\sim 25.8$. The lack of faint objects requires a break near this point. The best-fit rolling power-law with parameters $(\alpha,\alpha',\Sigma_{23}) = (0.74,-0.06,0.46)$ is an acceptable fit to the data with $P(\Delta>\Delta_{obs})=0.1$. The best-fit broken power-law model is an acceptable description of the data with $P(\Delta>\Delta_{OBS})=0.1$. The log-likelihood ratio implies that the broken power-law model of Equation \[eq:Fraser2008\] is preferred over the rolling power-law at the 90% significance level. The best-fit has $\alpha_1=0.7$ and breaks at $D_b=36$ km. Because the F08b$_{i>5}$ sample has no detections faint-ward of the break, both the break diameter and small-object slope of this sample are poorly constrained. The maximum likelihood is found with a break at $m(R)=25.8$ with no objects faint-ward of this point. ie. $\alpha_2=-\infty$. Note: we consider slopes, $\alpha_2>-6$ to avoid numerical errors in the likelihood calculations.
Presented in Figures \[fig:a1N\_highi\], \[fig:a2Db\_highi\], and \[fig:diff\_highi\] are the likelihood contours of the best-fit of Equation \[eq:Fraser2008\] to the F08b$_{i>5}$ sample and the histogram of those data. As can be seen, the LF is well described by a power-law for $m(R)\lesssim 25.8$. The lack of faint objects requires that a break must exist at $D_b\lesssim 100$ km.
The faint-object slope is highly uncertain, with a 1-sigma upper-limit of $\alpha_2\sim0.6$. This fit however is consistent with that of the F08b sample best-fit at the $\sim1$-sigma level. Though the inclinations of the detections we present in this survey are inaccurate, they hint that the break in the LF for the F08b$_{i>5}$ sample is not as sharp as the fit would suggest. The detections from this survey are consistent with the same break diameter (36 km) that breaks to a much flatter faint object slope similar to the 1-sigma upper-limit of the fit $(\alpha_2\sim0)$. The fit to the F08b$_{i>5}$ sample however, constrains the break to be brightward of $m(R)\lesssim 26.8$. Additional data brightward of $m(R)\sim27.5$ with accurate inclinations is required before the break location can be accurately constrained.
F08b$_{i<5}$ Sample
-------------------
The low-inclination LF is sufficiently described by a power-law. The best-fit power-law is an acceptable description of the sample, with parameters $(\alpha,m_o)=(0.59,24.0)$ and an Anderson-Darling statistic, $P(\Delta>\Delta_{obs})=0.15$. Indeed the best-fit of Equation \[eq:Fraser2008\] is found when the break occurs at the bright-end limit of the data, suggesting there is no strong evidence for a break in the F08b$_{i<5}$ data.
Presented in Figures \[fig:amo\_lowi\] and \[fig:diff\_lowi\] are the likelihood contours of the fit to the low-inclination LF, and the histogram of the observations. As can be seen, there is no apparent evidence for a break in the magnitude range of the observations.
To test whether or not the excited and cold samples have different LFs, we utilized the Anderson-Darling test described above (see Section \[sec:analysis\]) to determine whether or not the broken LF fit of the F08b$_{i>5}$ sample is an adequate description of the F08b$_{i<5}$ sample. We however, set $\alpha_2=0$ rather than the best-fit value, as this is a more realistic faint object slope than that of the best-fit of the F08b$_{i>5}$ sample $(\alpha_2\sim-\infty)$, as suggested by our survey. This slope is still within the 1-sigma credible region of the fit (see Section \[sec:highi\_fit\]). We found that for the F08b$_{i<5}$ sample, an LF with $(\alpha_1,\alpha_2,m_b,\log A) =(0.7,0,26.2,22.7)$ has $P(\Delta>\Delta_{obs})=0.4$, and is a sufficient description of the cold sample LF. We therefore conclude that we have no evidence that those KBOs with $i>5$ have a different LF than those with $i<5$.
@Bernstein2004 found that the LFs of the cold and excited populations were different. @Fuentes2008 found similar results for their survey alone, and concluded that the low inclination population LF exhibits a bright-end slope $\alpha\sim0.8-1.5$, steeper than $\alpha_1\sim0.7$ found when they considered objects of all $i$ in their survey. When they considered all available data - the F08b sample, but not including the survey presented here, and including their own search - they found that the low-inclination group exhibited a significantly steeper bright-object slope than the high-inclination sample. This finding is intriguing, as we find no evidence for a difference in the SDs of the low and high inclination groups. It is possible that their results are a consequence of not considering calibration and density offsets when performing the likelihood calculations, as we have done here. Clearly however, more observations are needed which probe the entire range of current observations before these results can be clarified.
Discussion
==========
The best-fit parameters for the LF presented in Equation \[eq:Fraser2008\] imply that the size distribution is a power-law with slope $q_1\sim 4.8$ for large objects, which breaks to a shallower slope at diameter $D_b\sim 50-95$ km assuming 6% albedos. Comparison of this size distribution to models which evolve a population of planetesimals and track their size distribution as a function of time in the Kuiper belt region can place a constraint on the formation processes and the duration of accretion in that region [@Kenyon2001; @Kenyon2002; @Kenyon2004].
In the early stages of formation, run-away growth occurs, and very rapidly grow objects as large as $\sim 10^3$ km in the Kuiper belt [@Kenyon2002]. During this process, a steep-sloped large object size distribution develops, which flattens with time as more objects become “large”. Calculations presented in @Kenyon2002, which simulate planet accretion for conditions in the Kuiper belt expected for the early solar system, imply that, in the absence of influences from Neptune, the modern-day large object slope would be shallower than that observed if accretion lasted the age of the solar system. They find that for an initial Kuiper belt mass similar to that predicted from the minimum-mass solar nebula model [@Hayashi1981] - much more massive than the current belt [@Fuentes2008] - accretion must have been halted after $\sim 100$ Myr. If KBOs are weak (easier to disrupt), then accretion might have gone on for as long as 1 Gyr [see Figures 9, and 10 from @Kenyon2002]. Some event(s) must have halted this process by clearing the majority of initial mass out of the belt before the slope became too shallow.
@Kenyon2004 has demonstrated that weaker bodies will exhibit a size distribution with a larger break diameter $D_b$ than stronger bodies would if they underwent the same evolutionary history. They calculated the size distribution expected from a Kuiper belt under the gravitational influence of Neptune which evolved for the age of the solar system. In this model, break diameters as large as 60 km were produced only for the weakest bodies they considered. This model however, produced a large object slope much too shallow to be consistent with the observations, implying that accretion over the age of the solar system cannot have occurred.
The existence of such a large break diameter in the KBO SD implies that, KBOs must be quite weak (strengthless rubble piles), and have undergone a period of increased collisional evolution. This suggests a scenario in which the event responsible for the early end to accretion, and the clearing of the majority of the mass in the Kuiper belt, also increased the rate of collisional evolution for a time, pushing the break diameter to the large value we see today.
Collisional disruption would be substantially increased for a period of time during the scattering of planetesimals by a rogue planet [@Gladman2006], or during a close stellar passage [@Ida2000; @Levison2004]. During these scenarios, relative velocities of KBOs are increased, causing collisions between bodies to result in catastrophic disruption, rather than accretion as would otherwise be the case. The combination of scattering and collisional grinding would produce a substantial mass loss in the Kuiper belt region, and produce the large knee observed currently. Increased bombardment would also occur if Neptune migrated outwards. In such a scenario, Neptune formed closer to the Sun than its current location. Gravitational scattering of small planetesimals transfer angular momentum to Neptune, causing the gas-giant to migrate outwards. During this process, some planetesimals have their orbits excited, and are thrust into the modern day Kuiper belt creating the orbital distribution we see today. The remaining planetesimals are cleared from the system during close encounters with Neptune causing a rapid and substantial mass-loss during the migration process. Various incarnations of this attractive scenario can provide the necessary collisional bombardment of KBOs, and simultaneously account for some of the other observed features of the Kuiper belt [@Malhotra1993; @Levison2003; @Hahn2005; @Levison2008].
The Neptune migration scenario predicts that the more excited population will have a more evolved size distribution, with a shallower large-object slope - objects which originated closer to the Sun have stronger encounters with Neptune, and thus have the most excited final orbits. These now excited objects would have originated from a more dense region, and hence underwent more rapid accretionary evolution, than objects which originated further from the Sun producing the shallower large object slope. We do not see any differences in the large object slopes of the cold and excited populations. Our findings suggest that, assuming migration of Neptune occurred, the total distance travelled by Neptune was sufficiently small such that the rate of planet formation and hence the size distribution of all objects scattered by Neptune and implaced in the Kuiper belt could not be substantially different.
Conclusions
===========
We have performed a survey of the Kuiper covering $\sim 1/3$ a square degree of the sky using Suprime-cam on the Subaru telescope, to a limiting magnitude of $m_{50}(R)\sim 26.8$ and have found 36 new KBOs. Using the likelihood technique of @Fraser2008 which accounts for calibration errors and sky density variations between separate observations, we have combined the observations of this survey with previous observations, and have found that the luminosity function of the Kuiper belt is inconsistent with a power-law with slope $\alpha_1=0.75$, but must break to a shallower slope $\alpha_2\sim0.2$ at magnitudes $m(R)\sim 24.1-25.3$. The luminosity function is consistent with a size distribution with large object slope $q_1\sim4.8$ that breaks to a shallower slope $q_2\sim1.9$ at a diameter of $\sim 50-95$ km assuming 6% albedos. We have found no conclusive evidence that the size distribution of KBOs with $i<5$ is different than that of those with $i>5$.
Acknowledgements
================
This project was funded by the National Science and Engineering Research Council and the National Research Council of Canada. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. This article is based in part on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
[lccccc]{} Night & Date & Seeing (”) & Number of Exposures & $\alpha$ & $\delta$\
1 & April 22 2007 & $\sim0.4-0.7$ & 55 $\times$ 200 s. & 13:46:57.7 & -10:44:00\
2 & May 8 2007 & $\sim0.4-0.8$ & 53 $\times$ 200 s. & 13:46:57.7 & -10:44:00\
[ccc]{} Chip & $Z$ (mag.) & $C$\
00 & 27.48 & -0.040\
01 & 27.81 & -0.050\
02 & 27.78 & -0.065\
03 & 27.85 & -0.020\
04 & 27.73 & -0.095\
05 & 27.89 & -0.020\
06 & 27.82 & -0.045\
07 & 27.83 & -0.040\
08 & 27.86 & -0.040\
09 & 27.75 & -0.065\
[lccccc]{} Object & $ r_{SDSS}$ (mag.) & $n$ & $\Delta$ & $i$ ($^o$) & Follow-up\
c9a2 & $24.038\pm0.0424$ & 3 & $40\pm3$ & $22\pm14$ & n\
c1a3 & $24.368\pm0.0575$ & 3 & $44\pm3$ & $9\pm12$ & n\
c6a3 & $24.394\pm0.0585$ & 6 & $45\pm2$ & $5\pm2$ & y\
c1a1 & $24.991\pm0.1018$ & 3 & $44\pm3$ & $1\pm7$ & n\
c3a1 & $25.179\pm0.1212$ & 3 & $42\pm3$ & $0\pm2$ & n\
c5a2 & $25.245\pm0.1293$ & 3 & $41\pm3$ & $3.3\pm10$ & n\
c5a5 & $25.373\pm0.1444$ & 3 & $49\pm3$ & $3\pm14$ & n\
c8a2 & $25.397\pm0.1485$ & 3 & $47\pm4$ & $15\pm15$ & n\
c4a1 & $25.432\pm0.1527$ & 3 & $45\pm3$ & $9\pm13$ & n\
c9a1 & $25.434\pm0.1527$ & 3 & $35\pm3$ & $28\pm16$ & n\
c9a4 & $25.438\pm0.1541$ & 3 & $48\pm3$ & $2\pm13$ & n\
c8a1 & $25.458\pm0.1569$ & 3 & $43\pm3$ & $2\pm11$ & n\
c2a1 & $25.462\pm0.1569$ & 3 & $48\pm3$ & $7\pm14$ & n\
c4a3 & $25.569\pm0.1737$ & 3 & $38\pm3$ & $5\pm10$ & n\
c4a5 & $25.823\pm0.2187$ & 3 & $42\pm3$ & $5\pm11$ & n\
c5a9 & $25.875\pm0.2311$ & 3 & $45\pm3$ & $5\pm12$ & n\
c6a1 & $26.019\pm0.2629$ & 6 & $37\pm3$ & $2\pm9$ & y\
c0a2 & $26.068\pm0.2753$ & 6 & $45\pm2$ & $1.1\pm0.3$ & y\
c2a3 & $26.279\pm0.3340$ & 4 & $43\pm3$ & $5\pm12$ & y\
c1a4 & $26.282\pm0.3340$ & 3 & $44\pm3$ & $4\pm12$ & n\
c0a1 & $26.334\pm0.3498$ & 6 & $45\pm3$ & $14\pm5$ & y\
c6a2 & $26.382\pm0.3663$ & 6 & $44\pm2$ & $3\pm1$ & y\
c2a2 & $26.583\pm0.4404$ & 6 & $39\pm2$ & $10\pm4$ & y\
c4a6 & $26.610\pm0.4527$ & 3 & $43\pm4$ & $23\pm17$ & n\
c4a2 & $26.673\pm0.4784$ & 3 & $45\pm3$ & $12\pm13$ & n\
c1a5 & $26.680\pm0.4828$ & 2 & $37\pm3$ & $27\pm17$ & n\
c5a3 & $26.688\pm0.4873$ & 2 & $44\pm3$ & $7\pm12$ & n\
c5a8 & $26.737\pm0.5103$ & 3 & $47\pm3$ & $7\pm13$ & n\
c5a7 & $26.858\pm0.5699$ & 3 & $44\pm3$ & $1\pm9$ & n\
c4a7 & $26.859\pm0.5699$ & 3 & $34\pm3$ & $9\pm9$ & n\
c9a3 & $26.883\pm0.5805$ & 1 & $47\pm5$ & $75\pm29$ & n\
c7a1 & $26.925\pm0.6079$ & 3 & $45\pm3$ & $16\pm15$ & n\
c5a1 & $27.120\pm0.7241$ & 3 & $43\pm3$ & $1\pm8$ & n\
c5a6 & $27.276\pm0.8391$ & 3 & $47\pm3$ & $6\pm14$ & n\
c1a2 & $27.353\pm0.8950$ & 2 & $55\pm4$ & $12\pm18$ & n\
c5a4 & $28.807\pm3.4344$ & 3 & $42\pm3$ & $0\pm1$ & n\
[lcccccc]{}
& $(\alpha,m_o)$ & $P(\Delta>\Delta_{obs})$ & $(\Sigma_{23}, \alpha, \alpha')$ & $P(\Delta>\Delta_{obs})$ & $(\log A,\alpha_1,\alpha_2,m_b)$ & $P(\Delta>\Delta_{obs})$\
& & & & & $(\log A, q_1, q_2, D_b)$ &\
F08b & $(0.58,23.31)$ & $<0.04$ & $(0.82,0.80,-0.06)$ & 0.4 & $(23.6,0.76, 0.18, 24.9)$ & 0.4\
& & & & & $(23.6,4.8, 1.9, 62)$ &\
F08b$_{i<5}$ & $(0.59,24.0)$ & 0.15 & $-$ & - & - & -\
F08b$_{i>5}$ & $(0.55,23.81)$ & $<0.04$ & $(0.46,0.74,-0.06)$ & 0.1 & $(22.7,0.70,-6,26.2)$ & 0.1\
& & & & & $(22.7,4.5,-\infty,36)$ &\
|
---
abstract: 'Generative adversarial networks (GANs) are an expressive class of neural generative models with tremendous success in modeling high-dimensional continuous measures. In this paper, we present a scalable method for unbalanced optimal transport (OT) based on the generative-adversarial framework. We formulate unbalanced OT as a problem of simultaneously learning a transport map and a scaling factor that push a source measure to a target measure in a cost-optimal manner. We provide theoretical justification for this formulation, showing that it is closely related to an existing static formulation by [@liero2018optimal]. We then propose an algorithm for solving this problem based on stochastic alternating gradient updates, similar in practice to GANs, and perform numerical experiments demonstrating how this methodology can be applied to population modeling.'
author:
- |
Karren D. Yang & Caroline Uhler\
Laboratory for Information & Decision Systems\
Institute for Data, Systems and Society\
Massachusetts Institute of Technology\
Cambridge, MA, USA\
`{karren, cuhler}@mit.edu`\
bibliography:
- 'iclr2019\_conference.bib'
title: Scalable Unbalanced Optimal Transport using Generative Adversarial Networks
---
Introduction
============
We consider the problem of unbalanced optimal transport: given two measures, find a cost-optimal way to transform one measure to the other using a combination of mass variation and transport. Such problems arise, for example, when modeling the transformation of a source population into a target population (Figure \[fig:pop\_example\]a). In this setting, one needs to model mass transport to account for the features that are evolving, as well as local mass variations to allow sub-populations to become more or less prominent in the target population [@schiebinger2017reconstruction].
Classical optimal transport (OT) considers the problem of pushing a source to a target distribution in a way that is optimal with respect to some transport cost without allowing for mass variations. Modern approaches are based on the Kantorovich formulation [@kantorovich1942translocation], which seeks the optimal probabilistic coupling between measures and can be solved using linear programming methods for discrete measures. Recently, [@cuturi2013sinkhorn] showed that regularizing the objective using an entropy term allows the dual problem to be solved more efficiently using the Sinkhorn algorithm. Stochastic methods based on the dual objective have been proposed for the continuous setting [@genevay2016stochastic; @seguy2017large; @arjovsky2017wasserstein]. Optimal transport has been applied to many areas, such as computer graphics [@ferradans2014regularized; @solomon2015convolutional] and domain adaptation [@courty2014domain; @courty2017optimal].
In many applications where a transport cost is not available, transport maps can also be learned using generative models such as generative adversarial networks (GANs) [@goodfellow2014generative], which push a source distribution to a target distribution by training against an adversary. Numerous transport problems in image translation [@mirza2014conditional; @zhu2017unpaired; @yi2017dualgan], natural language translation [@he2016dual], domain adaptation [@bousmalis2017unsupervised] and biological data integration [@amodio2018magan] have been tackled using variants of GANs, with strategies such as conditioning or cycle-consistency employed to enforce correspondence between original and transported samples. However, all these methods conserve mass between the source and target and therefore cannot handle mass variation. Several formulations have been proposed for extending the theory of OT to the setting where the measures can have unbalanced masses [@chizat2015unbalanced; @chizat2018interpolating; @kondratyev2016new; @liero2018optimal; @frogner2015learning]. In terms of numerical methods, a class of scaling algorithms [@chizat2016scaling] that generalize the Sinkhorn algorithm for balanced OT have been developed for approximating the solution to *optimal entropy-transport* problems; this formulation of unbalanced OT by @liero2018optimal corresponds to the Kantorovich OT problem in which the hard marginal constraints are relaxed using divergences to allow for mass variation. In practice, these algorithms have been used to approximate unbalanced transport plans between discrete measures for applications such as computer graphics [@chizat2016scaling], tumor growth modeling [@chizat2017tumor] and computational biology [@schiebinger2017reconstruction]. However, while optimal entropy-transport allows mass variation, it cannot explicitly model it, and there are currently no methods that can perform unbalanced OT between *continuous* measures.
\[fig:pop\_example\]
[**Contributions.**]{} Inspired by the recent successes of GANs for high-dimensional transport problems, we present a novel framework for unbalanced optimal transport that directly models mass variation in addition to transport. Concretely, our contributions are the following:
- We propose to solve a Monge-like formulation of unbalanced OT, in which the goal is to learn a stochastic transport map and scaling factor to push a source to a target measure in a cost-optimal manner. This generalizes the unbalanced Monge OT problem by @chizat2015unbalanced.
- By relaxing this problem, we obtain an alternative form of the optimal entropy-transport problem by @liero2018optimal, which confers desirable theoretical properties.
- We develop scalable methodology for solving the relaxed problem. Our derivation uses a convex conjugate representation of divergences, resulting in an alternating gradient descent method similar to GANs [@goodfellow2014generative].
- We demonstrate in practice how our methodology can be applied towards population modeling using the MNIST and USPS handwritten digits datasets, the CelebA dataset, and a recent single-cell RNA-seq dataset from zebrafish embrogenesis.
In addition to these main contributions, for completeness we also propose a new scalable method (Algorithm \[alg:1\]) in the Appendix for solving the optimal-entropy transport problem by @liero2018optimal in the continuous setting. The algorithm extends the work of @seguy2017large to unbalanced OT and is a scalable alternative to the algorithm of @chizat2016scaling for very large or continuous datasets.
Preliminaries
=============
[**Notation.**]{} Let ${\pazocal{X}}, {\pazocal{Y}}\subseteq \mathbb{R}^n$ be topological spaces and let ${\pazocal{B}}$ denote the Borel $\sigma$-algebra. Let ${\mathcal{M}_+^1}({\pazocal{X}}), {\mathcal{M}}_+({\pazocal{X}})$ denote respectively the space of probability measures and finite non-negative measures over ${\pazocal{X}}$. For a measurable function $T$, let $T_{\#}$ denote its pushforward operator: if $\mu$ is a measure, then $T_{\#} \mu$ is the pushforward measure of $\mu$ under $T$. Finally, let $\pi^{\pazocal{X}}$, $\pi^{\pazocal{Y}}$ be functions that project onto ${\pazocal{X}}$ and ${\pazocal{Y}}$; for a joint measure $\gamma \in {\mathcal{M}}_+({\pazocal{X}}\times {\pazocal{Y}})$, $\pi^{\pazocal{X}}_\#\gamma$ and $\pi^{\pazocal{Y}}_\#\gamma$ are its marginals with respect to ${\pazocal{X}}$ and ${\pazocal{Y}}$ respectively.
[**Optimal transport (OT)**]{} addresses the problem of transporting between measures in a cost-optimal manner. @monge1781memoire formulated this problem as a search over deterministic transport maps. Specifically, given $\mu \in {\mathcal{M}_+^1}({\pazocal{X}}), \nu \in {\mathcal{M}_+^1}({\pazocal{Y}})$ and a cost function $c: {\pazocal{X}}\times {\pazocal{Y}}\rightarrow \mathbb{R}^+$, Monge OT seeks a measurable function $T: {\pazocal{X}}\rightarrow {\pazocal{Y}}$ minimizing $$\inf_{T} \int_{{\pazocal{X}}} c(x,T(x)) ~d\mu(x)$$ subject to the constraint $T_{\#}\mu = \nu$. While the optimal $T$ has an intuitive interpretation as an optimal transport map, the Monge problem is non-convex and not always feasible depending on the choices of $\mu$ and $\nu$. The *Kantorovich OT* problem is a convex relaxation of the Monge problem that formulates OT as a search over probabilistic transport plans. Given $\mu \in {\mathcal{M}_+^1}({\pazocal{X}}), \nu \in {\mathcal{M}_+^1}({\pazocal{Y}})$ and a cost function $c: {\pazocal{X}}\times {\pazocal{Y}}\rightarrow \mathbb{R}^+$, Kantorovich OT seeks a joint measure $\gamma \in {\mathcal{M}_+^1}({\pazocal{X}}\times {\pazocal{Y}})$ subject to $\pi^{\pazocal{X}}_\#\gamma=\mu$ and $\pi^{\pazocal{Y}}_\#\gamma=\nu$ minimizing $$\label{eq:MK}
\begin{split}
W(\mu, \nu) := \inf_{\gamma} \int_{{\pazocal{X}}\times {\pazocal{Y}}} c(x,y) ~d\gamma(x,y).
\end{split}$$ Note that the conditional probability distributions $\gamma_{y|x}$ specify stochastic maps from ${\pazocal{X}}$ to ${\pazocal{Y}}$ and can be considered a “one-to-many" version of the deterministic map from the Monge problem. In terms of numerical methods, the relaxed problem is a linear program that is always feasible and can be solved in $O(n^3)$ time for discrete $\mu, \nu$. [@cuturi2013sinkhorn] recently showed that introducing entropic regularization results in a simpler dual optimization problem that can be solved efficiently using the Sinkhorn algorithm. Based on the entropy-regularized dual problem, [@genevay2016stochastic] and [@seguy2017large] proposed stochastic algorithms for computing transport plans that can handle continuous measures.
[**Unbalanced OT.**]{} Several formulations that extend classical OT to handle mass variation have been proposed [@chizat2015unbalanced; @chizat2018interpolating; @kondratyev2016new]. Existing numerical methods are based on a Kantorovich-like formulation known as *optimal-entropy transport* [@liero2018optimal]. This formulation is obtained by relaxing the marginal constraints of (\[eq:MK\]) using divergences as follows: given two positive measures $\mu \in {\mathcal{M}}_+({\pazocal{X}})$ and $\nu \in {\mathcal{M}}_+({\pazocal{Y}})$ and a cost function $c: {\pazocal{X}}\times {\pazocal{Y}}\rightarrow \mathbb{R}^+$, optimal entropy-transport finds a measure $\gamma \in {\mathcal{M}}_+({\pazocal{X}}\times {\pazocal{Y}})$ that minimizes $$\label{eq:transport}
\begin{split}
W_{ub}(\mu, \nu) := \inf_\gamma \int_{{\pazocal{X}}\times {\pazocal{Y}}} c(x,y) ~d\gamma(x,y) + D_{\psi_1}(\pi^{\pazocal{X}}_\#\gamma| \mu)
+ D_{\psi_2}(\pi^{\pazocal{Y}}_\#\gamma| \nu),
\end{split}$$ where $D_{\psi_1}$, $D_{\psi_2}$ are $\psi$-divergences induced by $\psi_1, \psi_2$. The $\psi$-divergence between non-negative finite measures $P,Q$ over ${\pazocal{T}}\subseteq \mathbb{R}^d$ induced by a lower semi-continuous, convex entropy function $\psi: \mathbb{R} \rightarrow \mathbb{R} \cup \{ \infty \}$ is $$\label{eq:div}
D_{\psi}(P | Q) := \psi'_{\infty}P^\perp({\pazocal{T}}) + \int_{{\pazocal{T}}} \psi \left(\frac{dP}{dQ} \right) dQ,$$ where $\psi'_{\infty} := \lim_{s \rightarrow \infty} \frac{\psi(s)}{s}$ and $\frac{dP}{dQ} Q + P^\perp$ is the Lebesgue decomposition of $P$ with respect to $Q$. Note that mass variation is allowed since the marginals of $\gamma$ are not constrained to be $\mu$ and $\nu$. In terms of numerical methods, the state-of-the-art in the discrete setting is a class of iterative scaling algorithms [@chizat2016scaling] that generalize the Sinkhorn algorithm for computing regularized OT plans [@cuturi2013sinkhorn]. There are no practical algorithms for unbalanced OT between continuous measures, especially in high-dimensional spaces.
\[fig:syn\_data\]
Scalable Unbalanced OT using GANs {#sec:met2}
=================================
In this section, we propose the first algorithm for unbalanced OT that directly models mass variation and can be applied towards transport between high-dimensional continuous measures. The starting point of our development is the following Monge-like formulation of unbalanced OT, in which the goal is to learn a stochastic transport map and scaling factor to push a source to a target measure in a cost-optimal manner.
[**Unbalanced Monge OT.**]{} Let $c_1: {\pazocal{X}}\times {\pazocal{Y}}\rightarrow \mathbb{R}^+$ be the cost of transport and $c_2: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ the cost of mass variation. Let the probability space $({\pazocal{Z}}, {\pazocal{B}}({\pazocal{Z}}), \lambda)$ be the source of randomness in the transport map $T$. Given two positive measures $\mu \in {\mathcal{M}}_+({\pazocal{X}})$ and $\nu \in {\mathcal{M}}_+({\pazocal{Y}})$, we seek a transport map $T: {\pazocal{X}}\times {\pazocal{Z}}\rightarrow {\pazocal{Y}}$ and a scaling factor $\xi: {\pazocal{X}}\rightarrow \mathbb{R}^+$ minimizing $$\begin{aligned}
\label{eq:ubOT_exact}
&L(\mu, \nu) := \inf_{T, \xi} \int_{\pazocal{X}}\int_{\pazocal{Z}}c_1(x, T(x,z)) d\lambda(z) \xi(x) d\mu(x) + \int_{\pazocal{X}}c_2(\xi(x)) d\mu(x),\end{aligned}$$ subject to the constraint $T_{\#}(\xi\mu \times \lambda) = \nu$. Concretely, the first and second terms of (\[eq:ubOT\_exact\]) penalize the cost of mass transport and variation respectively, and the equality constraint ensures that $(T, \xi)$ pushes $\mu$ to $\nu$ exactly. A special case of (\[eq:ubOT\_exact\]) is the unbalanced Monge OT problem by @chizat2015unbalanced, which employs a deterministic transport map (Figure \[fig:pop\_example\]b). We consider the more general case of stochastic (i.e. one-to-many) maps because it is a more suitable model for many practical problems. For example, in cell biology, it is natural to think of one cell in a source population as potentially giving rise to multiple cells in a target population. In practice, one can take ${\pazocal{Z}}= \mathbb{R}^n$ and $\lambda$ to be the standard Gaussian measure if a stochastic map is desired; otherwise $\lambda$ can be set to a deterministic distribution. The following are examples of problems that can be modeled using unbalanced Monge OT.
Suppose the objective is to model the transformation from a source measure (Column 1) to the target measure (Column 2), which represent a population of interest at two distinct time points. The transport map $T$ models the transport/movement of points from the source to the target, while the scaling factor $\xi$ models the growth (replication) or shrinkage (death) of these points. Different models of transformation are optimal depending on the relative costs of mass transport and variation (Columns 3-6).
Suppose the objective is to transport points from a source measure (1st panel, color) to a target measure (1st panel, grey) in the presence of class imbalances. A pure transport map would muddle together points from different classes, while an unbalanced transport map with a scaling factor is able to ameliorate the class imbalance (2nd panel). In this case, the scaling factor tells us explicitly how to downweigh or upweigh samples in the source distribution to balance the classes with the target distribution (3rd panel).
[**Relaxation.**]{} From an optimization standpoint, it is challenging to satisfy the constraint $T_{\#}(\xi\mu \times \lambda) = \nu$. We hence consider the following relaxation of (\[eq:ubOT\_exact\]) using a divergence penalty in place of the equality constraint: $$\label{eq:ubOT}
L_\psi(\mu, \nu) := \inf_{T, \xi} \int_{\pazocal{X}}\int_{\pazocal{Z}}c_1(x, T(x,z)) d\lambda(z) \xi(x) d\mu(x) + \int_{\pazocal{X}}c_2(\xi(x)) d\mu(x) + D_\psi(T_{\#}(\xi\mu \times \lambda) | \nu),$$ using an appropriate choice of $\psi$ that satisfies the requirements of Lemma \[lem:div\] in the Appendix[^1].This relaxation is the Monge-like version of the optimal-entropy transport problem (\[eq:transport\]) by @liero2018optimal. Specifically, $(T, \xi)$ specifies a joint measure $\gamma \in {\mathcal{M}}_+({\pazocal{X}}\times {\pazocal{Y}})$ given by $$\gamma(C) := \int_{\pazocal{X}}\int_{\pazocal{Z}}\mathbbm{1}_C(x,T(x,z)) d\lambda(z) \xi(x) d\mu(x), \quad \forall C \in {\pazocal{B}}({\pazocal{X}}) \times {\pazocal{B}}({\pazocal{Y}}),$$ and by reformulating (\[eq:ubOT\]) in terms of $\gamma$ instead of $(T, \xi)$, one obtains the objective function for optimal-entropy transport. The main difference between the formulations is their search space, since not all joint measures $\gamma \in {\mathcal{M}}_+({\pazocal{X}}\times {\pazocal{Y}})$ can be specified by some choice of $(T, \xi)$. For example, if $T$ is a deterministic transport map, then $\gamma$ is necessarily restricted to the set of deterministic couplings. Even if $T$ is sufficiently random, it is generally not possible to specify all joint measures $\gamma \in {\mathcal{M}}_+({\pazocal{X}}\times {\pazocal{Y}})$: in the asymmetric Monge formulation (\[eq:ubOT\]), all the mass transported to ${\pazocal{Y}}$ must come from somewhere within the support of $\mu$, since the scaling factor $\xi$ allows mass to grow but not to materialize outside of its original support. Therefore equivalence can be established in general only when restricting the support of $\gamma$ to $supp(\mu) \times {\pazocal{Y}}$ as described in the following lemma, whose proof is given in the Appendix.
\[lem:opt-entropy\] Let ${\pazocal{G}}$ be the set of joint measures supported on $supp(\mu) \times {\pazocal{Y}}$, and define $$\label{eq:OET}
\tilde W_{c, \psi_1, \psi_2}(\mu,\nu):= \inf_{\gamma \in {\pazocal{G}}} \int c ~d\gamma + D_{\psi_1}(\pi^{\pazocal{X}}_\#\gamma | \mu) + D_{\psi_2}(\pi^{\pazocal{Y}}_\#\gamma | \nu).$$ If $({\pazocal{Z}}, {\pazocal{B}}({\pazocal{Z}}), \lambda)$ is atomless and $c_2$ is an entropy function, then $L_\psi(\mu, \nu) = \tilde W_{c_1, c_2, \psi}(\mu, \nu)$.
Based on the relation between (\[eq:transport\]) and (\[eq:ubOT\]), several theoretical results for (\[eq:ubOT\]) follow from the analysis of optimal entropy-transport by @liero2018optimal. Importantly, one can show the following theorem, namely that for an appropriate and sufficiently large choice of divergence penalty, solutions of the relaxed problem (\[eq:ubOT\]) converge to solutions of the original problem (\[eq:ubOT\_exact\]). The proof is given in the Appendix.
\[prop:relaxed\] Suppose $c_1, c_2, \psi$ satisfy the existence assumptions of Proposition \[prop:wellposed\] in the Appendix, and let $({\pazocal{Z}}, {\pazocal{B}}({\pazocal{Z}}), \lambda)$ be an atomless probability space. Furthermore, let $\psi$ be uniquely minimized at $\psi(1)=0$. Then for a sequence $0 < \zeta^1< \cdots < \zeta^k < \cdots$ diverging to $\infty$ indexed by $k$, $\lim_{k \rightarrow \infty} L_{\zeta^k \psi}(\mu, \nu) = L(\mu,\nu)$. Additionally, let $\gamma^k$ be the joint measure specified by a minimizer of $L_{\zeta^k \psi}(\mu, \nu)$. If $L(\mu,\nu)<\infty$, then up to extraction of a subsequence, $\gamma^k$ converges weakly to $\gamma$, the joint measure specified by a minimizer of $L(\mu,\nu)$.
[**Algorithm.**]{} Using the relaxation of unbalanced Monge OT in (\[eq:ubOT\]), we now show that the transport map and scaling factor can be learned by stochastic gradient methods. While the divergence term cannot easily be minimized using the definition in (\[eq:div\]), we can write it as a penalty witnessed by an *adversary* function $f: {\pazocal{Y}}\rightarrow (-\infty, \psi'_{\infty}]$ using the convex conjugate representation (see Lemma \[lem:vc2\]): $$\label{eq:var-div}
D_\psi(T_{\#}(\xi\mu \times \lambda) | \nu) = \sup_f \int_{\pazocal{X}}\int_{\pazocal{Z}}f(T(x, z)) d\lambda(z) \xi(x) d\mu(x) - \int_{\pazocal{Y}}\psi^*(f(y)) d\nu(y),$$ where $\psi^*$ is the convex conjugate of $\psi$. The objective in (\[eq:ubOT\]) can now be optimized using alternating stochastic gradient updates after parameterizing $T$, $\xi$, and $f$ with neural networks; see Algorithm \[alg:TAG\] [^2]. The optimization procedure is similar to GAN training and can be interpreted as an adversarial game between $(T, \xi)$ and $f$:\
- $T$ takes a point $x \sim \mu$ and transports it from ${\pazocal{X}}$ to ${\pazocal{Y}}$ by generating $T(x, z)$ where $z \sim \lambda$.
- $\xi$ determines the importance weight of each transported point.
- Their shared objective is to minimize the divergence between transported samples and real samples from $\nu$ that is measured by the adversary $f$.
- Additionally, cost functions $c_1$ and $c_2$ encourage $T, \xi$ to find the most cost-efficient strategy.
Initial parameters $\theta$, $\phi$, $\omega$; step size $\eta$; normalized measures $\tilde \mu$, $\tilde \nu$, constants $c_\mu, c_\nu$.
Updated parameters $\theta$, $\phi$, $\omega$.
Sample $x_1, \cdots, x_n$ from $\tilde \mu$, $y_1, \cdots, y_n$ from $\tilde \nu$, $z_1, \cdots, z_n$ from $\lambda$; $$\begin{aligned}
\label{eq:loss}
\ell(\theta, \phi, \omega):= \frac{1}{n} \sum_{i=1}^n [ c_\mu c_1(x_i, T_\theta(x_i, z_i))\xi_\phi(x_i) &+ c_\mu c_2(\xi_\phi(x_i)) \\
&+ c_\mu \xi_\phi(x_i) f_\omega(T_\theta(x_i, z_i)) - c_\nu \psi^*(f_\omega(y_i)). ] \nonumber
\end{aligned}$$ Update $\omega$ by gradient descent on $-\ell(\theta, \phi, \omega)$. Update $\theta, \phi$ by gradient descent $\ell(\theta, \phi, \omega)$.
\[alg:TAG\]
Table \[tab:div\] in the Appendix provides some examples of divergences with corresponding entropy functions and convex conjugates that can be plugged into (\[eq:var-div\]). Further practical considerations for implementation and training are discussed in Appendix \[sec:prac\].
[**Relation to other approaches.**]{} The probabilistic Monge-like formulation (\[eq:ubOT\]) is similar to the Kantorovich-like entropy-transport problem (\[eq:transport\]) in theory, but they result in quite different numerical methods in practice. Algorithm \[alg:TAG\] solves the non-convex formulation (\[eq:ubOT\]) and learns a transport map $T$ and scaling factor $\xi$ parameterized by neural networks, enabling scalable optimization using stochastic gradient descent. The networks are immediately useful for many practical applications; for instance, it only requires a single forward pass to compute the transport and scaling of a point from the source domain to the target. Furthermore, the neural architectures of $T, \xi$ imbue their function classes with a particular structure, and when chosen appropriately, enable effective learning of these functions in high-dimensional settings. Due to the non-convexity of the optimization problem, however, Algorithm \[alg:TAG\] is not guaranteed to find the global optimum. In contrast, the scaling algorithm of @chizat2016scaling based on (\[eq:transport\]) solves a convex optimization problem and is proven to converge, but is currently only practical for discrete problems and has limited scalability. For completeness, in Section \[sec:met1\] of the Appendix, we propose a *new stochastic method* based on the same dual objective as @chizat2016scaling that can handle transport between continuous measures (Algorithm \[alg:1\] in the Appendix). This method generalizes the approach of @seguy2017large for handling transport between continuous measures and overcomes the scalability limitations of @chizat2016scaling. However, the output is in the form of the dual solution, which is less interpretable for practical applications compared to the output of Algorithm \[alg:TAG\]. In particular, while one can compute a deterministic transport map known as a barycentric projection from the dual solution, it is unclear how best to obtain a scaling factor or a stochastic transport map that can generate samples outside of the target dataset. In the numerical experiments of Section \[sec:exp\], we show the advantage of directly learning a transport map and scaling factor using Algorithm \[alg:TAG\].
The problem of learning a scaling factor (or weighting factor) that “balances" measures $\mu$ and $\nu$ also arises in causal inference. Generally, $\mu$ is the distribution of covariates from a control population and $\nu$ is the distribution from a treated population. The goal is to scale the importance of different members from the control population based on how likely they are to be present in the treated population, in order to eliminate selection biases in the inference of treatment effects. [@kallus2018deepmatch] proposed a generative-adversarial method for learning the scaling factor, but they do not consider transport.
Numerical Experiments {#sec:exp}
=====================
In this section, we illustrate in practice how Algorithm \[alg:TAG\] performs unbalanced OT, with applications geared towards population modeling.
[**MNIST-to-MNIST.**]{} We first apply Algorithm \[alg:TAG\] to perform unbalanced optimal transport between two modified MNIST datasets. The source dataset consists of regular MNIST digits with the class distribution shown in column 1 of Figure \[fig:mnist\_data\]a. The target dataset consists of either regular (for the experiment in Figure \[fig:mnist\_data\]b) or dimmed (for the experiment in Figure \[fig:mnist\_data\]c) MNIST digits with the class distribution shown in column 2 of Figure \[fig:mnist\_data\]a. The class imbalance between the source and target datasets imitates a scenerio in which certain classes (digits 0-3) become more popular and others (6-9) become less popular in the target population, while the change in brightness is meant to reflect population drift. We evaluated Algorithm \[alg:TAG\] on the problem of transporting the source distribution to the target distribution, enforcing a high cost of transport (w.r.t. Euclidean distance). In both cases, we found that the scaling factor over each of the digit classes roughly reflects its ratio of imbalance between the source and target distributions (Figure \[fig:mnist\_data\]b-c). These experiments validate that the scaling factor learned by Algorithm \[alg:TAG\] reflects the class imbalances and can be used to model growth or decline of different classes in a population. Figure \[fig:mnist\_data\]d is a schematic illustrating the reweighting that occurs during unbalanced OT.
\[fig:mnist\_data\]
[**MNIST-to-USPS.**]{} Next, we apply unbalanced OT from the MNIST dataset to the USPS dataset. As before, these two datasets are meant to imitate a population sampled at two different time points, this time with a large degree of evolution. We use Algorithm \[alg:TAG\] to model the evolution of the MNIST distribution to the USPS distribution, taking as transport cost the Euclidean distance between the original and transported images. A summary of the unbalanced transport is visualized in Figure \[fig:num\_data\]a. Each arrow originates from a real MNIST image and points towards the predicted appearance of this image in the USPS dataset. The size of the image reflects the scaling factor of the original MNIST image, i.e. whether it is relatively increasing or decreasing in prominence in the USPS dataset compared to the MNIST dataset according to the unbalanced OT model. Even though the Euclidean distance is not an ideal measure of correspondence between MNIST and USPS digits, many MNIST digits were able to preserve their likeness during the transport (Figure \[fig:num\_data\]b). We analyzed which MNIST digits were considered as increasing or decreasing in prominence by the model. The MNIST digits with higher scaling factors were generally brighter (Figure \[fig:num\_data\]c) and covered a larger area of pixels (Figure \[fig:num\_data\]d) compared to the MNIST digits with lower scaling factors. These results are consistent with the observation that the target USPS digits are generally brighter and contain more pixels.
\[fig:num\_data\]
[**CelebA-Young-to-CelebA-Aged.**]{} We applied Algorithm \[alg:TAG\] on the CelebA dataset to perform unbalanced OT from the population of young faces to the population of aged faces. This synthetic problem imitates a real application of interest, which is modeling the transformation of a population based on samples taken from two timepoints. Since the Euclidean distance between two faces is a poor measure of semantic similarity, we first train a variational autoencoder (VAE) [@kingma2013auto] on the CelebA dataset and encode all samples into the latent space. We then apply Algorithm \[alg:TAG\] to perform unbalanced OT from the encoded young to the encoded aged faces, taking the transport cost to be the Euclidean distance in the latent space. A summary of the unbalanced transport is visualized in Figure \[fig:celebA\_data\]a. Each arrow originates from a real face from the young population and points towards the predicted appearance of this face in the aged population. Generally, the transported faces retain the most salient features of the original faces (Figure \[fig:celebA\_data\]b), although there are exceptions (e.g. gender swaps) which reflects that some features are not prominent components of the VAE encodings. Interestingly, the young faces with higher scaling factors were significantly enriched for males compared to young faces with lower scaling factors; 9.6% (9,913/103,287) of young female faces had a high scaling factor as compared to 18.5% (8,029/53,447) for young male faces (Figure \[fig:celebA\_data\]c, top, $p = 0$). In other words, our model predicts growth in the prominence of male faces compared to female faces as the CelebA population evolves from young to aged. After observing this phenomenon, we confirmed based on checking the ground truth labels that there was indeed a strong gender imbalance between the young and aged populations: while the young population is predominantly female, the aged population is predominantly male (Figure \[fig:celebA\_data\]c, bottom).
[**Zebrafish embroygenesis.**]{} A problem of great interest in biology is lineage tracing of cells between different developmental stages or during disease progression. This is a natural application of transport in which the source and target distributions are unbalanced: some cells in the earlier stage are more poised to develop into cells seen in the later stage. To showcase the relevance of learning the scaling factor, we apply Algorithm \[alg:TAG\] to recent single-cell gene expression data from two stages of zebrafish embryogenesis [@farrelleaar3131]. The source population is from a late stage of blastulation and the target population from an early stage of gastrulation (Figure \[fig:zebrafish\]a). The results of the transport are plotted in Figure \[fig:zebrafish\]b-c after dimensionality reduction by PCA and T-SNE [@maaten2008visualizing]. To assess the scaling factor, we extracted the cells from the blastula stage with higher scaling factors (i.e. over 90th percentile) and compared them to the remainder of the cells using differential gene expression analysis, producing a ranked list of upregulated genes. Using the GOrilla tool [@eden2009gorilla], we found that the cells with higher scaling factors were significantly enriched for genes associated with differentiation and development of the mesoderm (Figure \[fig:zebrafish\]d). This experiment shows that analysis of the scaling factor can be applied towards interesting and meaningful biological discovery.
![[**Unbalanced OT on Zebrafish Single-Cell Gene Expression Data.**]{} (a) Illustration of the blastula and gastrula stages of zebrafish embryogenesis. (b) T-SNE plot [@maaten2008visualizing] of the unbalanced OT results for a subset of datapoints. The color of the transported points indicates the relative magnitude of the scaling factor (black = high, white = low). (c) Same plot as (b), where we have colored the source points instead of the transported points. (d) GOrilla output of significantly enriched processes [@eden2009gorilla] based on ranked list of enriched genes in cells with high scaling factors (black points from (c)) from a differential gene expression analysis. Processes in the graph are organized from more general (upstream) to more specific (downstream). []{data-label="fig:zebrafish"}](zebrafish_figure.png)
Acknowledgements {#acknowledgements .unnumbered}
----------------
Karren Yang was supported by an NSF Graduate Fellowship and ONR (N00014-17-1-2147). Caroline Uhler was partially supported by NSF (DMS-1651995), ONR (N00014-17-1-2147 and N00014-18-1-2765), IBM, and J-WAFS.
[^1]: Note that $D_\psi(T_{\#}(\xi\mu \times \lambda) | \nu) = 0 \nRightarrow T_{\#}(\xi\mu \times \lambda) = \nu$ in general since the total mass of the transported measure is not constrained; for the relaxation to be valid, $\psi(s)$ should attain a unique minimum at $s=1$ (see Lemma \[lem:div\]).
[^2]: We assume that one has access to samples from $\mu, \nu$, and in the setting where $\mu, \nu$ are not normalized, then samples to the normalized measures $\tilde \mu, \tilde \nu$ as well as the normalization constants $c_\mu, c_\nu$. These are reasonable assumptions for practical applications: for example, in a biological assay, one might collect $c_\mu$ cells from time point 1 and $c_\nu$ cells from time point 2. In this case, the samples are the measurements of each cell and the normalization constants are $c_\mu, c_\nu$.
|
---
author:
- Marina Bastea
title: 'Comment on “Metallization of Fluid Nitrogen and the Mott Transition in Highly Compressed Low-Z Fluids”'
---
The physical behavior and microscopic nature of highly compressed fluids continue to be unsolved and actively debated topics. The experimental data presented by Chau et al. on the electrical conductivity of nitrogen at high pressures are a welcome addition to the field. Unfortunately, in their desire to achieve a unifying picture of the behavior of low-Z fluids the authors oversimplify the interpretation of their data and [*de facto*]{} overlook the unique features of the systems they discuss.
The main hypothesis of the “low Z-systematics” introduced by Chau et al. is that nitrogen, hydrogen and oxygen are completely dissociated in the shock reverberation experiments that lead to their metallization. This is hardly a reasonable assumption for oxygen. The results and analysis of electrical conductivity experiments presented in Ref. [@bastea-O2] indicate that metallization of fluid oxygen occurs in the molecular phase, and not in an atomic regime as stated in [@chau-N2]. This conclusion is supported by recent [*ab initio*]{} simulations of fluid oxygen at the conditions of the shock reverberation experiments [@galli-O2]. The simulations also reveal unique features of fluid oxygen under pressure e.g., the role of the triplet spin state in the metallization transition and an unusual behavior of the molecular bond [@galli-O2].
Complete dissociation of nitrogen at $80 GPa$, the lowest pressure in the experiments of [@chau-N2], is not in fact supported by previous work. The authors arguments are loosely based on two theoretical calculations but overlook their explicit caveats and disagreements with experiments. The QMD simulations invoked in [@chau-N2] report “noticeable disagreement from the second shock” experiments in the dissociation region [@LANL-N2reshock]. It is therefore clear that any extrapolation of this work to the multiple-shock states relevant for the conductivity experiments has to be done with caution [@LANL-N2reshock]. As additional evidence for an atomic state assumption the authors use a dissociation energy calculated by Ross [@ross-N2] along with an empirical criterion introduced in [@nellis-h2]. However, by consistently following the treatment of [@ross-N2], the first shock ($14 GPa$) and reshock ($41 GPa$) dissociation fractions corresponding to the $80 GPa$ final pressure can be estimated to about $10^{-10}$ and $3 \times 10^{-3}$ respectively. Since temperature reaches over $90\%$ of its final value upon reshock it is reasonable to assume that the fluid retains a significant molecular component. After all even on the Hugoniot at $80 GPa$ and temperatures over $1.2eV$ in both theoretical studies the fluid appears to be $30-40\%$ molecular [@LANL-N2reshock; @ross-N2].
A discussion of the hydrogen metallization experiments and Mott scaling analysis can be found in [@nellis-96], where evidence is presented in support of a molecular fluid, in contradiction with the approach taken in [@chau-N2].
We would also like to point out several other inconsistencies of the “low-Z fluids systematics” presented in [@chau-N2]. For example, we calculate a Mott scaling parameter of $.27$ for nitrogen at the metallization conditions defined in the paper, in disagreement with the .35 value quoted by the authors. This comes simply from an average distance between atoms of $1.81 \AA$ at a density of $3.9g/cm^3$, and a Bohr radius (location of the maximum in the valence charge distribution $r^2\psi\psi^{\star}$ averaged over the solid angle) of $.91 bohr$, Ref. \[26\] of [@chau-N2]. The authors use of an empirical atomic radius meant to fit interatomic distances in a variety of crystalline compounds, see Ref. \[25\] of [@chau-N2], is puzzling and unnecessary. For example the quoted reference gives a $.5 bohr$ radius for hydrogen, in large disagreement with the accepted $1 bohr$ value. It should also be noted that although using zero pressure charge distributions to gain insight into the behavior at high pressure is perhaps reasonable, shifting the curves in the radial direction as done in Fig. 3 of [@chau-N2] in order to compare their spatial extent is a rather meaningles exercise. Scaling all distributions to unity would allow the intended comparison and make it more apparent that the radial extent of hydrogen exceeds that of atomic nitrogen in contradiction with the statements by Chau et al.
Nitrogen is a complex system with one of the largest dissociation energies known in its diatomic state, a multitude of chemical bonding configurations and a high pressure polymeric phase [@hemN2-mcmah]. An accurate interpretation of electrical conductivity data for nitrogen around its high pressure metallization transition will probably need to account for this complexity.
This work was performed under the auspices of the U. S. Department of Energy by University of California Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48.
[99]{} R. Chau et al, Phys. Rev. Lett. [**90**]{}, 245501 (2003). M. Bastea et al, Phys. Rev. Lett. [**86**]{}, 3108 (2001). B. Millitzer et al, to appear in Phys. Rev. Lett. (2003). S. Mazevet et al, Phys. Rev. B [**65**]{}, 014204 (2001). M. Ross, J. Chem. Phys. [**86**]{}, 7110 (1987). W.J.Nellis Phys. Rev. Lett. [**89**]{}, 165502 (2002). S.T. Weir et al Phys. Rev. Lett. [**76**]{}, 1860 (1996); W.J. Nellis et al Phys. Rev. B [**59**]{}, 3434 (1999). M.I. Eremets et al, Nature [**411**]{}, 170 (2001); A.K.McMahan et al, Phys. Rev. Lett., [**54**]{}, 1929 (1985).
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abstract: 'With the accumulation of many years of solar and atmospheric neutrino oscillation data, the approximate form of the $3 \times 3$ neutrino mixing matrix is now known. What is not known is the (presumably Majorana) neutrino mass matrix ${\cal M}_\nu$ itself. In this chapter, the approximate form of ${\cal M}_\nu$ is derived, leading to seven possible neutrino mass patterns: three have the normal hierarchy, two have the inverse hierarchy, and two have three nearly degenerate masses. The generalization of this to allow $U_{e3} \neq 0$ with maximal CP violation is also discussed. A specific automatic realization of this ${\cal M}_\nu$ from radiative corrections of an underlying non-Abelian discrete $A_4$ symmetry in the context of softly broken supersymmetry is presented.'
---
UCRHEP-T344\
August 2002
[**Neutrino Mass Matrix a la Mode\
**]{}
——————
Contribution to a special issue of the Proceedings of the Indian National Science Academy
With the recent addition of the SNO (Sudbury Neutrino Observatory) neutral-current data [@sno], the overall picture of solar neutrino oscillations [@sol] is becoming quite clear. Together with the well-established atmospheric neutrino data [@atm], the $3 \times 3$ neutrino mixing matrix is now determined to a very good first approximation by $$\pmatrix {\nu_e \cr \nu_\mu \cr \nu_\tau} = \pmatrix {\cos \theta & -\sin
\theta & 0 \cr \sin \theta/\sqrt 2 & \cos \theta/\sqrt 2 & -1/\sqrt 2 \cr
\sin \theta/\sqrt 2 & \cos \theta/\sqrt 2 & 1/\sqrt 2} \pmatrix {\nu_1 \cr
\nu_2 \cr \nu_3},$$ where $\nu_{1,2,3}$ are assumed to be Majorana neutrino mass eigenstates. In the above, $\sin^2 2 \theta_{atm} = 1$ is already assumed and $\theta$ is the solar mixing angle which is now known to be large but not maximal [@many], i.e. $\tan^2 \theta \simeq 0.4$. The $U_{e3}$ entry has been assumed zero but it is only required experimentally to be small [@react], i.e. $|U_{e3}| < 0.16$.
Denoting the masses of $\nu_{1,2,3}$ as $m_{1,2,3}$, the solar neutrino data [@sno; @sol] require that $m_2^2 > m_1^2$ with $\theta < \pi/4$, and in the case of the favored large-mixing-angle solution [@many], $$\Delta m^2_{sol} = m_2^2 - m_1^2 \simeq 5 \times 10^{-5}~{\rm eV}^2.$$ The atmospheric neutrino data [@atm] require $$|m_3^2 - m_{1,2}^2| \simeq 2.5 \times 10^{-3}~{\rm eV}^2,$$ without deciding whether $m_3^2 > m_{1,2}^2$ or $m_3^2 < m_{1,2}^2$.
The big question now is what the neutrino mass matrix itself should look like. Of course it may be obtained by using Eq. (1), i.e. $${\cal M}_\nu = \pmatrix {c^2 m_1 + s^2 m_2 & sc(m_1-m_2)/\sqrt 2 & sc(m_1-m_2)
/\sqrt 2 \cr sc(m_1-m_2)/\sqrt 2 & (s^2 m_1 + c^2 m_2 + m_3)/2 & (s^2 m_1 +
c^2 m_2 - m_3)/2 \cr sc(m_1-m_2)/\sqrt 2 & (s^2 m_1 + c^2 m_2 - m_3)/2 &
(s^2 m_1 + c^2 m_2 + m_3)/2},$$ where $c \equiv \cos \theta$ and $s \equiv \sin \theta$. However this is not very illuminating theoretically. Instead it has been proposed [@ma02] that it be rewritten in the form $${\cal M}_\nu = \pmatrix {a+2b+2c & d & d \cr d & b & a+b \cr d & a+b & b}.$$ To satisfy $m_2^2 > m_1^2$ for $\theta < \pi/4$, there are 2 cases to be considered.
\(I) For $a+2b+c > 0$ and $c < 0$, $$\begin{aligned}
m_1 &=& a+2b+c - \sqrt {c^2 + 2d^2}, \\
m_2 &=& a+2b+c + \sqrt {c^2 + 2d^2}, \\
m_3 &=& -a,\\
\tan \theta &=& \sqrt {2} d/ (\sqrt {c^2 + 2d^2} - c).\end{aligned}$$ (II) For $a+2b+c < 0$ and $c > 0$, $$\begin{aligned}
m_1 &=& a+2b+c + \sqrt {c^2 + 2d^2}, \\
m_2 &=& a+2b+c - \sqrt {c^2 + 2d^2}, \\
m_3 &=& -a,\\
\tan \theta &=& (\sqrt {c^2 + 2d^2} - c)/ \sqrt {2} d.\end{aligned}$$
Note that $\theta$ depends only on the ratio $d/c$, which must be of order unity. This shows the advantage for adopting the parametrization of Eq. (5). The constraints of Eqs. (2) and (3) are then realized by the following 7 different conditions on $a$, $b$, and $c$.
\(1) $||a+2b+c|-\sqrt{c^2+2d^2}| << |a+2b+c| << |a|$, i.e. $|m_1| << |m_2| <<
|m_3|$.
\(2) $\sqrt{c^2+2d^2} << |a+2b+c| << |a|$, i.e. $|m_1| \simeq |m_2| << |m_3|$.
\(3) $|a+2b+c| << \sqrt{c^2+2d^2} << |a|$, i.e. $|m_1| \simeq |m_2| << |m_3|$.
\(4) $|a|,~\sqrt{c^2+2d^2} << |a+2b+c|$, i.e. $|m_3| << |m_1| \simeq |m_2|$.
\(5) $|a|,~|a+2b+c| << \sqrt{c^2+2d^2}$, i.e. $|m_3| << |m_1| \simeq |m_2|$.
\(6) $\sqrt{c^2+2d^2} << ||a+2b+c|-|a|| << |a|$, i.e. $|m_1| \simeq |m_2|
\simeq |m_3|$.
\(7) $|a+2b+c| << \sqrt{c^2+2d^2} \simeq |a|$, i.e. $|m_1| \simeq |m_2|
\simeq |m_3|$.
Cases (1) to (3) have the normal hierarchy. Cases (4) and (5) have the inverse hierarchy. Cases (6) and (7) have 3 nearly degenerate masses. The versatility of Eq. (5) has clearly been demonstrated.
The above 7 cases encompass all models of the neutrino mass matrix that have ever been proposed which also satisfy Eq. (1). They are also very useful for discussing the possibility of neutrinoless double beta ($\beta
\beta_{0\nu}$) decay in the context of neutrino oscillations [@bbmass]. The effective mass $m_0$ measured in $\beta \beta_{0\nu}$ decay is $|a+2b+2c|$. However, neutrino oscillations constrain $|a+2b+c|$ and $\sqrt{c^2+2d^2}$, as well as $|d/c|$. Using $$|a+2b+2c| = ||a+2b+c| \pm |c|| = ||a+2b+c| \pm \cos 2 \theta \sqrt{c^2+2d^2}|,$$ the following conditions on $m_0$ are easily obtained: $$\begin{aligned}
(1) && m_0 \simeq \sin^2 \theta |m_2| \simeq \sin^2 \theta \sqrt
{\Delta m^2_{sol}}, \\
(2) && m_0 \simeq |m_{1,2}| << \sqrt {\Delta m^2_{atm}}, \\
(3) && m_0 \simeq \cos 2 \theta |m_{1,2}| << \cos 2 \theta \sqrt
{\Delta m^2_{atm}}, \\
(4) && m_0 \simeq \sqrt {\Delta m^2_{atm}}, \\
(5) && m_0 \simeq \cos 2 \theta \sqrt {\Delta m^2_{atm}}, \\
(6) && m_0 \simeq |m_{1,2,3}|, \\
(7) && m_0 \simeq \cos 2 \theta |m_{1,2,3}|.\end{aligned}$$ If $m_0$ is measured [@klapdor] to be significantly larger than 0.05 eV, then only Cases (6) and (7) are allowed. However, as Eqs. (20) and (21) show, the true mass of the three neutrinos is still subject to a two-fold ambiguity, which is a well-known result.
The underlying symmetry of Eq. (5) which results in $U_{e3} = 0$ is its invariance under the interchange of $\nu_\mu$ and $\nu_\tau$. Its mass eigenstates are then separated according to whether they are even $(\nu_{1,2})$ or odd $(\nu_3)$ under this interchange, as shown by Eq. (1). To obtain $U_{e3} \neq 0$, this symmetry has to be broken. One interesting possibility is to rewrite Eq. (5) as $${\cal M}_\nu = \pmatrix {a+2b+2c & d & d^* \cr d & b & a+b \cr d^* & a+b & b},$$ where $a,b,c$ are real but $d$ is complex. This reduces to Eq. (4) if $Im d = 0$, but if $Im d \neq 0$, then $U_{e3} \neq 0$.
To obtain $U_{e3}$ in a general way, first rotate to the basis spanned by $\nu_e, (\nu_\mu+\nu_\tau)/\sqrt 2$, and $(\nu_\tau-\nu_\mu)/\sqrt 2$, i.e. $${\cal M}_\nu = \pmatrix {a+2b+2c & \sqrt2 Red & -\sqrt2 i Imd \cr \sqrt2 Red
& a+2b & 0 \cr -\sqrt 2 i Imd & 0 & -a}$$ Whereas ${\cal M}_\nu$ is diagonalized by $$U {\cal M}_\nu U^T = \pmatrix {m_1 & 0 & 0 \cr 0 & m_2 & 0 \cr 0 & 0 & m_3},$$ ${\cal M}_\nu {\cal M}_\nu^\dagger$ is diagonalized by $$U ({\cal M}_\nu {\cal M}_\nu^\dagger) U^\dagger = \pmatrix {|m_1|^2 & 0 & 0
\cr 0 & |m_2|^2 & 0 \cr 0 & 0 & |m_3|^2}.$$ Here $${\cal M}_\nu {\cal M}_\nu^\dagger = \pmatrix {(a+2b+2c)^2 + 2|d|^2 &
2 \sqrt2 (a+2b+c) Red & 2 \sqrt2 i (a+b+c) Imd \cr 2 \sqrt2 (a+2b+c) Red &
(a+2b)^2 + 2(Red)^2 & 2 i Red Imd \cr -2 \sqrt2 i (a+b+c) Imd & -2 i Red Imd
& a^2 + 2(Imd)^2}.$$ To obtain $U_{e3}$ for small $Imd$, consider the matrix $$A = {\cal M}_\nu {\cal M}_\nu^\dagger - [a^2 + 2(Imd)^2] I,$$ where $I$ is the identity matrix. Now $A$ is diagonalized by $U$ as well and $U_{e3}$ is simply given by $$U_{e3} \simeq {A_{e3} \over A_{ee}} = {2 \sqrt 2 i (a+b+c) Imd \over
(a+2b+2c)^2 - a^2 + 2 (Red)^2}$$ to a very good approximation and leads to $$\begin{aligned}
(1), (2), (3) && U_{e3} \simeq {-\sqrt 2 i Imd \over a}, \\
(4), (6) && U_{e3} \simeq {i Imd \over \sqrt 2 b}, \\
(5) && U_{e3} \simeq {\sqrt2 i Imd \over c}, \\
(7) && U_{e3} \simeq {\sqrt2 i (a+c) Imd \over c^2 - a^2 + 2(Red)^2}.\end{aligned}$$ In all cases, the magnitude of $U_{e3}$ can be as large as the present experimental limit [@react] of 0.16 and its phase is $\pm \pi/2$. Thus the CP violating effect in neutrino oscillations is predicted to be maximal by Eq. (22), which is a very desirable scenario for future long-baseline neutrino experiments.
The above analysis shows that for $U_{e3} = 0$ and $\sin^2 2 \theta_{atm} = 1$, the seven cases considered cover all possible patterns of the $3 \times 3$ Majorana neutrino mass matrix, as indicated by present atmospheric and solar neutrino data. Any successful model should predict Eq. (5) at least as a first approximation. One such example already exists [@a4], where $b=c=d=0$ corresponds to the non-Abelian discrete symmetry $A_4$, i.e. the finite group of the rotations of a regular tetrahedron. This leads to Case (6), i.e. three nearly degenerate masses, with the common mass equal to that measured in $\beta \beta_{0\nu}$ decay. It has also been shown recently [@bmv] that starting with this pattern, the correct mass matrix, i.e. Eq. (22) with the complex phase in the right place, is automatically obtained with the most general application of radiative corrections. In particular, if soft supersymmetry breaking is assumed to be the origin of these radiative corrections, then the neutrino mass matrix is correlated with flavor violation in the slepton sector, and may be tested in future collider experiments.
Suppose that at some high energy scale, the charged lepton mass matrix and the Majorana neutrino mass matrix are such that after diagonalizing the former, i.e. $${\cal M}_l = \pmatrix {m_e & 0 & 0 \cr 0 & m_\mu & 0 \cr 0 & 0 & m_\tau},$$ the latter is of the form $${\cal M}_\nu = \pmatrix {m_0 & 0 & 0 \cr 0 & 0 & m_0 \cr 0 & m_0 & 0}.$$ From the high scale to the electroweak scale, one-loop radiative corrections will change ${\cal M}_\nu$ as follows: $$({\cal M}_\nu)_{ij} \to ({\cal M}_\nu)_{ij} + R_{ik} ({\cal M}_\nu)_{kj}
+ ({\cal M}_\nu)_{ik} R_{kj},$$ where the radiative correction matrix is assumed to be of the most general form, i.e. $$R = \pmatrix {r_{ee} & r_{e\mu} & r_{e\tau} \cr r_{e\mu}^* & r_{\mu\mu} &
r_{\mu\tau} \cr r_{e\tau}^* & r_{\mu\tau}^* & r_{\tau\tau}}.$$ Thus the observed neutrino mass matrix is given by $${\cal M}_\nu = m_0 \pmatrix {1+2r_{ee} & r_{e\tau} + r_{e\mu}^* & r_{e\mu} +
r_{e\tau}^* \cr r_{e\mu}^* + r_{e\tau} & 2r_{\mu\tau} & 1+r_{\mu\mu}+
r_{\tau\tau} \cr r_{e\tau}^* + r_{e\mu} & 1+r_{\mu\mu}+r_{\tau\tau} &
2r_{\mu\tau}^*}.$$ Now $r_{\mu\tau}$ may be chosen real by absorbing its phase into $\nu_\mu$ and $\nu_\tau$. Then using the redefinitions: $$\begin{aligned}
&& \delta_0 \equiv r_{\mu\mu} + r_{\tau\tau} - 2r_{\mu\tau}, \\
&& \delta \equiv 2r_{\mu\tau}, \\
&& \delta' \equiv r_{ee} - {1 \over 2} r_{\mu\mu} - {1 \over 2} r_{\tau\tau}
- r_{\mu\tau}, \\
&& \delta'' \equiv r_{e\mu}^* + r_{e\tau},\end{aligned}$$ the neutrino mass matrix becomes $${\cal M}_\nu = m_0 \pmatrix{1+\delta_0+2\delta+2\delta' & \delta'' &
\delta''^* \cr \delta'' & \delta & 1+\delta_0+\delta \cr \delta''^* &
1+\delta_0+\delta & \delta},$$ which is exactly that of Eq. (22). In other words, starting with Eq. (34), the correct ${\cal M}_\nu$ is automatically obtained. \[To simplify Eq. (42) without any loss of generality, $\delta_0$ will be set equal to zero from here on.\]
The successful derivation of Eq. (42) depends on having Eqs. (33) and (34). To be sensible theoretically, they should be maintained by a symmetry. At first sight, it appears impossible that there can be a symmetry which allows them to coexist. Here is where the non-Abelian discrete symmetry $A_4$ comes into play [@a4]. The key is that $A_4$ has three inequivalent one-dimensional representations , $'$, $''$, and one three-dimensional reprsentation , with the decomposition $$\underline {3} \times \underline {3} = \underline {1} + \underline {1}' +
\underline {1}'' + \underline {3} + \underline {3}.$$ This allows the following natural assignments of quarks and leptons: $$\begin{aligned}
&& (u_i,d_i)_L, ~~ (\nu_i,e_i)_L \sim \underline {3}, \\
&& u_{1R}, ~~ d_{1R}, ~~ e_{1R} \sim \underline {1}, \\
&& u_{2R}, ~~ d_{2R}, ~~ e_{2R} \sim \underline {1}', \\
&& u_{3R}, ~~ d_{3R}, ~~ e_{3R} \sim \underline {1}''.\end{aligned}$$ Heavy fermion singlets are then added [@bmv]: $$U_{iL(R)}, ~~ D_{iL(R)}, ~~ E_{iL(R)}, ~~ N_{iR} \sim \underline {3},$$ together with the usual Higgs doublet and new heavy singlets: $$(\phi^+,\phi^0) \sim \underline {1}, ~~~~ \chi^0_i \sim \underline {3}.$$ With this structure, charged leptons acquire an effective Yukawa coupling matrix $\bar e_{iL} e_{jR} \phi^0$ which has 3 arbitrary eigenvalues (because of the 3 independent couplings to the 3 inequivalent one-dimensional representations) and for the case of equal vacuum expectation values of $\chi_i$, i.e. $$\langle \chi_1 \rangle = \langle \chi_2 \rangle = \langle \chi_3 \rangle = u,$$ the unitary transformation $U_L$ which diagonalizes ${\cal M}_l$ is given by $$U_L = {1 \over \sqrt 3} \pmatrix {1 & 1 & 1 \cr 1 & \omega & \omega^2 \cr
1 & \omega^2 & \omega},$$ where $\omega = e^{2\pi i/3}$. This implies that the effective neutrino mass operator, i.e. $\nu_i \nu_j \phi^0 \phi^0$, is proportional to $$U_L^T U_L = \pmatrix {1 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0},$$ exactly as desired [@a4; @bmv].
To derive Eq. (52), the validity of Eq. (50) has to be proved. This is naturally accomplished in the context of supersymmetry. Let $\hat \chi_i$ be superfields, then its superpotential is given by $$\hat W = {1 \over 2} M_\chi (\hat \chi_1 \hat \chi_1 + \hat \chi_2 \hat \chi_2
+ \hat \chi_3 \hat \chi_3) + h_\chi \hat \chi_1 \hat \chi_2 \hat \chi_3.$$ Note that the $h_\chi$ term is invariant under $A_4$, a property not found in $SU(2)$ or $SU(3)$. The resulting scalar potential is $$V = |M_\chi \chi_1 + h_\chi \chi_2 \chi_3|^2 + |M_\chi \chi_2 + h_\chi \chi_3
\chi_1|^2 + |M_\chi \chi_3 + h_\chi \chi_1 \chi_2|^2.$$ Thus a supersymmetric vacuum $(V=0)$ exists for $$\langle \chi_1 \rangle = \langle \chi_2 \rangle = \langle \chi_3 \rangle = u
= -M_\chi /h_\chi,$$ proving Eq. (50), with the important additional result that the spontaneous breaking of $A_4$ at the high scale $u$ does not break the supersymmetry.
(270,110)(0,0) (0,50)(50,50) (270,50)(220,50) (50,50)(170,50) (170,50)(220,50) (110,50)(60,0,180)[4]{} (110,40)\[\][$\tilde w$]{} (25,40)\[\][$\nu_\mu$]{} (195,60)\[\][$\nu_\tau$]{} (245,60)\[\][$\nu_\mu$]{} (55,105)\[\][$\tilde \mu_L$]{} (165,105)\[\][$\tilde \tau_L$]{} (110,110)\[\][$\times$]{} (170,10)(220,50)[4]{} (270,10)(220,50)[4]{} (170,0)\[\][$\phi_2^0$]{} (270,0)\[\][$\phi_2^0$]{}
(270,110)(0,0) (0,50)(75,50) (270,50)(195,50) (0,50)(270,50) (135,10)(135,50)[4]{} (195,10)(195,50)[4]{} (135,50)(60,0,180)[4]{} (105,40)\[\][$\tilde w$]{} (165,40)\[\][$\tilde \phi_2$]{} (37,40)\[\][$\nu_\mu$]{} (235,40)\[\][$\nu_\mu$]{} (135,0)\[\][$\phi_2^0$]{} (195,0)\[\][$\phi_2^0$]{} (80,105)\[\][$\tilde \mu_L$]{} (190,105)\[\][$\tilde \tau_L$]{} (135,110)\[\][$\times$]{}
To generate the proper radiative corrections which will result in a realistic Majorana neutrino mass matrix, $A_4$ is assumed broken also by the soft supersymmetry breaking terms. In particular, the mass-squared matrix of the left sleptons will be assumed to be arbitrary. This allows $r_{\mu\tau}$ to be nonzero through $\tilde \mu_L - \tilde \tau_L$ mixing, from which the parameter $\delta$ may be evaluated, as shown in Figs. 1 and 2. For illustration, using the approximation that $\tilde m_1^2 >> \mu^2 >>
M_{1,2}^2 = \tilde m_2^2$, where $\mu$ is the Higgsino mass and $M_{1,2}$ are gaugino masses, I find $$\delta = {\sin \theta \cos \theta \over 16 \pi^2} \left[ (3g_2^2-g_1^2) \ln
{\tilde m_1^2 \over \mu^2} - {1 \over 4} (3g_2^2+g_1^2) \left( \ln
{\tilde m_1^2 \over \tilde m_2^2} - {1 \over 2} \right) \right].$$ Using $\Delta m_{32}^2 = 2.5 \times 10^{-3}$ eV$^2$ from the atmospheric neutrino data, this implies that $$\left[ \ln {\tilde m_1^2 \over \mu^2} - 0.3 \left( \ln {\tilde m_1^2 \over
\tilde m_2^2} - {1 \over 2} \right) \right] \sin \theta \cos \theta
\simeq 0.535 \left( {0.4 ~{\rm eV} \over m_0} \right)^2.$$ To the extent that the factor on the left cannot be much greater than unity, this means that $m_0$ cannot be much smaller than about 0.4 eV [@klapdor].
In the presence of $Im \delta''$, as shown by Eq. (30), $$U_{e3} \simeq {i Im \delta'' \over \sqrt 2 \delta},$$ and the previous expressions for the neutrino mass eigenvalues are still approximately valid with the replacement of $\delta'$ by $\delta' +
(Im \delta'')^2 /2 \delta$ and of $\delta''$ by $Re \delta''$. There is also the relationship $$\left[ {\Delta m_{12}^2 \over \Delta m_{32}^2} \right]^2 \simeq \left[
{\delta' \over \delta} + |U_{e3}|^2 \right]^2 + \left[ {Re \delta'' \over
\delta} \right]^2.$$ Using $\Delta m_{12}^2 \simeq 5 \times 10^{-5}$ eV$^2$ from solar neutrino data and $|U_{e3}| < 0.16$ from reactor neutrino data [@react], I find $$\begin{aligned}
&& Im \delta'' < 8.8 \times 10^{-4} ~(0.4~{\rm eV}/m_0)^2, \\
&& Re \delta'' < 7.8 \times 10^{-5} ~(0.4~{\rm eV}/m_0)^2.\end{aligned}$$
In conclusion, recent experimental progress on neutrino oscillations points to a neutrino mixing matrix which can be understood in a systematic way [@ma02] in terms of an all-purpose neutrino mass matrix, i.e. Eq. (5), and its simple extension, i.e. Eq. (22), to allow for a nonzero and $imaginary$ $U_{e3}$, i.e. Eq. (28). Seven possible cases have been identified, each with a different prediction for $\beta \beta_{0\nu}$ decay, i.e. Eqs. (15) to (21). A specific example is that of an underlying $A_4$ symmetry at some high energy scale, which allows the observed Majorana neutrino mass matrix to be derived from radiative corrections. It has been shown [@bmv] that this automatically leads to $\sin^2 2 \theta_{atm} = 1$ and a large (but not maximal) solar mixing angle. Using neutrino oscillation data, and assuming radiative corrections from soft supersymmetry breaking, the effective mass measured in neutrinoless double beta decay is predicted to be not much less than 0.4 eV.
This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837.
[99]{} Q. R. Ahmad [*et al.*]{}, SNO Collaboration, Phys. Rev. Lett. [**89**]{}, 011301, 011302 (2002). S. Fukuda [*et al.*]{}, Super-Kamiokande Collaboration, Phys. Rev. Lett. [**86**]{}, 5656 (2001) and references therein; Q. R. Ahmad [*et al.*]{}, SNO Collaboration, Phys. Rev. Lett. [**87**]{}, 071301 (2001). S. Fukuda [*et al.*]{}, Super-Kamiokande Collaboration, Phys. Rev. Lett. [**85**]{}, 3999 (2000) and references therein. V. Barger [*et al.*]{}, Phys. Lett. [**B537**]{}, 179 (2002); A. Bandyopadhyay [*et al.*]{}, Phys. Lett. [**B540**]{}, 14 (2002); J. N. Bahcall [*et al.*]{}, JHEP [**07**]{}, 054 (2002); P. Aliani [*et al.*]{}, hep-ph/0205053; P. C. de Holanda and A. Yu. Smirnov, hep-ph/0205241; A. Strumia [*et al.*]{}, hep-ph/0205261; G. L. Fogli [*et al.*]{}, hep-ph/0206162. M. Apollonio [*et al.*]{}, Phys. Lett. [**B466**]{}, 415 (1999); F. Boehm [*et al.*]{}, Phys. Rev. [**D64**]{}, 112001 (2001). E. Ma, hep-ph/0207352. See for example H. V. Klapdor-Kleingrothaus and U. Sarkar, Phys. Lett. [**B532**]{}, 71 (2002); S. Pascoli and S. T. Petcov, hep-ph/0205022; and references therein. Evidence for $m_0$ in the range 0.11 to 0.56 eV has been claimed by H. V. Klapdor-Kleingrothaus [*et al.*]{}, Mod. Phys. Lett. [**A16**]{}, 2409 (2001). See also comments by C. E. Aalseth [*et al.*]{}, hep-ex/0202018, and the reply by H. V. Klapdor-Kleingrothaus, hep-ph/0205228. E. Ma and G. Rajasekaran, Phys. Rev. [**D64**]{}, 113012 (2001); E. Ma, Mod. Phys. Lett. [**A17**]{}, 289 (2002); E. Ma, Mod. Phys. Lett. [**A17**]{}, 627 (2002). K. S. Babu, E. Ma, and J. W. F. Valle, hep-ph/0206292.
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author:
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[^1]\
Physics Department, Brookhaven National Laboratory, Upton, NY 11973 USA\
E-mail:
title: On the trace anomaly in 2+1 flavor QCD
---
Introduction
============
At high temperatures strongly interacting matter undergoes an transition to a new state where quark and gluons are not subject to confinement; it shows up in a qualitative change in the equation of state (see [@Petreczky:2012rq] for a review). The most common way to calculate the QCD equation of state is the integral method. The pressure is calculated as an integral of the trace of the energy momentum tensor $\varepsilon-3p$ or the trace anomaly $$\frac{p}{T^4}-\frac{p_0}{T_0^4}=\int_{T_0}^T
dT'\frac{\varepsilon - 3p}{T'^5}.$$ The study of the trace anomaly is also interesting in its own right. At high temperature it is determined by the running of the QCD coupling constant $\alpha_s$, and at leading order it is proportional $\alpha_s^2 T^4$. Thus it provides a sensitive test of the weakly coupled nature of the quark gluon plasma. For this reason it is also often called the interaction measure. Thermodynamic quantities are expected to have large cutoff effects at high temperatures when calculated using an unimproved fermion formulation, even if the lattice spacing is small in absolute units. These cutoff effects arise from the distortion of the quark dispersion relation on the lattice and can be cured by using an improved action [@heller; @hegde]. On the other hand, at low and intermediate temperatures the cutoff effects in thermodynamic quantities can be understood in terms of the cutoff effects of the hadron spectrum. In particular, the breaking of the so-called taste symmetry in the staggered fermion formulation could be the dominant source of cutoff effects when using staggered quarks. To control cutoff effects in pressure and other thermodynamic quantities, one first needs to understand the cutoff dependence of the trace anomaly. In this contribution we will study in detail the cutoff dependence of the trace anomaly and compare our lattice results with the hadron resonance gas model at low temperature and with resummed perturbation theory at high temperatures.
{width="49.50000%"}{width="46.20000%"}
\[fig\_Tr1\]
Numerical results
=================
We calculate the interaction measure in 2+1 flavor QCD using a tree-level improved action for gauge fields and the highly improved staggered quark (HISQ) action [@HPQCD]. This combination of gauge and quark action is called the HISQ/tree action. It can largely reduce cutoff effects in thermodynamic quantities both at low and high temperatures [@proc10; @tc].
The trace anomaly can be written in terms of the expectation values of gauge action and quark condensates as $$\label{tram}
\frac{\varepsilon-3p}{T^4} = R_\beta[\langle S_{gauge}\rangle_0-
\langle S_{gauge}\rangle_T]
-R_\beta R_m[2m_l(\langle \bar l l\rangle_0-
\langle \bar l l\rangle_T)+m_s(\langle \bar s s \rangle_0-
\langle \bar s s\rangle_T)],$$ where the subscript “$0$” refers to $T=0$ and “$T$” to finite temperature for observables evaluated at the same value of the cutoff. Furthermore, $R_\beta$ and $R_m$ are the lattice beta function and mass anomalous dimension that are defined below. The above expression is free of ultraviolet divergences. On a hypercubic lattice $N_s^3\times N_\tau$ the physical temperature is set by the size of the temporal dimension and the lattice spacing as $T=1/(N_\tau a)$. For $T=0$ calculations we use $N_\tau \geqslant N_s$, and for $T>0$ we keep $N_s/N_\tau=4$ and at fixed $N_\tau$ vary the lattice spacing $a$ by varying the gauge coupling $\beta=10/g^2$. The continuum limit in this setup corresponds to $N_\tau\to\infty$. Therefore, we carried out this study on lattices with $N_\tau=4,~6$, $8$, $10$ and $12$. The strange quark mass $m_s$ is tuned to the physical value, while the two degenerate light quarks have masses $m_l=m_s/20$, slightly heavier than physical ($m_l\simeq m_s/27$). In the continuum limit these light quark masses correspond to the pion mass of about $160$ MeV. To set the lattice spacing in physical units (fm) we use the $r_1$ scale defined through static quark potential $$r^2 \frac{d V}{d r}|_{r=r_1}=1.$$ We use the value $r_1=0.3106$ fm [@MILC_r1]. Alternatively, the kaon decay constant, $f_K=156.1$ MeV is used to set the scale. >From Fig. \[fig\_Tr1\] (left) one can see how the choice of reference observables affects the conversion of temperature from lattice units to MeV. Over the temperature range of interest on $N_\tau=6$ lattices the difference is within $9$ MeV, and on $N_\tau=12$ within $2$ MeV. Another effect of using different scales is related to the $\beta$-function and mass anomalous dimension that enter in Eq. (\[tram\]) $$R_\beta(\beta)=-a\frac{d\beta}{da},\,\,\,\,\,
R_m(\beta)=\frac{1}{m_s(\beta)}\frac{dm_s(\beta)}{d\beta},$$ where $m_s(\beta)$ defines a line of constant physics (LCP), *i.e.*, the combination of the gauge coupling and strange quark mass such that the kaon mass (in MeV) stays approximately constant in the whole $\beta$ range used in the simulation.
![Comparison of the HISQ/tree interaction measure on $N_\tau=4,~6$, $8$, $10$ and $12$ lattices with the stout continuum estimate [@stout_eos] (left), $N_\tau=6$ and $10$ HISQ/tree data (right). Filled (open) symbols in the right panel correspond to the $r_1$ ($f_K$) scale, see text. ](figures/e-3p_hisq_0921.eps "fig:"){width="49.50000%"}![Comparison of the HISQ/tree interaction measure on $N_\tau=4,~6$, $8$, $10$ and $12$ lattices with the stout continuum estimate [@stout_eos] (left), $N_\tau=6$ and $10$ HISQ/tree data (right). Filled (open) symbols in the right panel correspond to the $r_1$ ($f_K$) scale, see text. ](figures/e-3p_106.eps "fig:"){width="49.50000%"}
\[fig\_EoS\_hisq\]
$R_\beta$ for $r_1$ and $f_K$ scale is shown in Fig. \[fig\_Tr1\] (right), together with the 2-loop perturbative result.
The present status of the interaction measure with the HISQ/tree action is shown in Fig. \[fig\_EoS\_hisq\] together with the stout continuum estimate of Ref [@stout_eos]. For temperature $T<180$MeV we see good agreement between our HISQ/tree results and the stout continuum estimate. At higher temperatures we see significant discrepancies between HISQ/tree results and the stout results, though the discrepancies seem to disappear at temperatures $T>350$MeV. The height of the peak decreases slightly when going from $N_{\tau}=8$ to $N_{\tau}=10$ and $12$. Thus one may expect that the discrepancies between HISQ/tree and stout actions in the peak region will be reduced once the continuum extrapolation for HISQ/tree is performed. It should be noted that in the previous calculations with p4 and asqtad actions [@rbc08; @hoteos; @rbc10] the trace anomaly was significantly smaller due to large cutoff effects.
Next we study the effect of the scale setting on the trace anomaly. In Fig. \[fig\_EoS\_hisq\] (right) the results for $N_\tau=6$ and $10$ are shown using $r_1$ and $f_K$ to set the scale. Different scale settings affect the $N_{\tau}=6$ results, in particular in the peak region the trace anomaly is smaller if $f_K$ is used to set the scale. However, for $N_{\tau}=10$ there is almost no difference between the two scale setting procedures.
We also calculated the trace anomaly using the asqtad action. The expression for the trace anomaly in terms of expectation values of local operators is given in Ref. [@hoteos]. In Fig. \[fig\_hisq\_12\] we compare our results obtained with HISQ/tree action for $N_\tau=8$ and $N_\tau=12$ using the scale set with both $r_1$ (filled symbols) and $f_K$ (open symbols). If $r_1$ is used to set the scale, there is a significant discrepancy between the HISQ/tree and the asqtad result for $T<200$ MeV for $N_{\tau}=8$. As discussed above this is due to large cutoff effects at low temperatures when the asqtad action is used. The discrepancy is significantly smaller for $N_{\tau}=12$, as expected. If we use $f_K$ to set the scale, the discrepancies between HISQ/tree results and asqtad results are greatly reduced. Note, however, that the peak height in the $N_{\tau}=8$ asqtad data is reduced when $f_K$ is used to set the scale. Fig. \[fig\_hisq\_12\] also shows that the difference between the two scale setting procedures for the asqtad action is reduced in the case of $N_{\tau}=12$ data. This is again expected. There is, however still a significant reduction in the peak height if $f_K$ scale is used.
{width="49.50000%"}{width="49.50000%"} {width="49.50000%"}{width="49.50000%"}
\[fig\_hisq\_12\]
The low-temperature behavior of the interaction measure is shown in Fig. \[fig\_hisq\_high\] (left) and compared to the hadron resonance gas (HRG) calculation shown as the black line. At low temperatures, cutoff effects arising from the taste symmetry breaking of the staggered fermion formulation could be significant. To take into account possible cutoff effects in the comparison with HRG we replaced the contribution of each pion and kaon with a contribution averaged over the sixteen different tastes of pseudo-scalar mesons. The pseudo-scalar meson masses for each taste as function of the lattice spacing have been previously estimated by the HotQCD collaboration [@tc]. In Fig. \[fig\_hisq\_high\] we show the HRG results with the modified pseudo-scalar meson sector as colored lines for each value of $N_{\tau}$. We use the same color coding for the lattice results and the modified HRG result. Despite the fact that even for HISQ/tree action the effects of taste breaking in the pseudo-scalar sector are significant, the resulting cutoff effects are surprisingly small. They are of the same order or smaller than the statistical errors for all $N_{\tau}$ values, including $N_{\tau}=6$. Fig. \[fig\_hisq\_high\] shows that the lattice result starts to disagree with the HRG model at $T\sim150-160$ MeV.
Finally, let us compare our results for the trace anomaly with earlier calculations using the p4 action [@rbc08; @hoteos] as well as with resummed perturbation theory [@mike]. In Fig. \[fig\_hisq\_high\] (right) we compare our results for the HISQ/tree action with previous results obtained with p4 action and the stout continuum estimate. For $T>400$ MeV different lattice data agree well with each other. The only exception is the $N_{\tau}=4$ p4 data, which show the expected cutoff effects at high temperatures. The $N_{\tau}=4$ HISQ/tree data should show similar cutoff dependence, but interestingly enough this is not observed. It is important to stress that at high temperatures the cutoff effects for p4 action should be small for $N_{\tau} \ge 6$. Thus the agreement between p4 results and HISQ/tree results is expected. We see quite good agreement with the resummed perturbative results [@mike].
Conclusion
==========
We studied the trace anomaly in QCD using HISQ/tree and asqtad actions and lattices with temporal extent $N_{\tau}=4,~,6,~8,~10$ and $12$. Using different observables to set the scale, $r_1$ and $f_K$, allows for a crude estimate of the magnitude of cutoff effects, which seem to be small at the finest $N_\tau=12$ lattices both for HISQ/tree and asqtad actions. Clearly to get reliable continuum results for the trace anomaly and thus for the equation of state the calculations on $N_\tau=10$ and $12$ lattices need to be significantly extended. We also compared our lattice results with HRG at low temperatures and resummed perturbative results at high temperatures. The trace anomaly is well described by HRG for $T<150$MeV. We also find a good agreement between lattice and resummed perturbative results at high temperatures.
{width="49.50000%"}{width="49.50000%"}
\[fig\_hisq\_high\]
Acknowledgments {#acknowledgments .unnumbered}
===============
This work has been supported by contract DE-AC02-98CH10886 with the U.S. Department of Energy. The numerical simulations have been performed on BlueGene/L computers at Lawrence Livermore National Laboratory (LLNL), the New York Center for Computational Sciences (NYCCS) at Brookhaven National Laboratory, US Teragrid (Texas Advanced Computing Center), Cray XE6 at the National Energy Research Scientific Computing Center (NERSC), and on clusters of the USQCD collaboration in JLab and FNAL. The calculations for asqtad actions have been performed on BG/P computers of John von Neumann center in Jülich.
[00]{}
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[^1]: for HotQCD Collaboration: A. Bazavov, T. Bhattacharya, M. Buchoff, M. Cheng, N. Christ, C. DeTar, H.-T. Ding, S. Gottlieb, R. Gupta, P. Hegde, U. Heller, C. Jung, F. Karsch, E. Laermann, L. Levkova, Z. Lin, R. Mawhinney, S. Mukherjee, P. Petreczky, D. Renfrew, C. Schmidt, C. Schroeder, W. Soeldner, R. Soltz, R. Sugar, D. Toussaint, P. Vranas
|
---
abstract: 'We present results from an X-ray and radio study of the merging galaxy cluster Abell 115. We use the full set of 5 Chandra observations taken of A115 to date (360 ks total integration) to construct high-fidelity temperature and surface brightness maps. We also examine radio data from the Very Large Array at 1.5 GHz and the Giant Metrewave Radio Telescope at 0.6 GHz. We propose that the high X-ray spectral temperature between the subclusters results from the interaction of the bow shocks driven into the intracluster medium by the motion of the subclusters relative to one another. We have identified morphologically similar scenarios in Enzo numerical N-body/hydrodynamic simulations of galaxy clusters in a cosmological context. In addition, the giant radio relic feature in A115, with an arc-like structure and a relatively flat spectral index, is likely consistent with other shock-associated giant radio relics seen in other massive galaxy clusters. We suggest a dynamical scenario that is consistent with the structure of the X-ray gas, the hot region between the clusters, and the radio relic feature.'
author:
- 'Eric J. Hallman$^{\dagger}$, Brian Alden, David Rapetti, Abhirup Datta, Jack O. Burns'
bibliography:
- 'references.bib'
title: 'Probing the Curious Case of a Galaxy Cluster Merger in Abell 115 with High Fidelity Chandra X-ray Temperature and Radio Maps'
---
Introduction
============
In a cold dark matter dominated universe, structures form hierarchically, leading to mergers of smaller gravitationally bound objects into bigger ones. When the most massive structures – galaxy clusters – merge, they result in the most energetic events in the universe since the Big Bang. Observational evidence of shocked gas in merging galaxy clusters is now relatively common [@bullet; @Markevitch:07; @canning; @emery]. In particular, X-ray surface brightness and temperature maps show features that strongly suggest they result from shocks driven by the supersonic motion of merging subclusters. A shock in the intracluster medium (ICM) will result in both a density and temperature discontinuity in the gas, which creates an X-ray excess and a spectral temperature jump. Shocks can also manifest observationally in the form of radio “relics", arc-like structures in the outskirts. They are believed to be the result of shocks induced by mergers, compression of radio lobes, or the remnants of radio galaxies [for recent reviews, see @bruggenReview; @bruJones]. Except in a few cases [e.g., @finoguenov; @Datta:14] shocks in the ICM associated with radio relics are not detectable by their X-ray emission, due to their location far from the cluster center (r $>$ 1 Mpc) where the X-ray surface brightness is too low to reliably detect an enhancement [@hoeft07]. Prior work [@Botteon:16] suggests that Abell 115 may also be an example where there is an X-ray shock at the location of the relic.
A reasonable expectation is that all merging galaxy clusters will contain shocks [@ryu03; @skillman08; @vazza11], some of which may be observable through X-ray observations, modulo orientation effects that may smear out the contrast across the steep, narrow pressure discontinuity, and the X-ray surface brightness in the local gas.
Abell 115
---------
Abell 115 is a well-known massive (total virial mass $M_v \approx 3\times10^{15} M_\odot$) galaxy cluster at redshift $z = 0.192$, with double peaked structure in the X-ray surface brightness [@Forman:81; @Shibata:99; @Gutierrez:05]. Optical studies [@Barrena:07] indicate two distinct, redshift-separable components in the local galaxy population, roughly coincident with the two X-ray peaks. Both subclusters have disturbed morphology in the X-ray, and host cool cores, where some of the cool gas appears to be in the process of being stripped from the cluster core. X-ray and optical observational data, therefore, are consistent with the interpretation that the two subclusters are in the process of a merger. In addition, the northern subcluster hosts a 3C radio source at its center, 3C28 [@forman10]. As has been noted by prior studies, to the northeast of 3C28, there exists extended, diffuse radio emission in an arc-like structure that seems consistent with the appearance of other so-called cluster radio “relics" [@govoni:01; @Botteon:16]. While some of the radio emission appears to come directly from discrete radio galaxies, there is significant radio structure stretching between these individual sources. One possible interpretation of the extended radio structure is that this emission results from particle acceleration at a shock, or shocks, driven into the ICM by the motion of the subclusters relative to one another. Arc-like relics exist in other massive galaxy clusters [e.g., @rottgering; @bonafede; @giovannini; @sausage], and in many cases, a convincing argument has been made that the emission results from synchrotron radiation of a Fermi-accelerated particle population in the ambient intracluster magnetic field. Indeed, prior work on A115 has made a similar argument [@Botteon:16], citing evidence in the X-ray data for a shock feature coincident with the location of the extended radio structure to the northeast of 3C28.
Until now, a detailed description of the dynamics of A115 that is consistent with all of the multi-wavelength observations of A115 has not been offered. In this work we deduce the dynamics of A115 using evidence from the X-ray, radio and optical observations, in addition to comparison with numerical simulations. In Section \[sec:Chandra\], we discuss the data reduction we used for the Chandra X-ray observations of A115. In Section \[sec:RadioData\], we describe the data reduction of the VLA and GMRT radio observations. In Section \[sec:StrucAndDyn\], we give a description of the X-ray temperature structure and the likely dynamics of the cluster. In Section \[sec:RadioRelic\], we combine our evidence from both X-ray and radio data to interpret the location and morphology of the radio relic. In Section \[sec:Discussion\], we summarize the results and suggest future work.
Chandra X-ray Data Reduction {#sec:Chandra}
============================
We used multiple *Chandra* observations in our analysis. The *Chandra* observations (IDs 3233, 13458, 13459, 15578, 15581) were obtained from the Chandra Data Archive. The observations for 3233 were taken in 2002, while the later 4 were taken in November of 2012. Exposure times were $\sim$50 ks, $\sim$115 ks, $\sim$100 ks, $\sim$65 ks and $\sim$30 ks respectively, for a combined total of roughly 360 ks across the 5 observations. All were observed in VFAINT mode.
{width="70.00000%"}
The X-ray “Pypeline" for Data Reduction and Temperature Maps
------------------------------------------------------------
The X-ray data reduction is described in detail in @Schenck:14 and @Datta:14. The data reduction described in these studies has been aggregated into a data pipeline, which is designed to take *Chandra* observations and generate high resolution adaptive circular binned (ACB) temperature maps. Once *Chandra* observation IDs are given as input, the pipeline automatically downloads the data using *CIAO*[^1] and merges the multiple observations into a single image. Currently, the end user needs to provide a SAO DS9[^2] region file containing the point sources for exclusion. Once the sources file is given to the pipeline, it removes the sources from the images. The pipeline then generates light curves for each observation and removes flares. The user has the ability to inspect and customize this process to ensure accuracy. Response files are then generated for each observation using *specextract*. To create the ACB temperature map, $~\mathcal{O}(10^4)$ spectral fits are generated. This part of the process can be done in parallel on a supercomputer to drastically reduce the time required to complete the map.
The ACB temperature maps were produced using a method adapted from @Randall:08 [@Randall:10]. In these two papers, spectra were extracted from circular regions which were just large enough to reach some threshold of counts. The fitted temperature of each region was assigned to the pixel at the center of the circle. The circles are allowed to overlap, so some pixels will share counts with other pixels and the fitted temperatures will not be independent from one another.
The pipeline itself is written in Python and available for the community at large as an open source project on GitHub[^3]. Future desired functionality, includes native python multi-threading support, an automatic source finder, and graphical user interface improvements. The pipeline is currently in beta.
X-ray Surface Brightness and Temperature Image Generation
---------------------------------------------------------
The *Chandra* data were calibrated using *CIAO* 4.7 and *CALDB* 4.7, the most up-to-date versions at the time of analysis. Bad pixels and cosmic rays were removed using *acis*\_*remove*\_*hotpix* and charge transfer inefficiency (CTI) corrections were made using *acis*\_*process*\_*events*. Intervals of background flaring were excluded using light curves in the full band and the 9-12 keV band. The light curves were binned at 259 seconds per bin, the binning used for the blank-sky backgrounds. Count rates greater than 3-$\sigma$ from the mean were removed using *deflare*. We visually inspected the light curves to ensure flares were effectively removed. We used the blank-sky backgrounds in *CALDB* 4.7. The backgrounds were reprojected and processed to match the observations. Figure \[fig:fluxFig\] shows the combined, background-subtracted, point source excised 0.5-8.0 keV surface brightness image.
The ACB temperature map is created by fitting a thermal plasma model to the multiple extracted X-ray spectra. For *Chandra*, the source and background spectra were extracted using *dmextract* and the weighted response was extracted using *specextract*. We rescaled the background spectra using the ratio of the high-energy counts (9.5-12 keV) in the source and the background. We expect the counts to be predominantly from the background at these high energies.
{width="49.00000%"} {width="49.00000%"}
The spectra were then fit using an *APEC* thermal plasma model in *XSPEC*[^4]. We included photoelectric absorption, but the Galactic hydrogen column density was not fit. Instead, it was frozen at a value of 5.2$\times$ $10^{20}$ cm$^{-2}$ [@starkHI]. The resulting temperature map, with X-ray surface brightness contours, is shown in Figure \[fig:acbMap\].
Radio Data {#sec:RadioData}
==========
So far, the published radio map for A115 has been by @Botteon:16 for VLA C+D array data (also produced here as red contours in Figure \[fig:SBandRadio\]). Here, we have used archival multi-frequency radio observations of Abell 115 (see Table \[obs\_journal\]) in order to investigate on the nature of the diffuse radio emission in the cluster.
GMRT 610 MHz
------------
GMRT 610 MHz archival data on A115 were analysed in *CASA* (see Table \[obs\_journal\]). The data were downloaded in FITS format. We converted the FITS file into *CASA* MS format through the *CASA* task *importgmrt*. First, non-functional antennas were flagged based on the observing log. Then we used *AOFlagger*[^5] [@Offringa:12] for radio frequency interference (RFI) flagging. *AOFlagger* is a framework that implements several methods to deal with RFI. About 30% of the data were flagged in *AOFlagger*. Then in the output MS, further manual flagging was done. 12 out of 30 antennas and 73 out of 256 channels were flagged including the frequency band edges. After flagging some outliers and clipping some bad data, the rest were calibrated with the standard calibrator 3C48 (used as both flux and phase calibrator). Then, we separated the calibrated target data with the task *split*.
------- ------- ------- --------- --------- ---------------
Array $\nu$ BW Project Date of PI
(MHz) (MHz) Code Obs.
GMRT 610 32 20-004 30.6.11 A. Bonafede
VLA-D 1420 50 AF0349 26.3.99 L. Feretti
VLA-B 1420 64 15A-270 15.2.15 R. van Weeren
------- ------- ------- --------- --------- ---------------
: Summary of the Archival Radio Observations[]{data-label="obs_journal"}
We tried to recover diffuse emission from the GMRT 610 MHz data at the region between 3C28 and the head tail source as seen in Figure \[fig:SBandRadio\]. However, at the fullest resolution of the GMRT 610 MHz data we could hardly recover any diffuse emission. Hence, we have selected only the $0$ to $\sim$15 $k\lambda$ u-v range of GMRT data at 610 MHz for imaging, which recovered some amount of diffuse emission in the bridge region as now seen in Figure \[fig:VLA-GMRT\]. The imaging was performed by the *CASA* task *clean* choosing a cell size of 9$\arcsec$. Briggs weighting was again used with robust parameter -1. Wide-field imaging was done in *CASA* using w-projection algorithm with 512 w-projection planes. The image was restored with a beam 45$\arcsec$ $\times$ 45$\arcsec$. The RMS noise near the center of the field is $\approx$ 1.4 mJy/beam.
VLA L-band D-configuration
--------------------------
The A115 VLA L-band D configuration archival data were analyzed in *CASA*[^6]. The data were converted into measurement set (MS) format via task *importvla*. Then in the output MS file we have applied manual flagging. 7 out of 27 antennas were flagged. Then the calibration was completed using standard flux calibrator 3C48 and phase calibrator 0119+321. The calibrated target field data was separated from the multi-source dataset by the task *split* choosing only RR and LL correlations. Imaging was performed by the *CASA* task *clean* with the imaging mode ‘channel’ as this archival data contains single channel per IF. The VLA D-configuration synthesized beam size in L-band is 45$\arcsec$, so we chose the cell size to be 9$\arcsec$. Briggs weighting was used with robust parameter -1. The image was restored with a Gaussian beam with a 45$\arcsec$ FWHM. The RMS noise is $\approx$500 $\mu$Jy/beam near the center of the field. The results from VLA-D array L-band are shown in Figures \[fig:VLA-GMRT\] and \[fig:SBandRadio\]. In order to derive spectral index between any two images at two different radio frequencies, the images from both radio frequencies need to be of same angular resolution. Since the restoring beam of GMRT 610 MHz image is 45$\arcsec$ $\times$ 45$\arcsec$, we needed to match that resolution in VLA-D array L band as well.
VLA L-band B-configuration
--------------------------
The Abell 115 L band B array VLA data were first run through the standard *CASA* calibration and editing processes. The editing tasks like *flagdata* and *rflag* were used to flag the bad data. 3 out of 27 antennas along with some channels were flagged. The calibration was done using tasks *setjy* and *gaincal*. The gain solutions were checked using *plotcal*. The calibrated target data were separated from the multi-source data set with the task *split*. The imaging was done by the *CASA* task *clean* choosing a cell size of 1 arcsec. Briggs weighting was used with robust parameter -1. Wide-field imaging was done in *CASA* using the w-projection algorithm with 128 w-projection planes. The image was restored with restoring beam 3.2$\arcsec \times 2.8\arcsec$ with a beam p.a. of 78$^o$. The RMS noise is $\approx$45 $\mu$Jy/beam near the center of the image. The results from VLA-B array L-band are shown in Figure \[fig:SBandRadio\]. It is evident from this Figure that the resolution of the VLA-B array helps us to resolve the structure in the two sources: the well-known 3C28 (in the west) and the head-tail radio source (J0056+2627) to the northeast.
{width="70.00000%"}
Radio Spectral Index and Mach Number
------------------------------------
It is evident from the Figure \[fig:VLA-GMRT\] that the diffuse emission is best captured with the VLA-D array and GMRT 610 MHz analysis with broader restoring beam. We then calculate the spectral index in the for the bridge of radio emission between 3C28 and the head-tail radio source (J0056+2627) to the northeast, we get the average spectral index to be $-1.1 \pm 0.2 $. With a 45$\arcsec$ restoring beam, the bridge is unresolved in the transverse direction. However, the quality of the archival radio data prevents us from going any further with this analysis and creating a spectral index map of the region. In order to do so, we need new radio observations of this field at L and P bands with the VLA. The upgraded capabilities of the VLA will allow us to get a spectral index map of this ’bridge’.
If we are viewing a shock front oriented edge-on, the radio spectral index ($\alpha$) should be sensitive to the prompt emission from the shock front [@skillman13], given by $\alpha=\alpha_{prompt} = (1-s)/2$, where $s$ is the spectral index of the accelerated electrons given by $n_e(E) \propto E^{-s}$ [@hoeft07]. The theory of diffusive shock acceleration (DSA) for planar shocks, at the linear test-particle regime, predicts that this radio spectral index is related to the shock Mach number by: $$\label{eq:radmach}
M^2=\frac{2\alpha-3}{2\alpha+1}$$ where $\alpha$ is the radio spectral index ($S_\nu \propto \nu^\alpha$) [@blandford87; @hoeft07; @ogrean13]. Given the orientation of the features, we estimate a Mach number for the edge-on case. For prompt emission case and a spectral index within the relic region of A115 ($\alpha = -1.1$), the resulting Mach number is $2.1$. This is consistent with the estimates of the shock Mach number at the relic position computed by @Botteon:16 using the X-ray data.
X-ray Temperature Structure and Dynamics {#sec:StrucAndDyn}
========================================
The X-ray surface brightness morphology, coupled with the high-fidelity temperature map, strongly suggest a likely dynamical scenario, at least as projected on the sky. The addition of the optical redshifts of the member galaxies indicates the relative line of sight motion as well. @Barrena:07 analyzed optical redshift and photometric data for 115 galaxies, all of which were members of A115-N and A115-S. Their analysis strongly indicates two separate distributions of galaxies, A115-N with a velocity dispersion of $\sigma_v \approx 1000$ km s$^{-1}$, and A115-S with $\sigma_v \approx 800$ km s$^{-1}$. The analysis suggests the subclusters are moving toward one another along the line of sight with relative velocity $V_r \approx 1600$ km s$^{-1}$.
The defining features of the X-ray temperature map, shown in Figure \[fig:acbMap\], are the two cold, bullet-like structures in the north and south, with temperatures in the cold gas of $3.5 \leq T_X \leq 5$ keV, and the hot, amorphous region between them, with X-ray temperatures as fit in the ACB map ranging up to 15-20 keV. The overall dynamical picture seems to be that the two subclusters are moving both toward one another along the line of sight (as suggested by the optical data), and moving past each other in the plane of the sky (as suggested by the X-ray data). Given the elongated X-ray appearance, the northern subcluster appears to be moving to the west and slightly south, while the southern is moving to the north and east. We explored this interpretation by extracting spectra from the data.
In Figure \[fig:regions\], we show a set of regions, guided by features in the ACB map, that we have extracted spectra from (using *specextract*), and fit with *XSPEC*. The regions were chosen to isolate areas of interest in the ACB temperature map. Regions A and G contain the cold, X-ray bright cores of A115-N and A115-S respectively. Regions B and H were chosen to cover what appears to be cold gas stripped from the clusters by their motion and pressure effects, and indicates the direction of relative motion. Region D covers the very hot central region between the clusters. And Regions C, E, and F were chosen in order to quantify the temperature profile between A115-N and A115-S on either side of the hot region in the center. The spectra were fit with an *APEC* model including photoelectric absorption from galactic neutral hydrogen. The $N_H$ column density was fixed at the value from @starkHI, 5.2$\times10^{20}$cm$^{-2}$, as in the ACB map fits. The metallicity was left as a free parameter, except in the case of the hot central region, where only poor fits to the metallicity could be obtained. With the exception of the hot central region (region D in Figure \[fig:regions\]) whose spectral fit we describe later in this section, we fit the spectrum in the energy range 0.7-8.0 keV. The spectral fits are shown in Table \[XrayFits\].
{width="70.00000%"}
In light of the dynamical picture we described above, it is not straightforward to interpret the faint, hot region between the subclusters. Morphologically, this feature is not obviously consistent with either other observed shock features in galaxy cluster X-ray observations, or the appearance of shocks in numerical simulations of galaxy clusters. Interestingly, while working on this manuscript, an X-ray study of Abell 141 appeared that shows morphological similarity to A115 [@caglar]. It should be noted that though morphologically this hot region does not appear shock-like, its relative position compared to the two subclusters is where we might expect a shock given the likely dynamical scenario. Since this feature does not appear as prototypically shock-like, we have looked in detail at this feature. We used the ACB temperature map as a guide to discover this, and other features. To verify the temperature in that region, we also have extracted a spectrum from an elliptical region covering the hot region in between the clusters. In the hot region, we modified the spectral fitting slightly, in that we fixed the metallicity at $Z_{\odot}$=0.2, but the temperature fit was relatively insensitive to the choice within the range of the local fitted values around it. Additionally, we fit the spectrum in the 0.7-5.0 keV band, as we see flattening of the spectrum at high energy due to the contribution of the noise. We fit a bulk temperature in this region (marked region “D” in Figure \[fig:regions\]) of $T_X=11.03 \pm 1.74$ keV.
\[XrayFits\]
Region Name T$_X$ (keV) Z$_{\odot}$
-------- ------------- ----------------- -----------------
A N Core 3.01 $\pm$ 0.03 0.29 $\pm$ 0.02
B N Tail 5.30 $\pm$ 0.10 0.22 $\pm$ 0.03
C N Inter 6.90 $\pm$ 0.45 0.15 $\pm$ 0.06
D Hot Central 11.03 $\pm$ 1.7 fixed
E E Middle 8.82 $\pm$ 0.78 0.46 $\pm$ 0.09
F S Middle 7.44 $\pm$ 0.42 0.31 $\pm$ 0.07
G S Core 4.31 $\pm$ 0.15 0.29 $\pm$ 0.05
H S Tail 4.00 $\pm$ 0.08 0.28 $\pm$ 0.03
: Spectral Temperature Fits
---------------------------------------------------------------------------------
Spectral fit for temperature and chemical abundance
relative to solar for the regions shown in Figure \[fig:regions\]. X-ray
spectral fitting for these regions is described in Section \[sec:StrucAndDyn\].
---------------------------------------------------------------------------------
: Spectral Temperature Fits
{width="70.00000%"}
We have also examined the X-ray emission in the region of the radio relic, as was done in @Botteon:16. Figure \[fig:SBandRadio\] shows the X-ray surface brightness and radio contours, overlaid on the optical data. One can clearly see the location of 3C28, and the head-tail radio source (J0056+2627). The red contours, extending from west to east away from the 3C28 at the center of A115-N, are 1.5 GHz radio contours from the VLA B+D array, showing the location of the extended emission of the radio relic. If the relic is indeed being illuminated because of shock acceleration of a pre-existing population of particles, we need to understand how the presence of a shock at that location is consistent with the dynamics of the cluster. Shock-accelerated particle populations cool quickly by synchrotron radiation, and so the expectation is that the location of the relic will be almost precisely coincident with the shock accelerating the particles. While we do not see strong evidence for a shock at that location in the X-ray data, it is admittedly very faint. @Botteon:16 make an argument for evidence of a shock from the X-ray data near the radio relic. However, what is not immediately obvious is what dynamical scenario should result in a shock at that location. We address the dynamics in later sections.
Shocks and Temperature Features in the ICM {#sec:DynamicsAndShocks}
------------------------------------------
{width="\textwidth"}
We have run an automated shock finder on the surface brightness and spectral temperature maps, identical to the procedure described in @Datta:14 and @Schenck:14. High-quality surface brightness and spectral temperature maps can be probed using this technique, which is adapted from a similar calculation used in numerical simulations [@ryu03; @skillman08]. In numerically simulated clusters, we can find shocks using the full three-dimensional properties of the gas. The conditions for determining whether a given volumetric element of the simulation is the location of a shock are
$$\begin{aligned}
\label{eq:jump_cond}
\nabla \cdot {\bf v} & < & 0 \nonumber \\
\nabla{T} \cdot \nabla{K_S} & > & 0 \\
T_2 &>& T_1 \nonumber \\
\rho_2 & > & \rho_1 \nonumber\end{aligned}$$
where ${\bf v}$ is the velocity field, $T$ is the temperature, $\rho$ is the density and $K_S=T/\rho^{\gamma -1}$ is the entropy. The Mach number of the shock is then defined by the temperature jump, using the Rankine-Hugoniot shock jump conditions, to be
$$\label{eq:tempjump}
\frac{T_2}{T_1}=\frac{(5M^2 - 1)(M^2+3)}{16M^2}.$$
In X-ray images, all shock observables are projected on the sky. Therefore, for observational data, we must use 2-dimensional projected X-ray surface brightness and temperature maps. Therefore, the technique is modified to account for that. In this modified scenario, the shock-finder calculates the jump in temperature and surface brightness in $N$ evenly placed directions centering on a given pixel. The shock-finder then accepts those pixel-pairs between which the conditions $T_2>T_1$ and $S_{X2} > S_{X1}$ (where $S_{X2}$ and $S_{X1}$ represent the downstream and upstream X-ray surface brightness respectively, and the ratio is a proxy for $\rho_2 > \rho_1$ in observations) in equation \[eq:jump\_cond\] are satisfied. The other two conditions, $\nabla \cdot {\bf v} < 0$ and $\nabla{T} \cdot \nabla{K_S} > 0$, in equation \[eq:jump\_cond\] cannot be used in the case of observations. The Mach number for each successful pixel-pair is noted. The Mach number with the maximum value is chosen to be the resultant Mach number for that given pixel. The full details of the method are described in @Datta:14 and @Schenck:14.
The automated shock finder identifies only one region where there are pixels consistent with a shock, and that is in the area we interpret as leading the motion of A115-N. The shock finder identifies a number of pixels in the image as shocks, with Mach numbers ranging from $M\approx1$ to $M\approx3$. This area of the map has very low surface brightness, and the temperature map, by the nature of the method, has aggregated pixels over some relatively large area of the map. Again, as before, using the ACB map and shock finder as a guide, below we have explored more deeply with extractions in those regions.
We explored the region identified as a shock in the above analysis. First, we used a projected pressure map derived from the X-ray surface brightness and temperature maps. The projected pressure maps were generated by taking the square root of the X-ray surface brightness as a proxy for projected density, and multiplied it by the temperature fit to the X-ray spectra, represented by the ACB map. We then generated a projected pressure profile, across the region identified as a shock, and it is shown in the orange-boxed region and line in Figure \[fig:pressureProfiles\]. In the region upstream of the northern subcluster, we see a steep drop in pressure.
Other parts of the X-ray map where we might have expected to find evidence of a shock are the region to the northeast of A115-S (leading its motion), and northeast of both subclusters, in the location of the radio relic. We explored the pressure profile across the northern radio relic in the area where @Botteon:16 suggest they detect a shock. In Figure \[fig:pressureProfiles\], we show the projected pressure profile across that northern radio region (in green). In front of the southern subcluster (the region shown by the blue box and line), we see no such discontinuity, just a gradual decline in pressure. This is also true across the northern radio relic region. However, as noted in earlier sections, orientation and projection effects can act to diminish the observed surface brightness and X-ray temperature contrast at the location of a shock.
In the direction of motion of A115-N, where the shock detector identifies some pixels as part of a shock, the X-ray surface brightness is quite low. As a consequence, the region from which pixels are selected for an X-ray spectral fit by both the WVT and ACB methods (both using a signal-to-noise threshold of 50 for their regions) is quite large. Therefore it is difficult to extract a reasonable spectral fit to the regions either in front of or behind the identified shock location. However, we can quite easily extract a surface brightness profile across this region ahead of the deduced motion of A115-N, and that is shown in Figure \[fig:radialProfile\]. The regions from which the surface brightness is extracted are shown in the annular wedge regions in the map also in Figure \[fig:radialProfile\]. The fifth region in from the outer part of the shock feature shows a steep increase in surface brightness, by a factor of roughly two. This is the location where the shock finder identifies a shock, and the profile shows that location to be plausible. Better X-ray data may allow a good spectral fit, spatially resolved in this region. However, the current data do not permit that at this time.
{width="49.00000%"} {width="49.00000%"}
{width="95.00000%"}
These three panels are incremented from left to right chronologically. $\delta$t = 220 Myr, and image represents a projected size of about 2$h^{-1}$ Mpc. Note the cold subcluster to the right of each image, moving upward, driving a shock. The combination of shocks from the relative motion of the main and subcluster results in a hot region between them with relatively flat surface brightness, similar to A115. In the third panel, note the extended bow shock outside the X-ray visible region, similar to what we might be seeing in A115, as described in the text. \[fig:simulatedcluster\]
Comparing to Numerical Simulations
----------------------------------
For comparison, and to help deduce the dynamics of A115, we show synthetic observations of galaxy clusters simulated in a cosmological context using the cosmological N-body/hydrodynamic simulation code `Enzo`[^7] [@enzo]. The merger dynamics in the simulations are an excellent guide for determining the observable consequences of typical subcluster interactions. The simulations used for this purpose are described in detail in @jeltema. The simulation used here is of a volume of the Universe 128 $h^{-1}$ Mpc (comoving) on a side, on a $256^{3}$ root grid. The simulation is evolved in a $\Lambda$CDM cosmological model from an @eisenstein power spectrum from $z=99$ to $z=0$, with a maximum of five levels of adaptive mesh refinement (AMR). This results in a peak spatial resolution of 15.6$h^{-1}$ comoving kpc. We refine on both dark matter and baryon overdensity of 8.0. This particular simulation does not include the effects of metal line cooling, as has been done in other prior work. In this case, we are using the simulated clusters to understand the larger scale dynamics and observable effects outside the cluster core, where typically cooling times are long. Such simulations serve us as a guide for understanding the range of merger interactions we expect in a cosmological context.
In @jeltema, we extracted 16 simulated clusters at $z=0$ whose mass exceeded $M_{200} \geq 3 \times 10^{14} M_{\odot}$. The objects of highest virial mass in the simulation are similar in mass to A115. For the current study, we make use of the synthetic observations of these 16 clusters. For each of the identified clusters at $z=0$, we extracted a volume around that cluster in a series of 132 snapshots in time, equally spaced ($\delta t = 0.22$ Gyr) in the redshift interval $0 \leq z \leq 0.9$. For each of these time outputs, using the `yt`[^8] toolkit [@yt], we created synthetic 0.3-8.0 keV X-ray images and spectroscopic-like temperature ($T_{sl}$) maps [see @rasia]. The X-ray emission is calculated using the *CLOUDY*[^9] software [@ferland]. The projections are of an 8h$^{-3}$ Mpc$^3$ volume centered on the cluster, thus each image is an 8h$^{-2}$ Mpc$^2$ field. This exposes not only the merger activity within the cluster virial radius, but also the larger cosmological environment of the cluster, allowing us to understand how various observable effects originate.
{width="70.00000%"} {width="70.00000%"}
### Similarities to A115 in Numerical Galaxy Clusters
In the simulation data we see features that are consistent with the inferred dynamics of A115, as well as with the temperature and surface brightness substructure. When two merging subclusters pass each other in the simulations, both drive bow shocks into the other subclusters ICM. Additionally, these shocks then interact as they travel outward from the subclusters and collide. This combination of the two subcluster bow shocks can heat the gas between the clusters, leaving an observable structure even after the leading edge of the bow shock has propagated into a less dense, X-ray faint location ahead of the cluster. Figure \[fig:simulatedcluster\] shows the evolution of such a feature, using the synthetic X-ray temperature maps generated in our prior work. The surface brightness contours represent a dynamic range of 1000 from brightest to faintest, which is similar to the dynamic range of the *Chandra* X-ray images. The outer edge of the bow shock from the subcluster on the right of the images is in a very faint region of the image, and so would be very difficult to detect. The three panels from left to right are three stages in the cluster evolution, separated by approximately 220 Myr in time. To the right of each image, there is a cooler subcluster traveling toward the top of the image, with a heated region in front of it, representing shock heated gas. Between the subclusters, you can see a hotter region, with a relatively flat surface brightness profile, very similar to what we see in A115.
Although this is not a model of A115, it may represent a similar dynamical scenario based upon the morphological similarities that we see. These features are quite common in cluster mergers in numerical cosmological simulations, and we believe that a similar scenario is very likely responsible for the hot region between A115-N and A115-S. It is likely that the interaction of the edge of the two interfering bow shocks has heated the gas to high temperature, and that the upstream parts of the shock, as well as those away from the center of the merger, are in locations where the X-ray surface brightness is too low to detect them.
Establishing the Location of the Bow Shock in A115-N {#sec:north}
----------------------------------------------------
In order to determine the plausibility of this hypothesis, here we use estimates of the shock Mach number and stand-off distance to see if the location of the central hot region is consistent with this interpretation. We calculate the location of the bow shock around the northern subcluster of A115 using Moeckel’s method (M49)[^10], which is also summarized for cluster purposes in Appendix B of @Vikhlinin:01 [hereafter V01]. We employ the shock finder’s map of A115 to draw an asymptotic line to the part of the bow shock detected in this manner. This can be seen in the top panel of Figure \[fig:estimateShock\]. The curved cyan contours in this panel represent the shock contours. The dashed black line represents the asymptotic line of the bow shock. Using the surface brightness map (represented as white contours), we determine the subcluster direction of movement (solid black line). This allows us to obtain the angle $\phi \sim 36$ deg for the northern cluster (for reference, see Fig. 9 in V01), which provides us with a Mach number of the subcluster, $M\sim1.7$, using the relation $\phi=atan(M^2-1)^{-1/2}$.
To calculate the hyperbola function representing the bow shock we require both $M$ and the stand-off distance $x_0$. To obtain the latter, we proceed as follows. Since this object does not have a well-defined shoulder, we can locate the so-called body sonic point $S_B$ (marked with a green star in each panel of Figure \[fig:estimateShock\]; and being the point on the surface of the body where the flow speed equals the speed of sound), by using the relation between the angle $\theta$ (formed between the line of movement and the asymptote to the shoulder) and $M$, as found in Fig. 4 of M49. Having this angle, we find $S_B (x_{sb}=549$ kpc$, y_{sb}=103$ kpc$)$ on the surface of the subcluster. The coordinate origin is where the three black lines meet.
From Fig. 7 in M49, selecting the curve that assumes an axially symmetric body with respect to the line of movement, as well as the continuity method, we obtain $L/y_{sb}\sim 1$ (which is only a function of the body speed, i.e. $M$, calculated above to be $\sim1.7$). We can then find the shock detachment distance $L=x_{sb}-x_0$, and finally the shock stand-off distance $x_0 = 445$ kpc.
Given $M$ and $x_0$, we can now model the bow shock as the hyperbola shown by the solid, white line in the [**top**]{} panel of Figure \[fig:estimateShock\]. Due to the uncertainty in the assumption on the shape of the moving object, we repeat the calculation choosing instead the two-dimensional body, plus continuity method, curve from Fig. 7 of M49. We then obtain $x_0 = 272$ kpc and the dashed, white line in the top panel of Figure \[fig:estimateShock\]. We use these two cases to bracket a region (shaded white) where we might expect to find the true, underlying bow shock. [@Gutierrez:05] also drew a bow shock estimation from the surface brightness map in their Fig. 7, which is roughly consistent with our result.
Note that we encouragingly find that the bow shock region south of the northern object goes through the hot spot between the two subclusters. We can also obtain the sound speed in the cluster assuming a temperature of 10 keV, $\sim 1594$ km/s, providing $\sim 2710$ km/s for the northern subcluster (using $M \sim 1.7$) (or $1127$ km/s and $1916$ km/s, respectively, assuming 5 keV). Recall that from [@Barrena:07] the colliding line of sight velocity between the subclusters is $\sim 1600$ km/s.
Estimating the Location of the Assumed Bow Shock in A115-S {#sec:south}
----------------------------------------------------------
Since for the southern subcluster we do not have any visible residual of the bow shock, we cannot follow the first part of the method used for A115-N. Let us assume though that our argument holds and that the asymptotic line for the southern subcluster is aligned with that of the northern (dashed, black line over the central, hot region between subclusters), as shown in the bottom panel of Figure \[fig:estimateShock\]. We also morphologically establish the direction of movement as going through the subcluster as indicated by the solid, black line crossing it. Having these two lines, from this point on we can proceed as described in the previous subsection. For this subcluster, we then obtain $\phi\sim50$ deg and calculate $M\sim 1.3$, which is lower than that of the northern and thus consistent with the dynamical measurements of [@Barrena:07].
To obtain the stand-off distance $x_0$, from Fig. 4 of M49 and using $M\sim 1.3$, we have $\theta\sim 25$ deg. Positioning this angle in the bottom panel of Figure \[fig:estimateShock\] (between the solid, black and cyan lines) we obtain $x_{sb}=616$ kpc and $y_{sb}=59$ kpc, and following the same calculation and assumptions as before, $x_0= 528$ kpc. The solid, white line shows the hyperbola corresponding to the assumed bow shock in A115-S, for this being an axially symmetric object. Assuming instead a two-dimensional shape of the subcluster, we find $x_0 = 219$ kpc and the dashed, white line. This and the previous hyperbola bracket the white, shaded region where we might find the bow shock.
The Shock Hypothesis
--------------------
In summary, the calculations of shock stand-off distance and location strengthen our hypothesis, derived from morphological similarity to numerical simulations, that the location of the central hot spot can plausibly be attributed to the presence of the two bow shocks from the motion of the two subclusters. Between the subclusters, where there is high enough gas density to generate significant X-ray emission, we can see the hot gas. In the outer parts of the subclusters, and also upstream of the subcluster motion, the gas density is lower, and the X-ray features we expect for a shock are not visible.
The Radio Relic and its Relationship to the X-ray {#sec:RadioRelic}
=================================================
Prior work discussing the radio relic in A115 describes two possible scenarios for the origin of the extended radio emission. In @govoni:01 and @Gutierrez:05, the initial hypothesis was that the extended radio emission is coming from accelerated particles ejected from 3C28 and other nearby radio galaxies, which has subsequently been left behind by the motion of the cluster, and associated galaxies. The other scenario, described in @Botteon:16, is that the relic is associated with a shock, accelerating the particles to higher energy, causing them to radiate. These scenarios are not exclusive. This feature could result from a particle population that was ejected from the local radio galaxies and then re-accelerated by a merger shock. Shock acceleration theory (and recent observational evidence) suggests that accelerating a pre-existing population of particles with a nonthermal distribution is more efficient than accelerating particles out of the tail of a thermal distribution [@bruJones; @reaccelVanWeeren].
What is most relevant to this work is determining the plausibility of either scenario, based on the apparent dynamics of the cluster. Below, we explain the presence of a shock at the location and orientation of the radio relic based on a reasonable dynamical scenario. Also, we investigate whether the radio data support the shock hypothesis, or are consistent with the idea that this is simply stripped radio plasma, with no re-acceleration process.
{width="70.00000%"}
The morphology of shocks in merging numerical clusters seems to be in some cases consistent with the shape and location we see in A115. This feature could in fact be consistent with the dynamical picture deduced from X-ray and optical data. In Figure \[fig:simulatedcluster\], one can also see the outer part of the bow shock from the cold subcluster. This region would very likely not be visible in the X-ray observation of such a cluster, as it is in a region of very low X-ray surface brightness. However, should that portion of the bow shock propagate through a region of accelerated particles, one might expect Fermi processes to re-accelerate those particles to higher energy, and they may become visible as synchrotron sources.
As to the second question – whether the radio data support the shock re-acceleration scenario – unfortunately the quality of the data prohibit a definitive analysis. What we expect in a shock accelerated scenario is that the radio spectral index will vary spatially, depending on the exact location of the shock. At the location of the shock, the spectral index will be more flat, and behind the shock, as the highest energy particles radiate their energy away more quickly than those at lower energy, the spectral index will steepen. There are examples from other radio relics, when there is high-quality, high resolution radio data available. If a shock is sweeping through the radio emitting plasma, we would expect a spectral index gradient perpendicular to the long axis of the relic feature [see e.g., @sausage]. However, if this radio emitting plasma is simply stripped from the radio galaxies, and passively advected away, what we would expect is that the spectral index gradient will be parallel to the long axis of the feature, and in the direction of motion of the radio galaxies.
Summary and Conclusions {#sec:Discussion}
=======================
As discussed in [@Barrena:07], there are a number of features in the optical and X-ray data that support the description of the initial stages of a major merger occurring between the two subclusters of Abell 115. We are able to further corroborate this scenario, consistent with the prior X-ray observations of A115 [e.g., @Forman:81; @Shibata:99; @Gutierrez:05; @Botteon:16], through more detailed X-ray and radio observations, as well as comparing our results with relevant hydrodynamical simulations.
The dynamical scenario we have described here is shown pictorially in Figure \[fig:cartoon\]. The subclusters are orbiting in each other in the process of merging. Red lines represent the location of shocks, black arrows represent the past and current motion of the subclusters, blue ellipses represent the region of each subcluster core. The radio relic results from the Fermi acceleration of a relic population of cosmic rays ejected from the local radio galaxies, and advected behind the motion of A115-N. The curved remnant of the earlier bow shock, similar to what we see in simulations, continues to propagate into the relic plasma after A115-N has turned its motion. The interaction of the bow shocks of A115-N and A115-S produces a hot region between the subclusters.
Both prior work, and this analysis, suggest that the A115 merger results in a hot X-ray region between the clusters, which we interpret as a feature resulting from interacting bow shocks. High-energy particles, accelerated in 3C28 and the head-tail radio source (J0056+2627), are spread out behind A115-N, along the direction of motion. It has been suggested by @forman17 that hydrodynamic forces associated with the motion of the radio galaxies through the ICM are responsible for the morphology. Here, we suggest, consistent with the interpretation of the X-ray data by @Botteon:16, that a shock at the location of the radio relic, in our scenario as part of the leading bow shock of A115-N, has re-accelerated these particles accounting for the radio emission along the arc-like structure between and behind the two radio galaxies.
Though these conclusions are highly plausible, there are some additional observations that could make this case more definitively. Primarily, the addition of high quality, high resolution, multi-frequency radio observations of the relic could strongly constrain whether the shock acceleration scenario is correct. In particular, the availability of high-quality, wide bandwidth 350 MHz and 1.4 GHz observations, using the B, C, and D array configurations of the JVLA, would allow for resolved spectral index maps of the relic. These could be used to determine whether the spectral index gradient is perpendicular to the putative shock, or whether the spectral index variation is more consistent with a scenario where the radio plasma is ejected from the radio galaxies and simply ages via synchrotron cooling as it advects away.
Also, deep X-ray observations of the region around the relic, as well as in the regions where one might expect the outer parts of the bow shocks resulting in the X-ray hot spot, might clear up that interpretation as well.
[^1]: http://cxc.harvard.edu/ciao/
[^2]: http://ds9.si.edu/
[^3]: https://github.com/bcalden/xray-tmap-pypeline
[^4]: https://heasarc.gsfc.nasa.gov/xanadu/xspec/
[^5]: http://aoflagger.sourceforge.net
[^6]: https://casa.nrao.edu/
[^7]: http://enzo-project.org
[^8]: http://yt-project.org/
[^9]: http://www.nublado.org/
[^10]: Technical note available at https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930082597.pdf
|
---
author:
- |
*H.S.Abdel-Aziz* and *M.Khalifa Saad*[^1]\
[Math. Dept., Faculty of Science, Sohag Univ., 82524 Sohag, Egypt]{}
title: |
A study on special curves of AW($k$)-type\
in the pseudo-Galilean space
---
**Abstract.** This paper is devoted to the study of AW$(k)$-type $\left( 1\leq k\leq 3\right) $ curves according to the equiform differential geometry of the pseudo-Galilean space $G_{3}^{1}$. We show that equiform Bertrand curves are circular helices or isotropic circles of $G_{3}^{1}$. Also, there are equiform Bertrand curves of AW$(3)$ and weak AW$(3)$-types. Moreover, we give the relations between the equiform curvatures of these curves. Finally, examples of some special curves are given and plotted.
***M.S.C. 2010*:** 53A04, 53A35, 53C40.
**Key Words:** Frenet curves, Bertrand curves, curves of AW$(k)$-type, equiform differential geometry, pseudo-Galilean space.
Introduction
============
As it is well known, geometry of space is associated with mathematical group. The idea of invariance of geometry under transformation group may imply that, on some spacetimes of maximum symmetry there should be a principle of relativity which requires the invariance of physical laws without gravity under transformations among inertial systems [@1]. Besides, theory of curves and the curves of constant curvature in the equiform differential geometry of the isotropic spaces $I_{3}^{1}$ , $I_{3}^{2}$ and the Galilean space $G_{3}$ are described in [@2] and [3]{}, respectively. The pseudo-Galilean space is one of the real Cayley-Klein spaces. It has projective signature $(0,0,+,-)$ according to [@2]. The absolute of the pseudo-Galilean space is an ordered triple $\{w,f,I\}$ where $w$ is the ideal plane, $f$ a line in $w$ and $I$ is the fixed hyperbolic involution of the points of $f$. In [@4], from the differential geometric point of view, K. Arslan and A. West defined the notion of AW(k)-type submanifolds. Since then, many works have been done related to AW(k)-type submanifolds (see, for example, [@5; @6; @7; @8; @9; @10]). In [@9], Özgür and Gezgin studied a Bertrand curve of AW$(k)$-type and furthermore, they showed that there is no such Bertrand curve of AW$(1)$ and AW$(3)$-types if and only if it is a right circular helix. In addition, they studied weak AW$(2)$-type and AW$(3)$-type conical geodesic curves in Euclidean 3-space $E^{3}$. Besides, In $3$-dimensional Galilean space and Lorentz space, the curves of AW$(k)$-type were investigated in [@6; @8]. In [@7], the authors gave curvature conditions and characterizations related to AW$(k)$-type curves in $E^{n}$ and in [@10], the authors investigated curves of AW$(k)$-type in the $3$-dimensional null cone.
In this paper, to the best of author’s knowledge, Bertrand curves of AW$(k)$-type have not been presented in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$ in depth. Thus, the study is proposed to serve such a need. Our paper is organized as follows. In Section $2$, the basic notions and properties of a pseudo-Galilean geometry are reviewed. In Section $3$, properties of the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$ are given. Section $4$ contains a study of AW$(k)$-type equiform Frenet curves. Equiform Bertrand curves of AW$(k)$-type in $G_{3}^{1}$ included in section $5$.
Pseudo-Galilean geometric meanings
==================================
In this section, let us first recall basic notions from pseudo-Galilean geometry [@11; @12]. In the inhomogeneous affine coordinates for points and vectors (point pairs) the similarity group $H_{8}$ of $G_{3}^{1}$ has the following form $$\begin{aligned}
\bar{x}& =a+b.x, \notag \\
\bar{y}& =c+d.x+r.\cosh \theta .y+r.\sinh \theta .z, \notag \\
\bar{z}& =e+f.x+r.\sinh \theta .y+r.\cosh \theta .z,\end{aligned}$$where $a,b,c,d,e,f,r$ and $\theta $ are real numbers. Particularly, for $b=r=1,$ the group $(2.1)$ becomes the group $B_{6}\subset
H_{8}$ of isometries (proper motions) of the pseudo-Galilean space $G_{3}^{1}
$. The motion group leaves invariant the absolute figure and defines the other invariants of this geometry. It has the following form $$\begin{aligned}
\bar{x}& =a+x, \notag \\
\bar{y}& =c+d.x+\cosh \theta .y+\sinh \theta .z, \notag \\
\bar{z}& =e+f.x+\sinh \theta .y+\cosh \theta .z.\end{aligned}$$According to the motion group in the pseudo-Galilean space, there are non-isotropic vectors $A(A_{1},A_{2},A_{3})$ (for which holds $A_{1}\neq 0$) and four types of isotropic vectors: spacelike ($A_{1}=0,$ $A_{2}^{2}-A_{3}^{2}>0$), timelike ($A_{1}=0,$ $A_{2}^{2}-A_{3}^{2}<0$) and two types of lightlike vectors ($A_{1}=0,A_{2}=\pm A_{3}$). The scalar product of two vectors $u=(u_{1},u_{2},u_{3})$ and $v=(v_{1},v_{2},v_{3})$ in $G_{3}^{1}$ is defined by $$\left\langle u,v\right\rangle =\left\{
\begin{array}{c}
u_{1}v_{1},\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if }u_{1}\neq 0\text{ or }v_{1}\neq 0, \\
u_{2}v_{2}-u_{3}v_{3}\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if \ }u_{1}=0\text{ and }v_{1}=0.\end{array}\right\}$$We introduce a pseudo-Galilean cross product in the following way $$u\times _{G_{3}^{1}}v=\left\vert
\begin{array}{ccc}
0 & -j & k \\
u_{1} & u_{2} & u_{3} \\
v_{1} & v_{2} & v_{3}\end{array}\right\vert ,$$where $j=(0,1,0)$ and $k=(0,0,1)$ are unit spacelike and timelike vectors, respectively. Let us recall basic facts about curves in $G_{3}^{1}$, that were introduced in [@15].
A curve $\gamma (s)=(x(s),y(s),z(s))$ is called an admissible curve if it has no inflection points $(\dot{\gamma}\times \ddot{\gamma}\neq 0)$ and no isotropic tangents $(\dot{x}\neq 0)$ or normals whose projections on the absolute plane would be lightlike vectors $(\dot{y}\neq \pm \dot{z})$. An admissible curve in $G_{3}^{1}$ is an analogue of a regular curve in Euclidean space [@12].
For an admissible curve $\gamma :I\subseteq \mathbb{R}\rightarrow G_{3}^{1},$ the curvature $\kappa (s)$ and torsion $\tau (s)$ are defined by$$\kappa (s)=\frac{\sqrt{\left\vert \ddot{y}(s)^{2}-\ddot{z}(s)^{2}\right\vert
}}{(\dot{x}(s))^{2}},\text{ }\tau (s)=\frac{\ddot{y}(s)\dddot{z}(s)-\dddot{y}(s)\ddot{z}(s)}{\left\vert \dot{x}(s)\right\vert ^{5}\cdot \kappa ^{2}(s)},\text{\ }$$expressed in components. Hence, for an admissible curve $\gamma :I\subseteq
\mathbb{R}\rightarrow G_{3}^{1}$ parameterized by the arc length $s$ with differential form $ds=dx$, given by $$\gamma (x)=(x,y(x),z(x)),$$the formulas $(2.3)$ have the following form $$\kappa (x)=\sqrt{\left\vert y^{^{\prime \prime }}(x)^{2}-z^{^{\prime \prime
}}(x)^{2}\right\vert },\text{ }\tau (x)=\frac{y^{^{\prime \prime
}}(x)z^{^{\prime \prime \prime }}(x)-y^{^{\prime \prime \prime
}}(x)z^{^{\prime \prime }}(x)}{\kappa ^{2}(x)}.$$The associated trihedron is given by $$\begin{aligned}
\mathbf{e}_{1}& =\gamma ^{\prime }(x)=(1,y^{^{\prime }}(x),z^{^{\prime
}}(x)), \notag \\
\mathbf{e}_{2}& =\frac{1}{\kappa (x)}\gamma ^{^{\prime \prime }}(x)=\frac{1}{\kappa (x)}(0,y^{^{\prime \prime }}(x),z^{^{\prime \prime }}(x)), \notag \\
\mathbf{e}_{3}& =\frac{1}{\kappa (x)}(0,\epsilon z^{^{\prime \prime
}}(x),\epsilon y^{^{\prime \prime }}(x)),\end{aligned}$$where $\epsilon =+1$ or $\epsilon =-1$, chosen by criterion det$(e_{1},e_{2},e_{3})=1$, that means $$\left\vert y^{^{\prime \prime }}(x)^{2}-z^{^{\prime \prime
}}(x)^{2}\right\vert =\epsilon (y^{^{\prime \prime }}(x)^{2}-z^{^{\prime
\prime }}(x)^{2})\text{.}$$The curve $\gamma $ given by $(2.4)$ is timelike (resp. spacelike) if $\mathbf{e}_{2}(s)$ is a spacelike (resp. timelike) vector. The principal normal vector or simply normal is spacelike if $\epsilon =+1$ and timelike if $\epsilon =-1$. For derivatives of the tangent $\mathbf{e}_{1}$, normal $\mathbf{e}_{2}$ and binormal $\mathbf{e}_{3}$ vector fields, the following Frenet formulas in $G_{3}^{1}$ hold: $$\begin{aligned}
\mathbf{e}_{1}^{\prime }(x)& =\kappa (x)\mathbf{e}_{2}(x), \notag \\
\mathbf{e}_{2}^{\prime }(x)& =\tau (x)\mathbf{e}_{3}(x), \notag \\
\mathbf{e}_{3}^{\prime }(x)& =\tau (x)\mathbf{e}_{2}(x).\end{aligned}$$
Frenet formulas according to the equiform geometry of $G_{3}^{1}$
=================================================================
This section contains some important facts about equiform geometry. The equiform differential geometry of curves in the pseudo-Galilean space $G_{3}^{1}$ has been described in [@11]. In the equiform geometry a few specific terms will be introduced. So, let $\gamma (s):I\rightarrow
G_{3}^{1} $ be an admissible curve in the pseudo-Galilean space $G_{3}^{1}$, the equiform parameter of $\gamma $ is defined by $$\sigma :=\int \frac{1}{\rho }ds=\int \kappa ds,$$where $\rho =\frac{1}{\kappa }$ is the radius of curvature of the curve $\gamma $. Then, we have $$\frac{ds}{d\sigma }=\rho .$$Let $h$ be a homothety with the center in the origin and the coefficient $
\mu $. If we put $\bar{\gamma}=h(\gamma )$, then it follows $$\bar{s}=\mu s\text{ \ and \ }\bar{\rho}=\mu \rho ,\text{ }$$where $\bar{s}$ is the arc-length parameter of $\bar{\gamma}$ and $\bar{\rho}$ the radius of curvature of this curve. Therefore, $\sigma $ is an equiform invariant parameter of $\gamma $ [@11].
The functions $\kappa $ and $\tau $ are not invariants of the homothety group, then from $(2.3)$ it follows that $\bar{\kappa}=\frac{1}{\mu }\kappa $ and $\bar{\tau}=\frac{1}{\mu }\tau $.
From now on, we define the Frenet formulas of the curve $\gamma $ with respect to its equiform invariant parameter $\sigma $ in $G_{3}^{1}.$ The vector $$\mathbf{T}=\frac{d\gamma }{d\sigma },$$is called a tangent vector of the curve $\gamma .$ From $(2.6)$ and $(3.1)$ we get $$\mathbf{T}=\frac{d\gamma }{ds}\frac{ds}{d\sigma }=\rho \cdot \frac{d\gamma }{ds}=\rho \cdot \mathbf{e}_{1}.$$Further, we define the principal normal vector and the binormal vector by $$\mathbf{N}=\rho \cdot \mathbf{e}_{2},\text{ \ }\mathbf{B}=\rho \cdot \mathbf{e}_{3}.$$It is easy to show that $\left\{ \mathbf{T},\mathbf{N},\mathbf{B}\right\} $ is an equiform invariant frame of $\gamma .$ On the other hand, the derivatives of these vectors with respect to $\sigma $ are given by $$\left[
\begin{array}{c}
\mathbf{T} \\
\mathbf{N} \\
\mathbf{B}\end{array}\right] ^{\prime }=\left[
\begin{array}{ccc}
\dot{\rho} & 1 & 0 \\
0 & \dot{\rho} & \rho \tau \\
0 & \rho \tau & \dot{\rho}\end{array}\right] \left[
\begin{array}{c}
\mathbf{T} \\
\mathbf{N} \\
\mathbf{B}\end{array}\right] .$$ The functions $\mathcal{K}:I\rightarrow \mathbb{R}$ defined by $\mathcal{K}=\dot{\rho}$ is called the equiform curvature of the curve $\gamma $ and $\mathcal{T}:I\rightarrow \mathbb{R}$ defined by $\mathcal{T}=\rho \tau =\frac{\tau }{\kappa }$ is called the equiform torsion of this curve. In the light of this, the formulas $(3.4)$ analogous to the Frenet formulas in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$ can be written as $$\left[
\begin{array}{c}
\mathbf{T} \\
\mathbf{N} \\
\mathbf{B}\end{array}\right] ^{\prime }=\left[
\begin{array}{ccc}
\mathcal{K} & 1 & 0 \\
0 & \mathcal{K} & \mathcal{T} \\
0 & \mathcal{T} & \mathcal{K}\end{array}\right] \left[
\begin{array}{c}
\mathbf{T} \\
\mathbf{N} \\
\mathbf{B}\end{array}\right] .$$The equiform parameter $\sigma =\int \kappa (s)ds$ for closed curves is called the total curvature, and it plays an important role in global differential geometry of Euclidean space. Also, the function $\frac{\tau }{\kappa }$ has been already known as a conical curvature and it also has interesting geometric interpretation.
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve in the equiform geometry of the $G_{3}^{1}$, the following statements are true $($ see for details [@11; @13] $)$:
1. If $\gamma (s)$ is an isotropic logarithmic spiral in $G_{3}^{1}$. Then, $\mathcal{K=}$const$.\neq 0$ and $\mathcal{T}=0,$
2. If $\gamma (s)$ is a circular helix in $G_{3}^{1}$. Then, $\mathcal{K=}0$ and $\mathcal{T}=$const$.\neq 0,$
3. If $\gamma (s)$ is an isotropic circle in $G_{3}^{1}$. Then, $\mathcal{K=}0$ and $\mathcal{T}=0.$
AW($k$)-type curves in the equiform geometry of $G_{3}^{1}$
===========================================================
Let $\gamma :I\rightarrow G_{3}^{1}$ be a curve in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$. The curve $\gamma $ is called a Frenet curve of osculating order $l$ if its derivatives $\gamma ^{\prime
}(s),\gamma ^{\prime \prime }(s),\gamma ^{\prime \prime \prime
}(s),...,\gamma ^{(l)}(s)$ are linearly dependent and $\gamma ^{\prime
}(s),\gamma ^{\prime \prime }(s),\gamma ^{\prime \prime \prime
}(s),...,\gamma ^{(l+1)}(s)$ are no longer linearly independent for all $s\in I$ . To each Frenet curve of order $3$ one can associate an orthonormal $3$-frame $\left\{ \mathbf{T},\mathbf{N},\mathbf{B}\right\} $ along $\gamma $, such that $\gamma ^{\prime }(s)=\frac{1}{\rho }\mathbf{T}$, called the equiform Frenet frame (Eqs. $(3.5)$).
Now, we consider equiform Frenet curevs of osculating order $3$ in $G_{3}^{1}
$ and start with some important results.
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve in the equiform geometry of the pseudo-Galilean space. By the use of Frenet formulas $(3.5)$, we obtain the higher order derivatives of $\gamma $ as follows $$\begin{aligned}
\gamma ^{\prime }(s)& =\frac{d\gamma }{d\sigma }\frac{d\sigma }{ds}=\frac{1}{\rho }\mathbf{T}, \\
\gamma ^{\prime \prime }(s)& =\frac{1}{\rho ^{2}}\mathbf{N}, \\
\gamma ^{\prime \prime \prime }(s)& =\frac{1}{\rho ^{3}}\left( -\mathcal{K}\mathbf{N}\mathcal{+T}\mathbf{B}\right) , \\
\gamma ^{\prime \prime \prime \prime }(s)& =\frac{1}{\rho ^{4}}[(2\mathcal{K}^{2}\mathcal{+\mathcal{T}}^{2}-\mathcal{K}^{\prime })\mathbf{N}+(\mathcal{T}^{\prime }-3\mathcal{KT})\mathbf{B}].\end{aligned}$$
Let us write $$\begin{aligned}
Q_{1}& =\frac{1}{\rho ^{2}}\mathbf{N}, \\
Q_{2}& =\frac{1}{\rho ^{3}}\left( -\mathcal{K}\mathbf{N}\mathcal{+T}\mathbf{B}\right) , \\
Q_{3}& =\frac{1}{\rho ^{4}}[(2\mathcal{K}^{2}\mathcal{+\mathcal{T}}^{2}-\mathcal{K}^{\prime })\mathbf{N}+(\mathcal{T}^{\prime }-3\mathcal{KT})\mathbf{B}].\end{aligned}$$
$\gamma ^{\prime }(s),\gamma ^{\prime \prime }(s),\gamma ^{\prime \prime
\prime }(s)$ and $\gamma ^{\prime \prime \prime \prime }(s)$ are linearly dependent if and only if $Q_{1},Q_{2}$ and $Q_{3}$ are linearly dependent.
Frenet curves (of osculating order $3$) in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$ are called [@5]:
1. of type equiform AW$(1)$ if they satisfy $Q_{3}=0,$
2. of type equiform AW$(2)$ if they satisfy $\left\Vert Q_{2}\right\Vert
^{2}$ $Q_{3}=\langle Q_{3}\left( s\right) ,Q_{2} \rangle
Q_{2} ,$
3. of type equiform AW$(3)$ if they satisfy $\left\Vert Q_{1}\right\Vert
^{2}$ $Q_{3}=\langle Q_{3} ,Q_{1}\left( s\right) \rangle
Q_{1} ,$
4. of type weak equiform AW$(2)$ if they satisfy$$Q_{3}=\left\langle Q_{3},Q_{2}^{\ast }\right\rangle Q_{2}^{\ast },$$
5. of type weak equiform AW$\left( 3\right) $ if they satisfy$$Q_{3}=\left\langle Q_{3},Q_{1}^{\ast }\right\rangle Q_{1}^{\ast },$$where$$\begin{aligned}
Q_{1}^{\ast } &=&\frac{Q_{1}}{\left\Vert Q_{1}\right\Vert }, \notag \\
Q_{2}^{\ast } &=&\frac{Q_{2}-\left\langle Q_{2},Q_{1}^{\ast }\right\rangle
Q_{1}^{\ast }}{\left\Vert Q_{2}-\left\langle Q_{2},Q_{1}^{\ast
}\right\rangle Q_{1}^{\ast }\right\Vert }.\end{aligned}$$
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve $($of osculating order $3)$ in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$,
\(i) $\gamma $ is of type weak equiform AW$(2)$ if and only if $$2\mathcal{K}^{2}+\mathcal{T}^{2}-\mathcal{K}^{\prime }=0,$$
\(ii) $\gamma $ is of type weak equiform AW$(2)$ if and only if $$\mathcal{T}^{\prime }-3\mathcal{KT(}s\mathcal{)}=0.$$
According to Definition 4.1 and Notation 4.1, the proof is obvious.
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve $($of osculating order $3)$ in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$. Then $\gamma $ is of type equiform AW$(2)$ if and only if$$-\mathcal{K}^{\prime }+2\mathcal{K}^{2}+\mathcal{T}^{2}=0,$$$$\text{\ }3\mathcal{KT-T}^{\prime }=0.$$
Since $\gamma $ is of type equiform AW$(2)$, then from $(4.3)$, we obtain$$\frac{1}{\rho ^{4}}[(2\mathcal{K}^{2}\mathcal{+\mathcal{T}}^{2}(s)-\mathcal{K}^{\prime })\mathbf{N}+(\mathcal{T}^{\prime }-3\mathcal{KT})\mathbf{B}]=0.$$As we know, the vectors $\mathbf{N}$ and $\mathbf{B}$ are linearly independent, so we can write$$2\mathcal{K}^{2}\mathcal{+\mathcal{T}}^{2}-\mathcal{K}^{\prime }=0\text{ and
\ }\mathcal{T}^{\prime }-3\mathcal{KT}=0.$$The converse statement is straightforward and therefore the proof is completed.
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve $($of osculating order $3)$ in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$. Then $\gamma $ is of type equiform AW$(2)$ if and only if$$\mathcal{K}^{2}\mathcal{T}-\mathcal{KT}^{\prime }+\mathcal{TK}^{\prime }-\mathcal{T}^{3}=0.$$
Assuming that $\gamma $ is a Frenet curve in the equiform geometry of $G_{3}^{1}$ , then from $(4.2)$ and $(4.3)$, one can write$$\begin{aligned}
Q_{2}& =a_{11}\mathbf{N}+a_{12}\mathbf{B}, \\
Q_{3}& =a_{21}\mathbf{N}+a_{22}\mathbf{B},\end{aligned}$$where $a_{11}$,$a_{12}$, $a_{21}$ and $a_{22}$ are differentiable functions. Since $Q_{2}$ and $Q_{3}$ are linearly dependent, coefficients determinant equals zero and hence $$\left\vert
\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22}\end{array}\right\vert =0,$$where $$\begin{aligned}
a_{11} &=&\frac{-1}{\rho ^{3}}\mathcal{K},\text{ }a_{12}=\frac{1}{\rho ^{3}}\mathcal{T}, \notag \\
a_{21} &=&\frac{1}{\rho ^{4}}[-\mathcal{K}^{\prime }+2\mathcal{K}^{2}+\mathcal{T}^{2}],\text{ } \notag \\
a_{22} &=&\frac{1}{\rho ^{4}}[-3\mathcal{KT+T}^{\prime }].\end{aligned}$$From $(4.11)$ and $(4.12)$, we obtain $(4.10)$. It can be easily shown that the converse assertion is also true.
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve (of osculating order3) in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$,
\(i) If $\gamma $ is an isotropic logarithmic spiral in $G_{3}^{1}$, then $\gamma $ is of equiform AW$(2)$-type curve.
\(ii) If $\gamma $ is an equiform space or timelike general (circular) helix in $G_{3}^{1}$, then it is not of equiform AW$(k)$, weak AW$(2)$ and weak AW$(3)$-types.
Let $\gamma :I\rightarrow G_{3}^{1}$ be a Frenet curve (of osculating order $3$) in the equiform geometry of $G_{3}^{1}$. Then $\gamma $ is of equiform AW$\left( 3\right) $-type if and only if $$\mathcal{T}^{\prime }-3\mathcal{KT}=0.$$
Using Definition $4.1$ and Eqs. $(4.1)$ and $(4.3)$, we obtain $(4.13)$. The converse direction is obvious, hence our Theorem is proved.
Bertrand curves of AW$(k)$-type
===============================
A curve $\gamma :I\rightarrow G_{3}^{1}$ with equiform curvature $\mathcal{K}=0$ is called an equiform Bertrand curve if there exist a curve $\bar{\gamma}:I\rightarrow G_{3}^{1}$ with equiform curvature $\mathcal{\bar{K}}=0$ such that the principal normal lines of $\gamma $ and $\bar{\gamma}$ are parallel at the corresponding points. In this case $\bar{\gamma}$ is called an equifrm Bertrand mate of $\gamma $ and vise versa.
By Definition $5.1$, we can say that for given an equiform Bertrand pair $\left( \gamma ,\bar{\gamma}\right) $, there exist a functional relation $\bar{s}=\bar{s}(s)$ such that $\lambda (\bar{s}(s)=\lambda (s)$, then the equiform Bertrand mate of $\gamma $ is given by
$$\bar{\gamma}(s)=\gamma (s)+\lambda \mathbf{N}.$$
If $\left( \gamma ,\bar{\gamma}\right) $ is an equiform Bertrand pair in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$, then
(i)
: The function $\lambda $ is constant.
(ii)
: $\gamma $ with non-zero constant equiform torsion is a circular helix in $G_{3}^{1}$.
(iii)
: $\gamma $ with zero equiform torsion is an isotropic circle of $G_{3}^{1}$
Along $\gamma $ and $\bar{\gamma}$, let $\{\mathbf{T},\mathbf{N},\mathbf{B}\}
$ and $\{\mathbf{\bar{T}},\mathbf{\bar{N}},\mathbf{\bar{B}}\}$ be the Frenet frames according to the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$, respectively. Differentiate $(5.1)$ with respect to $s$, we obtain $$\mathbf{\bar{T}}=\mathbf{T}+\lambda \mathbf{N}^{\prime }+\lambda ^{\prime }\mathbf{N}.$$By using $(3.5)$, we have $$\mathbf{\bar{T}}=\mathbf{T}+\left( \lambda \mathcal{K+\lambda }^{\prime
}\right) \mathbf{N}+\lambda \mathcal{T}\mathbf{B}\text{.}$$Since $\mathbf{\bar{N}\ }$is parallel to $\mathbf{N}$, we get $$\lambda \mathcal{K+\lambda }^{\prime }\mathcal{=}0,$$it follows that $$\lambda =const.$$
If $\gamma $ has a non-zero constant equiform torsion, then $\gamma $ is characterized by $$\kappa =const.\neq 0,~\tau =const.\neq 0,$$and therefore $\tau /\kappa =const.$holds.
On the other hand, whenever $\mathcal{T}=0$, the natural equations of $\gamma $ is given by $$\kappa =const.\neq 0,~\tau =0,$$and so, the curve $\gamma $ is an isotropic circle in $G_{3}^{1}$ [@14]. Thus the proof is completed.
If $\left( \gamma ,\bar{\gamma}\right) $ is a Bertrand pair in the equiform geometry of the pseudo-Galilean space $G_{3}^{1}$, then the angle between tangent vectors at corresponding points is constant.
To prove that the angle is constant, we need to show that $\left\langle
\mathbf{\bar{T}},\mathbf{T}\right\rangle ^{\prime }=0$. For this purpose using $(3.5)$ to obtain $$\begin{aligned}
\left\langle \mathbf{\bar{T}},\mathbf{T}\right\rangle ^{\prime }&
=\left\langle \mathbf{\bar{T}}^{\prime },\mathbf{T}\right\rangle
+\left\langle \mathbf{\bar{T}},\mathbf{T}^{\prime }\right\rangle \notag \\
& =\left\langle \mathcal{\bar{K}}\mathbf{\bar{T}}+\mathbf{\bar{N}},\mathbf{T}\right\rangle +\left\langle \mathbf{\bar{T}},\mathcal{K}\mathbf{T}+\mathbf{N}\right\rangle \notag \\
& =\mathcal{\tilde{K}}\left\langle \mathbf{\bar{T}},\mathbf{T}\right\rangle
+\left\langle \mathbf{\bar{N}},\mathbf{T}\right\rangle +\mathcal{K}\left\langle \mathbf{\bar{T}},\mathbf{T}\right\rangle \notag \\
& \ \ \ +\left\langle \mathbf{\bar{T}},\mathbf{N}\right\rangle .\text{ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\end{aligned}$$Because of $\mathbf{\bar{N}}$ is parallel to $\mathbf{N},$ then$$\left\langle \mathbf{\bar{N}},\mathbf{T}\right\rangle =0,\left\langle
\mathbf{\bar{T}},\mathbf{N}\right\rangle =0.$$Since $\left( \gamma ,\bar{\gamma}\right) $ is a Berrand pair in the equiform geometry of $G_{3}^{1}$, then from Theorem $5.1$, we have$$\mathcal{K=}0\text{ and }\mathcal{\bar{K}=}0.$$After substituting $(5.4)$ and $(5.5)$ into $(5.3)$, we get $$\left\langle \mathbf{\bar{T}},\mathbf{T}\right\rangle ^{\prime }=0.$$In the light of $(5.6)$ the angle between $\mathbf{\bar{T}},\mathbf{T}$ is constant. Thus this completes the proof.
Let $\gamma (s):I\rightarrow G_{3}^{1}$ be a Bertrand curve in the equiform geometry of $G_{3}^{1}.$ Then
\(i) $\gamma $ is a weak equiform AW$(3)$-type but not a weak equiform AW$(2)$-type.
\(ii) $\gamma $ is equiform AW$(3)$-type but not equiform AW$(1)$ and AW$(2)$-types.
Examples
========
We consider some examples (timelike and spacelike curves [@11; @12]) which characterize equiform general (circular) helices with respect to the Frenet frame $\left\{ \mathbf{T},\mathbf{N},\mathbf{B}\right\} $ in the equiform geometry of $G_{3}^{1}$ which satisfy some conditions of equiform curvatures ($\mathcal{K=K}(s),\mathcal{T=T}(s);~\mathcal{K=}const.\neq 0,\mathcal{T=}const.\neq 0;~\mathcal{K=}const.\neq 0,\mathcal{T=}0$).
Consider the equiform **timelike** general helix $\mathbf{r}:I\longrightarrow G_{3}^{1},I\subseteq \mathbb{R}$ parameterized by the arc length $s$ with differential form $ds=dx,$ given by $$\mathbf{r}(x)=(x,y(x),z(x)),$$where $$\begin{aligned}
x(s) &=&s, \\
y(s) &=&\frac{e^{-as}}{\left( a^{2}-b^{2}\right) ^{2}}\left( \left(
a^{2}+b^{2}\right) \cosh \left( bs\right) +2ab\sinh \left( bs\right) \right)
, \\
z(s) &=&\frac{e^{-as}}{\left( a^{2}-b^{2}\right) ^{2}}\left( 2ab\cosh \left(
bs\right) +\left( a^{2}+b^{2}\right) \sinh \left( bs\right) \right) ; \\
a,b&\in&\mathbb{R}-\left\{ 0\right\}.\end{aligned}$$The corresponding derivatives of $\mathbf{r}$** **are as follows $$\begin{aligned}
\mathbf{r}^{\prime } &=&\left( 1,\frac{-e^{-as}}{\left( a^{2}-b^{2}\right) }\left( a\cosh \left( bs\right) +b\sinh \left( bs\right) \right) ,\frac{e^{-as}}{\left( b^{2}-a^{2}\right) }\left( b\cosh \left( bs\right) +a\sinh
\left( bs\right) \right) \right) , \\
\mathbf{r}^{\prime \prime } &=&\left( 0,e^{-as}\cosh \left( bs\right)
,e^{-as}\sinh \left( bs\right) \right) , \\
\mathbf{r}^{\prime \prime \prime } &=&\left( 0,e^{-as}\left( -a\cosh \left(
bs\right) +b\sinh \left( bs\right) \right) ,e^{-as}\left( b\cosh \left(
bs\right) -a\sinh \left( bs\right) \right) \right) .\end{aligned}$$
First of all, we find that the tangent vector of $\mathbf{r}$ has the form $$\begin{aligned}
\mathbf{e}_{1} &=&\left( x^{\prime },y^{\prime },z^{\prime }\right) \\
&=&\left( 1,\frac{-e^{-as}}{\left( a^{2}-b^{2}\right) }\left( a\cosh \left(
bs\right) +b\sinh \left( bs\right) \right) ,\frac{e^{-as}}{\left(
b^{2}-a^{2}\right) }\left( b\cosh \left( bs\right) +a\sinh \left( bs\right)
\right) \right) .\end{aligned}$$
Then the two normals (normal and binormal) of the curve are, respectively$$\begin{aligned}
\mathbf{e}_{2} &=&\left( 0,\cosh \left( bs\right) ,\sinh \left( bs\right)
\right) , \\
\mathbf{e}_{3} &=&\left( 0,\sinh \left( bs\right) ,\cosh \left( bs\right)
\right) ;~~\det [\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}]=1.\end{aligned}$$
Thus the computations of the coordinate functions of $\mathbf{r}$ lead to$$\kappa =e^{-as},~\tau =b~.$$
From the equiform Frenet formulas $(3.5)$ we can express vector fields $\mathbf{T},\mathbf{N},\mathbf{B}$ as follows $$\begin{aligned}
\mathbf{T} &=&\left( e^{as},\frac{-1}{\left( a^{2}-b^{2}\right) }\left(
a\cosh \left( bs\right) +b\sinh \left( bs\right) \right) ,\frac{1}{\left(
b^{2}-a^{2}\right) }\left( b\cosh \left( bs\right) +a\sinh \left( bs\right)
\right) \right) , \\
\mathbf{N} &=&\left( 0,e^{as}\cosh \left( bs\right) ,e^{as}\sinh \left(
bs\right) \right) , \\
\mathbf{B} &=&\left( 0,e^{as}\sinh \left( bs\right) ,e^{as}\cosh \left(
bs\right) \right) ,\end{aligned}$$respectively. In the light of this, the equiform curvatures are given by$$~\mathcal{K}=ae^{as},\mathcal{T}=-be^{as}.$$
{width="5cm"} \[fig:timelikeg2\]
Let $\mathbf{r}:I\longrightarrow G_{3}^{1},I\subseteq \mathbb{R}$ be the equiform **spacelike** general helix, given by $$\mathbf{r}(x)=(x,y(x),z(x)),$$where $$\begin{aligned}
x(s) &=&s, \\
y(s) &=&\frac{e^{-as}}{\left( a^{2}-b^{2}\right) ^{2}}\left( 2ab\cosh \left(
bs\right) +\left( a^{2}+b^{2}\right) \sinh \left( bs\right) \right) , \\
z(s) &=&\frac{e^{-as}}{\left( a^{2}-b^{2}\right) ^{2}}\left( \left(
a^{2}+b^{2}\right) \cosh \left( bs\right) +2ab\sinh \left( bs\right) \right)
; \\
a,b &\in &\mathbb{R}-\left\{ 0\right\} .\end{aligned}$$For the coordinate functions of $\mathbf{r}$, we have $$\begin{aligned}
\mathbf{r}^{\prime } &=&\left( 1,\frac{e^{-as}}{\left( b^{2}-a^{2}\right) }\left( b\cosh \left( bs\right) +a\sinh \left( bs\right) \right) ,\frac{-e^{-as}}{\left( a^{2}-b^{2}\right) }\left( a\cosh \left( bs\right) +b\sinh
\left( bs\right) \right) \right) , \\
\mathbf{r}^{\prime \prime } &=&\left( 0,e^{-as}\sinh \left( bs\right)
,e^{-as}\cosh \left( bs\right) \right) , \\
\mathbf{r}^{\prime \prime \prime } &=&\left( 0,e^{-as}\left( b\cosh \left(
bs\right) -a\sinh \left( bs\right) \right) ,e^{-as}\left( b\sinh \left(
bs\right) -a\cosh \left( bs\right) \right) \right) .\end{aligned}$$Also, the associated trihedron is given by $$\begin{aligned}
\mathbf{e}_{1} &=&\left( 1,\frac{e^{-as}}{\left( b^{2}-a^{2}\right) }\left(
b\cosh \left( bs\right) +a\sinh \left( bs\right) \right) ,\frac{-e^{-as}}{\left( a^{2}-b^{2}\right) }\left( a\cosh \left( bs\right) +b\sinh \left(
bs\right) \right) \right) , \\
\mathbf{e}_{2} &=&\left( 0,\sinh \left( bs\right) ,\cosh \left( bs\right)
\right) , \\
\mathbf{e}_{3} &=&\left( 0,-\cosh \left( bs\right) ,-\sinh \left( bs\right)
\right) .~\end{aligned}$$
The curvature and torsion of this curve are $$\kappa =e^{-as},~\tau =-b~.~$$
Furthermore, the tangent, normal and binormal vector fields in the equiform geometry of $G_{3}^{1}$ are obtained as follows $$\begin{aligned}
\mathbf{T} &=&\left( e^{as},\frac{1}{\left( b^{2}-a^{2}\right) }\left(
b\cosh \left( bs\right) +a\sinh \left( bs\right) \right) ,\frac{-1}{\left(
a^{2}-b^{2}\right) }\left( a\cosh \left( bs\right) +b\sinh \left( bs\right)
\right) \right) , \\
\mathbf{N} &=&\left( 0,e^{as}\sinh \left( bs\right) ,e^{as}\cosh \left(
bs\right) \right) , \\
\mathbf{B} &=&\left( 0,-e^{as}\cosh \left( bs\right) ,-e^{as}\sinh \left(
bs\right) \right) ,\end{aligned}$$respectively.
The equiform curvatures of $\mathbf{r}$ are $$~\mathcal{K}=ae^{as},\mathcal{T}=-be^{as}.$$
{width="6.5cm"} \[fig:spacelikeg2\]
In this example, let us consider the equiform timelike** circular** helix $\mathbf{r}:I\longrightarrow G_{3}^{1}$ given by $$\mathbf{r}(x)=(x,y(x),z(x)),$$where $$\begin{aligned}
x(s) &=&s, \\
y(s) &=&\frac{a^{3}s}{b\left( b^{2}-a^{2}\right) }\left( b\sinh \left( \frac{b}{a}\ln (as)\right) -a\cosh \left( \frac{b}{a}\ln (as)\right) \right) , \\
z(s) &=&\frac{a^{3}s}{b\left( b^{2}-a^{2}\right) }\left( b\cosh \left( \frac{b}{a}\ln (as)\right) -a\sinh \left( \frac{b}{a}\ln (as)\right) \right) ; \\
a,b &\in &\mathbb{R}-\left\{ 0\right\} .\end{aligned}$$For this curve, the equiform vector fields are obtained as follows $$\begin{aligned}
\mathbf{T} &=&\left( \frac{s}{a},\frac{as}{b}\cosh \left( \frac{b}{a}\ln
(as)\right) ,\frac{as}{b}\sinh \left( \frac{b}{a}\ln (as)\right) \right) , \\
\mathbf{N} &=&\left( 0,\frac{s}{a}\sinh \left( \frac{b}{a}\ln (as)\right) ,\frac{s}{a}\cosh \left( \frac{b}{a}\ln (as)\right) \right) , \\
\mathbf{B} &=&\left( 0,\frac{s}{a}\cosh \left( \frac{b}{a}\ln (as)\right) ,\frac{s}{a}\sinh \left( \frac{b}{a}\ln (as)\right) \right) ,\end{aligned}$$respectively.
It follows that$$\mathcal{K}=\frac{1}{a},\mathcal{T}=\frac{-b}{a^{2}}.$$
{width="6cm"} \[fig:circulart\]
Let the equiform **spacelike** circular helix $\mathbf{r}:I\longrightarrow G_{3}^{1},I\subseteq \mathbb{R}$ in the form $$\mathbf{r}(x)=(x,y(x),z(x)),$$where $$\begin{aligned}
x(s) &=&s, \\
y(s) &=&\frac{a^{3}s}{b\left( b^{2}-a^{2}\right) }\left( b\cosh \left( \frac{b}{a}\ln (as)\right) -a\sinh \left( \frac{b}{a}\ln (as)\right) \right) , \\
z(s) &=&\frac{a^{3}s}{b\left( b^{2}-a^{2}\right) }\left( b\sinh \left( \frac{b}{a}\ln (as)\right) -a\cosh \left( \frac{b}{a}\ln (as)\right) \right) ; \\
a,b &\in &\mathbb{R}-\left\{ 0\right\} .~\end{aligned}$$Here, the equiform differntial vectors are respectively, as follows $$\begin{aligned}
\mathbf{T} &=&\left( \frac{s}{a},\frac{as}{b}\sinh \left( \frac{b}{a}\ln
(as)\right) ,\frac{as}{b}\cosh \left( \frac{b}{a}\ln (as)\right) \right) , \\
\mathbf{N} &=&\left( 0,\frac{s}{a}\cosh \left( \frac{b}{a}\ln (as)\right) ,\frac{s}{a}\sinh \left( \frac{b}{a}\ln (as)\right) \right) , \\
\mathbf{B} &=&\left( 0,-\frac{s}{a}\sinh \left( \frac{b}{a}\ln (as)\right) ,-\frac{s}{a}\cosh \left( \frac{b}{a}\ln (as)\right) \right) .\end{aligned}$$Equiform curvature and equiform torsion are calculated as follows$$\mathcal{K}=\frac{1}{a},\mathcal{T}=\frac{b}{a^{2}}.$$
{width="5cm"} \[fig:circulars\]
If we consider the equiform **timelike** isotropic logarithmic spiral $\mathbf{r}:I\longrightarrow G_{3}^{1},I\subseteq \mathbb{R}$ parameterized by the arc length $s$ with differential form $ds=dx,$ given by$$\mathbf{r}(x)=(x,y(x),0),$$where $$\begin{aligned}
x(s) &=&s, \\
y(s) &=&\frac{as+b}{a^{2}}\left( \ln (as+b)-1\right) , \\
z(s) &=&0; \\
a,b &\in &\mathbb{R}-\left\{ 0\right\} .\end{aligned}$$For this curve, we get$$\begin{aligned}
\mathbf{r}^{\prime } &=&\left( 1,\frac{\ln (as+b)}{a},0\right) , \\
\mathbf{r}^{\prime \prime } &=&\left( 0,\frac{1}{as+b},0\right) , \\
\mathbf{r}^{\prime \prime \prime } &=&\left( 0,\frac{-a}{\left( as+b\right)
^{2}},0\right) ,\end{aligned}$$and $$\begin{aligned}
\mathbf{e}_{1} &=&\left( 1,\frac{\ln (as+b)}{a},0\right) , \\
\mathbf{e}_{2} &=&\left( 0,1,0\right) , \\
\mathbf{e}_{3} &=&\left( 0,0,1\right) ;~\kappa =\frac{1}{as+b},~\tau =0.\end{aligned}$$In this case, equiform Frenet vectors and equiform curvatures are as follows $$\begin{aligned}
\mathbf{T} &=&\left( as+b,\frac{\left( as+b\right) \ln (as+b)}{a},0\right) ,
\\
\mathbf{N} &=&\left( 0,as+b,0\right) , \\
\mathbf{B} &=&\left( 0,0,as+b\right) ,~\mathcal{K}=a,\mathcal{T}=0.\end{aligned}$$respectively.
{width="5cm"} \[fig:spiral11\]
From aforementioned calculations, according to (**Proposition** $\mathbf{4.2}$** and Theorems** $\mathbf{4.1-4.3}$), examples $1-4$ are not characterize curves of equiform AW$(k)$, weak equiform AW$(2)$ and weak equiform AW$(3)$-types. On the other hand, the last example shows that the curve is of equiform AW$(2)$ and AW$(3)$-types and it is not of equiform AW$(1)$-type. Also, it is of weak equiform AW$(2)$ and not of weak equiform AW$(3)$-types.
Conclusion
==========
In this paper, we have considered some special curves of equiform AW$(k)$-type of the pseudo-Galilean $3$-space. Also, using the equiform curvature conditions of these curves, the necessary and sufficient conditions for them to be equiform AW$(k)$ and weak equiform AW$(k)$-types are given. Furthermore, several examples to confirm our main results have been given and illustrated.
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M. Külahci, M. Bektas and M. Ergüt, On harmonic curvatures of null curves of the AW$(k)$-type in Lorentzian space, Z. Naturforsch, 63 a (2008), 248-252.
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C. Özgür and F. Gezgin, On some curves of AW(k)-type, Differ. Geom. Dyn. Syst., 7(2005), 74-80.
D. W. Yoon, General Helices of AW$(k)$-Type in the Lie Group, Journal of Applied Mathematics, Article ID 535123, (2012), 1-10.
Z. Erjavec and B. Divjak, The equiform differential geometry of curves in the pseudo-Galilean space, Math. Communications, 13(2008), 321-332.
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B. Divjak, Geometrija pseudogalilejevih prostora, Ph. D. thesis, University of Zagreb, 1997.
B. Divjak, Curves in pseudo-Galilean geometry, Annales Univ. Sci. Budapest 41(1998), 117-128.
[^1]: E-mail address: mohamed\[email protected]
|
---
abstract: 'We consider symmetric Markov chains on $\Z^d$ where we do [**not**]{} assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of normalized symmetric Markov chains to converge to a diffusion on $\R^d$ corresponding to an elliptic operator in divergence form.'
author:
- 'Richard F. Bass [^1] and Takashi Kumagai [^2]'
title: 'Symmetric Markov chains on $\Z^d$ with unbounded range'
---
Introduction
============
Let $X_n$ be a symmetric Markov chain on $\Z^d$. We say that $X_n$ has [*bounded*]{} range if there exists $K>0$ such that $\P(X_{n+1}=y \mid X_n=x)=0$ whenever $| y-x|\geq K$. The range is [*unbounded*]{} if for every $K$ there exists $x$ and $y$ (depending on $K$) with $|x-y|>K$ such that $\P(X_{n+1}=y \mid X_n=x)>0$. There is a great deal known about Markov chains on graphs when the chains have bounded range. The purpose of this paper is to obtain results for Markov chains on $\Z^d$ that have unbounded range.
Suppose $C_{xy}$ is the conductance between $x$ and $y$. We impose a condition on $C_{xy}$ (see (A3) below) which essentially says that the $C_{xy}$ satisfy a uniform second moment condition. Let $Y_t$ be the continuous time Markov chain on $\Z^d$ determined by the $C_{xy}$, while $X_n$ is the discrete time Markov chain determined by these conductances. The transition probabilities for the Markov chain $X$ are defined by $$\P^x(X_1=y)=\frac{C_{xy}}{\sum_z C_{xz}},$$ while the process $Y_t$ is the Markov chain that has the same jumps as $X$ but where the times between jumps are independent exponential random variables with parameter 1. When (A3) holds, together with two very mild regularity conditions, we obtain upper bounds on the transition probabilities of the form $$\P(Y_t=y\mid Y_0 =x) \leq ct^{-d/2}$$ and some corresponding lower bounds when $x$ and $y$ are not too far apart. Unlike the case of bounded range, reasonable universal bounds of Gaussian type need not hold when the range is unbounded. We also obtain bounds on the exit probabilities $\P(\sup_{s\leq t} |Y_s-x|>\lambda t^{1/2})$.
We say a uniform Harnack inequality holds for $X$ if whenever $h$ is nonnegative and harmonic for the Markov chain $X$ in the ball $B(x_0,R)$ of radius $R>1$ about a point $x_0$, then $$h(x)\leq Ch(y), \qquad |x-x_0|, |y-x_0|<R/2,$$ where $C$ is independent of $R$. Even when $X_n$ is a random walk, i.e., the increments $X_n-X_{n-1}$ form an independent identically distributed sequence, a uniform Harnack inequality need not hold. However, if we impose an additional strong assumption (see (A4)) on the conductances, then we can prove such a Harnack inequality.
We prove that if we have Markov chains $X^{(n)}$ on $\Z^d$ satisfying Assumption (A3) uniformly in $n$, the sequence of processes $X^{(n)}_t=X_{[nt]}/\sqrt n$ is tight in the space $D[0,\infty)$ of right continuous, left limit functions, and all subsequential limit points are continuous processes. Under an additional condition on the conductances (A5) (different than the one needed for the Harnack inequality), we then show that the $X^{(n)}_\cdot$ converge weakly as processes to the law of the diffusion corresponding to an elliptic operator $${{\cal L}} f(x)=\sum_{i,j=1}^d
\frac{\partial}{\partial x_i} \Big( a_{ij}(\cdot) \frac{\partial f}{\partial x_j}(\cdot)\Big)
(x)$$ in divergence form. The exact statement is given by Theorem \[clt\].
In the case of bounded range Markov chains on $\Z^d$ some of our estimates have been obtained by [@SZ], and we obviously owe a debt to that paper. Not all of their methods extend to the unbounded case, however. In particular,
1. New techniques were needed to obtain the exit probability estimates.
2. A new method was needed to obtain lower bounds for the process killed on exiting a ball. This method should apply in many other instances, and is of interest in itself.
3. Harnack inequalities in the case of unbounded range are quite a bit more subtle, and this section is all new.
4. In the proof of the central limit theorem, new methods were needed to handle the case of unbounded range. Moreover, even in the bounded range case our result covers more general situations.
There are many versions of central limit theorems that investigate the asymptotic behavior of $ \sum_{i=1}^n f(X_i) $ when $X_n$ is a symmetric Markov chain on a graph. These are quite different from the central limit theorem of this paper. Our formulation has much more in common with the work of Stroock and Varadhan [@SV], Chapter 11. There they consider certain non-symmetric chains and show convergence to the law of a diffusion corresponding to an operator in nondivergence form: $$\sL f(x)=\sum_{i,j=1}^d a_{ij}(x) \frac{\partial^2 f}{\partial x_i \partial x_j} (x)
+\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x).$$ Our result is the analogue for symmetric chains and operators in divergence form.
The next section sets up the notation and framework and states the assumptions we need. Section 3 has the exit time and hitting time estimates, Section 4 has the lower bounds, and Section 5 discusses the Harnack inequality. Our central limit theorem is proved in Section 6.
The letter $c$ with or without subscripts and primes will denote finite positive constants whose exact value is unimportant and which may change from line to line.
Framework
=========
We let $|\cdot|$ be the Euclidean norm and $B(x,r):=\{y\in \Z^d: |x-y|< r\}$. We sometimes write $|A|$ for the cardinality of a set $A\subset \Z^d$.
For each $x,y\in \Z^d$ with $x\ne y$, let $C_{xy}\in [0,\infty)$ be such that $C_{xy}=C_{yx}$. We call $C_{xy}$ the [*conductance*]{} between $x$ and $y$. We assume the following;
(A1) There exist $c_1,c_2>0$ such that $$c_1\le \nu_x:=\sum_{y\in \Z^d}C_{xy}\le c_2\fal x\in \Z^d.$$
(A2) There exist $M_0\ge 1,
\delta>0$ such that the following holds: for any $x,y\in \Z^d$ with $|x-y|=1$, there exist $N\ge 2$ and $z_1,\cdots, x_N\in B(x,M_0)$ such that $x_1=x$, $x_N=y$ and $C_{x_ix_{i+1}}\ge \delta$ for $i=1,\cdots, N-1$.
(A3) There exists a decreasing function $\vp: \mathbb{N}\to \R_+$ with $\sum_{i=1}^{\infty}i^{d+1}\vp(i)<\infty$ and $\vp(2i)\le c\;\vp (i)$ for all $i\in \mathbb{N}$ such that $$C_{xy}\le \vp (|x-y|)\qquad \mbox{for all }~~x, y\in \Z^d.$$
Note that (A1) and (A2) are very mild regularity conditions. (A1) prevents degeneracies, while (A2) says, roughly speaking, that the chain is locally irreducible in a uniform way. (A3) is the substantive assumption and says that the $C_{xy}$ satisfy a uniform finite second moment condition. In fact, (A3) implies the following: there exists $C_0>0$ such that $$\label{eq:fin2mo}
\sup_{x\in \Z^d}\sum_{y\in \Z^d}|x-y|^2
C_{xy}\le C_0.$$ To see this, $$\begin{aligned}
\sum_{y\in \Z^d}|x-y|^2
C_{xy} & \le
\sum_{y\in \Z^d}|x-y|^2\vp(|x-y|)\label{eq:22a}\\
&=\sum_{i=0}^\infty\sum_{i< |x-y|\leq i+1} |x-y|^2 \vp(|x-y|)\nonumber \\
&\le c_3\sum_i (i+1)^2\vp (i) (i+1)^{d-1}
<\infty \nonumber\end{aligned}$$ for all $x\in \Z^d$, where (A3) is used in the last inequality.
Define a symmetric Markov chain by $$\P^x(X_1=y)=\frac {C_{xy}}{\nu_x}\fal
x,y\in \Z^d.$$ Define $p_n(x,y):=\P^x(X_n=y)$ and $\bp_n(x,y)=p_n(x,y)/\nu_y$. Note that $\bp_n(x,y)=\bp_n(y,x)$. By (A1), the ratio of $p_n(x,y)$ to $\bp_n(x,y)$ is bounded above and below by positive constants.
Let $\mu_x\equiv 1$ for all $x\in \Z^d$ and for each $A\subset \Z^d$, define $\mu (A)=\sum_{y\in A}\mu_y=|A|$ and $\nu (A)=\sum_{y\in A}\nu_y$. Note that $L^2(\Z^d, \mu)=L^2(\Z^d, \nu)$ by (A1). Now, for each $f\in L^2(\Z^d,\mu)$, define $$\begin{aligned}
\ce (f,f)&=&\tfrac 12 \sum_{x,y\in \Z^d}(f(x)-f(y))^2C_{xy},\\
\cf&=&\{f\in L^2(\Z^d,\mu): \ce(f,f)<\infty\}.\end{aligned}$$
It is easy to check $(\ce,\cf)$ is a regular Dirichlet form on $L^2(\Z^d,\mu)$ and the generator is $$\sum_{x,y\in \Z^d}(f(y)-f(x))
C_{xy}.$$
Let $Y_t$ be the corresponding continuous time $\mu$-symmetric Markov chain on $\Z^d$. Let $\{U_i^x: i\in \N, x\in \Z^d\}$ be an independent sequence of exponential random variables, where the parameter for $U_i^x$ is $\nu_x$, and that is independent of $X_n$ and define $T_0=0, T_n=\sum_{k=1}^n U_k^{X_{k-1}}$. Define $T_0=0, T_n=\sum_{k=1}^n U_k$. Set $\widetilde Y_t=X_n$ if $T_n\le t<T_{n+1}$; it is well known that the laws of $\widetilde Y$ and $Y$ are the same, and hence $\widetilde Y$ is a realization of the continuous time Markov chain corresponding to (a time change of) $X_n$. Note that by (A1), the mean exponential holding time at each point for $\widetilde Y$ can be controlled uniformly from above and below by a positive constant. Let $p(t,x,y)$ be the transition density for $Y_t$ with respect to $\mu$.
We now introduce several processes related to $Y_t$, needed in what follows. For each $D\ge 1$, let $\cs=D^{-1}\Z^d$ and define the rescaled process as $V_t=D^{-1}Y_{D^2t}$. Let $\mu^D$ be a measure on $\cs$ defined by $\mu^D(A)=D^{-d}\mu (DA)=D^{-d}|A|$ for $A\subset \cs$. We can easily show that the Dirichlet form corresponding to $V_t$ is $$\ce^D (f,f)=\tfrac 12 \sum_{x,y\in \cs}(f(x)-f(y))^2 D^{2-d}C_{Dx,Dy},$$ and the infinitesimal generator of $V_t$ is $$\sA^D f(x)=\sum_{y\in \cs} (f(y)-f(x)
C_{Dx,Dy}D^2
=\sum_{x,y\in \cs}(f(y)-f(x))\frac{C_{Dx,Dy}D^{2-d}}{\mu^D_x},$$ for each $f\in L^2(\cs,\mu^D)$, where we denote $\mu^D_x:=\mu^D(\{x\})=D^{-d}$ for each $x\in \cs$. The heat kernel $p^D(t,x,y)$ for $V_t$ with respect to $\mu^D$ can be expressed as $$\label{eq:heatrel}
p^D(t,x,y)=D^dp(D^2t, Dx,Dy)\fal x,y\in \cs, t>0.$$ For $\lambda\ge 1$, let $W^{\lambda}_t$ be a process on $\cs$ with the large jumps of $V_t$ removed. More precisely, $W^{\lambda}_t$ is a process whose Dirichlet form and infinitesimal generator are $$\begin{aligned}
\ce^{D,\lambda} (f,f)&=&\tfrac 12 \sum_{{x,y\in \cs}\atop
{|x-y|\le\lambda^{1/2}}}(f(x)-f(y))^2 D^{2-d}C_{Dx,Dy},\\
\calam f(x)&=&\sum_{{y\in \cs}\atop
{|x-y|\le\lambda^{1/2}}}(f(y)-f(x))\frac{C_{Dx,Dy}D^2}{\mu_{Dx}}.\end{aligned}$$ for each $f\in L^2(\cs,\mu^D)$. We denote the heat kernel for $W_t^\lambda$ by $p^{D,\lambda}(t,x,y)$, $x,y\in \cs$.
Heat kernel estimates
=====================
Nash inequality
---------------
For $f\in L^2(\Z^d, \mu)$, let $$\ce_{NN}(f,f)=\tfrac 12 \sum_{{x,y\in \Z^d};
{|x-y|=1}}(f(x)-f(y))^2,$$ which is the Dirichlet form for the simple symmetric random walk in $\Z^d$. We will prove the following Nash inequality.
Assume (A2). There exists $c_1>0$ such that for any $f\in L^2(\Z^d, \mu)$, $$\label{eq:nash1}
\|f\|_2^{2(1+2/d)}\le c_1 \ce(f,f)\|f\|_1^{4/d}.$$ In particular, $$\begin{aligned}
p(t,x,y)\le c_1t^{-d/2}&\fal & x,y\in
\Z^d, t>0,\label{eq:nash2}\\
p^D(t,x,y)\le c_1t^{-d/2}&\fal & x,y\in\cs, t>0.\label{eq:nash3}\end{aligned}$$
\[remark1\] [Since $p(t,x,y)=\P^x(Y_t=y)/\mu_y$, we have $p(t,x,y)\le 1/\mu_y$, so (\[eq:nash2\]) is a crude estimate for small $t$. However, we will continue to use it since we are mainly interested in the large time asymptotics. ]{}
Note that the equivalence of (\[eq:nash1\]) and (\[eq:nash2\]) is a well-known fact (see [@CKS]).
The Markov chain corresponding to $\ce_{NN}$ is a (continuous time) simple random walk; let $r_t$ be its transition probabilities. Since, as is well known, we have $r_t(x,x)\leq c t^{-d/2}$, then by [@CKS] we have $$\|f\|_2^{2(1+2/d)}\le c_1 \ce_{NN}(f,f)\|f\|_1^{4/d}
\fal f\in L^2(\Z^d, \mu).$$ See also [@SZ]. By (A2), there exists $c_2>0$ such that $$\ce_{NN}(f,f)\le c_2\ce(f,f)\fal f\in L^2(\Z^d, \mu).$$ Using these facts and (\[eq:heatrel\]), we have the desired result.
Exit time probability estimates
-------------------------------
In this subsection, we will obtain some exit time estimates. The argument presented here was first established in [@BL1] and then extended and simplified in [@CK], [@HK].
\[thm:kern2\] There exists $c_1>0$ such that $$\label{kern3}
p^{D,\lambda}(t,x,y)\leq c_1\;t^{-\frac{d}{2}}
\;\exp\left( -\lambda^{-\frac 12} |x-y|\right)$$ for all $t\in(0,1]$, $x, y\in \cs$ and $\lambda\ge M_0$, where $M_0$ is given in (A2).
Since $\lambda\ge M_0$, by (A2), we have $\ce_{NN}(f,f)\le c\ce^{D,\lambda} (f,f)$ for all $f\in L^2(\Z^d, \mu)$. So we have (\[eq:nash1\]) where $\ce(f,f)$ is replaced by $\ce^{1,\lambda} (f,f)$, and by a scaling argument we have $$p^{D,\lambda}(t,x,y)\le c_1t^{-d/2}\fal x,y\in\cs, t>0.$$ Thus by Theorem (3.25) of [@CKS], we have $$\label{kern4}
p^{D,\lambda}(t,x,y)\leq c_1\;t^{-\frac{d}2}
\;\exp\left(-E(2t,x,y)\right)$$ for all $t\le 1$ and $x,y\in \cs$, where $$\begin{aligned}
E(t,x,y)&=&\sup\{|\psi(y)-\psi(x)|-t\; \Lambda(\psi)^2 :
\Lambda(\psi)<\infty\},\\
\Lambda(\psi)^2&=& \|e^{-2\psi}\Gamma_\lambda[e^\psi]\|_\infty \vee
\|e^{2\psi}\Gamma_\lambda[e^{-\psi}]\|_\infty,\end{aligned}$$ and $\Gamma_{\lambda}$ is defined by $$\label{densi}
\Gamma_{\lambda}[v](\xi)=
\sum_{{\eta,\xi\in\cs}\atop
{|\xi-\eta|\le\lambda^{1/2}}}(v(\eta)-v(\xi))^2
\frac{C_{D\eta,D\xi}D^2}{\mu_{D\xi}},\qquad
\xi\in\cs.$$ Now let $\psi(\xi)=\lambda^{-1/2}(|\xi-x|\wedge |x-y|)$. Then, $|\psi(\eta)-\psi(\xi)|\le \lambda^{-1/2}|\eta-\xi|$, so that $$(e^{\psi(\eta)-\psi(\xi)}-1)^2\le |\psi(\eta)-\psi(\xi)|^2
e^{2|\psi(\eta)-\psi(\xi)|}\le c\lambda^{-1}|\eta-\xi|^2$$ for $\eta,\xi\in \cs$ with $|\eta-\xi|\le \lambda^{1/2}$. Hence $$\begin{aligned}
e^{-2\psi(\xi)}\Gamma_\lambda[e^\psi](\xi)&=&
\sum_{{\eta\in\cs}\atop
{|\xi-\eta|\le\lambda^{1/2}}}(e^{\psi(\eta)-\psi(\xi)}-1)^2
\frac{C_{D\eta,D\xi}D^2}{\mu_{D\xi}}\\
&\le &\lambda^{-1}\sum_{{\eta'\in \Z^d}
\atop{|\xi'-\eta'|\le D\lambda^{1/2}}}
|\eta'-\xi'|^2\frac{C_{\eta',\xi'}}{\mu_{\xi'}}\le C'\end{aligned}$$ for all $\xi\in \cs$ where (\[eq:fin2mo\]) is used in the last inequality. We have the same bound when $\psi$ is replaced by $-\psi$, so $\Lambda(\psi)^2\leq {C'}^2$. Noting that $|\psi(y)-\psi(x)|\le \lambda^{-\frac 12}\;
|x-y|$, we see that (\[kern3\]) follows from (\[kern4\]).
We now prove the following exit time estimate for the process. For $A\subset \Z^d$ and a process $Z_t$ on $\Z^d$, let $$\tau=\tau_A(Z):=\inf\{t\ge 0:
Z_t\notin A\}, \qquad T_A=T_A(Z):=\inf\{t\ge 0: Z_t\in A\}.$$
\[thm:4.2\] For $A> 0$ and $0<B<1$, there exists $\gamma_i =\gamma_i(A,B)\in (0,1)$, $i=1,2$, such that for every $D>0$ and $x\in
\Z^d$, $$\begin{aligned}
\P^x \left( \tau_{B(x,\, AD)}(Y)<\gamma_1 \, D^2 \right)
&\le &B,\label{eq:tighty}\\
\P^x \left( \tau_{B(x,\, AD)}(X)<\gamma_2 \, D^2 \right)
&\le &B.\label{eq:tightx}\end{aligned}$$
It follows from Lemma \[thm:kern2\] that for $t\in [1/4, \, 1]$ and $r >0$, $$\label{a6}
\P^x \left( |W^{\lambda}_t-x|\geq r\right)
= \sum_{ y\in \cs : \, |y-x|>r} p^{D,\lambda}(t, x, y) \mu^D_y \leq c_1 \,I_{r,\lambda},$$ where $I_{r,\lambda}:=e^{-\frac r2 \lambda^{-\frac 12}}$. Define $\sigma_r:=\inf \{t\geq 0: \,
|W_t^{\lambda}-W_0^{\lambda}|
\geq r \}$. Then by (\[a6\]) and the strong Markov property of $W^{\lambda}$ at time $\sigma_r$, $$\begin{aligned}
\P^x \left(\sigma_r \leq 1/2 \right)
&\leq & \P^x \left(\sigma_r \leq 1/2 \hbox{ and }
|W^{\lambda}_{1}-x|\leq r/2 \right)
+ \P^x \left(|W^{\lambda}_{1}-x| > r /2 \right) \\
&\leq & \P^x \left(\sigma_r \leq 1/2 \hbox{ and }
|W^{\lambda}_{1}-W^{\lambda}_{\sigma_r}|
) > r /2 \right) + c_1\, I_{r/2,\lambda} \\
&= & \mathbb{P}^x \left(1_{\{\sigma_r \leq 1/2\}}
\mathbb{P}^{W^{\lambda}_{\sigma_r}}\left( |W^{\lambda}_{1-{\sigma_r}}-
W^{\lambda}_{0}| > r /2 \right)\right) + c_1\, I_{r/2,\lambda} \\
&\leq & \sup_{y\in B(x,r)^c}\sup_{s\le 1/2} \mathbb{P}^y
\left( |W^{\lambda}_{1-s}-y| > r /2 \right)
+c_1\, I_{r/2,\lambda}\\\end{aligned}$$ Here in the second and the last inequalities, we used (\[a6\]). By the strong Markov property of $W^{\lambda}$, for every $r > 0$, $$\begin{aligned}
\P^x \left( \sup_{s\leq 1} |W^{\lambda}_{s}-
W^{\lambda}_0 | > r \right)
&\leq & \P^x (\sigma_r \leq 1/2) + \P^x (1/2 <\sigma_r \leq 1)
\nonumber\\
&\leq & c_2\, I_{r/2,\lambda} +
\mathbb{P}^x(\sigma_{r/2}\le 1/2)+
\mathbb{P}^x(\sigma_{r/2}> 1/2,\sigma_r \leq 1)\nonumber\\
&\leq & c_2\, I_{r/2,\lambda} +
\mathbb{P}^x(\sigma_{r/2}\le 1/2)+
\mathbb{E}^x \left[ \mathbb{P}^{W^{\lambda}_{1/2} } (\sigma_{r /2} \leq
1/2 ) \right]\nonumber\\
&\leq & c_3\, I_{r/4,\lambda}.\label{eq:kern5}\end{aligned}$$ The constants $c_1, c_2, c_3>0$ above are independent of $D \ge 1$, $x\in \cs$ and $\lambda \ge M_0$.
Now, define $B^{\lambda}$ to be the infinitesimal generator of $V_t$ with small jumps removed: $$\label{blambda}
B^{\lambda} v(\xi)=
\sum_{{\eta\in \cs}\atop
{|\eta-\xi|>\lambda^{1/2}}}(f(\eta)-f(\xi))\frac{C_{D\eta,D\xi}D^2}{\mu_{D\xi}}.$$ Recall that $\calam$ is the generator of $W^\lambda$. We see that $\calam+B^\lambda$ is the generator for $V_t$. Hence, if $Q^V_t$ and $Q^{W^\lambda}_t$ are the semigroups associated with $V_t$ and $W_t^\lambda$ respectively, we have that $$\label{eq:semi}
Q^V_t v=Q^{W^\lambda}_tv +\sum_{k=1}^\infty S_k^{\lambda}(t)v,\qquad
v\in L^{\infty}(\cs,\mu^D),$$ where $$\label{eq:sk}
S_k^{\lambda}(t)v=\int_0^t Q^{W^\lambda}_{t-s} B^{\lambda}
S_{k-1}^{\lambda}(s)v\, ds,\qquad k\geq1$$ with $S_0^{\lambda}(t):=Q^{W^\lambda}_t$ (see, for example, Theorem $2.2$ in [@Le]). Note that the series in (\[eq:semi\]) defines a bounded linear operator on $L^\infty(\cs,\mu^D)$ for each $t>0$; this can be seen as follows. First, by (\[eq:22a\]) and a simple calculation, we have $$\label{sekiin}
\sum_{{\eta\in \cs}\atop
{|\eta-\xi|>\lambda^{1/2}}}\frac{C_{D\eta,D\xi}D^2}{\mu_{D\xi}}
\le c_4\sum_{{y\in \Z^d}\atop
{|y-x|>D\lambda^{1/2}}}{C_{x,y}D^2}
\le \frac {c_5}{D^2\lambda}\sum_{y\in \Z^d}|x-y|^2\vp(|x-y|)D^2
\le \frac{c_6}{\lambda}.$$ Using this, we see that there exists $c_7>0$ independent of $\lambda$ such that $$\|B^{\lambda} v\|_\infty\leq \frac{c_7}{\lambda}\|v\|_\infty.$$ Noting that $\|Q^{W^\lambda}_tv\|_\infty\leq \|v\|_\infty$, by induction we have from (\[eq:sk\]) that $$\label{sknorm}
\|S_k^{\lambda}(t)v\|_\infty\leq \frac{(c_8\lambda^{-1}
\;t)^k}{k!} \|v\|_\infty, \quad t>0,\; k\geq 1,$$ and so the series above is bounded from $L^\infty(\cs,\mu^D)$ to $L^\infty(\cs,\mu^D)$ for each $t>0$.
We will apply the above with $\lambda=M_0$. By (\[sknorm\]), for any bounded function $f$ on $\cs$, we have $$\| Q^V_t f - Q^{W^\lambda}_t f \|_\infty
\leq \sum_{k=1}^\infty \frac{(c_8\lambda^{-1} \, t)^k} {k !} \, \| f \|_\infty
\leq c_9 \, t \, e^{c_9 t } \, \| f \|_\infty.$$ Applying this with $f$ equal to the indicator of $(\overline{B(\xi,r)})^c$, it follows that there is a constant $c_{10}>0$ that is independent of $D \ge 1$ such that for every $\xi\in \cs$ and every $t\le 1$, $$\label{eqn:a11}
\P^\xi \left( | V_t - \xi | > r \right) \leq
\P^\xi \left( | W^{M_0}_t -\xi | > r \right)
+ c_{10} \, t .$$ Applying the same argument we used in deriving (\[eq:kern5\]), we conclude there are positive constants $c_{11}, c_{12}$ such that for $\xi\in \cs$, $$\label{eqn:a9}
\P^\xi \left( \sup_{s\leq t} | V_s - \xi|> r
\right) \leq c_{11} e^{-c_{12} r }+c_{11} \,t \qquad \hbox{for
every } r >0 \hbox{ and } t\leq 1.$$ This implies that for every $x\in \Z^d$, $D'\ge 1$ and $r >0$, $$\label{eqn:a10}
\P^x \left( \sup_{s\leq {D'}^2 \, t}| Y_s - x | > r\, D'
\right) \leq c_{11} e^{-c_{12} r }+c_{11} \,t
\qquad \hbox{for every } r >0 \hbox{ and }
t\leq 1.$$ For $A>0$ and $B\in (0, \, 1)$, we choose $r_0$ and $t_0$ so that $c_{11} e^{-c_{12} r_0 }+c_{11} \,t_0<B$ and take $D=r_0D'/A$. Then, by (\[eqn:a10\]), $$\P^x \left( \sup_{s\leq \gamma_1\,D^2 } |Y_s-x| \geq
A\,D \right) \leq B \qquad \hbox{for every }
D\ge r_0/A,$$ where $\gamma_1=(A/r_0)^2t_0$. For $D< r_0/A$, we have $$\label{eq:r_0/a}
\P(U_1>\gamma_1\frac {r_0^2}{A^2})\le \P(U_1>\gamma_1D^2)\le
\P^x \left( \sup_{s\leq \gamma_1\,D^2 } |Y_s-x| <
A\,D \right),$$ where $U_1$ is an exponential random variable with parameter 1. By (A1), the left hand side of (\[eq:r\_0/a\]) is greater than $1-B$ if $\gamma_1$ is taken to be small. Thus, (\[eq:tighty\]) is proved.
Now (\[eq:tightx\]) can be proved in the same way as Theorem $2.8$ in [@BL1].
Lower bounds and regularity for the heat kernel
===============================================
We now introduce the space-time process $Z_s:=(U_s, V_s)$, where $U_s=U_0+s$. The filtration generated by $Z$ satisfying the usual conditions will be denoted by $\{ \widetilde \cf_s; \, s\geq 0\}$. The law of the space-time process $s\mapsto Z_s$ starting from $(t, x)$ will be denoted as $\P^{(t, x)}$. We say that a non-negative Borel measurable function $q(t,x)$ on $[0, \infty)\times \cs$ is [*parabolic*]{} in a relatively open subset $B$ of $[0, \infty)\times \cs$ if for every relatively compact open subset $B_1$ of $B$, $q(t, x)=\E^{(t,x)} \left[ q (Z_{\tau_{B_1}}) \right]$ for every $(t, x)\in B_1$, where $\tau_{B_1}=\inf\{s> 0: \, Z_s\notin B_1\}$.
We denote $\gamma:=\gamma(1/2, 1/2)<1$ the constant in (\[eq:tighty\]) corresponding to $A=B=1/2$. For $t\ge 0$ and $r>0$, we define $$Q^D(t,x,r):=[t,t+\gamma r^{2}]\times (B(x,r)\cap \cs),$$ where $B(x,r)=\{y\in \R^d: |x-y|\le r\}$.
It is easy to see the following (see, for example, Lemma 4.5 in [@CK] for the proof).
\[4.6\] For each $t_0>0$ and $x_0\in \Z^d$, $q^D(t,x):=p^D (t_0-t, x,x_0)$ is parabolic on $[0,t_0)\times \cs$.
The next proposition provides a lower bound for the heat kernel and is the key step for the proof of the Hölder continuity of $p^D (t, x, y)$.
\[szhol\] There exists $c_1>0$ and $\theta\in (0,1)$ such that if $|x-y|\leq t^{1/2}$, $x,y\in \Z^d$ and $r>t^{1/2}/\theta$, then $$\P^x(Y_t=y, \tau_{B(x,r)}>t)\geq c_1t^{-d/2}.$$
To prove this we first need some preliminary propositions. A version of the following weighted Poincaré inequality can be found in Lemma 1.19 of [@SZ]; we give an alternate proof.
\[1.19\] For $D\geq1$ and $l\in \Z^d$, let $$g_D(l)= c_0D^d \prod_{i=1}^d e^{-|l_i|/D},$$ where $c_0$ is determined by the equation $\sum_{l\in \Z^d}g_D(l)=1$. Then there exists $c_1>0$ such that $$c_1\Big<(f-\langle f \rangle_{g_D})^2\Big>_{g_D}\le D^{2-d}\sum_{l\in \Z^d}g_D(l)\sum_{i=1}^d(f(l+e^i)-f(l))^2,
\qquad f\in L^2(\cs),$$ where $$\langle f \rangle_{g_D}=D^{-d}\sum_{l\in \Z^d}f(D^{-1}l)g_D(l)$$ and $e^i$ is the element of $\Z^d$ whose $j$-th component is $1$ if $j=i$ and $0$ otherwise.
A scaling argument shows that it suffices to consider only the $D=1$ case. Because of the product structure, it is enough to consider the case when $d=1$.
The weighted Poincaré inequality restricted to integers in $[-10,10]$, i.e., where the sums are restricted to being over $\{-10, \ldots, 10\}$, follows easily from the usual Poincaré inequality. We will prove our weighted Poincaré inequality for positive $k$ and the same argument works for negative $k$. These facts together with the weighted Poincaré inequality on $[-10,10]$ and standard techniques as in [@Je] give us the weighted Poincaré inequality for all of $\Z$. So we restrict attention to nonnegative $k$. Therefore all our sums below are over nonnegative integers.
Let $$\begin{aligned}
I&=& \sum_{k,\ell} (f(k)-f(\ell))^2 e^{-k} e^{-\ell},\\
J_k&=& \sum_{\ell >k} \sum_{m=k}^{\ell-1} \sum_{n=k}^{\ell-1} \dfm\, \dfn e^{-\ell},\\
K&=&\sum_n \dfn^2 e^{-n}.\\\end{aligned}$$ Note $$I=\sum_k (f(k)-\langle{f}\rangle_{g_D})^2 e^{-k},$$ so we need to show $I\leq c_2K$. We have, since $f(k)-f(\ell)=0$ when $k=\ell$, $$\begin{aligned}
I&=&2\sum_k \sum_{\ell>k} (f(k)-f(\ell))^2 e^{-k} e^{-\ell}\\
&=&2\sum_k \sum_{\ell>k} \Big(\sum_{m=k}^{\ell-1} \dfm\Big)\Big(\sum_{n=k}^{\ell-1} \dfn
\Big) e^{-k}e^{-\ell}\\
&=&2\sum_k J_k e^{-k}.\end{aligned}$$ We see that $$\begin{aligned}
J_k&=&\sum_{m\geq k}\sum_{n\geq k} \sum_{\ell> m\lor n} e^{-\ell} \dfm\, \dfn\\
&\leq &\sum_{m\geq k} \sum_{n\geq k} e^{-m\lor n} \dfm\, \dfn\\
&=&2\sum_{m\geq k} \sum_{n\geq m} e^{-n} \dfm\, \dfn \\
&=&2\sum_{n\geq k} \sum_{m=k}^{n-1} e^{-n} \dfm\, \dfn\\
&&\qquad\qquad+2 \sum_{n\geq k} e^{-n} \dfn^2\\
&\leq& 2\sum_{n\geq k} e^{-n} \dfn\, (f(n)-f(k))+2K.\end{aligned}$$ Hence $$\begin{aligned}
I&\leq& c_3\sum_k \sum_{n\geq k} e^{-n} \dfn\, (f(n)-f(k)) e^{-k} +\sum_k 2e^{-k}K\\
&\leq& c_4\Big(\sum_k \sum_{n\geq k} e^{-n} e^{-k} \dfn^2\Big)^{1/2}
\Big(\sum_k\sum_{n\geq k} e^{-n} [f(n)-f(k)]^2 e^{-k}\Big)^{1/2}\\
&&\qquad\qquad +c_4K\\
&\leq &c_4K^{1/2} I^{1/2} +c_4K.\end{aligned}$$ This implies $$I\leq c_5K$$ as required.
The proof of the following lemma is similar to that of (1.16) in [@SZ], but since we need some modifications, we will give the proof.
\[1.20\] There is an $\eps>0$ such that $$\label{1.16}
p^D(t,D^{-1}k,D^{-1}m)\ge \eps t^{-d/2},$$ for all $D\ge 1$, $(t,k,m)\in (D^{-1},\infty)\times \cs\times \cs$ with $|D^{-1}k-D^{-1}m|\le 2t^{1/2}$.
First, note that it is enough to prove the following: there is an $\eps>0$ such that $$\label{1.25}
D^{-d}\sum_{l\in \Z^d}\log \Big(p^D(\tfrac 12, D^{-1}k,D^{-1}(l+m))\Big)g_D(l)\ge \tfrac 12 \log\eps,$$ for all $D\ge 1$ and $k,m\in \Z^d$ with $|D^{-1}(k-m)|\le 2$. Indeed, by the Chapman-Kolmogorov equation, symmetry, and the fact $g_D(j)\le 1$ for all $j\in \Z^d$, $$p^D(1,D^{-1}k,D^{-1}m)\ge D^{-d}\sum_jp^D(\tfrac 12, D^{-1}k,D^{-1}(j+k))p^D(\tfrac 12, D^{-1}m,
D^{-1}(j+k))g_D(j).$$ Thus, by Jensen’s inequality, (\[1.25\]) gives $$p^D(1,D^{-1}k,D^{-1}l)\ge \eps\qquad D\ge 1, |D^{-1}k-D^{-1}l|\le 2.$$ By a simple scaling argument, this gives (\[1.16\]).
So we will prove (\[1.25\]). Set $u_t(l)=p^D(t,D^{-1}k,D^{-1}(l+m))$ and let $$G(t)=D^{-d}\sum_{l\in \Z^d}\log(u_t(l))g_D(l).$$ By Jensen’s inequality, we see that $G(t)\le 0$. Further, $$G'(t)=D^{-d}\sum_{l\in \Z^d}\frac{\partial u}{\partial t}
(l)\frac{g_D(l)}{u_t(l)}
=-\ce^D(u_t(D\,\cdot),\frac{g_D(D\,\cdot)}{u_t(D\,\cdot)}).$$ Next, note that the following elementary inequality holds (see (1.23) of [@SZ] for the proof). $$\Big(\frac db -\frac ca\Big)(b-a)\le -\frac{c\wedge d}2 (\log b-\log a)^2+\frac{(d-c)^2}{2(c\wedge d)},\qquad
a,b,c,d>0.$$ Hence $$\begin{aligned}
G'(t)&=&-\frac{D^{2-d}}2
\sum_{l\in \Z^d}\sum_{e\in \Z^d}\Big(\frac{g_D(l+e)}{u_t(l+e)}-\frac{g_D(l)}{u_t(l)}\Big)
\Big(u_t(l+e)-u_t(l)\Big)C_{l,l+e}\\
&\ge & \frac{D^{2-d}}2\sum_{l\in \Z^d}\sum_{e\in \Z^d}\frac{g_D(l+e)\wedge g_D(l)}2\Big(
\log u_t(l+e)-\log u_t(l)\Big)^2C_{l,l+e}\\
&&\ \ -\frac{D^{2-d}}2\sum_{l\in \Z^d}\sum_{e\in \Z^d}\frac{|g_D(l+e)-g_D(l)|^2}{2(g_D(l+e)\wedge g_D(l))}C_{l,l+e}\\
&\ge & cD^{2-d}\sum_{l\in \Z^d}\sum_{j=1}^d(g_D(l+e^j)\wedge g_D(l))
\Big(\log u_t(l+e^j)-\log u_t(l)\Big)^2\\
&&\ \ -D^{2-d}\sum_{l\in \Z^d}\sum_{e\in \Z^d}\frac{|g_D(l+e)-g_D(l)|^2}{4(g_D(l+e)\wedge g_D(l))}C_{l,l+e},\end{aligned}$$ where the last inequality is due to (A2) and the definition of $g_D$ (here recall that $e^i$ is in the element of $\Z^d$ whose $j$-th component is $1$ if $j=i$ and $0$ otherwise). Note $|g_D(l+e)-g_D(l)|\le c_1D^{-1}|e| (g_D(l+e)\wedge g_D(l))$. Thus $$\begin{aligned}
D^{2-d}&\sum_{l\in \Z^d}\sum_{e\in \Z^d}\frac{|g_D(l+e)-g_D(l)|^2}{4(g_D(l+e)\wedge g_D(l))}C_{l,l+e}
\le c_2D^{-d}\sum_{l}\sum_{e}C_{l,l+e}|e|^2 (g_D(l+e)\wedge g_D(l))\\
&\le c_3\Big(\sup_{l}\sum_{e}C_{l,l+e}|e|^2\Big)\cdot D^{-d}\sum_{l}g_D(l)
= c_3\Big(\sup_{l}\sum_{e}C_{l,l+e}|e|^2\Big)<c_4,\end{aligned}$$ where we used (A3) in the last inequality. Note also $\min_{1\le i\le d} g_D(l+e^i)\ge c_5g_D(l)$. Combining these, we have $$\begin{aligned}
G'(t)&\ge &c_6D^{2-d}\sum_{l\in \Z^d}\sum_{j=1}^d
\Big(\log u_t(l+e^j)-\log u_t(l)\Big)^2g_D(l)-c_4\\
&\ge & c_7D^{-d}\sum_l(\log u_t(l)-G(t))^2g_D(l)-c_4,\end{aligned}$$ where we used Lemma \[1.19\] in the last inequality.
Next, for $\sigma>0$, set $A_t(\sigma)=\{l\in \Z^d: u_t(l)\ge e^{-\sigma}\}$. Then, writing $f^+$ and $f^-$ for the positive and negative parts of $f$, we have for each $\sigma>0$, $$\begin{aligned}
D^{-d}&\sum_l(\log u_t(l)-G(t))^2g_D(l)\ge
D^{-d}\sum_l(-(\log u_t)^-(l)-G(t))^2g_D(l)\\
&\ge \frac{G(t)^2}{2D^d}\sum_{l\in A_t(\sigma)}g_D(l)-\sigma^2,\end{aligned}$$ where we used the elementary inequality $(A+B)^2\ge (A^2/2)-B^2$, $A,B\in \R$, in the last inequality. Thus, we have $$\label{v2.v}
G'(t)\ge c_8I_{t,\sigma}G(t)^2-(c_4+\sigma^2),$$ where we let $I_{t,\sigma}=D^{-d}\sum_{l\in A_t(\sigma)}g_D(l)$. On the other hand, by (\[eq:tighty\]) and scaling, we can find $r_0>2$ such that $$D^{-d}\sum_{|D^{-1}l|\le r_0}p^D(t,D^{-1}k,D^{-1}(l+m))\ge 1/2,\qquad D\ge 1,
t\le 1, \mbox{ and } |D^{-1}(k-m)|\le 2.$$ In particular, if $\beta$ is the smallest value of $e^{-2U}$ on $[-r_0,r_0]$, then for each $t\in [1/4,1]$, $$1/2\le D^{-d}\sum_{|D^{-1}l|\le r_0}u_t(l)\le e^{-\sigma}r_0^d+(\sup_k |u_t(D^{-d}k)|)\cdot
\frac {I_{t,\sigma}}{\beta^d}.$$ Thus by taking $\sigma=(4r_0^d)$ and using (\[eq:nash3\]), we obtain $I_{t,\sigma}\ge c\beta^d$. Combining this with (\[v2.v\]), there exists $0<\delta<1$ such that $$\label{1.26s}
G'(t)\ge \delta G(t)^2-\delta^{-1},\qquad D\ge 1, t\in [1/4,1], \mbox{ and }
|D^{-1}(k-m)|\le 2.$$
Now, by (\[1.26s\]) and the mean value theorem, $$\label{1.26ws}
G(1/2)-G(t)\ge -(4\delta)^{-1},\qquad t\in [1/4,1].$$ We may assume $G(1/2)\le -5/(2\delta)$, since otherwise (\[1.25\]) is clear. Then, by (\[1.26ws\]) we have $G(t)\le -2\delta^{-1}$. So $\delta G(t)^2/2-\delta^{-1}\ge \delta^{-1}>0$. So, by (\[1.26s\]) again, $$G'(t)\ge \delta G(t)^2/2,\qquad t\in [1/4,1].$$ But this means that $$G(\tfrac 12)^{-1}\le G(\tfrac 12)^{-1}-G(\tfrac 14)^{-1}=-\int_{1/4}^{1/2}\frac {G'(s)}{G^2(s)}ds
\le -\frac {\delta}8,$$ and therefore $G(1/2)\ge -8\delta^{-1}$. Thus (\[1.25\]) holds with $\eps^{1/2}=\frac 12 \exp (-8\delta^{-1})$.
\[offdiag\] Given $\delta>0$ there exists $\kappa$ such that if $x,y\in \Z^d$ and $C\subset \Z^d$ with $\mbox{\rm dist}\,(x,C)$ and $\mbox{\rm dist}\,(y,C)$ both larger than $\kappa t^{1/2}$, then $$\P^x(Y_t=y, T_C\leq t)\leq \delta t^{-d/2}.$$
By the strong Markov property we have $$\begin{aligned}
\P^x(Y_t=y, T_C\leq t/2)&=&
\P^x(1_{\{T_C\leq t/2\}}\P^{Y_{T_C}}(Y_{t-{T_C}}=y))\\
&\leq& c_1 (t/2)^{-d/2} \P^x(T_C\leq t/2).\end{aligned}$$ In Proposition \[thm:4.2\] let us choose $A=1$ and $B=\delta/(4c_1 2^{d/2})$. If we take $\kappa> (2\gamma_1)^{-1/2}$, then Proposition \[thm:4.2\] tells us that $$\P^x (T_C\leq t/2)\leq \P^x(\tau_{B(x,\kappa t^{1/2})}\leq t/2)\leq B,$$ and then $$\label{off1}
\P^x(Y_t=y, T_C\leq t/2)\leq \frac{\delta}{2} t^{-d/2}.$$
We now consider $ \P^x(Y_t=y, t/2\leq T_C\leq t).$ If the first hitting time of $C$ occurs between time $t/2$ and time $t$, then the last hitting time of $C$ before time $t$ happens after time $t/2$. So if $S_C=\sup\{s\leq t: Y_s\in C\}$, then $$\P^x(Y_t=y, t/2\leq T_C\leq t)\leq \P^x(Y_t=y, t/2\leq S_C\leq t).$$ We claim that by time reversal, $$\label{offtime} \P^x(Y_t=y, t/2\leq S_C\leq t)
=\P^y(Y_t=x, T_C\leq t/2).$$ To see this, observe by the symmetry of the heat kernel $p$, we have that if $t_i=(t/2)+ it/(2n)$, then $$\begin{aligned}
&&\P^x(Y_{t_k}=z_k, \ldots, Y_{t_{n-1}}=z_{n-1}, Y_{t_n}=y)\\
&&=p(t_k,x,z_k) p(t/(2n),z_k,z_{k+1})\cdots p(t/(2n),z_{n-1},y)\\
&&=\P^y(Y_{t/(2n)}=z_{n-1}, \ldots, Y_{t-t_k}=z_k, Y_t=x).\end{aligned}$$ If we sum over $z_k\in C$ and $z_{k+1}, \ldots, z_{n-1}\notin C$, we have $$\begin{aligned}
\P^x( &Y_{t_k}\in C, Y_{t_{k+1}}\notin C, \ldots, Y_{t_{n-1}}\notin C, Y_t=y)\\
&= \P^y(Y_{t/(2n)}\notin C, \ldots, Y_{t-t_{k+1}}\notin C, Y_{t-t_k}\in C, Y_t=x).\end{aligned}$$ If we sum over $k$, this yields $$\P^x(t/2\leq S_n'\leq t, Y_t=y)=\P^y(0\leq T_n'\leq t/2, Y_t=x),$$ where $S_n'=\sup\{t_k: Y_{t_k}\in C\}$ and $T_n'=\inf\{t_k: Y_{t_k}\in C\}$. Letting $n\to \infty$ proves (\[offtime\]).
Arguing as in the first part of the proof, $$\P^y(Y_t=x, T_C\leq t/2)\leq \frac{\delta}{2} t^{-d/2}.$$ Therefore $$\P^x(Y_t=y, t/2\leq T_C\leq t)\leq \frac{\delta}{2} t^{-d/2},$$ and combining with (\[off1\]) proves the proposition.
[Proof of Proposition \[szhol\].]{} We have from Lemma \[1.20\] that there exists $\varepsilon$ such that $$p(t,x,y)\geq \varepsilon t^{-d/2}$$ if $|x-y|\leq 2t^{1/2}$. If we take $\delta=\varepsilon/2$ in Lemma \[offdiag\], then provided $r>\kappa t^{1/2}$, we have $$\P^x(Y_t=y, \tau_{B(x,r)}\leq t)\leq \frac{\eps}{2} t^{-d/2}.$$ Subtracting, $$\P^x(Y_t=y, \tau_{B(x,r)}>t)\geq \frac{\eps}{2} t^{-d/2}$$ if $|x-y|\leq 2t^{1/2}$, which is equivalent to what we want.
As a corollary of Proposition \[szhol\] we have
\[szholcor\] For each $0<\eps<1$, there exists $\theta=\theta(\eps)>0$ with the following property: if $D\geq 1$, $x,y\in \cs$ with $|x-y|<t^{1/2}$, $r>0$, $t\in [0, (\theta r)^2)$, and $\Gamma \subset B(y,t^{1/2})\cap \cs$ satisfies $\mu^D(\Gamma)t^{-d/2}\ge \eps$, then $$\label{ga55.2}
\P^{ x} ( V_t\in \Gamma \mbox{ and } \tau_{B(x,r)}>t)\geq c_1\eps.$$
\[krysaf\] For each $0<\delta<1$, there exists $\gamma=\gamma_\delta\in (0,1)$ such that for $t>0$, $r>0$ and $x\in \cs$, if $A\subset
Q^D_\gamma(t,x,r):=[t,t+\gamma_{\delta} r^{2}]\times (B(x,r)\cap\cs)$ satisfies $m\otimes \mu^D (A)/m\otimes \mu^D (Q^D_\gamma (t,x,r))\ge \delta$, then $$\P^{(t, x)} ( T_A(Z) <
\tau_{Q^D_\gamma (t,x,r)}(Z)) \geq c_1\delta.$$
For each $\delta>0$, take $\gamma=\theta(\delta/4)^2$. Note that there exists $s=s_r\in [t+\delta\gamma r^2/4,t+\gamma r^2)$ such that $$\label{elpar}
\mu^D (A_s)\ge \delta r^d/4\,\,\ge \frac {\delta}4 \Big(\frac{s-t}{\gamma}\Big)^{d/2}
\ge \frac {\delta}4 (s-t)^{d/2},$$ where $A_s=\{(s,z)\in [0,\infty)\times \cs: (s,z)\in A\}$. Indeed, if not then $$m\otimes \mu^D (A)\le \delta\gamma r^{2+d}/4+(\gamma -\delta\gamma/4)\cdot (\delta/4)\cdot r^{2+d}
\le \delta\gamma r^{2+d}/2,$$ which contradicts $m\otimes \mu^D (A)\ge \delta m\otimes \mu^D (Q^D_\gamma
(t,x,r))=\delta \gamma r^{2+d}$. Now, using this fact and Corollary \[szholcor\] (with $\eps=\delta/4$), we have $$\begin{aligned}
\P^{(t, x)} ( T_A(Z) < \tau_{Q^D_\gamma(t,x,r)}(Z)) &\geq &
\P^{(t, x)} ( V_{s-t}\circ\theta_t\in A_s \mbox{ and } \tau_{B(x,r)}\circ\theta_t>s-t)\\
&\geq & c_1\delta/4, \end{aligned}$$ which completes the proof.
We will also use the following Lévy system formula for $Y$ (cf. Lemma 4.7 in [@CK]).
\[4.9\] Let $f$ be a non-negative measurable function on $\R_+\times \cs\times \cs$, vanishing on the diagonal. Then for every $t\geq 0 $, $x\in \cs$ and a stopping time $T$ of $\{\cf_t \}_{t\geq 0}$, $$\E^x \left[\sum_{s\le T}f((s,V_{s-}, V_s)) \right]
=\E^x \left[\int_0^T \sum_{y\in \cs} f((s,V_s, y))
\frac{D^2C_{DV_s, Dy}}{\mu_{Y_{D^2s}}}\;ds \right]$$
Now we prove that the heat kernel $p^D (t, x, y)$ is Hölder continuous in $(t, x, y)$, uniformly over $D$. For $(t, x)\in [0, \infty)\times \cs$ and $r>0$ let $Q^D(t, x, r):=[t, \, t+\gamma r^2]
\times (B(x, R)\cap\cs)$, where $\gamma:=\gamma(1/2, 1/2)\wedge \gamma_{1/3}<1$. Here $\gamma(1/2, 1/2)$ is the constant in (\[eq:tighty\]) corresponding to $A=B=1/2$ and $\gamma_{1/3}$ is the constant in Lemma \[krysaf\] corresponding to $\delta=1/3$.
The following theorem can be proved similarly to Theorem 4.1 in [@BL2] and Theorem 4.14 in [@CK]. We will write down the proof for completeness.
\[4.15\] There are constants $c>0$ and $\beta>0$ (independent of $R,D$) such that for every $0<R$, every $D\ge 1$, and every bounded parabolic function $q$ in $Q^D(0, x_0, 4R)$, $$\label{eqn:holder1}
|q(s, x) -q(t, y)| \leq c \, \| q \|_{\infty, R} \,
R^{-\beta} \, \left( |t-s|^{1/2} + |x-y|
\right)^\beta$$ holds for $(s, x), \, (t, y)\in Q^D(0, x_0, R)$, where $\| q \|_{\infty, R}:=\sup_{(t,y)\in [0, \, \gamma (4R)^2 ] \times \cs } |q(t,y)|$. In particular, for the transition density function $p^D (t, x, y)$ of $V$, $$\label{eqn:holder2}
|p^D (s, x_1, y_1) -p^D (t, x_2, y_2)| \leq c \,
t_0^{-(d+\beta)/2}
\left( |t-s|^{1/2} + |x_1-x_2|+|y_1-y_2| \right)^\beta,$$ for any $0<t_0<1$, $t, \, s \in [t_0, \, \infty)$ and $(x_i, y_i)\in
\cs\times \cs$ with $i=1, 2$.
Recall that $Z_s=(U_s, V_s)$ is the space-time process of $V$, where $U_s=U_0+s$. In the following, we suppress the superscript $D$ from $Q^D(\cdot,\cdot,\cdot)$. Without loss of generality, assume that $0\leq q (z) \leq \| q \|_{\infty, R} =1$ for $z\in [0, \, \gamma\, (4R)^2 ] \times \cs$. By Lemma \[krysaf\], there is a constant $c_1>0$ such that if $x\in \cs$, $0<r<1$ and $A\subset Q(t, x, r/2)$ with $\frac {m\otimes\mu^D (A)}{ m\otimes\mu^D (Q(t, x, r/2))} \geq 1/3$, then $$\label{4.14}
\P^{(t, x)} ( T_A(Z) < \tau_r(Z)) \geq c_1,$$ where $\tau_r:=\tau_{Q(t, x, r)}$. By Lemma \[4.9\] with $f(s, y, z)=1_{B(x, r)}(y) \, 1_{\cs\setminus B(x, s)}(z)$ and $T=\tau_r$, there is a constant $c_2>0$ such that if $s\geq 2 r$, $$\label{eqn:4.15}
\P^{(t, x)} (V_{\tau_r} \notin B(x, s) )
= \E^{(t, x)} \left[ \int_0^{\tau_r}
\sum_{y\in\cs\setminus \overline{B(x, s)}}
\frac{D^2C_{DV_v, Dy}}{\mu_{Y_{D^2v}}}\, dv \right]
\leq \frac{c_2}{s^2} \E^{(t, x)}[\tau_r]\le \frac{c_2r^2}{s^2}.$$ The first inequality of (\[eqn:4.15\]) is due to the following computation. $$\begin{aligned}
\sup_{z\in B(x,r)\cap\cs}D^2\sum_{y\in\cs\setminus \overline{B(x, s)}}
C_{Dz,Dy}&\le &\sup_{z'\in B(Dx,Dr)}D^2\sum_{|z'-y'|\ge Ds/2}C_{z'y'}\le
D^2\sum_{i>Ds/2}\vp (i)i^{d-1}\\
&\le &\frac 4{s^2}\sum_{i}\vp (i)i^{d+1}\le \frac c{s^2},\end{aligned}$$ where (A3) is used in the last inequality. The last inequality of (\[eqn:4.15\]) is due to the fact $\E^{(t, x)}[\tau_r]\le r^2$; this is clearly true since the time interval for $Q(t,x,r)$ is $\gamma r^2$, which is less than $r^2$. ($\E^x \tau_{B(x_0,r)}\leq c_1 r^2$ is also true – see Lemma \[harnexp\] (a).) Let $$\eta= 1-\frac{c_1}4 \quad \hbox{and} \quad
\rho= \tfrac12 \wedge \left( \frac{\eta}2\right)^{1/2}
\wedge \left( \frac{c_1 \, \eta} { 8 \, c_2} \right)^{1/2}.$$ Note that for every $(t, x)\in Q(0, x_0, R)$, $q$ is parabolic in $Q(t, x, R)\subset Q(0, \, x_0, \, 2R )$. We will show that $$\label{eqn:4.16}
\sup_{Q(t, x, \rho^k R )} q - \inf_{Q(t, x, \rho^k R )} q
\leq \eta^k \qquad \hbox{for all } k.$$
For notational convenience, we write $Q_i$ for $Q(t, x, \rho^i R)$ and $\tau_i$ for $\tau_{Q(t, x, \rho^i R)}$. Define $$a_i = \inf_{Q_i} q \quad \hbox{and} \quad
b_i =\sup_{Q_i} q.$$ Clearly $b_i-a_i\leq 1 \leq \eta^i$ for all $i\leq 0$. Now suppose that $b_i-a_i\leq \eta^i$ for all $i\leq k$ and we are going to show that $b_{k+1}-a_{k+1} \leq \eta^{k+1}$. Observe that $Q_{k+1}\subset Q_k$ and so $a_k \leq q \leq b_k$ on $Q_{k+1}$. Define $$A':=\{ z\in Q_{k+1}: \, q(z) \leq (a_k+b_k)/2 \}.$$ We may suppose $\frac {m\otimes\mu^D (A')}{ m\otimes\mu^D (Q_{k+1} )} \geq 1/2$, for if not we use $1-q$ instead of $q$. Let $A$ be a compact subset of $A'$ such that $\frac {m\otimes\mu^D (A)}{ m\otimes\mu^D (Q_{k+1} )} \geq 1/3$. For any given $\varepsilon >0$, pick $z_1, z_2 \in Q_{k+1}$ so that $q(z_1)\geq b_{k+1}-\varepsilon$ and $q(z_2) \leq a_{k+1} +
\varepsilon$. Then by (\[4.14\])-(\[eqn:4.16\]), $$\begin{aligned}
b_{k+1} - a_{k+1}-2 \varepsilon
&\leq& q(z_1)- q(z_2)\\
&=& \E^{z_1} \left[ q(Z_{T_A \wedge \tau_{k+1}})-q(z_2)
\right] \\
&=& \E^{z_1} \left[ q(Z_{T_A}) -q(z_2); \,
T_A< \tau_{k+1} \right]\\
&&\qquad + \E^{z_1} \big[ q(Z_{\tau_{k+1}})-q(z_2); \,
T_A> \tau_{k+1}, \\
&&\qquad\qquad Z_{\tau_{k+1}}\in Q_k \big]\\
&&\qquad + \sum_{i=1}^\infty \E^{z_1} \big[ q(Z_{\tau_{k+1}})-q(z_2); \,
T_A> \tau_{k+1}, \\
&& \qquad\qquad Z_{\tau_{k+1} }\in Q_{k-i}
\setminus Q_{k+1-i} \big] \\
&\leq& \left( \frac{a_k+b_k}2 -a_k \right) \P^{z_1} ( T_A< \tau_{k+1})\\
&&\qquad + (b_k-a_k) \P^{z_1} ( T_A> \tau_{k+1} ) \\
&&\qquad + \sum_{i=1}^\infty (b_{k-i}-a_{k-i}) \P^{z_1}
( Z_{\tau_{k+1}} \notin Q_{k+1-i}) \\
&\leq& (b_k-a_k) \left( 1-\frac{\P^{z_1} ( T_A< \tau_{k+1})}2 \right)
+ \sum_{i=1}^\infty c_2 \, \eta^k (\rho^2
/ \eta )^i \\
&\leq& ( 1-\frac{c_1}2)\, \eta^k + 2c_2 \eta^{k-1} \rho^2 \\
&\leq& ( 1-\frac{c_1}2) \eta^k + \frac{c_1}4 \eta^k \\
&=& \eta^{k+1}.\end{aligned}$$ Since $\varepsilon$ is arbitrary, we have $b_{k+1}-a_{k+1}\leq \eta^{k+1}$ and this proves (\[eqn:4.16\]).
For $z=(s, x)$ and $w=(t, y)$ in $Q(0, x_0, R)$ with $s\leq t$, let $k$ be the largest integer such that $|z-w|:= (\gamma^{-1} |t-s|)^{1/2}+|x-y| \leq \rho^k R$. Then $\log (|z-w|/R) \geq (k+1) \log \rho $, $w\in Q(s, x, \rho^k R)$ and $$\begin{aligned}
|q(z)-q(w)| \leq \eta^k = e^{k \log \eta} \leq
c_3 \left( \frac{|z-w|}{R} \right)^{\log \eta / \log \rho }.\end{aligned}$$ This proves (\[eqn:holder1\]) with $\beta =
\log \eta / \log \rho$.
By (\[eq:nash2\]) and Lemma \[4.6\], for every $0<t_0< 1$, $T_0\ge 2$ and $y \in \cs$, $q(t, x):=p^D (T_0-t , x, y)$ is a parabolic function on $[0, T_0-\frac{t_0}2 ] \times \cs$ bounded above by $c_4 \, t_0^{-d/2}$.
For each fixed $t_0\in (0, \, 1)$ and $T_0\ge 2$, take $R$ such that $\gamma R^2 = t_0/2$. Let $s, t\in [t_0, T_0]$ with $s>t$ and $x_1, x_2 \in \cs$. Assume first that $$\label{eqn:4.47}
|s-t|^{1/2}+|x_1-x_2| <
\gamma^{1/2}\, R = (t_0/ 2)^{1/2}$$ and so $(T_0-t, x_2)\in Q(T_0-s, x_1, R)\subset
[0, T_0-\frac{t_0}2) \times \cs$. Applying (\[eqn:holder1\]) to the parabolic function $q(t, x)$ with $(T_0-s, x_1)$, $(T_0-t, x_2)$ and $Q(T_0-s, x_1, R)$ in place of $(s, x)$, $(t, y)$ and $Q (0, x_0, R)$ there respectively, we have $$\label{eqn:holder3}
|p^D (s , x_1, y) - p^D (t , x_2, y) | \leq c \, t_0^{-(d+\beta) /2}
(|t-s|^{1/2} + |x_1-x_2| )^\beta.$$ By (\[eq:nash3\]), the inequality (\[eqn:holder3\]) is true when (\[eqn:4.47\]) does not hold. So (\[eqn:holder3\]) holds for every $t, s \in [t_0, T_0]$ and $x_1, x_2 \in \cs$ for all $T_0\ge 2$. Inequality (\[eqn:holder2\]) now follows from (\[eqn:holder3\]) by the symmetry of $p(t, x, y)$ in $x$ and $y$.
Harnack inequality
==================
A function $h$ defined on $\Z^d$ is harmonic on a subset $A$ of $\Z^d$ with respect to the Markov chain $X$ if $$\sum_z h(z) \P^x(X_1=z) =h(x), \qquad x\in A.$$ Because the Markov chain may not have bounded range, $h$ must be defined on all of $\Z^d$. In order to avoid $h$ possibly being infinite in $A$, we will assume that $h$ is bounded on $\Z^d$, but in what follows, the constants do not depend at all on the $L^\infty$ bound on $h$. We say $h$ is harmonic with respect to $Y$ if $h(Y_{t\land \tau_A})$ is a $\P^x$-martingale for each $x\in \Z^d$, where $\tau_A=\inf\{t: Y_t\notin A\}$. It is not hard to see that a function is harmonic for $X$ if and only if it is harmonic for $Y$, since the hitting probabilities of $X$ and $Y$ are the same. Also, because the state space is discrete, it is routine to see that a function is harmonic in a domain $A$ if and only if $\ce(h,f)=0$ for all bounded $f$ supported in $A$; we will not use this latter fact.
In this section we first give an example of a symmetric random walk, i.e., where $\{X_{n+1}-X_n\}$ are symmetric i.i.d. random variables, for which a uniform Harnack inequality fails. Note that the Harnack inequality does hold for each ball of radius $n$, but not with a constant independent of $n$. Let $e^j$ be the unit vector in the $x_j$ direction, $j=1, \ldots, d$.
Let $b_n=n^{n^n}$ (or any other quickly growing sequence), let $a_n$ be a sequence of positive numbers tending to 0, subject only to $\sum a_n\leq 1/32$ and $\sum_n a_n b_n^2<\infty$. Let $\eps=\frac12 \sum a_n$. Let ${\xi}_i$ be an i.i.d. sequence of random vectors on $\Z^d$ with $\P^0({\xi}_1=\pm e^j)=(1-\eps)/(2d)$. Let $\P^0({\xi}_1=\pm b_n e^1)=a_n$. Let $X_n=\sum_{i=1}^n {\xi}_i$.
Now let $\delta\in(0,1)$, $r_n=(1-\delta) b_n$, $z_n=(b_n,0)$, $B_n=B(0,r_n)$, $\tau_n=\min\{k: X_k\notin B_n\}$, and $T_0=\min\{k:
X_k=0\}$. Define $$h_n(x)=\P^x(X_{\tau_n}=z_n).$$ Each $h_n$ is a harmonic function in $B_n$. If a uniform Harnack inequality were to hold, there would exist $C$ not depending on $n$ such that $$h_n(0)/h_n(y)\leq C, \qquad y\in B(0, r_n/2).$$
Since $\delta b_n \gg b_{n-1}$ for $n$ large, the only way $X_{\tau_n}$ can equal $z_n$ is if the random walk jumps from 0 to $z_n$. So for $y_n\in B_n$, $y_n\ne 0$, $$h_n(y_n)=\P^{y_n}(T_0<\tau_n)h(0).$$
But we claim that if $y_n\sim r_n/4$, then $\P^{y_n}(T_0<\tau_n)$ will tend to 0 when $n\to \infty$, and then $h_n(0)/h_n(y_n)\to \infty$. So no uniform Harnack inequality exists.
The claim is true is all dimensions greater than or equal to 2, but is easier to prove when $d\geq 3$, so we concentrate on this case. We have $$\begin{aligned}
\P^{y_n}(T_0<\tau_n) & \leq &
\P^{y_n}(T_0<\infty)=
\P^{y_n}(T_0<r_n^{1/4}) +\P^{y_n}(T_0\geq r_n^{1/4})\\
& \leq & \P^{y_n}(\max_{i\leq r_n^{1/4}} |X_i-X_0|\geq |y_n|)
+\sum_{i=[r_n^{1/4}]}^\infty \P^{y_n}(X_i=0).\end{aligned}$$ The first term on the last line goes to 0 by Doob’s inequality (applied to each $(X_i, e^j)$, $j=1, \ldots, d$). By Spitzer [@Spi], p. 75, the sum above is bounded by $$c\sum_{i=[r_n^{1/4}]}^\infty \frac{1}{i^{d/2}}\leq c' (r_n^{1/4})^{1-(d/2)},$$ which goes to 0 as $n\to \infty$.
Note that by taking $a_n$ tending to 0 fast enough, ${\xi}_1$ can be made to be sub-Gaussian, or have even better tails.
Lawler [@Law] proved that the Harnack inequality holds for a class of symmetric random walks with bounded range and also for a class of Markov chains with bounded range which are in general not reversible. The content of the next proposition is that this continues to be true for symmetric Markov chains with bounded range.
\[harnbdd\] Suppose the Markov chain has range bounded by $K$. Let $x_0\in \Z^d$. There exist constants $c_1$ and $\theta$ not depending on $x_0$ such that if $r\geq 4K(\theta^{-1}+1)$ and $h$ is nonnegative and bounded on $\Z^d$ and harmonic on $B(x_0,r)$, then $$h(x)\leq c_1 h(y), \qquad x,y\in B(x_0,\theta r).$$
First let us suppose that $d\geq 3$; we will remove this restriction at the end of the proof. Let $$G_B(x,y)=\E^x\int_0^{\tau_B} 1_{\{y\}}(Y_s)\, ds,$$ where $\tau_B=\inf\{t: Y_t\notin B(x_0,r)\}$. $G_B$ is the Green function for the process $Y$ killed on exiting $B(x_0,r)$. Since we are assuming $d\geq 3$ and $p(t,x,y)$ is always bounded by some constant, then by (\[eq:nash2\]) we see that $G_B$ is bounded, say by $c_2$.
It follows by Proposition \[szhol\] that there exists $\kappa$ such that $\P^x(Y_t=y, \tau_B>t)$ is bounded below by $c_3t^{-d/2}$ provided $|x-x_0|, |y-x_0|\leq t^{1/2}$ and $r>\kappa t^{1/2}$. Set $\theta=1/(4\kappa)$. So integrating over $t\in [4\theta^2r^2,8\theta^2r^2]$, we see $G_B(x,y)\geq c_5$ for $x,y\in B(x_0, 2\theta r)$.
Define $\overline h(x)=\E^x[h(Y_{T}); T<\tau_B],$ where $T=\inf\{t: Y_t\in B(x_0,2\theta r)\}$. It is routine that $\overline h$ is equal to $h$ on $B'=B(x_0,\theta r)$, is 0 outside of $B(x_0, r)$, and is excessive with respect to the process $Y_t$ killed on exiting $B(x_0,r)$. The fact that $X$ has bounded range and $r>4K(\theta^{-1}+1)$ is what allows us to assert that $\overline h$ is equal to $h$ in $B(x_0, \theta r)$. See [@FOT], p. 319, for the definition of excessive. By [@FOT], Theorem 2.2.1, there exists a measure $\pi$ supported on $\overline {B(x_0, r)}$ such that $$\widetilde \ce (\overline h,v)=\int v(x) \, \pi(dx)$$ for all continuous $v$ with support contained in $B(x_0,r)$, where $\widetilde \ce$ is the Dirichlet form for $Y_t$ killed on exiting $B(x_0,r)$. An easy approximation argumen shows that we also have $$\widetilde \ce(G_B \pi, v)=\int v(x) \, \pi(dx)$$ for such $v$, and we conclude $\overline h=G_B \pi$. Since $\overline h$ is harmonic in $B(x_0, \theta r)$ and in $B(x_0,r)\setminus B(x_0, 2\theta r)$, it is not hard to see that $\pi$ in fact is supported in $\overline{B(x_0,2\theta r)}\setminus B(x_0,\theta r)$. So for $x,y\in B(x_0,\theta r)$, the upper and lower bounds on $G_B(x,y)$ imply $$\begin{aligned}
h(x)&=\sum_z G_B(x,z) \pi(\{z\}) \leq c_2 \pi(\Z^d)\\
&=\frac{c_2}{c_5} c_5\pi(\Z^d) \leq \frac{c_2}{c_5} \sum_z G_B(y,z) \pi(\{z\})\\
&=\frac{c_2}{c_5} h(y).\end{aligned}$$ This proves the theorem when $d\geq 3$.
When $d=2$, define a Markov chain $X'$ on $\Z^3$ by setting $C'_{(x_1,x_2,x_3),(y_1,y_2,y_3)}$ to be equal to $C_{(x_1,x_2), (y_1,y_2)}$ if $x_3=y_3$; equal to 1 if $x_1=y_1, x_2=y_2$, and $x_3= y_3 \pm 1$; and equal to 0 otherwise. Suppose $h$ is harmonic with respect to $X$ on $A\subset\Z^2$. If we define $h'(x_1,x_2,x_3)=
h(x_1,x_2)$, it is routine to check that $h'$ is harmonic with respect to $X'$ on $A\times \Z\subset \Z^3$. The Harnack inequality we just proved above applies to $h'$, and a Harnack inequality for $h$ then follows immediately.
As the example at the beginning of this section shows, a uniform Harnack inequality need not hold when the range is unbounded, so an additional assumption is needed to handle this case. The assumption is modeled after [@BK] and the proof is similar to the one in [@BL2]. We assume
(A4)There exists a constant $c_1$ such that $C_{xy}\leq c_1 C_{xy'}$ whenever $|y-y'|\leq |x-y|/3$.
\[harnunbdd\] Suppose (A1)–(A3) hold and in addition (A4) holds. Suppose $x_0\in \Z^d$ and $R>M_0$, where $M_0$ is defined in (A2). There exists a constant $c_1$ such that if $h$ is nonnegative and bounded on $\Z^d$ and harmonic on $B(x_0,2R)$, then $$\label{harin}
h(x)\leq c_1 h(y), \qquad x,y\in B(x_0,R).$$
Before proving Theorem \[harnunbdd\] we prove a lemma. Note that (A4) is not needed for this lemma.
\[harnexp\] (a) $\E^x \tau_{B(x_0,r)}\leq c_1 r^2$.
\(b) There exist $\theta\in (0,1)$ and $c_1, c_2>0$ such that if $r> M_0/\theta$, then $\P^x (\tau_{B(x_0,r)}\geq r^2)\geq c_2$
and $\E^x \tau_{B(x_0,r)}\geq c_3 r^2$ if $x\in B(x_0,\theta r)$.
If $p(t,x,y)$ denotes the transition densities for $Y_t$, we know $$p(t,x,y)\leq c_4 t^{-d/2}.$$ So if we take $t=c_5r^2$ for large enough $c_5$, then $$\P^x(Y_t\in B(x_0,r))=\sum_{z\in B(x_0,r)} p(t,x,z)\leq c_6t^{-d/2} |B(x_0,r)| \leq \tfrac12.$$ This implies $$\P^x(\tau_{B(x_0,r)}> t)\leq \tfrac12.$$ By the Markov property, for $m$ a positive integer $$\P^x(\tau_{B(x_0,r)}> (m+1)t)\leq \E^x[\P^{Y_{mt}}(\tau_{B(x_0,r)}> t); \tau_{B(x_0,r)}>mt]\leq \tfrac12 \P^x(\tau_{B(x_0,r)}>mt).$$ By induction, $$\P^x(\tau_{B(x_0,r)}>mt)\leq 2^{-m},$$ and the first part of (a) follows.
We also know by Proposition \[szhol\] that there exists $\kappa>1$ such that $$\P^x(Y_t=y, \tau_{B(x_0,r)}>t)\geq c_6 t^{-d/2}$$ if $|x-x_0|, |y-x_0|\leq t^{1/2}$ and $r>\kappa t^{1/2}$. Therefore taking $t=r^2/\kappa^2$, $$\P^x(\tau_{B(x_0,r)}>t)\geq \P^x(Y_t\in B(x_0,t^{1/2}), \tau_{B(x_0,r)}>t)\geq c_6 t^{-d/2} |B(x_0,t^{1/2})|\geq c_7$$ if $x\in B(x_0, r/\kappa)$. Let $\theta=1/\kappa$. So $\E^x \tau_{B(x_0,r)}\geq t\P^x(\tau_{B(x_0,r)}>t)\geq c_7 r^2$, which proves (b).
[Proof of Theorem \[harnunbdd\]:]{} Let $\kappa$ and $\theta $ be as in Lemma \[harnexp\]. That a Harnack inequality inequality holds for each finite $R$ is easy, provided $R\le 16M_0/\theta$, so it suffices to assume $R>16 M_0/\theta$.
First of all, if $z_1\in \Z^d$ and $w\notin B(z_1,2r)$, by the Lévy system formula, $$\E^x \sum_{s\leq \tau_{B(z_1,r)}\land t} 1_{(Y_{s-}\in B(z_1,r), Y_s=w)}=\E^x \int_0^{\tau_{B(z_1,r)}\land t} C_{Y_s,w}\, ds.$$ Letting $t\to \infty$, we have $$\P^x(Y_{\tau_{B(z_1,r)}}=w)=\E^x\int_0^{\tau_{B(z_1,r)}} C_{Y_s,w}\, ds.$$ By (A4) the right hand side is bounded above by the quantity $c_{2} C_{z_1w}\E^x \tau_{B(z_1,r)}$ and below by the quantity $c_{3} C_{z_1w}\E^x \tau_{B(z_1,r)}$. By Lemma \[harnexp\], if $x,y\in B(z_1,\theta r)$, then $\E^x \tau_{B(z_1,r)}
\leq c_{4} \E^y \tau_{B(z_1,r)}$. We conclude $$\P^x(Y_{\tau_{B(z_1,r)}}=w)\leq c_{5} \P^y(Y_{\tau_{B(z_1,r)}}=w).$$ Taking linear combinations, if $H$ is a bounded function supported in $B(z_1,2r)^c$, then $$\label{3.2}
\E^x H(Y_{\tau_{B(z_1, r)}})\leq c_{5} \E^y H(Y_{\tau_{B(z_1,r)}}), \qquad
x,y\in B(z_1, \theta r).$$
Choose $r_0=8M_0/\theta$. If $r\geq r_0$, then setting $t=r^2/\kappa^2$, $$\P^x(Y_t=y, \tau_{B(z_1,r)}>t)\geq c_{6} t^{-d/2}, \qquad x,y\in B(z_1, \theta r).$$ Summing over $A\subset B(z_1, \theta r)$, we see that $$\label{hhit}
\P^x(T_A<\tau_{B(z_1,r)})\geq\P^x(Y_t\in A, \tau_{B(z_1,r)}>t)\geq c_{6} |A| t^{-d/2}
=c_{6}|A|r^{-d}, \qquad x\in B(z_1, \theta r).$$ In particular, note that if $C\subset B(z_1,r)$ and $|C|/|B(z_1,r)|\geq 1/3$, then $$\label{addlabel1}
\P^x(T_C<\tau_{B(z_1,r)})\geq c_{7}, \qquad x\in B(z_1, \theta r).$$
Next suppose $x,y\in B(z_1, \theta r_0)$. In view of (A2) $$\P^x(T_{\{y\}} <\tau_{B(z_1, r_0)})\geq c_{8}.$$ By optional stopping, $$\begin{aligned}
h(x)& \geq \E^x [h(Y_{T_{\{y\}}}); T_{\{y\}}<\tau_{B(z_1,r_0)}]\\
&=h(y)\P^x(T_{\{y\}}<\tau_{B(z_1,r_0)})\\
&\geq c_{8} h(y). \end{aligned}$$
By looking at a constant multiple of $h$, we may assume $\inf_{B(x_0,R)} h=1$. Choose $z_0\in B(x_0,R)$ such that $h(z_0)= 1$. We want to show that $h$ is bounded above in $B(x_0,R)$ by a constant not depending on $h$.
Let $$\label{3.5}
\eta=\frac{c_{7}}{3}, \qquad \zeta=\frac13\land (c_{5}^{-1}\eta)\land c_{8}.$$
Now suppose there exists $x\in B(x_0,R)$ with $h(x)=K$ for some $K$ large. Let $r$ be chosen so that $$\label{3.6}
2R^d/(c_{6}\zeta K)\leq
|B(x_0,\theta r)|\leq
4R^d/(c_{6}\zeta K).$$ Note this implies $$\label{3.7}
r\leq c_9K^{-1/d}R.$$ Without loss of generality we may assume $K$ is large enough that $r\leq \theta R/4$. Let $$\label{3.75}
A=\{w\in B(x,\theta r): h(w)\geq \zeta K\}.$$ By (\[hhit\]) and optional stopping, $$\begin{aligned}
1\geq h(z_0)&\geq \E^{z_0}[h(Y_{T_A\land \tau_{B(x_0,2R)}});
T_A<\tau_{B(x_0,2R)}]\\
&\geq \zeta K \P^{z_0}(T_A<\tau_{B(x_0,2R)})\\
&\geq c_{6}\zeta K |A|R^{-d},\\\end{aligned}$$ hence $$\label{3.77}
\frac{|A|}{|B(x,\theta r)|}\leq
\frac{R^d}{c_{6}\zeta K |B(x,\theta r)|}\leq \frac12.$$ Let $C$ be a set contained in $B(x,\theta r)\setminus A$ such that $$\label{3.8}
\frac{|C|}{|B(x,\theta r)|}\geq \frac13.$$
Let $H=h1_{B(x,2r)^c}$. We claim $$\E^x[h(Y_{\tau_{B(x,r)}}); Y_{\tau_{B(x,r)}}\notin B(x,2r)]\leq \eta K.$$ If not $$\E^x H(Y_{\tau_{B(x,r)}})> \eta K,$$ and by (\[3.2\]), for all $y\in B(x,\theta r)$, $$\begin{aligned}
h(y)&\geq \E^y h(Y_{\tau_{B(x,r)}})\geq \E^y[h(Y_{\tau_{B(x,r)}}); Y_{\tau_{B(x,r)}}\notin B(x,2r)]\\
&\geq c_{5}^{-1} \E^x H(Y_{\tau_{B(x,r)}})\geq c_{5}^{-1}\eta K\\
&\geq \zeta K, \\ \end{aligned}$$ contradicting (\[3.8\]) and the definition of $A$.
Let $N=\sup_{B(x,2r)} h(z)$. We then have $$\begin{aligned}
K&=h(x)=\E^x[h(Y_{T_{C}}); T_C<\tau_{B(x,r)}]+
\E^x[h(Y_{\tau_{B(x,r)}}); \tau_{B(x,r)}<T_{C}, Y_{\tau_{B(x,r)}}\in B(x,2r)]\\
&\qquad \qquad +\E^x[h(Y_{\tau_{B(x,r)}}); \tau_{B(x,r)}<T_{C}, Y_{\tau_{B(x,r)}}\notin B(x,2r)]\\
&\leq \zeta K\P^x(T_{C}<\tau_{B(x,r)})+N\P^x(\tau_{B(x,r)}<T_{C})+\eta K\\
&=\zeta K\P^x(T_C<\tau_{B(x,r)})+N(1-\P^x(T_C<\tau_{B(x,r)}))+\eta K,\\\end{aligned}$$ or $$\frac{N}{K}\geq \frac{1-\eta-\zeta\P^x(T_C<\tau_{B(x,r)})}{1-\P^x(T_C<\tau_{B(x,r)})}.$$
Using (\[addlabel1\]) there exists $\beta>0$ such that $N\geq K(1+\beta)$. Therefore there exists $x'\in B(x,2r)$ with $h(x')\geq K(1+\beta)$.
Now suppose there exists $x_1\in B(x_0,R)$ with $h(x_1)=K_1$. Define $r_1$ and $A_1$ in terms of $K_1$ analogously to (\[3.6\]) and (\[3.75\]). Using the above argument (with $x_1$ replacing $x$ and $x_2$ replacing $x'$), there exists $x_2\in B(x_1, 2r_1)$ with $h(x_2)=K_2\geq (1+\beta)K_1$. We continue and obtain $r_2$ and $A_2$ and then $x_3,K_3,r_3, A_3$, etc. Note $x_{i+1}\in B(x_i,2r_i)$ and $K_i\geq (1+\beta)^{i-1}K_1$. In view of (\[3.7\]), $\sum_i |x_{i+1}-x_i|\leq c_{10} K_1^{-1/d}R$. If $K_1$ is big enough, we have a sequence $x_1,x_2, \ldots$ contained in $B(x_0,3R/2)$ Since $K_i\geq (1+\beta)^{i-1} K_1$ and $r_i\leq c_{11} K_i^{-1/d}R$, there will be a first integer $i$ for which $r_i<2r_0$. But for all $y\in B(x_i, \theta r_i)$ we have $h(y)\geq c_{8} h(x_i)$, so then $A_i= B(x_i, \theta r_i)$, a contradiction to (\[3.77\]).
\[jayrosen\] Let $\xi_i$ be an i.i.d. sequence of symmetric random vectors taking values in $\Z^d$ with finite second moments. Let $X_n=\sum_{i=1}^n
\xi_i$ and suppose $X_n$ is aperiodic. Suppose there exists $c_1$ such that $$\P(\xi_1=y)\leq c_1 \P(\xi_1=y')$$ whenever $|y-y'|\leq |y|/3$. Then there exists $c_2$ and $R_0$ such that for all $R$ larger than $R_0$ and any $w\notin B(x_0, R)$, $$\P^x(X_{\tau_{B(x_0,R)}}=w)\leq c_2 \P^y(X_{\tau_{B(x_0,R)}}=w),
\qquad x,y\in B(x_0,R/2).$$
We let $C_{xy}=\P(\xi_1=y-x)$. Since the $\xi_i$ are symmetric, then the $X_n$ form a symmetric Markov chain, and it is easy to see that (A1)–(A4) are satisfied. We then apply Theorem \[harnunbdd\] to $h(x)=\P^x(Y_{\tau_{B(x_0,R)}}=w)$.
Central limit theorem
=====================
Suppose we have a sequence $C_{xy}^n$ of conductances satisfying (A1), (A2), and (A3) with constants and $\vp$ independent of $n$. Let $Y^{(n)}_t$ be the corresponding continuous time Markov chains on $\Z^d$ and set $$Z^{(n)}_t=Y^{(n)}_{nt}/\sqrt n.$$
As noted previously, the Dirichlet form corresponding to the process $Z^{(n)}$ is $$\label{cltform}
\sE_n(f,f)=n^{2-d} \sum_{x,y\in n^{-1}\Z^d} (f(y)-f(x))^2 C^n_{nx,ny}.$$ We will also need to discuss the form $$\label{cltform2}
\sE^R_n(f,f)=n^{2-d} \sum_{x,y\in n^{-1}\Z^d} (f(y)-f(x))^2 C^{n,R}_{nx,ny},$$ where $C^{n,R}_{k,l}, k,l\in \Z^d$ is equal to $C^n_{k,l}$ if $|k-l|\leq nR$ and 0 otherwise.
Since the state space of $Z^{(n)}$ is $n^{-1}\Z^d$ while the limit process will have $\R^d$ as its state space, we need to exercise some care with the domains of the functions we deal with. First, if $g$ is defined on $\R^d$, we define $R_n(g)$ to be the restriction of $g$ to $n^{-1}\Z^d$: $$R_n(g)(x)= g(x), \qquad x\in n^{-1}\Z^d.$$ If $g$ is defined on $n^{-1}\Z^d$, we next define an extension of $g$ to $\R^d$. The one we use is defined as follows. For $k\in \Z^d$, let $$Q_n(k)=\prod_{j=1}^d [n^{-1}k_j, n^{-1}(k_j+1)].$$ When $d=1$, we define the extension, $E_n(g)$, to be linear in each $Q_n(k)$ and to agree with $g$ on the endpoints of each interval $Q_n(k)$. For $d>1$ we define $E_n(g)$ inductively. We use the definition in the $(d-1)$-dimensional case to define $E_n(g)$ on each face of each $Q_n(k)$. We define $E_n(g)$ in the interior of a $Q_n(k)$ so that if $L$ is any line segment contained in the $Q_n(k)$ that is parallel to one of the coordinate axes, then $E_n(g)$ is linear on $L$. For example, when $d=2$, $n=1$, and $k=(0,0)$, then $$\begin{aligned}
E_n(g)(s,t)&=g(0,0)(1-s)(1-t)+g(0,1)(1-s)t+g(1,0)s(1-t)\\
&\qquad +g(1,1)st, \qquad \qquad 0\leq s,t\leq1. \end{aligned}$$
Recall $e^j$ is the unit vector in the $x_j$ direction and let $(x,y)$ denote the inner product in $\R^d$. If $k=(k_1, \ldots, k_d)\in \Z^d$, let $\sP(k)$ be the union of the line segment from 0 to $(k_1, 0, \ldots, 0)$, the line segment from $(k_1, 0, \ldots, 0)$ to $(k_1, k_2, 0, \ldots, 0)$, ..., and the line segment from $(k_1, \ldots, k_{d-1},0)$ to $k$. For $z\in \Z^d$ and $1\leq i\leq d$, let $$L^i_z=\{(y,k)\in (n^{-1}\Z^d)^2: y+ n^{-1}\sP(nk) \mbox{ contains the line segment from $z$ to }
z+n^{-1}e^i\}.$$ We note that $(x,k)\in L^i_z$ for $z\in n^{-1}\Z^d$ if and only if $(x+k)_l=z_l$ for $l=1,...,i-1$, $x_l=z_l$ for $l=i+1,...,d$ and $z_i\in [x_i\wedge
(x+k)_i, x_i\vee (x+k)_i)$. So, for each $k$, the number of $x$ that satisfies $(x,k)\in L^i_z$ is at most $n|k_i|$.
Recall $\sgn r$ is equal to 1 if $r>0$, equal to 0 if $r=0$, and equal to $-1$ if $r<0$. We define a map $a^n$ from $\R^d$ into $\cal M$, the collection of $d\times d$ matrices as follows: Fix $R$. If $x\in n^{-1}\Z^d$, let the $(i,j)$-th element of $a^n$ be given by $$\label{defan}
\big(a^n(x)\big)_{ij}
=\sum_{(y,k)\in L^i_x} C_{ny, n(y+k)}^{n,R} nk_{j}\,\sgn k_i.$$ For general $x=(x_i)_{i=1}^d\in\R^d$, we define $a^n(x):=a^n([x]_n)$, where we set $[x]_n=(n^{-1}[nx_i])_{i=1}^d$. $a^n(x)$ is not symmetric in general, but under (A5), we see that $(a^n(x))_{ij}$ is bounded for all $i,j$, (which can be proved similarly to (\[aolacop\]) below) and when $n$ is large, we can use Cauchy-Schwarz, etc., as in the symmetric case. Note that if $C_{xy}^n=0$ for $|x-y|>1$ (i.e., the nearest neighbor case), then the expression in (\[defan\]) is equal to $2C_{nx,nx+e^i}^n$ if $i=j$ and equal to 0 if $i\ne j$. (In particular, $a^n(x)$ is symmetric in this case.)
We make the following assumption.
(A5) There exist $R>0$ and a Borel measurable $a:\R^d\to\cal M$ such that $a$ is symmetric and uniformly elliptic, the map $x\to a(x)$ is continuous, and $a^n$ converges to $a$ uniformly on compacts sets.
We will see from the proofs below that if (A5) holds for one $R$, then it holds for every $R>1$ and the limit $a$ is independent of $R$.
Since $a$ is uniformly elliptic, if we define $$\ce_a(f,f)=\int_{\R^d}(\nabla f(x), a(x)\nabla f(x))dx,$$ then $(\ce_a, H^1(\R^d))$ is a regular Dirichlet form on $L^2(\R^d, dx)$ where $H^1(\R^d)$ is the Sobolev space of square integrable functions with one square integrable derivative. Further, it is well-known that the corresponding heat kernel $p^a(t,x,y)$ satisfies the following estimate, $$\label{aroest}
c_1t^{-d/2}\exp\Big(-c_2\frac{|x-y|^2}t\Big)\le p^a(t,x,y)\le
c_3t^{-d/2}\exp\Big(-c_4\frac{|x-y|^2}t\Big),$$ for all $t>0$ and all $x,y\in \R^d$. As a consequence, the corresponding diffusion (which we denote by $\{Z_t\}$) can be defined without ambiguity from any starting point.
In this section we prove the following central limit theorem. Let $C([0,t_0]; \R^d)$ be the collection of continuous paths from $[0,t_0]$ to $\R^d$.
\[clt\] Suppose (A1)-(A3) and (A5) hold.
\(a) Then for each $x$ and each $t_0$ the $\P^{[x]_n}$-law of $\{Z^{(n)}_t; 0\leq t\leq t_0\}$ converges weakly with respect to the topology of the space $D([0,t_0], \R^d)$. The limit probability gives full measure to $C([0,t_0], \R^d)$.
\(b) If $Z_t$ is the canonical process on $C([0,\infty), \R^d)$ and $\P^x$ is the weak limit of the $\P^{[x]_n}$-laws of $Z^{(n)}$, then the process $\{Z_t, \P^x\}$ has continuous paths and is the symmetric process corresponding to the Dirichlet form $\ce_a$.
Before giving the proof, we discuss three examples. First, suppose each $X^{(n)}$ is the sum of i.i.d. random vectors. Then the $C^{n}_{xy}$ will depend only on $y-x$, and so the $a^n(x)$ will be constant in the variable $x$. Therefore, if convergence holds, the limit $a(x)$ will be constant in $x$. This means that the limit is a linear transformation of $d$-dimensional Brownian motion, as one would expect.
For another example, suppose the $X^{(n)}$ are nearest neighbor Markov chains, i.e., $C^{n}_{xy}=0$ if $|x-y|\ne 1$. Then in this case the result of [@SZ] is included in our Corollary \[d1corsz\] and \[cltcor\].
Third, suppose $C^n_{xy}=C_{xy}$ does not depend on $n$. Unless $C_{xy}$ is a function only of $y-x$, then (2.6) of [@SZ] (which is (\[szstrassmp\]) below) will not be satisfied, and this situation is covered by Theorem \[clt\] but not by the results of [@SZ]. To be fair, the goal of [@SZ] was not to obtain a general central limit theorem, but instead to come up with a way of approximating diffusions by Markov chains. Condition (A5) is restrictive. For this $C^n_{xy}=C_{xy}$ case, if we further assume that $C_{xy}=0$ for $|x-y|>1$, then $a(x)$ is always a constant matrix. Indeed, in this case the expression in (\[defan\]) is equal to $2C_{nx,nx+e^i}\delta_{ij}$, which converges to $(a(x))_{ij}$ uniformly on compacts as $n\to\infty$ by (A5). So, for any $m\in \N$, the limit of $a^n(x/m)$ is equal to $a(x)$, i.e., $a(x/m)=a(x)$. Since $a$ is continuous, we conclude $a(x)=a(0)$ for all $x\in \R^d$.
Before we prove Theorem \[clt\], we prove a proposition showing tightness of the laws of $Z^{(n)}$.
\[clttight\] Suppose $\{n_j\}$ is a subsequence. Then there exists a further subsequence $\{n_{j_k}\}$ such that
\(a) For each $f$ that is $C^\infty$ on $\R^d$ with compact support, $E_{n_{j_k}}(P_t^{n_{j_k}} R_{n_{j_k}}(f))$ converges uniformly on compact subsets; if we denote the limit by $P_t f$, then the operator $P_t$ is linear and extends to all continuous functions on $\R^d$ with compact support and is the semigroup of a symmetric strong Markov process on $\R^d$ with continuous paths.
\(b) For each $x$ and each $t_0$ the $\P^{[x]_{n_{j_k}}}$ law of $\{Z_t^{(n_{j_k})}; 0\leq t \leq t_0\}$ converges weakly to a probability $\P^x$ giving full measure to $C([0,t_0]; \R^d)$.
Let $t_0>0$ and $\eta>0$. Let $\tau_n$ be stopping times bounded by $t_0$ and let $\delta_n\to 0$. Then by Proposition \[thm:4.2\] and the strong Markov property, $$\limsup_{n \to \infty }\P( |{ Z}^{(n)}_{\tau_n+\delta_n}-{ Z}^{(n)}_{\tau_n}|>\eta)=0.$$ This, Proposition \[thm:4.2\], and [@A] imply that the laws of the $\{{Z}^{(n)}\}$ are tight in $D[0,t_0]$ for each $t_0$.
Fix $t_0$ and $\eta>0$. ${Z}^{(n)}$ will have a jump of size larger than $\eta$ before time $t_0$ only if $|Y^{(n)}_{t}-Y^{(n)}_{t-}|\geq \eta\sqrt n$ for some $t\leq nt_0$. By the Lévy system formula, the probability of this is bounded by $$\begin{aligned}
\E^x\sum_{s\leq nt_0} 1_{(|Y^{(n)}_s-Y^{(n)}_{s-}|\geq \eta\sqrt n)}&=\E^x \int_0^{nt_0} \sum_{|x-Y^{(n)}_s|\geq
\eta\sqrt n} C^n_{Y^{(n)}_sx}\, ds\\
&\leq c_1(nt_0) \sum_{i\geq \eta \sqrt n} \vp(i) i^{d-1}\\
&\leq c_1 t_0 \eta^{-2} \sum_{i\geq \eta \sqrt n} \vp(i) i^{d+1}, \end{aligned}$$ which tends to 0 by dominated convergence as $n\to \infty$. Since this is true for each $t_0$ and $ \eta>0$ we conclude that any subsequential limit point of the sequence ${Z}^{(n)}$ will have continuous paths.
From this point on the argument is fairly standard. We give a sketch, leaving the details to the reader. Take a countable dense subset $\{t_i\}$ of $[0,\infty)$ and a countable dense subset $\{f_m\}$ of the $C^\infty$ functions on $\R^d$ with compact support. Let $P_t^n$ be the semigroup for $Z^{(n)}$. In view of Theorem \[4.15\], $E_{n_{j}}(P_{t_i}^{{n_{j}}}(R_{n_{j}}(f_m)))$ will be equicontinuous. By a diagonalization argument, we can find a subsequence $\{n_{j_k}\}$ of $\{n_j\}$ such that for each $i$ and $m$, as $n_{j_k}\to \infty$, these functions converge uniformly on compact sets. Call the limit $P_{t_i} f_m$. Using the equicontinuity, we can define $P_t f_m$ by continuity for all $t$, and because the norm of each $P_t$ is bounded by 1, we can also define $P_t f$ by continuity for $f$ continuous with compact support. Using the equicontinuity yet again, it is easy to see that the $P_t$ satisfy the semigroup property and that $P_t$ maps continuous functions with compact support into continuous functions. One can thus construct a strong Markov process that has $P_t$ as its semigroup. The symmetry of $P^{(n)}_t$ leads to the symmetry of $P_t$.
For each $x$, the $\P^{[x]_{n_j}}$ laws of $\{Z^{(n_{j})}_t; 0\leq t\leq t_0\}$ are tight. Fix $x$, let $\{n'\}$ be any subsequence of $\{n_{j_k}\}$ along which the $\P^{[x]_{n'}}$ converge weakly, and let $\P$ be the weak limit of the subsequence $\P^{[x]_{n'}}$. Suppose $F$ is a continuous functional on $C([0,t_0]; \R^d)$ of the form $F(\omega)=\prod_{\ell=1}^L g_i(\omega(s_i))$, where the $g_i$ are continuous with compact support and $0\leq s_1<\cdots < s_L\leq t_0$. When $L=1$, then $$\begin{aligned}
\E g_1(Z_{s_1})&=\lim \E^{[x]_{n'}} R_{n'}(g_1)(Z_{s_1}^{(n')})\\
&=\lim P_{s_1}^{n'} R_{n'}(g_1)([x]_{n'})\\
&=P_{s_1} g_1(x).\end{aligned}$$ Thus the one-dimensional distributions of a subsequential limit point of the $\P^{[x]_{n_{j_k}}}$ do not depend on the subsequence $\{n'\}$. Using the Markov property of $Z^{(n)}$ and the equicontinuity, a similar argument shows that the same is true of the $L$-dimensional distributions. Therefore there must be weak convergence along the subsequence $\{n_{j_k}\}$. As proved above, the weak limit is concentrated on the set of continuous paths.
[Proof of Theorem \[clt\]:]{} We denote the Dirichlet form for the process $Z^{(n)}$ by $\ce_n$. Suppose $f, g$ are $C^\infty$ on $\R^d$ with compact support. Let $U^n_\lam$ be the $\lam$-resolvent for $Z^{(n)}$; this means that $$U_\lam^n h(x)=\E^x\int_0^\infty e^{-\lam t} h(Z^{(n)}_t)\, dt$$ for $x\in n^{-1}\Z^d$ and $h$ having domain $n^{-1}\Z^d$. We write $P_t^n$ for the semigroup for $Z^{(n)}$.
Using Proposition \[clttight\], we need to show that if we have a subsequential limit point of the $P^n_t$ in the sense of that proposition, then the limiting process corresponds to the Dirichlet form $\ce_a$. Let $\{n'\}$ be a subsequence of $\{n\}$ for which the subsequence converges in the sense of Proposition \[clttight\], and let $U_\lam$ be the $\lam$-resolvent of the limiting process.
Let $F_{n'}=U^{n'}_\lam (R_{n'}(f))$. Then $$\label{cltA}
\ce_{n'}(F_{n'}, R_{n'}(g))=(R_{n'}(f), R_{n'}(g))-\lam (F_{n'}, R_{n'}(g)),$$ where we let $(h_1,h_2)=\sum_{x\in n^{-1}\Z^d} h_1(x)h_2(x)\mu^D_x$ for functions defined on $n^{-1}\Z^d$. (Recall that our base measure is $\mu^D$.) Let $H_n=E_n(F_n)$ and $H=U_\lam f$. The equicontinuity result of Theorem \[4.15\] and Proposition \[clttight\] shows that the $H_{n'}$ converges uniformly on compacts to $H$. If we can show $$\label{clt1}
\ce_a(H,g)=(f,g)-\lam (H,g),$$ this will show that the $\lam$-resolvent for the limiting process is the same as the $\lam$-resolvent for the process corresponding to $\ce_a$, and the proof will be complete; we also use $(h_1,h_2)$ to denote $\int h_1(x) h_2(x)\, dx$ when $h_1, h_2$ are functions defined on $\R^d$.
Next, since $f\in L^2(\R^d)$ and $f$ is $C^\infty$, then $R_n(f)\in L^2(d\mu_n)$. Standard Dirichlet form theory shows that $$\| U^n_\lam (R_n(f))\|_2\leq \frac{1}{\lam}\| R_n(f)\|_2,$$ that is, the $L^2$ norm of $F_n$ is bounded in $n$. We see that $$\label{clt2.5}
\int |\nabla H_n(x)|^2 \, dx \leq c_1 \ce_n(F_n,F_n)=c_1((R_n(f),F_n)-\lam(F_n,F_n))$$ is bounded in $n$. By the compact imbedding of $W^{1,2}$ into $L^2$, we conclude that $\{H_n\}$ is a compact sequence in $L^2(\R^d)$; here $W^{1,2}$ is the space of functions whose gradient is square integrable. Since $H_{n'}$ converges on compacts to $H$, it follows that $H_{n'}$ converges in $L^2$ to $H$. We note also that by (\[cltA\]) $$\label{clt2.7}
\ce_n(F_n,F_n)=(R_n(f),F_n)-\lambda (F_n, F_n)$$ is uniformly bounded in $n$.
We need to know that $$\label{clt2}
|\sE^{R}_n(F_n,R_n(g))-\ce_{a^n}(H_n,g)|\to 0$$ as $n\to \infty$. The proof of this is a bit lengthy and we defer it to Lemma \[cltlem\] below.
We also need to show that $$\label{clt2q}
|\sE_n(F_n,R_n(g))-\sE^{R}_n(F_n,R_n(g))|\to 0$$ as $n\to \infty$. This follows because by Cauchy-Schwarz, we have $$\begin{aligned}
\Bigl| \sum_{x,y\in n^{-1}\Z^d}& (F_n(y)-F_n(x))
n^{2-d}C^{n}_{nx,ny}
(R_n(g)(y)-R_n(g)(x))\\
&-\sum_{x,y\in n^{-1}\Z^d} (F_n(y)-F_n(x))
n^{2-d} C^{n,R}_{nx,ny} (R_n(g)(y)-R_n(g)(x))\Bigr|\\
\leq & c\ce_n(F_n,F_n)^{1/2}
\Big[\sum_{x,y\in n^{-1}\Z^d}
n^{2-d}(C^{n}_{nx,ny}-C^{n,R}_{nx,ny}) (R_n(g)(y)-R_n(g)(x))^2\Big]^{1/2}.\end{aligned}$$ The term within the brackets on the last line is bounded by $$c\| \nabla g\|^2_\infty
\sup_{x\in n^{-1}\Z^d}\sum_{y\in \Z^d, |x-y|>nR} |x-y|^2 C^n_{xy}
\leq c'\sum_{i>nR} i^{d-1} i^2 \vp(i),$$ which will be less than $\eps^2$ if $n$ is large.
Using (\[cltA\]), (\[clt1\]), (\[clt2\]), and (\[clt2q\]), we see that it suffices to show $$\label{clt3}
\ce_{a^{n'}}(H_{n'},g)\to \ce_a(H,g).$$ Now $$\label{clt4}
|\ce_{a^{n'}}(H_{n'},g)-\ce_a(H_{n'},g)|=\Bigl|\int \nabla H_{n'} \cdot (a^{n'}-a) \nabla g\Bigr|.$$ Since $\nabla g$ is bounded with compact support and $|\nabla H_{n'}|$ is bounded in $L^2$, then (A5) and the Cauchy-Schwarz inequality tell us that the right hand side of (\[clt4\]) tends to 0 as $n\to \infty$. Therefore we need to show $$\label{clt5}
\ce_a(H_{n'},g)\to \ce_a(H,g).$$ But if $\nabla h$ is bounded with compact support, then $$\label{clt6}
\int (\nabla H_{n'})\, h=-\int H_{n'} \nabla h\to -\int H\, \nabla h=\int (\nabla H)\, h.$$ If we take the supremum over such $h$ that also have $L^2$ norm bounded by 1, then Fatou’s lemma and the Cauchy-Schwarz inequality show that $\nabla H$ is in $L^2$. If $h$ is bounded with compact support, let $\varepsilon>0$ and approximate $h$ by a $C^1$ function $\tilde h$ with compact support such that $\|h-\tilde h\|_2\leq \varepsilon$. Since $|\nabla H_n|$ is bounded in $L^2$, then $|\int \nabla H_{n'}(h-\tilde h)|\leq c_1\varepsilon$ and $|\int \nabla H(h-\tilde h)|\leq c_1\varepsilon$. So by (\[clt6\]) $$\limsup_{n'\to \infty} \Bigl|\int \nabla H_{n'}h-\int \nabla H \, h\bigr|\leq 2c_1\varepsilon.$$ Because $\varepsilon $ is arbitrary, we have $$\label{clt7}
\int \nabla H_{n'}\, h\to \int \nabla H\, h.$$ If we apply (\[clt7\]) with $h=a\nabla g$, we obtain (\[clt5\]).
To complete the proof we have
\[cltlem\] With the notation of the above proof, $$|\sE^{R}_n(F_n,R_n(g))-\ce_{a^n}(H_n,g)|\to 0$$ as $n\to \infty$.
Let $\eps, \eta_1, \eta_2, \delta>0$ and let $\{\sS_m\}$ be a collection of cubes with disjoint interiors whose union contains the support of $g$ and such that the oscillation of $a$ on each $\sS_m$ is less than $\eta_1$ and the oscillation of $\nabla g$ on each $\sS_m$ is less than $\eta_2$. One way to construct such a collection is to take a cube large enough to contain the support of $g$, divide it into $2^d$ equal subcubes, and then divide each of the subcubes and so on until the oscillation restrictions are satisfied.
[*Step 2.*]{} Let $\sS_m'$ be the cube with the same center as $\sS_m$ but side length $(1-2\delta)$ times as long. Let $A=\cup_m(\sS_m-\sS_m')$. We claim it suffices to show that $$\begin{aligned}
\Bigl| \int_{A^c} &\nabla H_n(x) \cdot a^n(x) \nabla g(x)\, dx\nonumber\\
&- \sum_{x\notin A, x\in n^{-1}\Z^d}\ \sum_{y\in n^{-1}\Z^d} (F_n(y)-F_n(x))
n^{2-d}C^{n,R}_{nx,ny} (R_n(g)(y)-R_n(g)(x))\Bigr|\nonumber\\
&\qquad\to 0 \label{t7A}\end{aligned}$$ as $n\to \infty$. To see this, note first that by Cauchy-Schwarz and (\[clt2.5\]) $$\begin{aligned}
\int_{A} \nabla H_n(x) \cdot a^n(x) \nabla g(x)\, dx
&\leq \ce_{a^n}(H_n,H_n)^{1/2} \Big(\int_{A} \nabla g(x) \cdot a^n(x) \nabla g(x)\, dx\Big)^{1/2}\\
&\leq c\ce_{a^n}(H_n,H_n)^{1/2} \| \nabla g\|_\infty |A|^{1/2}\end{aligned}$$ will be less than $\eps$ if $\delta$ is taken sufficiently small. Next note that for any $x\in n^{-1}\Z^d$, $$\begin{aligned}
\sum_{y\in n^{-1}\Z^d}n^{2-d}C^{n,R}_{nx,ny} (R_n(g)(y)-R_n(g)(x))^2
&\leq n^{-d} \|\nabla g\|_\infty^2
\sum_{y\in n^{-1}\Z^d} C^{n,R}_{nx,ny} |ny-nx|^2\\
&\leq cn^{-d}.\end{aligned}$$ So by Cauchy-Schwarz and (\[clt2.7\]) $$\begin{aligned}
\sum_{x\in A, x\in n^{-1}\Z^d} \sum_{y\in n^{-1}\Z^d} &(F_n(y)-F_n(x))
n^{2-d}C^{n,R}_{nx,ny} (R_n(g)(y)-R_n(g)(x))\nonumber\\
&\leq \ce_n (F_n,F_n)^{1/2} \Big(
\sum_{x\in A, x\in n^{-1}\Z^d}\sum_{y\in n^{-1}\Z^d}n^{2-d}C^{n,R}_{nx,ny}
(R_n(g)(y)-R_n(g)(x))^2\Big)^{1/2}\nonumber\\
&\leq c\ce_n (F_n,F_n)^{1/2} \Big(n^{-d}\, \mbox{\rm card}\, (A\cap n^{-1}\Z^d)\Big)^{1/2},
\label{crdel}\end{aligned}$$ which will be less than $\eps$ if $\delta $ is taken small enough and $n$ is large.
[*Step 3.*]{} Let $x_m$ be the center of $\sS_m$. Define $\ol g$ by requiring $\ol g$ to be linear on each $\sS_m$ and satisfying $\ol g(x_m)=g(x_m)$, $\nabla\ol g(x_m)=\nabla g(x_m)$. We claim it suffices to show that $$\begin{aligned}
\Bigl| \int_{A^c}& \nabla H_n(x) \cdot a^n(x) \nabla \ol g(x)\, dx\nonumber\\
&- \sum_{x\notin A, x\in n^{-1}\Z^d}\ \sum_{y\in n^{-1}\Z^d} (F_n(y)-F_n(x))
n^{2-d}C^{n,R}_{nx,ny} (R_n(\ol g)(y)-R_n(\ol g)(x))\Bigr|\nonumber\\
&\qquad \to 0 \label{t7B}\end{aligned}$$ To see this, note that $$\begin{aligned}
\Bigl| \int_{A^c} \nabla H_n(x) \cdot a^n(x) \nabla \ol g(x)\, dx-
&\int_{A^c} \nabla H_n(x) \cdot a^n(x) \nabla g(x)\, dx \Bigr|\\
&\leq \ce_{a^n} (H_n, H_n)^{1/2}
\Big( \int_{A^c} \nabla (\ol g-g)(x) \cdot a^n(x) \nabla (\ol g-g)(x)\, dx\Big)^{1/2}\\
&\leq c\ce_{a^n} (H_n, H_n)^{1/2} \eta_2,\end{aligned}$$ which will be less than $\eps$ if $\eta_2$ is chosen small enough. A similar argument shows that the difference between the second term in (\[t7B\]) and the corresponding term with $\ol g$ replaced by $g$ is small; cf. Step 2.
[*Step 4.*]{} Let $\ol C^n_{xy}=C^{n,\delta/2}_{xy}$ and define $\ol a^n(x)$ by $(\ol a^n(x))_{ij}
=\sum_{(y,k)\in L^i_x} \ol C_{ny, n(y+k)}^{n} nk_{j}\sgn k_i$. We claim it suffices to show that $$\begin{aligned}
\Bigl| \int_{A^c}& \nabla H_n(x) \cdot \ol a^n(x) \nabla \ol g(x)\, dx\nonumber\\
&- \sum_{x\notin A, x\in n^{-1}\Z^d}\ \sum_{y\in n^{-1}\Z^d} (F_n(y)-F_n(x))
n^{2-d}\ol C^{n}_{nx,ny} (R_n(\ol g)(y)-R_n(\ol g)(x))\Bigr|\nonumber\\
&\qquad \to 0 \label{t7C}\end{aligned}$$ To prove this, we first note that the following can be proved in the same way as (\[clt2q\]).
$$\begin{aligned}
\Bigl|\sum_{x\notin A, x\in n^{-1}\Z^d}& \sum_{y\in n^{-1}\Z^d} (F_n(y)-F_n(x))
n^{2-d}\ol C^{n}_{nx,ny} (R_n(\ol g)(y)-R_n(\ol g)(x))\\
&-\sum_{x\notin A, x\in n^{-1}\Z^d}\ \sum_{y\in n^{-1}\Z^d} (F_n(y)-F_n(x))
n^{2-d} C^{n,R}_{nx,ny} (R_n(\ol g)(y)-R_n(\ol g)(x))\Bigr|\to 0,\end{aligned}$$
as $n\to\infty$. Next, $$\begin{aligned}
\Bigl| \int_{A^c}& \nabla H_n(x) \cdot \ol a^n(x) \nabla \ol g(x)\, dx
-\int_{A^c} \nabla H_n(x) \cdot a^n(x) \nabla \ol g(x)\, dx\Bigr|\nonumber\\
&\le c\Bigl(\int_{A^c} (\nabla H_n(x))^2dx\Bigr)^{1/2}
\Bigl(\int_{A^c}(\ol a^n(x)- a^n(x))(\nabla \ol g(x))^2\Bigr)^{1/2}.\label{aola}\end{aligned}$$ We can estimate $$\begin{aligned}
\Bigl|(\ol a^n(x)- a^n(x))_{ij}\Bigr|&\le \sum_{(y,k)\in L^i_x}
|\ol C_{ny, n(y+k)}^{n}-C_{ny, n(y+k)}^{n}|nk_{j}\nonumber\\
&\le c_1\sup_{x\in \Z^d}\sum_{y\in \Z^d, |x-y|>n\delta/2} |x-y|^2 C^n_{xy}
\leq c_2\sum_{i>n\delta/2} i^{d-1} i^2 \vp(i),\label{aolacop}\end{aligned}$$ where in the second inequality, we used the fact that for each $k$, the number of $y$ that satisfies $(y,k)\in L^i_z$ is at most $n|k_i|$ (as mentioned when we defined $L^i_z$). So the right hand side of (\[aola\]) will be less than $\eps$ if $n$ is large.
[*Step 5.*]{} We have chosen the $\sS_m$ so that the oscillation of $a$ on each $\sS_m$ is at most $\eta_1$. Since we have that the $a^n$ converge to the $a$ uniformly on compacts and there are only finitely many $\sS_m$’s, then for $n$ large the oscillation of $a^n$ on any $\sS_m$ will be at most $2\eta_1$.
[*Step 6.*]{} We will now prove (\[t7C\]). By Step 3, $\ol g$ is linear on each $\sS_m$, so it is enough to discuss the case where $\ol g(x)=x_{j_0}$ on $\sS'_m$ for some $j_0$ and then use a linearity argument. Noting that $H_n=F_n$ on $n^{-1}\Z^d$, define $$\ce_n^{\sS'_m}(H_n,\ol g):=
\sum_{x\in \sS'_m\cap n^{-1}\Z^d}\
\sum_{y\in n^{-1}\Z^d} (H_n(y)-H_n(x))
n^{2-d}\ol C^{n}_{nx,ny} (R_n(\ol g)(y)-R_n(\ol g)(x)).$$ Since there is no term involving different $\sS'_m$’s, we will consider each $\sS'_m$ separately. We will fix an $x_0\in \sS'_m$ and look at the terms involving $H_n(x_0+n^{-1} e_i)-H_n(x_0)$. First, by an elementary computation using the definition of the linear extension map $E_n$, we have $$\label{t71}
\int_{Q_n(x_0)}\frac{\partial H_n}{\partial x_i}dx=\frac 1{2^{d-1}n^{d-1}}\sum_{z\in
V_i(x_0)}(H_n(z+n^{-1}e_i)-H_n(z))$$ where $V_i(x_0)$ is the collection of vertices of the face of $Q_n(x_0)$ perpendicular to $e_i$ and with the smaller $e_i$ component. (E.g., for a square, $V_1(x_0)$ is the two leftmost corners, $V_2(x_0)$ is the two lower corners.) So $$\begin{aligned}
\int_{Q_n(x_0)}(\nabla H_n,\ol a^n\nabla \ol g)dx&=\sum_{i,j=1}^d\int_{Q_n(x_0)}
\frac{\partial}{\partial x_i} H_n \ol a^n_{ij}\frac{\partial}{\partial x_j} \ol g\, dx=
\sum_i\ol a^n_{ij_0}(x_0)\int_{Q_n(x_0)}\frac{\partial}{\partial x_i} H_n dx\\
&=\sum_{i=1}^d\ol a^n_{ij_0}(x_0)\frac 1{2^{d-1}n^{d-1}}\sum_{z\in
V_i(x_0)}(H_n(z+n^{-1}e_i)-H_n(z)).\end{aligned}$$ Summing over all cubes that contains $H_n(x_0+n^{-1}e^i)-H_n(x_0)$, the coefficient in front of $H_n(x_0+n^{-1}e^i)-H_n(x_0)$ will be $$\label{t72}
\frac{n^{1-d}}{2^{d-1}}
\sum_{z\in V_i(x_0+n^{-1}e^i-e_*)}\ol a^n_{ij_0}(z),$$ where $e_*=(1/n,...,1/n)$.
We next look at $\ce_n^{\sS'_m}(H_n,\ol g)$. Since $\ol g(x+k)-\ol g(x)=k_{j_0}$ where $k=(k_1,...,k_d)$, we have $$\ce_n^{\sS'_m}(H_n,\ol g)=n^{2-d}
\sum_{{x\in \sS'_m\cap n^{-1}Z^d,}\atop{k\in n^{-1}Z^d}}
(H_n(x+k)-H_n(x))
\ol C^{n}_{nx,n(x+k)}k_{j_0}.$$ Let us fix $x$ and $k$ and replace $(H_n(x+k)-H_n(x))$ by the sum $\sum_{m=1}^{|k|}
(H_n(z_{m+1})-H_n(z_m))$ (here $|k|:=|k_1|+...+|k_d|$ and $|z_{m+1}-z_m|=1/n$) so that the union of the line segments belongs to $x+ n^{-1} \sP(k)$. We will get a term of the form $H_n(x_0+n^{-1}e_i)-H_n(x_0)$ if $z_m=x_0$ and $z_{m+1}=x_0+n^{-1}e_i$ (we get $H_n(x_0)-H_n(x_0+n^{-1}e_i)$ if $z_{m+1}=x_0$ and $z_{m}=x_0+n^{-1}e_i$), so the contribution will be $$n^{2-d}\ol C^n_{nx,n(x+k)}k_{j_0}(\sgn~ k_i).$$ Summing over $x\in \sS'_m\cap n^{-1}Z^d,k\in n^{-1}\Z^d$, we have that the coefficient in front of $H_n(x_0+n^{-1}e^i)-H_n(x_0)$ for $\ce_n^{\sS'_m}(H_n,\ol g)$ is $$\label{t73}
n^{2-d}\sum_{{x\in \sS'_m\cap n^{-1}Z^d,}\atop {(x,k)\in L^i_{x_0}}}
\ol C^n_{nx,n(x+k)}k_{j_0}(\sgn~ k_i).$$ On the other hand, by the definition of $\ol a^n$, we have $$\label{t73q}
n^{2-d}\sum_{(x,k)\in L^i_{x_0}} \ol C^n_{nx,n(x+k)}k_{j_0}(\sgn~ k_i)=n^{1-d}
\ol a^n_{ij_0}(x_0).$$ Let $\sS_m''$ be the cube with the same center as $\sS_m'$ but side length $(1-2\delta)$ times as long. If $x_0\in \sS_m''\cap n^{-1}\Z^d$, then the expressions in (\[t73\]) and (\[t73q\]) are equal, because $\ol C^n_{nx,n(x+k)}=0$ for $x\notin \sS_m'\cap n^{-1}\Z^d, (x,k)\in L^i_{x_0} $. Since the oscillation of $a^n$ on each $\sS'_m$ is less that $2\eta_1$ as in Step 5, by (\[aolacop\]) the oscillation of $\ol a^n$ on each $\sS'_m$ is less that $3\eta_1$. Thus, when $x_0\in \sS_m''\cap n^{-1}\Z^d$, we see that the absolute value of the difference between (\[t72\]) and (\[t73\]) is bounded by $3\eta_1n^{1-d}$. (Note that $\mbox{\rm card}\, V_i(x_0+n^{-1}e^i-e_*)=2^{d-1}$ is used here.) Now, if $x_0\in (\sS'_m-\sS_m'')\cap n^{-1}\Z^d$, then the difference between (\[t72\]) and (\[t73\]) is bounded by $c_*n^{1-d}$, because similarly to (\[aolacop\]) we have $$\sum_{(x,k)\in L^i_{x_0}} \ol C^n_{nx,n(x+k)}nk_{j_0}(\sgn~ k_i)
\le c_1\sup_{x\in \Z^d}\sum_{y\in \Z^d} |x-y|^2 C^n_{xy}
\leq c_2\sum_{i} i^{d-1} i^2 \vp(i)=:c_*.$$ Denote $H_{x_0,i}:=H_n(x_0+n^{-1}e^i)-H_n(x_0)$, $A':=(\cup_m(\sS_m'-\sS_m''))\cap n^{-1}\Z^d$ and $B:=(\cup_m\sS_m'')$$\cap n^{-1}\Z^d$. Using the Cauchy-Schwarz inequality, we have $$\begin{aligned}
&\Big|\int_{\cup_m\sS'_m}(\nabla H_n,\ol a^n\nabla \ol g)dx-\sum_m\ce_n^{\sS'_m}(H_n,\ol g)\Big|
\label{compcdis}\\
\le &\eta_1n^{1-d}\sum_{x_0\in B,i=1,\cdots, d}|H_{x_0,i}|
+c_*n^{1-d}\sum_{x_0\in A',i=1,\cdots, d}|H_{x_0,i}|\nonumber\\
\le& c_1\eta_1\Big(n^{-d}\mbox{\rm card}\, B\Big)^{1/2}
\Big(n^{2-d}\sum_{x_0\in n^{-1}\Z^d,i}(H_{x_0,i})^2\Big)^{1/2}
\\&~~~~+c_*\Big(n^{-d}\mbox{\rm card}\, A'\Big)^{1/2}
\Big(n^{2-d}\sum_{x_0\in n^{-1}\Z^d,i}(H_{x_0,i})^2\Big)^{1/2}\nonumber\\
\le &c_2(\eta_1+\eps)\Big(n^{2-d}\sum_{x_0\in n^{-1}\Z^d,i}(H_{x_0,i})^2\Big)^{1/2}
\le c_3(\eta_1+\eps),\nonumber\end{aligned}$$ if $\delta $ is taken small enough and $n$ is large. We thus complete the proof of (\[t7C\]).
When $d=1$, Lemma \[cltlem\] can be proved under much milder conditions.
(A6) There exists $R>0$ and a Borel measurable $a:\R^d\to\cal M$ such that for each $r>0$ $$\label{intcov}
\lim_{n\to\infty}\int_{|x|\le r}|a^n(x)-a(x)|dx=0.$$
\[d1cor\] Let $d=1$ and suppose (A1)-(A3) and (A6) hold. Then the conclusions of Theorem \[clt\] hold.
[Proof:]{} The proof is similar to the proof of Theorem \[clt\]. Let us point out the places where we need modifications. First, we can prove that there exist $c_1,c_2>0$ such that $c_1\le a^n(x)\le c_2$ for all $x\in \R^d$ and $n\in \N$. Indeed, by (A2) the lower bound is guaranteed and the upper bound can be proved similarly to (\[aolacop\]). So, we know $\ce_{a^n}(f,f)$ is bounded whenever $f\in L^2$. For the proof that the right hand side of (\[clt4\]) goes to $0$ as $n\to\infty$, we use (\[intcov\]). (To be more precise, the convergence of $a^n$ to $a$ locally in $L^2$ is used there, which is guaranteed by (\[intcov\]) and the fact that the $a^n$ are uniformly bounded.) Noting these facts, the proofs of Theorem \[clt\] and Proposition \[clttight\] go the same way as above. For the proof of Lemma \[cltlem\], in Step 1, we do not need to control the oscillation of $a$ on each $\sS_m$. Step 5 is not needed. We have that the expression (\[t72\]) is equal to $\ol a^n_{ij_0}(x_0)$, and this is equal to the expression in (\[t73q\]). (This is a key point; because of this we do not have to worry about the oscillation of $a$ and $a^n$.) Finally, in the computation of (\[compcdis\]), the difference on the set $B$ is $0$ due to the fact just mentioned, and we can prove that (\[compcdis\]) is small directly.
We now give an extension of the result in [@SZ] to the case of unbounded range. Assume
(A7) There exists $R>0$ such that for each $r>1$ $$\label{szstrassmp}
\lim_{n\to\infty}\sum_{k\in \Z^d}\sup_{|y|\le nr}
\sup_{|x-y|\le nR}
\Big|C^{n,R}_{x,x+k}-C^{n,R}_{y,y+k}\Big|=0.$$ Let the $(i,j)$-th element of $b^n$ be given by $$\label{defsza}
\big(b^n(x)\big)_{ij}
=\sum_{k\in n^{-1}\Z^d} C_{nx, n(x+k)}^{n,R}n^2 k_ik_{j},\qquad x\in n^{-1}\Z^d.$$ For general $x=(x_i)_{i=1}^d\in\R^d$, define $b^n(x):=b^n([x]_n)$. Assume the $b^n$ version of (A6);
(A8) There exists $R>0$ and a Borel measurable $a:\R^d\to\cal M$ such that for each $r>0$ $$\label{intcovw2}
\lim_{n\to\infty}\int_{|x|\le r}|b^n(x)-a(x)|dx=0.$$
We can recover and generalize the convergence theorem given in [@SZ] as follows.
\[d1corsz\] Suppose that (A1)-(A3), (A7), and (A8) hold. Then the conclusions of Theorem \[clt\] hold.
[Proof:]{} For each $\eps>0$, let $R'=R'(\eps)>0$ be an integer that satisfies $\sum_{s\ge R'}\vp (s)s^2<\eps$. Note that $C^{n,R}_{x,y}=
C^{n,R'/n}_{x,y}+1_{\{|x-y|>R'\}}C^{n,R}_{x,y}$. Then, for any $r\ge 1$, any $x\in n^{-1}\Z^d$ such that $|x|\le r$, and any $n\ge R'/R$, we have $$\begin{aligned}
\Big|\big(a^n(x)\big)_{ij}-\big(b^n(x)\big)_{ij}\Big|
&\le &\sum_{k'\in \Z^d}
\Big|\sum_{y:(y,k')\in L_x^{i,*}}C_{ny, ny+k'}^{n,R}\sgn k'_i-C_{nx, nx+k'}^{n,R}k'_i\Big|k'_j\\
&\le &R'^2\Big(\sum_{k'\in \Z^d}\sup_{|y'|\le nr}\sup_{|x'-y'|\le R'}
\Big|C^{n,R'/n}_{x',x'+k'}-C^{n,R'/n}_{y',y'+k'}\Big|\Big)
+2\sum_{s\ge R'}\vp (s)s^2\\
&\le &R'^2\Big(\sum_{k'\in \Z^d}\sup_{|y'|\le nr}\sup_{|x'-y'|\le nR}
\Big|C^{n,R}_{x',x'+k'}-C^{n,R}_{y',y'+k'}\Big|\Big)
+2\eps,\end{aligned}$$ where $L^{i,*}_z=\{(y,k')\in (n^{-1}\Z^d)\times \Z^d:
y+n^{-1}\sP(k') \mbox{ contains the line segment from $z$ to } z+n^{-1}e^i\}$. In the second inequality, we used the fact that if $(y,k')\in L^{i,*}_x$ and $x'=nx, y'=ny$, then $|x'-y'|=n|x-y|\le n|k'/n|=k'\le
n\cdot R'/n=R'$. Using (\[szstrassmp\]) in (A7), the right hand side converges to $0$ as $n\to\infty$. In other words, $$\label{aasznear}
|(a^n(x))_{ij}-b^n(x))_{ij}|\to 0~~\mbox{uniformly on compacts as } n\to\infty.$$ Similarly, for any $r\ge 1$, we can prove $$\label{aasz0}
|(b^n(x))_{ij}-(b^n(y))_{ij}|\to 0~~\mbox{ as }~~n\to\infty,~~
|x-y|\le n^{-1}R,~ |x|\le r.$$ Now the proof of this corollary goes similarly to the proofs above. As before we point out places where we need modifications. First, as in Corollary \[d1cor\], we can prove that there exist $c_1,c_2>0$ such that $c_1I\le b^n(x)\le c_2I$ for all $x\in \R^d$ and $n\in \N$. So we know $\ce_{b^n}(f,f)$ is bounded whenever $f\in L^2$. As in Corollary \[d1cor\], we use (\[intcovw2\]) to show that the right hand side of (\[clt4\]) goes to $0$ as $n\to\infty$. Noting these facts, the proofs of Theorem \[clt\] and Proposition \[clttight\] go in the same way as before. For the proof of Lemma \[cltlem\], in Step 1, we do not need to control the oscillation of $a$ on each $\sS_m$. Step 4 with respect to $b^n$ works due to (\[aasznear\]). Step 5 is not needed. Thanks to (\[aasznear\]) and (\[aasz0\]), the difference between the expression in (\[t72\]) (with $a$ replaced by $b$) and the expression in (\[t73q\]) is small. (This is again the key point; because of this we do not have to worry about the oscillation of $a$ and $b^n$.) Finally, in the computation of (\[compcdis\]), the difference on the set $B$ is small due to the fact just mentioned.
\[remark2\] [If (A7) does not hold, $b^n$ need not be the right approximation in general. Indeed, here is an example where $a^n$ converges to $a$, but $b^n$ does not as $n\to\infty$. Suppose $d=1$ and let $C^n_{k, k+i}$ equal $r_i$ if $k$ is odd, $s_i$ if $k$ is even, $i=1,2$. Then, we have $$\begin{aligned}
b^n(k/n)
&=&\left\{\begin {array}{ll} r_1+s_1+8r_2,\qquad \mbox{if $k$ is odd},\\
r_1+s_1+8s_2,\qquad \mbox{if $k$ is even}.\end{array}\right.\\
a^n(k/n)
&=&\left\{\begin {array}{ll} 2r_1+4(r_2+s_2),\qquad \mbox{if $k$ is odd},\\
2s_1+4(r_2+s_2),\qquad \mbox{if $k$ is even}.\end{array}\right.\end{aligned}$$ Suppose $r_1=s_1$ and $r_2\ne s_2$. Then, the value of $b^n(k/n)$ depends on whether $k$ is odd or even, so $b^n$ does not converge locally in $L^2$ as $n\to\infty$, whereas $a^n(k/n)=2r_1+4(r_2+s_2)$ is constant. In this case, the assumption of Theorem \[clt\] (and Corollary \[d1cor\]) holds and $a(x)=2r_1+4(r_2+s_2)$. ]{}
Theorem \[clt\] gives a central limit theorem for the processes $Y^{(n)}$. Note that the base measure for $Y^{(n)}$ is the uniform measure, which converges with respect to Lebesgue measure on $\R^d$. We finally discuss the convergence of the discrete time Markov chains $X^{(n)}$. Let $Y^\nu_t$ be the continuous time $\nu$-symmetric Markov chain on $\Z^d$ which corresponds to $(\ce,\cf)$. It is a time change of $Y_t$ and it can be defined from $X_n$ as follows. Let $\{U_i: i\in \N, x\in \Z^d\}$ be an independent collection of exponential random variables with parameter 1 that are independent of $X_n$. Define $T_0=0, T_n=\sum_{k=1}^n U_k$. Set $\widetilde Y^\nu=X_n$ if $T_n\le t<T_{n+1}$; then the laws of $\widetilde Y^\nu$ and $Y^\nu$ are the same. Let $\nu^D$ be a measure on $\cs$ defined by $\nu^D(A)=D^{-d}\nu (DA)$ for $A\subset \cs$. Since $\cs\subset \R^d$, we will regard $\nu^D$ as a measure on $\R^d$ from time to time. By (A1), we see that $c_1\mu^D(A)\le \nu^D(A)\le c_2\mu^D(A)$ for all $A\subset \cs$ and all $d$. So $\{\nu^D\}_D$ is tight and there is a convergent subsequence. We assume the following.
(A9) There exists a Borel measure ${\bar \nu}$ on $\R^d$ such that $\nu^D$ converges weakly to ${\bar \nu}$ as $D\to\infty$.
Let $Z^{\bar \nu}_t$ be the diffusion process corresponding to the Dirichlet form $\ce_a$ considered on $L^2(\R^d,{\bar \nu})$. It is a time changed process of $Z_t$ in Theorem \[clt\]. Note that by (A1), ${\bar \nu}$ is mutually absolutely continuous with respect to Lebesgue measure on $\R^d$ so it charges no set of zero capacity. Further, the heat kernel for $Z^{\bar \nu}_t$ still enjoys the estimates (\[aroest\]).
Now we have a corresponding theorem for the discrete time Markov chains $X^{(n)}$. Define $$W^{(n)}_t=X^{(n)}_{[nt]}/\sqrt n.$$
\[cltcor\] Suppose (A1)-(A3), (A5), and (A9) hold.
\(a) Then for each $x$ and each $t_0$ the $\P^{[x]_n}$-law of $\{W^{(n)}_t; 0\leq t\leq t_0\}$ converges weakly with respect to the topology of the space $D([0,t_0], \R^d)$. The limit probability gives full measure to $C([0,t_0], \R^d)$.
\(b) If $Z^{\bar \nu}_t$ is the canonical process on $C([0,\infty), \R^d)$ and $\P^x$ is the weak limit of the $\P^{[x]_n}$-laws of $W^{(n)}$, then the process $\{Z^{\bar \nu}_t, \P^x\}$ has continuous paths and is the symmetric process corresponding to the Dirichlet form $\ce_a$ considered on $L^2(\R^d,{\bar \nu})$.
[Proof:]{} Let $Y^{(n),\nu}_t$ be the continuous time Markov chains on $\Z^d$ corresponding to $\ce_n$ considered on $L^2(\Z^d,\nu)$, and set $Z^{(n),\nu}_t=Y^{(n),\nu}_{nt}/\sqrt n$. Then, by changing the measure $\mu^D$ to $\nu^D$ in the proof, we have the results corresponding to Theorem \[clt\] for $Z^{(n),\nu}_t$ and $Z^{\bar \nu}_t$. So it suffices to show that there is a metric for $D([0,t_0], \R^d)$ with respect to which the distance between $W^{(n)}$ and $Z^{(n),\nu}$ goes to 0 in probability, where in the definition of $Z^{(n),\nu}$ we use the realization of $Y^{(n),\nu}$ given in terms of the $X^{(n)}$ by means of independent exponential random variables of parameter 1.
We use the $J_1$ topology of Skorokhod; see [@Bi]. The paths of $Y^{(n),\nu}$ agree with those of $X^{(n)}$ except that the times of the jumps do not agree. Note that $X^{(n)}$ jumps at times $k/n$, while $Y^{(n),\nu}$ jumps at times $T_k/n$. So it suffices to show that if $T_k$ is the sum of i.i.d. exponentials with parameter 1, then for each $\eta>0$ and each $t_0$ $$\P(\sup_{k\leq [nt_0]} |T_k-k|\geq n\eta)\to 0$$ as $n\to \infty$. But by Doob’s inequality, the above probability is bounded by $$\frac{4\Var T_{[nt_0]}}{n^2\eta^2}=\frac{4[nt_0]}{n^2\eta^2}\to 0$$ as desired.
\[remark3\] [We remark that the definition of $a^n$, and hence the statement of (A5), depends on the definition of $\sP(k)$ and of the extension operator $E_n$. It would be nice to have a central limit theorem with a more robust statement. ]{}
\[remark4\] [ We make a few comments comparing the central limit theorem in our paper and the convergence theorem in [@SZ] in the case of bounded range. The result in [@SZ] requires a smoothness condition on the conductances $C^n_{xy}$, while we require smoothness instead on the $a^n$. Thus our theorem has weaker hypotheses, and as Remark \[remark2\] shows, there are examples where one set of hypotheses holds and the other set does not. On the other hand, if (A1)-(A3) hold, then the $\{b^n\}$ will automatically be symmetric, equi-bounded and equi-uniformly elliptic; if in addition $b^n\to a$, then $a$ will be bounded and uniformly elliptic and this does not need to be assumed. ]{}
[10]{}
D. Aldous. Stopping times and tightness, Ann. Probab., [**6**]{} (1978), 335–340.
M.T. Barlow, R.F. Bass and C. Gui, The Liouville property and conjecture of De Giorgi, Comm. Pure Appl. Math., [**53**]{} (2000), 1007–1038.
R.F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc. **357**, (2005) 837–850.
R.F. Bass and D.A. Levin, Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc. **354**, no. 7 (2002), 2933–2953.
R.F. Bass and D.A. Levin, Harnack inequalities for jump processes, Potential Anal. **17** (2002), 375–388.
P. Billingsley, Convergence of Probability Measures, 2nd ed., John Wiley, New York, 1999.
E.A. Carlen and S. Kusuoka and D.W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. Henri Poincaré-Probab. Statist. **23** (1987), 245-287.
Z.Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on $d$-sets, Stochastic Process Appl. [**108**]{} (2003), 27-62.
M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, deGruyter, Berlin, 1994.
J. Hu and T. Kumagai, Nash-type inequalities and heat kernels for non-local Dirichlet forms, Kyushu J. Math., to appear.
D. Jerison, The weighted Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. [**53**]{} (1986), 503–523.
G.F. Lawler, Estimates for differences and Harnack inequality for difference operators coming from random walks with symmetric, spatially inhomogeneous, increments, Proc. London Math. Soc. [**63**]{} (1992), 552–568.
T. Leviatan, Perturbations of Markov processes, J. Funct. Anal. [**10**]{} (1972), 309–325.
P.-A. Meyer, Renaissance, recollements, mélanges, ralentissement de processus de Markov, Ann. Inst. Fourier [**25**]{} (1975), 464–497.
F. Spitzer, Principles of Random Walk, Springer-Verlag, New York, 1976.
D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin, 1979.
D.W. Stroock and W. Zheng, Markov chain approximations to symmetric diffusions, Ann. Inst. Henri. Poincaré-Probab. Statist. **33** (1997), 619-649.
$\begin{array}{l}
\mbox{Richard F. Bass}\\
\mbox{Department of Mathematics}\\
\mbox{University of Connecticut}\\
\mbox{Storrs, CT 06269, U.S.A.}\\
\mbox{E-mail: {\tt [email protected]}}\\
\mbox{\ }\\
\mbox{Takashi Kumagai}\\
\mbox{Research Institute for Mathematical Sciences}\\
\mbox{Kyoto University}\\
\mbox{Kyoto 606-8502, Japan}\\
\mbox{E-mail: {\tt [email protected]}}\\
\end{array}$
[^1]: Research partially supported by NSF grant DMS0244737.
[^2]: Research partially supported by Ministry of Education, Japan, Grant-in-Aid for Scientific Research for Young Scientists (B) 16740052.
|
---
author:
- Paula Cerejeiras
- Uwe Kähler
- Teppo Mertens
- Frank Sommen
- Adrian Vajiac
- Mihaela Vajiac
title: A Primer on Script Geometry
---
Introduction
============
In the last two decades one can observe an increasing interest in the analysis of discrete structures. On the one hand this increasing interest is based on the fact that increased computational power is nowadays available to everybody and that computers can essentially work only with discrete values. This means that one requires discrete structures which are inspired by the usual continuous structures. On the other hand, the increased computational power also means that problems in physics which are traditionally modeled by means of continuous analysis are more and more directly studied on the discrete level, the principal example being the Ising model from statistical physics as opposed to the continuous Heisenberg model. Another outstanding example can be seen in the change of the philosophy of the Finite Element Method. The classical point of view of the Finite Element Method is to start from the variational formulation of a partial differential equation and to apply a Galerkin-Petrov or a Galerkin-Bubnov method via a neste sequence of finite-dimensional subspaces. These are created by discretizing the continuous domain by a mesh and to construct the basis functions of the finite-dimensional spaces as functions over the mesh. The modern approach lifts the problem and its finite element modelation directly on to the mesh resulting in the so-called Finite Element Exterior Calculus. The basic idea behind this discrete exterior calculus is that large classes of mixed finite element methods can be formulated on Hilbert complexes where one solves the variational problem on finite-dimensional subcomplexes. This not only represents a more elegant way of looking at finite element methods, but it also has two practical advantages. First of all it allows a better characterization of stable discretizations by requiring two hypotheses: they can be written as a subcomplex of a Hilbert complex and there exists a bounded cochain projection from that complex to the subcomplex [@Arnold2]. This was later on extended to abstract Hilbert complexes [@Stern]. Secondly, it mimics the engineer’s approach of directly performing finite element modeling on the mesh.
The principal example of this approach is the Hodge-deRham complex for approximating manifolds. Maybe it is worthwile to point out that the underlying ideas are much older. Whitney introduced his complex of Whitney forms in 1957 [@Whitney]. Among other things, he used them to identity the de Rham cohomology with simplicial cohomology. While these was done with purely geometric applications in mind later in it was shown that Whitney forms are finite elements on the deRham complex. Nevertheless, it is interesting to note that the original idea was purely geometric in nature.
Another example of this approach can be found in computational modeling [@Desbrun]. There, a discrete exterior calculus based on simplicial co-chains is introduced. One of the advantages is that it avoids the need for interpolation of forms and many important tools could be obtain like discrete exterior derivative, discrete boundary and co-boundary operator. An important step consisted also in the establishment of a discrete Poincaé lemma. It states that given a closed $k$-cochain $\omega$ on a (logically) star-shaped complex, i.e. $d\omega=0$ there exists a $(k-1)$-chochain $\alpha$ such that $\omega=d\alpha$.
While standard Whitney forms are linked to barycentric coordinates and, therefore, can be easily adapted to more general meshes, a large part of the above mentioned applications of Hodge theory to discrete structures are linked to simplicial complexes which are not that easily adapted to more general meshes.
To overcome this problem we are going to present a new type of algebraic topology based on the concept of scripts. A priori scripts are based on complexes, but more general than simplicial complexes. It is based on more geometrical constructions which also makes this concept rather intuitive. To make that clear and to make understanding easy we provide many concrete (classic) examples, including the torus, Klein bottle, and projective plane. Also, newly introduced notions and operations will always be accompanied by concrete examples so as to make understanding easier for the reader. As will be seen many of these notions and operations are rather intuitive while at the same time provide a more geometric understanding than classic approaches.
One of the key points in this theory is the concept of tightness which replaces the need for the establishment of a Poincaré lemma. Hereby, tighness imposes cells and chains to be minimal which is in fact what the geometric meaning of the Poincaré lemma represents. This can easily be seen if one notices that tightness means that the local homology at the level of cells is trivial which corresponds to the Poincaré lemma for manifolds which says that each point has a neighborhood with trivial homology.
In Chapter 2 we introduce the basic concepts, including the geometrical offprint of a script, equivalent and unitary scripts. The geometric offprint or skeleton of a script as a the support of a boundary chain will provide us with all the necessary geometrical information so as to represent the geometric boundary of a chain.
In Chapter 3 we are going to discuss the geometrical properties of scripts. This is closely linked to minimising and uniqueness properties of scripts. In particular the question of the skeleton being a unique minimal script will lead us to the central notion of tightness. A variety of examples will show that tightness is indeed a geometric and intuitive notion.
One of the essential parts in possible applications is the possibility to manipulate scripts. In Chapter 4 we present and discuss basic operations, such as creation and cleaning (removing) operations as well as identification operations. Again a variety of examples will be given.
In Chapter 5 we introduce the necessary concepts of metrics on scripts, dual scripts, and the corresponding Dirac and Laplace operators. This will be the groundwork for a function theory of monogenic and harmonic functions on our discrete structures.
Finally, Chapter 6 will be dedicated to the question of Cartesian products on scripts. We will give two types of Cartesian products on scripts and discuss tighness in this context. As always examples will be provided. Additionally, the introduction of a Cartesian product also allows us to give a notion of discrete curvature in the two-dimensional case which is much more intuitive than the standard notion.
Scripts in general
==================
Complexes
---------
A [*complex*]{} is a (finite or infinite) sequence of modules together with boundary maps $\partial_i:\mathcal{M}_i\longrightarrow\mathcal{M}_{i+1}$ such that $\partial_{i+1}\circ\partial_{i}=0$.
The starting point for scripts is the idea of a complex of free modules over ${{\mathbb Z}}$ together with boundary maps, with certain properties. A script is a special sequence of modules: $$\label{def:script}
\mathcal{M}_{-2} \longleftarrow \mathcal{M}_{-1} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1} \overset{\partial}{\longleftarrow}\mathcal{M}_{2}
\overset{\partial} {\longleftarrow} \cdots $$ whereby $\partial : \mathcal{M}_{k} \rightarrow \mathcal{M}_{k-1}$ is a linear map called [*boundary map*]{} satisfying to $\partial \circ \partial = \partial^2 =0$. We have the following terminology:
- $\mathcal{M}_{-2}=\{0\}$;
- $\mathcal{M}_{-1}=\mathbb{Z}$ is the [*accumulator module*]{}, generated by $1$, which is called [*accumulator*]{};
- $\mathcal{M}_k$ is [*the module of $k$–chains*]{}, defined as a free $\mathbb{Z}$–module generated over a set $\mathcal{C}_k = \{ C^k_j \}_{j\in J}$ of so-called [*$k$–cells*]{}. An element of $\mathcal{M}_k$ is called a [*$k$–chain*]{}, thus we write: $$\label{module_M_k}
\mathcal{M}_{k} = \left\{ C^k=\sum_{j\in J} \lambda_j C^k_j : \lambda_j \in \mathbb{Z}, C^k_j \in \mathcal{C}_k \right\}.$$
The lower index spaces of $k$–cells have special terminology:
- $\mathcal{C}_0 = \{p_j=C^0_j \}_{j\in J}$ is called the set of [*points*]{};
- $\mathcal{C}_1 = \{ \ell_j=C^1_j \}_{j\in J}$ is called the set of [*lines*]{};
- $\mathcal{C}_2 = \{v_j=C^2_j \}_{j\in J}$ is called the set of [*planes*]{}.
Using this notation, we write, for example: $$\label{module_M_0}
\mathcal{M}_{0} = \left\{ \sum_{j\in J} \lambda_j p_j : \lambda_j \in \mathbb{Z}, p_j \in \mathcal{C}_0 \right\}.$$ In conclusion, we define a script as follows.
A [*script*]{} is a complex of free modules $\mathcal{M}_k$ over ${{\mathbb Z}}$ of type: $$\begin{aligned}
\label{def:script1}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1} \overset{\partial}{\longleftarrow}\mathcal{M}_{2}
\overset{\partial} {\longleftarrow} \cdots $$ generated by the spaces of $k$–chains $\mathcal{C}_k$ together with the boundary map $\partial$ at each level. The [*dimension of the script*]{} is the largest $n$ for which $\mathcal{M}_n\neq \emptyset$. If $\mathcal{M}_n\neq \emptyset$ for all $n$, then the script is said to be an [*infinite script*]{}.
Immediate examples
------------------
\[example:addition\_Z\] Consider the $0$–dimensional script $$\begin{aligned}
\label{addition_Z_complex}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\end{aligned}$$ with $\partial p_j =1$ for all $p_j \in \mathcal{C}_0$. In this case we have $\displaystyle\partial\left( \sum_{j\in J} \lambda_j p_j \right) =
\sum_{j\in J} \lambda_j,$ so it represents the usual addition in $\mathbb{Z}$.
\[example:interval\] The following $1$–dimensional script $$\begin{aligned}
\label{interval_complex}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0} \overset{\partial}{\longleftarrow} \mathcal{M}_{1}
\end{aligned}$$ where $\mathcal{C}_0 = \{ p, q\}, \mathcal{C}_1 = \{ \ell \},$ $\partial p = \partial q =1,$ and $\partial \ell = p-q$, represents an interval.
\[example:circles\_spheres\] The $1$–dimensional script $$\begin{aligned}
\label{circle_complex}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1}
\end{aligned}$$ with $\mathcal{C}_0 = \{ p_1, p_2 \}$, $\mathcal{C}_1 = \{ \ell_1, \ell_2 \}$, $\partial p_j =1$, and $\partial \ell_j = p_1-p_2, ~j=1,2$, represents a circle. The extension of this script to $$\begin{aligned}
\label{sphere_complex}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1} \overset{\partial}{\longleftarrow} \mathcal{M}_{2}
\end{aligned}$$ with $\mathcal{C}_0, \mathcal{C}_1$ as before, $\mathcal{M}_2 = \{ v_1, v_2 \},$ and extra relations $\partial v_j = \ell_1- \ell_2, ~j=1,2,$ represents a $2$–sphere in an elementary form.
In general, the extension $$\begin{aligned}
\label{m_sphere_complex}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1} \overset{\partial}{\longleftarrow}
\cdots \overset{\partial}{\longleftarrow} \mathcal{M}_{k} \overset{\partial}{\longleftarrow}
\cdots \overset{\partial}{\longleftarrow} \mathcal{M}_{m}
\end{aligned}$$ where $\mathcal{C}_k = \{ C^k_1, C^k_2 \}$ and $\partial C^k_j = C^{k-1}_2 - C^{k-1}_1$ represents a $m$–sphere.
\[example\_simplexes\] Regular simplexes are a special case of scripts: consider the sets of $k$–cells (points, lines, etc.) as follows: $$\begin{aligned}
\mathcal{C}_0&= \{ [0], [1], \cdots, [m] \}, \\
\mathcal{C}_1&= \{ [i,j], i,j =0, \cdots, m \}, \\
\vdots \\
\mathcal{C}_k&= \{ [\alpha_0, \cdots, \alpha_k]: 0\leq \alpha_0 < \cdots < \alpha_k \leq m\}, \\
\vdots \\
\mathcal{C}_m &= \{ [0, 1, \cdots, m] \}
\end{aligned}$$ and define the boundary map by: $$\begin{aligned}
\label{simplex_boundary}
\partial [\alpha_0, \cdots, \alpha_k] = \sum_{j=0}^k (-1)^{j} [\alpha_0, \cdots, \alpha_k]_j^{\hat ~}
= \sum_{j=0}^k (-1)^{j} [\alpha_0, \cdots, \alpha_{j-1}, \alpha_{j+1}, \cdots, \alpha_k].
\end{aligned}$$ It is easy to see that the script defined this way represents a regular simplex.
The geometrical offprint of a script
------------------------------------
Let $C^k = \displaystyle\sum_{j\in J} \lambda_j C^k_j$ be a general $k$–chain. The [*support*]{} of a single $k$–cell is itself and, in general, it is denoted by $$\begin{aligned}
{{\operatorname{supp}C^k}} = \{C_j^k \,\big|\, \lambda_j \not=0 \}.
\end{aligned}$$ Moreover, we denote by ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k =$ $\partial C^k$ the so–called [*geometrical boundary*]{} of the chain $C^k.$
Therefore there are natural maps $$\begin{aligned}
\label{geo_offprint}
\{ 1 \} \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} {\mathcal{P}}(\mathcal{C}_0) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} {\mathcal{P}}(\mathcal{C}_1)
\overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} {\mathcal{P}}(\mathcal{C}_2) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow}
\cdots \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} {\mathcal{P}}(\mathcal{C}_k) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \cdots\end{aligned}$$ which represent so–to–speak the [*geometrical offprint of the script*]{}.
Subscripts
----------
Consider a script and let $$\label{Eq:1.011}
0 \longleftarrow {{\mathbb Z}}\overset{\partial^\prime}{\longleftarrow} \mathcal{M}^\prime_{0}
\overset{\partial^\prime}{\longleftarrow} \mathcal{M}_{1}^\prime \overset{\partial^\prime} {\longleftarrow}
\cdots $$ be another script for which $$\label{Eq:1.012}
\mathcal{C}_k' \subset \mathcal{C}_k, \quad k=0, 1, \ldots $$ and such that if $C^{'k}_j \in \mathcal{C}_k'$ then ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^{'k}_j \subset \mathcal{C}_{k-1}'$ and $\partial' C^{'k}_j =\partial C^{'k}_j$, for all $k=0, 1, \ldots$ Then we call this new script a [*subscript*]{} of the original script.
In particular for a $k$–cell $C^k_j$ we may consider $$\label{Eq:1.013}
\mathcal{C}_k' = \{ C^{k}_j \}, \quad \mathcal{C}_{k-1}' = {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^{k}_j, \quad \dots, \quad\mathcal{C}_{k-l}' = {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^l C^{k}_j,\quad \dots$$ then the corresponding script $$\label{Eq:1.014}
0 \overset{\partial}{\longleftarrow} \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}(\mathcal{C}^\prime_0)
\overset{\partial}{\longleftarrow} \mathcal{M}_{1}(\mathcal{C}^\prime_1)\cdots
\overset{\partial}{\longleftarrow} \{ \lambda C^k_j\,\big|\,\lambda \in \mathbb{Z}\}$$ is called the *subscript generated by* $\{ C^k_j \}.$
More general, for a subset $A \subset \mathcal{C}^k$ we may consider the subscript for which $$\label{Eq:1.015}
\mathcal{C}^\prime_k = A, \quad \mathcal{C}^\prime_{k-1} = {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}A
:= \bigcup_j {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k_j, \, C^k_j \in A, \dots, \mathcal{C}^\prime_{k-l} = {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^l A.$$ This is called the [*subscript generated by*]{} $A.$
\[Ex:1.0005\] A subscript of a symplex is called a simplicial complex. It can be generated by a subset $A$ of a $k$–dimensional subsimplexes of the overall $m$–symplex $[0,\dots, m].$
Equivalent scripts
------------------
Consider a cell $C^k_j \in \mathcal{C}_k$ and replace $\mathcal{C}_k$ by $\mathcal{C}_k^{\prime} = (\mathcal{C}_k \setminus \{ C^k_j \}) \cup
\{ C'^{\,k}_j \}, $ where we set $C'^{\,k}_j = \pm C^{k}_j$ and $\partial C'^{\,k}_j = \pm \partial C^{k}_j,$ and whenever $C^{k}_j \in {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^{k+1}_l$ and $$\partial C^{k+1}_l = \lambda_j C^k_j + \sum_{i\neq j} \lambda_i C^k_i$$ we replace $\partial C^{k+1}_l$ by $\pm \lambda_j C'^{\,k}_j +\displaystyle\sum_{i\neq j} \lambda_i C^k_i$.
Then the newly obtained script is called an [*equivalent script*]{}. Note that, by iteration, this definition includes permutation of indices as well since it just corresponds to changing the names of objects. Clearly, the ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}$–maps for equivalent scripts are essentially the same and they have the same geometrical offprint. The converse is usually not true.
Unitary scripts
---------------
A script is called [*unitary*]{} if for every $k$–cell $C_j^k,$ the boundary map $\partial C^{k}_j = \displaystyle\sum_i \lambda_i C^{k-1}_i$ only involves the values $\lambda_i = \pm 1$ (that is, whenever $\lambda_i \not=0$ ).
Given a candidate for the geometrical offprint of a unitary script $$\label{Eq:1.017}
0 \overset{\partial}{\longleftarrow} \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1} \overset{\partial}{\longleftarrow} \cdots$$ it may happen that any other unitary script with the same geometrical offprint is equivalent to this script. In those cases the script is determined by its geometrical offprint up to equivalence. This property motivates the previous definition of equivalence of scripts. In fact a unitary cell $C^k_j$ has a boundary $\partial C^k_j$ that can be seen as a surface ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k_j$ with an orientation on it. More general, we may also consider [*unitary chains*]{} $C^k = \displaystyle\sum_j \lambda_j C^k_j$, with $\lambda_j = \pm 1.$
Cycles, boundary, and homology
------------------------------
A $k$–chain $C^k \in \mathcal{M}_k(\mathcal{C}_k)$ that is [ *closed*]{}, i.e. $\partial C^k=0$, is called a $k$–[*cycle*]{}. By $\mathcal{Z}_k({\mathcal{C}_k})$ we denote the module of all $k$–cycles.
A $k$–cycle $C^k \in \mathcal{Z}_k(\mathcal{C}_k)$ is called a $k$–[*boundary*]{} if for some $C^{k+1} \in \mathcal{M}_{k+1}(\mathcal{C}_{k+1})$ we have $C^k = \partial C^{k+1}$. By $\mathcal{B}_k(\mathcal{C}_k)$ we denote the module of $k$–boundaries. The $k$–th homology space of the script is given by $$\begin{aligned}
\label{k_homology_space}
\mathcal{H}_k(\mathcal{C}_k) := \mathcal{Z}_k(\mathcal{C}_k) /
\mathcal{B}_k(\mathcal{C}_k).
\end{aligned}$$
One can also define local modules: let $\mathcal{U} \subset \mathcal{C}_k$, then $\mathcal{M}_k(\mathcal{U}), \mathcal{Z}_k(\mathcal{U}),
\mathcal{B}_k(\mathcal{U}),$ and $\mathcal{H}_k(\mathcal{U})$ denote the modules of $k$–chains, $k$–cycles, $k$–boundaries, and $k$–homology of $\mathcal{U}$, respectively.
One can also define relative homology. For that we extend the boundary ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}$ (which up to now is only defined for $k$–cells and sets of $k$–cells) to $k$–chains. For $C^k \in \mathcal{M}_k(\mathcal{C}_k)$ we set $$\label{Eq:1.018}
{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k) := {{\operatorname{supp}(}}\partial C^k) \subset \mathcal{C}_{k-1}.$$ Next, let $\mathcal{U} \subset {{\mathcal C}}_k, \mathcal{V} \subset {{\mathcal C}}_{k-1},$ then by $\mathcal{Z}_k(\mathcal{U}, \mathcal{V})$ we denote the module of $k$–chains $C^k \in \mathcal{M}_k(\mathcal{U})$ for which ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k) \subset \mathcal{V}$ or also $\partial C^k \in \mathcal{B}_{k-1}(\mathcal{V}).$
By $\mathcal{U}_{\mathcal{V}}$ we denote the subset of $k$–cells $C^k_j \in \mathcal{U}$ for which ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k_j) \subset \mathcal{V}$ and we denote by $\mathcal{B}_k(\mathcal{U}, \mathcal{V})$ the module of $k$–chains $C^k \in \mathcal{Z}_k(\mathcal{U}, \mathcal{V})$ of the form $$\label{Eq:1.019}
C^k = C'^k+C''^k, \quad C'^k \in \mathcal{B}_k(\mathcal{U}),
C''^k \in \mathcal{M}_k(\mathcal{U}_{\mathcal{V}}).$$ Clearly, also $\mathcal{M}_k(\mathcal{U}_{\mathcal{V}}) \subset
\mathcal{Z}_k(\mathcal{U}, \mathcal{V})$. By $\mathcal{H}_k(\mathcal{U}, \mathcal{V}) = \mathcal{Z}_k(\mathcal{U},
\mathcal{V}) / \mathcal{B}_k(\mathcal{U}, \mathcal{V})$ we denote the homology module of $\mathcal{U}$ relative to $\mathcal{V}.$ In this way everything is naturally defined and above all, crystal clear.
Other rings
-----------
We presented the theory of scripts over the ring $\mathbb{Z}$ of integers. Sometimes it will be useful to allow more values like the field of rational numbers $\mathbb{Q}$ (e.g. to study invertible morphisms). Moreover, one can also consider the scripts over other rings like $\mathbb{Z}/n{{\mathbb Z}}$ ($n \in \mathbb{Z}$), or polynomials.
For any script over $\mathbb{Z}$ $$0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow}
\mathcal{M}_{0} \overset{\partial}{\longleftarrow} \mathcal{M}_{1}
\overset{\partial}{\longleftarrow} \cdots$$ we can consider the script over $\mathbb{Z}/n{{\mathbb Z}}$: $$\label{Eq:1.020}
0 \longleftarrow \mathbb{Z}/n{{\mathbb Z}}\overset{\partial_n}{\longleftarrow} \Pi_n(\mathcal{M}_{0}) \overset{\partial_n}{\longleftarrow}
\Pi_n(\mathcal{M}_{1}) \overset{\partial_n}{\longleftarrow} \cdots$$ whereby $\Pi_n : \mathbb{Z} \rightarrow \mathbb{Z}/n{{\mathbb Z}}$ is the natural projection and $\partial_n = \Pi_n \circ \partial.$
In case $n=3$, $\Pi_3$ leaves unitary scripts invariant because then $\Pi_3(\partial C^k_j) = \partial C^k_j$ for every $k$–cell $C^k_j$. Moreover, every script over $\mathbb{Z}/3{{\mathbb Z}}$ is by definition unitary and $n=3$ is the lowest case for which every cell has 2 states of orientation.
We note that not all scripts over $\mathbb{Z}/n{{\mathbb Z}}$ ($n\ge 4$) are unitary. We will see later the script for the Klein bottle (\[fig:klein\]) is not unitary.
Yet one can also consider $\mathbb{Z}/2{{\mathbb Z}}$ and the projection $\Pi_2$. In this case orientability is no longer an issue and in fact every $k$–chain has the form $C^k = \displaystyle\sum_{j \in A} C^k_j,$ $A \subset \mathcal{C}_k,$ so that the map $C^k \rightarrow {{\operatorname{supp}C^k}}$ from $\mathcal{M}_k(\mathcal{C}_k)$ to $\mathcal{P}(\mathcal{C}_k)$ is bijective. In particular, for every cell $C^k_j$, its boundary $\partial C^k_j$ is mapped bijectively on ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k_j)$ and so there is a one–to–one correspondence between a $\mathbb{Z}/2{{\mathbb Z}}$–script and its geometrical offprint.
Also, for every $C^k \in \mathcal{M}_k$, the boundary $\partial C^k$ may be identified with ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k = {{\operatorname{supp}\partial C^k}}$. Moreover, $${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}({{\operatorname{supp}C^k}}) = \bigcup_j {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k_j),\quad C^k_j \in {{\operatorname{supp}C^k}}$$ and one obtains: $${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}({{\operatorname{supp}C^k}}) = \bigcup_j {{\operatorname{supp}(}}\partial C^k_j),$$ (with $C^k_j \in \mbox{supp } C^k$) which is bigger than: $${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k)={{\operatorname{supp}(}}\partial C^k)={{\operatorname{supp}}}\left(\partial\sum_j\lambda_j
C^k_j\right) ={{\operatorname{supp}}}\left(\sum_j\lambda_j \partial C^k_j\right).$$ Thus, we have the following:
${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k)\subset {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\mbox{supp } C^k)$. In general, the inclusion is strict.
Whereby, it is best to not fully identify $C^k$ with ${{\operatorname{supp}C^k}}$. However, for a unitary script over $\mathbb{Z},$ the operator $\Pi_2$ may be identified with the projection on the geometrical offprint, or skeleton; $\Pi_2$ is a kind of Röntgen image.
Clifford algebra
----------------
A way to encode simplexes is given by a Clifford algebra of the appropriate dimension as follows. Consider $m+1$ points and attach to them the $m+1$ basis elements ${{\mathbf{e}}}_0, \cdots, {{\mathbf{e}}}_m$ generating the Clifford algebra $\mathbb{R}_{m+1,0}$ with relations $$\label{Eq:1.021}
{{\mathbf{e}}}_j {{\mathbf{e}}}_k + {{\mathbf{e}}}_k {{\mathbf{e}}}_j = 2\delta_{j,k}, \quad j,k \in \{ 0, \dots, m\}.$$ Then every basis element of $\mathbb{R}_{m+1,0}$ has the form $$\label{Eq:1.022}
{{\mathbf{e}}}_A = {{\mathbf{e}}}_{j_0} \cdots {{\mathbf{e}}}_{j_k}, \quad A = \{ j_0, \dots, j_k \}, ~s.t. ~ 1\leq j_0 < \cdots < j_k \leq m.$$ We now identify basis elements with symplexes $$\label{Eq:1.023}
{{\mathbf{e}}}_A = {{\mathbf{e}}}_{j_0} \cdots {{\mathbf{e}}}_{j_k} \quad \rightarrow \quad [A]= [ j_0, \dots, j_k ].$$ Then any $k$–vector $\displaystyle\sum_{|A|=k} \lambda_A {{\mathbf{e}}}_A$ is mapped isomorphically on the $k$–chain $\displaystyle\sum_{|A|=k} \lambda_A [A] \in \mathcal{M}_k
(\mathcal{C}_k).$
Next, let ${{\mathbf{e}}}= {{\mathbf{e}}}_0+{{\mathbf{e}}}_1 + \cdots +{{\mathbf{e}}}_m;$ then $$\label{Eq:1.024}
{{\mathbf{e}}}\cdot {{\mathbf{e}}}_A := \left(\sum_{j=0}^m {{\mathbf{e}}}_{j} \right) \cdot {{\mathbf{e}}}_{A}
= [{{\mathbf{e}}}~ {{\mathbf{e}}}_A]_{k-1} \longrightarrow \partial [ A] = \partial [ j_0, \dots, j_k ].$$ This is called the Clifford algebra representation of simplicial complexes which it turns out to be very useful. Note that here $\cdot$ denotes the inner product, not the regular Clifford product and the boundary operator is well defined.
Geometrical properties of scripts
=================================
Minimization
------------
General complexes are too general for the sake of their intrinsic geometries and there are a number of elementary properties one may assume. We say that a cell $C^k_j$ is in [*minimal state*]{} if $$\partial C^k_j = \sum_{l} \lambda_j^l C^{k-1}_l$$ with $\gcd_l (\lambda_j^l) =1$ (where $\gcd_l$ is the usual greatest common divisor w.r.t. the index $l$).
\[Lm:2.001\] Every complex has a canonical minimization.
Assume we already minimized $\mathcal{C}_0, \dots,\mathcal{C}_{k-1},$ and let $C^k_j \in \mathcal{C}_k.$ If $\partial C^k_j =0$ we remove $C^k_j$ from $\mathcal{C}_k$ and also from any $\partial C^{k+1}_s$ in which it occurs. Let $g =\gcd_l (\lambda_j^l) >1$ then replace $C^k_j$ by $C^{\prime k}_j =\displaystyle \frac{1}{g}C^k_j$ in $\mathcal{C}_k$ and by $g C^{\prime k}_j$ in any $\partial C^{k+1}_s$ where it occurs. Note that this may change $C^{k+1}_s$ from minimal to non–minimal.
Without too much loss of generality one may hence assume scripts to be minimal. Unitary scripts *are already* minimal.
The skeleton problem
--------------------
Let $$0 \longleftarrow \mathbb{Z}
\overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_1
\overset{\partial}{\longleftarrow} \cdots$$ be a minimal script and let $$\label{Eq:1.025}
\cdots \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \mathcal{P}(\mathcal{C}_{k-1})
\overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \mathcal{P}(\mathcal{C}_{k}) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \cdots$$ be its skeleton (or geometrical offprint). In general there may exist other scripts with the same skeleton. This leads to the following:
**Problem:** When does it happen that a skeleton corresponds to a unique [*minimal*]{} script (up to equivalence)?
In what follows we may assume that $\mathcal{C}_k$ has no redundant cells, i.e. cells $C^k_j$ that do not appear in any ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^{k+1}_j)$. In this case the skeleton has the form $$\label{Eq:1.026}
\mathcal{P}(\mathcal{C}_m) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longrightarrow} \mathcal{P}({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\mathcal{C}_{m}))
= \mathcal{P}(\mathcal{C}_{m-1}) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longrightarrow} \mathcal{P}({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^2(\mathcal{C}_{m}))
= \mathcal{P}(\mathcal{C}_{m-2}){\longrightarrow} \cdots$$
The notion of tightness provides an answer to this problem.
Tight scripts: definitions
--------------------------
Let $\mathcal{U} \subset \mathcal{C}_k$; then $\mathcal{U}$ is called [*set tight*]{} and we write [*$s$–tight*]{} if $\mathcal{Z}_k(\mathcal{U})$ is generated by a single cycle $C^k$. By definition such a cycle will be minimal.
A cycle $C^k$ is called [*cycle tight*]{} and we write [ *$c$–tight* ]{}if ${{\operatorname{supp}C^k}}$ is $s$–tight and $C^k$ also generates $\mathcal{Z}_k({{\operatorname{supp}C^k}})$. In this case $C^k$ is minimal and ${{\operatorname{supp}C^k}}$ is $s$–tight.
A single cell $C^k_j$ is called [*tight*]{} if $\mathcal{Z}_k({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k_j)$ is $s$–tight and generated by $\partial C^k_j$. The interested reader will see that this means that $C^k_j$ is minimal.
A script is [*tight*]{} if and only if each of its cells is tight.
Let $\mathcal{U} \subset \mathcal{C}_k$ and $\mathcal{V} \subset \mathcal{C}_{k-1}$ then $\mathcal{U}$ is [*$s$–tight relative*]{} to $\mathcal{V}$ if $\mathcal{Z}_k(\mathcal{U}, \mathcal{V})$ is generated by a single chain $C^k.$
A chain $C^k$ is called [*$c$–tight* ]{} if $C^k$ generates $\mathcal{Z}_k({{\operatorname{supp}C^k}}, {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k))$ i.e. ${{\operatorname{supp}C^k}}$ is $s$–tight relative to ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k)$ and $C^k$ is minimal.
We use the same notation in the two definitions since in the case where a chain $C^k$ is a cycle we have that ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k)=\emptyset$ and the definitions agree.
Elementary properties of tight scripts
--------------------------------------
First note that in a minimal script we may assume that for every point $p \in \mathcal{C}_0, \partial p = 1$ so that $\partial : \mathcal{M}_0 \rightarrow \mathbb{Z}$ corresponds to integration (summation).
We prove the following structure theorem for cells of dimension $1$ in tight scripts:
\[Lm:2.002\] In a tight script every line $\ell \in \mathcal{C}_1$ may be interpreted as an oriented line from a point $p$ to another point $q$, i.e. $\partial \ell = q-p, ~p, q \in \mathcal{C}_0.$
The case $\partial \ell =0$ is pathological and the case ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}\ell = \{ p \}$ does not occur since $\mathcal{Z}_0(\{ p \}) =0.$ Also in case ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}\ell = \{p, q, r, \ldots \},$ $\mathcal{Z}_0({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}\ell)$ has at least 2 generators $r-q$ and $q-p.$ Therefore we must have that ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}\ell = \{p, q \},$ where $p, q \in \mathcal{C}_0, p\not=q,$ and $\mathcal{Z}_0(\{p, q \})$ is obviously generated by $q-p.$
In the case of $2-$cells, we first define the notion of [*polygon*]{}:
\[def:2.001\] Let $\ell_1, \cdots, \ell_n$ be $n$ distinct lines for which $\partial \ell_j = p_j-p_{j-1}, j=1, \dots, n-1, \partial \ell_n =
p_0-p_{n-1},$ for some set $\{p_0, p_1, \dots, p_{n-1} \} \subset \mathcal{P}_0$ of distinct points. Then the cycle $\ell_1+\ell_2+\cdots+\ell_n$ is called an [*$n-$polygon*]{}, $n\geq 2.$
The following structure theorem for $2-$cells in tight scripts follows:
\[Th:2.001\] Let $v \in \mathcal{C}_2$ be tight $2-$cell; then there exists $n\ge 2$ such that $\partial v = \ell_1+\ell_2+\cdots+\ell_n$ is an $n-$polygon.
Pick $\ell_1 \in {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(v),$ with $\partial \ell_1 = p_1-p_0.$ Then there exist a point $p_2\neq p_1$ and a line $\ell_2 \not= \ell_1$ for which $\partial \ell_2 = p_2-p_1$ (otherwise, we would have $p_1 \in \mbox{supp }(\partial\partial v) ).$
If $p_2=p_0$ we have a $2-$gon inside ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(v).$ If $p_2\neq p_0$ then $p_2 \notin \{ p_0, p_1 \}$ and there exist $p_3\neq p_2$ and $\ell_3$ for which $\partial \ell_3 = p_3-p_2$, $\ell_3\neq
\ell_2$. We also have that $\ell_3\neq\ell_1$ since $p_2\notin\{p_0,p_1\}$. Now, if $p_3 \in \{ p_0, p_1 \}$ the tightness condition requires that $p_3=p_0$ (otherwise $\ell_2$ and $\ell_3$ will form a $2-$gon). In this case $\{ \ell_1, \ell_2, \ell_3 \}$ is a $3-$gon.
If $p_3 \notin \{ p_0, p_1, p_2 \}$ we repeat the process and there exist $p_4\neq p_3$ and $\ell_4 \notin \{ \ell_1, \ell_2, \ell_3 \}$ with $\partial \ell_4 = p_4-p_3,$ and the proof follows inductively.
After finitely many steps we create a polygon $\ell'_1+\ell'_2+\cdots+\ell'_n$ inside ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(v).$ Due to tightness, $\ell'_1+\ell'_2+\cdots+\ell'_n$ generates $\mathcal{Z}_1({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(v))$ or $=\pm \partial v.$
\[Cor:2.001\] A tight $2-$dimensional script is always unitary.
Following the previous theorem, any two cell will have an $n-$polygon as boundary. Therefore the script is unitary.
Tight scripts provide a solution to the skeleton problem (\[Eq:1.025\]):
\[Th:2.002\] Let $$\label{Eq:1.0027}
0 \overset{\partial}{\longleftarrow} \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0} \overset{\partial}{\longleftarrow} \cdots$$ be a tight script with skeleton: $$\label{Eq:1.0028}
\cdots \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \mathcal{P}(\mathcal{C}_{k-1})
\overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \mathcal{P}(\mathcal{C}_{k}) \overset{{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}}{\longleftarrow} \cdots .$$ Then any minimal script with the same skeleton is equivalent to the original script.
This clearly holds for $0 \overset{\partial}{\longleftarrow} \mathbb{Z}
\overset{\partial}{\longleftarrow} \mathcal{M}_{0}.$ Assume the property for $$0 \overset{\partial}{\longleftarrow} \mathbb{Z}
\overset{\partial}{\longleftarrow}
\mathcal{M}_{0}\overset{\partial}{\longleftarrow} \cdots
\overset{\partial}{\longleftarrow} \mathcal{M}_{k-1}$$ and let $C^k_j \in \mathcal{C}_k.$ Since the script is tight, we have that $\mathcal{Z}_{k-1}({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k_j)$ has one generator (up to sign). Let us call this generator $\sum_l \lambda_l C^{k-1}_l,$ then we can choose $\partial C^k_j = \lambda \sum_l \lambda_l C^{k-1}_l$ and that fixes $\mathcal{M}_{k-1} \overset{\partial}{\longleftarrow}
\mathcal{M}_{k}$ because $\lambda=\pm1$ when $\partial C^k_j$ is minimal.
We expect the converse to be true as well, we leave the proof to the interested student.
Tight scripts also solve the [*assignment problem*]{}. Suppose given $$0 \overset{\partial}{\longleftarrow} \mathbb{Z}
\overset{\partial}{\longleftarrow}
\mathcal{M}_{0}\overset{\partial}{\longleftarrow} \cdots
\overset{\partial}{\longleftarrow} \mathcal{M}_{k-1}$$ and for $C^k_j$ we also know ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C^k_j = \{ C^{k-1}_l, \, \text{some} \,\, l's \}.$ How to actually find the coefficients $\lambda_l$ for which $\partial C^k_j$ eventually equals $\sum_l \lambda_l C^{k-1}_l$?
First of all, one $\lambda_l$ may be freely chosen. Next, one has the equation $0 = \partial^2 C^k_j = \sum_l \lambda_l \partial C^{k-1}_l$ which for a tight cell has a unique solution up to a constant. As we need a cell to be minimal, the constant is $\pm 1.$ Therefore the solution is unique up to sign hence the script obtained is unique up to equivalence.
\[Def:2.002\] Let $\ell_1, \cdots, \ell_n$ be $n$ distinct lines for which $\partial \ell_1 = p_1-p, \partial \ell_j = p_j-p_{j-1}, j=2,
\cdots, n-1, \partial \ell_n = q-p_{n-1},$ whereby $\{p, p_0, p_1, \cdots, p_{n-1}, q \} \subset \mathcal{P}_0$ are distinct points. Then the chain $\ell_1+\ell_2+\dots+\ell_n$ is called a [*simple curve*]{} from $p$ to $q$ with length $n.$
\[Th:2.003\] Let $\ell$ be a tight one-chain of length $n$ inside a tight script; then $\ell$ is either a cycle or a simple curve of same length $n$ between two points.
The chain $\ell$ together with $\mbox{supp } \ell$ and ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\mbox{supp } \ell)$ defines a $1-$dimensional graph with lines in $\mbox{supp } \ell$ and points in ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\mbox{supp } \ell).$ Every line connects 2 points. Since the script is tight, this graph must be connected or else $\mathcal{Z}_1(\mbox{supp } \ell, {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell))$ wold have more than one generator.
If $\ell$ is not a cycle, then the graph contains no loops, so it is actually a tree.
Finally, should ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell)$ have 3 points $p, q, r$ or more, then $p$ can be connected to $q$ by a (simple) curve $\ell_1,$ and $q$ to $r$ by another curve $\ell_2.$ Then the curves $\ell_1, \ell_2 \in \mathcal{Z}_1(\mbox{supp } \ell, {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell))$ are different, contradicting the tightness of $\ell$. Hence ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell)$ can have at most two points ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell) = \{p, q \} (p\not=q)$ and since ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell)$ is connected there exists a simple curve from $p$ to $q$ inside the tree. This would be the single generator of $\mathcal{Z}_1(\mbox{supp } \ell, {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\ell)),$ therefore the tree is this simple curve.
CW-complexes as Scripts
-----------------------
A CW-complex is a Hausdorff space $X$ together with a partition of $X$ into open cells (of varying dimension) that satisfies two properties:
(i) for each $n-$ dimensional open cell $C$ there is a continuous map $f$ from the closed ball $\mathbb{B}\subset {\mathbb{R}^n}$ to $X$ such that
1. the restriction of $f$ to $\overset{\circ}{\mathbb{B}}$ is a homeomorphism onto cell $C;$
2. the image of the sphere $\partial \mathbb{B}$ is equal to the union of finitely many cells of dimension less than $n.$
(ii) A CW-complex is [*regular*]{} if the map $f$ is a homeomorphism on the closed ball.
Next let $\mathcal{C}_k$ be the set of all $k-$dimensional cells in a regular CW-complex; then for each $C^k_j \in \mathcal{C}_k$ we put ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k_j) = \{ C^{k-1}_l : C^{k-1}_l \subset f(\partial \mathbb{B})
\};$ we must have that $$f(\partial \mathbb{B}) \subset {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k_j) \cup {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^2 (C^k_j) \cup
\cdots \cup {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^k (C^k_j).$$
In this way we obtain a skeleton in which every cell is basically a $k-$dimensional polyhedron. Now, a polyhedron is always the skeleton of a tight unitary script therefore the skeleton of a CW-complex is the skeleton of a tight unitary script. This means that this tight script is the only minimal script attached to this skeleton. So we have proven:
\[Th:2.004\] To a given regular CW-complex corresponds a unique tight unitary script.
The converse is not true as there exist tight unitary scripts that do not correspond to a CW-complex. Clearly, every 2D-tight script is CW-complex.
A $2-$torus
-----------
We have $$\begin{gathered}
\mathcal{C}_{0}= \{ p_0, p_1, p_2, p_3 \}, \quad \partial p_j =1, j=0, 1, 2, 3. \\
\mathcal{C}_{1} = \{ \ell_1, \ell_2, \ell_3, \ell_4, \ell_5, \ell_6, \ell_7, \ell_8 \}, \\
\partial \ell_1 = p_1-p_0, \quad \partial \ell_2 = p_0 -p_1, \quad \partial \ell_3 = p_2 -p_0, \quad \partial \ell_4 = p_0 -p_2 \\
\partial \ell_5 = p_3-p_2, \quad \partial \ell_6 = p_2 -p_3, \quad \partial \ell_7 = p_3 -p_1, \quad \partial \ell_8 = p_1 -p_3 \\
\mathcal{C}_{2} = \{ v_1, v_2, v_3, v_4 \}, \\
\partial v_1 = \ell_5+\ell_8-\ell_1-\ell_4, \quad \partial v_2 = \ell_6+\ell_4-\ell_2-\ell_8, \\
\partial v_3 = \ell_1+\ell_7-\ell_5-\ell_3, \quad \partial v_4 = \ell_2+\ell_3-\ell_6-\ell_7, \\
\mathcal{C}_{3}= \{ C \}, \quad \partial C =v_1+v_2+v_3+v_4.\end{gathered}$$
Hence $\partial C =v_1+v_2+v_3+v_4$ is a unitary and tight script but is not the image of a sphere so its no CW-complex.
\[fig:torus\]
A Klein bottle
--------------
We take $$\begin{gathered}
\mathcal{C}_{0}= \{ p_0, p_1, p_2, p_3 \}, \quad \partial p_j =1, j=0, 1, 2, 3. \\
\mathcal{C}_{1} = \{ \ell_1, \ell_2, \ell_3, \ell_4, \ell_5, \ell_6, \ell_7, \ell_8 \}, \\
\partial \ell_1 = p_1-p_0, \quad \partial \ell_2 = p_0 -p_1, \quad \partial \ell_3 = p_2 -p_0, \quad \partial \ell_4 = p_0 -p_2 \\
\partial \ell_5 = p_3-p_1, \quad \partial \ell_6 = p_1 -p_3, \quad \partial \ell_7 = p_3 -p_2, \quad \partial \ell_8 = p_2 -p_3 \\
\mathcal{C}_{2} = \{ v_1, v_2, v_3, v_4 \}, \\
\partial v_1 = \ell_5+\ell_8-\ell_2-\ell_3, \quad \partial v_2 = \ell_6-\ell_1-\ell_4-\ell_8, \\
\partial v_3 = -\ell_1+\ell_3+\ell_7-\ell_5, \quad \partial v_4 =
\ell_4-\ell_2-\ell_6-\ell_7.\end{gathered}$$
\[fig:klein\]
Notice that $$\partial v_1 + \partial v_2 + \partial v_3 + \partial v_4 = -2\ell_1 -2\ell_2.$$ So if we add a new cell $v_5, ~\partial v_5 = \ell_1 + \ell_2,$ to the script we obtain a cycle for the “extended Klein bottle”: $$\partial v_1 + \partial v_2 + \partial v_3 + \partial v_4 + 2 \partial v_5 = 0.$$ Hence, $v_1 + v_2 + v_3 + v_4 + 2 v_5 $ is a tight cycle but it is not unitary.
A projective plane
------------------
We take $$\begin{gathered}
\mathcal{C}_{0}= \{ p_1, p_2, p_3 \}, \quad \partial p_j =1, j=1, 2, 3. \\
\mathcal{C}_{1} = \{ \ell_1, \ell_2, \ell_3, \ell_4, \ell_5, \ell_6 \}, \\
\partial \ell_1 = p_2-p_1, \quad \partial \ell_2 = p_1 -p_2, \quad \partial \ell_3 = p_3 -p_1, \quad \partial \ell_4 = p_1 -p_3 \\
\partial \ell_5 = p_3-p_2, \quad \partial \ell_6 = p_2 -p_3, \\
\mathcal{C}_{2} = \{ v_1, v_2, v_3, v_4 \}, \\
\partial v_1 = -\ell_2+\ell_5-\ell_3, \quad \partial v_2 = -\ell_1-\ell_4-\ell_5, \\
\partial v_3 = -\ell_1+\ell_3+\ell_6, \quad \partial v_4 =
\ell_4-\ell_2-\ell_6.\end{gathered}$$
\[fig:proj\]
Also here $\partial v_1 + \partial v_2 + \partial v_3 + \partial v_4 = -2\ell_1
-2\ell_2$ and we can add a fifth cell $v_5, ~\partial v_5 = \ell_1 + \ell_2.$ Then we have a tight $2-$cycle which is not unitary: $v_1 + v_2 + v_3 + v_4 + 2 v_5 $ with $$\partial v_1 + \partial v_2 + \partial v_3 + \partial v_4 + 2 \partial v_5 = 0.$$
**Remarks:**
i) Scripts can in fact always be made unitary by attaching extra cells. In the above example we can make an extra cell $v_6, ~\partial v_6 = \ell_1 + \ell_2.$ Then we get a unitary cycle $\partial v_1 + \partial v_2 + \partial v_3 + \partial v_4 +
\partial v_5 + \partial v_6 = 0$ but that cycle is no longer tight. Also, using a Gauss method one may extend cells to make things tight, but that would generally not be unitary except on the level of the 2-cycles (boundaries of 3-cells).
Hence, tight and unitary scripts are very special and deserve a new name. We define a tight and unitary script to be a [*geometrical script*]{} or *geoscript*.
ii) This section is not nearly complete and there are many interesting problems such as
**Problem:** let $C$ be a tight chain; can one prove that $\partial C$ is also tight? In particular, if $C$ is a tight $2-$chain, can we prove that $\partial C$ is a polygon?
The answer to both questions is an obvious no, as the union of two disjoint circles realized as the boundary of a script forming a cylinder seen as a single $2-$dimensional tight chain is a counterexample to both.
This kind of problems will turn out quite important later on, in particular when we study extensions. But first we need more constructive methods.
**Remark:** A cell $C^k_j$ is tight iff ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C^k_j)$ has trivial homology. Tightness hence means that the local homology at the level of cells is trivial. For manifolds this would correspond to a version of Poincaré’s lemma which means that each point has a neighborhood with trivial homology.
Basic Operations on Scripts
===========================
An essential part of working with scripts is the possibility to modify them. Here we discuss three different types of operations on scripts, cleaning or removing operations, operations of creation, and operations of identification.
Cleaning Operations
-------------------
There are two types of cleaning operators:
1. Floating Cells
2. Free arcs (or domes)
Here are the definitions of the two:
A [*floating cell*]{} is a cell for which $\partial C_j^k=0$. If it appears in $\partial C_l^{k+1}$ then $ C_l^{k+1}$ is usually not tight and one can in fact remove $C_j^k$ from the script by simple cancellation and replace it by $0$ if it appears in $C_l^{k+1}$.
A [*free arc*]{} is a cell $C_j^k$ that does not appear in any boundary $\partial C_l^{k+1}$. It may thus be removed from the script by simple cancellation.
We will see from Chapter 5 that these operations are each other’s dual. However, removing floating cells may be essential on the way to a tight script while removing free arcs is not and it could make the situation worse, therefore we will not go through this process automatically.
Removing a free arc does not affect tightness, as tightness works from a higher dimensional cell to a lower dimensional cell, therefore, as a free arc does not appear in any boundary, removing it will not affect tightness.
Operations of Creation
----------------------
1. Creating New Cells
2. Pulling cells together
One can [*create new cells*]{} through the following method. Let $\mathcal{C}^{k-1}$ be a $k-1$ cycle, then we may add a new cell $C_j^k$ to the script for which $\partial C_j^k=\mathcal{C}^{k-1}$. It is as if one creates an arc or a dome above a cycle. Creating new cells is the inverse of removing free arcs.
The action of [*pulling cells together*]{} is the dual of the above and it corresponds to introducing a new cell $C_j^{k-1}$ with $\partial C_j^{k-1}=0$ that appears in a number of $\partial C_l^k$ whereby $C_l^k$ form a cocycle (see Chapter 5). As $\partial C_j^{k-1}=0$ this operation worsens tightness and it is usually not done.
Operations of Identification
----------------------------
1. Glueing cells together
2. Melting cells together
3. Cutting cells open
4. Expanding cells
For [*glueing cells together*]{} let $C_1^k$ and $C_2^k$ be cells for which $\partial C_1^k= \partial C_2^k$ then we may add the constraint $ C_1^k= C_2^k$ to the script. This equation can be solved by replacing $C_2^k$ by $C_1^k$ wherever it occurs in $\partial C_l^{k+1}$ for some $C_l^{k+1}$ and to remove $C_2^k$ from the script.
This operation may turn $C_l^{k+1}$ into a floating cell, so a cleaning may follow a glueing operation.
Notice that points always satisfy $\partial p_1=1=\partial p_2$, so glueing together points is “free”. Once this is done one can glue together lines, then planes, etc., as one likes. Also, if one has the relations $\partial \lambda C_1^k=\partial \mu C_2^k$, one may glue $\lambda C_1^k=\mu C_2^k$.
The process of [*melting cells together*]{} is the dual of glueing. For example, let $C_1^k$ and $C_2^k$ be $k-$cells such that, if they do appear in the script, they only appear inside $\partial C_l^{k+1}$ as a sum $C_1^k+C_2^k$ or as a linear combination $\lambda C_1^k+\mu C_2^k$. Then we create a new cell $C_0^k$ and add the equations $C_0^k=C_1^k+C_2^k$ (or $C_0^k=\lambda C_1^k+\mu C_2^k$).
This equation can be solved by replacing $C_1^k+C_2^k$ by $C_0^k$ in every $\partial C_l^{k+1}$ whenever it occurs. After this operation $C_1^k$ and $C_2^k$ become free cells and they can be removed.
As a warning, even when completing this operation there may appear free cells in $\partial C_1^k$ or $\partial C_2^k$ that one may consider to remove them as well.
Notice that for the highest dimension $k=m$, the operation of melting cells is free (and optional); after completing this operation one may melt lower dimensional cells.
The third operation of identification is [*cutting cells open*]{} which is the inverse of glueing cells together and it involves the creation of a new cell $C_2^k$ with $\partial C_2^k=\partial C_1^k$, together with a possible replacement of $C_1^k$ by $C_2^k$ inside $\partial C_l^{k+1}$. Here one starts with the higher dimensions and then works down through the dimensions of the cells.
The last operation of identification, the expanding of cells is defined as the opposite of melting.
In conclusion, hereby one replaces a cell $C_0^k$ by a chain $$C_1^k+\dots + C_l^k$$ such that $\partial C_0^k=\partial C_1^k+\dots + \partial
C_l^k$. Whenever $C_0^k$ appears in $\partial C_1^{k+1}$, one replaces it as well.
To enable an expansion one may have to create extra lower dimensional objects that may be needed to build the boundaries $\partial C_1^k,\dots,\partial C_l^k$, starting with extra points to create zero lines, etc... An expansion is called free if $\cup {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_j^k)\setminus {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_0^k)$, $\cup {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^2(C_j^k)\setminus {\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}^2(C_0^k)$, etc. consist entirely of new cells.
Further details we leave as an exercise to the reader.
Examples of basic operations on scripts
---------------------------------------
We start from the rectangle in Figure \[fig:moebius\] and we have:
\[fig:moebius\]
$$\mathcal{C}_0=\{p_1,p_2, p_3, p_4, p_5, p_6\},$$ with $\partial p_j=1$, $$\mathcal{C}_1=\{l_1,l_2, l_3, l_4, l_5, l_6, l_7\},$$ with $\partial l_1=p_2-p_1$, $\partial l_2=p_4-p_3$, $\partial l_3=p_3-p_1$, $\partial l_4=p_5-p_3$, $\partial l_5=p_4-p_2$, $\partial l_6=p_6-p_4$, and $\partial l_7=p_6-p_5$, $$\mathcal{C}_2=\{v_1,v_2\},$$ with $\partial v_1=l_3+l_2-l_5-l_1$ and $\partial v_2=l_4+l_7-l_6-l_2$.
Next we glue points $p_5=p_2$ and $p_6=p_4$ thus removing $p_6$ and $p_5$. This effectuates the following changes in the script: $\partial l_4=p_2-p_3$, $\partial l_6=p_1-p_4$, $\partial l_7=p_1-p_2=-\partial l_1$.
Next we glue line $l_7=-l_1$, which works since $\partial l_7=-\partial l_1$ and this makes the following changes in the script: $$\partial v_2=l_4-l_1-l_6-l_2,$$ leading to Moebius strip.
Reconsider the Moebius Strip with: $\partial l_1=p_2-p_1$, $\partial l_2=p_4-p_3$, $\partial l_3=p_3-p_1$, $\partial l_4=p_2-p_3$, $\partial l_5=p_4-p_2$, $\partial l_6=p_1-p_4$, and with: $\partial v_1=l_3+l_2-l_5-l_1$ and $\partial v_2=l_4-l_1-l_6-l_2$.
\[fig:projM\]
Attach a new cell $v_3$ to the boundary of Moebius Script ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}v_3=\{l_3, l_4, l_5, l_6\}$. Tightness leads to: $$\partial v_3=l_5+l_6+l_3+l_4$$ and $$\partial v_1+\partial v_2+ \partial v_3=2(l_3+l_4-l_1)$$ so we obtain a projective plane.
We start with two Moebius strips (see Figure \[fig:2moebius\]).
\[fig:2moebius\]
Equations: exercise.
Next we glue points $p_6=p_1$, $p_5=p_2$, $p_7=p_4$, and $p_8=p_3$. This gives rise to the following new relations: $$\partial l_{11}=p_3-p_1, \qquad \partial l_{12}=p_2-p_3,\qquad
\partial l_9=p_4-p_2,\qquad \partial l_{10}=p_1-p_4.$$ This allows us to glue further: $$l_{11}=l_3,\qquad l_{12}=l_5,\qquad l_9=l_4,\qquad l_{10}=l_6.$$ This leads to the following script in Figure \[fig:klein2\].
\[fig:klein2\]
Next if we glue points $p_6=p_1$, $p_5=p_2$, $p_7=p_4$, and $p_8=p_3$. This also gives rise to the following new relations: $$\partial l_{1}=p_2-p_1, \qquad \partial l_{2}=p_4-p_3,\qquad
\partial l_7=p_1-p_2,\qquad \partial l_{8}=p_3-p_4,$$ $$\partial l_{3}=p_3-p_1, \qquad \partial l_{4}=p_4-p_2,\qquad
\partial l_5=p_2-p_3,\qquad \partial l_{6}=p_1-p_4,$$
$$\partial v_{1}=l_3+l_2-l_4-l_1, \qquad \partial
v_{2}=l_5-l_1-l_6-l_2,\qquad \partial v_3=l_4+l_8-l_3-l_7,\qquad
\partial v_{4}=l_6-l_7-l_5-l_8.$$ This script is not equivalent to the Klein bottle script introduced earlier, but they do have a common refinement (expansion). To that end we first introduce new lines $l_4'$ and $l_3'$ with: $$\partial l_4'=p_3-p_2,\qquad \partial l_3'=p_4-p_1.$$
\[fig:klein\_refinement\]
Then we consider the expansions: $$v_1=v_{11}+v_{12},\qquad v_3=v_{31}+v_{32},$$ $$\partial v_{11}=l_3-l_4'-l_1,\qquad \partial v_{12}=l_4'+l_2-l_4,$$ $$\partial v_{31}=l_4-l_3'-l_7,\qquad \partial v_{32}=l_3'+l_8-l_3.$$ Now we melt $v_{11}$ to $v_{32}$ and $v_{31}$ to $v_{12}$, i.e.: $$v_1'=v_{11}+v_{32}, \qquad v_3'=v_{31}+v_{12}.$$ We then get rid of $l_3$ and $l_4$: $$\partial v_1'=l_3'+l_8-l_4-l_1,\qquad \partial v_3'=l_4'+l_2-l_3'-l_7.$$ This leads to the new script:
\[fig:klein3\]
This is equivalent (up to the names, of course) to the first Klein script introduced earlier. For CW-complexes topology is available and the two Klein bottles are topologically equivalent, but not as scripts.
The Thomson addition of two projective planes is known to be a Klein bottle. It is obtained by deleting a disk from each projective plane and glueing the edges together. Now, removing a disk from a projective plane gives a Moebius strip. So to carry out the Thomson addition of projective planes we, in fact, have to glue together two Moebius strips.
3D World with 2D–disk portal
----------------------------
The main idea here is to take two copies (top and bottom) of the $\mathbb{R}^3\setminus \mathbb{D}$ where $\mathbb{D}$ is the two–dimensional disk. These two copies are glued together by two disks at the portal and two hemispheres at infinity.
\[fig:2Dportal\]
The elements are:
- The points at infinity $pi_1, pi_2$ and the points at the portal $p_1, p_2$
- The lines at infinity $li_1, li_2$ and the lines at the portal $lp_1, lp_2$
- The lines for the top-space $lt_1, lt_2$ and for the bottom space $lb_1, lb_2$
- The planes at infinity $vi_1, vi_2$ and the planes at the portal $vp_1, vp_2$
- The planes for the top-space $vt_1, vt_2$ and for the bottom space $vb_1, vb_2$
- $3-$D worlds for the top space $wt_1, wt_2$ and for the bottom space $wb_1, wb_2$.
Here is the script (world equations) for the lines: $$\partial li_1=pi_2-pi_1, \qquad \partial li_2=pi_2-pi_1, \qquad
\partial lp_1=p_2-p_1,\partial lp_2=p_2-p_1,$$
$$\partial lt_1=p_1-pi_1, \qquad \partial lt_2=pi_2-p_2, \qquad \partial
lb_1=\partial lt_1,\partial lb_2=\partial lt_2.$$
Here is the script for the planes: $$\partial vi_1=li_2-li_1, \qquad \partial vi_2=li_2-li_1,$$
$$\partial vp_1=lp_2-lp_1=\partial vp_2,$$
$$\partial vt_1=lt_1+lp_1+lt_2-li_1,\qquad \partial vt_2=li_2-lt_2-lp_2-lt_1,$$
$$\partial vb_1=lb_1+lp_1+lb_2-li_1,\qquad \partial vb_2=li_2-lb_2-lp_2-lb_1.$$
Here are the scripts for the $3-$D worlds in this case: $$\partial wt_1=vp_1+vt_1+vt_2-vi_1,\qquad \partial wt_2=vi_2-vp_2-vt_1-vt_2,$$
$$\partial wb_1=vp_2+vb_1+vb_2-vi_1,\qquad \partial wb_2=vi_2-vp_1-vb_1-vb_2.$$
Closed portal equations (open at $\infty$, closed portal and open at infinity): $$\partial wt_1+\partial wt_2-\partial wb_1-\partial wb_2=2(vp_1-vp_2)$$
Open portal equations: $$\partial wt_1+\partial wt_2+\partial wb_1+\partial wb_2=2(vi_2-vi_1).$$
“Infinity” is like a second portal.
[**PORTAL DISCONNECTION:**]{}
$$\partial vp_3=\partial vp_4=\partial vp_1=\partial vp_2$$
$$\partial wt_2\longrightarrow vi_2-vp_3-vt_1-vt_2$$
$$\partial wb_2\longrightarrow vi_2-vp_4-vb_1-vb_2$$
[**THIS IS CUTTING ALONG THE PORTAL:**]{}
Re-glueing $vp_1=vp_3,\qquad vp_4=vp_2$, then no more portal.
The portal is an example of a $3-$D geometry that is a non-orientable $3-$manifold that generalizes the Klein bottle.
Metrics, Duals of Scripts, Dirac Operators
==========================================
In this section we will define the notion of the dual of a script, which may not be a script in general. This will also allow us to introduce a Dirac operator and monogenic scripts.
Metrics
-------
\[definition metric\] For any script we can define the following metric, called the [*Kronecker metric*]{}: $${\left\langle C^k_i, C^l_j \right\rangle } = \delta_{k,l} \delta_{i,j};$$ where $C^k_i\in\mathcal{C}_k$ and $C^l_i\in\mathcal{C}_l$.
This extends through linearity to an inner product on the modules of chains. Chains in modules of different dimensions are orthogonal.
This represents the most canonical example of an inner product. More general ones can be defined when needed, however this metric allows an easy and straight-forward way of realising the notion of script duality which follows.
Duality
-------
Here we will describe the notion of a dual of a script and start with the definition of the dual of the $\partial$ operator with respect to the metric above. This dual of the $\partial$ operator will be denoted by $d$. Just as in the classical case and, via the Stokes formula, the dual boundary operator becomes:
The [*dual boundary operator*]{} $d:\mathcal{C}_k\to\mathcal{C}_{k+1}$ is given by: $${\left\langle d C^k_i, C^{k+1}_j \right\rangle }={\left\langle C^k_i,\partial C^{k+1}_j \right\rangle }.$$
Since $\partial^2=0$ it is easy to see that $d^2=0$ as well, where $d^2:\mathcal{C}_k\to\mathcal{C}_{k+2}.$
As expected, the dual boundary operator will take the accumulator into a sum of points, a point into a sum of lines, and so forth.
For a script: $$\begin{aligned}
0 \longleftarrow \mathbb{Z} \overset{\partial}{\longleftarrow} \mathcal{M}_{0}
\overset{\partial}{\longleftarrow} \mathcal{M}_{1} \overset{\partial}{\longleftarrow}\mathcal{M}_{2}
\overset{\partial} {\longleftarrow} \cdots \overset{\partial} {\longleftarrow} \mathcal{M}_{k}\end{aligned}$$
we now have, using the $d$ boundary operator above: $$\begin{aligned}
0 \longrightarrow \mathbb{Z} \overset{d}{\longrightarrow} \mathcal{M}_{0}
\overset{d}{\longrightarrow} \mathcal{M}_{1} \overset{d}{\longrightarrow}\mathcal{M}_{2}
\overset{d} {\longrightarrow} \cdots \overset{d} {\longrightarrow} \mathcal{M}_{k}=\mathbb{Z}^n,\end{aligned}$$ where n is the number of connected components of $\mathcal{M}_{k}$.
The [*dual of the script*]{} is given by : $$\begin{aligned}
\label{script-dual}
\mathbb{Z}^n \overset{\partial'}{\longleftarrow} \mathcal{M'}_{0}
\overset{\partial'}{\longleftarrow} \mathcal{M'}_{1} \overset{\partial'}{\longleftarrow}\mathcal{M'}_{2}
\overset{\partial'} {\longleftarrow} \cdots \overset{\partial'} {\longleftarrow}\mathcal{M'}_{k}=\mathbb{Z}.\end{aligned}$$ where $\partial'=d$ and $\mathcal{M'}_{l}=\mathcal{M}_{k-l-1}$.
The dual of a script will not, in general, be a script by itself. For example, one must remove free arcs as they are in the kernel of the $d$ operator and $\mathcal{M}_{k}$ shuld be generated by a single cell so that it becomes the accumulator of the dual script $\mathcal{M'}_{0}$.
We will show that the dual of a script is also a script if and only if the original script is orientable. In this case we have that the resulting dual script will be orientable as well, since the dual of the accumulator for each connected component will be the equivalent of a “volume cell” in the dual.
\[lemma unitary script\] The dual of a unitary script is unitary.
For each $C_i^l\in \mathcal{C}_l$ with $l\neq m $, where $m$ is the dimension of the script, we can determine $d C_i^l$ by calculating the inner product for each $C_j^{l+1}$
$$\begin{aligned}
\langle d C_i^l, C_j^{l+1} \rangle &= \langle C_i^l, \partial C_j^{l+1} \rangle\\
&= \sum_{m} \lambda_m^{(j)} \langle C_i^l, C_m^{l} \rangle\\
&= \lambda_i^{(j)}\end{aligned}$$
Hence $d C_i^l = \sum_{j} \lambda_{i}^{(j)} C_{j}^{l+1}$.
We have
$$0 \leftarrow \mathbb{Z} \overset{\partial}{\leftarrow} \mathcal{M}_0$$
with $\partial p_j = 1$ for all $p_j\in \mathcal{C}_0$.\
The dual of this script is given by:
$$0 \leftarrow \mathcal{M}_0 \overset{d}{\leftarrow} \mathbb{Z}$$
where $d 1 = \sum_{j} p_j$ and $dp_j = 0$. The dual is a script if and only if $\mathcal{C}_0 = \{p\}$ and thus we have $d 1 = p$, $dp = 0$.
The following script
$$0 \leftarrow \mathbb{Z} \overset{\partial}{\leftarrow} \mathcal{M}_0 \overset{\partial}{\leftarrow} \mathcal{M}_1$$
where $\mathcal{C}_0 = \{p,q\}$, $\mathcal{C}_1 = \{\ell\}$, $\partial p = \partial q = 1$ and $\partial \ell = p-q$, represents an interval.\
The dual of this script is given by
$$0 \leftarrow \mathcal{M}_1 \overset{d}{\leftarrow} \mathcal{M}_0 \overset{d}{\leftarrow} \mathbb{Z}$$
where, by following the proof of Lemma \[lemma unitary script\], we have $d p = l$, $d q = -l$, $d1 = p+q$ and $d\ell = 0$.
The script
$$0 \leftarrow \mathbb{Z} \overset{\partial}{\leftarrow} \mathcal{M}_0 \overset{\partial}{\leftarrow} \mathcal{M}_1 \overset{\partial}{\leftarrow} \ldots \overset{\partial}{\leftarrow} \mathcal{M}_k \overset{\partial}{\leftarrow} \ldots \overset{\partial}{\leftarrow} \mathcal{M}_m$$
where $\mathcal{C}_l = \{C_1^l,C_2^l\}$, $\partial C_1^0 = \partial C_2^0 = 1$ and $\partial C_j^{k} = C_{1}^{k-1}-C_{2}^{k-1}$ for $k=1,\ldots,m$, $l=0,\ldots,m$, represents an $m$-sphere.
The dual of this script won’t be a script itself. But we can still calculate its dual:
$$0 \leftarrow \mathcal{M}_{m} \overset{d}{\leftarrow} \ldots \overset{d}{\leftarrow} \mathcal{M}_2 \overset{d}{\leftarrow} \mathcal{M}_1 \overset{d}{\leftarrow} \mathcal{M}_0 \overset{d}{\leftarrow} \mathbb{Z}$$
with $d 1 = C_1^0+C_2^0$, $dC_{1}^k=C_1^{k+1}+C_2^{k+1}$, $dC_{2}^k=-C_1^{k+1}-C_2^{k+1}$ where $k = 0, \ldots,m-1$ and $dC_1^m = dC_2^m = 0$.
If we want the dual to be a script we add the volume $\mathcal{C}_{m+1} = \{C_{1}^{m+1}\}$, with $\partial C_j^{m+1} = C_{1}^{m}-C_{2}^{m}$, of the $m$-sphere so that we have the script of the $m$-dimensional ball.
Its dual script is then given by
$$0 \leftarrow \mathcal{M}_{m+1} \overset{d}{\leftarrow} \ldots \overset{d}{\leftarrow} \mathcal{M}_2 \overset{d}{\leftarrow} \mathcal{M}_1 \overset{d}{\leftarrow} \mathcal{M}_0 \overset{d}{\leftarrow} \mathbb{Z}$$
with $d 1 = C_1^0+C_2^0$, $dC_{1}^k=C_1^{k+1}+C_2^{k+1}$, $dC_{2}^k=-C_1^{k+1}-C_2^{k+1}$ where $k = 0, \ldots,m-1$, $dC_1^m = C_1^{m+1}$, $dC_2^m = -C_1^{m+1}$ and $dC_1^{m+1} = 0$.
We have the following script
$$\begin{aligned}
\mathcal{C}_0 &= \{[0],[1],\ldots,[m]\}\\
\mathcal{C}_1 &= \{[i,j]\vert 0\leq i<j\leq m\}\\
\vdots\\
\mathcal{C}_k &= \{[\alpha_0,\ldots,\alpha_k]\vert 0\leq \alpha_0<\alpha_1<\ldots<\alpha_k\leq m\}\\
\vdots\\
\mathcal{C}_m &= \{[0,1,\ldots,m]\}\\\end{aligned}$$
where the boundary map is defined as:
$$\partial [\alpha_0,\ldots,\alpha_k] = \sum_{j=0}^k (-1)^j [\alpha_0,\ldots,\alpha_k]_j = \sum_{j=0}^k (-1)^j [\alpha_0,\ldots, \alpha_{j-1}, \alpha_{j+1},\ldots,\alpha_k]$$
The dual boundary operator is
$$\begin{aligned}
d [\alpha_0,\ldots,\alpha_k] =& \sum_{j=0}^{\alpha_0-1} [j,\alpha_0,\ldots,\alpha_k] - \sum_{j=\alpha_0+1}^{\alpha_1-1} [\alpha_0,j,\alpha_1,\ldots,\alpha_k] \\
&+ \ldots + (-1)^k \sum_{j=\alpha_{k-1}+1}^{\alpha_k-1} [\alpha_0,\ldots,\alpha_{k-1},j,\alpha_k]\\
&+ (-1)^{k+1}\sum_{j=\alpha_k+1}^{m} [\alpha_0,\ldots,\alpha_k, j]\end{aligned}$$
If $\alpha_l = \alpha_{l+1}-1$ then we leave out the sum $\sum_{j=\alpha_{l}+1}^{\alpha_{l+1}-1}$.
We have the following script for a $2-$torus:
$$\begin{gathered}
\mathcal{C}_0 = \{p_0, p_1, p_2, p_3\}, \quad \partial p_j = 1, j = 0, 1, 2, 3.\\
\mathcal{C}_1 = \{\ell_1, \ell_2, \ell_3, \ell_4, \ell_5, \ell_6, \ell_7, \ell_8\},\\
\partial \ell_1 = p_1 - p_0, \quad \partial \ell_2 = p_0 - p_1, \quad \partial \ell_3 = p_2 - p_0, \quad \partial \ell_4 = p_0 - p_2\\
\partial \ell_5 = p_3 - p_2, \quad \partial \ell_6 = p_2 - p_3, \quad \partial \ell_7 = p_3 - p_1, \quad \partial \ell_8 = p_1 - p_3\\
\mathcal{C}_2 = \{v_1, v_2, v_3, v_4\},\\
\partial v_1 = \ell_5 + \ell_8 - \ell_1 - \ell_4, \quad \partial v_2 = \ell_6 + \ell_4 - \ell_2 - \ell_8,\\
\partial v_3 = \ell_1 + \ell_7 - \ell_5 - \ell_3, \quad \partial v_4 = \ell_2 + \ell_3 - \ell_6 - \ell_7,\\
\mathcal{C}_3 = \{C\}, \quad \partial C = v_1 + v_2 + v_3 + v_4\end{gathered}$$
Here we compute the dual of the $2-$torus described in section 3.6. We have that:
$$\begin{gathered}
d 1 = p_0+p_1+p_2+p_3, \\
dp_0= \ell_2 - \ell_1+ \ell_4- \ell_3 ;\quad dp_1= \ell_1 - \ell_2+ \ell_8- \ell_7 ;\quad dp_2= \ell_3 - \ell_4+ \ell_6- \ell_5 ;\quad dp_3= \ell_5 - \ell_6+ \ell_7- \ell_8 \\
d\ell_1=v_3-v_1; \quad d\ell_2=v_4-v_2; \quad d\ell_3=v_4-v_3; \quad d\ell_4=v_2-v_1; \\
d\ell_5=v_1-v_3; \quad d\ell_6=v_2-v_4; \quad d\ell_7=v_3-v_4; \quad d\ell_8=v_1-v_2; \\
d v_1 = d v_2 = d v_3 = d v_4 = C ; \\
dC=0.\end{gathered}$$
With the change $\partial'=d$ and $\mathcal{C}'_k=\mathcal{C}_{2-k-1}$ and $v'=p$, $1'=C$, $p'=l$ and $l'=v$ we see that the dual of the torus is represented by the torus itself, with a suitable change of indices.
We look at the following script of a Klein bottle described in section 3.7.
$$\begin{gathered}
\mathcal{C}_0 = \{p_0, p_1, p_2, p_3\}, \quad \partial p_j = 1, j = 0, 1, 2, 3.\\
\mathcal{C}_1 = \{\ell_1, \ell_2, \ell_3, \ell_4, \ell_5, \ell_6, \ell_7, \ell_8\},\\
\partial \ell_1 = p_1 - p_0, \quad \partial \ell_2 = p_0 - p_1, \quad \partial \ell_3 = p_2 - p_0, \quad \partial \ell_4 = p_0 - p_2\\
\partial \ell_5 = p_3 - p_1, \quad \partial \ell_6 = p_1 - p_3, \quad \partial \ell_7 = p_3 - p_2, \quad \partial \ell_8 = p_2 - p_3\\
\mathcal{C}_2 = \{v_1, v_2, v_3, v_4\},\\
\partial v_1 = \ell_5 + \ell_8 - \ell_2 - \ell_3, \quad \partial v_2 = \ell_6 - \ell_1 - \ell_4 - \ell_8,\\
\partial v_3 = -\ell_1 + \ell_3 + \ell_7 - \ell_5 , \quad \partial v_4 = \ell_4 - \ell_2 - \ell_6 - \ell_7,\end{gathered}$$
Here we compute the dual of the Klein bottle, which will not be a script. We have that:
$$\begin{gathered}
d 1 = p_0+p_1+p_2+p_3, \\
dp_0= \ell_2 - \ell_1+ \ell_4- \ell_3 ;\quad dp_1= \ell_1 - \ell_2+ \ell_6- \ell_5 ;\quad dp_2= \ell_3 - \ell_4+ \ell_8- \ell_7 ;\quad dp_3= \ell_5 - \ell_6+ \ell_7- \ell_8 \\
d\ell_1=-v_2-v_3; \quad d\ell_2=-v_1-v_4; \quad d\ell_3=v_3-v_1; \quad d\ell_4=v_4-v_2; \\
d\ell_5=v_1-v_3; \quad d\ell_6=v_2-v_4; \quad d\ell_7=v_3-v_4; \quad d\ell_8=v_1-v_2; \\
d v_1 = d v_2 = d v_3 = d v_4 = 0 .\end{gathered}$$
As mentioned in section 3.7, we can add a new cell $v_5$, with $\partial v_5 = \ell_1 + \ell_2$ to the script we obtain a script for the “extended Klein bottle", which is tight, but not unitary. Just adding this cell won’t make the dual into a script:
$$\begin{gathered}
d1 = p_0+p_1+p_2+p_3\\
d p_0 = -\ell_1 + \ell_2 - \ell_3 +\ell_4, \quad d p_1 = \ell_1 -\ell_2 - \ell_5 + \ell_6,\\
d p_2 = \ell_3 - \ell_4 - \ell_7 + \ell_8, \quad d p_3 = \ell_5 - \ell_6 + \ell_7 - \ell_8\\
d \ell_1 = -v_2 - v_3 + v_5, \quad d \ell_2 = -v_1 - v_4 + v_5, \quad d \ell_3 = -v_1 + v_3, d \ell_4 = -v_2 + v_4\\
d \ell_5 = v_1 - v_3, \quad d \ell_6 = v_2 - v_4, \quad d \ell_7 = v_3 - v_4, \quad d \ell_8 = v_1 - v_2\\
d v_1 = dv_2 = dv_3 = dv_4 = dv_5 = 0\end{gathered}$$
But if we add a cell $C$, with $\partial C = v_1+v_2+v_3+v_4+2v_5$, then its dual will be a script, but the script won’t be unitary any more. The dual operator will be the same for the points and lines, but it changes for the planes:
$$\begin{gathered}
d v_1 = dv_2 = dv_3 = dv_4 = C, \quad d v_5 = 2C\\
dC=0\end{gathered}$$
As $dv_5 = 2C$, we have that $v_5$ is not a tight cell in the dual script. Hence the dual is not tight and not unitary.
As in section 3.8 we start from the following script:
$$\begin{gathered}
\mathcal{C}_0 = \{p_1,p_2,p_3\}, \quad \partial p_j = 1, j=1,2,3\\
\mathcal{C}_1 = \{ \ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6\},\\
\partial \ell_1 = p_2 - p_1, \quad \partial \ell_2 = p_1 - p_2, \quad \partial \ell_3 = p_3 - p_1, \quad \partial \ell_4 = p_1 - p_2\\
\partial \ell_5 = p_3 - p_2, \quad \partial \ell_6 = p_2 - p_3,\\
\mathcal{C}_2 = \{v_1,v_2,v_3,v_4\}\\
\partial v_1 = -\ell_2 - \ell_3 + \ell_5, \quad \partial v_2 = -\ell_1 - \ell_4 - \ell_5\\
\partial v_3 = -\ell_1 + \ell_3 + \ell_6, \quad \partial v_4 = -\ell_2 + \ell_4 - \ell_6\end{gathered}$$
Here we compute the dual of the projective plane, which won’t be a script. We have that:
$$\begin{gathered}
d 1 = p_1+p_2+p_3, \\
dp_1= \ell_2 - \ell_1+ \ell_4- \ell_3 ;\quad dp_2= \ell_1 - \ell_2+ \ell_6- \ell_5 ;\quad dp_3= \ell_3 - \ell_4+ \ell_5- \ell_6 \\
d\ell_1=-v_2-v_3; \quad d\ell_2=-v_1-v_4; \quad d\ell_3=v_3-v_1; \quad d\ell_4=v_4-v_2; \\
d\ell_5=v_1-v_2; \quad d\ell_6=v_3-v_4; \\
d v_1 = d v_2 = d v_3 = d v_4 = 0.\end{gathered}$$
We can do the same trick as with the Klein bottle and add a $v_5$, with $\partial v_5 = \ell_1 + \ell_2$ to the script we obtain a script for the “extended projective plane". Calculating its dual yields
$$\begin{gathered}
d1 = p_1 + p_2 + p_3\\
dp_1 = \ell_2 - \ell_1 + \ell_4 - \ell_3, \quad dp_2 = \ell_1 - \ell_2 - \ell_5 + \ell_6, \quad dp_3 = \ell_3 - \ell_4 + \ell_5 - \ell_6\\
d\ell_1 = -v_2-v_3+v_5, \quad d \ell_2 = -v_1 - v_4 + v_5, \quad d \ell_3 = -v_1 + v_3\\
d\ell_4 = -v_2 + v_4, \quad d\ell_5 = v_1-v_2, \quad d\ell_6 = v_3-v_4\\
dv_1 = dv_2 = dv_3 = dv_4 = dv_5 = 0\end{gathered}$$
which still isn’t a script. So once again adding a new cell $C\in\mathcal{C}_3$, $\partial C = v_1+v_2+v_3+v_4+2v_5$, then the dual will be a script. The dual operator acts the same on points and lines but for planes and for $C$ we have
$$\begin{gathered}
dv_1 = dv_2 = dv_3 = dv_4 = C, \quad dv_5 = 2C\\
dC=0\end{gathered}$$
After the glueing of the cells done in section 4.4, we get the following script:
$$\begin{gathered}
\mathcal{C}_0 = \{p_1,p_2,p_3,p_4\}, \quad \partial p_j = 1, j=1,2,3,4\\
\mathcal{C}_1 = \{ \ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6\},\\
\partial \ell_1 = p_2 - p_1, \quad \partial \ell_2 = p_4 - p_3, \quad \partial \ell_3 = p_3 - p_1 \\
\partial \ell_4 = p_2 - p_3, \quad \partial \ell_5 = p_4 - p_2, \quad \partial \ell_6 = p_1 - p_4,\\
\mathcal{C}_2 = \{v_1,v_2\}\\
\partial v_1 = -\ell_1 + \ell_2 + \ell_3 - \ell_5, \quad \partial v_2 = -\ell_1 - \ell_2 + \ell_4 - \ell_6\end{gathered}$$
Note that the dual won’t be a script, nevertheless it is given by
$$\begin{gathered}
d1 = p_1 + p_2 + p_3 + p_4\\
dp_1 = - \ell_1 - \ell_3 + \ell_6, \quad dp_2 = \ell_1 + \ell_4 - \ell_5,\\
dp_3 = -\ell_2 + \ell_3 - \ell_4, \quad dp_4 = \ell_2 + \ell_5 - \ell_6\\
d\ell_1 = -v_1 - v_2, \quad d \ell_2 = v_1 - v_2, \quad d \ell_3 = v_1\\
d\ell_4 = v_2, \quad d\ell_5 = -v_1, \quad d\ell_6 = -v_2\\
dv_1 = dv_2 = 0\end{gathered}$$
Reconsider the Moebius strip with an extra cell $v_3$:
$$\begin{gathered}
\mathcal{C}_0 = \{p_1,p_2,p_3,p_4\}, \quad \partial p_j = 1, j=1,2,3,4\\
\mathcal{C}_1 = \{ \ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6\},\\
\partial \ell_1 = p_2 - p_1, \quad \partial \ell_2 = p_4 - p_3, \quad \partial \ell_3 = p_3 - p_1 \\
\partial \ell_4 = p_2 - p_3, \quad \partial \ell_5 = p_4 - p_2, \quad \partial \ell_6 = p_1 - p_4,\\
\mathcal{C}_2 = \{v_1,v_2,v_3\}\\
\partial v_1 = -\ell_1 + \ell_2 + \ell_3 - \ell_5, \quad \partial v_2 = -\ell_1 - \ell_2 + \ell_4 - \ell_6\\
\partial v_3 = \ell_5+\ell_6+\ell_3+\ell_4\end{gathered}$$
Here we compute the dual of the other projective plane model described in section 4.4... [**insert picture**]{}. $$\begin{gathered}
d 1 = p_1+p_2+p_3+p_4, \\
dp_1= - \ell_1 - \ell_3+ \ell_6 ;\quad dp_2= \ell_1 + \ell_4- \ell_5 ;\quad dp_3= - \ell_2 + \ell_3- \ell_4;\quad dp_4= \ell_2 + \ell_5 - \ell_6 \\
d\ell_1=-v_1-v_2; \quad d\ell_2= v_1-v_2; \quad d\ell_3=v_1+v_3; \quad d\ell_4=v_2+v_3; \\
d\ell_5=-v_1+v_3; \quad d\ell_6=-v_2+v_3; \\
d v_1 = d v_2 = d v_3 = 0.\end{gathered}$$
We will compute the dual of the Klein bottle, described in section 4.4. Its script is
$$\begin{gathered}
\mathcal{C}_0 = \{p_1,p_2,p_3,p_4\}, \quad \partial p_j = 1, j=1,2,3,4\\
\mathcal{C}_1 = \{ \ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6,\ell_7,\ell_8\},\\
\partial \ell_1 = p_2 - p_1, \quad \partial \ell_2 = p_4 - p_3, \quad \partial \ell_3 = p_3 - p_1, \quad \partial \ell_4 = p_4 - p_2,\\
\partial \ell_5 = p_2 - p_3, \quad \partial \ell_6 = p_1 - p_4, \quad \partial \ell_7 = p_1 - p_2, \quad \partial \ell_8 = p_3 - p_4\\
\mathcal{C}_2 = \{v_1,v_2,v_3,v_4\}\\
\partial v_1 = -\ell_1 + \ell_2 + \ell_3 - \ell_4, \quad \partial v_2 = -\ell_1 - \ell_2 + \ell_5 - \ell_6\\
\partial v_3 = -\ell_3 + \ell_4 - \ell_7 + \ell_8, \quad \partial v_4 = -\ell_5 + \ell_6 - \ell_7 - \ell_8\end{gathered}$$
Note that the dual won’t be a script, nevertheless it is given by
$$\begin{gathered}
d1 = p_1 + p_2 + p_3 + p_4\\
dp_1 = - \ell_1 - \ell_3 + \ell_6 + \ell_7, \quad dp_2 = \ell_1 - \ell_4 + \ell_5 - \ell_7,\\
dp_3 = -\ell_2 + \ell_3 - \ell_5 + \ell_8, \quad dp_4 = \ell_2 + \ell_4 - \ell_6 - \ell_8\\
d\ell_1 = -v_1 - v_2, \quad d \ell_2 = v_1 - v_2, \quad d \ell_3 = v_1 - v_3, \quad d\ell_4 = -v_1 + v_3,\\
d\ell_5 = v_2 - v_4, \quad d\ell_6 = -v_2 + v_4, \quad d\ell_7 = -v_3 - v_4, \quad d\ell_8 = v_3 - v_4\\
dv_1 = dv_2 = dv_3 = dv_4 = 0\end{gathered}$$
\[example without portal\]
Here is the script without the 2D-disk portal:
$$\begin{gathered}
\mathcal{C}_0 = \{pi_1,pi_2,p_1,p_2\}\\
\partial pi_1 = \partial pi_2 = \partial p_1 = \partial p_2 = 1\\
\hline
\mathcal{C}_1 = \{li_1,li_2,lt_1,lb_1,lt_2,lb_2\}\\
\partial li_1 = pi_2-pi_1 = \partial li_2, \quad
\partial lp_1 = p_2-p_1 = \partial lp_2,\\
\partial lt_1=p_1-pi_1 = \partial
lb_1, \quad \partial lt_2 = pi_2-p_2 = \partial lb_2\\
\hline
\mathcal{C}_2 = \{vi_1,vi_2,vp_1,vp_2,vt_1,vt_2,vb_1,vb_2\}\\
\partial vi_1=li_2-li_1 = \partial vi_2, \quad \partial vp_1=lp_2-lp_1=\partial vp_2\\
\partial vt_1=lt_1+lp_1+lt_2-li_1,\quad \partial vt_2=li_2-lt_2-lp_2-lt_1\\
\partial vb_1=lb_1+lp_1+lb_2-li_1,\quad \partial vb_2=li_2-lb_2-lp_2-lb_1\\
\hline
\mathcal{C}_3 = \{wt_1,wt_2,wb_1,wb_2\}\\
\partial wt_1=vp_1+vt_1+vt_2-vi_1,\quad \partial wt_2=vi_2-vp_2-vt_1-vt_2\\
\partial wb_1=vp_2+vb_1+vb_2-vi_1,\quad \partial wb_2=vi_2-vp_1-vb_1-vb_2\end{gathered}$$
Its dual won’t be a script, but it is given by
$$\begin{gathered}
d1 = pi_1+pi_2+p_1+p_2,\\
\hline
dpi_1 = -li_1 - li_2 - lt_1 - lb_1, \quad dpi_2 = li_1 + li_2 + lt_2 + lb_2,\\
dp_1 = -lp_1 - lp_2 + lt_1 + lb_1, \quad dp_2 = lp_1 + lp_2 - lt_2 - lb_2,\\
\hline
dli_1 = -vi_1 - vi_2 - vt_1 - vb_1, \quad dli_2 = vi_1 + vi_2 + vt_2 + vb_2,\\
dlp_1 = -vp_1 - vp_2 + vt_1 + vb_1, \quad dlp_2 = vp_1 + vp_2 - vt_2 - vb_2,\\
dlt_1 = vt_1 - vt_2, \quad dlb_1 = vb_1 - vb_2, \quad dlt_2 = vt_1 - vt_2, \quad dlb_2 = vb_1 - vb_2,\\
\hline
dvi_1 = -wt_1 - wb_1, \quad dvi_2 = wt_2 + wb_2, \quad dvp_1 = wt_1 - wb_2, \qquad dvp_2 = -wt_2 + wb_1\\
dvt_1 = wt_1 - wt_2, \quad dvt_2 = wt_1 - wt_2, \quad dvb_2 = wb_1 - wb_2, \quad dvb_1 = wb_1 - wb_2\\
\hline
dwt_1 = dwt_2 = dwb_1 = dwb_2 = 0\end{gathered}$$
We add two cells $vp_3$ and $vp_4$ to the script of Example \[example without portal\] such that $\partial vp_3 = \partial vp_4= \partial vp_1 = \partial vp_2$ and change the boundary of $wt_1$ and $wb_2$:
$$\begin{aligned}
\partial wt_2 &= vi_2-vp_3-vt_1-vt_2\\
\partial wb_2 &= vi_2-vp_4-vb_1-vb_2\end{aligned}$$
Its dual still won’t be a script, but it is given by
$$\begin{gathered}
d1 = pi_1+pi_2+p_1+p_2,\\
\hline
dpi_1 = -li_1 - li_2 - lt_1 - lb_1, \quad dpi_2 = li_1 + li_2 + lt_2 + lb_2,\\
dp_1 = -lp_1 - lp_2 + lt_1 + lb_1, \quad dp_2 = lp_1 + lp_2 - lt_2 - lb_2,\\
\hline
dli_1 = -vi_1 - vi_2 - vt_1 - vb_1, \quad dli_2 = vi_1 + vi_2 + vt_2 + vb_2,\\
dlp_1 = -vp_1 - vp_2 - vp_3 - vp_4 + vt_1 + vb_1, \\
dlp_2 = vp_1 + vp_2 + vp_3 + vp_4 - vt_2 - vb_2,\\
dlt_1 = vt_1 - vt_2, \quad dlb_1 = vb_1 - vb_2, \quad dlt_2 = vt_1 - vt_2, \quad dlb_2 = vb_1 - vb_2,\\
\hline
dvi_1 = -wt_1 - wb_1, \quad dvi_2 = wt_2 + wb_2, \quad dvp_1 = wt_1, \qquad dvp_2 = wb_1\\
dvt_1 = wt_1 - wt_2, \quad dvt_2 = wt_1 - wt_2, \quad dvb_2 = wb_1 - wb_2, \quad dvb_1 = wb_1 - wb_2\\
dvp_3 = -wt_2, \quad dvp_4 = -wb_2\\
\hline
dwt_1 = dwt_2 = dwb_1 = dwb_2 = 0\end{gathered}$$
Let $\mathbb{Z}_5$ contain the elements $\{1,2,3,4,5\}$. We start from the following script:
$$\begin{gathered}
\mathcal{C}_0 = \{p_j\mid j\in \mathbb{Z}_{5}\} \cup \{p_j\mid j\in \mathbb{Z}_{5}\}, \quad \partial p_j = \partial q_j = 1\\
\mathcal{C}_1 = \{k_j\mid j\in \mathbb{Z}_{5}\} \cup \{l_j\mid j\in \mathbb{Z}_{5}\} \cup \{m_j\mid j\in \mathbb{Z}_{5}\}\\
\partial k_j = p_{j+1} - p_j, \quad \partial l_j = p_j - q_j, \quad \partial m_j = q_{j+2} - q_j, \quad j \in\mathbb{Z}_5\\
\mathcal{C}_2 = \{v_j\mid j\in \mathbb{Z}_{5}\}\cup\{v\}\\
\partial v_j = l_j + k_j + k_{j+1} - l_{j+2} - m_j, \quad j\in\mathbb{Z}_5\\
\partial v = m_1 + m_2 + m_3 + m_4 + m_5\end{gathered}$$
The dual won’t be a script, nevertheless it is given by
$$\begin{gathered}
d1 = \sum_{j=1}^5 (p_j + q_j)\\
dp_j = k_{j-1} - k_j + l_j, \quad dq_j = m_{j-2} - m_j - l_j, \quad j\in\mathbb{Z}_5\\
dk_j = v_j + v_{j-1}, \quad dl_j = v_j - v_{j-2}, \quad dm_j = v - v_j, \quad j\in \mathbb{Z}_5\\
dv_1 = dv_2 = dv_3 = dv_4 = dv_5 = dv = 0\end{gathered}$$
Duality and Orientability in general:
-------------------------------------
1) If the script is generated by a single cell of dim $n+1$ then $\partial C^{n+1}$ will be an orientation on $C^n.$ The dual complex will map any $n-$cell on $\pm C^{n+1}$ which means that $C^{n+1}$ acts as accumulator for the dual complex which is now a script. Necessary for this is that the script is orientable (counter-example: Klein bottle)
Observations concerning metrics and duality in the two-dimensional case:
1. Every tight unitary 2D-script can be identified with a CW-complex and, therefore, can be given a unique topology.
2. Every line has two end points and every plane element is a polygon.
3. Suppose that a 2D-script is orientable. We then can create a 3-cell $C^3$ such that $\partial C^3 = \sum_j C^2_j.$ In this case we know that this 3-cell will become the accumulator of the dual complex while also the accumulator of the original complex will become an extra 3-cell of the dual complex.
We know come to a key theorem for three-dimensional scripts arising from two-dimensional scripts with an extra 3D-cell attached.
If a three-dimensional script consisting of a single 3-cell is unitary and tight and the dual script is also unitary and tight then the CW complex associated to the $2-$dimensional subscript constructed above becomes an orientable, connected, compact (topological) manifold of dimension $2$.
Orientability and connectedness follow from $\partial C^3$ being tight (generated by a unitary cycle which unique up to the sign).
From the assumption that $\partial C^3$ is tight follows the orientability and connectedness, indeed, in this case $\partial C^3$ is generated by a unitary cycle which is unique up to the sign.
We also have that the 2D-subscript is a CW complex due to the fact that it is tight. Remains to be proven that every point in this CW-complex has a local neighborhood homeomorphic to the unit disk.
Case 1. The point belongs to a $2-$dimensional face $C^2$ of the CW-complex. In this case since $C^2$ is tight its boundary is homeomorphic to a polygon so its cell is homeomorphic to a disk containing that point.
Case 2. The point belongs to one of the lines $C^1_j$. Due to the tightness of the dual boundary operator $d$ acting on $C^1_j$, $C^1_j$ connects two two-dimensional faces. Therefore, $$C^1_j \in \partial C^2_k \quad and \quad C^1_j \in \partial C^2_l, ~k\not= l.$$
Clearly, the union of $C^1_j$ and these two faces is a neighborhood homeomorphic to the disk.
Case 3. The point is one of the points $C^0_j.$ In that case every point is the dual of a solid polygon. This means that there are a number of lines and $2-$cells issuing from that point that can be ordered in the following form: line $\ell_1,$ face $v_1,$ line $\ell_2,$ face $v_2,$ …line $\ell_{k-1},$ face $v_{k-1},$ line $\ell_1.$
They form a dual of a polygon which is also a disk. This concludes the proof.
**Remark:** Every tightness condition has been used in the proof.
Clearly, the Klein bottle and the projective plane are non-orientable and the dual of their complex is also not a tight script.
Discrete Dirac Operators on Scripts
-----------------------------------
The [*Dirac Operator*]{} in this case is given by $\slashed\partial =d+\partial$.
Here the Dirac operator acts on sums of chains of different dimensions.
The [*discrete Hodge-Laplace Operator*]{} in this case is given by $\slashed\partial^2=d\partial +\partial d$.
In this case the [*harmonic and monogenic functions respectively*]{} are solutions of $\slashed\partial^2 F=0$ and $\slashed\partial F=0$ respectively. They will be linear combinations of all $C_j^k$’s or subsets of them.
In particular there could be harmonic and monogenic functions corresponding to chains of a given dimension $k$, for example a $k-$chain $F$ is monogenic iff it satisfies the Hodge system: $$dF=\partial F=0.$$ However, note that not all monogenic chains are sums of solutions of the Hodge system.
The [*sound*]{} of a script is the sum all eigenvalues of the Hodge Laplacian.
As both the boundary operator $\partial$ and its dual $d$ are linear operators, we can describe them using matrices. Doing so, we can determine each entry of these matrices using the metric defined in Definition \[definition metric\], which implies that the matrix representation for $d$ is the transpose of the matrix for $\partial$.\
Hence if we are looking for monogenic or harmonic functions, we can calculate the eigenvalues and eigenvectors of the Dirac operator and the Laplace operator.
We have the following:
$$\begin{aligned}
\langle \slashed{\partial}^2 C_i^k,C_j^l\rangle &=\langle \slashed{\partial} C_i^k,\slashed{\partial} C_j^l\rangle\\
&=\langle \partial C_i^k,\partial C_j^l\rangle + \langle d C_i^k,d C_j^l\rangle\\\end{aligned}$$
This is only non-zero if $k = l$, this means that the Laplace operator sends $k$-chains to $k$-chains, i.e. the matrix representation of $\slashed{\partial}^2$ will be a block diagonal matrix.
In the general case where the dual isn’t a script, we had $d1 = \sum_{j}p_j$ and $\partial p_j = 1$. Thus we have:
$$\begin{aligned}
\slashed{\partial}(\mu + \sum_{j=1}^n \lambda_j p_j) &= (d+\partial)(\mu + \sum_{j=1}^n \lambda_j p_j)\\
&= \underbrace{\left(\begin{matrix}
0 & 1 & 1 & \cdots & 1\\
1 & 0 & 0 & \cdots & 0\\
1 & 0 & 0 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
1 & 0 & 0 & \cdots & 0
\end{matrix}\right)}_{:=A}\left(\begin{matrix}
\mu\\
\lambda_1\\
\lambda_2\\
\vdots\\
\lambda_n\\
\end{matrix}\right)\end{aligned}$$
It has eigenvalues $0,\sqrt{n}, -\sqrt{n}$ with multiplicities $n-1, 1,1$ respectively. Thus all corresponding eigenvectors of eigenvalue 0 (Monogenic functions) are linear combinations of $p_1 - p_j$ for $j=2,\ldots,n$.\
For harmonic functions we need to look at the eigenvalues of
$$A^2 = \left(\begin{matrix}
n & 0 & 0 & \cdots & 0\\
0 & 1 & 1 & \cdots & 1\\
0 & 1 & 1 & \cdots & 1\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 1 & 1 & \cdots & 1
\end{matrix}\right)$$
which are $0,n$ with multiplicities $n-1, 2$ respectively and the same eigenvectors. The sound here is $2n$.
We have $dp = -dq = \ell$, $d1 = p+q$, $\partial \ell = p-q$ and $\partial p = \partial q = 1$. This yields
$$\begin{aligned}
\slashed{\partial}(\mu + \lambda_1 p + \lambda_2 q + \nu \ell) &= (d+\partial)(\mu + \lambda_1 p + \lambda_2 q + \nu \ell)\\
&= \underbrace{\left(\begin{matrix}
0 & 1 & 1 & 0\\
1 & 0 & 0 & 1\\
1 & 0 & 0 & -1\\
0 & 1 & -1 & 0
\end{matrix}\right)}_{:=A}\left(\begin{matrix}
\mu\\
\lambda_1\\
\lambda_2\\
\nu
\end{matrix}\right)\end{aligned}$$
which has eigenvalues $\sqrt{2}, -\sqrt{2}$ both with multiplicity 2. Hence the only monogenic function is 0. Squaring this matrix yields
$$A^2 = \left(\begin{matrix}
2 & 0 & 0 & 0\\
0 & 2 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 2
\end{matrix}\right)$$
It is now trivial to see that the sound is 8.
We have the following matrix representation of the Dirac operator for the $m$-sphere
$$\begin{aligned}
A = \left(\begin{matrix}
& 1 & 1\\
1 & & & 1 & 1 \\
1 & & & -1 & -1 \\
& 1 & -1 & & & \ddots\\
& 1 & -1 & & & & \ddots\\
& & & \ddots & & & & 1 & 1 &\\
& & & & \ddots & & & -1 & -1 &\\
& & & & & 1 & -1\\
& & & & & 1 & -1
\end{matrix}\right)\end{aligned}$$
where the empty spaces are zeros. Simple calculation yields
$$A^2 = \left(\begin{matrix}
2\\
& 3 & -1\\
& -1 & 3\\
& & & 4\\
& & & & \ddots\\
& & & & & 4\\
& & & & & & 2 & 2\\
& & & & & & 2 & 2
\end{matrix}\right)$$
Hence it is easy to see that $A^2$ only has 0,4,2 as eigenvalues with multiplicity $1,2m, 2$. The sound is thus $8m+4$. Moreover the only monogenic and harmonic functions are multiples of $C_1^m-C_2^m$. These are the only functions where $\partial F = dF = 0$.\
If we add the extra cell in order to make the dual a script, we get the following matrix for the Dirac operator on the $m$-dimensional ball:
$$\begin{aligned}
A = \left(\begin{matrix}
& 1 & 1\\
1 & & & 1 & 1 \\
1 & & & -1 & -1 \\
& 1 & -1 & & & \ddots\\
& 1 & -1 & & & & \ddots\\
& & & \ddots & & & & 1 & 1 &\\
& & & & \ddots & & & -1 & -1 &\\
& & & & & 1 & -1 & & & 1\\
& & & & & 1 & -1 & & & -1\\
& & & & & & & 1 & -1
\end{matrix}\right)\end{aligned}$$
The matrix for the Laplace operator becomes
$$A^2 = \left(\begin{matrix}
2\\
& 3 & -1\\
& -1 & 3\\
& & & 4\\
& & & & \ddots\\
& & & & & 4\\
& & & & & & 3 & 1\\
& & & & & & 1 & 3\\
& & & & & & & & 2
\end{matrix}\right)$$
which has eigenvalues 2,4 with multiplicity $4,2m$ respectively. The sound is $8m+8$ and there are no non-zero monogenics or harmonics.
Note that for a $m$-simplex, we have that the matrix representation of $\slashed{\partial}$ will only contain $0,1,-1$. In order to calculate the eigenvalues of the Laplace operator we will look at
$$\alpha_{i,j}^k = \langle C_i^k,\slashed{\partial}^2 C_j^k\rangle =\langle C_i^k,d\partial C_j^k\rangle + \langle C_i^k,\partial d C_j^k\rangle$$
If $i=j$, then
$$\alpha_{i,j}^k = (k + 1) + (m-k) = m+1$$
If $[\alpha_0,\ldots,\alpha_k] = C_i^k\neq C_j^k = [\beta_0,\ldots,\beta_k]$, then we have two possibilities:
(i) $|\{\alpha_0,\ldots,\alpha_k\}\setminus\{\beta_0,\ldots,\beta_k\}| \geq 2$. Applying $d$ results in a linear combination of $C_j^k$ with 1 extra number and $\partial$ is a linear combination of $C_j^k$ with 1 number less, so $\partial d$ will change at most only 1 $\alpha_l$ and the same for $d \partial$, thus we have $\langle C_i^k,d\partial C_j^k\rangle = \langle C_i^k,\partial d C_j^k\rangle = 0$.
(ii) $|\{\alpha_0,\ldots,\alpha_k\}\setminus\{\beta_0,\ldots,\beta_k\}|=1$, thus there is a unique $l,l'$ such that $\alpha_{l} \neq \beta_{l'}$. Using the same reasoning as in (ii), we only need to look at the term where we deleted $\beta_{l'}$ and replaced it with $\alpha_l$. Hence we have
$$\begin{aligned}
\alpha_{i,j}^k =& \langle \partial C_i^k,\partial C_j^k\rangle + \langle C_i^k,\partial d C_j^k\rangle\\
=& (-1)^{l+l'} + \langle C_{i}^k, \partial d C_j^k \rangle\\
=& (-1)^{l+l'} + (-1)^{l+l'+1}\\
=& 0\end{aligned}$$
Hence the matrix representation of $\slashed{\partial}^2$ is a diagonal matrix with $m+1$ on the diagonal. Hence there are no harmonic or monogenic functions except for 0. The sound of an $m$-simplex is $2^{m+1}(m+1)$.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & 1 & -1 & 1 & 0 & 0 & 0 & 0\\
1 & -1 & 0 & 0 & 0 & 0 & -1 & 1\\
0 & 0 & 1 & -1 & -1 & 1 & 0 & 0\\
0 & 0 & 0 & 0 & 1 & -1 & 1 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
-1 & 0 & 1 & 0\\
0 & -1 & 0 & 1\\
0 & 0 & -1 & 1\\
-1 & 1 & 0 & 0\\
1 & 0 & -1 & 0\\
0 & 1 & 0 & -1\\
0 & 0 & 1 & -1\\
1 & -1 & 0 & 0
\end{matrix}\right)\\
A_3&=\left(\begin{matrix}
1\\
1\\
1\\
1
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0 & 0\\
A_0^T & 0 & A_1 & 0 & 0\\
0 & A_1^T & 0 & A_2 & 0\\
0 & 0 & A_2^T & 0 & A_3\\
0 & 0 & 0 & A_3^T & 0
\end{matrix}\right)$$
Which has eigenvalues $-2\sqrt{2},-2,0,2,2\sqrt{2}$ with multiplicity 2,6,2,6,2 respectively. The monogenic functions are linear combinations of $\ell_1+\ell_2+\ell_5+\ell_6$ and $\ell_3+\ell_4+\ell_7+\ell_8$.\
The action of $\slashed{\partial}^2$ will have the same eigenvectors with the square of the corresponding eigenvalue. Hence the sound is equal to 80.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & 1 & -1 & 1 & 0 & 0 & 0 & 0\\
1 & -1 & 0 & 0 & -1 & 1 & 0 & 0\\
0 & 0 & 1 & -1 & 0 & 0 & -1 & 1\\
0 & 0 & 0 & 0 & 1 & -1 & 1 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
0 & -1 & -1 & 0\\
-1 & 0 & 0 & -1\\
-1 & 0 & 1 & 0\\
0 & -1 & 0 & 1\\
1 & 0 & -1 & 0\\
0 & 1 & 0 & -1\\
0 & 0 & 1 & -1\\
1 & -1 & 0 & 0
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0\\
A_0^T & 0 & A_1 & 0\\
0 & A_1^T & 0 & A_2\\
0 & 0 & A_2^T & 0\\
\end{matrix}\right)$$
where the zeros are zero matrices. It has eigenvalues $-2\sqrt{2},-2,-\sqrt{2},0,\sqrt{2},2,2\sqrt{2}$ with multiplicity $2,4,2,1,2,4,2$ respectively. The monogenic functions are multiples of $\ell_3+\ell_4+\ell_5+\ell_6$.\
The action of $\slashed{\partial}^2$ will have the same eigenvectors with the square of the corresponding eigenvalue. Hence the sound is equal to 72.
If we just add the cell $v_5$ with $\partial v_5 = \ell_1+\ell_2$, then the only thing that changes with respect to the previous example is the matrix $A_2$:
$$A_2=\left(\begin{matrix}
0 & -1 & -1 & 0 & 1\\
-1 & 0 & 0 & -1 & 1\\
-1 & 0 & 1 & 0 & 0\\
0 & -1 & 0 & 1 & 0\\
1 & 0 & -1 & 0 & 0\\
0 & 1 & 0 & -1 & 0\\
0 & 0 & 1 & -1 & 0\\
1 & -1 & 0 & 0 & 0
\end{matrix}\right)$$
Now calculating the eigenvalues of Dirac operator we get $-2\sqrt{2}, -2, -\sqrt{2},0,\sqrt{2},2,2\sqrt{2}$ with multiplicities $2,5,1,2,1,5,2$. The monogenic functions are linear combinations of $\ell_3+\ell_4+\ell_5+\ell_6$ and $v_1+v_2+v_3+v_4+2v_5$. An easy calculations shows that the sound is 76.\
Adding an extra cell $C$ such that $\partial C = v_1 + v_2 + v_3 + v_4 + 2v_5$, yields an extra matrix:
$$A_3 = \left(\begin{matrix}
1\\
1\\
1\\
1\\
2
\end{matrix}\right)$$
In this case, the total matrix representation of the Dirac operator becomes
$$A=\left(\begin{matrix}
0 & A_0 & 0 & 0 & 0\\
A_0^T & 0 & A_1 & 0 & 0\\
0 & A_1^T & 0 & A_2 & 0\\
0 & 0 & A_2^T & 0 & A3\\
0 & 0 & 0 & A_3^T & 0
\end{matrix}\right)$$
Now we have Eigenvalues $-2\sqrt{2},-2,-\sqrt{2},0,\sqrt{2},2,2\sqrt{2}$ with multiplicities $3,5,1,1,1,5,3$. The sound here is equal to 92 and the monogenic and harmonic functions are multiples of $\ell_3+\ell_4+\ell_5+\ell_6$.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & 1 & -1 & 1 & 0 & 0\\
1 & -1 & 0 & 0 & -1 & 1\\
0 & 0 & 1 & -1 & 1 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
0 & -1 & -1 & 0\\
-1 & 0 & 0 & -1\\
-1 & 0 & 1 & 0\\
0 & -1 & 0 & 1\\
1 & -1 & 0 & 0\\
0 & 0 & 1 & -1
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0\\
A_0^T & 0 & A_1 & 0\\
0 & A_1^T & 0 & A_2\\
0 & 0 & A_2^T & 0\\
\end{matrix}\right)$$
where the zeros are zero matrices. It has eigenvalues $-\sqrt{6},-\sqrt{3},-\sqrt{2},\sqrt{2},\sqrt{3},\sqrt{6}$ with multiplicity $3,1,3,3,1,3$ respectively. There are no non-zero monogenic functions.\
The eigenvalues of $\slashed{\partial}^2$ will be the square of the eigenvalues of $\slashed\partial$. Hence the sound is equal to 54.
If we just add the cell $v_5$ with $\partial v_5 = \ell_1+\ell_2$, then the only thing that changes is the matrix $A_2$:
$$A_2=\left(\begin{matrix}
0 & -1 & -1 & 0 & 1\\
-1 & 0 & 0 & -1 & 1\\
-1 & 0 & 1 & 0 & 0\\
0 & -1 & 0 & 1 & 0\\
1 & -1 & 0 & 0 & 0\\
0 & 0 & 1 & -1 & 0
\end{matrix}\right)$$
Now calculating the eigenvalues of Dirac operator we get $-\sqrt{6},-2,-\sqrt{3},-\sqrt{2},0,\sqrt{2},\sqrt{3},2,\sqrt{6}$ with multiplicities $3,1,1,2,1,2,1,1,3$. The monogenic functions are multiples of $v_1+v_2+v_3+v_4+2v_5$. An easy calculations shows that the sound is 58.
Adding an extra cell $C$ such that $\partial C = v_1 + v_2 + v_3 + v_4 + 2v_5$, yields an extra matrix:
$$A_3 = \left(\begin{matrix}
1\\
1\\
1\\
1\\
2
\end{matrix}\right)$$
and the total matrix representation of the Dirac operator becomes
$$A=\left(\begin{matrix}
0 & A_0 & 0 & 0 & 0\\
A_0^T & 0 & A_1 & 0 & 0\\
0 & A_1^T & 0 & A_2 & 0\\
0 & 0 & A_2^T & 0 & A3\\
0 & 0 & 0 & A_3^T & 0
\end{matrix}\right)$$
Now we have Eigenvalues $-2\sqrt{2},-\sqrt{6},-2,-\sqrt{3},-\sqrt{2},\sqrt{2},\sqrt{3},2,\sqrt{6},2\sqrt{2}$ with multiplicities $1,3,1,1,2,2,1,1,3,1$. The sound here is equal to 74.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & 0 & -1 & 0 & 0 & 1\\
1 & 0 & 0 & 1 & -1 & 0\\
0 & -1 & 1 & -1 & 0 & 0\\
0 & 1 & 0 & 0 & 1 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
-1 & -1\\
1 & -1\\
1 & 0\\
0 & 1\\
-1 & 0\\
0 & -1
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0\\
A_0^T & 0 & A_1 & 0\\
0 & A_1^T & 0 & A_2\\
0 & 0 & A_2^T & 0\\
\end{matrix}\right)$$
where the zeros are zero matrices. It has eigenvalues $-2,0,2$ with multiplicity $6,1,6$ respectively. The monogenic functions are multiples of $\ell_3 + \ell_4 + \ell_5 + \ell_6$.\
The action of $\slashed{\partial}^2$ has eigenvalues $0,4$ with multiplicity $1,12$ respectively. The harmonic functions are also multiples of $\ell_3 + \ell_4 + \ell_5 + \ell_6$. The sound is equal to 48.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & 0 & -1 & 0 & 0 & 1\\
1 & 0 & 0 & 1 & -1 & 0\\
0 & -1 & 1 & -1 & 0 & 0\\
0 & 1 & 0 & 0 & 1 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
-1 & -1 & 0\\
1 & -1 & 0\\
1 & 0 & 1\\
0 & 1 & 1\\
-1 & 0 & 1\\
0 & -1 & 1
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0\\
A_0^T & 0 & A_1 & 0\\
0 & A_1^T & 0 & A_2\\
0 & 0 & A_2^T & 0\\
\end{matrix}\right)$$
where the zeros are zero matrices. It has eigenvalues $-2,2$ with multiplicity $7,7$ respectively. There are no non-zero monogenic functions.\
The action of $\slashed{\partial}^2$ has eigenvalue $4$ with multiplicity $14$. There are no non-zero harmonic functions. The sound is equal to 56.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & 0 & -1 & 0 & 0 & 1 & 1 & 0\\
1 & 0 & 0 & -1 & 1 & 0 & -1 & 0\\
0 & -1 & 1 & 0 & -1 & 0 & 0 & 1\\
0 & 1 & 0 & 1 & 0 & -1 & 0 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
-1 & -1 & 0 & 0\\
1 & -1 & 0 & 0\\
1 & 0 & -1 & 0\\
-1 & 0 & 1 & 0\\
0 & 1 & 0 & -1\\
0 & -1 & 0 & 1\\
0 & 0 & -1 & -1\\
0 & 0 & 1 & -1
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0\\
A_0^T & 0 & A_1 & 0\\
0 & A_1^T & 0 & A_2\\
0 & 0 & A_2^T & 0\\
\end{matrix}\right)$$
where the zeros are zero matrices. It has eigenvalues $-\sqrt{6},-2,-\sqrt{2},0,\sqrt{2},2,\sqrt{6}$ with multiplicity $4,2,2,1,2,2,4$ respectively. The monogenic functions are multiples of $\ell_3 + \ell_4 + \ell_5 + \ell_6$.\
The action of $\slashed{\partial}^2$ has eigenvalues $0,2,4,6$ with multiplicity $1,4,4,8$ respectively. The harmonic functions are also multiples of $\ell_3 + \ell_4 + \ell_5 + \ell_6$. The sound is equal to 72.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
$$\begin{aligned}
A_0&=\left(\begin{matrix}
1 & 1 & 1 & 1
\end{matrix}\right)\\
A_1&=\left(\begin{matrix}
-1 & -1 & 0 & 0 & -1 & -1 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & -1 & -1 & 1 & 1 & 0 & 0\\
0 & 0 & 1 & 1 & 0 & 0 & -1 & -1
\end{matrix}\right)\\
A_2&=\left(\begin{matrix}
-1 & -1 & 0 & 0 & -1 & -1 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & -1 & -1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 1 & 0 & -1 & 0 & -1\\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\
0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1
\end{matrix}\right)\\
A_3&=\left(\begin{matrix}
-1 & 0 & -1 & 0\\
0 & 1 & 0 & 1\\
1 & 0 & 0 & -1\\
0 & -1 & 1 & 0\\
1 & -1 & 0 & 0\\
1 & -1 & 0 & 0\\
0 & 0 & 1 & -1\\
0 & 0 & 1 & -1
\end{matrix}\right)\end{aligned}$$
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0 & 0\\
A_0^T & 0 & A_1 & 0 & 0\\
0 & A_1^T & 0 & A_2 & 0\\
0 & 0 & A_2^T & 0 & A_3\\
0 & 0 & 0 & A_3^T & 0
\end{matrix}\right)$$
where the zeros are zero matrices. It has eigenvalues $-2\sqrt{2},-2,-\sqrt{2},0,\sqrt{2},2,2\sqrt{2}$ with multiplicity $4,6,2,1,2,6,4$ respectively. The monogenic functions are multiples of $lt_1 - lb_1 - lt_2 + lb_2$.\
The action of $\slashed{\partial}^2$ has eigenvalues $0,2,4,8$ with multiplicity $1,4,12,8$ respectively. The harmonic functions are also multiples of $lt_1 - lb_1 - lt_2 + lb_2$. The sound is equal to 120.
In this case $A_2$ and $A_3$ change to
$$\begin{aligned}
A_2&=\left(\begin{matrix}
-1 & -1 & 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0\\
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1\\
0 & 0 & -1 & -1 & -1 & -1 & 1 & 0 & 1 & 0\\
0 & 0 & 1 & 1 & 1 & 1 & 0 & -1 & 0 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1
\end{matrix}\right)\\
A_3&=\left(\begin{matrix}
-1 & 0 & -1 & 0\\
0 & 1 & 0 & 1\\
1 & 0 & 0 & 0\\
0 & 0 & 1 & 0\\
0 & -1 & 0 & 0\\
0 & 0 & 0 & -1\\
1 & -1 & 0 & 0\\
1 & -1 & 0 & 0\\
0 & 0 & 1 & -1\\
0 & 0 & 1 & -1
\end{matrix}\right)\end{aligned}$$
So now the Dirac operator has eigenvalues
$$\begin{aligned}
\begin{array}{l|l}
\text{Eigenvalues} & \text{multiplicity}\\
\hline
-\sqrt{8+2\sqrt{2}} & 1\\
-\sqrt{7} & 1\\
-\sqrt{8-2\sqrt{2}} & 1\\
-2\sqrt{2} & 2\\
-\sqrt{5} & 1\\
-2 & 4\\
-\sqrt{3} & 1\\
-1 & 1\\
0 & 3\\
1 & 1\\
\sqrt{3} & 1\\
2 & 4\\
\sqrt{5} & 1\\
2\sqrt{2} & 2\\
\sqrt{8-2\sqrt{2}} & 1\\
\sqrt{7} & 1\\
\sqrt{8+2\sqrt{2}} & 1
\end{array}\end{aligned}$$
The monogenic functions are linear combinations of $lt_1 - lb_1 - lt_2 + lb_2$, $-vi_1 + vi_2-vp_1-vp_2+vp_3+vp_4$ and $-2vi_1+2vi_2-4vp_2+4vp_3-vt_1-vt_2+vb_1+vb_2$.\
The eigenvalues of the Laplace operator are the square of the eigenvalues of the Dirac operator. The harmonic functions are also linear combinations of $lt_1 - lb_1 - lt_2 + lb_2$, $-vi_1 + vi_2-vp_1-vp_2+vp_3+vp_4$ and $-2vi_1+2vi_2-4vp_2+4vp_3-vt_1-vt_2+vb_1+vb_2$. The sound is equal to 128.
Let $A_k$ denote the matrix representation of $\partial$ applied on $\mathcal{C}_k$, we have:
Hence for the matrix representation of $\slashed{\partial}$ we get
$$A:=\left(\begin{matrix}
0 & A_0 & 0 & 0\\
A_0^T & 0 & A_1 & 0\\
0 & A_1^T & 0 & A_2\\
0 & 0 & A_2^T & 0
\end{matrix}\right)$$
where the zeros are zero matrices. It has the following eigenvalues
$$\begin{aligned}
\begin{array}{l|l}
\text{Eigenvalues} & \text{multiplicity}\\
\hline
-\sqrt{10} & 1\\
-\sqrt{5+\sqrt{5}} & 3\\
-\sqrt{5} & 4\\
-\sqrt{5-\sqrt{5}} & 3\\
-\sqrt{2} & 5\\
\sqrt{2} & 5\\
\sqrt{5-\sqrt{5}} & 3\\
\sqrt{5} & 4\\
\sqrt{5+\sqrt{5}} & 3\\
\sqrt{10} & 1
\end{array}\end{aligned}$$
There are no non-zero monogenic functions.\
The action of $\slashed{\partial}^2$ has eigenvalues $2,5-\sqrt{5},5,5+\sqrt{5},10$ with multiplicities 10,6,8,6,2 respectively. Hence there are no non-zero harmonic functions and the sound is equal to 140.
Cartesian Products of Scripts
=============================
There are two interesting ways to define the cartesian products of scripts depending on whether or not one uses the accumulator $"1"$
Cubic cartesian product
-----------------------
To define the “cubic” cartesian product we start from two scripts:
$$\mathcal{S}:\qquad 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0)\leftarrow^{\partial} \dots \leftarrow M_k(\mathcal{C}_k)$$
$$\mathcal{S'}:\qquad 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0')\leftarrow^{\partial} \dots \leftarrow M_k(\mathcal{C}_k'),$$
and consider the truncated complexes:
$$\mathcal{S}^{\bullet}:\qquad 0 \leftarrow M_0(\mathcal{C}_0)\leftarrow^{\partial} \dots$$
$$\mathcal{S'}^{\bullet}:\qquad 0\leftarrow M_0(\mathcal{C}_0')\leftarrow^{\partial} \dots$$
Then the cubic cartesian product $\mathcal{S}^{\cdot}\times \mathcal{S'}^{\cdot}$ is defined as the complex
$$\mathcal{S}^{\bullet}\times \mathcal{S'}^{\bullet}:\qquad 0\leftarrow M_0(\mathcal{C}_0'')\leftarrow^{\partial} \dots \leftarrow M_k(\mathcal{C}_k'')$$ where
$$\mathcal{C}_k''=\bigcup_{s=0}^k \mathcal{C}_s\times \mathcal{C}_{k-s}'$$
and $$\mathcal{C}_s\times \mathcal{C}_{k-s}'=\{(C_j^s,C_l^{'k-s})\, C_j^s\in \mathcal{C}_s\, C_l^{'k-s}\in\mathcal{C}_{k-s}'\}\, .$$
Hereby, we define
$$\partial (C_j^s,C_l^{'k-s})=(\partial C_j^s,C_l^{'k-s})+(-1)^s(C_j^s,\partial C_l^{'k-s})$$ and also assume linearity $$(\lambda C_i^s+\mu C_j^s, C^l)=\lambda (C_i^s, C^l)+\mu(C_j^s, C^l).$$
Notice that $$\partial (\partial C_j^s,C_l^{'k-s})=(-1)^{s-1}(\partial C_j^s,\partial C_l^{'k-s})$$ and $$\partial (C_j^s,\partial C_l^{'k-s})=(\partial C_j^s,\partial C_l^{'k-s})$$
from which one can readily obtain that $\partial^2=0$ and, since $M_k(\mathcal{C}_k'')$ is the free $\mathbb{Z}-$module over $\mathcal{C}_k''$, we obtain a new complex.
In a similar way, one can also introduce longer cartesian products such as
$$\mathcal{S}_1^{\bullet}\times \mathcal{S}_2^{\bullet}\times \cdots \times \mathcal{S}_l^{\bullet},$$
where $$\mathcal{S}_j^{\bullet}:\qquad 0 \leftarrow M_0(\mathcal{C}_{0,j})\leftarrow \dots \leftarrow M_k(\mathcal{C}_{k,j})\, .$$
The product complex is given by:
$$\prod_{j=1}^{l}\mathcal{S}_j^{\bullet}:= 0 \leftarrow M_0(\mathcal{C}''_{0})\leftarrow \dots \leftarrow M_k(\mathcal{C}''_{k})\, ,$$ whereby $$\mathcal{C}_k''=\bigcup_{k_1+\cdots+k_l=k} \mathcal{C}_{k_1,1}\times \cdots \times \mathcal{C}_{k_l,l}\, ,$$ and $$\mathcal{C}_{k_1,1}\times \cdots \times \mathcal{C}_{k_l,l}\,$$
is the set of $l-$tuples of the form $$(C_{j1}^{k_1, 1},\cdots, C_{jl}^{k_l, l})\, ,$$ with $C_{js}^{k_s, s}\in \mathcal{C}_{k_s,s}$. The boundary operator is given by $$\partial (C_{j1}^{k_1, 1},\cdots, C_{jl}^{k_l, l})=(\partial C_{j1}^{k_1, 1},\cdots, C_{jl}^{k_l, l})+(-1)^{k_1}(C_{j1}^{k_1, 1},\partial C_{j2}^{k_2, 2}\cdots, C_{jl}^{k_l, l})+\cdots
+(-1)^{k_1+\cdots+k_{l-1}}(C_{j1}^{k_1, 1},\cdots,\partial C_{jl}^{k_l, l}).$$
Examples
--------
We will elaborate on the following examples:
- The $l-$dimensional cube $\mathcal{Q}_l^{\bullet}$
- The multicube $\mathcal{Q}_{l_1,\dots, l_s}^{\bullet}$
- The semigrid $\mathbb{N}^m$.
- The grid $\mathbb{Z}^m$
- The torus $\mathbb{Z}_{k_1,\dots,k_s}$
[**i)**]{} The one dimensional cube is the script:
$$\mathcal{Q}_1:\qquad 0 \leftarrow \mathbb{Z} \leftarrow M_0(\{p_0,p_1\})\leftarrow M_1(\{ l\})\, ,$$
with $\partial p_0=\partial p_1=1$ and $\partial l=p_1-p_0$.
The $l-$ dimensional cube $\mathcal{Q}_l^{\bullet}$ is then defined to be:
$$\mathcal{Q}_l^{\bullet}=\mathcal{Q}_1^{\bullet}\times\mathcal{Q}_1^{\bullet}\times \cdots \times\mathcal{Q}_1^{\bullet}\, ,$$ for example $\mathcal{Q}_3^{\bullet}=\mathcal{Q}_1^{\bullet}\times\mathcal{Q}_1^{\bullet}\times\mathcal{Q}_1^{\bullet}\,$ consists of the following elements:
- 8 points: $$\mathcal{C}_0''=\{(p_{j_1}, p_{j_2}, p_{j_2})|\, j_1,\,j_2,\,j_3\in\{0,1\}\}\, ,$$
- 12 lines: $$\mathcal{C}_1''=\{(l,p_{j_1}, p_{j_2})|\, j_1,\,j_2\in\{0,1\}\}\bigcup \{(p_{j_1}, l, p_{j_2})|\, j_1,\,j_2\in\{0,1\}\}\bigcup \{(p_{j_1}, p_{j_2}, l)|\, j_1,\,j_2\in\{0,1\}\}\, ,$$
- 6 planes: $$\mathcal{C}_2''=\{(l,l,p_{j})|\, j\in\{0,1\}\}\bigcup \{(l,p_{j}, l)|\, j\in\{0,1\}\}\bigcup \{(l,l,p_{j})|\, j\in\{0,1\}\} \, ,$$
- 1 volume element: $$\mathcal{C}_3''=\{(l, l, l)\}\, .$$
The boundary maps $\partial$ for the cube are evident. For example, we have: $$\partial (l, l, l)=(p_1-p_0,l,l)-(l,p_1-p_0,l)+(l,l,p_1-p_0)\, ,$$
$$\partial (l,l,p_j)=(p_1-p_0,l,p_j)-(l,p_1-p_0,p_j)\, ,$$
$$\partial (l,p_{j_1},p_{j_2})=(p_1-p_0,p_{j_1},p_{j_2})\, ,$$ and $\partial (p_{j_1},p_{j_2},p_{j_3})=0$ inside $\mathcal{Q}_3^{\bullet}$ while $\partial (p_{j_1},p_{j_2},p_{j_3})=1$ inside the extended $\mathcal{Q}_3$.
[**ii)**]{} For the multicube $\mathcal{Q}_{l_1,\dots, l_s}^{\cdot}$, we start from the script $\mathbb{Z}_k^{\circ}$ with points: $\{ p_0,p_1,\cdots, p_k \}$ and lines $\{ l_1,l_2,\cdots, l_k \}$, and equations $\partial l_j=p_j-p_{j-1}$.
We then define the multicube as $$\mathcal{Q}_{l_1,\dots, l_s}^{\cdot}=\mathbb{Z}_{l_1}^{\circ}\times\cdots\times \mathbb{Z}_{l_s}^{\circ}.$$
[**iii)**]{} For the semigrid $\mathbb{N}^m$ we first denote $\mathbb{N}$ to be the script with points $\{ p_0,p_1,\cdots, p_k,\cdots \}$ and lines $\{ l_1,l_2,\cdots, l_k,\cdots \}$, and relations $\partial l_j=p_j-p_{j-1}$.
Then we can define $\mathbb{N}^m=\mathbb{N}\times \mathbb{N}\times\cdots \mathbb{N}$, where the cartesian product is taken $m$ times.
[**iv)**]{} For the grid $\mathbb{Z}^m$ we have as starting point the script $\mathbb{Z}$ with points $\{p_j|\, j\in\mathbb{Z}\}$, lines $\{l_j|\, j\in\mathbb{Z}\}$ and relations $\partial l_j=p_j-p_{j-1}$.
We then define $\mathbb{Z}^m=\mathbb{Z}\times\mathbb{Z}\times\cdots\times \mathbb{Z}\, .$
One should also note that $\mathbb{N}^{m_1}\times \mathbb{Z}^{m_2}$ exists.
[**v)**]{} For the torus $\mathbb{Z}_{k_1,\dots,k_s}$ we first define $\mathbb{Z}_k$ to be the $k-$polygon with points $\{ p_0,p_1,\cdots, p_k \}$, lines $\{ l_1,l_2,\cdots, l_k, \}$, relations $\partial l_j=p_j-p_{j-1}$, and glueing constraints $p_k=p_0$, i.e. $\partial l_k=p_0-p_{k-1}$.
The torus is then defined as $$\mathbb{Z}_{k_1,\dots,k_s}=\mathbb{Z}_{k_1}\times\mathbb{Z}_{k_2}\times\cdots \times\mathbb{Z}_{k_s}.$$
Tightness in Cubic Cartesian Products
-------------------------------------
Consider a script:
$$\mathcal{S}:\qquad 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0)\leftarrow^{\partial} M_1(\mathcal{C}_1)\leftarrow \dots$$
and its truncated version:
$$\mathcal{S}^{\bullet}:\qquad 0\leftarrow^{\partial} M_0(\mathcal{C}_0)\leftarrow^{\partial} M_1(\mathcal{C}_1) \leftarrow^{\partial} \dots$$
If script $\mathcal{S}$ is tight then every line $l\in\mathcal{C}_1$has two points $\partial l=q-p$. However, within the truncated complex $\mathcal{S}^{\bullet}$, we have $\partial p=\partial q=0$ and line $l$ is no longer tight because $Z_0(\{p,q\})=\mathbb{Z}^2$. We see then that the accumulator is quite essential in script geometry, it implies that tight lines consist of two points.
Consider a second script:
$$\mathcal{S}':\qquad 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0')\leftarrow^{\partial} M_1(\mathcal{C}_1')\leftarrow \dots$$
Even if $\mathcal{S}$ and $\mathcal{S}'$ are tight, $\mathcal{S}\times \mathcal{S}'$ won’t be tight. We need to “attach” an accumulator.
Consider the extension $S^{\prime\prime}$ of $S\times S^{\prime}$ given by $$S^{\prime\prime}: 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0^{\prime\prime})\leftarrow M_1(\mathcal{C}_1^{\prime\prime})\leftarrow \ldots$$ with $\partial(p,q)=1$ for $(p,q)\in \mathcal{C}_0^{\prime\prime}$. Then $S^{\prime\prime}$ is still a complex.
We have to prove that for every line $L\in \mathcal{C}_1^{\prime\prime}$, $\partial^2 L=0$. But lines $L$ come in two-forms $L=(l,p^\prime)$ or $L=(p, l^\prime)$and $\partial L$ is given by $(\partial l, p^\prime)=(p-q, p^\prime)$, resp. $(p,\partial l^\prime)=(p,q^\prime-p^\prime)$. Clearly, $\partial(p-q.p^\prime)=\partial((p,p^\prime)-(q,p^\prime))=1-1=0$ and similarly $\partial(p,q^\prime-p^\prime)=0$.
We now come to the main theorem.
Let $S$ and $S^\prime$ be tight scripts then the extended cubic Cartesian product $S^{\prime\prime}$ is also tight.
Let $C_j^k\in \mathcal{C}_k, C^{\prime,l}_k\in \mathcal{C}^{\prime}_l$. Then $\partial(C_j^k,C_i^{\prime, l})=(\partial C_j^k,C_i^{\prime,l})+(-1)^k(C_j^k,\partial C_i^{\prime,l})$ and so ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_j^k,C_i^{\prime, l})$ consists of two parts ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_j^k,C_i^{\prime, l})=({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C_j^k,C_i^{\prime, l})\cup ((C_j^k,{\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}C_i^{\prime, l})$.
Now, consider a cycle $\mathcal{C}_{k+l-1}$ with support in ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_j^k,C_i^{\prime,l})$. Then $$\mathcal{C}_{k+l-1}= \{ (\mathcal{C}_{k-1}, C_i^{\prime,l})+(C^{k}_j,\mathcal{C}_{l-1}^{\prime}) \}$$ for some chains $\mathcal{C}_{k-1}\in M_{k-1}({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_j^k))$ and $\mathcal{C}^{\prime}_{l-1}\in M_{l-1}({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(C_i^{\prime, l}))$.
Now, $$\begin{aligned}
0=\partial \mathcal{C}_{k+l} & = & (\partial \mathcal{C}_{k-1}, C_i^{\prime,l})+(-1)^{k-1}(\mathcal{C}_{k-1},\partial C_i^{\prime, l})\\
&&+(\partial C_j^k, \mathcal{C}^{\prime}_{l-1})+(-1)^k(C_j^k,\partial \mathcal{C}^{\prime}_{l-1})\end{aligned}$$ implies that the parts of different dimensions in this sum are also zero. In particular, $$(\partial \mathcal{C}_{k-1}, C_i^{\prime, l})=(C_j^k, \partial \mathcal{C}^{\prime}_{l-1})=0$$ and so $\partial \mathcal{C}_{k-1}=0\; \& \; \partial \mathcal{C}^{\prime}_{l-1}=0$.
As $S$ and $S^\prime$ are tight, there are constants $\lambda,\mu$, such that $ \mathcal{C}_{k-1}=\lambda \partial C_j^k$, $ \mathcal{C}^{\prime}_{l-1}=\mu \partial C_i^l$. We also have that $$\begin{aligned}
0 & = & (-1)^{k-1}( \mathcal{C}_{k-1},\partial C_i^{\prime, l})+(\partial C_j^k, \mathcal{C}^{\prime}_{l-1})\\
& = & (-1)^{k-1}\lambda (\partial C_j^{k}, \partial C_i^{\prime, l})+\mu (\partial C_j^k,\partial C_i^{\prime, l})\end{aligned}$$ so that, in fact $\mu=(-1)^k\lambda$.
But that implies that $$\begin{aligned}
\mathcal{C}_{k+l} & = & \lambda(\partial C_j^k,C_i^{\prime,l})+(-1)^k\lambda (C_j^k,\partial C_i^{\prime, l})\\
& = & \lambda \partial (C_j^k, C_i^{\prime, l})\end{aligned}$$ which means that $(C_j^k,C_i^{\prime, l})$ is tight.
The 3D Lie sphere $LS^3$
------------------------
Analysis:
The $3-$dimensional Lie sphere is the non-oriented $3-$dimensional manifold consisting of points in $\mathbb{C}^3$ of the form $e^{i\theta}\underline{\omega}, \theta\in [0,\pi[,\underline{\omega}\in S^2$.
In polar coordinates we put $$\underline{\omega}=\cos\psi(\cos\varphi e_1+\sin\varphi e_2)+\sin\psi e_3.$$
We now create a script for $S^2$ putting $$\begin{aligned}
p_1 & \leftrightarrow & e_1, p_2\leftrightarrow -e_1\\
l_1 & \leftrightarrow & \cos\varphi e_1+\sin\varphi e_2, \varphi\in ]0,\pi[\\
l_2 & \leftrightarrow & \cos\varphi e_1-\sin\varphi e_2, \varphi\in ]0,\pi[\\
v_1 & \leftrightarrow & \cos\psi(\cos\varphi e_1+\sin\varphi e_2)+\sin\psi e_3, \varphi\in [0,2\pi [, \psi\in ]0, \frac{\pi}{2}]\\
v_1 & \leftrightarrow & \cos\psi(\cos\varphi e_1+\sin\varphi e_2)-\sin\psi e_3.\end{aligned}$$ Clearly, it makes sense to consider the script with $C_0=\{p_1,p_2\}, C_1=\{l_1,l_2\}, C_2=\{v_1,v_2\}$ and relations $$\begin{aligned}
& \partial l_1=p_2-p_1, &\partial l_2=p_2-p_1 \\
&\partial v_1=l_2-l_1, & \partial v_2=l_2-l_1\end{aligned}$$ and orientation $v_2-v_1$.
For the set of points $e^{i\theta}, \theta\in [0, \pi-\epsilon]$, we name “1”$\leftrightarrow e^{i0}=1$, “2”$\leftrightarrow e^{i(\pi-\epsilon)}$, $I\leftrightarrow \mbox{ set }e^{i\theta}, \theta\in ]0,\pi-\epsilon[$ and together they form the script $J$ with $$C_0=\{1,2\}, C_1=\{I\}, \partial I =``2''-``1''.$$ Next we form the cartesian product of scripts $S^2\times J$ and a script for $LS^3$ will emerge if we take the limit $\epsilon\rightarrow 0$ and make the necessary identifications on gluings for the end-cells.
In extenso for $S^2\in J$ we have 4 points ($(p_1,1),(p_1,2),(p_2,1),(p_2,2)$), 6 lines ($(l_1,1),(l_1,2),(l_2,1),(l_2,2),(p_1,I),(p_2,I)$), and 6 planes ($(v_1,1),(v_2,1),(v_1,2),(v_2,2),(l_1,I),(l_2,I)$), and 2 3D-cells ($(v_1,I),(v_2,I)$).
The script relations are given by $$\begin{aligned}
\partial (l_j,k) & = & (p_2,k)-(p_1,k), k=1,2,\\
\partial (v_j,k) & = & (l_2,k)-(l_1,k),\\
\partial (p_j, I) & = & (p_j,2)-(p_j,1),\\
\partial (l_j,I) & = & (p_2,I)-(p_1,I)-(l_j,2)+(l_j,1),\\
\partial (v_j, I) & = & (l_2,I)-(l_1,I)+(v_j,2)-(v_j,1).\end{aligned}$$
We taking limit $\epsilon\rightarrow 0$ it seems to we have to glue:
stage 1: $(p_1,2)=(p_2,1), (p_2,2)=(p_1,1)$
stage 2: $(l_1,2)= - (l_2,1), (l_2,2)=-(l_1,1)$
indeed $\partial (l_1,2)=(p_2,2)-(p_1,2)=(p_1,2)-(p_2,1)=-\partial (l_2,1)$ confirms this
stage 3: $(v_1,2)=(v_2,1)$, $(v_2,2)=(v_1,1)$
To simplify the notation we put $$(p_j,1)=p_j, (l_j,1)=l_j, (v_j,1)=v_j$$ and we arrive at the following script for the Lie sphere with 2 points ($p_1,p_2$), 4 lines ($l_1,l_2,(p_1,I),(p_2,I)$), 4 planes ($v_1,v_2,(l_1,I),(l_2,I)$), and 2 volumes ($(v_1,I),(v_2,I)$).
The script equations are those we have for $S^2:\partial l_j=p_2-p_1,\partial v_j=l_2-l_1$ together with new equations $$\begin{aligned}
\partial(p_1,I) & = & (p_1,2)-(p_1,1)=p_2-p_1\\
\partial(p_2,I) & = & p_1-p_2,\\
\partial(l_1,I) & = & (p_2,I)-(p_1,I)-(l_1,2)+(l_1,1)\\
\partial(l_2,I) & = & \mathrm{ibid}\\
\partial(v_1,I) & = & (l_2,I)-(l_1,I)+v_2-v_1\\
\partial(v_2,I) & = & (l_2,I)-(l_1,I)+v_1-v_2\\
\partial(v_2,I)-\partial(v_1,I)=2(v_1-v_2)\end{aligned}$$
Synthesis: By this we mean the reconstruction of the geometry from the script.
In this case it won’t be possible because the script is not tight. We have $$\partial(l_1,I)=(p_2,I)-(p_1,I)+l_2+l_1$$ and $\partial(p_2,I)+\partial l_2=p_1-p_2+p_2-p_1=0$ so $(p_2,I)+l_2$ and $-(p_1,I)+l_1$ are linearly independent cycles inside ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(l_1,I)$ and $H_1({\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(l_1,I))=\mathbb{Z}$. So the cells $(l_1,I), (l_2,I)$ are not tight and the canonical 2D-script is no CW complex. The geometrical offprint of the script does not determine the script and hence also not the geometry of the Lie sphere. For examples when comparing the Lie sphere $(l_1,I)$ has to be a rectangle of which the corners are glued together in opposite way (like the Möbius band) but the sides are not glued together. It can be seen that $(l_1,I)$ and $(L_2,I)$ together form a 2D-torus which corresponds to the embedding of Lie sphere $LS^2$ inside $LS^3$. But the script itself does not lead to this interpretation; there is not enough information.
In fact we could introduce new tight $2-$cells $V_1,V_2,W_1,W_2$ with $$\begin{aligned}
& \partial V_1=(p_2,I)+l_2, & \partial V_2=(p_1,I)-l_1\\
& \partial W_1=(p_2,I)+l_1, & \partial W_2=(p_1,I)-l_2\end{aligned}$$ and then put $(l_1,I)=V_1-V_2$, $(l_2,I)=W_1-W_2$.
But that contradicts the Lie sphere because there $(p_2,I)+l_2$ is homologically non-trivial. In fact $(p_2,I)+l_2$ is homologous to $(p_1,I)-l_2$ and the double cycle $(p_2,I)+l_2+(p_1,I)-l_1$ is homologous to the circle $LS^1\leftrightarrow (p_1,I)+(p_2,I)$ ($l_2-l_1=\partial v_1$): the set of points $e^{i\theta},\theta\in[0,2\pi[$.
On the level of planes we see that $(l_1,I)-(l_2,I)$ is a cycle that is in fact $LS^2$ and its homologous to $v_2-v_1$, i.e. $S^2$ and also to $v_1-v_2$ for indeed $2(v_1-v_2)=\partial(v_2-v_1,I)$ is a boundary. But the sphere $v_2-v_1$ itself is not a boundary and hence $LS^2\leftrightarrow (l_1-l_2,I)$ is also not a boundary. It seems that the homological information included in the script is in full agreement with the Lie sphere $LS^3$ in spite of the fact that the script is not tight.
To arrive at a tight script for $LS^3$ we can replace interval $J$ by the “double interval” $J_2$ with $C_0=\{"1","2","3"\}, C_1=\{I_1,I_2\}$, $\partial I_1="2"-"1"$, and $\partial I_2="3"-"2"$..
First we consider the cartesian product $S^2\times J_2$, followed by the list of identifications (and new notations): $$\begin{aligned}
(p_j,1)=p_j, & (l_j,1)=l_j, & (v_j,1)=v_j\\
(p_j,2)=ip_j, & (l_j,2)=il_j, & (v_j,2)=iv_j\\
(p_1,3)=p_2, & (p_2,3)=p_1 & \\
(l_1,3)=-l_2, & (l_2,3)=-l_1 & \\
(v_1,3)=v_2, & (v_2,3)=v_1. &\end{aligned}$$
Apart from the spheres $S^2=\{p_j,l_j,v_j\}$, $iS^2=\{ip_j,il_j,iv_j\}$ we have:
extra lines $(p_j,I_1),(p_j,I_2)$
extra spheres $(l_j,I_1), (l_j, I_2)$
extra volumes $(v_j,I_1), (v_j,I_2)$
Moreover, apart from the relations for the scripts $S^2$ and $iS^2$ we have $$\begin{aligned}
& \partial(p_j,I_1)=ip_j-p_j, & \partial(p_j,I_2)=p_{3-j}-ip_j\\
& \partial(l_j,I_1)=(p_2-p_1, I_1)-il_j+l_j, & \\
& \partial(l_j,I_2)=(p_2-p_1,I_2)+l_{3-j}+il_j & \\
& \partial(v_j,I_1)=(l_2-l_1,I_1)+iv_j-v_j, & \\
& \partial(v_j,I_2)=(l_2-l_1,I_2)+v_{3-j}-iv_j. & \\\end{aligned}$$
Now the geometry of $LS^3$ is fully determined by the script which is also tight. As an exercise one can study homology of Lie sphere, but that gives no new information compared to the previous non-tight scripts. This examples shows that it is not a good idea to systematically demand tightness; non-tight scripts may be much simpler and still relevant. So the idea of script geometry goes beyond CW-complexes and it is more than just a generalization.
A discrete curvature model
--------------------------
\[fig:curvature\]
What is represented in this figure is the geometric offprint of a 2D-script which is unitary and tight. Hence the above figure determines the script up to equivalence. It appears to be a curved or bent 2D-surface with
- a curvature point $P$ with negative curvature $-1$
- a curvature point $Q$ with positive curvature $+1$
All the other points, lines, and spheres are like in the standard grid $\mathbb{Z}^2$ that is flat.
In Einstein’s theory, curvature is linked to gravity and in script geometry, curvature is a property of the geometric offprint alone, not of the actual script itself, that seems closer related to electromagnetism.
But of course we want to have at least one script for which this model is the geometric offprint we will realize it as 2D-surface inside $\mathbb{Z}^3$ where $\mathbb{Z}$ consists of points $a\in\{\ldots,-2,-1,0,1,2,\ldots\}$ and lines $I_j, j\in \mathbb{Z}$ with $\partial I_j="j+1"-"j"$. Hence, in $\mathbb{Z}^3$ itself we have points $(a_1,a_2,a_3)\in\mathbb{Z}^3$, lines $(I_j,a_2,a_3), (a_1,I_j,a_3), (a_1,a_2,I_j)$ with e.g. $\partial(I_j,a_2,a_3)=(j+1,a_2,a_3)-(j,a_2,a_3)$ and planes $(I_j,I_k,a_3)$, $(I_j,a_2,I_k)$, $(a_1,I_j,I_k)$ with e.g. $\partial(I_j,I_k,a_3)=(j+1,I_k,a_3)-(j,I_k,a_3)-(I_j,k+1,a_3)+(I_j,k,a_3)$ and volumes $(I_j,I_k,I_l)$ but we won’t be needing theses.
Hence, the script equations follow from the cartesian product $\mathbb{Z}^3$ and all we have to do is to determine how the curvature model fits into $\mathbb{Z}^3$.
We have decomposed it into 6 overlapping zones consisting of points
- Zone 1: $(a_1,a_2,0) \; \& \; a_1\leq 0, a_2\geq 0$
- Zone 2: $(a_1,a_2,0)\; \& \; a_1\leq 0, a_2\leq 0$
- Zone 3: $(a_1,a_2,0) \; \& \; a_1\geq 0, a_2\leq 0$
- Zone 4: $(a_1,0,a_3) \; \& \; a_1\geq 0, a_3=0,1,2,3$
- Zone 5: $(0,a_2,a_3) \; \& \; a_2\geq 0, a_3=0,1,2,3$
- Zone 6: $(a_1,a_2,3) \; \& \; a_1\geq 0, a_2\ge 0$
Also the lines can be computed zone by zone, of course these will be overlapping.
This leads to the list:
- Lines in Zone 1: $$\begin{aligned}
(I_j,a_2,0) & \& & j<0,a_2\geq 0\\
(a_1,I_j,0) & \& & j\geq 0, a_1\leq 0\end{aligned}$$
- Lines in Zone 2: $$\begin{aligned}
(I_j,a_2,0) & \& & j<0,a_2\leq 0\\
(a_1,I_j,0) & \& & j< 0, a_1\leq 0\end{aligned}$$
- Lines in Zone 3: $$\begin{aligned}
(I_j,a_2,0) & \& & j\geq 0,a_2\leq 0\\
(a_1,I_j,0) & \& & j< 0, a_1\geq 0\end{aligned}$$
- Lines in Zone 4: $$\begin{aligned}
(I_j,0,a_3) & \& & j\geq 0,a_3=0,1,2,3\\
(a_1,0,I_j) & \& & a_1\geq 0, j=0,1,2\end{aligned}$$
- Lines in Zone 5: $$\begin{aligned}
(0,I_j,a_3) & \& & j\geq 0,a_3=0,1,2,3\\
(0,a_2,I_j) & \& & a_2\geq 0, j=0,1,2\end{aligned}$$
- Lines in Zone 6: $$\begin{aligned}
(I_j,a_2,3) & \& & j\geq 0,a_2\geq 0\\
(a_1,I_j,3) & \& & j\geq 0, a_1\geq 0\end{aligned}$$
Similarly, we have the plane elements (no overlapping)
- Planes in Zone 1: $$(I_j,I_k,0)\; ; \; j<0 \; \& \; k\geq 0$$
- Planes in Zone 2: $$(I_j,I_k,0)\; ; \; j<0 \; \& \; k< 0$$
- Planes in Zone 3: $$(I_j,I_k,0) \; ; \; j\geq 0\; \& \; k<0$$
- Planes in Zone 4: $$(I_j,0,I_k)\; ; \; j\geq 0 \; \& \; k= 0,1,2$$
- Planes in Zone 5: $$(0,I_j,I_k)\; ; \; j\geq 0 \; \& \; k= 01,2,3$$
- Planes in Zone 6: $$(I_j,I_k,3)\; ; \; j\geq 0 \; \& \; k\geq 0$$
This fixes the whole script because the script relations follow from the cartesian product. In script geometry much of the creativity lies in finding the best algorithms to describe something. Scripts also involve gravity and electromagnetism combined whereby everything is expressed in terms of chains and their supports.
The “simplicial” cartesian product
----------------------------------
We again start from two scripts $$\begin{aligned}
S & : & 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0)\overset{\partial}{\longleftarrow} \cdots\\
S^\prime & : & 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_0^\prime)\overset{\partial}{\longleftarrow}\cdots\\\end{aligned}$$ Then the “simplicial” cartesian product is defined as the complex $$S\times S^\prime : 0\leftarrow\mathbb{Z}=M_{-2}({1})\overset{\partial}{\longleftarrow}M_{-1}(\mathcal{C}_{-1}^{\prime\prime})\overset{\partial}{\longleftarrow}M_{0}(\mathcal{C}_{0}^{\prime\prime})\overset{\partial}{\longleftarrow}\cdots$$ whereby this time for $k\geq -2$ $$\mathcal{C}_k^{\prime\prime}=\bigcup_{s=-1}^{k+1} \mathcal{C}_s\times\mathcal{C}_{k-s}^\prime$$ and as before $$\mathcal{C}_s\times \mathcal{C}_{k-s}^\prime=\{(C_j^s,C_l^{\prime, k-s}:\ldots)\}$$ so for example $$\begin{aligned}
\mathcal{C}_{-2}^{\prime\prime} & = & \{(1,1)\}=\{1\}\\
\mathcal{C}_{-1}^{\prime\prime} & = & \{(p_j,1),(1,p_j^\prime):p_j\in \mathcal{C}_0,p_j^\prime\in \mathcal{C}_0^\prime\}\\
\mathcal{C}_{0}^{\prime\prime} & = & \{(p_j,p_l^\prime):p_j\in \mathcal{C}_0,p_j^\prime\in \mathcal{C}_0^\prime\}\cup \{(1,l_j^\prime):l_j^\prime\in \mathcal{C}_1^\prime\}\cup \{(l_j,1):l_j\in C_j\}\end{aligned}$$ Just like before the $\partial$-operator is defined by $$\partial (C_j^s,C_l^{\prime, k-s})=(\partial C_j^s,C_l^{\prime, k-s})+(-1)^s(C_j^s,\partial C_l^{\prime, k-s})$$ whereby this time $s\geq -1, k\geq -2$.
So for example $$\begin{aligned}
\partial (p_j,1) & = & (1,1)=1\\
\partial (1,p_j^\prime) & = & -(1,\partial p_j^\prime)=-(1,1)=-1\\
\partial (p_j,p_l^\prime) & = & (1,p_l^\prime)+(p_j,1)\\
\partial (1,l_j^\prime) & = & -1(1,\partial l_j^\prime)=-(1,p_\alpha^\prime-p_\beta^\prime)\end{aligned}$$ and so on.
So the main difference with the cubic case is that we make explicit use of the accumulations of $S$ and $S^\prime$ within the $\partial$-operator. This means that we have extra cells $$(C_j^k,1), (1,C_j^{\prime, k}).$$ Also the elements $(p_j,p_l^\prime)$ behave like lines rather than points while the sets $$\mathcal{C}_0\times\{1\}=\{(p_j,s):\cdots\}, \{ 1 \}\times \mathcal{C}_0^\prime=\{(1,p_j^\prime):\cdots\}$$ are like two sets of points for which $$\partial (p_j,1)=1, \partial (1,p_j^\prime)=-1.$$ This may seem questionable because normally the boundary of a point is +1. But one can always introduce $-(1,p_j^\prime)$ as “new points”. Also the dimensions of cells seem to have a shift $-1$, it is $$\mathrm{dim}1=-2, \qquad \mathrm{dim}(p_j,1)=-1,\qquad \mathrm{dim}(p_j,p_l^\prime)=0.$$ while one would rather expect $$\mathrm{dim}1=-1, \qquad \mathrm{dim}(p_j,1)=0,\qquad \mathrm{dim}(p_j,p_l^\prime)=1.$$ One can of course redefine the dimensions of the cells in this way. But that gives problems when defining longer symplicial cartesian products like $S_1\times S_2\times\ldots\times S_l$, $$\begin{aligned}
S_j & : & 0\leftarrow \mathbb{Z}\leftarrow M_0(\mathcal{C}_{0,j})\overset{\partial}{\longleftarrow}\ldots\\
\times_{j=1}^l S_j & : & 0\leftarrow \mathbb{Z}=M_{-l}(\mathcal{C}_{-l}^{\prime\prime})\overset{\partial}{\longleftarrow} M_{-l+1}(\mathcal{C}_{-l+1}^{\prime\prime})\leftarrow\ldots \overset{\partial}{\longleftarrow} M_{0}(\mathcal{C}_{0}^{\prime\prime})\leftarrow\ldots\end{aligned}$$ whereby for $k\geq -l$ $$\mathcal{C}_k^{\prime\prime}=\bigcup_{k_1+\ldots+k_l=k}\mathcal{C}_{k_1,1}\times \ldots \times \mathcal{C}_{k_l,l}$$ and the boundary operators is still given by $$\begin{aligned}
&& \partial (C_{j_1}^{k_1,1},\ldots,C_{j_l}^{k_l,l})\\
& = & (\partial C_{j_1}^{k_1,1},\ldots,C_{j_l}^{k_l,l})+\ldots +(-1)^{k_1+\ldots+k_l}(C_{j_1}^{k_1,1},\ldots,\partial C_{j_l}^{k_l,l}),\quad k_1,\ldots,k_l=-1,0,\ldots.\end{aligned}$$ Again, one can consider the elements $$(p_j^1,1,\ldots,1),(1,p_j^2,\ldots,1), \ldots, (1, \ldots,p_j^l)$$ as a partition of the whole set of points and one could redefine the dimension of objects $$\mathrm{dim} (\mbox{ object })\rightarrow \mathrm{dim} (\mbox{ object })+l-1$$ to be in agreement with the general theory of scripts and to renormalize the points, lines, etc. so that $\partial$ point$=+1$. But these are rather cosmetic changes one does not need to make.
Let $$S:0\leftarrow \mathbb{Z}\leftarrow M_0(\{p\})$$ the script of a single point. Then $S\times S\times S\times S$ has cells $$\begin{aligned}
k=-4 & : & (1,1,1,1)=1\\
k=-3 & : & (p,1,1,1),(1,p,1,1),(1,1,p,1),(1,1,1,p)\\
k=-2 & : & (p,p,1,1), (p,1,p,1),(p,1,1,p),(1,p,p,1),(1,p,1,p),(1,1,p,p)\\
k=-1 & : & (p,p,p,1),(p,p,1,p),(p,1,p,p),(1,p,p,p)\\
k=0 & : & (p,p,p,p)\end{aligned}$$
The boundary map in this script follows from the general theory. For example, $$\begin{aligned}
\partial (1,p,1,1) & = &-1\\
\partial (p,1,p,1) & = & (1,1,p,1)-(p,1,1,1)\\
\partial (p,p,1,p) & = & (1,p,1,p)+(p,1,1,p)-(p,p,1,1), \mbox{ etc. }\end{aligned}$$ The geometric offprint of this script is the same as that of a symplex (up to dimensional shift). Hence, the script is tight and hence the script is equivalent to a symplex.
Let $S: 0\leftarrow \mathbb{Z}\leftarrow M_0(\{ p,q\})\leftarrow M_1(\{l\}), \partial l=p-q.$. Prove that $S\times S$ is also a 3D symplex.
Concerning tightness we have:
Let $S_1,S_2$ be tight scripts then the symplicial cartesian product $S_1\times S_2$ is also tight.
Adapt the cubic case (exercise).
A similar result holds for $S_1 \times S_2 \times \ldots \times S_l$; in fact one can use associativity $(S_1\times S_2)\times S_3=S_1\times (S_2\times S_3)=S_1\times S_2\times S_3$.
The simplicial refinement
-------------------------
In this subsection we start from a script $$S:0\leftarrow \mathbb{Z}\leftarrow M_0\overset{\partial}{\longleftarrow} \ldots$$ and first consider the $1$-point script $P$ $$C_0=\{p\},\partial p=1.$$ The idea is to construct a canonical symplicial complex such that to every cell $C_j^k$ corresponds a unique chain of symplexes $\sigma(C_j^k)$ so that ${\mbox{\kern -1.3em {{
\renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss}
\renewcommand\encodingdefault{OT2} \normalfont \selectfont }b}}}(\partial C_j^k)=supp \partial^2C_j^k=\{ 0\}$ which implies $\partial\sigma (C_j^k)=\sigma (\partial C_j^k)$, i.e. the script can be replaced via $\sigma$ by a symplicial complex.
The idea is based on the idea that, within $S\times P$, $$\partial (C_j^k,p)=(\partial C_j^k,p)+(-1)^k (C_j^k,1)$$ so that $(C_j^k,1)$ is cobordant with $(-1)^{k-1}(C_j^k,p)$.
The algorithm is recursive and goes in stages.
- Stage 0: Identify $S$ with $(S,1)$ consisting of cells $(C_j^k,1)$ of which the dimension is shifted to $k-1$. In particular $(1,1)$ has dimension $-2$. Next for every point $(p_j,1)\in (S,1)$ we put $\sigma(p_j,1)=-(1,p_j)$.
- Stage $k$: Suppose that we have completed stage $k-1$ and let $(C_j^k,1)\in (C^k,1)$. For each such element we create a new point $p_j^k$ and define $$\sigma (C_j^k)=(-1)^{k-1}(\partial C_j^k,p_j^k)\sim (C_j^k,1)$$ where $\sim$ denotes the cobordism.
The dimensions are the same, it is a chain of symplexes because $\partial C_j^k=\sum$ symplexes and (symplex, point) is a symplex.
Also $\partial\sigma (C_j^k)=(\partial C_j^k,\partial p_j^k)=(\partial C_j^k,1)=\sigma (\partial C_j^k)$ whatever. Finally for every $C_j^{k+1}$, if $C_l^k$ occurs in $\partial C_j^{k+1}$, replace $C_j^k$ by $\sigma(C_l^k)$ and raise the dimension of $\sigma(C_l^k)$ by $+1$.
- Last stage: Cancel all elements $(C_j^k,1)$.
Consider the disc $C_0=\{p_1,p_2\}, C_1=\{l_1,l_2\}, C_2=\{v\}, \partial l_1=\partial l_2=p_2-p_1, \partial v=l_2-l_1$. First we are to replace $p_j\rightarrow -(1,p_j)$ then $\partial l_1=\partial l_2=-(1,p_2)+(1,p_1)$. Next we introduce points $q_1,q_2$ then we replace $$\sigma : l_j\rightarrow +(-(1,p_2)+(1,p_1),q_j)=-((1,p_2),q_j)+((1,p_1),q_j)$$ now we replace $$\partial v=l_2-l_1\rightarrow -((1,p_2),q_2)+((1+p_1),q_2)+((1,p_2),q_1)-((1,p_1),q_1).$$ So taking new point $q$, we obtain $$\sigma: v\rightarrow -((1,p_2),q_2)+((1,p_1),q_2)+((1,p_2),q_1)-((1,p_1,q_1),q);$$ it is the sum of 4 triangles
\[fig:curvature1\]
It is important to know that every script can be refined to a simplicial complex. Note that the refinement of the sphere $S^2$ is an octahedron. For higher dimensions: similar story.
[99]{}
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P. Cerejeiras, U. Kähler, F. Sommen, A. Vajiac, Script Geometry, in: S. Bernstein, U. Kähler, I. Sabadini F. Sommen, Modern Trends in Hypercomplex Analysis, Birkhäuser, Basel, 2016, 79-110.
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Affilations {#affilations .unnumbered}
-----------
Paula Cerejeiras\
CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics,University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal\
[email protected]\
Uwe Kähler\
CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics,University of Aveiro, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal\
[email protected]\
Teppo Mertens\
Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Ghent, Belgium\
[email protected]\
Frank Sommen\
Clifford Research Group, Department of Mathematical Analysis, Ghent University, Galglaan 2, B-9000 Ghent, Belgium\
[email protected]\
Adrian Vajiac\
CECHA - Center of Excellence in Complex and Hypercomplex Analysis, Chapman University, One University Drive, Orange CA 92866\
[email protected]\
MihaelaVajiac\
CECHA - Center of Excellence in Complex and Hypercomplex Analysis, Chapman University, One University Drive, Orange CA 92866\
[email protected]\
|
---
abstract: |
We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters $$(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\ (lq-l+1)q^{l-2})$$ for integers $q$ and $l$ ($q, l\ge 2$), and $$(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1)$$ for all prime powers $q$ and $l\in \{1, 2, \dots, q\}$. The new graphs given by these constructions have parameters $(36, 12, 5, 2, 5)$, $(54, 18, 7, 4,
7)$, $(72, 24, 10, 4, 10)$, $(96, 24, 7, 3, 7)$, $(108, 36, 14, 8,
14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on “Parameters of directed strongly regular graphs," at ${http://homepages.cwi.nl/^\sim aeb/math/dsrg/dsrg.html}$ by S. Hobart and A. E. Brouwer. We review these constructions and show how our methods may be used to construct other infinite families of directed strongly regular graphs.
address:
- 'Department of Mathematics, Iowa State University, Ames, Iowa, 50011, U. S. A.'
- 'Department of Mathematics, Iowa State University, Ames, Iowa, 50011, U. S. A.'
author:
- Oktay Olmez
- 'Sung Y. Song'
title: Construction of directed strongly regular graphs using finite incidence structures
---
[ Affine planes; group-divisible designs; partial geometries.]{}
Introduction and preliminaries {#sec:intro}
==============================
Directed strongly regular graphs were introduced by Duval [@Du] in 1988 as directed versions of strongly regular graphs. There are numerous sources for these graphs. Some of known constructions of these graphs use combinatorial block designs [@FK], coherent algebras [@FK; @KM], finite geometries [@FK0; @FK; @GH; @KP], matrices [@Du; @EH; @GH], and regular tournaments [@EH; @Jo]. Some infinite families of these graphs also appear as Cayley graphs of groups [@DI; @HS; @Jo; @Jo2; @KM].
In this paper, we construct some new infinite families of directed strongly regular graphs by using certain finite incidence structures, such as, non-incident point-block pairs of a divisible design and anti-flags of a partial geometry.
In Section 2, we describe a construction of a directed strongly regular graph with parameters $(v, k, t, \lambda, \mu)$ given by $$(l(q-1)q^l,\ l(q-1)q^{l-1},\ (lq-l+1)q^{l-2},\ (l-1)(q-1)q^{l-2},\
(lq-l+1)q^{l-2}),$$ for integers $l\ge 2$ and $q\ge 2$. The graph is defined on the set of non-incident point-block pairs of group divisible design GD$(l, q^{l-2}, q; ql)$. Among the feasible parameters listed in [@BH], our construction realizes the feasibility of the parameter sets $(36, 12, 5, 2, 5)$ and $(96, 24,
7, 3, 7)$; and then $(72, 24, 10, 4, 10)$ and $(108, 36, 15, 6, 15)$ by applying for a construction of Duval [@Du].
In section 3, we construct a directed strongly regular graph on the set of anti-flags of a partial geometry. In particular, if we use the partial geometry obtained from the affine plane of order $q$ by considering all $q^2$ points and taking the $ql$ lines from exactly $l$ pencils (parallel classes) of the plane, we obtain a directed strongly regular graph with parameters $$(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1).$$ Thus, for example, it follows the feasibility of the parameter sets $(54, 18, 7, 4, 7)$ and $(108, 36, 14, 8, 14)$ among those listed in [@BH].
In section 4, we construct two families of directed strongly regular graphs with parameters $$(ql(l-1),\ q(l-1),\ q,\ 0,\ q)$$ and $$(ql(l-1),\ 2q(l-1)-1,\ ql-1,\ ql-2,\ 2q)$$ for integers $q\ge 2\mbox{ and } l\ge 3$. These graphs are not new. J[ø]{}rgensen [@Jo2] showed that the graph with parameters $(ql(l-1),\ q(l-1),\ q,\ 0,\ q)$ is unique for every integers $l\ge
2$ and $q\ge 2$. Godsil, Hobart and Martin constructed the second family of graphs in [@GH Corollary 6.3]. Our construction for the second family is essentially similar to the one in [@GH]. However, both families of the graphs are constructed by using the almost trivial incidence structure obtained from a partition of a set of $ql$ elements into $l$ mutually disjoint $q$-element subsets.
While there is a number of families of directed strongly regular graphs constructed by using flags of finite geometries (cf. [@BH; @FK; @GH; @KM; @KP]), there are few results for using anti-flags or partial geometries. The results presented here can only begin to illustrate some constructions using anti-flags of partial geometries or divisible designs. More work is needed to explore more general incidence structures and address the existence and non-existence problems of directed strongly regular graphs and their related incidence structures.
In the remainder of this section, we recall the definition and some properties of directed strongly regular graphs.
A loopless directed graph $D$ with $v$ vertices is called *directed strongly regular graph* with parameters $(v, k, t,
\lambda, \mu)$ if and only if $D$ satisfies the following conditions:
- Every vertex has in-degree and out-degree $k$.
- Every vertex $x$ has $t$ out-neighbors, all of which are also in-neighbors of $x$.
- The number of directed paths of length two from a vertex $x$ to another vertex $y$ is $\lambda$ if there is an edge from $x$ to $y$, and is $\mu$ if there is no edge from $x$ to $y$.
Another definition of a directed strongly regular graph, in terms of its adjacency matrix, is often conveniently used. Let $D$ be a directed graph with $v$ vertices. Let $A$ denote the adjacency matrix of $D$, and let $I=I_v$ and $J=J_v$ denote the $v\times v$ identity matrix and all-ones matrix, respectively. Then $D$ is a directed strongly regular graph with parameters $(v, k, t, \lambda,
\mu)$ if and only if (i) $JA=AJ=kJ$ and (ii) $A^2=tI+\lambda A+\mu
(J-I-A)$.
Duval observed that if $t=\mu$ and $A$ satisfies above equations (i) and (ii), then so does $A\otimes J_m$ for every positive integer $m$; and so, we have:
\[pro1\] [@Du] If there exists a directed strongly regular graph with parameters $(v,k,t,\lambda,\mu)$ and $t=\mu$, then for each positive integer $m$ there exists a directed strongly regular graph with parameters $(mv,mk,mt,m\lambda,m\mu)$.
Throughout the paper, we will write $x\rightarrow y$ if there is an edge from a vertex $x$ to another vertex $y$, and $x\nrightarrow y$ if there is no edge from $x$ to $y$. We will also write $x\leftrightarrow y$ if and only if both $x\rightarrow y$ and $y\rightarrow x$.
Construction of graphs using certain divisible designs {#sec:CBG}
======================================================
The first family of directed strongly regular graphs, which we shall describe in this section, use non-incident point-block pairs of group divisible designs GD$(l,q^{l-2}, q; ql)$ for integers $q\ge 2$ and $l\ge 2$.
Let $P$ be a $ql$-element set with a partition ${\mathcal}{P}$ of $P$ into $l$ parts (‘*groups*’) of size $q$. Let ${\mathcal}{P}=\{S_1, S_2,
\dots, S_l\}$. Let $${\mathcal}{B}=\{B\subset P:\ |B\cap S_i|=1 \mbox{ for all } i=1,2, \dots,
l\}.$$ Then ${\mathcal}{B}$ consists of $q^l$ subsets of size $l$ called *blocks*. The elements of $P$ will be called *points* of the incidence structure $(P, {\mathcal}{B})$ with the natural point-block incidence relation $\in$. This structure has property that any two points from the same group never occur together in a block while any two points from different groups occur together in $q^{l-2}$ blocks. It is known as a group-divisible design GD$(l,q^{l-2}, q; ql)$.
\[D1\] Let $(P, {\mathcal}{B})$ be the incidence structure defined as above. Let $D=D(P, {\mathcal}{B})$ be the directed graph with its vertex set $$V(D)=\{(p, B)\in P\times {\mathcal}{B}:\ p\notin B\},$$ and directed edges given by $(p, B)\rightarrow (p', B')$ if and only if $p\in B'$.
\[dsrg1\] Let $D$ be the graph $D(P,{\mathcal}{B})$ defined in Definition \[D1\]. Then $D$ is a directed strongly regular graph with parameters $$\begin{array}{l} v=lq^l(q-1),\\
k= lq^{l-1}(q-1),\\ t=\mu=q^{l-2}(lq-l+1),\\
\lambda=q^{l-2} (l-1)(q-1).\\ \end{array}$$
It is easy to verify the values of $v$ and $k$. To find the value of $t$, let $(p, B)\in V(D)$ with $B=\{b_1, b_2, \dots, b_l\}$, and let $N^+((p,B))$ and $N^-((p,B))$ denote the set of out-neighbors and that of in-neighbors of $(p,B)$, respectively. Then for $t$ we need to count the elements of $$N^+((p,B))\cap N^-((p,B))
\ = \ \{(p^*, B^*)\in V(D):\ p\in B^* ,\ p^*\in B\}.$$ Without loss of generality, suppose that $p\in S_1$. Then condition $p\in B^*$ requires that $b_1^*$ must be $p$. The second condition $p^*\in B$ requires that $p^*=b_i$ for some $i$. With the choice of $p^*=b_1$, there are $q^{l-1}$ blocks $B^*=\{b_1^*, b_2^*, \dots, b_l^*\}$ with $b_1^*=p$ can be paired with $p^*=b_1$. On the other hand, with $p^*=b_j$ for $j\neq 1$, we have $(q-1)q^{l-2}$ choices for $B^*$ with $b_1^*=p$, $b_j^*\in S_j\setminus\{b_j\}$ and $b_i^*\in S_i$ for all $i\in \{2, 3, \dots, l\}\setminus \{j\}$. Hence $t =
q^{l-1}+(l-1)(q-1)q^{l-2}$.
We now claim that the number of directed paths of length two from a vertex $(p, B)$ to another vertex $(p', B')$ depends only on whether there is an edge from $(p,B)$ to $(p',B')$ or not.
Suppose that $(p, B)\rightarrow (p', B')$; that is, $p\in B'$. Then, without loss of generality, we may assume that $p=b_1'$, and have $$\begin{array}{ll} \lambda &=\ |N^+((p,B))\cap N^-((p',B'))|\\
& = |\{(p^*, B^*)\in V(D):\ p=b_1'\in B^*,\ p^*\in B'\}|\\
&= |\{(p^*, \{p, b_2^*,\dots, b_l^*\}): \ p^*\in
B'\setminus\{b_1'\},\ b_i^*\in S_i\setminus\{p^*\}\mbox{ for } i=2,
3, \dots, l\}|\\
& =(l-1)(q-1)q^{l-2}.\end{array}$$
For $\mu$, suppose $(p, B)\nrightarrow (p', B')$. Then $\mu=|\{(p^*,B^*): p\in B^*,\ p^*\in B'\}|$ with $p\notin B'$. Without loss of generality, we assume that $p\in S_1$, and have $$\begin{array}{ll} \mu &
=\ |\{(b_1',\{p, b_2^*, \dots, b_l^*\}):\ b_i^*\in S_i \mbox{ for
}i=2, 3,\dots, l\}|\\
&\ +\ \sum\limits_{i=2}^l |\{(b_i', \{p, b_2^*,\dots, b_l^*\}):
b_i^*\in S_i\setminus\{b_i'\},\ b_j^*\in S_j\mbox{ for } j\in\{2,
3,\dots,
l\}\setminus\{i\} \}|\\
& =\ q^{l-1}+(l-1)(q-1)q^{l-2}.\end{array}$$ This completes the proof.
By Proposition \[pro1\], there are directed strongly regular graphs with parameters $$(mlq^l(q-1),\ mlq^{l-1}(q-1),\ mq^{l-2}(lq-l+1),\ mq^{l-2}(l-1)(q-1),\
mq^{l-2}(lq-l+1))$$ for all positive integers $m$. These graphs can be constructed directly by replacing all edges by multiple edges with multiplicity $m$ in the above construction.
In the case of $l=2$ ($m=1$) in the above construction, we obtain the graphs with parameters $$(2q^2(q-1), \ 2q(q-1),\ 2q-1,\ q-1,\ 2q-1).$$ These graphs are constructed on the sets of non-incident vertex-edge pairs of complete bipartite graphs $K_{q,q}$. If we use complete bipartite multigraph with multiplicity $m$ for each edge as in the above remark, we obtain directed strongly regular graphs with parameters, $$(2mq^2(q-1), \ 2mq(q-1),\ m(2q-1),\ m(q-1),\ m(2q-1)).$$ With several combinations of small $m$ and $q$, we obtain new directed strongly regular graphs with parameter sets $(36, 12, 5, 2, 5)$, $(96, 24, 7, 3, 7)$, $(72, 24, 10, 4, 10)$ and $(108, 36, 15, 6,
15)$. So the feasibility of these parameter sets which are listed in [@BH] has been realized.
Construction of graphs using partial geometries {#sec:AP}
===============================================
In this section, we use partial geometries to construct a family of directed strongly regular graphs. The concept of a partial geometry was introduced by R. C. Bose in connection with his study of large cliques of more general strongly regular graphs in [@Bo].
A partial geometry ${\textsl}{pg}(\kappa, \rho, \tau)$ is a set of points $P$, a set of lines ${\mathcal}{L}$, and an incidence relation between $P$ and ${\mathcal}{L}$ with the following properties:
1. Every line is incident with $\kappa$ points ($\kappa\ge 2$), and every point is incident with $\rho$ lines ($\rho\ge 2$).
2. Any two points are incident with at most one line.
3. If a point $p$ and a line $L$ are not incident, then there exists exactly $\tau$ ($\tau\ge 1$) lines that are incident with $p$ and incident with $L$.
Here we use parameters $(\kappa, \rho, \tau)$ instead of more traditional notations, $(K, R, T)$ or $(1+t, 1+s, \alpha)$ used in [@Bo; @BC] or [@Th]. In what follows, we often identify a line $L$ as the set of $\kappa$ points that are incident with $L$; so, write “$p\in
L$," as well as “$p$ is on $L$," and “$L$ passes through $p$" when $p$ and $L$ are incident.
\[dpg\] Let $D=D({\textsl}{pg}(\kappa, \rho, \tau))$ be the directed graph with its vertex set $$V(D)=\{(p, L)\in P\times {\mathcal}{L}:\ p\notin L\},$$ and directed edges given by $(p, L)\rightarrow (p', L')$ if and only if $p\in L'$.
\[pg\] Let $D$ be the graph $D({\textsl}{pg}(\kappa, \rho,\tau))$ defined as above. Then $D$ is a directed strongly regular graph with parameters $$\begin{array}{l} v=\frac{\kappa\rho(\kappa-1)(\rho-1)}{\tau}
\left(1+\frac{(\kappa-1)(\rho-1)}{\tau}\right),\\
k=\frac{\kappa\rho(\kappa-1)(\rho-1)}{\tau},\\
t=\mu= \kappa\rho-\tau,\\
\lambda= (\kappa-1)(\rho-1).\\ \end{array}$$
The value $v$ is clear as it counts the number of anti-flags of ${\textsl}{pg}(\kappa,\rho,\tau)$ which has $\kappa\left(1+\frac{(\kappa-1)(\rho-1)}{\tau}\right)$ points and $\rho\left(1+\frac{(\kappa-1)(\rho-1)}{\tau}\right)$ lines. Also it is clear that $$k=|\{(p',L'): p\in
L'\}|=\rho(v-\kappa)=\kappa\rho(\kappa-1)(\rho-1)/\tau.$$
For $t$, given a vertex $(p, L)\in V(D)$, we count the cardinality of $$N^+((p,L))\ \cap\ N^-((p,L))=\{(p', L')\in V(D):\ p\in L' ,\ p'\in
L\}.$$ Among the $\rho$ lines passing through $p$, $\rho-\tau$ lines are parallel to $L$. If $L'$ is parallel to $L$, then all $\kappa$ points on $L$ can make legitimate non-incident point-line pairs $(p', L')$ with given $L'$. In the case when $L'$ is not parallel to $L$, all points on $L$ except for the common incident point of $L$ and $L'$, can form desired pairs. Hence we have $$t=(\rho-\tau)\kappa + \tau (\kappa-1)= \kappa\rho-\tau.$$
For $\lambda$, suppose $(p, L)\rightarrow (p',L')$. It is clear that $$\lambda=|\{(p^*,L^*): p\in L^*,\ p^*\in L'\}|=(\rho-1)(\kappa-1),$$ because each of $\rho-1$ lines passing through $p$ (excluding $L'$) can be paired with any of $\kappa-1$ points on $L'\setminus\{p\}$.
For given $(p, L)$ and $(p',L')$ with $p\notin L'$, $$\mu=|\{(p^*,L^*): p\in L^*,\ p^*\in L'\}|=(\rho-\tau)\kappa
+\tau(\kappa-1)=\kappa\rho-\tau,$$ since among the $\rho$ lines passing through $p$, the ones that are parallel to $L'$ can form desired pairs with any of $\kappa$ points on $L'$, while each of the remaining $\tau$ lines can be paired with $\kappa-1$ points on $L'$.
Partial geometries are ubiquitous. For example, partial geometries with $\tau =1$ are generalized quadrangles, those with $\tau=\kappa
-1$ are transversal designs, and those with $\tau=\rho-1$ are known as nets. In order to have some concrete examples of new directed strongly regular graphs of small order, we consider a special class of partial geometries that are obtained from finite affine planes.
Let $AP(q)$ denote the affine plane of order $q$. Let ${\overline}{AP}^l(q)$ denote the partial geometry obtained from $AP(q)$ by considering all $q^2$ points and taking the lines of $l$ parallel classes of the plane. Then ${\overline}{AP}^l(q)$ inherits the following properties from $AP(q)$: (i) every point is incident with $l$ lines, and every line is incident with $q$ points, (ii) any two points are incident with at most one line, (iii) if $p$ and $L$ are non-incident point-line pair, there are exactly $l-1$ lines containing $p$ which meet $L$. That is, ${\overline}{AP}^l(q)={\textsl}{pg}(q,l,l-1)$.
Let $D=D({\overline}{AP}^l(q))$ be the directed graph $D({\textsl}{pg}(q, l, l-1))$ defined as in Definition \[dpg\]. Then $D$ is a directed strongly regular graph with parameters $$(v,k, t, \lambda, \mu)=(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\
lq-l+1).$$
It immediately follows from Theorem \[pg\].
In particular, if $l=q$, ${\overline}{AP}^q(q)={\textsl}{pg}(q,q,q-1)$ is a transversal design TD$(q,q)=(P, \mathcal{G}, \mathcal{L})$ of order $q$, block size $q$, and index 1 in the following sense.
1. $P$ is the set of $q^2$ points of ${\overline}{AP}^q(q)$;
2. ${\mathcal}{G}$ is the partition of $P$ into $q$ classes (groups) such that each class consists of $q$ points that were collinear in $AP(q)$ but not in ${\overline}{AP}^q(q)$;
3. ${\mathcal}{L}$ is the set of $q^2$ lines (blocks);
4. every unordered pair of points in $P$ is contained in either exactly one group or in exactly one block, but not both.
Let $D$ be the graph $D({\overline}{AP}^q(q))$ defined as above. Then $D$ is a directed strongly regular graph with parameters $$(v,k, t, \lambda, \mu)=(q^3(q-1),\ q^2(q-1),\ q^2-q+1,\ (q-1)^2,\
q^2-q+1).$$
This method produces new directed strongly regular graphs with parameters $(54, 18, 7, 4, 7)$ when $q=3$; and so $(108,
36, 14, 8, 14)$ by Proposition \[pro1\].
Construction of graphs using partitioned sets {#sec:DSGN}
=============================================
In this section we construct directed strongly regular graphs for two parameter sets, $$(ql(l-1),\ q(l-1),\ q,\ 0,\ q)$$ and $$(ql(l-1),\ 2q(l-1)-1,\
ql-1,\ ql-2,\ 2q)$$ for all positive integers $q\ge 1$ and $l\ge 3$. The incidence structure which will be used here may be viewed as a degenerate case of those used in earlier sections. The graphs produced by this construction are not new. The latter family of graphs have been constructed by Godsil, Hobart and Martin in [@GH Corollary 6.3]. Nevertheless, we introduce our construction to illustrate a variation of constructions which may be applied to produce new families of directed strongly regular graphs.
Let $P$ be a set of $ql$ elements (‘points’), and let $S_1, S_2,
\dots, S_l$ be $l$ mutually disjoint $q$-element subsets of $P$ (‘blocks’). We denote the family of blocks by ${\mathcal}{S}=\{S_1, S_2,
\dots, S_l\}$. We will say that point $x\in P$ and block $S\in
{\mathcal}{S}$ is a non-incident point-block pair if and only if $x\notin
S$.
For the directed strongly regular graph with the first parameter set, let $D=D(P,{\mathcal}{S})$ be the directed graph with its vertex set $$V(D)=\{(x, S)\in P\times {\mathcal}{S}:\ x\notin
S \},$$ and directed edges defined by $(x,S)\rightarrow (x',S')$ if and only if $x\in S'$.
Let $P$, ${\mathcal}{S}$ and $D(P,{\mathcal}{S})$ be as the above. Then $D(P,{\mathcal}{S})$ is a directed strongly regular graph with parameters $$(ql(l-1),\ q(l-1),\ q,\ 0,\ q).$$
Easy counting arguments give the values for parameters.
J[ø]{}rgensen [@Jo2] showed that the directed strongly regular graph with parameters $(ql(l-1),\ q(l-1),\ q,\ 0,\ q)$ is unique for all positive integers $l$ and $q$.
For the directed strongly regular graph with the second parameter set, let $G=G(P,{\mathcal}{S})$ be the directed graph with its vertex set $$V(G)=\{(x, S)\in P\times {\mathcal}{S}:\ x\notin S \},$$ where $P$ and ${\mathcal}{S}$ are as the above, and let edges on $V(G)$ be defined by: $(x,S)\rightarrow (x',S')$ if and only if $$\left \{\begin{array}{l} (1)\ x\in S' ; \mbox{ or} \\
(2)\ S=S'\mbox{ and } x\neq x'.\\ \end{array}\right .$$
Let $G$ be the graph $G(P,{\mathcal}{S})$ defined as above. Then $G$ is a directed strongly regular graph with parameters $$(ql(l-1),\ 2q(l-1)-1,\
ql-1,\ ql-2,\ 2q).$$
Clearly $v=ql(l-1)$ as $|V(G)|=|V|\cdot (|{\mathcal}{S}|-1)$.
Given a vertex $(x,S)$, let $$\begin{array}{l}
N_1((x, S)):=\{(x', S'): x\in S' \mbox{ and } x'\in S\}\\
N_2((x, S)):=\{(x', S'): S=S' \mbox{ and }
x\neq x'\}\\
N_3((x, S)):=\{(x', S'): x\in S'\mbox{ and } x'\notin S\}\\
\end{array}$$ Then by simple counting, we have $$\begin{array}{l} |N_1((x, S))|=q\\ |N_2((x, S))|= q(l-2)+q-1\\
|N_3((x, S)|=q(l-2). \\ \end{array}$$ Hence, $t=|N_1((x, S))|+
|N_2((x, S))|= ql-1$, and $k=t+|N_3((x,S))|=2ql-2q-1$.
Finally, we count the number of vertices $(x^*, S^*)$ that belongs to $N^+((x,S))\cap N^-((x',S'))$ for $\lambda$ and $\mu$.
For $\lambda$, given an edge $(x,S)\rightarrow (x',S')$, we shall need to consider the following two cases separately: (Case 1) when $x\in S'$, and (Case 2) when $S=S'$ and $x\neq x'$.\
Case (1). Suppose $x\in S'$. Then unless $S^*=S$, $S^*$ must contain $x$ which forces $S^*=S'$. With $S^*=S'$, all points that are not belong to $S'\cup \{x'\}$ can be chosen to be $x^*$; and so, there are $q(l-1)-1$ choices for $x^*$. With $S^*=S$, the $q-1$ points of the set $S'\setminus\{x\}$ are possible for $x^*$. Together there are $ql-2$ vertices $(x^*, S^*)$ belonging to $N^+((x,S))\cap N^-((x',S'))$.\
Case (2). Let $S=S'$. Then with the choice of $S^*=S=S'$, $x^*$ can be chosen from $P\setminus (S\cup \{x, x'\})$; and so, there are $q(l-1)-2$ choices for $x^*$. Also with the choice of $S^*$ being the block containing $x$, any point in $S$ can be chosen as $x^*$; and so, there are $q$ possible choices for $x^*$. Together we have $ql-2$ choices for $(x^*, S^*)$ as well. Hence we have $\lambda=ql-2$.
For $\mu$, suppose $S\neq S'$ and $x\notin S'$. Then it is clear that $$\mu=|\{(x^*, S^*): x\in S^*, x^*\in S'\}\cup \{(x^*, S^*): S^*=S,
x\in S'\}|=q+q.$$ This completes the proof.
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abstract: 'We show that a topological phase supporting Majorana fermions can form in a 2D electron gas (2DEG) adjacent to an interdigitated superconductor-ferromagnet structure. An advantage of this setup is that the 2DEG can induce the required Zeeman splitting and superconductivity from a single interface, allowing one to utilize a wide class of 2DEGs including the surface states of bulk InAs. We demonstrate that the interdigitated device supports a robust topological phase when the finger spacing $\lambda$ is smaller than half of the Fermi wavelength $\lambda_F$. In this regime, the electrons effectively see a “smeared" Zeeman splitting and pairing field despite the interdigitation. The topological phase survives even in the opposite limit $\lambda>\lambda_F/2$, though with a reduced bulk gap. We describe how to electrically generate a vortex in this setup to trap a Majorana mode, and predict an anomalous Fraunhofer pattern that provides a sharp signature of chiral Majorana edge states.'
author:
- 'Shu-Ping Lee$^1$, Jason Alicea$^2$, and Gil Refael$^{1,3}$'
title: 'Electrical manipulation of Majorana fermions in an interdigitated superconductor-ferromagnet device'
---
Topological superconductors have attracted considerable recent interest because they may provide the first unambiguous realization of Majorana fermions in any physical setting. The pursuit of these elusive objects in condensed matter [@BeenakkerMajoranaReview; @J.A.MajoranaReview] is motivated largely by the non-Abelian statistics [@Read-Green-2D-P-wave-superconductor; @Ivanov-2001-Non-Abelian; @alicea2011non] that they underpin, which is widely sought for quantum computation [@RevModPhys.80.1083]. Although much attention recently has focused on finding Majorana fermions in 1D systems [@Kitaev-Unpaired-Majorana-1D-wire; @Fu-Kane-1D-Fractional-Josephson-2009; @PhysRevLett.105.077001; @PhysRevLett.105.177002; @Topological-Insulator-Nanoribbons; @Leo-Kouwenhoven-Majorana-Science], 2D platforms [@Read-Green-2D-P-wave-superconductor; @PhysRevLett.100.096407; @Sato-Fujimoto-Cold-atom; @PhysRevLett.104.040502; @Patrick-Lee-P-wave-suerpconductor; @PhysRevB.81.125318; @SCZhang-Half-Metal] offer some unique virtues such as the ability to perform interferometry [@Stern-Halperin-Non-Abelian-Quantum-Hall-State; @Bonderson-Kitaev-Shtengel-Non-Abelian-Statistics-Fractional-Quantum-Hall-State; @PhysRevLett.102.216403; @PhysRevLett.102.216404; @JDSau-Detection-Majorana; @PhysRevB.83.104513; @Eytan-Detect-MajoranaFermion] to probe non-Abelian statistics. One promising 2D scheme involves a quantum well sandwiched between an $s$-wave superconductor and a magnetic insulator [@PhysRevLett.104.040502]. Fabricating this device is, however, rather nontrivial as one must synthesize high quality interfaces on both sides of the quantum well—which is typically buried in a heterostructure. One can avoid a multilayered architecture by invoking a specific type of 2D electron gas (2DEG) with appreciable Rashba and Dresselhaus coupling [@PhysRevB.81.125318], but the candidate materials for this proposal are limited.
In this manuscript we introduce a new 2D Majorana platform (Fig. \[SCFMDevice\]) consisting of interdigitated superconductor/ferromagnet insulator strips deposited on a 2DEG (periodically modulated 1D topological superconductors were considered in Ref. [@Avoidance-of-Majorana-Resonances]). The proposed setup exhibits several virtues. For one, our device requires interface engineering on only one side of the 2DEG—alleviating one experimental challenge with previous semiconductor-based proposals. Because of this feature one can also employ a wider variety of 2DEGs, including surface states of bulk semiconductors such as InAs [@PhysRevB.61.15588; @PhysRevB.67.165329; @PhysRevB.82.235303]. Meanwhile, this structure naturally allows one to *electrically* generate vortices to trap Majorana zero-modes, potentially allowing Majorana fermions to be braided using currents similar to the proposal of Ref. [@Romito-Alicea-Refael-Oppen-Manipulating-Majorana-fermions-using-supercurrents]. We further show that in our device (as well as any 2D topological superconductor) Majorana edge states can be detected by observing an anomalous shift of the zeros in the Fraunhofer pattern measured in a long Josephson junction.
![Schematic of the proposed interdigitated superconductor-ferromagnet 2DEG architecture. The device supports a robust topological phase when the finger spacing $\lambda$ is smaller than half of the Fermi wavelength (*i.e.* $\lambda<\lambda_F/2$). A topological phase can also appear in the regime $\lambda>\lambda_F/2$, though generally with a suppressed gap. Table I provides values of $\lambda_F$ for select 2DEGs when the chemical potential is set to $\mu = 0$.[]{data-label="SCFMDevice"}](SCFMDevice){width="7cm"}
2DEG $\alpha$ \[eV${\rm \AA}$\] $m/m_e$ $\lambda_F$ \[$\mu$m\]
---------------------------------------------------------------------------------- ---------------------------- --------- ------------------------
InGaAs/InAlAs [@PhysRevLett.89.046801] 0.05 0.04 1.19
InSb/InAlSb [@PhysRevB.79.235333] 0.14 0.0139 1.22
Bulk InAs surface [@PhysRevB.61.15588; @PhysRevB.67.165329; @PhysRevB.82.235303] 0.11 0.03 0.72
: Effective mass $m$ in units of the electron mass $m_e$, Rashba coupling strength $\alpha$, and Fermi wavelength $\lambda_F$ evaluated at $\mu = 0$ for the 2DEG’s listed in the left column.
We model the semiconductor in this device with the following Hamiltonian, $$\begin{aligned}
H &=& \int d^2{\bf r}\bigg{\{}\psi^{\dagger} \left[-\frac{\hbar^2\nabla^2}{2m}-\mu-i\alpha(\sigma^x\partial_y-\sigma^y\partial_x) \right]\psi
\nonumber\\
&+& V_z({\bf r})\psi^{\dagger}\sigma^z \psi+
\left[\Delta({\bf r}) \psi_{\uparrow} \psi_{\downarrow}+{\rm H.c.}\right]\bigg{\}},
\label{2}\end{aligned}$$ where $\psi^\dagger_\sigma$ creates an electron with spin $\sigma$ and effective mass $m$, $\mu$ is the chemical potential, $\alpha$ is the Rashba coupling strength, and $\sigma^{a}$ are Pauli matrices that contract with the spin indices. The spatially varying Zeeman and pairing fields induced by the alternating ferromagnetic and superconducting strips are respectively denoted by $V_z({\bf r})$ and $\Delta({\bf r})$. For simplicity we will retain only their maximal Fourier components and take $V_z({\bf r})=2\overline V_z\sin^2(\frac{1}{2}Qx)$ and $\Delta({\bf r})=2\overline \Delta\cos^2(\frac{1}{2}Qx)$. Here $\overline V_z$ and $\overline\Delta$ are the spatial average of these quantities, which modulate at wavevector $Q=2\pi/\lambda$ with $\lambda$ the finger spacing shown in Fig. \[SCFMDevice\]. This choice is expected to not only quantitatively capture the effects of interdigitation, but as we will see also leads to an intuitive physical picture for the device’s behavior.
As a primer it is worth recalling the physics of the sandwich structure originally proposed by Sau *et al.* [@PhysRevLett.104.040502], where a uniform Zeeman field $V_z^{\rm unif}$ opens a chemical potential window in which only one Fermi surface is present. Incorporating $s$-wave pairing with strength $\Delta^{\rm unif}$ in this regime effectively drives the 2DEG into a topological $p+ip$ superconductor due to the interplay with spin-orbit coupling. [@PhysRevLett.104.040502; @PhysRevB.81.125318; @J.A.MajoranaReview] Quantitatively, the topological phase appears provided $(V_z^{\rm unif})^2>(\Delta^{\rm unif})^2+\mu^2$. In our interdigitated setup it is natural to expect that when the Fermi wavelength $\lambda_F$ for the semiconductor greatly exceeds the finger spacing $\lambda$, electrons in the 2DEG effectively experience ‘smeared’ Zeeman and pairing fields with strength $\overline V_z$ and $\overline \Delta$. Similar physics to the uniform case ought to then emerge—in particular, a topological phase when $\overline V_z^2 \gtrsim \overline \Delta^2+\mu^2$.
To confirm this intuition and extract the phase diagram for arbitrary $\lambda_F/\lambda$, we study the quasiparticle spectrum for Eq. (\[2\]). Defining a Nambu spinor $\Psi_{\bf k}=[\psi_{\uparrow}({\bf k}),\psi_{\downarrow}({\bf k}),\psi^{\dagger}_{\downarrow}(-{\bf k}),-\psi^{\dagger}_{\uparrow}(-{\bf k})]^{T}$, the Hamiltonian can be written in momentum space as $$\begin{aligned}
H&=&\int \frac{d^2{\bf k}}{(2\pi)^2} (\overline{\mathcal{H}}_{\bf k}+\delta \mathcal{H}_{\bf k})
\label{MomentumBdGeq2}
\\
\overline{\mathcal{H}}_{\bf k}&=&\Psi_{\bf k}^{\dagger} \bigg{[}\left(\frac{\hbar^2k^2}{2m}-\mu\right)\tau^z+\alpha\left(k_y\sigma^x-k_x\sigma^y\right)\tau^z
\nonumber \\
&+& \overline V_z\sigma^z+\overline \Delta\tau^x\bigg{]}\Psi_{\bf k}
\\
\delta \mathcal{H}_{\bf k}&=&\Psi_{\bf k}^{\dagger} \left[-\frac{\overline V_z}{2} \sigma^z+\frac{\overline \Delta}{2}\tau^x\right]\Psi_{{\bf k} + Q{\bf \hat{x}}}+{\rm H.c.}\end{aligned}$$ with $\tau^a$ Pauli matrices that act in particle-hole space. The Hamiltonian $\overline{\mathcal{H}}_{\bf k}$ describes a semiconductor proximate to a uniform superconductor and ferromagnet and is precisely the model studied in Ref. [@PhysRevLett.104.040502]. The bulk excitation spectrum obtained from $\overline{\mathcal{H}}_{\bf k}$ in the topological phase with $\mu=0$, $\overline{V_z}=1.5\overline{\Delta}$, $m\alpha^2=3\overline{\Delta}$ and $k_y = 0$ appears in the red dashed lines of Fig. \[BulkBdG\]; roughly, the gap at $k_x = 0$ is set by $\overline{V}_z$ while $\overline\Delta$ determines the gap at the Fermi wavevector $k_F = 2\pi/\lambda_F$. Our interdigitated structure produces a new term $\delta \mathcal{H}_{\bf k}$ that couples spinors with wavevectors ${\bf k}$ and ${\bf k} \pm Q{\bf \hat{x}}$. As we ‘turn on’ these couplings the spectrum of $\overline{\mathcal{H}}_{\bf k}$ evolves very similarly to the band structure of free electrons in a weak periodic potential [@ashcroft1976solid]. In particular, the dominant effect of $\delta \mathcal{H}_{\bf k}$ is to open a gap in the excitation spectrum whenever the energies cross Bragg planes at $k_x = \pm Q/2 = \pm\pi/\lambda$ (modulo reciprocal lattice vectors). For momenta away from these values $\delta \mathcal{H}_{\bf k}$ couples states that are far from resonant and hence perturbs these only weakly.
![Bulk quasiparticle spectrum versus $k_x$ in a uniform structure (dashed curve) and interdigitated device with $\lambda_F/\lambda=2.6$ (solid curve). In both cases we use parameters $k_y = 0$, $\mu=0$, $\overline{V_z}=1.5\overline{\Delta}$, and $m\alpha^2=3\overline{\Delta}$. The red arrow indicates the pairing gap at the Fermi momentum $k_F\equiv\frac{2\pi}{\lambda_F}\approx2m\alpha/\hbar^2$, while the blue arrow denotes the degeneracy gap opened at $k_x=\frac{\pi}{\lambda}$ due to the interdigitation. Note that the excitation spectra for the uniform and interdigitated systems differ appreciably only at rather higher energies here.[]{data-label="BulkBdG"}](BulkBdG2){width="8cm"}
It follows that for $\lambda_F/\lambda \gg 1$ the periodic modulation modifies the quasiparticle spectrum appreciably only at very high energies. This point is illustrated by the solid curves in Fig. \[BulkBdG\], which display the numerically obtained spectrum for the full Hamiltonian in Eq. (\[MomentumBdGeq2\]) in a repeated zone scheme, using the same parameters as above but now with $\lambda_F/\lambda=2.6$. Even for this ratio of $\lambda_F/\lambda$, the spectrum is nearly identical to that of the uniform case away from $k_x = \pm \pi/\lambda$. When $\lambda_F/\lambda \gg 1$ one can clearly incorporate $\delta\mathcal{H}_{\bf k}$ while essentially leaving the bulk excitation gap exhibited by the uniform system intact. Thus by adiabatic continuity our interdigitated device supports a topological phase in this limit provided $\overline V_z^2 \gtrsim \overline \Delta^2+\mu^2$, consistent with the intuition provided earlier. As further evidence, Figs. \[BdG\](a) and (b) display the quasiparticle spectrum as a function of $k_y$ in a system with open boundary conditions along the $x$ direction. The data correspond to $\mu=0$, $m\alpha^2=3\overline{\Delta}$ and $\lambda_F/\lambda=2.5$, while the Zeeman energy changes from $\overline{V_z}=0.5\overline{\Delta}$ in (a) to $\overline{V_z}=2\overline{\Delta}$ in (b). In (a) a trivial gapped state clearly emerges due to the weak Zeeman energy. The larger $\overline{V}_z$ value in (b), however, satisfies our topological criterion, and one indeed sees the signature gapless chiral Majorana edge states inside of the bulk gap.
![Quasiparticle spectrum in various regimes for an interdigitated device with periodic boundary conditions along $y$ but open boundary conditions along $x$. In all parts we take $\mu=0$ and $m\alpha^2=3\overline{\Delta}$, while the finger spacing and Zeeman energy vary as (a) $\lambda_F/\lambda = 2.5$, $\overline{V_z}=0.5\overline{\Delta}$, (b) $\lambda_F/\lambda = 2.5$, $\overline{V_z}=2\overline{\Delta}$, and (c) $\lambda_F/\lambda = 1.5$, $\overline{V_z}=2\overline{\Delta}$. A trivial state appears in (a) while the larger Zeeman field in (b) drives a topological phase supporting chiral Majorana edge states within the bulk gap. Interestingly, the topological phase and associated Majorana edge states survive even in (c) despite the relatively small ratio of $\lambda_F/\lambda$.[]{data-label="BdG"}](BdGEdgeState){width="8.6cm"}
As one reduces the ratio $\lambda_F/\lambda$ to a value of order one or smaller, the physics becomes considerably more subtle. Indeed, once $\lambda_F/\lambda \approx 2$ the Bragg plane at $k_x = Q/2$ approaches the Fermi wavevector, and the pairing gap can then be dramatically altered by the interdigitation. We ascertain the global phase diagram of our device by numerically computing the minimum excitation gap $\delta$ for a system on a torus as a function of $\lambda_F/\lambda$ and $\overline{V}_z/\overline\Delta$. Figure \[PhaseDiagram2\] shows the results for $\mu=0$ and spin-orbit energies of $m\alpha^2=3.2\overline{\Delta}$ in (a) and $m\alpha^2=1.3\overline{\Delta}$ in (b). The following points are noteworthy here: 1) At ‘large’ $\lambda_F/\lambda$ topological superconductivity appears when $\overline{V_z} \gtrsim \overline\Delta$, in line with our results above. 2) The topological phase survives over a range of parameters even for rather small values of $\lambda_F/\lambda$, though the gap is generally reduced compared to the large $\lambda_F/\lambda$ limit. Figure \[BdG\](c) illustrates the spectrum in the $\lambda_F/\lambda<2$ regime for a system with open boundary conditions along $x$; just as in Fig. \[BdG\](b) the characteristic chiral edge states again appear here. 3) Interestingly, for $\lambda_F/\lambda\sim 1$ the critical value of $\overline{V}_z$ required to generate the topological phase *decreases* compared to the uniform case.
![Phase diagrams for $\mu = 0$ and spin-orbit energies (a) $m\alpha^2=3.2\overline{\Delta}$ and (b) $m\alpha^2=1.3\overline{\Delta}$. The horizontal axis represents ratio of the Fermi wavelength $\lambda_F$ to the finger spacing $\lambda$, while the vertical axis is the Zeeman energy normalized by pairing strength. The shading indicates the bulk gap $\delta$ normalized by $\overline{\Delta}$. Red dashed lines denote the boundary between topological phase and trivial phases. []{data-label="PhaseDiagram2"}](PhaseDiagram){width="8cm"}
![(a) Scheme to electrically stabilize a vortex binding a Majorana zero-mode. Here the singular phase winding is induced by current flowing from contact 1 to contact 2, rather than from a magnetic field. In (b) we illustrate the probability density extracted from the near-zero energy mode generated by a current-induced vortex at the center of the device (parameters are $\mu=0$, $m\alpha^2=1.3\overline{\Delta}$, $\overline{V}_z=2\overline{\Delta}$, and $\lambda=\lambda_F/4$). The large central peak corresponds to the Majorana bound to the vortex, which hybridizes weakly with the outer Majorana running along the perimeter. []{data-label="TunnelingCurrent"}](TunnelingCurrent12){width="8.5cm"}
Having numerically demonstrated that our device exhibits a topological phase with an edge state, we now describe how the interdigitated structure naturally allows us to *electrically* generate vortices to trap Majorana zero-modes. Consider the setup of Fig. \[TunnelingCurrent\]. Supercurrent flowing from contact 1 to contact 2 produces a winding in the superconducting phase $\theta({\bf r})$ across the fingers in the device. When the phase difference between the contacts approaches $2\pi$, a vortex forms near the center of the system to minimize the energy $E\propto\int d^2{\bf r}(\nabla\theta)^2$. The spatial profile of the phase follows from the supercurrent $ {\bf j}({\bf r})\propto\Delta^*({\bf r})\nabla\Delta({\bf r})-\Delta({\bf r})\nabla\Delta^*({\bf r})$, with $\Delta({\bf r})=\Delta_{SC}({\bf r})e^{i\theta({\bf r})}$, where the pairing potential’s magnitude satisfies $\Delta_{SC}({\bf r})=2\overline{\Delta}$ beneath the superconductors \[blue regions in Fig. \[TunnelingCurrent\](a)\] and goes to zero under the ferromagnets \[pink regions in Fig. \[TunnelingCurrent\](a)\]. In particular, one can extract $\theta({\bf r})$ by iterating the current conservation equation $\nabla\cdot{\bf j}({\bf r})=0$ subject to boundary conditions along the system’s perimeter. With this phase in hand, one can diagonalize the Hamiltonian in the presence of a current-induced vortex and extract the wavefunctions for each quasiparticle state. Figure \[TunnelingCurrent\](b) illustrates the resulting probability distribution for the near-zero-energy state in the spectrum; the large central peak corresponds to a localized Majorana mode bound to the vortex core, while the outer peak represents a second Majorana mode running along the edge.
Finally, we discuss the detection of Majorana edge states in the topological phase exhibited by our device (or, equivalently, any other realization) via an unconventional Fraunhofer pattern. [^1] Consider a pair of topological superconductors forming a long Josephson junction of width $w$ pierced by a magnetic field \[see Fig. \[FraunhoferPatternForMajorana\](a)\]. At low energies it suffices to focus only on the chiral edge states, which can be modeled by an effective Hamiltonian $H = H_t + H_b + H_{\rm tunneling}$. [@Read-Green-2D-P-wave-superconductor; @PhysRevB.75.045317; @Eytan-Detect-MajoranaFermion] The first two terms $H_{t/b}=\pm iv\hbar\int dx \gamma_{t/b}\partial_x \gamma_{t/b}$ describe the kinetic energy for the top/bottom edge states, with $\gamma_{t/b}$ Majorana operators and $v$ the edge velocity. The last term incorporates inter-edge tunneling with strength $t$ at the interface and reads $H_{\rm tunneling}=it\int_{-w/2}^{w/2} dx \gamma_t\gamma_b\cos[\theta(x)/2]$, where $\theta(x)$ is the local superconducting phase difference across the junction induced by the applied field. Neglecting the magnetic field that is produced from the tunneling current, $\theta(x)$ is determined by the external magnetic flux $\Phi$ according to $\theta(x)=\theta_0+2\pi\frac{\Phi}{\Phi_0}\frac{x}{w}$ ($\Phi_0$ is the flux quantum and $\theta_0$ is the phase difference at the junction’s center).
The Majorana-mediated contribution to the local current density flowing across the junction follows from $j(x)=\frac{et}{\hbar}\sin[\theta(x)/2]i\gamma_t\gamma_b$. We calculate the current perturbatively in $t$ assuming the weak-tunneling limit $\frac{tw}{2\pi\hbar v}<1$ where the hybridization energy is smaller than the level spacing. In this case the physics depends sharply on whether, at $t = 0$, Majorana zero-modes exist at each edge. If neither edge supports a zero-mode, then the current vanishes to first order in $t$. However, if zero-modes exist at both edges (due to an odd number of vortices in their bulk) then a finite current $\langle j(x)\rangle=\frac{et}{\hbar L}\sin[\theta(x)/2]$ emerges, where $L$ is the superconductors’ perimeter. Integrating over the junction width yields a total Majorana-mediated current $$I_M=\frac{twe}{\hbar L}\sin\left(\frac{\theta_0}{2}\right)\left[\frac{\sin(\frac{\pi}{2}\frac{\Phi}{\Phi_0})}{\frac{\pi}{2}\frac{\Phi}{\Phi_0}}\right].$$ The solid black curve in Fig. \[FraunhoferPatternForMajorana\](b) illustrates $|I_M|$ as a function of $\Phi$; remarkably, the zeros occur at *even* multiples of $\Phi/\Phi_0$ in contrast to the conventional Fraunhofer pattern shown for comparison in the red dashed curve. For a sample of size $5\mu m\times 5 \mu m$ with the coupling energy $t=0.025$meV [@Eytan-Detect-MajoranaFermion], we estimate that the typical magnitude of $I_M$ is $\sim 1.5$nA. This result is valid when the edge velocity obeys $v>3\times10^4 m/s$ so that the weak tunneling limit is satisfied. It is important to keep in mind, however, that the experimentally observed current will not be given by $I_M$ alone—a potentially much larger conventional current $I_s$ flows in parallel. The magnitude of the total current $I_{\rm tot} = I_s + I_M$ is sketched by the blue curve in Fig. \[FraunhoferPatternForMajorana\](b). One can infer the existence of $I_M$ by the unconventional Fraunhofer pattern that exhibits shifted zeros as shown in the figure. We note that very recently an experiment of this type has been performed in a long Josephson junction formed at the surface of a 3D topological insulator [@DGG-FraunhoferPattern], though the findings are rather different from what we predict here.
In conclusion, we have shown that our interdigitated structure exhibits a topological phase that is particularly robust when the finger spacing is smaller than half of the Fermi wavelength. There the bulk gap can be comparable to that in a uniform system; furthermore, additional perturbations induced by the interdigitation (such as variations in chemical potential and Rashba strength) should play a minor role. To access this regime the finger spacing should be $\lesssim 600$nm for the quantum wells listed in Table I and $\lesssim 400$nm for the surface state of bulk InAs. We also note that since electrons effectively see ‘smeared’ fields in this limit, the specific interdigitated pattern studied here is by no means required—similar physics should arise, *e.g.*, in checkerboard arrangements. An interesting feature of our setup is that vortices can be generated by applying currents. This mechanism may eventually provide a practical means of manipulating and braiding vortices for quantum computation. We also pointed out that chiral Majorana edge states produce an anomalous Fraunhofer pattern that can be observed in any realization of topological $p + ip$ superconductivity.
![(a) Long Josephson junction formed by adjacent topological superconductors. A magnetic field $\vec{B}$ orients perpendicular to the plane and uniformly penetrates through the junction. (b) The solid black curve represents the magnitude of the tunneling current arising from coupled Majorana zero-modes at the edge. This contribution exhibits zeros at even multiples of the flux quantum in sharp contrast to the Fraunhofer pattern exhibited by ordinary $s$-wave superconductor junctions (red dashed curve). The blue curve represents the anomalous Fraunhofer pattern that would arise in an experiment due to the Majorana-mediated component and a parallel conventional current contribution.[]{data-label="FraunhoferPatternForMajorana"}](FraunhoferPatternForMajorana2){width="8.5cm"}
We are indebted to Charles M. Marcus for proposing the interdigitated structure studied here, and to Julia Meyer for discussions on the Fraunhofer pattern. We also thank E. Grosfeld, K. T. Law, J. Eisenstein, P. T. Bhattacharjee, S. Iyer, and D. Nandi for illuminating discussions. We are grateful to the Packard Foundation (GR), Humboldt Foundation and to the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the Gordon and Betty Moore Foundation. JA gratefully acknowledges funding from the National Science Foundation through grant DMR-1055522 and the Alfred P. Sloan Foundation.
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[^1]: This idea has been independently proposed by Julia Meyer and Manuel Houzet.
|
---
abstract: 'This short example shows a contrived example on how to format the authors’ information for [*IJCAI–PRICAI–20 Proceedings*]{} using LaTeX.'
author:
- 'First Author$^1$[^1]'
- Second Author$^2$
- |
Third Author$^{2,3}$Fourth Author$^4$\
$^1$First Affiliation\
$^2$Second Affiliation\
$^3$Third Affiliation\
$^4$Fourth Affiliation\
{first, second}@example.com, [email protected], [email protected]
title: 'IJCAI–PRICAI–20 Example on typesetting multiple authors'
---
Introduction
============
This short example shows a contrived example on how to format the authors’ information for [*IJCAI–PRICAI–20 Proceedings*]{}.
Author names
============
Each author name must be followed by:
- A newline [\\\\]{} command for the last author.
- An [\\And]{} command for the second to last author.
- An [\\and]{} command for the other authors.
Affiliations
============
After all authors, start the affiliations section by using the [\\affiliations]{} command. Each affiliation must be terminated by a newline [\\\\]{} command. Make sure that you include the newline on the last affiliation too.
Mapping authors to affiliations
===============================
If some scenarios, the affiliation of each author is clear without any further indication (*e.g.*, all authors share the same affiliation, all authors have a single and different affiliation). In these situations you don’t need to do anything special.
In more complex scenarios you will have to clearly indicate the affiliation(s) for each author. This is done by using numeric math superscripts [\${\^$i,j, \ldots$}\$]{}. You must use numbers, not symbols, because those are reserved for footnotes in this section (should you need them). Check the authors definition in this example for reference.
Emails
======
This section is optional, and can be omitted entirely if you prefer. If you want to include e-mails, you should either include all authors’ e-mails or just the contact author(s)’ ones.
Start the e-mails section with the [\\emails]{} command. After that, write all emails you want to include separated by a comma and a space, following the same order used for the authors (*i.e.*, the first e-mail should correspond to the first author, the second e-mail to the second author and so on).
You may “contract" consecutive e-mails on the same domain as shown in this example (write the users’ part within curly brackets, followed by the domain name). Only e-mails of the exact same domain may be contracted. For instance, you cannot contract “[email protected]" and “[email protected]" because the domains are different.
[^1]: Contact Author
|
---
abstract: 'The moduli space $M(c_2)$, of stable rank two vector bundles of degree one on a very general quintic surface $X\subset {{\mathbb P}}^3$, is irreducible for all $c_2\geq 4$ and empty otherwise.'
address:
- |
Laboratoire J. A. Dieudonné, CNRS UMR 7351\
Université de Nice-Sophia Antipolis\
06108 Nice, Cedex 2, France
- |
Laboratoire J. A. Dieudonné, CNRS UMR 7351\
Université de Nice-Sophia Antipolis\
06108 Nice, Cedex 2, France
author:
- Nicole Mestrano
- Carlos Simpson
title: Irreducibility of the moduli space of stable vector bundles of rank two and odd degree on a very general quintic surface
---
[^1]
Introduction
============
Let $X\subset {{\mathbb P}}^3_{{{\mathbb C}}}$ be a very general quintic hypersurface. Let $M(c_2):= M_X(2,1,c_2)$ denote the moduli space [@HuybrechtsLehn] of stable rank $2$ vector bundles on $X$ of degree $1$ with $c_2(E)=c_2$. Let $\overline{M} (c_2):= \overline{M}_X(2,1,c_2)$ denote the moduli space of stable rank $2$ torsion-free sheaves on $X$ of degree $1$ with $c_2(E)=c_2$. Recall that $\overline{M} (c_2)$ is projective, and $M(c_2)\subset \overline{M} (c_2)$ is an open set, whose complement is called the [*boundary*]{}. Let $\overline{M(c_2)}$ denote the closure of $M(c_2)$ inside $\overline{M} (c_2)$. This might be a strict inclusion, as will in fact be the case for $c_2\leq 10$.
In [@MestranoSimpson1] we showed that $M(c_2)$ is irreducible for $4\leq c_2\leq 9$, and empty for $c_2\leq 3$. In [@MestranoSimpson2] we showed that the open subset $M(10)^{\rm sn}\subset M(10)$, of bundles with seminatural cohomology, is irreducible. In 1995 Nijsse [@Nijsse] showed that $M(c_2)$ is irreducible for $c_2\geq 16$.
In the present paper, we complete the proof of irreducibility for the remaining intermediate values of $c_2$.
\[maintheorem\] For any $c_2\geq 4$, the moduli space of bundles $M(c_2)$ is irreducible.
For $c_2\geq 11$, the moduli space of torsion-free sheaves $\overline{M} (c_2)$ is irreducible. On the other hand, $\overline{M} (10)$ has two irreducible components: the closure $\overline{M(10)}$ of the irreducible open set $M(10)$; and the smallest stratum $M(10,4)$ of the double dual stratification corresponding to torsion-free sheaves whose double dual has $c'_2=4$. Similarly $\overline{M}(c_2)$ has several irreducible components when $5\leq c_2\leq 9$ too.
The moduli space $\overline{M} (c_2)$ is good for $c_2\geq 10$, generically smooth of the expected dimension $4c_2-20$, whereas for $4\leq c_2\leq 9$, the moduli space $M(c_2)$ is not good. For $c_2\leq 3$ it is empty.
Yoshioka [@YoshiokaAppli; @YoshiokaK3; @YoshiokaAbelian], Gomez [@GomezThesis] and others have shown that the moduli space of stable torsion-free sheaves with irreducible Mukai vector (which contains, in particular, the case of bundles of rank $2$ and degree $1$) is irreducible, over an abelian or K3 surface. Those results use the triviality of the canonical bundle, leading to a symplectic structure and implying among other things that the moduli spaces are smooth [@Mukai]. Notice that the case of K3 surfaces includes degree $4$ hypersurfaces in ${{\mathbb P}}^3$.
We were motivated to look at a next case, of bundles on a quintic or degree $5$ hypersurface in ${{\mathbb P}}^3$ where $K_X={{\mathcal O}}_X(1)$ is ample but not by very much. This paper is the third in a series starting with [@MestranoSimpson1; @MestranoSimpson2] dedicated to Professor Maruyama who, along with Gieseker, pioneered the study of moduli of bundles on higher dimensional varieties [@Gieseker; @GiesekerCons; @Maruyama; @MaruyamaET; @MaruyamaTransform]. Recall that the moduli space of stable bundles is irreducible for $c_2\gg 0$ on any smooth projective surface [@GiesekerLi; @Li; @OGradyIrred; @OGradyBasic], but there exist surfaces, such as smooth hypersurfaces in ${{\mathbb P}}^3$ of sufficiently high degree [@Mestrano], where the moduli space is not irreducible for intermediate values of $c_2$.
Our theorem shows that the irreducibility of the moduli space of bundles $M(c_2)$, for all values of $c_2$, can persist into the range where $K_X$ is ample. On the other hand, the fact that $\overline{M} (10)$ has two irreducible components, means that if we consider all torsion-free sheaves, then the property of irreducibility in the good range has already started to fail in the case of a quintic hypersurface.
A possible application of our theorem to the case of Calabi-Yau varieties could be envisioned, by noting that a general hyperplane section of a quintic threefold in ${{\mathbb P}}^4$ will be a quintic surface $X\subset {{\mathbb P}}^3$.
[*Outline of the proof*]{}
Our technique is to use O’Grady’s method of deformation to the boundary [@OGradyIrred; @OGradyBasic], as it was exploited by Nijsse [@Nijsse] in the case of a very general quintic hypersurface. We use, in particular, some of the intermediate results of Nijsse who showed, for example, that $\overline{M} (c_2)$ is connected for $c_2\geq 10$. Application of these results is made possible by the explicit description of the moduli spaces $M(c_2)$ for $4\leq c_2\leq 9$ obtained in [@MestranoSimpson1] and the partial result for $M(10)$ obtained in [@MestranoSimpson2].
The boundary $\partial \overline{M} (c_2):= \overline{M} (c_2)-M(c_2)$ is the set of points corresponding to torsion-free sheaves which are not locally free. We just endow $\partial \overline{M} (c_2)$ with its reduced scheme structure. There might in some cases be a better non-reduced structure which one could put on the boundary or onto some strata, but that won’t be necessary for our argument and we don’t worry about it here.
We can further refine the decomposition $$\overline{M} (c_2)=M(c_2)\sqcup \partial \overline{M} (c_2)$$ by the [*double dual stratification*]{} [@OGradyBasic]. Let $M (c_2; c_2')$ denote the locally closed subset, again with its reduced scheme structure, parametrizing sheaves $F$ which fit into an exact sequence $$0\rightarrow F \rightarrow F^{\ast\ast} \rightarrow S \rightarrow 0$$ such that $F\in \overline{M} (c_2)$ and $S$ is a coherent sheaf of finite length $d=c_2-c_2'$ hence $c_2(F^{\ast\ast})=c_2'$. Notice that $E=F^{\ast\ast}$ is also stable so it is a point in $M(c'_2)$. The stratum can be nonempty only when $c'_2\geq 4$, which shows by the way that $\overline{M}(c_2)$ is empty for $c_2\leq 3$. The boundary now decomposes into locally closed subsets $$\partial \overline{M} (c_2) = \coprod _{4 \leq c_2' < c_2} M (c_2;c'_2).$$ Let $\overline{M (c_2,c_2')}$ denote the closure of $M(c_2,c_2')$ in $\overline{M} (c_2)$. Notice that we don’t know anything about the position of this closure with respect to the stratification; its boundary will not in general be a union of strata. We can similarly denote by $\overline{M(c_2)}$ the closure of $M(c_2)$ inside $\overline{M} (c_2)$, a subset which might well be strictly smaller than $\overline{M} (c_2)$.
The construction $F\mapsto F^{\ast\ast}$ provides, by the definition of the stratification, a well-defined map $$M (c_2;c'_2)\rightarrow M(c'_2).$$ The fiber over $E\in M(c'_2)$ is the Grothendieck ${\rm Quot}$-scheme ${\rm Quot}(E;d)$ of quotients of $E$ of length $d:=c_2-c'_2$.
It follows from Li’s theorem [@Li Proposition 6.4] that if $M(c_2')$ is irreducible, then $M(c_2;c'_2)$ and hence $\overline{M(c_2;c'_2)}$ are irreducible, with ${\rm dim} (M(c_2;c'_2) )= {\rm dim} ( M(c_2') )+ 3 (c_2-c_2')$. See Corollary \[forward\] below. From the previous papers [@MestranoSimpson1; @MestranoSimpson2], we know the dimensions of $M(c_2')$, so we can fill in the dimensions of the strata, as will be summarized in Table \[maintable\]. Furthermore, by [@MestranoSimpson1] and Li’s theorem, the strata $M(c_2;c'_2)$ are irreducible whenever $c'_2\leq 9$.
Nijsse [@Nijsse] proves that $\overline{M} (c_2)$ is connected whenever $c_2\geq 10$, using O’Grady’s techniques [@OGradyIrred; @OGradyIrred]. By [@MestranoSimpson1], the moduli space $\overline{M} (c_2)$ is [*good*]{}, that is to say it is generically reduced of the expected dimension $4c_2-20$, whenever $c_2\geq 10$. In particular, the dimension of the Zariski tangent space, minus the dimension of the space of obstructions, is equal to the dimension of the moduli space. The Kuranishi theory of deformation spaces implies that $\overline{M} (c_2)$ is locally a complete intersection. Hartshorne’s connectedness theorem [@HartshorneConnectedness] now says that if two different irreducible components of $\overline{M} (c_2)$ meet at some point, then they intersect in a codimension $1$ subvariety. This intersection has to be contained in the singular locus.
The singular locus in $M(c_2)$ contains a subvariety denoted $V(c_2)$, which is the set of bundles $E$ with $h^0(E)>0$. For $c_2\geq 10$, the locus $V(c_2)$ has dimension $3c_2-11$, and $V(c_2)$ may be described as the space of bundles fitting into an exact sequence $$0\rightarrow {{\mathcal O}}_X\rightarrow E\rightarrow J_P(1)\rightarrow 0$$ where $P$ satisfies Cayley-Bacharach for quadrics. For $c_2\geq 11$ it is a general set of points, and the extension class is general, from which one can see that the closure of $V(c_2)$ meets the boundary. For $c_2=10$, $V(10)$ is also irreducible but its general point parametrizes bundles corresponding to subschemes $P$ consisting of $10$ general points on a smooth quadric section $Y\subset X$ (i.e. the intersection of $X$ with a divisor in $|{{\mathcal O}}_{{{\mathbb P}}^3}(2)|$). A generization then has seminatural cohomology so is contained in the component constructed in [@MestranoSimpson2], in particular meeting the boundary. On the other hand, any other irreducible components of the singular locus have strictly smaller dimension [@MestranoSimpson1].
These properties of the singular locus and $V(c_2)$, together with the connectedness statement of [@Nijsse], allow us to show that any irreducible component of $\overline{M} (c_2)$ meets the boundary. O’Grady proves furthermore an important lemma, that the intersection with the boundary must have pure codimension $1$.
We explain the strategy for proving irreducibility of $M(10)$ and $M(11)$ below, but it will perhaps be easiest to explain first why this implies irreducibility of $M(c_2)$ for $c_2\geq 12$. Based on O’Grady’s method, this is the same strategy as was used by Nijsse who treated the cases $c_2\geq 16$.
Suppose $c_2\geq 12$ and $Z\subset \overline{M} (c_2)$ is an irreducible component. Suppose inductively we know that $M(c_2-1)$ is irreducible. Then $\partial Z:= Z\cap \partial \overline{M} (c_2)$ is a nonempty subset in $Z$ of codimension $1$, thus of dimension $4c_2-21$. However, by looking at Table \[maintable\], the boundary $\partial \overline{M} (c_2)$ is a union of the stratum $M(c_2,c_2-1)$ of dimension $4c_2-21$, plus other strata of strictly smaller dimension. Therefore, $\partial Z$ must contain $M(c_2,c_2-1)$. But, the general torsion-free sheaf parametrized by a point of $M(c_2,c_2-1)$ is the kernel $F$ of a general surjection $E\rightarrow S$ from a stable bundle $E$ general in $M(c_2-1)$, to a sheaf $S$ of length $1$. We claim that $F$ is a smooth point of the moduli space $\overline{M} (c_2)$. Indeed, if $F$ were a singular point then there would exist a nontrivial co-obstruction $\phi : F\rightarrow F(1)$, see [@Langer; @MestranoSimpson1; @Zuo]. This would have to come from a nontrivial co-obstruction $E\rightarrow E(1)$ for $E$, but that cannot exist because a general $E$ is a smooth point since $M(c_2-1)$ is good. Thus, $F$ is a smooth point of the moduli space. It follows that a given irreducible component of $M(c_2,c_2-1)$ is contained in at most one irreducible component of $\overline{M} (c_2)$. On the other hand, by the induction hypothesis $M(c_2-1)$ is irreducible, so $M(c_2,c_2-1)$ is irreducible. This gives the induction step, that $M(c_2)$ is irreducible.
The strategy for $M(10)$ is similar. However, due to the fact that the moduli spaces $M(c'_2)$ are not good for $c'_2\leq 9$, in particular they tend to have dimensions bigger than the expected dimensions, there are several boundary strata which can come into play. Luckily, we know that the $M(c'_2)$, hence all of the strata $M(10,c'_2)$, are irreducible for $c'_2\leq 9$.
The dimension of $M(c_2)$, equal to the expected one, is $20$. Looking at the row $c_2=10$ in Table \[maintable\] below, one may see that there are three strata $M(10,9)$, $M(10,8)$ and $M(10,6)$ with dimension $19$. These can be irreducible components of the boundary $\partial Z$ if we follow the previous argument. More difficult is the case of the stratum $M(10,4)$ which has dimension $20$. A general point of $M(10,4)$ is not in the closure of $M(10)$, in other words $M(10,4)$, which is closed since it is the lowest stratum, constitutes a separate irreducible component of $\overline{M} (10)$. Now, if $Z\subset M(10)$ is an irreducible component, $\partial Z$ could contain a codimension $1$ subvariety of $M(10,4)$.
The idea is to use the main result of [@MestranoSimpson2], that the moduli space $M(10)^{\rm sn}$ of bundles with seminatural cohomology, is irreducible. To prove that $M(10)$ is irreducible, it therefore suffices to show that a general point of any irreducible component $Z$, has seminatural cohomology. From [@MestranoSimpson2] there are two conditions that need to be checked: $h^0(E)=0$ and $h^1(E(1))=0$. The first condition is automatic for a general point, since the locus $V(10)$ of bundles with $h^0(E)>0$ has dimension $3\cdot 10-11=19$ so cannot contain a general point of $Z$. For the second condition, it suffices to note that a general sheaf $F$ in any of the strata $M(10,9)$, $M(10,8)$ and $M(10,6)$ has $h^1(F(1))=0$; and to show that the subspace of sheaves $F$ in $M(10,4)$ with $h^1(F(1))>0$ has codimension $\geq 2$. This latter result is treated in Section \[lowest\], using the dimension results of Ellingsrud-Lehn for the scheme of quotients of a locally free sheaf, generalizing Li’s theorem. This is how we will show irreducibility of $M(10)$.
The full moduli space of torsion-free sheaves $\overline{M} (10)$ has two different irreducible components, the closure $\overline{M(10)}$ and the lowest stratum $M(10,4)$. This distinguishes the case of the quintic surface from the cases of abelian and K3 surfaces, where the full moduli spaces of stable torsion-free sheaves were irreducible [@YoshiokaAbelian; @YoshiokaK3; @GomezThesis].
For $M(11)$, the argument is almost the same as for $c_2\geq 12$. However, there are now two different strata of codimension $1$ in the boundary: $M(11,10)$ coming from the irreducible variety $M(10)$, and $M(11,4)$ which comes from the other $20$-dimensional component $M(10,4)$ of $\overline{M} (10)$. To show that these two can give rise to at most a single irreducible component in $M(11)$, completing the proof, we will note that they do indeed intersect, and furthermore that the intersection contains smooth points.
Preliminary facts
=================
The moduli space $\overline{M} (c_2)$ is locally a fine moduli space. The obstruction to existence of a Poincaré universal sheaf on $\overline{M} (c_2)\times X$ is an interesting question but not considered in the present paper. A universal family exists etale-locally over $\overline{M} (c_2)$ so for local questions we may consider $\overline{M} (c_2)$ as a fine moduli space.
The Zariski tangent space to $\overline{M} (c_2)$ at a point $E$ is ${\rm Ext}^1(E,E)$. If $E$ is locally free, this is the same as $H^1({\rm End}(E))$. The [*space of obstructions*]{} ${\rm obs}(E)$ is by definition the kernel of the surjective map $${\rm Tr}: {\rm Ext}^2(E,E)\rightarrow H^2({{\mathcal O}}_X).$$ The [*space of co-obstructions*]{} is the dual ${\rm obs}(E)^{\ast}$ which is, by Serre duality with $K_X={{\mathcal O}}_X(1)$, equal to ${\rm Hom}^0(E,E(1))$, the space of maps $\phi : E\rightarrow E(1)$ such that ${\rm Tr}(E)=0$ in $H^0({{\mathcal O}}_X(1))\cong {{\mathbb C}}^4$. Such a map is called a [*co-obstruction*]{}.
Since a torsion-free sheaf $E$ of rank two and odd degree can have no rank-one subsheaves of the same slope, all semistable sheaves are stable, and Gieseker and slope stability are equivalent. If $E$ is a stable sheaf then ${\rm Hom}(E,E)= {{\mathbb C}}$ so the space of trace-free endomorphisms is zero. Notice that $H^1({{\mathcal O}}_X)=0$ so we may disregard the trace-free condition for ${\rm Ext}^1(E,E)$. An Euler-characteristic calculation gives $${\rm dim}(Ext^1(E,E)) - {\rm dim}({\rm obs}(E)) =4c_2-20,$$ and this is called the [*expected dimension*]{} of the moduli space. The moduli space is said to be [*good*]{} if the dimension is equal to the expected dimension.
\[lci\] If the moduli space is good, then it is locally a complete intersection.
Kuranishi theory expresses the local analytic germ of the moduli space $\overline{M} (c_2)$ at $E$, as $\Phi ^{-1}(0)$ for a holomorphic map of germs $\Phi : ({{\mathbb C}}^a ,0) \rightarrow ({{\mathbb C}}^b,0)$ where $a={\rm dim}(Ext^1(E,E))$ (resp. $b={\rm dim}({\rm obs}(E))$). Hence, if the moduli space has dimension $a-b$, it is a local complete intersection.
We investigated closely the structure of the moduli space for $c_2\leq 9$, in [@MestranoSimpson1].
\[leq9\] The moduli space $M(c_2)$ is empty for $c_2\leq 3$. For $4\leq c_2\leq 9$, the moduli space $M(c_2)$ is irreducible. It has dimension strictly bigger than the expected one, for $4\leq c_2\leq 8$, and for $c_2=9$ it is generically nonreduced but with dimension equal to the expected one; it is also generically nonreduced for $c_2=7$. The dimensions of the moduli spaces, the dimensions of the spaces of obstructions at a general point, and the dimensions $h^1(E(1))$ for a general bundle $E$ in $M(c_2)$, are given in the following table.
$c_2$ $4$ $5$ $6$ $7$ $8$ $9$
------------------------ ----- ----- ----- ----- ------ ------
${\rm dim}(M)$ $2$ $3$ $7$ $9$ $13$ $16$
${\rm dim}({\rm obs})$ $6$ $3$ $3$ $3$ $1$ $1$
$h^1(E(1))$ $0$ $1$ $0$ $0$ $0$ $0$
: \[leq9table\] Moduli spaces for $c_2\leq 9$
The bundles $E$ occuring for $c_2\leq 9$ always fit into an extension of $J_P(1)$ by ${{\mathcal O}}_X$. As in the previous paper, we apologize again for the change of notation with respect to [@MestranoSimpson1] where we considered bundles of degree $-1$, but the indexing by second Chern class remains the same. The subscheme $P\subset X$ is locally a complete intersection of length $c_2$ and satisfies the Cayley-Bacharach condition for quadrics.
In [@MestranoSimpson1], we considered $c$ the number of conditions imposed by $P$ on quadrics. This is related to $h^1(E(1))$ by the exact sequences $$H^0({{\mathcal O}}_X(2))\rightarrow H^0({{\mathcal O}}_P(2))\rightarrow H^1(J_{P/X}(2)) \rightarrow 0$$ and $$0\rightarrow H^1(E(1)) \rightarrow H^1(J_{P/X}(2))\rightarrow H^2({{\mathcal O}}_X(1)) \rightarrow 0$$ where $H^2(E(1))= H^0(E(1))^{\ast}=0$ by stability, and $H^2({{\mathcal O}}_X(1))=H^2(K_X)={{\mathbb C}}$. The number $c$ is the rank of the evaluation map of $H^0({{\mathcal O}}_X(2))$ on $P$, so $h^1(J_{P/X}(2))= c_2-c$, and by the second exact sequence we have $h^1(E(1))=c_2-c-1$. This helps to extract the remaining values of the last row of the table from the discussion of [@MestranoSimpson1].
Be careful that [@MestranoSimpson1 Lemma 5.2] doesn’t discuss $h^1(E(1))$ but rather speaks of $h^1(E)$ in our notation.
For the column $c_2=9$, see [@MestranoSimpson1], Theorem 6.1 and Proposition 7.2 for the dimension $16$ and general obstruction space of dimension $1$. The proof of Proposition 7.2 starts out by ruling out, for a general point of an irreducible component, all cases of Proposition 7.1 except case (d), for which $c=8$. Thus $h^1(E(1))=9-8-1=0$ for a general bundle.
For the column $c_2=8$, see [@MestranoSimpson1 Theorem 7.2] for irreducibility and the dimension. The family constructed in [@MestranoSimpson1 Section 6.2] has obstruction space of dimension $1$, and the dimension of the whole space is $13$, strictly greater than the expected dimension, so it follows that the generic space of obstructions has dimension $1$. Note that $c=7$ for the $13$-dimensional family considered in [@MestranoSimpson1 Section 7.4], so $h^1(E(1))=8-7-1=0$.
For $c_2=6,7$ the general bundle corresponds to a subscheme $P$ contained in, and spanning a unique plane. In this case, the space of obstructions has dimension $3$ by [@MestranoSimpson1 Lemma 5.5].
For the column $c_2=7$, see [@MestranoSimpson1 Proposition 7.3] where the dimension is $9$. In the proof there, the biggest stratum of the moduli space corresponds to $c=6$, giving $h^1(E(1))=7-6-1=0$.
For the column, $c_2=6$, see [@MestranoSimpson1 Proposition 7.4] where the dimension is $7$ and $c=5$, so again $h^1(E(1))=6-5-1=0$.
For the columns $c_2=4,5$, note that the subscheme $P$ is either $4$ or $5$ points contained in a line. Both of these configurations impose $c=3$ conditions on quadrics, since $h^0({{\mathcal O}}_{{{\mathbb P}}^1}(2))=3$. This gives values of $4-3-1=0$ and $5-3-1=1$ for $h^1(E(1))$ respectively. The moduli space is generically smooth and its dimension is equal to $c_2-2$ by [@MestranoSimpson1 Lemma 7.7], and we get the dimension of the space of co-obstructions by subtracting the expected dimension.
We also proved that the moduli space is good for $c_2\geq 10$, known by Nijsse [@Nijsse] for $c_2\geq 13$.
\[singularlocus\] For $c_2\geq 10$, the moduli space $M(c_2)$ is good. The singular locus $M(c_2)^{\rm sing}$ is the union of the locus $V(c_2)$ consisting of bundles with $h^0(E)>0$, which has dimension $3c_2-11$, plus other pieces of dimension $\leq 13$ which in particular have codimension $\geq 6$.
Following O’Grady’s and Nijsse’s terminology $V(c_2)$ denotes the locus which which is the image of the moduli space of bundles together with a section, called $\Sigma _{c_2}$. See [@MestranoSimpson1], Theorem 7.1. Any pieces of the singular locus corresponding to bundles which are not in $V(c_2)$, have dimension $\leq 13$ by [@MestranoSimpson1 Corollary 5.1].
Let $M(10)^{\rm sn}\subset M(10)$ denote the open subset of bundles $E\in M(10)$ which have [*seminatural cohomology*]{}, that is where for any $m$ at most one of $h^i(E(m))$ is nonzero for $i=0,1,2$. Then $E\in M(10)^{\rm sn}$ if and only if $h^0(E)=0$ and $h^1(E(1))=0$. The moduli space $M(10)^{\rm sn}$ is irreducible.
See [@MestranoSimpson2], Theorem 0.2 and Corollary 3.5.
The double dual stratification {#strata}
==============================
Our proofs will make use of O’Grady’s techniques [@OGradyIrred; @OGradyBasic], as they were recalled and used by Nijsse in [@Nijsse]. The main idea is to look at the boundary of the moduli spaces. His first main observation is the following [@OGradyBasic Proposition 3.3]:
\[codim1\] The boundary of any irreducible component (or indeed, of any closed subset) of $M(c_2)$ has pure codimension $1$, if it is nonempty.
The boundary is divided up into Uhlenbeck strata corresponding to the “number of instantons”, which in the geometric picture corresponds to the number of points where the torsion-free sheaf is not a bundle, counted with correct multiplicities. A boundary stratum denoted $M(c_2,c_2-d)$ parametrizes torsion-free sheaves $F$ fitting into an exact sequence of the form $$0\rightarrow F\rightarrow E \stackrel{\sigma}{\rightarrow} S \rightarrow 0$$ where $E\in M(c_2-d)$ is a stable locally free sheaf of degree $1$ and $c_2(E)=c_2-d$, and $S$ is a finite coherent sheaf of finite length $d$ so that $c_2(F)=c_2$. In this case $E=F^{\ast\ast}$. We may think of $M(c_2,c_2-d)$ as the moduli space of pairs $(E,\sigma )$. Forgetting the quotient $\sigma$ gives a smooth map $$M(c_2,c_2-d) \rightarrow M(c_2-d),$$ sending $F$ to its double dual. The fiber over $E$ is the Grothedieck ${\rm Quot}$ scheme ${\rm Quot}(E, d)$ parametrizing quotients $\sigma$ of $E$ of length $d$.
Since we are dealing with sheaves of degree $1$, all semistable points are stable and our objects have no non-scalar automorphisms. Hence the moduli spaces are fine, with a universal family existing etale-locally and well-defined up to a scalar automorphism. We may view the double-dual map as being the relative Grothendieck ${\rm Quot}$ scheme of quotients of the universal object $E^{\rm univ}$ on $M(c_2-d)\times X/M(c_2-d)$. Furthermore, locally on the ${\rm Quot}$ scheme the quotients are localized near a finite set of points, and we may trivialize the bundle $E^{\rm univ}$ near these points, so $M(c_2,c_2-d)$ has a covering by, say, analytic open sets which are trivialized as products of open sets in the base $M(c_2-d)$ with open sets in ${\rm Quot}(E,d)$ for any single choice of $E$. This is all to say that the map $M(c_2,c_2-d)\rightarrow M(c_2-d)$ may be viewed as a fibration in a fairly strong sense, with fiber ${\rm Quot}(E,d)$.
Li shows in [@Li Proposition 6.4] that ${\rm Quot}(E,d)$ is irreducible with a dense open subset $U$ parametrizing quotients which are given by a collection of $d$ quotients of length $1$ supported at distinct points of $X$:
\[quotli\] Suppose $E$ is a locally free sheaf of rank $2$ on $X$. Then for any $d>0$, ${\rm Quot}(E,d)$ is an irreducible scheme of dimension $3d$, containing a dense open subset parametrizing quotients $E\rightarrow S$ such that $S\cong \bigoplus {{\mathbb C}}_{y_i}$ where ${{\mathbb C}}_{y_i}$ is a skyscraper sheaf of length $1$ supported at $y_i\in X$, and the $y_i$ are distinct. This dense open set maps to $X^{(d)}-{\rm diag}$ (the space of choices of distinct $d$-uple of points in $X$), with fiber over $\{ y_i\}$ equal to $\prod _{i=1}^d {{\mathbb P}}(E_{y_i})$.
See Propostion 6.4 in the appendix of [@Li]. Notice right away that $U$ is an open subset of ${\rm Quot}(F,d)$, and that $U$ fibers over the set $X^{(d)}-{\rm diag}$ of distinct $d$-uples of points $(y_1,\ldots , y_d)$ (up to permutations). The fiber over a $d$-uple $(y_1,\ldots , y_d)$ is the product of projective lines ${{\mathbb P}}(F_{y_i})$ of quotients of the vector spaces $F_{y_i}$. As $X^{(d)}-{\rm diag}$ has dimension $2d$, and $\prod _{i=1}^d {{\mathbb P}}(F_{y_i})$ has dimension $d$, we get that $U$ is a smooth open variety of dimension $3d$.
This theorem may also be viewed as a consequence of a more precise bound established by Ellingsrud and Lehn [@EllingsrudLehn], which will be stated as Theorem \[el\] below, needed for our arguments in Section \[lowest\].
\[forward\] We have $${\rm dim} (M(c_2;c'_2) )= {\rm dim} ( M(c_2') )+ 3 (c_2-c_2').$$ If $M(c_2')$ is irreducible, then $M(c_2;c'_2)$ and hence $\overline{M(c_2;c'_2)}$ are irreducible.
The fibration $M (c_2;c'_2)\rightarrow M(c'_2)$ has fiber the ${\rm Quot}$ scheme whose dimension is $3 (c_2-c_2')$ by the previous proposition. Furthermore, these ${\rm Quot}$ schemes are irreducible so if the base is irreducible, so is the total space.
Corollary \[forward\] allows us to fill in the dimensions of the strata $M (c_2;c'_2)$ in the following table. The entries in the second column are the expected dimension $4c_2-20$; in the third column the dimension of $M:= M(c_2)$; and in the following columns, ${\rm dim} M (c_2,c_2-d)$ for $d=1,2,\ldots $. The rule is to add $3$ as you go diagonally down and to the right by one.
$c_2$ e.d. ${\rm dim}(M)$ ${\scriptstyle d=1}$ ${\scriptstyle d=2}$ ${\scriptstyle d=3}$ ${\scriptstyle d=4}$ ${\scriptstyle d=5}$ ${\scriptstyle d=6}$ ${\scriptstyle d=7}$ ${\scriptstyle d=8}$
-------------------------- -------------------------- -------------------------- -------------------------- ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- ---------------------- ----------------------
$4$ $-4$ $2$ $-$ $-$ $-$ $-$ $-$ $-$ $-$ $-$
$5$ $0$ $3$ $5$ $-$ $-$ $-$ $-$ $-$ $-$ $-$
$6$ $4$ $7$ $6$ $8$ $-$ $-$ $-$ $-$ $-$ $-$
$7$ $8$ $9$ $10$ $9$ $11$ $-$ $-$ $-$ $-$ $-$
$8$ $12$ $13$ $12$ $13$ $12$ $14$ $-$ $-$ $-$ $-$
$9$ $16$ $16$ $16$ $15$ $16$ $15$ $17$ $-$ $-$ $-$
$10$ $20$ $20$ $19$ $19$ $18$ $19$ $18$ $20$ $-$ $-$
$11$ $24$ $24$ $23$ $22$ $22$ $21$ $22$ $21$ $23$ $-$
$12$ $28$ $28$ $27$ $26$ $25$ $25$ $24$ $25$ $24$ $26$
${\scriptstyle \geq 13}$ ${\scriptstyle 4c_2-20}$ ${\scriptstyle 4c_2-20}$ ${\scriptstyle 4c_2-21}$
: \[maintable\] Dimensions of strata
The first remark useful for interpreting this information, is that any irreducible component of $\overline{M} (c_2)$ must have dimension at least equal to the expected dimension $4c_2-20$. In particular, a stratum with strictly smaller dimension, must be a part of at least one irreducible component consisting of a bigger stratum. For $c_2\geq 11$, we have $${\rm dim} (M (c_2,c_2'))< {\rm dim}(\overline{M} (c_2)) = 4c_2-20.$$ Hence, for $c_2\geq 11$ the closures $\overline{M (c_2,c_2')}$ cannot themselves form irreducible components of $\overline{M} (c_2)$, in other words the irreducible components of $\overline{M} (c_2)$ are the same as those of $M (c_2)$. Notice, on the other hand, that $\overline{M} (10)$ contains two pieces of dimension $20$, the locally free sheaves in $M(10)$ and the sheaves in $M(10,4)$ whose double duals come from $M(4)$.
Recall from [@MestranoSimpson1] that the moduli spaces $M(c_2)$ are irreducible for $c_2=4,\ldots , 9$. It follows from Corollary \[forward\] that the strata $M(c_2,c_2')$ are irreducible, for any $c_2'\leq 9$. In particular, the piece $\overline{M (10,4)}$ is irreducible, and its general point, representing a non-locally free sheaf, is not confused with any point of $\overline{M(10)}$. Since the other strata of $\overline{M} (10)$ all have dimension $<20$, it follows that $\overline{M (10,4)}$ is an irreducible component of $\overline{M} (10)$. One similarly gets from the table that $\overline{M}(c_2)$ has several irreducible components when $5\leq c_2\leq 9$.
Hartshorne’s connectedness theorem {#hartshorne}
==================================
Hartshorne proves a connectedness theorem for local complete intersections. Here is the version that we need.
Suppose $Z$ is a local complete intersection of dimension $d$. Then, any nonempty intersection of two irreducible components of $Z$ has pure dimension $d-1$.
See [@HartshorneConnectedness; @SawantHC].
\[meetsing\] If the moduli space $\overline{M} $ is good, and has two different irreducible components $Z_1$ and $Z_2$ meeting at a point $z$, then $Z_1\cap Z_2$ has codimension $1$ at $z$ and the singular locus ${\rm Sing}(\overline{M} )$ contains $z$ and has codimension $1$ at $z$.
If $\overline{M} $ is good, then by Lemma \[lci\] it as a local complete intersection so Hartshorne’s theorem applies: $Z_1\cap Z_2$ has pure codimension $1$. The intersection of two irrreducible components is necessarily contained in the singular locus.
We draw the following conclusions.
\[meetalongV\] Suppose, for $c_2\geq 10$, that two different irreducible components $Z_1$ and $Z_2$ of $\overline{M} $ meet at a point $z$ not on the boundary. Then $c_2=10$ and both components contain the subscheme $V(c_2)= \{ E, h^0(E)>0 \}$ which in turn contains $z$ and has codimension $\leq 1$ at $z$.
We have seen in [@MestranoSimpson1 Theorem 7.1] that for $c_2\geq 10$, a codimension $1$ piece of ${\rm Sing}(M)$ has to be in $V(c_2)$, cf Proposition \[singularlocus\] above. On the other hand $V(c_2)$ is irreducible, see the proof of Corollary \[meetsboundary\] below, so a codimension $1$ piece of ${\rm Sing}(M)$ has to be equal to $V(c_2)$. This is contained in both irreducible components $Z_1,Z_2$ by Corollary \[meetsing\]. One may furthermore note that ${\rm dim}(V(c_2))=3c_2-11$ whereas the dimension of the moduli space is $4c_2-20$, thus for $c_2\geq 11$ the singular locus has codimension $\geq 2$, so the situation of the present corollary can only happen for $c_2=10$.
Next, recall one of Nijsse’s theorems, connectedness of the moduli space.
For $c_2\geq 10$, the moduli space $\overline{M} $ is connected.
See [@Nijsse], Proposition 3.2.
\[meetsboundary\] Suppose $Z$ is an irreducible component of $\overline{M} (c_2)$ for $c_2\geq 10$. Then $Z$ meets the boundary in a nonempty codimension $1$ subset.
The codimension $1$ property is given by Lemma \[codim1\], so we just have to show that $\overline{Z}$ contains a boundary component.
To start with, note that for $c_2\geq 10$, the first boundary stratum $M(c_2,c_2-1)$ has codimension $1$, so it must meet at least one irreducible component of $\overline{M(c_2)}$, call it $Z_0$. Suppose $Z\subset M(c_2)$ is another irreducible component with $c_2\geq 10$. By the connectedness of $\overline{M} (10)$, there exist a sequence of irreducible components $Z_0,\ldots , Z_k=\overline{Z}$ such that $Z_i\cap Z_{i+1}$ is nonempty. By Lemma \[meetalongV\], either $Z_{k-1}\cap Z_k$ meets the boundary, or else it contains $V(c_2)$. In the first case we are done.
In the second case, one must have $c_2=10$ by Corollary \[meetalongV\] (although here one could alternatively argue that for $c_2\geq 11$, $V(11)$ itself meets the boundary as was discussed in the proof of [@Nijsse Proposition 3.2]). At $c_2=10$, $V(10)$ is also irreducible. It parametrizes subschemes $P\subset X$ such that there exists a quadric section $Y\subset X$, a divisor in the linear system $|{{\mathcal O}}_X(2)|$, containing $P$. A general element of $V(10)$ corresponds to $10$ general points on a general smooth quadric section $Y$. One may see that for a general $E\in V(10)$, we have $h^1(E(1))=0$. This implies that any irreducible component of $M(10)$ containing $V$ parametrizes, generically, bundles with seminatural cohomology. Thus, any irreducible component of $M(10)$ containing $V$ must be the unique component constructed in [@MestranoSimpson2]. See the proof of Corollary \[seminat10\] below for more details of this argument. But that component meets the boundary, indeed the $19$-dimensional boundary strata $M(10,9)$ etc., are also contained in the same irreducible component—again see the proof of Corollary \[seminat10\] below. It follows that any irreducible component of $M(10)$ which contains $V(10)$, must meet the boundary. This completes the proof.
Seminaturality along the $19$-dimensional boundary components {#seminat19}
=============================================================
To treat the case $c_2=10$, we will apply the main result of our previous paper.
\[sn\] Suppose $Z$ is an irreducible component of $M(10)$. Suppose that $\overline{Z}$ contains a point corresponding to a torsion-free sheaf $F$ with $h^1(F(1))=0$. Then $Z$ is the unique irreducible component containing the open set of bundles with seminatural cohomology, constructed in [@MestranoSimpson2].
The locus $V(c_2)$ of bundles with $h^0(E)\neq 0$ has dimension $\leq 19$, so a general point of $Z$ must have $h^0(E)=0$. The hypothesis implies that a general point has $h^1(E(1))=0$. Thus, there is a nonempty dense open subset $Z'\subset Z$ parametrizing bundles with $h^0(E)=0$ and $h^1(E(1))=0$. By [@MestranoSimpson2 Corollary 3.5], these bundles have seminatural cohomology. Thus, our open set is $Z'= M(10)^{\rm sn}$, the moduli space of bundles with seminatural cohomology, shown to be irreducible in the main Theorem 0.2 of [@MestranoSimpson2].
Using this proposition, and since we know by Corollary \[meetsboundary\] that any irreducible component $Z$ meets the boundary in a codimension $1$ subset, in order to prove irreducibility of $M(10)$, it suffices to show that the torsion-free sheaves $F$ parametrized by general points on the various irreducible components of the boundary of $\overline{M(10)}$ have $h^1(F(1))=0$.
The dimension is ${\rm dim}(Z)=20$, so the boundary components will have dimension $19$. Looking at the line $c_2=10$ in Table \[maintable\], we notice that there are three $19$-dimensional boundary pieces, and a $20$-dimensional piece which must constitute a different irreducible component. Consider first the $19$-dimensional pieces, $$M(10,9), \;\;\;\; M(10,8) \mbox{ and } M(10,6).$$ Recall that $M(10,10-d)$ consists generically of torsion-free sheaves $F$ fitting into an exact sequence $$\label{ffs}
0\rightarrow F \rightarrow F^{\ast\ast} \rightarrow S \rightarrow 0$$ where $F^{\ast\ast}$ is a general point in the moduli space of stable bundles with $c_2=10-d$, and $S$ is a general quotient of length $d$.
\[firstthree\] For a general point $F$ in either of the three boundary pieces $M(10,9)$, $M(10,8)$ or $M(10,6)$, we have $h^1(F(1))=0$.
Notice that $\chi (F^{\ast\ast}(1)) =15-c_2(F^{\ast\ast}) \geq 6$ and by stability $h^2(F^{\ast\ast}(1))= h^0(F^{\ast\ast}(-1))=0$, so $F^{\ast\ast}$ has at least six linearly independent sections. In particular, for a general quotient $S$ of length $1$, $2$ or $4$, consisting of the direct sum $S=\bigoplus S_x$ of general rank $1$ quotients $E_x\rightarrow S_x$ at $1$, $2$ or $4$ distinct general points $x$, the map $$H^0(F^{\ast\ast}(1))\rightarrow H^0(S)$$ will be surjective.
For a general point $F^{\ast\ast}$ in either $M(9)$, $M(8)$ or $M(6)$, we have $h^1(F^{\ast\ast}(1))=0$. These results from [@MestranoSimpson1] were recalled in Proposition \[leq9\], Table \[leq9table\]. The long exact sequence associated to now gives $h^1(F(1))=0$.
This treats the $19$-dimensional irreducible components of the boundary. There remains the piece $\overline{M(10,4)}$ which has dimension $20$. This is a separate irreducible component. It could meet $\overline{M(10)}$ along a $19$-dimensional divisor, and we would like to show that $h^1(F(1))=0$ for the sheaves parametrized by this divisor. In particular, we are no longer in a completely generic situation so some further discussion is needed. This will be the topic of the next section.
The lowest stratum {#lowest}
==================
The lowest stratum is $M(10,4)$, which is therefore closed. We would like to understand the points in $\overline{M(10)} \cap M(10,4)$. These are singular, so our main tool will be to look at where the singular locus of $\overline{M} (10)$ meets $M(10,4)$. Denote this by $$M(10,4)^{\rm sing} := {\rm Sing}(\overline{M} (10)) \cap M(10,4).$$ In what follows, we give a somewhat explicit description of the lowest moduli space $M(4)$.
\[m4start\] For $E\in M(4)$ we have $h^1(E)=0$, $h^0(E)=h^2(E)=3$, $h^0(E(1)) =11$, and $h^1(E(1))=h^2(E(1))=0$.
Choosing an element $s\in H^0(E)$ gives an exact sequence $$\label{esE}
0\rightarrow {{\mathcal O}}_X \rightarrow E\rightarrow J_{P/X}(1)\rightarrow 0.$$ In [@MestranoSimpson1] we have seen that $P\subset X\cap \ell$ is a subscheme of length $4$ in the intersection of $X$ with a line $\ell \subset {{\mathbb P}}^3$. As $P$ spans $\ell$, the space of linear forms vanishing on $P$ is the same as the space of linear forms vanishing on $\ell$, so $H^0(J_{P/X}(1))\cong {{\mathbb C}}^2$. In the long exact sequence associated to , note that $H^1({{\mathcal O}}_X)=0$, giving $$0\rightarrow H^0({{\mathcal O}}_X)\rightarrow H^0(E)\rightarrow H^0(J_{P/X}(1))\rightarrow 0$$ hence $H^0(E)\cong {{\mathbb C}}^3$. By duality, $H^2(E)\cong {{\mathbb C}}^3$, and the Euler characteristic of $E$ is $6$, so $H^1(E)=0$.
For $E(1)$, note that $H^2(E(1))=0$ by stability and duality, and gives an exact sequence $$0\rightarrow H^1(E(1))\rightarrow H^1(J_{P/X}(2))\rightarrow H^2({{\mathcal O}}_X(1)) \rightarrow 0.$$ On the other hand, $H^1(J_{P/X}(2)) \cong {{\mathbb C}}$ corresponding to the length $4$ of $P$, minus the dimension $3$ of the space of sections of ${{\mathcal O}}_P(2)$ coming from global quadrics (since the space of quadrics on $\ell$ has dimension $3$). This gives $H^1(E(1))=0$. The Euler characteristic then gives $h^0(E(1))=11$. This is also seen in the first part of the exact sequence, where $H^0({{\mathcal O}}_X(1))={{\mathbb C}}^4$ and $H^0(J_{P/X}(2))\cong {{\mathbb C}}^7$.
If $p\in {{\mathbb P}}^3$, let $G\cong {{\mathbb C}}^3$ be the space of linear generators of the ideal of $p$, that is to say $G:=H^0(J_{p/{{\mathbb P}}^3}(1))$, and consider the natural exact sequence of sheaves on ${{\mathbb P}}^3$ $$0\rightarrow {{\mathcal O}}_{{{\mathbb P}}^3}(-1)\rightarrow {{\mathcal O}}_{{{\mathbb P}}^3}\otimes G^{\ast} \rightarrow {{\mathcal R}}_p \rightarrow 0.$$ Here the cokernel sheaf ${{\mathcal R}}_p$ is a reflexive sheaf of degree $1$, and $c_2({{\mathcal R}}_p)$ is the class of a line. The restriction ${{\mathcal R}}_p|_X$ therefore has $c_2=5$. If $p\in X$, it is torsion-free but not locally free, giving a point in $M(5,4)$. It turns out that these sheaves account for all of $M(4)$ and $M(5)$.
\[descrip45\] Suppose $E\in M(4)$. Then there is a unique point $p\in X$ such that $E$ is generated by global sections outside of $p$, and ${{\mathcal R}}_p|_X$ is isomorphic to the subsheaf of $E$ generated by global sections. This fits into an exact sequence $$0\rightarrow {{\mathcal R}}_p|_X \rightarrow E\rightarrow S\rightarrow 0$$ where $S$ has length $1$, in particular $E\cong ({{\mathcal R}}_p|_X)^{\ast\ast}$. The correspondence $E\leftrightarrow p$ establishes an isomorphism $M(4)\cong X$.
For $E'\in M(5)$ there exists a unique point $p\in {{\mathbb P}}^3-X$ such that $E'\cong {{\mathcal R}}_p|_X$. This correspondence establishes an isomorphism $\overline{M(5)}\cong {{\mathbb P}}^3$ such that the boundary component $M(5,4)\cap \overline{M(5)}$ is exactly $X\subset {{\mathbb P}}^3$. Note however that $M(5,4)$ itself is bigger and constitutes another irreducible component of $\overline{M} (5)$.
Consider the exact sequence . The space $H^0(J_{P/X}(1))$ consists of linear forms on $X$ (or equivalently, on ${{\mathbb P}}^3$), which vanish along $P$. However, a linear form which vanishes on $P$ also vanishes on $\ell$. In particular, elements of $H^0(J_{P/X}(1))$ generate $J_{X\cap \ell /X}(1)$, which has colength $1$ in $J_{P/X}(1)$.
Let $R\subset E$ be the subsheaf generated by global sections, and let $S$ be the cokernel in the exact sequence $$0\rightarrow R \rightarrow E\rightarrow S\rightarrow 0.$$ We also have the exact sequence $$0\rightarrow J_{X\cap \ell /X}(1)\rightarrow J_{P/X}(1)\rightarrow S\rightarrow 0$$ so $S$ has length $1$. It is supported on a point $p$. The sheaf $R$ is generated by three global sections so we have an exact sequence $$0\rightarrow L \rightarrow {{\mathcal O}}_X^3\rightarrow R\rightarrow 0.$$ The kernel is a saturated subsheaf, hence locally free, and by looking at its degree we have $L={{\mathcal O}}_X(-1)$. Thus, $R$ is the cokernel of a map ${{\mathcal O}}_X(-1)\rightarrow {{\mathcal O}}_X^3$ given by three linear forms; these linear forms are a basis for the space of forms vanishing at the point $p$. We see that $R$ is the restriction to $X$ of the sheaf ${{\mathcal R}}_p$ described above, hence $E\cong ({{\mathcal R}}_p |_X)^{\ast\ast}$. The map $E\mapsto p$ gives a map $M(4)\rightarrow X$, with inverse $p\mapsto ({{\mathcal R}}_p |_X)^{\ast\ast}$.
The second paragraph, about $\overline{M(5)}$, is not actually needed later and we leave it to the reader.
Even though the moduli space $M(4)$ is smooth, it has much more than the expected dimension, and the space of co-obstructions is nontrivial. It will be useful to understand the co-obstructions, because if $F\in M(10,4)$ is a torsion-free sheaf with $F^{\ast\ast}=E$ then co-obstructions for $F$ come from co-obstructions for $E$ which preserve the subsheaf $F\subset E$.
\[m4obs\] Suppose $E\in M(4)$. A general co-obstruction $\phi : E\rightarrow E(1)$ has generically distinct eigenvalues with an irreducible spectral variety in ${\rm Tot}(K_X)$.
It suffices to write down a map $\phi:E\rightarrow E(1)$ with generically distinct eigenvalues and irreducible spectral variety. To do this, we construct a map $\phi _R:R\rightarrow R(1)$ using the expression $R={{\mathcal R}}_p |_X$. The exact sequence defining ${{\mathcal R}}_p$ extends to the Koszul resolution, a long exact sequence $$0\rightarrow {{\mathcal O}}_{{{\mathbb P}}^3}(-1)\rightarrow {{\mathcal O}}_{{{\mathbb P}}^3}^3 \rightarrow
{{\mathcal O}}_{{{\mathbb P}}^3}(1)^3 \rightarrow J_{p/{{\mathbb P}}^3}(2)\rightarrow 0.$$ Thus ${{\mathcal R}}_p$ may be viewed as the image of the middle map. Without loss of generality, $p$ is the origin in an affine system of coordinates $(x,y,z)$ for ${{\mathbb A}}^3\subset {{\mathbb P}}^3$, and the coordinate functions are the three coefficients of the maps on the left and right in the Koszul sequence. The $3\times 3$ matrix in the middle is $$K:= \left(
\begin{array}{ccc}
0 & z & -y \\
-z & 0 & x \\
y & -x & 0
\end{array}
\right) .$$ Any $3\times 3$ matrix of constants $\Phi$ gives a composed map $$\phi _R : {{\mathcal R}}_p \hookrightarrow {{\mathcal O}}_{{{\mathbb P}}^3}(1)^3 \stackrel{\Phi}{\rightarrow} {{\mathcal O}}_{{{\mathbb P}}^3}(1)^3 \rightarrow {{\mathcal R}}_p(1).$$ Use the first two columns of $K$ to give a map $k:{{\mathcal O}}_{{{\mathbb P}}^3}^2\rightarrow {{\mathcal R}}_p$ which is an isomorphism over an open set. On the other hand, the projection onto the first two coordinates gives a map $q:{{\mathcal R}}_p\rightarrow {{\mathcal O}}_{{{\mathbb P}}^3}(1)^2$ which is, again, an isomorphism over an open set. The composition of these two is the map given by the upper $2\times 2$ square of $K$, $$qk= K_{2,2}:=
\left(
\begin{array}{cc}
0 & z \\
-z & 0
\end{array}
\right) .$$ We can now analyze the map $\phi_R$ by noting that $q\phi _R k = K_{2,3} \Phi K_{3,2}$ where $K_{2,3}$ and $K_{3,2}$ are respectively the upper $2\times 3$ and left $3\times 2$ blocks of $K$. Over the open set where $q$ and $k$ are isomorphisms, $$q\phi _R q^{-1} = q\phi _R k (qk)^{-1} = K_{2,3} \Phi K_{3,2} K_{2,2}^{-1}.$$ Now $$K_{3,2} K_{2,2}^{-1} =
\left(
\begin{array}{cc}
0 & z \\
-z & 0 \\
y & -x
\end{array}
\right)
\cdot
\left(
\begin{array}{cc}
0 & -1/z \\
1/z & 0
\end{array}
\right)
=
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
-x/z & -y/z
\end{array}
\right) .$$ Suppose $$\Phi =
\left(
\begin{array}{ccc}
\alpha & \beta & \gamma \\
\delta & \epsilon & \psi \\
\chi & \theta & \rho
\end{array}
\right)$$ then $$q\phi _R q^{-1} = K_{2,3} \Phi K_{3,2} K_{2,2}^{-1}$$ $$=
\left(
\begin{array}{ccc}
0 & z & -y \\
-z & 0 & x
\end{array}
\right)
\cdot
\left(
\begin{array}{ccc}
\alpha & \beta & \gamma \\
\delta & \epsilon & \psi \\
\chi & \theta & \rho
\end{array}
\right)
\cdot
\left(
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
-x/z & -y/z
\end{array}
\right)$$ $$=
\left(
\begin{array}{ccc}
0 & z & -y \\
-z & 0 & x
\end{array}
\right)
\cdot
\left(
\begin{array}{cc}
\alpha -\gamma x/z & \beta -\gamma y/z \\
\delta -\psi x/z & \epsilon -\psi y/z \\
\chi -\rho x/z & \theta -\rho y/z
\end{array}
\right)$$ $$=
\left(
\begin{array}{cc}
z\delta -\psi x - y\chi +\rho xy/z & z\epsilon -\psi y + y\theta -\rho y^2/z \\
-z\alpha +\gamma x- x\chi +\rho x^2/z & -z\beta +\gamma y + x\theta -\rho xy/z
\end{array}
\right) .$$ Notice that the trace of this matrix is $${\rm Tr}(\phi ) = x(\theta -\psi ) + y (\gamma -\chi ) + z (\delta -\beta ),$$ which is a section of $H^0({{\mathcal O}}_{{{\mathbb P}}^3}(1))$ vanishing at $p$. A co-obstruction should have trace zero, so we should impose three linear conditions $$\theta = \psi , \;\;\; \chi = \gamma \;\;\; \delta = \beta$$ which together just say that $\Phi$ is a symmetric matrix. Our expression simplifies to $$q\phi _R q^{-1} =
\left(
\begin{array}{cc}
\beta z -\psi x - \gamma y +\rho xy/z & \epsilon z -\rho y^2/z \\
-\alpha z +\rho x^2/z & -\beta z + \psi x +\gamma y -\rho xy/z
\end{array}
\right) .$$ Now, restrict ${{\mathcal R}}_p$ to $X$ to get the sheaf $R$, take its double dual to get $E=R^{\ast\ast}$, and consider the induced map $\phi : E\rightarrow E(1)$. Over the intersection of our open set with $X$, this will have the same formula. We can furthermore restrict to the curve $Y\subset X$ given by the intersection with the plane $y=0$. Note that $X$ is in general position subject to the condition that it contain the point $p$. Setting $y=0$ the above matrix becomes $$(q\phi q^{-1}) |_{y=0} =
\left(
\begin{array}{cc}
\beta z -\psi x & \epsilon z \\
-\alpha z +\rho x^2/z & -\beta z + \psi x
\end{array}
\right) .$$ Choose for example $\beta = \psi = 0$ and $\alpha = \rho = \epsilon = 1$, giving the matrix whose determinant is $${\rm det} \left(
\begin{array}{cc}
0 & z \\
x^2/z -z & 0
\end{array}
\right) = z^2 - x^2 = (z+x) (z-x) .$$ The eigenvalues of $\phi |_Y$ are therefore $\pm \sqrt{(z+x) (z-x)}$, generically distinct. For a general choice of the surface $X$, our curve $Y = X\cap (y=0)$ will intersect the planes $x=z$ and $x=-z$ transversally, so the two eigenvalues of $\phi |_Y$ are permuted when going around points in the ramification locus different from $p$. This provides an explicit example of $\phi$ for which the spectral variety is irreducible, completing the proof of the lemma. We included the detailed calculations because they look to be useful if one wants to write down explicitly the spectral varieties.
Turn now to the study of the boundary component $M(10,4)$ consisting of torsion-free sheaves in $\overline{M} (10)$ which come from bundles in $M(4)$. A point in $M(10,4)$ consists of a torsion-free sheaf $F$ in an exact sequence of the form $$0\rightarrow F \rightarrow E \stackrel{\sigma }{\rightarrow} S\rightarrow 0$$ where $E= F^{\ast\ast}$ is a point in $M(4)$, and $S$ is a length $6$ quotient.
The basic description of the space of obstructions as dual to the space of $K_X$-twisted endomorphisms still holds for torsion-free sheaves. Thus, the obstruction space for $F$ is ${\rm Hom}^o(F,F(1))^{\ast}$. A co-obstruction is a map $\phi : F\rightarrow F(1)=F\otimes K_X$ with ${\rm Tr}(\phi )=0$, which is a kind of [*Higgs field*]{}. Since the moduli space is good, a point $F$ is in ${\rm Sing}(\overline{M} (10))$ if and only if the obstruction space is nonzero, that is to say, if and only if there exists a nonzero trace-free $\phi : F\rightarrow F(1)$.
To give a map $\phi$ is the same thing as to give a map $\varphi :E \rightarrow E (1)$ compatible with the quotient map $E \rightarrow S$, in other words fitting into a commutative square with $\sigma$, for an induced map $\varphi _S:S\rightarrow S$. The maps $\varphi$, co-obstructions for $E$, were studied in Lemma \[m4obs\] above.
Let ${{\mathbb P}}(E)\rightarrow X$ denote the Grothendieck projective space bundle. A point in ${{\mathbb P}}(E)$ is a pair $(x,s)$ where $x\in X$ and $s: E_x\rightarrow S_x$ is a rank one quotient of the fiber. Suppose given a map $\varphi : E \rightarrow E(1)$. We can consider the [*internal spectral variety*]{} $${\rm Sp}_E(\varphi )\subset {{\mathbb P}}(E)$$ defined as the set of points $(x,s)\in {{\mathbb P}}(E)$ such that there exists a commutative diagram $$\begin{array}{ccc}
E_x & \stackrel{\varphi (x)}{\longrightarrow} & E_x \\
\downarrow && \downarrow \\
S_x & \longrightarrow & S_x .
\end{array}$$ The term ‘internal’ signifies that it is a subvariety of ${{\mathbb P}}(E)$ as opposed to the classical spectral variety which is a subvariety of the total space of $K_X$. Here, we have only given ${\rm Sp}_E(\varphi )$ a structure of closed subset of ${{\mathbb P}}(E)$, hence of reduced subvariety. It would be interesting to give it an appropriate scheme structure which could be non-reduced in case $\varphi$ is nilpotent, but that will not be needed here.
\[forusual\] Suppose $E\in M(4)$ and $\varphi : E\rightarrow E(1)$ is a general co-obstruction. Then the internal spectral variety ${\rm Sp}_E(\varphi )$ has a single irreducible component of dimension $2$. A quotient $E\rightarrow S$ consisting of a disjoint sum of rank one quotients $s_i:E_{x_i}\rightarrow S_i$ with $S=\bigoplus S_i$ and the points $x_i$ disjoint, is compatible with $\varphi$ if and only if the points $(x_i,s_i)\in {{\mathbb P}}(E)$ lie on the internal spectral variety ${\rm Sp}_E(\varphi )$.
Notice that $z\in X$ is a point such that $\varphi (z)=0$, then the whole fiber ${{\mathbb P}}(E)_z\cong {{\mathbb P}}^1$ is in ${\rm Sp}_E(\varphi )$. In particular, if such a point exists then the map ${\rm Sp}_E(\varphi )\rightarrow X$ will not be finite.
A first remark is that the zero-set of $\varphi$ is $0$-dimensional. Indeed, if $\varphi$ vanished along a divisor $D$, then $D\in |{{\mathcal O}}_X(n)|$ for $n\geq 1$ and $\varphi : F\rightarrow F(1-n)$. This is possible only if $n=1$ and $\varphi : F\rightarrow F$ is a scalar endomorphism (since $F$ is stable). However, the trace of the co-obstruction vanishes, so the scalar $\varphi$ would have to be zero, which we are assuming is not the case.
At an isolated point $z$ with $\varphi (z)=0$, the fiber of the projection ${\rm Sp}_E(\varphi )\rightarrow X$ contains the whole ${{\mathbb P}}(E_z)={{\mathbb P}}^1$. However, these can contribute at most irreducible components of dimension $\leq 1$ (although we conjecture that in fact these fibers are contained in the closure of the $2$-dimensional component so that ${\rm Sp}_E(\varphi )$ is irreducible).
Away from such fibers, the internal spectral variety is isomorphic to the external one, a two-sheeted covering of $X$, and by Lemma \[m4obs\], for a general $\varphi$ the monodromy of this covering interchanges the sheets so it is irreducible. Thus, ${\rm Sp}_E(\varphi )$ has a single irreducible component of dimension $2$, and it maps to $X$ by a generically finite ($2$ to $1$) map.
The second statement, that a quotient consisting of a direct sum of rank one quotients, is compatible with $\varphi$ if and only if the corresponding points lie on ${\rm Sp}_E(\varphi )$, is immediate from the definition.
\[defusual\] A triple $(E,\varphi , \sigma )$ where $E\in M(4)$, $\varphi : E\rightarrow E(1)$ is a non-nilpotent map, and $\sigma = \bigoplus s_x$ is a quotient composed of six rank $1$ quotients over distinct points, compatible with $\varphi$ as in the previous Corollary \[forusual\], leads to an obstructed point $F=F_{(E,\varphi , \sigma )}\in M(10,4)^{\rm sing}$ obtained by setting $F:= \ker (\sigma )$. Such a point will be called [*usual*]{}.
Ellingsrud and Lehn have given a very nice description of the Grothendieck quotient scheme of a bundle of rank $r$ on a smooth surface. It extends the basic idea of Li’s theorem which we already stated as Theorem \[quotli\] above, and will allow us to count dimensions of strata in $M(10,4)$.
\[el\] The quotient scheme parametrizing quotients of a locally free sheaf ${{\mathcal O}}_X^r$ of rank $r$ on a smooth surface $X$, located at a given point $x\in X$, and of length $\ell$, is irreducible of dimension $r\ell -1$.
See [@EllingsrudLehn]. We have given the local version of the statement here.
In our case, $r=2$ so the dimension of the local quotient scheme is $2\ell -1$.
A given quotient $E\rightarrow S$ decomposes as a direct sum of quotients $E\rightarrow S_i$ located at distinct points $x_i\in X$. Order these by decreasing length, and define the [*length vector*]{} of $S$ to be the sequence $(\ell _1,\ldots , \ell _k)$ of lengths $\ell _i=\ell (S_i)$ with $\ell _i \geq \ell _{i+1}$. This leads to a stratification of the ${\rm Quot}$ scheme into strata labelled by length vectors. By Ellingsrud-Lehn, the dimension of the space of quotients supported at a single (but not fixed) point $x_i$ and having length $\ell _i$, is $2\ell _i + 1$, giving the following dimension count.
For a fixed bundle $E$ of rank $2$, the dimension of the stratum associated to length vector $(\ell _1,\ldots , \ell _k)$ in the ${\rm Quot}$-scheme of quotients $E\rightarrow S$ with total length $\ell = \sum _{i=1}^k \ell _i$, is $$\sum (2\ell _i + 1) = 2\ell + k.$$
Recall that the moduli space $M(4)$ has dimension $2$, so the dimension of the stratum of $M(10,4)$ corresponding to a vector $(\ell _1,\ldots , \ell _k)$ is $14+k$. In particular, $M(10,4)$ has a single stratum $(1,1,1,1,1,1)$ of dimension $20$, corresponding to quotients which are direct sums of rank one quotients supported at distinct points, and a single stratum $(2,1,1,1,1)$ of length $19$. This yields the following corollary.
\[intstrata\] If $Z'\subset M(10,4)$ is any $19$-dimensional irreducible subvariety, then either $Z'$ is equal to the stratum $(2,1,1,1,1)$, or else the general point on $Z'$ consists of a direct sum of six rank $1$ quotients supported over six distinct points of $X$.
\[usual\] The singular locus $M(10,4)^{\rm sing}$ has only one irreducible component of dimension $19$. This irreducible component has a nonempty dense open subset consisting of the usual points (Definition \[defusual\]). For a usual point, the co-obstruction $\varphi$ is unique up to a scalar, so this open set may be viewed as the moduli space of usual triples $(E,\varphi , \sigma )$, which is irreducible.
Suppose $Z'\subset M(10,4)^{\rm sing}$ is an irreducible component. Consider the two cases given by Corollary \[intstrata\].
(i)—If $Z'$ contains an open set consisting of points which are direct sums of six rank $1$ quotients supported on distinct points of $X$, then this open set parametrizes usual triples. Furthermore, a point in this open set corresponds to a choice of $(E,\varphi )$ together with six points on the internal spectral variety ${\rm Sp}_E(\varphi )$. We count the dimension of this piece as follows.
Let $M'(4)$ denote the moduli space of pairs $(E,\varphi )$ with $E\in M(4)$ and $\varphi$ a nonzero co-obstruction for $E$. The space of co-obstructions for any $E\in M(4)$, has dimension $6$ and the family of these spaces forms a vector bundle over $M(4)$ (more precisely, a twisted vector bundle twisted by the obstruction class for existence of a universal family over $M(4)$). Thus, the moduli space of pairs has a fibration $M'(4)\rightarrow M(4)$ whose fibers are ${{\mathbb P}}^{5}$. In particular, $M'(4)$ is a smooth irreducible variety of dimension $7$.
For a general such $(E,\varphi )$ the moduli space of usual triples has dimension $\leq 12$, with a unique $12$ dimensional piece corresponding to a general choice of $6$ points on the unique $2$-dimensional irreducible component of ${\rm Sp}_E(\varphi )$. This gives the $19$-dimensional component of $M(10,4)^{\rm sing}$ mentionned in the statement of the proposition.
Suppose $(E, \varphi )$ is not general, that is to say, contained in some subvariety of $M'(4)$ of dimension $\leq 6$. Then, as $\varphi$ is nonzero, even though we no longer can say that it is irreducible, in any case the internal spectral variety ${\rm Sp}_E(\varphi )$ has dimension $2$ so the space of choices of $6$ general points on it has dimension $\leq 12$, and this contributes at most subvarieties of dimension $\leq 18$ in $M(10,4)^{\rm sing}$. This shows that in the first case (i) of Corollary \[intstrata\], we obtain the conclusion of the proposition.
(ii)—Suppose $Z'$ is equal to the stratum of $M(10,4)$ corresponding to length vector $(2,1,1,1,1)$. In this case, we show that a general point of $Z'$ has no non-zero co-obstructions, contradicting the hypothesis that $Z'\subset M(10,4)^{\rm sing}$ and showing that this case cannot occur.
Fix $E\in M(4)$. The space of co-obstructions of $E$ has dimension $6$. Suppose $E\rightarrow S_1$ is a quotient of length $2$. If it is just the whole fiber of $E$ over $x_1$, then it is automatically compatible with any co-obstruction. However, these quotients contribute only a $2$-dimensional subspace of the space of such quotients which has dimension $5$ by Ellingsrud-Lehn. Thus, these points don’t contribute general points. On the other hand, a general quotient of length $2$ corresponds to an infinitesimal tangent vector in ${{\mathbb P}}(E)$, and the condition that this vector be contained in ${\rm Sp}_E(\varphi )$ imposes two conditions on $\varphi$. Therefore, the space of co-obstructions compatible with $S_1$ has dimension $\leq 4$. Next, given a nonzero co-obstruction in that subspace, a general quotient $E\rightarrow S_2$ of length $1$ will not be compatible, so imposing compatibility with $S_1$ and $S_2$ leads to a space of co-obstructions of dimension $\leq 3$. Continuing in this way, we see that imposing the condition of compatibility of $\varphi$ with a general quotient $S=S_1\oplus \cdots \oplus S_5$ in the stratum $(2,1,1,1,1)$ leads to $\varphi =0$. Thus, a general point of this stratum has no non-zero co-obstructions as we have claimed, and this case (ii) cannot occur.
Hence, the only case from Corollary \[intstrata\] which can contribute a $19$-dimensional stratum, contributes the single irreducible component described in the statement of the proposition. One may note that $\varphi$ is uniquely determined for a general set of six points on its internal spectral variety, since the first $5$ points are general in ${{\mathbb P}}(E)$ and impose linearly independent conditions.
\[meetsingood\] Suppose $M(10,4)\cap \overline{M(10)}$ is nonempty. Then it is the unique $19$-dimensional irreducible component of usual triples in $M(10,4)^{\rm sing}$ identified by Proposition \[usual\].
By Hartshorne’s theorem, the intersection $M(10,4) \cap \overline{M(10)}$ has pure dimension $19$ if it is nonempty. This could also be seen from O’Grady’s lemma that the boundary of $\overline{M(10)}$ has pure dimension $19$. However, any point in this intersection is singular. By Proposition \[usual\], the singular locus $M(10,4)^{\rm sing}$ has only one irreducible component of dimension $19$, and it is the closure of the space of usual triples.
If the intersection $M(10,4)\cap \overline{M(10)}$ is nonempty, the torsion-free sheaves $F$ parametrized by general points satisfy $h^1(F(1))=0$. We show this by a dimension estimate using Ellingsrud-Lehn. The more precise information about $M(10,4)^{\rm sing}$ given in Proposition \[usual\], while not really needed for the proof at $c_2=10$, will be useful in treating the case of $c_2=11$ in Section \[sec11\].
\[codim2\] The subspace of $M(10,4)$ consisting of points $F$ such that $h^1(F(1))\geq 1$, has codimension $\geq 2$.
Use the exact sequence $$0\rightarrow F\rightarrow E\rightarrow S\rightarrow 0$$ where $E\in M(4)$. One has $h^1(E(1))=0$ for all $E\in M(4)$, see Lemma \[m4start\]. Therefore, $h^1(F(1))=0$ is equivalent to saying that the map $$\label{showsurj}
H^0(E(1))\rightarrow H^0(S(1)) \cong {{\mathbb C}}^6$$ is surjective.
Considering the theorem of Ellingsrud-Lehn, there are two strata to be looked at: the case of a direct sum of six quotients of rank $1$ over distint points, to be treated below; and the case of a direct sum of four quotients of rank $1$ and one quotient of rank $2$. However, this latter stratum already has codimension $1$, and it is irreducible. So, for this stratum it suffices to note that a general quotient $E\rightarrow S$ in it leads to a surjective map , which may be seen by a classical general position argument, placing first the quotient of rank $2$.
Consider now the stratum of quotients which are the direct sum of six rank $1$ quotients $s_i$ at distinct points $x_i\in X$. Fix the bundle $E$. The space of choices of the six quotients $(x_i,s_i)$ has dimension $18$. We claim that the space of choices such that is not surjective, has codimension $\geq 2$.
Note that $h^0(E(1))= 11$. Given six quotients $(x_i,s_i)$, if the map (with $S=\bigoplus S_i$) is not surjective, then its kernel has dimension $\geq 6$, so if we choose five additional points $(y_j,t_j)\in {{\mathbb P}}(E)$ with $t_j:E_{y_j}\rightarrow T_j$ for $T_i$ of length $1$, the total evaluation map $$\label{toteval}
H^0(E(1))\rightarrow \bigoplus _{i=1}^6 S_i(1) \oplus \bigoplus _{j=1}^5 T_j(1)$$ has a nontrivial kernel. Consider the variety $$W:= \{ (u, \ldots (x_i,s_i)\ldots , \ldots (y_j,t_j)\ldots ) \mbox{ s.t. }0\neq u\in H^0(E(1)), s_i(u)=0, t_j(u)=0 \}$$ with the nonzero section $u$ taken up to multiplication by a scalar.
Let $Q'_6(E)$ and $Q'_5(E)$ denote the open subsets of the quotient schemes of length $6$ and length $5$ quotients of $E$ respectively, open subsets consisting of quotients which are direct sums of rank one quotients over distinct points. Let $K\subset Q'_6(E)$ denote the locus of quotients $E\rightarrow S$ such that the kernel sheaf $F$ has $h^1(F(1))\geq 1$. It is a proper closed subset, since it is easy to see that a general quotient $E\rightarrow S$ leads to a surjection . The above argument with shows that $K\times Q'_5(E)\subset p(W)$ where $p:W\rightarrow Q'_6(E)\times Q'_5(E)$ is the projection forgetting the first variable $u$. Our goal is to show that $K$ has dimension $\leq 16$.
We claim that $W$ has dimension $\leq 32$ and has a single irreducible component of dimension $32$. To see this, start by noting that the choice of $u$ lies in the projective space ${{\mathbb P}}^{10}$ associated to $H^0(E(1))\cong {{\mathbb C}}^{11}$.
For a section $u$ which is special in the sense that its scheme of zeros has positive dimension, the locus of choices of $(x_i,s_i)$ and $(y_j,t_j)$ has dimension $\leq 22$, but might have several irreducible components depending on whether the points are on the zero-set of $u$ or not. However, the space of sections $u$ which are special in this sense, is equal to the space of pairs $u'\in H^0(E)$, $u''\in H^0({{\mathcal O}}_X(1))$ up to scalars for both pieces, and this has dimension $2+3=5$, which is much smaller than the dimension of the space of all sections $u$. Therefore, these pieces don’t contribute anything of dimension higher than $27$.
For a section $u$ which is not special in the sense of the previous paragraph, the space of choices of a single rank $1$ quotient $(x,s)$ which vanishes on the section, has a single irreducible component of dimension $2$. It might possibly have some pieces of dimension $1$ corresponding to quotients located at the zeros of $u$ (although we don’t think so). Hence, the space of choices of point in $W$ lying over the section $u$, has dimension $\leq 22$ and has a single irreducible component of dimension $22$.
Putting these together over ${{\mathbb P}}^{10}$, the dimension of $W$ is $\leq 32$ and it has a single irreducible component of dimension $32$, as claimed. Its image $p(W)$ therefore also has dimension $\leq 32$, and has at most one irreducible component of dimension $32$. Denote this component, if it exists, by $p(W)'$.
Suppose now that $K$ had an irreducible component $K'$ of dimension $17$. Then $K'\times Q'_5(E) \subset p(W)$, but ${\rm dim}(Q'_5(E))=15$ so $p(W)'$ would exist and would be equal to $K'\times Q'_5(E)$. However, $p(W)'$ is symmetric under permutation of the $11$ different variables $(x,s)$ and $(y,t)$, but that would then imply that $P(W)'$ was the whole of $Q'_6(E)\times Q'_5(E)$ which is not the case. Therefore, $K$ must have codimension $\geq 2$. This completes the proof of the proposition.
\[intsn\] Suppose $M(10,4)\cap \overline{M(10)}$ is nonempty. Then a general point of this intersection corresponds to a torsion-free sheaf with $h^1(F(1))=0$.
By Hartshorne’s or O’Grady’s theorem, if the intersection is nonempty then it has pure dimension $19$. However, the space of torsion-free sheaves $F\in M(10,4)$ with $h^1(F(1))>0$ has dimension $\leq 18$ by Proposition \[codim2\]. Thus, a general point in any irreducible component of $M(10,4)\cap \overline{M(10)}$ must have $h^1(F(1))=0$. In fact there can be at most one irreducible component, by Corollary \[meetsingood\].
Irreducibility for $c_2=10$ {#sec10}
===========================
\[forseminat\] Suppose $Z$ is an irreducible component of $M(10)$. Then, for a general point $F$ in any irreducible component of the intersection of $\overline{Z}$ with the boundary, we have $h^1(F(1))=0$.
By O’Grady’s lemma, the intersection of $\overline{Z}$ with the boundary has pure dimension $19$. By considering the line $c_2=10$ in the Table \[maintable\], this subset must be a union of some of the irreducible subsets $\overline{M(10,9)}$, $\overline{M(10,8)}$, $\overline{M(10,6)}$, and the unique $19$-dimensional irreducible component of $M(10,4)^{\rm sing}$ given by Proposition \[usual\]. Combining Proposition \[firstthree\] and Lemma \[intsn\], we conclude that $Z$ contains a point $F$ such that $h^1(F(1))=0$. Thus, $h^1(E(1))=0$ for a general bundle $E$ parametrized by a point of $Z$.
\[seminat10\] Suppose $Z$ is an irreducible component of $M(10)$. Then the bundle $E$ parametrized by a general point of $Z$ has seminatural cohomology, and $Z$ is the closure of the irreducible open set $M(10)^{\rm sn}$.
The closure of $Z$ meets the boundary in a nonempty subset, by Corollary \[meetsboundary\]. By the previous Corollary \[forseminat\], there exists a point $F$ in $\overline{Z}$ with $h^1(F(1))=0$, thus the general bundle $E$ in $Z$ also satisfies $h^1(E(1))=0$. By Proposition \[sn\], the irreducible moduli space $M(10)^{\rm sn}$ of bundles with seminatural cohomology is an open set of $Z$.
\[irred10\] The moduli space $M(10)$ of stable bundles of degree $1$ and $c_2=10$, is irreducible.
By Corollary \[seminat10\], any irreducible component of $M(10)$ contains a dense open set parametrizing bundles with seminatural cohomology. By the main theorem of [@MestranoSimpson2], there is only one such irreducible component.
\[tf10\] The full moduli space of stable torsion-free sheaves $\overline{M} (10)$ of degree $1$ and $c_2=10$, has two irreducible components, $\overline{M(10)}$ and $M(10,4)$ meeting along the irreducible component of usual triples in $M(10,4)^{\rm sing}$. These two components have the expected dimension, $20$, hence the moduli space is good and connected.
Recall that we know $M(10,4)$ is irreducible by the results of [@MestranoSimpson1]. Also $M(10)$ is irreducible. Any component has dimension $\geq 20$, and by looking at the dimensions in Table \[maintable\], these are the only two possible irreducible components. Since they have dimension $20$ which is the expected dimension, it follows that the moduli space is good.
It remains to be proven that these two components do indeed intersect in a nonempty subset, which then by Corollary \[meetsingood\] has to be the irreducible component of usual triples in $M(10,4)^{\rm sing}$. Notice that Corollary \[meetsingood\] did not say that the intersection was necessarily nonempty, since it started from the hypothesis that there was a meeting point. It is a consequence of Nijsse’s connectedness theorem that the intersection is nonempty, but this may be seen more concretely as follows.
Consider the stratum $M(10,5)$. Recall from [@MestranoSimpson1] that the moduli space $M(5)$ consists of bundles which fit into an exact sequence of the form $$0\rightarrow {{\mathcal O}}_X \rightarrow E \rightarrow J_{P/X}(1)\rightarrow 0,$$ such that $P= \ell \cap X$ for $\ell \subset {{\mathbb P}}^3$ a line. In what follows, choose $\ell$ general so that $P$ consists of $5$ distinct points.
The space of extensions ${\rm Ext}^1(J_{P/X}(1), {{\mathcal O}}_X)$ is dual to ${\rm Ext}^1({{\mathcal O}}_X, J_{P/X}(2))=H^1(J_{P/X}(2))$. We have the exact sequence $$H^0({{\mathcal O}}_X(2))\rightarrow H^0({{\mathcal O}}_P(2))\rightarrow H^1(J_{P/X}(2))\rightarrow 0.$$ However, $H^0({{\mathcal O}}_X(2))=H^0({{\mathcal O}}_{{{\mathbb P}}^3}(2))$ and the map to $H^0({{\mathcal O}}_P(2))$ factors through $H^0({{\mathcal O}}_{\ell}(2))$, the space of degree two forms on $\ell \cong {{\mathbb P}}^1$, which has dimension $3$. Hence, the cokernel $H^1(J_{P/X}(2))$ has dimension $2$. The extension classes which correspond to bundles, are the linear forms on $H^1(J_{P/X}(2))$ which don’t vanish on any of the images of the lines in $H^0({{\mathcal O}}_P(2))$ corresponding to the $5$ different points. Since $H^1(J_{P/X}(2))$ has dimension two, we can find a family of extension classes whose limiting point is an extension which vanishes on one of the lines corresponding to a point in $P$. This gives a degeneration towards a torsion-free sheaf with a single non-locally free point, still sitting in a nontrivial extension of the above form. We conclude that the limiting bundle is still stable, so we have constructed a degeneration from a point of $M(5)$, to the single boundary stratum $M(5,4)$.
Notice that the dimension of $M(5,4)$ is bigger than that of $M(5)$, so the set of limiting points is a strict subvariety of $M(5,4)$. We have $\overline{M} (5)=M(5) \cup M(5,4)$, and we have shown that the closures of these two strata have nonempty intersection. This fact is also a consequence of the more explicit description of $\overline{M(5)}$ stated in Theorem \[descrip45\] above (but where the proof was left to the reader).
Moving up to $c_2=10$, it follows that the closure of the stratum $M(10,5)$ intersects $M(10,4)$. However, $M(10,4)$ is closed, and the remaining strata of the boundary have dimension $\leq 19$, so all of the other strata in the boundary, in particular $M(10,5)$, are contained in the closure of the locus of bundles $\overline{M(10)}$. Thus, $\overline{M(10,5)}\subset \overline{M(10)}$, but $M(10,4)\cap \overline{M(10,5)}\neq \emptyset$, proving that the intersection $M(10,4)\cap \overline{M(10)}$ is nonempty.
[*Physics discussion:*]{} From this fact, we see that there are degenerations of stable bundles in $M(10)$, near to boundary points in $M(10,4)$. Donaldson’s Yang-Mills metrics then degenerate towards Uhlenbeck boundary points, metrics where $6$ instantons appear. However, these degenerations go not to all points in $M(10,4)$ but only to ones which are in the irreducible subvariety $M(10,4)^{\rm sing}\subset M(10,4)$ consisting of points on the internal spectral variety of a nonzero Higgs field $\varphi : E\rightarrow E\otimes K_X$. It gives a constraint of a global nature on the $6$-tuples of instantons which can appear in Yang-Mills metrics on a stable bundle $F\in M(10)$. It would be interesting to understand the geometry of the Higgs field which shows up, somewhat virtually, in the limit.
Irreducibility for $c_2\geq 11$ {#sec11}
===============================
Consider next the moduli space $\overline{M} (11)$ of stable torsion-free sheaves of degree one and $c_2=11$. The moduli space is good, of dimension $24$. From Table \[maintable\], the dimensions of the boundary strata are all $\leq 23$, so the set of irreducible components of $\overline{M} (11)$ is the same as the set of irreducible components of $M(11)$. Suppose $Z$ is an irreducible component. By Corollary \[meetsboundary\], $Z$ meets the boundary in a nonempty subset of codimension $1$, i.e. dimension $23$. From Table \[maintable\], the only two possibilities are $M(11,10)$ and $M(11,4)$. Note that $M(11,4)$ is closed since it is the lowest stratum; it is irreducible by Li’s theorem and irreducibility of $M(4)$. The stratum $M(11,10)$ is irreducible because of Theorem \[irred10\].
The intersection $M(11,4)\cap \overline{M(11,10)}$ is a nonempty subset containing, in particular, points which are torsion-free sheaves $F'$ entering into an exact sequence of the form $$0\rightarrow F' \rightarrow F \rightarrow S_x\rightarrow 0$$ where $F$ is a usual point of $M(10,4)^{\rm sing}$, $x\in X$ is a general point, and $F\rightarrow S_x$ is a general rank one quotient.
Theorem \[tf10\] shows that the intersection $M(10,4)\cap \overline{M(10)}$ is nonempty. It is the unique $19$-dimensional irreducible component of $M(10,4)^{\rm sing}$, containing the usual points. Starting with a general point $F\in M(10,4)\cap \overline{M(10)}$ and taking an additional general rank $1$ quotient $S_x$, the subsheaf $F'$ gives a point in $M(11,4)\cap \overline{M(11,10)}$.
Let $Y \subset M(10,4)$ be the unique $19$-dimensional irreducible component of the singular locus $M(10,4)^{\rm sing}$. It contains a dense open set where the quotient $S$ is a direct sum of six quotients $(x_i,s_i)$ of rank $1$. Choose a quasi-finite surjection $Y'\rightarrow Y$ such that $(x_i,s_i)$ are well defined as functions $Y'\rightarrow {{\mathbb P}}(E)$.
Forgetting the quotients and considering only the bundle $E$ gives a map $Y'\rightarrow M(4)$. Fix a bundle $E$ in the image of $Y'\rightarrow M(4)$. Let $Y'_E$ denote the fiber of $Y'$ over $E$, which has dimension $\geq 17$.
We claim that for any $0\leq k\leq 5$, there exists a choice of $k$ out of the $6$ points such that the map $Y'_E\rightarrow {{\mathbb P}}(E)^4$ is surjective. For $k=0$ this is automatic, so assume that $k\leq 4$ and it is known for $k$; we need to show that it is true for $k+1$ points. Reorder so that the $k$ points to be chosen, are the first ones. For a general point $q\in {{\mathbb P}}(E)^{k}$, let $Y'_{E,q}$ denote the fiber of $Y'_E\rightarrow {{\mathbb P}}(E)^{k}$ over $q$. We have ${\rm dim}(Y'_{E,q})\geq 17 -3k$. We get an injection $$Y'_{E,q}\rightarrow {{\mathbb P}}(E)^{6-k}.$$ Suppose that the image mapped into a proper subvariety of each factor; then it would map into a subvariety of dimension $\leq 2(6-k)$, which would give ${\rm dim}(Y'_{E,q})\leq 12-2k$. However, for $k\leq 4$ we have $12-2k < 17 - 3k$, a contradiction. Therefore, at least one of the projections must be a surjection $Y'_{E,q}\rightarrow {{\mathbb P}}(E)$. Adding this point to our list, gives a list of $k+1$ points such that the map $Y'_E\rightarrow {{\mathbb P}}(E)^{k+1}$ is surjective. This completes the induction, yielding the following lemma.
\[general5\] Suppose $Y\subset M(10,4)$ is as above. Then for a fixed bundle $E\in M(4)$ corresponding to some points in $Y$, and for a general point in the fiber $Y_E$ over $E$, some $5$ out of the $6$ quotients correspond to a general point of ${{\mathbb P}}(E)^5$.
$\Box$
\[seventh\] Suppose $F$ is the torsion-free sheaf parametrized by a general point of $Y$, and let $F'$ be defined by an exact sequence $$0\rightarrow F'\rightarrow F\stackrel{(x_7,s_7)}{\longrightarrow} S_7 \rightarrow 0$$ where $S_7$ has length $1$ and $(x_7,s_7)$ is general (with respect to the choice of $F$) in ${{\mathbb P}}(E)$. Then $F'$ has no nontrivial co-obstructions: ${\rm Hom}(F',F'(1))=0$.
The space of co-obstructions for the bundle $E$ has dimension $6$. Imposing a condition of compatibility with a general rank-$1$ quotient $(x_i,s_i)$ cuts down the dimension of the space of co-obstructions by at least $1$.
By Lemma \[general5\] above, we may assume after reordering that the first five points $(x_1,s_1),\ldots , (x_5,s_5)$ constitute a general vector in ${{\mathbb P}}(E)^5$. Adding the $7$th general point given by the statement of the proposition, we obtain a general point $(x_1,s_1),\ldots , (x_5,s_5),(x_7,s_7)$ in ${{\mathbb P}}(E)^6$. As this $6$-tuple of points is general with respect to $E$, it imposes vanishing on the $6$-dimensional space of co-obstructions, giving ${\rm Hom}(F',F'(1))=0$.
There exists a point $$F'\in \overline{M(11,10)} \cap M(11,4)$$ in the boundary of $\overline{M} (11)$, such that $F$ is a smooth point of $\overline{M} (11)$.
By Lemma \[seventh\], choosing a general quotient $(x_7,s_7)$ gives a torsion-free sheaf $F'$ with no co-obstructions, hence corresponding to a smooth point of $\overline{M} (11)$. By construction we have $F'\in \overline{M(11,10)} \cap M(11,4)$.
\[irred11\] The moduli space $\overline{M} (11)$ is irreducible.
Suppose $Z$ is an irreducible component. Then $Z$ meets the boundary in a codimension $1$ subset; but by looking at Table \[maintable\], there are only two possibilities: $\overline{M(11,10)}$ and $M(11,4)$. The co-obstructions vanish for general points of $M(10,4)$ since those correspond to $6$ general quotients of rank $1$, and the co-obstructions vanish for general points of $M(10)$ by goodness. It follows that there are no co-obstructions at general points of $\overline{M(11,10)}$ or $M(11,4)$, so each of these is contained in at most a single irreducible component of $\overline{M} (11)$. However, in the previous corollary, there is a unique irreducible component containing $F'$, which shows that the irreducible components containing $\overline{M(11,10)}$ and $M(11,4)$ must be the same. Hence, $\overline{M}(11)$ has only one irreducible component.
The cases $c_2\geq 12$ are now easy to treat.
\[irred12\] For any $c_2\geq 12$, the moduli space $\overline{M} (c_2)$ of stable torsion-free sheaves of degree $1$ and second Chern class $c_2$, is irreducible.
By Corollary \[meetsboundary\], any irreducible component of $\overline{M} (c_2)$ meets the boundary in a subset of codimension $1$. However, for $c_2\geq 12$, the only stratum of codimension $1$ is $M(c_2,c_2-1)$. By induction on $c_2$, starting at $c_2=11$, we may assume that $M(c_2,c_2-1)$ is irreducible. Furthermore, if $E$ is a general point of $M(c_2-1)$ then $E$ admits no co-obstructions, since $M(c_2-1)$ is good. Hence, a general point $F$ in $M(c_2,c_2-1)$, which is the kernel of a general length-$1$ quotient $E\rightarrow S$, doesn’t admit any co-obstructions either. Therefore, $\overline{M} (c_2)$ is smooth at a general point of $M(c_2,c_2-1)$. Thus, there is a unique irreducible component containing $M(c_2,c_2-1)$, which completes the proof that $\overline{M} (c_2)$ is irreducible.
We have finished proving our main statement, Theorem \[maintheorem\] of the introduction:
For $4\leq c_2\leq 9$, this is shown in [@MestranoSimpson1]. For $c_2=10$ it is Theorem \[irred10\], for $c_2=11$ it is Theorem \[irred11\], and $c_2\geq 12$ it is Theorem \[irred12\]. Note that for $c_2\geq 16$ it is Nijsse’s theorem [@Nijsse].
It was shown in [@MestranoSimpson1] that the moduli space is good for $c_2\geq 10$ (shown by Nijsse for $c_2\geq 13$), and from Table \[leq9table\] we see that it isn’t good for $4\leq c_2\leq 9$. The moduli space of torsion-free sheaves $\overline{M} (c_2)$ is irreducible for $c_2\geq 11$, as may be seen by looking at the dimensions of boundary strata in Table \[maintable\]. Whereas $M(4)=\overline{M}(4)$ is irreducible, the dimensions of the strata in Table \[maintable\] imply that $\overline{M}(c_2)$ has several irreducible components for $5\leq c_2\leq 9$, although we haven’t answered the question as to their precise number. By Theorem \[tf10\], $\overline{M} (10)$ has two irreducible components $\overline{M(10)}$ and $M(10,4)$.
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[^1]: This research project was initiated on our visit to Japan supported by JSPS Grant-in-Aid for Scientific Research (S-19104002)
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abstract: 'We present a cumulative framework for the problem of model order reduction in limited time and frequency intervals. The proposed algorithms construct the reduced order model in steps and continue to accumulate the interpolation conditions induced in each step. We show that if the reduced-order model accumulated after each step is time- or frequency-limited pseudo-optimal model for $\mathcal{H}_{2,T}$- or $\mathcal{H}_{2,\Omega}$-model reduction problem, the error decays monotonically after each step irrespective of the choice of interpolation points and tangential directions. Moreover, the proposed algorithms also generate the approximations of the time- or frequency-limited controllability Gramian and the time- or frequency-limited observability Gramian, which monotonically approaches the original solution irrespective of the choice of interpolation points and tangential directions. We show the efficacy of the proposed algorithms using two numerical examples.'
author:
- 'Umair Zulfiqar, Victor Sreeram, and Xin Du[^1] [^2]'
nocite: '[@benner2016frequency]'
title: 'A Cumulative Framework for $\mathcal{H}_2$-Model Order Reduction in Limited Time and Frequency Intervals with Monotonic Decay in Error'
---
[Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{}
Cumulative framework, $\mathcal{H}_2$-norm, Limited frequency, Limited time, Pseudo-optimal.
Introduction
============
order reduction (MOR) is a process of obtaining a reduced-order model (ROM), which accurately preserves the input-output dynamics and some desired properties of the original high-order model. The specific properties and dynamics to be preserved lead to various classes of the MOR procedures. In projection-based MOR techniques, the original high-order model is projected onto a reduced subspace to obtain a ROM, which preserves the desired and dominant characteristics of the original model. The ROM can then be used as a surrogate in the design and analysis, which provides significant numerical and computational advantages [@benner2005dimension].\
Moore presented one of the most widely used MOR techniques in [@moore1981principal]. Moore’s balanced truncation (BT) truncates the states which have an insignificant share in the overall energy transfer and thus provides a compact ROM, which contains the important states (and dynamics) of the original model. In many applications, a specific time or frequency interval is more important, i.e., the ROM should maintain a superior accuracy within that desired time or frequency interval. Gawronski and Juang generalized BT for time- and frequency-limited MOR scenarios in [@gawronski1990model]. In time-limited BT (TLBT) and frequency-limited BT (FLBT), the states which have an insignificant share in the overall energy transfer within the desired time and frequency intervals respectively are truncated. BT, TLBT, and FLBT require solutions of Lyapunov equations, which become computationally infeasible when the original model is a large-scale model. In [@balakrishnan2001efficient]-[@kurschner2018balanced], their applicability is extended to large-scale systems by using a low-rank approximate solution of the Lyapunov equations which can be efficiently computed.\
Iterative rational Krylov algorithm (IRKA) is another important algorithm that is as accurate as BT. IRKA is computationally efficient and does not require any solution of dense Lyapunov equation like that in BT. It generates local-optimum for the $\mathcal{H}_2$-MOR problem. The original algorithm was presented for single-input single-output (SISO) systems in [@gugercin2008h_2], which was later generalized for multi-input multi-output (MIMO) systems in [@beattie2014model]. IRKA is a tangential interpolation algorithm that generates a ROM, which interpolates the original transfer function at the mirror images of its poles in the directions of its input and output residuals. IRKA generally quickly converges for the SISO case even if the interpolation points and tangential directions are chosen randomly. The speed of convergence, however, slows down as the number of inputs and outputs are increased. In [@wolf2014h], an iteration-free algorithm is proposed which satisfies a subset of the first-order optimality conditions, which IRKA satisfies upon convergence. The algorithm is named as “Pseudo-optimal Rational Krylov (PORK)" algorithm. In [@panzer2014model], a cumulative reduction (CURE) scheme is proposed, which generates the ROM in steps, and the final ROM is the accumulation of all the ROMs generated in each step. The interpolation conditions induced at each step are retained and accumulated in the final ROM. If PORK is used to generate the ROM at each step, the $\mathcal{H}_2$-norm error continues to decay monotonically irrespective of the choice of interpolation points and the tangential directions. Moreover, the final ROM is also pseudo-optimal ROM, i.e., it satisfies a subset of the first-order optimality conditions for the $\mathcal{H}_2$-MOR problem [@wilson1970optimum].\
IRKA is generalized for the time- and frequency-limited scenarios in [@goyal2017towards] and [@vuillemin2013h2], respectively. The algorithms presented in [@sinani2019h2] and [@vuillemin2014frequency] yield a ROM which satisfies the first-order optimality conditions for the problems $\mathcal{H}_{2,T}$- and $\mathcal{H}_{2,\Omega}$-MOR. In [@zulfiqar2019time] and [@zulfiqar2019frequency], iteration-free algorithms are presented which constructs a ROM that satisfies a subset of the first-order optimality conditions for $\mathcal{H}_{2,T}$- and $\mathcal{H}_{2,\Omega}$-MOR problems, respectively. These algorithms are a generalization of PORK, i.e., namely time-limited PORK (TLPORK) and frequency-limited PORK (FLPORK).\
In this paper, we generalize CURE in a way that if the ROM accumulated at each step is obtained using TLPORK and FLPORK, the $\mathcal{H}_{2,T}$- and $\mathcal{H}_{2,\Omega}$-norm errors, respectively continue to decay monotonically irrespective of the choice of interpolation points and the tangential directions. Additionally, the proposed algorithms also provide an approximation of time-limited and frequency-limited Gramians in a computationally efficient way, which can be used to extend the applicability of TLBT and FLBT to large-scale systems. The approximate Gramians monotonically approach the original solution after each step of CURE. We establish the significance of the proposed algorithm by demonstrating its effectiveness on two numerical examples.
Preliminaries
=============
Consider a $n^{th}$ order linear-time invariant system with the following state-space realization $$\begin{aligned}
\dot{x}(t)=Ax(t)+Bu(t)&&y(t)=Cx(t)\label{eq:1}\end{aligned}$$ where $x(t)\in\mathbb{R}^{n\times 1}$, $u(t)\in\mathbb{R}^{1\times m}$, and $y(t)\in\mathbb{R}^{p\times 1}$ are state, input, and output vectors, respectively. $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{p\times n}$ are the state matrix, input matrix and output matrix, respectively. Let the system (\[eq:1\]) has the following transfer function representation $$\begin{aligned}
H(s)=C(sI_n-A)^{-1}B.\nonumber\end{aligned}$$ The MOR problem is to construct a $r^{th}$ order model such that $r<<n$ which approximates the original high-order model (\[eq:1\]). Let the ROM has the following state-space and transfer function representations $$\begin{aligned}
\dot{x}_r(t)=A_rx_r(t)+B_r&u(t),\hspace{2.5cm}y_r(t)=C_rx_r(t)\\
H_r(s)&=C_r(sI_r-A_r)^{-1}B_r\end{aligned}$$ where $A_r\in\mathbb{R}^{r\times r}$, $B_r\in\mathbb{R}^{r\times m}$, and $C_r\in\mathbb{R}^{p\times r}$. MOR aims to ensure that $y_r(t)\approx y(t)$ or $||H(s)-H_r(s)||$ is small in some defined sense depending on the nature of the problem. The time-limited MOR problem is to ensure $y_r(t)\approx y(t)$ is small within the desired time interval $[t_1,t_2]$. In frequency-limited MOR problem, a ROM is sought which ensures a small frequency domain error $||H(j\omega)-H_r(j\omega)||$ within the desired frequency interval $[\omega_1,\omega_2]$. The error within the desired time and frequency intervals can be quantified using time-limited and frequency-limited $\mathcal{H}_2$-norms, i.e., $\mathcal{H}_{2,T}$ and $\mathcal{H}_{2,\Omega}$, respectively. In projection-based MOR techniques, the ROM is obtained by projecting the original high-order model onto a reduced subspace such that the important properties of the original system are retained in the ROM. Let $V_r$ and $W_r$ be the input and output reduction subspaces, respectively. Then the ROM is obtained as $$\begin{aligned}
A_r=W_r^TAV_r,&& B_r=W_r^TB, && C_r=CV_r\end{aligned}$$ where $V\in \mathbb{R}^{n\times r}$ and $W\in \mathbb{R}^{n\times r}$. Let $P_T$ and $P_\Omega$ be the time-limited and frequency-limited controllability Gramians of the state-space realization (\[eq:1\]) within the desired time and frequency intervals $[t_1,t_2]$ and $[\omega_1,\omega_2]$, respectively. $P_T$ and $P_\Omega$ solve the following Lyapunov equations $$\begin{aligned}
AP_T+P_TA^T+e^{At_1}BB^Te^{A^Tt_1}-e^{At_2}BB^Te^{A^Tt_2}=0\\
AP_\Omega+P_\Omega A^T+\mathscr{F}(A)BB^T+BB^T\mathscr{F}(A)^T=0\end{aligned}$$ where $$\begin{aligned}
\mathscr{F}(A)=Re\Big(\frac{j}{2\pi}ln\big((j\omega_1I+A)^{-1}(j\omega_2I+A)\big)\Big).\end{aligned}$$ Let $Q_T$ and $Q_\Omega$ be the time-limited and frequency-limited observability Gramians of the state-space realization (\[eq:1\]) within the desired time and frequency intervals $[t_1,t_2]$ and $[\omega_1,\omega_2]$, respectively. $Q_T$ and $Q_\Omega$ solve the following Lyapunov equations $$\begin{aligned}
A^TQ_T+Q_TA+e^{A^Tt_1}C^TCe^{At_1}-e^{A^Tt_2}C^TCe^{At_2}=0\\
A^TQ_\Omega+Q_\Omega A+\mathscr{F}(A)^TC^TC+C^TC\mathscr{F}(A)=0\end{aligned}$$ The $\mathcal{H}_{2,T}$-norm of the error transfer function is defined as $$\begin{aligned}
||H(s)-H_r(s)||_{\mathcal{H}_{2,T}}&=\Big(||H(s)||^2_{\mathcal{H}_{2,T}}+||H_r(s)||^2_{\mathcal{H}_{2,T}}-2\Big\langle H(s), H_r(s)\Big\rangle_{\mathcal{H}_{2,T}}\Big)^{\frac{1}{2}}\nonumber\\
&=\Big(trace(CP_TC^T)+trace(C_rP_{r,T}C_r^T)-2~trace(CP_{2,T}C_r^T)\Big)^{\frac{1}{2}}\nonumber\\
&=\Big(trace(B^TQ_TB)+trace(B_r^TQ_{r,T}B_r)-2~trace(B^TQ_{2,T}B_r)\Big)^{\frac{1}{2}}\nonumber\end{aligned}$$ where $P_{r,T}$, $P_{2,T}$, $Q_{r,T}$, and $Q_{2,T}$ solve the following Lyapunov equations $$\begin{aligned}
A_rP_{r,T}+P_{r,T}A_r^T+e^{A_rt_1}B_rB_r^Te^{A_r^Tt_1}-e^{A_rt_2}B_rB_r^Te^{A_r^Tt_2}=0\\
AP_{2,T}+P_{2,T}A_r^T+e^{At_1}BB_r^Te^{A_r^Tt_1}-e^{At_2}BB_r^Te^{A_r^Tt_2}=0\\
A_r^TQ_{r,T}+Q_{r,T}A_r+e^{A_r^Tt_1}C_r^TC_re^{A_rt_1}-e^{A_r^Tt_2}C_r^TC_re^{A_rt_2}=0\\
A^TQ_{2,T}+Q_{2,T}A_r+e^{A^Tt_1}C^TC_re^{A_rt_1}-e^{A^Tt_2}C^TC_re^{A_rt_2}=0\end{aligned}$$ Similarly, the $\mathcal{H}_{2,\Omega}$-norm of the error transfer function is defined as $$\begin{aligned}
||H(s)-H_r(s)||_{\mathcal{H}_{2,\Omega}}&=\Big(||H(s)||^2_{\mathcal{H}_{2,\Omega}}+||H_r(s)||^2_{\mathcal{H}_{2,\Omega}}-2\Big\langle H(s), H_r(s)\Big\rangle_{\mathcal{H}_{2,\Omega}}\Big)^{\frac{1}{2}}\nonumber\\&=\Big(trace(CP_\Omega C^T)+trace(C_rP_{r,\Omega}C_r^T)-2~trace(CP_{2,\Omega}C_r^T)\Big)^{\frac{1}{2}}\nonumber\\
&=\Big(trace(B^TQ_\Omega B)+trace(B_r^TQ_{r,\Omega}B_r)-2~trace(B^TQ_{2,\Omega}B_r)\Big)^{\frac{1}{2}}\nonumber\end{aligned}$$ where $P_{r,\Omega}$, $P_{2,\Omega}$, $Q_{r,\Omega}$, and $Q_{2,\Omega}$ solve the following Lyapunov equations $$\begin{aligned}
A_rP_{r,\Omega}+P_{r,T}A_r^T+\mathscr{F}(A_r)B_rB_r^T+B_rB_r^T\mathscr{F}(A_r)^T=0\\
AP_{2,\Omega}+P_{2,\Omega}A_r^T+\mathscr{F}(A)BB_r^T+BB_r^T\mathscr{F}(A_r)^T=0\\
A_r^TQ_{r,\Omega}+Q_{r,\Omega}A_r+\mathscr{F}(A_r)^TC_r^TC_r+C_r^TC_r\mathscr{F}(A_r)=0\\
A^TQ_{2,\Omega}+Q_{2,\Omega}A_r+\mathscr{F}(A)^TC^TC_r+C^TC_r\mathscr{F}(A_r)=0\end{aligned}$$ When $[t_1,t_2]=[0,\infty]$ and $[\omega_1,\omega_2]=[-\infty,\infty]$, $P=P_T=P_\Omega$, $Q=Q_T=Q_\Omega$, and $||\cdot||_{\mathcal{H}_2}=||\cdot||_{\mathcal{H}_{2,T}}=||\cdot||_{\mathcal{H}_{2,\Omega}}$ where $P$, $Q$, and $\mathcal{H}_2$ are the standard controllability Gramian, observability Gramian, and $\mathcal{H}_2$-norm, respectively.
PORK
----
Let $V_r$ be the input reduction subspace which satisfies the following tangential interpolation condition for any output reduction subspace $W_r$ such that $W_r^TV_r=I$, i.e., $$\begin{aligned}
H(\sigma_i)b_i=H_r(\sigma_i)b_i\label{Eq:18A}\end{aligned}$$ where $\{\sigma_1,\cdots,\sigma_r\}$ and $\{b_1,\cdots,b_r\}$ are the interpolation points and the associated right tangential directions, respectively. This interpolation condition can be achieved if $V_r$ satisfies the following $$\begin{aligned}
Ran(V_r)=\underset {i=1,\cdots,r}{span}\{(\sigma_iI-A)^{-1}Bb_i\}.\end{aligned}$$ Choose any $W_r$ such that $W_r^TV_r=I$, for instance, $W_r=V_r(V_rV_r^T)^{-1}$ and define the following matrices $$\begin{aligned}
\tilde{A}&=W_r^TAV_r,\hspace*{3mm}\tilde{B}=W_r^TB,\hspace*{3mm}B_\bot=B-V_r\tilde{B},\\
L_r&=(B_\bot^TB_\bot)^{-1}B_\bot^T\big(AV_r-V_r\tilde{A}\big),\hspace*{3mm}S_r=\tilde{A}-\tilde{B}L_r.\end{aligned}$$ Then $V_r$ solves the following Sylvester equations $$\begin{aligned}
AV_r-V_rS_r-BL_r=0\label{Eq:22A}\\
AV_r-V_rA_r-B_{\perp}L_r=0\label{Eq:23A}\end{aligned}$$ where the interpolation points are the eigenvalues of $S_r$ and the right tangential directions are encoded in $L_r$.\
Let the ROM is constructed as the following $$\begin{aligned}
A_r=-P_rS_r^TP_r^{-1},&& B_r=-P_rL_r^T,&&C_r=CV_r\end{aligned}$$ where $P_r^{-1}$ solves the following Lyapunov equation $$\begin{aligned}
-S_r^TP_r^{-1}-P_r^{-1}S_r+L_r^TL_r=0.\end{aligned}$$ Let $H_r(s)$ be represented in its pole-residue form as the following $$\begin{aligned}
H_r(s)=\sum_{i=1}^{r}\frac{l_ir_i^T}{s-\lambda_i}.\end{aligned}$$ Then $H_r(s)$ satisfies the following interpolation condition $$\begin{aligned}
H(-\lambda_i)r_i=H_r(-\lambda_i)r_i,\end{aligned}$$ and a subset of the first-order optimality conditions for the problem $||H(s)-H_r(s)||^2_{\mathcal{H}_2}$, i.e., $$\begin{aligned}
||H(s)-H_r(s)||^2_{\mathcal{H}_2}=||H(s)||^2_{\mathcal{H}_2}-||H_r(s)||^2_{\mathcal{H}_2}.\end{aligned}$$ Thus $H_r(s)$ is called a pseudo-optimal ROM.
CURE
----
In CURE, the $r^{th}$-order $H_r(s)$ is not constructed in a single step, and it is constructed adaptively in $k$ steps. $(A_r,B_r,C_r)$, $S_r$, $L_r$, and $B_{\perp}$ are accumulated after each step such that the interpolation condition (\[Eq:18A\]), and the Sylvester equations (\[Eq:22A\]) and (\[Eq:23A\]) are satisfied after each step as well as after $k$ steps. In other words, at each step, new interpolation points and the tangential directions are added without disturbing the interpolation conditions induced by the previous interpolation points and tangential directions.\
Let $V^{(k)}$, $S^{(k)}$, $L^{(k)}$, $(\tilde{A}^{(k)},\tilde{B}^{(k)},\tilde{C}^{(k)})$, and $B_{\perp}^{(k)}$ be matrices of $k^{th}$ step for any $W^{(k)}$ such that $(W^{(k)})^TV^{(k)}=I$, i.e., $$\begin{aligned}
\tilde{A}^{(k)}&=(W^{(k)})^TAV^{(k)},&& \tilde{B}^{(k)}=(W^{(k)})^TB_{\perp}^{(k-1)},&\tilde{C}^{(k)}&=CV^{(k)}\nonumber\end{aligned}$$ where $$\begin{aligned}
AV^{(k)}-V^{(k)}S^{(k)}-B_{\perp}^{(k-1)}L^{(k)}=0\\
AV^{(k)}-V^{(k)}\tilde{A}^{(k)}-B_{\perp}^{(k)}L^{(k)}=0,\end{aligned}$$ $B_{\perp}^{(0)}=B$ and $B_{\perp}^{(k)}=B_{\perp}^{(k-1)}-V^{(k)}\tilde{B}^{(k)}$.\
After each step, the ROM can be accumulated and a $r^{th}$ order ROM can be obtained adaptively in $k$ steps, i.e., the order of the ROM grows after each step without affecting the interpolation conditions induced in the previous steps. The accumulated ROM and the associated matrices for $i=1,\cdots,k$ is given by $$\begin{aligned}
A_{tot}^{(i)}&=\begin{bmatrix}A_{tot}^{(i-1)}&0\\&\\\tilde{B}^{(i)}L_{tot}^{(i-1)}&\tilde{A}^{(i)}\end{bmatrix},&& B_{tot}^{(i)}=\begin{bmatrix}B_{tot}^{(i-1)}\\\\\tilde{B}^{(i)}\end{bmatrix},\nonumber\\
C_{tot}^{(i)}&=\begin{bmatrix}C_{tot}^{(i-1)}&&\tilde{C}^{(i)}\end{bmatrix},&&L_{tot}^{(i)}=\begin{bmatrix}L_{tot}^{(i-1)}&&L^{(i)}\end{bmatrix},\nonumber\\
S_{tot}^{(i)}&=\begin{bmatrix}S_{tot}^{(i-1)}&-B_{tot}^{(i-1)}L^{(i)}\\&\\0&S^{(i)}\end{bmatrix},&&V_{tot}^{(i)}=\begin{bmatrix}V_{tot}^{(i-1)}& V^{(i)}\end{bmatrix}\nonumber\end{aligned}$$ where $$\begin{aligned}
AV_{tot}^{(i)}-V_{tot}^{(i)}S_{tot}^{(i)}-BL_{tot}^{(i)}=0,\nonumber\end{aligned}$$ and $A_{tot}^{(0)}$, $B_{tot}^{(0)}$, $C_{tot}^{(0)}$, $L_{tot}^{(0)}$, $S_{tot}^{(0)}$, and $V_{tot}^{(0)}$ are all empty matrices.\
An important property of CURE is that if $(\tilde{A}^{(i)},\tilde{B}^{(i)},\tilde{C}^{(i)})$ is computed using PORK for $i=1,\cdots,k$, $(A_{tot}^{(i)},B_{tot}^{(i)},C_{tot}^{(i)})$ stays pseudo-optimal. This further implies that $(A_{tot}^{(i-1)},B_{tot}^{(i-1)},C_{tot}^{(i-1)})$ is a pseudo-optimal ROM of $(A_{tot}^{(i)},B_{tot}^{(i)},C_{tot}^{(i)})$ as $S_{tot}^{(i)}$ and $\hat{C}_{tot}^{(i)}$ contain the interpolation points and tangential directions encoded in $S_{tot}^{(i-1)}$ and $\hat{C}_{tot}^{(i-1)}$, respectively. Let $H_{tot}^{(i)}\big(s\big)$ be defined as $$\begin{aligned}
H_{tot}^{(i)}\big(s\big)=C_{tot}^{(i)}(sI-A_{tot}^{(i)})^{-1}B_{tot}^{(i)}.\nonumber\end{aligned}$$ If $H_{tot}^{(i)}\big(s\big)$ stays pesudo-optimal, the following holds $$\begin{aligned}
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_2}=||H(s)||^2_{\mathcal{H}_2}-||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_2}.\nonumber\end{aligned}$$ Thus $||H(s)||^2_{\mathcal{H}_2}\geq||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_2}$. Moreover, since $$\begin{aligned}
||H_{tot}^{(i)}\big(s\big)-H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_2}=||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_2}-||H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_2},\nonumber\end{aligned}$$ $||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_2}\geq ||H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_2}$. Therefore, $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_2}$ decays monotonically irrespective of the choice of interpolation points and tangential directions.
TLPORK
------
Let $\mathscr{G}(s)$ and $\mathscr{G}_r(s)$ be defined as the following $$\begin{aligned}
\mathscr{G}(s)&=C(sI_n-A)^{-1}\begin{bmatrix}e^{At_1}B &-e^{At_2}B\end{bmatrix}\\
\mathscr{G}_r(s)&=C_r(sI_r-A_r)^{-1}\begin{bmatrix}e^{A_rt_1}B_r &-e^{A_rt_2}B_r\end{bmatrix}.\end{aligned}$$ It is shown in [@zulfiqar2019time] that $H_r(s)$ satisfies a subset of the first-order optimality conditions for the problem $||H(s)-H_r(s)||^2_{\mathcal{H}_{2,T}}$ if the following interpolation conditions are satisfied, i.e., $$\begin{aligned}
\mathscr{G}(-\lambda_i)\hat{b}_i=\mathscr{G}_r(-\lambda_i)\hat{b}_i\label{Eq:33A}\end{aligned}$$ where $$\begin{aligned}
\begin{bmatrix}\hat{b}_1&\cdots&\hat{b}_r\end{bmatrix}=\begin{bmatrix}Re^{\Lambda t_1}\\Re^{\Lambda t_2}\end{bmatrix},\nonumber\end{aligned}$$ $\Lambda=diag(-\lambda_1,\cdots,-\lambda_r)$, and $R=\begin{bmatrix}r_1&\cdots&r_r\end{bmatrix}$. This can be achieved if the input reduction subspace is computed from the following Sylvester equation $$\begin{aligned}
AV_{r,T}-V_{r,T}S_r-e^{At_1}BL_re^{-S_rt_1}+e^{At_2}BL_re^{-S_rt_2}=0,\label{Eq:34A}\end{aligned}$$ and the ROM is computed as $$\begin{aligned}
A_r=-P_{r,T}S_r^TP_{r,T}^{-1},&& B_r=-P_{r,T}L_r^T, && C_r=CV_{r,T}\label{Eq:35A}\end{aligned}$$ where $P_{r,T}^{-1}$ satisfies the following Lyapunov equation $$\begin{aligned}
-S_r^TP_{r,T}^{-1}-P_{r,T}^{-1}S_r&+e^{-S_r^Tt_1}L_r^TL_re^{-S_rt_1}-e^{-S_r^Tt_2}L_r^TL_re^{-S_rt_2}=0.\label{Eq:36A}\end{aligned}$$ If the ROM is obtained as in (\[Eq:35A\]), the following condition is satisfied $$\begin{aligned}
||H(s)-H_r(s)||^2_{\mathcal{H}_{2,T}}=||H(s)||^2_{\mathcal{H}_{2,T}}-||H_r(s)||^2_{\mathcal{H}_{2,T}}.\end{aligned}$$
FLPORK
------
Let $\mathscr{H}(s)$ and $\mathscr{H}_r(s)$ be defined as the following $$\begin{aligned}
\mathscr{H}(s)&=C(sI_n-A)^{-1}\begin{bmatrix}B &\mathscr{F}(A)B\end{bmatrix}\\
\mathscr{H}_r(s)&=C_r(sI_r-A_r)^{-1}\begin{bmatrix}B_r &\mathscr{F}(A_r)B_r\end{bmatrix}.\end{aligned}$$ It is shown in [@zulfiqar2019frequency] that $H_r(s)$ satisfies a subset of the first-order optimality conditions for the problem $||H(s)-H_r(s)||^2_{\mathcal{H}_{2,\Omega}}$ if the following interpolation conditions are satisfied $$\begin{aligned}
\mathscr{H}(-\lambda_i)\bar{b}_i=\mathscr{H}_r(-\lambda_i)\bar{b}_i\label{Eq:40A}\end{aligned}$$ where $$\begin{aligned}
\begin{bmatrix}\bar{b}_1&\cdots&\bar{b}_r\end{bmatrix}=\begin{bmatrix}R\mathscr{F}(\Lambda)\\R\end{bmatrix}.\nonumber\end{aligned}$$ This can be achieved if the input reduction subspace is computed from the following Sylvester equation $$\begin{aligned}
AV_{r,\Omega}-V_{r,\Omega}S_r-\mathscr{F}(A)BL_r-BL_r\mathscr{F}(-S_r)=0,\label{Eq:41A}\end{aligned}$$ and the ROM is computed as $$\begin{aligned}
A_r=-P_{r,\Omega}S_r^TP_{r,\Omega}^{-1},&& B_r=-P_{r,\Omega}L_r^T, && C_r=CV_{r,\Omega}\label{Eq:42A}\end{aligned}$$ where $P_{r,\Omega}^{-1}$ satisfies the following Lyapunov equation $$\begin{aligned}
-S_r^TP_{r,\Omega}^{-1}-P_{r,\Omega}^{-1}S_r&+\mathscr{F}(-S_r)^TL_r^TL_r+L_r^TL_r\mathscr{F}(-S_r)=0.\label{Eq:43A}\end{aligned}$$ If the ROM is obtained as in (\[Eq:42A\]), the following condition is satisfied $$\begin{aligned}
||H(s)-H_r(s)||^2_{\mathcal{H}_{2,\Omega}}=||H(s)||^2_{\mathcal{H}_{2,\Omega}}-||H_r(s)||^2_{\mathcal{H}_{2,\Omega}}.\end{aligned}$$
Main Work
=========
In this section, we generalize CURE for time-limited and frequency-limited scenarios aiming to inherit the monotonic decay in the error property when TLPORK or FLPORK is used within CURE. Applying CURE on $\mathscr{G}(s)$ and $\mathscr{H}(s)$ can generate their ROMs with the monotonic decay in the error; however, the ROMs of $\mathscr{G}(s)$ and $\mathscr{H}(s)$ are not desired for the time-limited and frequency-limited $\mathcal{H}_2$-MOR problems. The problem under consideration is to obtain the ROMs of $H(s)$ such that $\mathscr{G}_r(s)$ and $\mathscr{H}_r(s)$ satisfy the interpolatory conditions (\[Eq:33A\]) and (\[Eq:40A\]), respectively. This is the same dilemma that time-limited IRKA (TLIRKA) [@goyal2017towards] and frequency-limited IRKA (FLIRKA) [@vuillemin2013h2] face. In TLIRKA and FLIRKA, the IRKA-type framework is used with the modification that the rational Krylov subspaces are computed differently than that in IRKA. We follow a similar approach for the problem under consideration. We retain the CURE-type framework but compute the reduction subspace differently. One problem, however, is that CURE is only valid if $V_r$ satisfies equations (\[Eq:22A\]) and (\[Eq:23A\]). In the sequel, we tackle this issue and generalize CURE to adaptively construct ROMs whose $\mathcal{H}_{2,T}$- and $\mathcal{H}_{2,\Omega}$-norm errors decay monotonically irrespective of the choice of interpolation points and tangential directions.\
Recall the ROM constructed by PORK is given by $$\begin{aligned}
A_r=-P_rS_r^TP_r^{-1},&& B_r=-P_rL_r^T,&&C_r=CV_r.\nonumber\end{aligned}$$ Since pseudo-optimality is a property of the transfer function, and it does not depend on the state-space realization, a state transformation can be applied using $P_r$ as the transformation matrix, i.e., $$\begin{aligned}
A_r^{(\mathbb{T})}=-S_r^T,&& B_r^{(\mathbb{T})}=-L_r^T,&&C_r^{(\mathbb{T})}=CV_rP_r.\nonumber\end{aligned}$$ On similar lines, the ROM constructed by TLPORK can be defined as $$\begin{aligned}
A_r^{(\mathbb{T})}=-S_r^T,&& B_r^{(\mathbb{T})}=-L_r^T,&&C_r^{(\mathbb{T})}=CV_{r,T}P_{r,T},\nonumber\end{aligned}$$ and the ROM constructed by FLPORK can be defined as $$\begin{aligned}
A_r^{(\mathbb{T})}=-S_r^T,&& B_r^{(\mathbb{T})}=-L_r^T,&&C_r^{(\mathbb{T})}=CV_{r,\Omega}P_{r,\Omega}.\nonumber\end{aligned}$$ Since $P_{r,T}$ and $P_{r,\Omega}$ satisfy equation (\[Eq:36A\]) and (\[Eq:43A\]), respectively, the following holds (see [@gawronski1990model] for details) $$\begin{aligned}
P_{r,T}&=\Big(e^{-S_r^Tt_1}P_r^{-1}e^{-S_rt_1}-e^{-S_r^Tt_2}P_r^{-1}e^{-S_rt_2}\Big)^{-1}\\
P_{r,\Omega}&=\Big(\mathscr{F}(-S_r)^TP_r^{-1}+P_r^{-1}\mathscr{F}(-S_r)\Big)^{-1}.\end{aligned}$$ It can also be noted that $P_r^{-1}$ is actually the controllability Gramian of the pair $(A_r^{(\mathbb{T})},B_r^{(\mathbb{T})})$. Similarly, the following relationships hold between $V_{r,T}$ and $V_{r,\Omega}$, and $V_r$, i.e., $$\begin{aligned}
V_{r,T}&=e^{At_1}V_re^{-S_rt_1}-e^{At_2}V_re^{-S_rt_2}\label{Eq:47A}\\
V_{r,\Omega}&=\mathscr{F}(A)V_r+V_r\mathscr{F}(-S_r).\label{Eq:48A}\end{aligned}$$ This can be verified by putting (\[Eq:47A\]) and (\[Eq:48A\]) in (\[Eq:34A\]) and (\[Eq:41A\]), respectively, and by noting that $Ae^{At}=e^{At}A$ and $A\mathscr{F}(A)=\mathscr{F}(A)A$ [@gawronski1990model]. Thus the time-limited and frequency-limited pseudo-optimality can be enforced on the pair $(A_r^{(\mathbb{T})},B_r^{(\mathbb{T})})$ by selecting $C_r^{(\mathbb{T})}$ as $CV_{r,T}P_{r,T}$ and $CV_{r,\Omega}P_{r,\Omega}$, respectively.\
It is shown in [@wolf2014h] that $P_{tot}^{(i)}$ can be obtained recursively if $(\tilde{A}^{(i)},\tilde{B}^{(i)},\tilde{C}^{(i)})$ is obtained at each step of CURE using PORK, i.e., $$\begin{aligned}
P_{tot}^{(i)}=\begin{bmatrix}P_{tot}^{(i-1)}&0\\0&\tilde{P}^{(i)}\end{bmatrix}\end{aligned}$$ where $$\begin{aligned}
\tilde{A}^{(i)}\tilde{P}^{(i)}+\tilde{P}^{(i)}(\tilde{A}^{(i)})^T+\tilde{B}^{(i)}(\tilde{B}^{(i)})^T=0\nonumber\\
(-S^{(i)})^T(\tilde{P}^{(i)})^{-1}+(\tilde{P}^{(i)})^{-1}(-S^{(i)})+(L^{(i)})^T(L^{(i)})=0.\nonumber\end{aligned}$$ One can note that $S_{tot}^{(i)}$ and $L_{tot}^{(i)}$ do not depend on $C_{tot}^{(i)}$ or $C_{tot}^{(i-1)}$ in each step of CURE. Thus, the basic structure of CURE is not affected if $C_{tot}^{(i)}$ is manipulated in each step. We now show that if the time-limited or frequency-limited pseudo-optimality is enforced at each step of CURE by appropriately constructing $C_{tot}^{(i)}$, $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}$ or $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}$ decays monotonically at each step irrespective of the choice of interpolation points and the tangential directions.\
Let $(\tilde{A}^{(i)},\tilde{B}^{(i)},\tilde{C}^{(i)})$ is constructed using PORK in each iteration of CURE, and let $H_{tot}^{(i)}\big(s\big)$ is constructed as the following $$\begin{aligned}
A_{tot}^{(i)}=(-S_{tot}^{(i)})^T,&& B_{tot}^{(i)}=(-L_{tot}^{(i)})^T,&& C_{tot}^{(i)}=C\mathbb{V}_{tot,T}^{(i)}\mathbb{P}_{tot,T}^{(i)}\label{Eq:50A}\end{aligned}$$ or $$\begin{aligned}
A_{tot}^{(i)}=(-S_{tot}^{(i)})^T,&& B_{tot}^{(i)}=(-L_{tot}^{(i)})^T,&& C_{tot}^{(i)}=C\mathbb{V}_{tot,\Omega}^{(i)}\mathbb{P}_{tot,\Omega}^{(i)}\label{Eq:51A}\end{aligned}$$ where $$\begin{aligned}
\mathbb{V}_{tot,T}^{(i)}&=e^{At_1}V_{tot}^{(i)}e^{-S_{tot}^{(i)}t_1}-e^{At_2}V_{tot}^{(i)}e^{-S_{tot}^{(i)}t_2},\\
\mathbb{V}_{tot,\Omega}^{(i)}&=\mathscr{F}(A)V_{tot}^{(i)}+V_{tot}^{(i)}\mathscr{F}(-S_{tot}^{(i)}),\\
\mathbb{P}_{tot,T}^{(i)}&=\Big(e^{(-S_{tot}^{(i)})^Tt_1}\big(P_{tot}^{(i)}\big)^{-1}e^{-S_{tot}^{(i)}t_1}-e^{(-S_{tot}^{(i)})^Tt_2}(P_{tot}^{(i)})^{-1}e^{-S_{tot}^{(i)}t_2}\Big)^{-1},\\
\mathbb{P}_{tot,\Omega}^{(i)}&=\Big(\mathscr{F}(-S_{tot}^{(i)})^T\big(P_{tot}^{(i)}\big)^{-1}+\big(P_{tot}^{(i)}\big)^{-1}\mathscr{F}(-S_{tot}^{(i)})\Big)^{-1}.\end{aligned}$$ Needless to say that $H_{tot}^{(i)}\big(s\big)$ is time-limited or frequency-limited pseudo-optimal ROM of $H(s)$ as the pseudo-optimality is judiciously enforced on $H_{tot}^{(i)}\big(s\big)$. Thus $$\begin{aligned}
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}&=||H(s)||^2_{\mathcal{H}_{2,T}}-||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}\textnormal{ and }\\
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}&=||H(s)||^2_{\mathcal{H}_{2,\Omega}}-||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}.\end{aligned}$$ Moreover, $H_{tot}^{(i-1)}\big(s\big)$ is a time-limited or frequency-limited pseudo-optimal ROM of $H_{tot}^{(i)}\big(s\big)$ as $\big(S_{tot}^{(i)},L_{tot}^{(i)}\big)$ contains the interpolation points and tangential directions of the pair $\big(S_{tot}^{(i-1)},L_{tot}^{(i-1)}\big)$. This is because the basic structure of CURE is not affected due to the choice of $H_{tot}^{(i)}\big(s\big)$ according to equations (\[Eq:50A\]) and (\[Eq:51A\]). Therefore, $$\begin{aligned}
||H_{tot}^{(i)}\big(s\big)-H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_{2,T}}&=||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}-||H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_{2,T}},\\
||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}&\geq ||H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_{2,T}},\\
||H_{tot}^{(i)}\big(s\big)-H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}&=||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}-||H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}},\\
||H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}&\geq ||H_{tot}^{(i-1)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}.\end{aligned}$$ Hence, $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}$ and $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}$ decay monotonically in each step irrespective of the choice of interpolation points and tangential directions. Also, note that $$\begin{aligned}
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}&=CP_TC^T-C\mathbb{V}_{tot,T}^{(i)}\mathbb{P}_{tot,T}^{(i)}(\mathbb{V}_{tot,T}^{(i)})^TC^T\nonumber\\
&=C\Big(P_T-\mathbb{V}_{tot,T}^{(i)}\mathbb{P}_{tot,T}^{(i)}(\mathbb{V}_{tot,T}^{(i)})^T\Big)C^T\nonumber\\
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}&=CP_\Omega C^T-C\mathbb{V}_{tot,\Omega}^{(i)}\mathbb{P}_{tot,\Omega}^{(i)}(\mathbb{V}_{tot,\Omega}^{(i)})^TC^T\nonumber\\
&=C\Big(P_\Omega-\mathbb{V}_{tot,\Omega}^{(i)}\mathbb{P}_{tot,\Omega}^{(i)}(\mathbb{V}_{tot,\Omega}^{(i)})^T\Big)C^T.\nonumber\end{aligned}$$ Therefore, $\mathbb{V}_{tot,T}^{(i)}\mathbb{P}_{tot,T}^{(i)}(\mathbb{V}_{tot,T}^{(i)})^T$ and $\mathbb{V}_{tot,\Omega}^{(i)}\mathbb{P}_{tot,\Omega}^{(i)}(\mathbb{V}_{tot,\Omega}^{(i)})^T$ monotonically approach to $P_T$ and $P_\Omega$ after each iteration of CURE. Hence, it also provides the approximations of $P_T$ and $P_\Omega$.
\[alg:1\] Initialize $B_{\perp}^{(0)}=B$, $S_{tot}^{(0)}=[\hspace{2mm}]$, $C_{tot}^{(0)}=[\hspace{2mm}]$, $L_{tot}^{(0)}=[\hspace{2mm}]$, $P_{tot}^{(0)}=[\hspace{2mm}]$, $\bar{B}_{tot}^{(0)}=[\hspace{2mm}]$, $V_{tot}^{(i)}=[\hspace{2mm}]$.\
**for** $i=1,\cdots,k$\
$Ran(V^{(i)})=\underset {j=1,\cdots,r_j}{span}\{(\sigma_jI-A)^{-1}Bb_j\}$.\
Set $W^{(i)}=V^{(i)}\big(V^{(i)}(V^{(i)})^T\big)^{-1}$, $\tilde{A}=(W^{(i)})^TAV^{(i)}$, $\tilde{B}=(W^{(i)})^TB_\bot^{(i-1)}$, $B_\bot^{(i)}=B_\bot^{i-1}-V^{(i)}\tilde{B}$, $L^{(i)}=\Big(\big(B_\bot^{(i)}\big)^TB_\bot^{(i)}\Big)^{-1}\big(B_\bot^{(i)}\big)^T\big(AV^{(i)}-V^{(i)}\tilde{A}\big)$, $S^{(i)}=\tilde{A}-\tilde{B}L^{(i)}$.\
Solve $\big(-S^{(i)}\big)^T\big(P^{(i)}\big)^{-1}-\big(P^{(i)}\big)^{-1}S^{(i)}+\big(L^{(i)}\big)^TL^{(i)}=0$.\
$P_{tot}^{(i)}=\begin{bmatrix}P_{tot}^{(i-1)}&0\\0&\tilde{P}^{(i)}\end{bmatrix}$, $\bar{B}_{tot}^{(i)}=\begin{bmatrix}\bar{B}_{tot}^{(i-1)}\\-P^{(i)}\big(L^{(i)}\big)^T\end{bmatrix}$, $S_{tot}^{(i)}=\begin{bmatrix}S_{tot}^{(i-1)}&-\bar{B}_{tot}^{(i-1)}L^{(i)}\\0&S^{(i)}\end{bmatrix}$, $L_{tot}^{(i)}=\begin{bmatrix}L_{tot}^{(i-1)}&&L^{(i)}\end{bmatrix}$, $A_{tot}^{(i)}=(-S_{tot}^{(i)})^T$, $B_{tot}^{(i)}=(-L_{tot}^{(i)})^T$, $V_{tot}^{(i)}=\begin{bmatrix}V_{tot}^{(i-1)}& V^{(i)}\end{bmatrix}$.\
$\mathbb{P}_{tot,T}^{(i)}=\Big(e^{(-S_{tot}^{(i)})^Tt_1}\big(P_{tot}^{(i)}\big)^{-1}e^{-S_{tot}^{(i)}t_1}-e^{(-S_{tot}^{(i)})^Tt_2}(P_{tot}^{(i)})^{-1}e^{-S_{tot}^{(i)}t_2}\Big)^{-1}$ or\
$\mathbb{P}_{tot,\Omega}^{(i)}=\Big(\mathscr{F}(-S_{tot}^{(i)})^T\big(P_{tot}^{(i)}\big)^{-1}+\big(P_{tot}^{(i)}\big)^{-1}\mathscr{F}(-S_{tot}^{(i)})\Big)^{-1}.$\
$\mathbb{V}_{tot,T}^{(i)}=e^{At_1}V_{tot}^{(i)}e^{-S_{tot}^{(i)}t_1}-e^{At_2}V_{tot}^{(i)}e^{-S_{tot}^{(i)}t_2}$ or $\mathbb{V}_{tot,\Omega}^{(i)}=\mathscr{F}(A)V_{tot}^{(i)}+V_{tot}^{(i)}\mathscr{F}(-S_{tot}^{(i)})$\
$C_{tot}^{(i)}=C\mathbb{V}_{tot,T}^{(i)}\mathbb{P}_{tot,T}^{(i)}$ or $C_{tot}^{(i)}=C\mathbb{V}_{tot,\Omega}^{(i)}\mathbb{P}_{tot,\Omega}^{(i)}$.\
**end for**.
We have so far just presented a V-type algorithm for time/frequency-limited CURE (TL/FL-CURE), i.e., only $\mathbb{V}_{tot,T}^{(i)}$ or $\mathbb{V}_{tot,\Omega}^{(i)}$ is computed for the construction of the ROM. A W-type algorithm can also be developed by analogy, which we now present without proof. The W-type algorithm for TL/FL-CURE also ensures that $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}$ or $||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}$ decays monotonically at each step irrespective of the choice of interpolation points and the tangential directions.
\[alg:1\] Initialize $C_{\perp}^{(0)}=C$, $S_{tot}^{(0)}=[\hspace{2mm}]$, $C_{tot}^{(0)}=[\hspace{2mm}]$, $L_{tot}^{(0)}=[\hspace{2mm}]$, $Q_{tot}^{(0)}=[\hspace{2mm}]$, $\bar{C}_{tot}^{(0)}=[\hspace{2mm}]$, $W_{tot}^{(i)}=[\hspace{2mm}]$.\
**for** $i=1,\cdots,k$\
$Ran(W^{(i)})=\underset {j=1,\cdots,r_j}{span}\{(\sigma_jI-A^T)^{-1}C^Tc_j^T\}$.\
Set $V^{(i)}=W^{(i)}$, $W^{(i)}=W^{(i)}\big(W^{(i)}(W^{(i)})^T\big)^{-1}$, $\tilde{A}=(W^{(i)})^TAV^{(i)}$, $\tilde{C}=C_\bot^{(i-1)}V^{(i)}$, $C_\bot^{(i)}=C_\bot^{i-1}-\tilde{C}(W^{(i)})^T$, $L^{(i)}=\Big((W^{(i)})^TA-\tilde{A}(W^{(i)})^T\Big)\big(C_\bot^{(i)}\big)^T\Big(C_\bot^{(i)}\big(C_\bot^{(i)}\big)^T\Big)^{-1}$, $S^{(i)}=\tilde{A}-L^{(i)}\tilde{C}$.\
Solve $-S^{(i)}\big(Q^{(i)}\big)^{-1}-\big(Q^{(i)}\big)^{-1}\big(S^{(i)}\big)^T+L^{(i)}\big(L^{(i)}\big)^T=0$.\
$Q_{tot}^{(i)}=\begin{bmatrix}Q_{tot}^{(i-1)}&0\\0&\tilde{Q}^{(i)}\end{bmatrix}$, $\bar{C}_{tot}^{(i)}=\begin{bmatrix}\bar{C}_{tot}^{(i-1)}\\-\big(L^{(i)}\big)^TQ^{(i)}\end{bmatrix}$, $S_{tot}^{(i)}=\begin{bmatrix}S_{tot}^{(i-1)}&0\\-L^{(i)}\bar{C}_{tot}^{(i-1)}&S^{(i)}\end{bmatrix}$, $L_{tot}^{(i)}=\begin{bmatrix}L_{tot}^{(i-1)}\\L^{(i)}\end{bmatrix}$, $A_{tot}^{(i)}=(-S_{tot}^{(i)})^T$, $C_{tot}^{(i)}=(-L_{tot}^{(i)})^T$, $W_{tot}^{(i)}=\begin{bmatrix}W_{tot}^{(i-1)}& W^{(i)}\end{bmatrix}$.\
$\mathbb{Q}_{tot,T}^{(i)}=\Big(e^{-S_{tot}^{(i)}t_1}\big(Q_{tot}^{(i)}\big)^{-1}e^{(-S_{tot}^{(i)})^Tt_1}-e^{-S_{tot}^{(i)}t_2}(Q_{tot}^{(i)})^{-1}e^{(-S_{tot}^{(i)})^Tt_2}\Big)^{-1}$ or\
$\mathbb{Q}_{tot,\Omega}^{(i)}=\Big(\mathscr{F}(-S_{tot}^{(i)})\big(Q_{tot}^{(i)}\big)^{-1}+\big(Q_{tot}^{(i)}\big)^{-1}\mathscr{F}(-S_{tot}^{(i)})^T\Big)^{-1}.$\
$\mathbb{W}_{tot,T}^{(i)}=e^{A^Tt_1}W_{tot}^{(i)}e^{(-S_{tot}^{(i)})^Tt_1}-e^{A^Tt_2}W_{tot}^{(i)}e^{(-S_{tot}^{(i)})^Tt_2}$ or $\mathbb{W}_{tot,\Omega}^{(i)}=\mathscr{F}(A)^TW_{tot}^{(i)}+W_{tot}^{(i)}\mathscr{F}(-S_{tot}^{(i)})^T$.\
$B_{tot}^{(i)}=\mathbb{Q}_{tot,T}^{(i)}(\mathbb{W}_{tot,T}^{(i)})^TB$ or $B_{tot}^{(i)}=\mathbb{Q}_{tot,\Omega}^{(i)}(\mathbb{W}_{tot,\Omega}^{(i)})^TB$.\
**end for**.
Note that $$\begin{aligned}
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,T}}&=B^TQ_TB-B^T\mathbb{W}_{tot,T}^{(i)}\mathbb{Q}_{tot,T}^{(i)}(\mathbb{W}_{tot,T}^{(i)})^TB\nonumber\\
&=B^T\Big(Q_T-\mathbb{W}_{tot,T}^{(i)}\mathbb{Q}_{tot,T}^{(i)}(\mathbb{W}_{tot,T}^{(i)})^T\Big)B\nonumber\\
||H(s)-H_{tot}^{(i)}\big(s\big)||^2_{\mathcal{H}_{2,\Omega}}&=B^TQ_\Omega B-B^T\mathbb{W}_{tot,\Omega}^{(i)}\mathbb{Q}_{tot,\Omega}^{(i)}(\mathbb{W}_{tot,\Omega}^{(i)})^TB\nonumber\\
&=B^T\Big(Q_\Omega-\mathbb{W}_{tot,\Omega}^{(i)}\mathbb{Q}_{tot,\Omega}^{(i)}(\mathbb{W}_{tot,\Omega}^{(i)})^T\Big)B.\nonumber\end{aligned}$$ Therefore, $\mathbb{W}_{tot,T}^{(i)}\mathbb{Q}_{tot,T}^{(i)}(\mathbb{W}_{tot,T}^{(i)})^T$ and $\mathbb{W}_{tot,\Omega}^{(i)}\mathbb{Q}_{tot,\Omega}^{(i)}(\mathbb{W}_{tot,\Omega}^{(i)})^T$ monotonically approach to $Q_T$ and $Q_\Omega$ after each iteration of CURE. Hence, it also provides the approximations of $Q_T$ and $Q_\Omega$.
Numerical Examples
==================
**Example 1:** Consider the following state-space matrices of the original $6^{th}$ order system: $$\begin{aligned}
A&=\begin{bmatrix}0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\\-5.4545&4.5455&0&-0.0545&0.0455&0\\10&-21&11&0.1&-0.21&0.11\\ 0&5.5&-6.5&0&0.055&-0.065\end{bmatrix},\nonumber\\
B&=\begin{bmatrix}0&0&0&0.0909&0.4&-0.5\end{bmatrix}^T,\hspace*{2mm}C=\begin{bmatrix}2&-2&3&0&0&0\end{bmatrix},\hspace*{2mm}D=0.\nonumber\end{aligned}$$ We construct $2^{nd}$ order ROMs using BT, FLBT, CURE, and FL-CURE. We use PORK within CURE. We also construct a ROM using FLBT with the approximate Gramians generated by FL-CURE, and we name it as “CUREd-FLBT". The desired frequency interval is specified as $[8,9]$ rad/sec. The interpolation points and (right and left) tangential directions used in the two steps are $\{1,2\}$ and $\{1,1\}$, respectively. The sigma plot of the error within $[8,9]$ rad/sec is shown in Figure \[fig1\]. Similarly, we construct ROMs using TLBT and TL-CURE. The desired time interval is specified as $[0,0.5]$ sec. The impulse responses within $[0,0.5]$ sec is shown in Figure \[fig2\]. The comparison of decay in error for CURE, FL-CURE, TL-CURE is given in Table \[tab1\].
![Error plot within $[8,9]$ rad/sec[]{data-label="fig1"}](Fig_1){width="10cm"}
![Impulse response within $[0,0.5]$ sec[]{data-label="fig2"}](Fig_2){width="10cm"}
Method Step $||H(s)-H^{(i)}_{tot}\big(s\big)||_{\mathcal{H}_2/\mathcal{H}_{2,\Omega}/\mathcal{H}_{2,T}}$
-------- ------ ----------------------------------------------------------------------------------------------
1 1.9959
2 1.9952
1 0.0243
2 0.0010
1 0.1223
2 0.0294
: Decay in Error[]{data-label="tab1"}
\
**Example 2:** Consider the following state-space matrices of the original $6^{th}$ order system: $$\begin{aligned}
A&=\begin{bmatrix}-9&-29&-100&-82&-19&-2\\
1&0&0&0&0&0\\
0&1&0&0&0&0\\
0&0&1&0&0&0\\
0&0&0&1&0&0\\
0&0&0&0&1&0\end{bmatrix},\hspace*{2mm}B=\begin{bmatrix}1\\0\\0\\0\\0\\0\end{bmatrix},\nonumber\\C&=\begin{bmatrix}2&-2&3&0&0&0\end{bmatrix},\hspace*{30mm}D=0.\nonumber\end{aligned}$$ We construct $3^{rd}$ order ROMs using BT, FLBT, CURE, and FL-CURE. We use PORK within CURE. We also construct a ROM using FLBT with the approximate Gramians generated by FL-CURE. The desired frequency interval is specified as $[15,16]$ rad/sec. The interpolation points and (right and left) tangential directions used in the two steps are $\{1,2,3\}$ and $\{1,1,1\}$, respectively. The sigma plot of the error within $[15,16]$ rad/sec is shown in Figure \[fig3\]. Similarly, we construct ROMs using TLBT and TL-CURE. The desired time interval is specified as $[0,1]$ sec. The impulse responses within $[0,1]$ sec is shown in Figure \[fig4\]. The comparison of decay in error for CURE, FL-CURE, TL-CURE is given in Table \[tab2\].
![Error plot within $[15,16]$ rad/sec[]{data-label="fig3"}](Fig_3){width="10cm"}
![Impulse response within $[0,1]$ sec[]{data-label="fig4"}](Fig_4){width="10cm"}
Method Step $||H(s)-H^{(i)}_{tot}\big(s\big)||_{\mathcal{H}_2/\mathcal{H}_{2,\Omega}/\mathcal{H}_{2,T}}$
-------- ------ ----------------------------------------------------------------------------------------------
1 $0.1252$
2 $1.1251$
3 $1.1248$
1 $3.29\times 10^{-7}$
2 $4.30\times 10^{-8}$
3 $1.04\times 10^{-8}$
1 $0.0018$
2 $9.95\times 10^{-4}$
3 $3.28\times 10^{-4}$
: Decay in Error[]{data-label="tab2"}
Conclusion
==========
In this paper, we presented a cumulative scheme for model order reduction in limited time and frequency intervals, which adaptively construct a reduced model such that the error decays monotonically in each iteration irrespective of the choice of interpolation points and tangential directions. Moreover, the algorithm also provides approximate time-limited and frequency-limited Gramians of the system which can be used for computing time-limited or frequency-limited balanced realization. The numerical results validate the theory developed in the paper.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is supported by National Natural Science Foundation of China under Grant (No. $61873336$, $61873335$), and supported in part by $111$ Project (No. D$18003$).
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P. Vuillemin, “Frequency-limited model approximation of large-scale dynamical models,” Ph.D. dissertation, 2014.
U. Zulfiqar, V. Sreeram, and X. Du, “Time-limited pseudo-optimal h2-model order reduction,” *arXiv preprint arXiv:1909.10275*, 2019.
——, “Frequency-limited pseudo-optimal rational krylov algorithm for power system reduction,” *arXiv preprint arXiv:1910.02374*, 2019.
[^1]: U. Zulfiqar and V. Sreeram are with the School of Electrical, Electronics and Computer Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009 (email: [email protected], [email protected]).
[^2]: X. Du is with the School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China, and also with the Shanghai Key Laboratory of Power Station Automation Technology, Shanghai University, Shanghai 200444, P.R. China (e-mail: [email protected])
|
---
abstract: |
Background
: Nuclear astrophysics centers on the role of nuclear physics in the cosmos. In particular, nuclear masses at the limits of stability are critical in the development of stellar structure and the origin of the elements.
Purpose
: To test and validate the predictions of recently refined nuclear mass models against the newly published AME2016 compilation.
Methods
: The basic paradigm underlining the recently refined nuclear mass models is based on existing state-of-the-art models that are subsequently refined through the training of an artificial neural network. Bayesian inference is used to determine the parameters of the neural network so that statistical uncertainties are provided for all model predictions.
Results
: We observe a significant improvement in the Bayesian Neural Network (BNN) predictions relative to the corresponding “bare" models when compared to the nearly 50 new masses reported in the AME2016 compilation. Further, AME2016 estimates for the handful of impactful isotopes in the determination of $r$-process abundances are found to be in fairly good agreement with our theoretical predictions. Indeed, the BNN-improved Duflo-Zuker model predicts a root-mean-square deviation relative to experiment of $\sigma_{\rm rms}\!\simeq\!400$keV.
Conclusions
: Given the excellent performance of the BNN refinement in confronting the recently published AME2016 compilation, we are confident of its critical role in our quest for mass models of the highest quality. Moreover, as uncertainty quantification is at the core of the BNN approach, the improved mass models are in a unique position to identify those nuclei that will have the strongest impact in resolving some of the outstanding questions in nuclear astrophysics.
author:
- 'R. Utama[^1]'
- 'J. Piekarewicz'
bibliography:
- 'AME2016vsBNN.bib'
title: 'Validating neural-network refinements of nuclear mass models'
---
Introduction {#intro}
============
As articulated in the most recent US long-range plan for nuclear science[@LongRangePlan] “nuclear astrophysics addresses the role of nuclear physics in our universe", particularly in the development of structure and on the origin of the chemical elements. In this context, fundamental nuclear properties such as masses, radii, and lifetimes play a critical role. However, knowledge of these nuclear properties is required at the extreme conditions of density, temperature, and isospin asymmetry found in most astrophysical environments. Indeed, exotic nuclei near the drip lines are at the core of several fundamental questions driving nuclear structure and astrophysics today: *what are the limits of nuclear binding?*, *where do the chemical elements come from?*, and *what is the nature of matter at extreme densities?*[@LongRangePlan; @national2012Nuclear; @QuarksCosmos:2003].
Although new experimental facilities have been commissioned with the aim of measuring nuclear masses, radii, and decays far away from stability, at present some of the required astrophysical inputs are still derived from often uncontrolled theoretical extrapolations. And even though modern experimental facilities of the highest intensity and longest reach will determine nuclear properties with unprecedented accuracy throughout the nuclear chart, it has been recognized that many nuclei of astrophysical relevance will remain beyond the experimental reach[@Mumpower:2015ova; @RocaMaza:2008ja; @Utama:2015hva]. Thus, reliance on theoretical models that extrapolate into unknown regions of the nuclear chart becomes unavoidable. Unfortunately, these extrapolations are highly uncertain and may ultimately lead to faulty conclusions[@Blaum:2006]. However, one should not underestimate the vital role that experiments play and will continue to play. Indeed, measurements of even a handful of exotic short-lived isotopes are of critical importance in constraining theoretical models and in so doing better guide the extrapolations.
Although no clear-cut remedy exists to cure such unavoidable extrapolations, we have recently offered a path to mitigate the problem[@Utama:2015hva; @Utama:2016rad; @Utama:2017wqe] primarily in the case of nuclear masses. The basic paradigm behind our two-pronged approach is to start with a robust underlying theoretical model that captures as much physics as possible followed by a *Bayesian Neural Network* (BNN) refinement that aims to account for the missing physics[@Utama:2015hva]. Several virtues were identified in such a combined approach. First, we observed a significant improvement in the predictions of those nuclear masses that were excluded from the training of the neural network—even for some of the most sophisticated mass models available in the literature[@Moller:1993ed; @Duflo:1995; @Goriely:2010bm]. Second, mass models of similar quality that differ widely in their predictions far away from stability tend to drastically and systematically reduce their theoretical spread after the implementation of the BNN refinement. Finally, due to the Bayesian nature of the approach, the refined predictions are always accompanied by statistical uncertainties. This philosophy was adopted in our most recent work[@Utama:2017wqe], which culminated with the publication of two refined mass models: the mic-mac model of Duflo and Zuker[@Duflo:1995] and the microscopic HFB-19 functional of Goriely and collaborators[@Goriely:2010bm].
As luck would have it, shortly after the submission of our latest manuscript[@Utama:2017wqe] we became aware of the newly published atomic mass evaluation AME2016[@AME:2016]. This is highly relevant given that the training of the neural network relied exclusively on a previous mass compilation (AME2012)[@AME:2012]. Thus, insofar as the nearly 50 new masses appearing in the newest compilation, ours are bonafide theoretical predictions. Confronting the newly refined mass models against the newly published data is the main goal of this brief report.
This short manuscript has been organized as follows. First, no further physics justification nor detailed account of the BNN framework are given here, as both were extensively addressed in our most recent publication[@Utama:2017wqe]. Second, the results presented in Sec.\[results\] are limited to those nuclei appearing in the AME2016 compilation whose masses were not reported previously or whose values, although determined from experimental trends of neighboring nuclides, have a strong impact on $r$-process nucleosynthesis. As we articulate below, the main outcome from this study is the validation of the novel BNN approach. Indeed, we conclude that the improvement reported in Ref.[@Utama:2017wqe] extends to the newly determined nuclear masses—which in the present case represent true model predictions. We end the paper with a brief summary in Sec.\[conclusions\].
Results
=======
In Ref.[@Utama:2017wqe] we published refined mass tables with the aim of taming the unavoidable extrapolations into unexplored regions of the nuclear chart that are critical for astrophysical applications. Specifically, we refined the predictions of both the Duflo-Zuker[@Duflo:1995] and HFB-19[@Goriely:2010bm] models using the AME2012 compilation in the mass region from ${}^{40}$Ca to ${}^{240}$U. The latest AME2016 compilation includes mass values for 46 additional nuclei in the ${}^{40}$Ca-${}^{240}$U region, and these are listed in Table\[Table1\] alongside predictions from various models. These include the “bare” models ([*i.e.,*]{} before BNN refinement) HFB-19[@Goriely:2010bm], Duflo-Zuker[@Duflo:1995], FRDM-2012[@Moller:2012], HFB-27[@Goriely:2013nxa], and WS3[@Liu:2011ama]. Also shown are the predictions from the BNN-improved Duflo-Zuker and HFB-19 models[@Utama:2017wqe]. The last column displays the total binding energy as reported in the AME2016 compilation[@AME:2016]; quantities displayed in parentheses in the last three columns represent the associated errors. Note that we quote *differences* between the model predictions and the experimental masses using only the central values. Finally, the last row contains root-mean-square deviations associated with each of the models. The corresponding information in graphical form is also displayed in Fig.\[Fig1\], but only for the five bare models discussed in the text.
[|cc||\*[5]{}[r|]{}|\*[2]{}[r|]{}|r|]{}
------------------------------------------------------------------------
Z & N & HFB-19 & DZ & FRDM-2012 & HFB-27 & WS3 & HFB19-BNN & DZ-BNN & AME2016\
------------------------------------------------------------------------
20 & 33 & 1.959 & 3.476 & 4.571 & 2.169 & 1.370 & 1.948(1.520) & 1.675(0.951) & 441.521(0.044)\
20 & 34 & 0.031 & 2.829 & 3.035 & 1.191 & 0.850 & 0.470(1.500) & 0.840(0.880) & 445.365(0.048)\
21 & 35 & -1.547 & 0.563 & 1.227 & -0.777 & -0.296 & -0.216(0.928) & -0.226(0.686) & 460.417(0.587)\
21 & 36 & -2.563 & 0.667 & 0.752 & -1.473 & -0.121 & -0.953(0.975) & -0.117(0.611) & 464.632(1.304)\
24 & 40 & -1.880 & 0.008 & -0.872 & -1.370 & 0.130 & -0.130(0.793) & -0.192(0.498) & 531.268(0.440)\
25 & 37 & -0.146 & 0.195 & 0.327 & -0.106 & 0.549 & 0.555(1.060) & 0.001(0.416) & 529.387(0.007)\
27 & 25 & 1.351 & 0.519 & -0.274 & -0.469 & 0.689 & 0.416(1.290) & 0.141(0.650) & 432.946(0.008)\
29 & 27 & 2.564 & 0.188 & 0.199 & -0.516 & 0.579 & 1.444(0.953) & 0.460(0.513) & 467.949(0.015)\
30 & 52 & -0.645 & -1.697 & -0.018 & -0.315 & -1.130 & -0.505(0.896) & -0.498(0.638) & 680.692(0.003)\
32 & 54 & -0.800 & -1.143 & 0.338 & -0.430 & -0.596 & -0.759(0.734) & -0.469(0.557) & 718.498(0.438)\
34 & 57 & -0.279 & -0.084 & 0.790 & 0.251 & 0.015 & 0.465(0.824) & 0.297(0.504) & 758.470(0.433)\
37 & 63 & 0.768 & -0.132 & -1.038 & -0.002 & -0.846 & 1.144(0.886) & 0.212(0.496) & 824.432(0.020)\
39 & 66 & 0.851 & 0.642 & -0.293 & 1.211 & 0.788 & 0.555(0.868) & 0.895(0.492) & 868.247(1.337)\
40 & 42 & -0.718 & -0.132 & 0.395 & 0.722 & -0.092 & -0.858(0.709) & -0.313(0.522) & 694.185(0.011)\
40 & 66 & -0.239 & -0.882 & -1.344 & 0.051 & -0.155 & -0.581(0.826) & -0.671(0.536) & 882.816(0.433)\
40 & 67 & 0.191 & -0.473 & -0.823 & 0.751 & 0.525 & -0.079(0.838) & -0.196(0.492) & 886.717(1.122)\
41 & 43 & 0.379 & 0.152 & 0.263 & 1.139 & 0.101 & 0.309(0.833) & 0.018(0.531) & 707.133(0.013)\
41 & 69 & 0.011 & -1.066 & -1.091 & 0.471 & -0.062 & -0.014(0.855) & -0.583(0.501) & 908.079(0.838)\
43 & 71 & -0.138 & -1.003 & -0.540 & 0.122 & -0.532 & 0.141(0.731) & -0.290(0.522) & 945.090(0.433)\
43 & 72 & -0.289 & -0.494 & -0.129 & 0.241 & 0.189 & 0.121(0.690) & 0.441(0.495) & 950.881(0.789)\
45 & 76 & -0.839 & -0.930 & -0.336 & -0.539 & 0.161 & -0.061(0.611) & 0.175(0.510) & 997.674(0.619)\
46 & 77 & -1.179 & -0.794 & -0.326 & -0.919 & -0.295 & -0.420(0.651) & 0.018(0.501) & 1017.214(0.789)\
48 & 81 & -1.411 & -0.911 & -1.165 & -1.351 & -0.976 & -0.458(0.801) & -0.560(0.444) & 1066.705(0.017)\
48 & 83 & -1.750 & -0.940 & -1.141 & -1.110 & -0.720 & -0.378(0.753) & -0.253(0.607) & 1075.009(0.102)\
51 & 87 & -2.058 & -0.724 & 0.133 & -0.528 & -0.872 & -0.395(0.824) & -0.208(0.570) & 1128.163(1.064)\
53 & 88 & -1.112 & -0.281 & 0.498 & -0.332 & -0.138 & -0.052(0.663) & 0.034(0.596) & 1156.518(0.016)\
56 & 93 & -0.899 & -0.150 & -0.415 & -0.529 & -0.157 & -0.139(0.650) & -0.074(0.420) & 1211.935(0.438)\
57 & 93 & -0.899 & -0.520 & -0.836 & -0.579 & -0.669 & -0.501(0.715) & -0.677(0.398) & 1222.234(0.435)\
57 & 94 & -1.209 & -0.460 & -0.745 & -0.889 & -0.635 & -0.690(0.713) & -0.581(0.388) & 1227.485(0.435)\
63 & 74 & -0.732 & -0.680 & -0.261 & -0.052 & 0.340 & -0.497(0.666) & -0.152(0.418) & 1116.629(0.004)\
81 & 109 & -0.167 & 0.302 & -0.298 & -0.357 & -0.149 & -0.187(0.470) & -0.225(0.289) & 1494.552(0.008)\
82 & 133 & -1.841 & 0.444 & 0.988 & -0.061 & 0.200 & -0.749(0.522) & 0.257(0.317) & 1666.838(0.052)\
83 & 111 & -0.269 & 0.764 & 0.000 & -0.219 & -0.506 & -0.280(0.457) & 0.247(0.301) & 1516.930(0.006)\
85 & 113 & -0.244 & 0.736 & 0.076 & -0.364 & -0.162 & -0.238(0.484) & 0.225(0.304) & 1538.336(0.006)\
87 & 110 & -0.443 & 0.995 & 0.061 & -0.233 & 0.282 & -0.287(0.680) & 0.394(0.475) & 1511.731(0.054)\
87 & 111 & -1.042 & 0.566 & -0.347 & -0.662 & -0.141 & -0.914(0.650) & 0.016(0.404) & 1520.483(0.032)\
87 & 115 & -0.144 & 0.293 & 0.126 & -0.164 & 0.099 & -0.113(0.571) & -0.215(0.317) & 1559.246(0.007)\
87 & 145 & -1.391 & 1.367 & 0.099 & -0.511 & 0.399 & 0.086(0.655) & 0.144(0.421) & 1758.409(0.014)\
87 & 146 & -1.538 & 1.850 & 0.183 & -0.618 & 0.667 & 0.082(0.746) & 0.426(0.586) & 1763.633(0.020)\
88 & 113 & -0.426 & 0.712 & 0.311 & -0.206 & 0.112 & -0.320(0.712) & 0.167(0.425) & 1541.551(0.020)\
89 & 116 & 0.075 & 0.042 & 0.354 & 0.215 & 0.271 & 0.153(0.739) & -0.473(0.389) & 1570.884(0.051)\
89 & 117 & -0.438 & -0.141 & 0.103 & -0.018 & 0.149 & -0.376(0.703) & -0.649(0.361) & 1579.583(0.050)\
92 & 123 & 0.408 & 0.268 & -0.346 & 0.248 & 0.319 & 0.443(0.747) & -0.215(0.437) & 1638.434(0.089)\
92 & 124 & 0.375 & 0.505 & -0.328 & 0.125 & 0.201 & 0.384(0.698) & 0.005(0.456) & 1648.362(0.028)\
92 & 129 & 0.652 & 1.641 & -0.285 & 0.622 & 0.095 & 0.504(0.677) & 0.978(0.536) & 1687.265(0.051)\
92 & 130 & 0.639 & 1.399 & -0.236 & 0.529 & 0.071 & 0.457(0.697) & 0.713(0.531) & 1695.584(0.052)\
& [**1.093**]{} & [**1.018**]{} & [**0.997**]{} & [**0.723**]{} & [**0.513**]{} & [**0.587**]{} & [**0.479**]{} &\
The trends displayed in Table\[Table1\] and even more clearly illustrated in Fig.\[Fig1\] are symptomatic of a well-known problem, namely, that theoretical mass models of similar quality that agree in regions where masses are experimentally known differ widely in regions where experimental data is not yet available[@Blaum:2006]. Given that sensitivity studies suggest that resolving the $r$-process abundance pattern requires mass-model uncertainties of the order of $\lesssim\!100$keV[@Mumpower:2015hva], the situation depicted in Fig.\[Fig1\] is particularly dire. However, despite the large scattering in the model predictions, which worsens as one extrapolates further into the neutron drip lines, significant progress has been achieved in the last few years. Indeed, in the context of density functional theory, the HFB-27 mass model of Goriely, Chamel, and Pearson predicts a rather small rms deviation of $\sim\!0.5$MeV for all nuclei with neutron and proton numbers larger than 8[@Goriely:2013nxa]. Further, in the case of the Weizsäcker-Skyrme WS3 model of Liu, Wang, Deng, and Wu, the agreement with experiment is even more impressive: the rms deviation relative to 2149 known masses is a mere $\sim\!0.34$MeV[@Liu:2011ama]. Although not as striking, the success of both models extends to their *predictions* of the 46 new masses listed in Table\[Table1\]: $\sigma_{\rm rms}\!=\!0.72\,{\rm MeV}$ and $\sigma_{\rm rms}\!=\!0.51\,{\rm MeV}$, respectively. This represents a significant improvement over earlier mass models that typically predict a rms deviation of the order of 1MeV; see Table\[Table1\] and Fig.\[Fig1\].
![Theoretical predictions for the total binding energy relative to experiment for the 46 nuclei in the ${}^{40}$Ca-${}^{240}$U region that appear in the latest AME2016 compilation but not in any of the earlier mass evaluations. The models shown here are representative of some of the most sophisticated mass models available in the literature. Quantities in parentheses denote the rms deviations.[]{data-label="Fig1"}](Fig1.pdf){width="0.5\columnwidth"}
However, our main focus is to assess the improvement in the predictions of two of these earlier mass models (HFB-19 and DZ) as a result of the BNN refinement. In agreement with the nearly a factor-of-two improvement reported in Ref.[@Utama:2017wqe], we observe a comparable gain in the predictions of the 46 nuclear masses listed in Table\[Table1\]; that is, $\sigma_{\rm rms}\!=\!(1.093\!\rightarrow\!0.587) \,{\rm MeV}$ and $\sigma_{\rm rms}\!=\!(1.018\!\rightarrow\!0.479) \,{\rm MeV}$ for HFB-19 and DZ, respectively. Of course, an added benefit of the BNN approach is the supply of theoretical error bars. Indeed, when such error bars are taken into account—as we do in Fig.\[Fig2\]—then *all* of the refined predictions are consistent with the experimental values at the 2$\sigma$ level. For reference, also included in Fig.\[Fig2\] are the impressive predictions of the WS3 model, albeit without any estimates of the theoretical uncertainties.
![Theoretical predictions for the total binding energy relative to experiment for the 46 nuclei in the ${}^{40}$Ca-${}^{240}$U region that appear in the latest AME2016 compilation but not in any of the earlier mass evaluations. The models shown include HFB-19 and Duflo-Zuker together with their corresponding BNN refinements (shown with error bars). For reference the WS3 model of Liu and collaborators is also shown. Quantities in parentheses denote the rms deviations.[]{data-label="Fig2"}](Fig2.pdf){width="0.5\columnwidth"}
[|cc||r|r|r|r||r|]{}
------------------------------------------------------------------------
Z & N & WS3 & FRDM & DZ & DZ-BNN & AME2016\
------------------------------------------------------------------------
48 & 84 & -1.101 & -1.704 & -1.384 & -0.542(0.761) & 1078.176\
48 & 85 & -1.090 & -1.273 & -1.524 & -0.556(0.954) & 1079.827\
48 & 86 & -1.252 & -1.322 & -1.664 & -0.611(1.170) & 1082.988\
49 & 84 & -0.910 & -0.775 & -0.676 & -0.198(0.574) & 1092.595\
49 & 85 & -0.806 & -0.256 & -0.738 & -0.049(0.695) & 1094.914\
49 & 86 & -1.162 & -0.590 & -1.124 & -0.243(0.849) & 1097.820\
49 & 87 & -1.124 & -0.178 & -1.223 & -0.190(1.030) & 1099.832\
49 & 88 & -1.487 & -0.531 & -1.607 & -0.476(1.240) & 1102.439\
50 & 86 & -0.785 & -0.190 & -0.445 & 0.135(0.631) & 1114.520\
50 & 88 & -1.259 & -0.246 & -1.043 & -0.054(0.848) & 1119.594\
& [**1.117**]{} & [**0.877**]{} & [**1.210**]{} & [**0.369**]{} &\
![Theoretical predictions for the total binding energy of those nuclei that have been identified as impactful in $r$-process nucleosynthesis[@Mumpower:2015hva]. All experimental values have been estimated from experimental trends of neighboring nuclides[@AME:2016]. Quantities in parentheses denote the rms deviations.[]{data-label="Fig3"}](Fig3.pdf){width="0.5\columnwidth"}
We close this section by addressing a particular set of nuclear masses that have been identified as “impactful" in sensitivity studies of the elemental abundances in $r$-process nucleosynthesis. These include a variety of neutron-rich isotopes in palladium, cadmium, indium, and tin; see Table I of Ref.[@Mumpower:2015hva]. As of today, none of these critical isotopes have been measured experimentally. However, many of them have been “flagged" (with the symbol “[**\#**]{}") in the AME2016 compilation to indicate that while not strictly determined experimentally, the provided mass estimates were obtained from *experimental trends of neighboring nuclides*[@AME:2016]. In Table\[Table2\] theoretical predictions are displayed for those isotopes that have been both labeled as impactful and flagged. Predictions are provided for the WS3[@Liu:2011ama], FRDM-2012[@Moller:2012], DZ[@Duflo:1995], and BNN-DZ[@Utama:2017wqe] mass models. Root-mean-square deviations of the order of 1MeV are recorded for all models, except for the improved Duflo-Zuker model where the deviation is only 369keV. This same information is depicted in graphical form in Fig.\[Fig3\]. The figure nicely encapsulates the spirit of our two-prong approach, namely, one that starts with a mass model of the highest quality (DZ) that is then refined via a BNN approach. The improvement in the description of the experimental data together with a proper assessment of the theoretical uncertainties are two of the greatest virtues of the BNN approach. Indeed, the BNN-DZ predictions are consistent with all masses of those impactful nuclei that have been determined from the experimental trends.
Conclusions
===========
Nuclear masses of neutron-rich nuclei are paramount to a variety of astrophysical phenomena ranging from the crustal composition of neutron stars to the complexity of $r$-process nucleosynthesis. Yet, despite enormous advances in experimental methods and tools, many of the masses of relevance to astrophysics lie well beyond the present experimental reach, leaving no option but to rely on theoretical extrapolations that often display large systematic variations. The current situation is particularly troublesome given that sensitivity studies require mass-model uncertainties to be reduced to about $\lesssim\!100$keV in order to resolve $r$-process abundances.
There are at least two different approaches currently used to alleviate this problem. The first one consists of painstakingly difficult measurements near the present experimental limits that aim to inform and constrain mass models. The second approach is based on a global refinement of existing mass models through the training of an artificial neural network. This is the approach that we have advocated in this short contribution. Given that the training of the neural network relied exclusively on the AME2012 compilation, our approach was validated by comparing our theoretical predictions against the new information provided in the most recent AME2016 compilation.
The comparison against the newly available AME2016 data was highly successful. For the nearly 50 new mass measurement reported in the ${}^{40}$Ca-${}^{240}$U region, the rms deviation of the two BNN-improved models explored in this work (Duflo-Zuker and HFB-19) was reduced by nearly a factor of two relative to the unrefined bare models. Further, for the masses of several impactful isotopes for the $r$-process, the predictions from the improved Duflo-Zuker model were fully consistent with the new AME2016 estimates and in far better agreement than some of the most sophisticated mass models available in the literature. Finally and as important, all nuclear-mass predictions in the BNN approach incorporate properly estimated statistical uncertainties. When these theoretical error bars are incorporated, then *all* of our predictions are consistent with experiment at the 2$\sigma$ level.
Ultimately, improvements in mass models require a strong synergy between theory an experiment. Next-generation rare-isotope facilities will produce new exotic nuclei that will help constrain the physics of weakly-bound nuclei. In turn, improved theoretical models will suggest new measurements on a few critical nuclei that will best inform nuclear models. We are confident that the BNN approach advocated here will play a critical role in this endeavor, particularly in identifying those nuclei that have the strongest impact in resolving some outstanding questions in nuclear astrophysics. We are hopeful that in the near future mass-model uncertainties—both statistical and systematic—will be reduced to less than $100$keV, which represents the elusive standard required to resolve the $r$-process abundance pattern.
We are thankful to Pablo Giuliani for many stimulating discussions. This material is based upon work supported by the U.S. Department of Energy Office of Science, Office of Nuclear Physics under Award Number DE-FG02-92ER40750.
[^1]: Present address: Cold Spring Harbor Laboratory, NY 11724
|
---
abstract: 'Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by means of computer algebra. On the other hand, sometimes closed-forms of such extensions can be derived by induction. It is expected that the method used to obtain the different recursive equations can be applied to extend other hypergeometric summation formulas given in the literature.'
author:
- 'J. L. González-Santander ORCID: 0000-0001-5348-4967'
- 'C/ Ovidi Montllor i Mengual 7, pta.9'
- |
46017, Valencia, Spain.\
[email protected]\
title: 'Recursive formulas for $_{2}F_{1}$ and $_{3}F_{2}$ hypergeometric series'
---
**Keywords**: generalized hypergeometric functions, hypergeometric summation formulas, contiguous hypergeometric identities, recursive hypergeometric formulas
**Mathematics Subject Classification**: 33C05, 33C20
Introduction
============
Hypergeometric and generalized hypergeometric functions have many applications as solutions of problems concerning mathematics [@Jason; @Branges] and physics [@Calabra; @Moch]. It is worth noting that whenever these functions can be expressed in terms of gamma functions, the results are very important in many applications and also from a theoretical point of view. However, few summation theorems are available in the literature [Andrews]{}. Thereby, in the last decades, many publications have devoted to broaden these classical results. For instance, it is well-known that by repeated application of the contiguous relations of the hypergeometric function $_{2}F_{1}\left( a,b;c;z\right) $ [@DLMF Sect. 15.5(ii)], any function $_{2}F_{1}\left( a+k,b+\ell ;c+m;z\right) $, in which $k$, $\ell $, and $m$ are integers, can be expressed as as a linear combination of $_{2}F_{1}\left( a,b;c;z\right) $ and any of its contiguous functions, with coefficients that are rational functions of $a$, $b$, $c$, and $z$. Also, by systematic exploitation of the relations between contiguous functions given by generalized hypergeometric functions $_{p}F_{q}$ [Rainville]{}, we can find in [@LavoieRathie] a generalization of Watson’s theorem, and in [@Lavoie] a generalization of Dixon’s theorem. However, these generalizations extend these theorems for a finite number of integer numbers summed to certain parameters of the corresponding hypergeometric functions.
In this paper, we provide recursive formulas in order to extend some known summation formulas of $_{2}F_{1}$ hypergeometric function at arguments $z\neq 1$, and $_{3}F_{2}$ hypergeometric function at argument $z=1$. These recursive formulas are very suitable for symbolic computation and numerical evaluation by using computer algebra, and they are not restricted to a finite number of cases, as in the papers stated above. Despite the fact that these recursive formulas might be automatically computed using creative telescoping (also called Wilf-Zeilberger’s theory [Andrews]{}) following the method describe in [@Koomwinder] for non-terminating series, we propose here a much more simple approach using contiguous relations. Also, it is sometimes possible to derive general expressions in closed-form by induction from the corresponding recursive equation. In fact, the method described in this paper is quite general and can be used for many other summation formulas given in the literature, so that here we present some selected examples.
This article is organized as follows. In Section \[Section: 2F1\] we derive some recursive formulas for $_{2}F_{1}$ hypergeometric function at arguments $z=\frac{1}{2},2,-1$. Section \[Section: 3F2\] extends some known $_{3}F_{2}$ hypergeometric summation formulas at argument unity, such as Pfaff-Saalschutz sum, Watson’s sum and Dixon’s sum, among others. Finally, the conclusions are collected in Section \[Section: Conclusions\].
$_{2}F_{1}$ recursive formulas\[Section: 2F1\]
==============================================
In this Section we consider Gauss’s hypergeometric series, defined as$$_{2}F_{1}\left( \left.
\begin{array}{c}
a,b \\
c\end{array}\right\vert z\right) =\sum_{m=0}^{\infty }\frac{\left( a\right) _{m}\left(
b\right) _{m}}{m!\left( c\right) _{m}}z^{m}, \label{2F1_def}$$where $\left( \alpha \right) _{m}=\Gamma \left( \alpha +m\right) /\Gamma
\left( \alpha \right) $ denotes the Pochhammer symbol. When $a$ or $b$ are negative integers, the series (\[2F1\_def\]) terminates, so that it converges. If it is not so, (\[2F1\_def\]) is absolutely convergent when $\left\vert z\right\vert <1$, except when the parameter $c=0,-1,-2,\ldots $ In the case in which $\left\vert z\right\vert =1$, the series is absolutely convergent when $\left( c-a-b\right) >0$, conditionally convergent when $-1<\,$$\left( c-a-b\right) \leq 0$ and $z\neq 1$; and divergent when $\left( c-a-b\right) \leq -1$.
In order to obtain the recursive formulas stated below, we will use known contiguous relations connecting Gauss‘s hypergeometric series $_{2}F_{1}\left( a,b;c;z\right) $ with other two hypergeometric series which vary their parameters $a$, $b$, $c$, $\pm 1$ unit. The idea is to particularize the parameters in the contiguous relation in such a way that we can define a recursive relation in which the initial iteration ($k=0$)is given by a known summation formula. Iterating the recursive relation and with the aid of computer algebra is quite easy to obtain summation formulas for other integers $k$.
First, we generalize Gauss’s second theorem. In [@LavoieWhimple] we can find an extension of Gauss’s second theorem for $j=0$ and some particular values of $k$.
If $k\in
\mathbb{N}
$ and $j=0,1$, then the following recursive equation holds true:$$G_{k}\left( a,b,j\right) =G_{k-1}\left( a,b,j\right) +\frac{a}{a+b+1+j}G_{k-1}\left( a+1,b+1,j\right) , \label{Recursive_Gauss_2nd}$$where$$G_{k}\left( a,b,j\right) =\,_{2}F_{1}\left( \left.
\begin{array}{c}
a,b+k \\
\frac{a+b+j+1}{2}\end{array}\right\vert \frac{1}{2}\right) ,$$and, according to Gauss’s second theorem [@DLMF Eqn. 15.4.28]$$G_{0}\left( a,b,0\right) =\frac{\sqrt{\pi }\Gamma \left( \frac{a+b+1}{2}\right) }{\Gamma \left( \frac{a+1}{2}\right) \Gamma \left( \frac{b+1}{2}\right) },\quad a+b\neq -1,-2\ldots ,$$and according to [@DLMF Eqn. 15.4.29]$$\begin{aligned}
&&G_{0}\left( a,b,1\right) \label{G0(a,b,j=1)} \\
&=&\frac{2\sqrt{\pi }}{a-b}\Gamma \left( \frac{a+b}{2}+1\right) \left[ \frac{1}{\Gamma \left( \frac{a}{2}\right) \Gamma \left( \frac{b+1}{2}\right) }-\frac{1}{\Gamma \left( \frac{a+1}{2}\right) \Gamma \left( \frac{b}{2}\right)
}\right] . \notag\end{aligned}$$
Consider the contiguous relation [@Lebedev Eqn. 9.2.13] $$_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta +1 \\
\gamma\end{array}\right\vert z\right) =\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta \\
\gamma\end{array}\right\vert z\right) +\frac{\alpha z}{\gamma }\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1 \\
\gamma +1\end{array}\right\vert z\right) ,$$and set $\alpha =a$, $\beta =b+k$, $\gamma =\frac{a+b+j+1}{2}$, and $z=\frac{1}{2}$.
From the above recursive equation (\[Recursive\_Gauss\_2nd\]), we prove the following identity.
If $k\in
\mathbb{N}
$, then$$_{2}F_{1}\left( \left.
\begin{array}{c}
a,a+k \\
a+1\end{array}\right\vert \frac{1}{2}\right) =2^{a}\left[ 2^{k-1}-\frac{k-1}{a+1}\,_{2}F_{1}\left( \left.
\begin{array}{c}
2-k,a+1 \\
a+2\end{array}\right\vert -1\right) \right] , \label{Identity_2F1}$$where notice that the hypergeometric sum on the RHS of (\[Identity\_2F1\]) is a finite sum.
Taking $b=a$ and $j=1$, (\[Recursive\_Gauss\_2nd\]) is reduced to$$G_{k}\left( a\right) =G_{k-1}\left( a\right) +\frac{a}{2\left( a+1\right) }G_{k-1}\left( a+1\right) , \label{Recursive_G_k(a)}$$where$$G_{k}\left( a\right) =\,_{2}F_{1}\left( \left.
\begin{array}{c}
a,a+k \\
a+1\end{array}\right\vert \frac{1}{2}\right) , \label{G_k(a)}$$and according to [@Prudnikov3 Eqn. 7.3.7(16)]$$G_{0}\left( a\right) =2^{a-1}a\left[ \psi \left( \frac{a+1}{2}\right) -\psi
\left( \frac{a}{2}\right) \right] , \label{G0(a)}$$where $\psi \left( z\right) $ denotes the digamma function. Now, from ([Recursive\_G\_k(a)]{}) and (\[G0(a)\]), and with the aid of computer algebra, perform the first iterations as:$$\begin{aligned}
G_{1}\left( a\right) &=&2^{a}, \\
G_{2}\left( a\right) &=&2^{a}\left( 2-\frac{1}{a+1}\right) , \\
G_{3}\left( a\right) &=&2^{a}\left( 4-\frac{2}{a+1}-\frac{2}{a+2}\right) , \\
G_{4}\left( a\right) &=&2^{a}\left( 8-\frac{3}{a+1}-\frac{6}{a+2}-\frac{3}{a+3}\right) ,\end{aligned}$$thus, we can establish the conjecture$$G_{k}\left( a\right) =2^{a}\left( 2^{k-1}-\sum_{i=1}^{k-1}\frac{\left(
k-1\right) !}{\left( i-1\right) !\left( k-i-1\right) !\left( a+i\right) }\right) , \label{Gk(a)_sum}$$which can be proved by induction. Finally, rewrite (\[Gk(a)\_sum\])expressing the sum therein as a hypergeometric function and match the result to (\[G\_k(a)\]), to obtain (\[Identity\_2F1\]).
Next, we generalized the result given in [@KimRathie], which is given in table form for $k=-1,-2,-3,-4,-5$.
If $k\in
\mathbb{Z}
^{-}$ and $n\in
\mathbb{Z}
^{+}$, then the following recursive equation holds true:$$\begin{aligned}
&&G_{k}\left( n,a\right) \label{Recursive_Srivastava} \\
&=&\frac{2\left( a+k-1\right) \left( 2a+k+n-1\right) }{k\left( 2a+k-1\right)
}G_{k+1}\left( n,a\right) -\frac{2a+k-2}{k}G_{k+1}\left( n,a-1\right) ,
\notag\end{aligned}$$where$$G_{k}\left( n,a\right) =\,_{2}F_{1}\left( \left.
\begin{array}{c}
-n,a \\
2a-1+k\end{array}\right\vert 2\right) ,$$and, according to [@Srivastava]$$\begin{aligned}
&&G_{0}\left( n,a\right) \\
&=&\frac{\Gamma \left( a-\frac{1}{2}\right) }{\sqrt{\pi }}\left[ \frac{1+\left( -1\right) ^{n}}{2}\frac{\Gamma \left( \frac{n+1}{2}\right) }{\Gamma
\left( a+\frac{n-1}{2}\right) }-\frac{1-\left( -1\right) ^{n}}{2}\frac{\Gamma \left( \frac{n}{2}+1\right) }{\Gamma \left( a+\frac{n}{2}\right) }\right] .\end{aligned}$$
Consider the following contiguous relations [@Lebedev Eqns. 9.2.7&14]$$\begin{aligned}
&&\gamma \left( \gamma +1\right) \,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta \\
\gamma\end{array}\right\vert z\right) \label{Lebedev_1} \\
&=&\gamma \left( \gamma -\alpha +1\right) \,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta +1 \\
\gamma +2\end{array}\right\vert z\right) \notag \\
&&+\alpha \left[ \gamma -\left( \gamma -\beta \right) z\right]
\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1 \\
\gamma +2\end{array}\right\vert z\right) , \notag\end{aligned}$$and$$\begin{aligned}
&&_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta +1 \\
\gamma +1\end{array}\right\vert z\right) -\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta \\
\gamma\end{array}\right\vert z\right) \label{Lebedev_2} \\
&=&\,\frac{\alpha \left( \gamma -\beta \right) z}{\gamma \left( \gamma
+1\right) }\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1 \\
\gamma +2\end{array}\right\vert z\right) . \notag\end{aligned}$$Therefore, eliminating the hypergeometric function on the RHS of ([Lebedev\_1]{}) and (\[Lebedev\_2\]), we arrive at$$\begin{aligned}
&&\gamma \,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta \\
\gamma\end{array}\right\vert z\right) -\frac{\left( \gamma -\alpha +1\right) \left( \gamma
-\beta \right) z}{\gamma +1}\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta +1 \\
\gamma +2\end{array}\right\vert z\right) \\
&=&\left[ \gamma -\left( \gamma -\beta \right) z\right] \,_{2}F_{1}\left(
\left.
\begin{array}{c}
\alpha ,\beta +1 \\
\gamma +1\end{array}\right\vert z\right) .\end{aligned}$$Substitute now $\alpha =-n$, $\beta =a-1$, $\gamma =2\left( a-1\right) +k$, and $z=2$ to obtain (\[Recursive\_Srivastava\]).
Finally, we generalize Kummer’s theorem. This case has been discussed more extensively in [@Choi].
If $k\in
\mathbb{N}
$, then$$G_{k+1}\left( a,b\right) =\frac{b+k}{k}G_{k}\left( a,b\right) -\frac{2b\left( a+k+1\right) }{k\left( a-b+1\right) }G_{k}\left( a+2,b+1\right) ,
\label{Recursive_Kummer}$$where$$G_{k}\left( a,b\right) =\,_{2}F_{1}\left( \left.
\begin{array}{c}
a+k,b \\
a-b+1\end{array}\right\vert -1\right) , \label{Gk(a,b)_Kummer}$$and, according to [@Choi]$$\begin{aligned}
&&G_{1}\left( a,b\right) \\
&=&\frac{\sqrt{\pi }\Gamma \left( a+1-b\right) }{2^{a+1}}\left( \frac{1}{\Gamma \left( \frac{a}{2}+1\right) \Gamma \left( \frac{a+1}{2}-b\right) }+\frac{1}{\Gamma \left( \frac{a+1}{2}\right) \Gamma \left( \frac{a}{2}-b+1\right) }\right) .\end{aligned}$$
From the identity [@DLMF Eqn. 15.5.20] and the differentiation formula [@DLMF Eqn. 15.5.1], setting $\alpha \rightarrow \alpha +1$, we arrive at$$\begin{aligned}
&&z\left( 1-z\right) \frac{\left( \alpha +1\right) \beta }{\gamma }\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha +2,\beta +1 \\
\gamma +1\end{array}\right\vert z\right) \\
&=&\left( \gamma -\alpha +1\right) \,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha ,\beta \\
\gamma\end{array}\right\vert z\right) +\left( \alpha +1-\gamma +\beta z\right)
\,_{2}F_{1}\left( \left.
\begin{array}{c}
\alpha +1,\beta \\
\gamma\end{array}\right\vert z\right) .\end{aligned}$$Thereby, taking $\alpha =a+k$, $\beta =b$, $\gamma =a-b+1$, and $z=-1$, we arrive at (\[Recursive\_Kummer\]).
Note that $\forall k=0$, (\[Gk(a,b)\_Kummer\]) reduces to Kummer’s summation formula [@DLMF Eqn. 15.4.26], but in this case, the recursive equation (\[Recursive\_Kummer\]) collapses. However, it is worth noting that in [@Choi] we find a closed-form of (\[Gk(a,b)\_Kummer\]), which reads as$$G_{k}\left( a,b\right) =\frac{\Gamma \left( 1+a-b\right) }{2\Gamma \left(
a+k\right) }\sum_{m=0}^{k}\binom{k}{m}\frac{\Gamma \left( \frac{a+k+m}{2}\right) }{\Gamma \left( \frac{a-k+m}{2}-b+1\right) }. \label{Gk_Choi}$$
As by-product, we can obtain an interesting identity, inserting ([Gk\_Choi]{}) in the recursive formula (\[Recursive\_Kummer\]). Direct substitution yields$$\begin{aligned}
&&\sum_{m=0}^{k+1}\binom{k+1}{m}\frac{\Gamma \left( \frac{a+k+m+1}{2}\right)
}{\Gamma \left( \frac{a-k+m+1}{2}-b\right) } \label{LHS_Choi} \\
&=&\sum_{m=0}^{k}\binom{k}{m}\left( a+k-\frac{bm}{k}\right) \frac{\Gamma
\left( \frac{a+k+m}{2}\right) }{\Gamma \left( \frac{a-k+m}{2}-b+1\right) }.
\notag\end{aligned}$$
Now, recast (\[LHS\_Choi\]) as $$\frac{\Gamma \left( \frac{a+k+1}{2}\right) }{\Gamma \left( \frac{a-k+1}{2}-b\right) }+\sum_{m=0}^{k}\binom{k}{m}\frac{\left( k+1\right) \left(
a+k+m\right) }{2\left( m+1\right) }\frac{\Gamma \left( \frac{a+k+m}{2}\right) }{\Gamma \left( \frac{a-k+m}{2}-b+1\right) },$$thus we obtain this interesting formula, $\forall k=1,2,\ldots $$$\begin{aligned}
&&\frac{\Gamma \left( \frac{a+k+1}{2}\right) }{\Gamma \left( \frac{a-k+1}{2}-b\right) } \label{Identity_Choi} \\
&=&\sum_{m=0}^{k}\binom{k}{m}\left( a+\frac{k-1}{2}-\frac{bm}{k}-\frac{\left( k+1\right) \left( a+k-1\right) }{2\left( m+1\right) }\right) \frac{\Gamma \left( \frac{a+k+m}{2}\right) }{\Gamma \left( \frac{a-k+m}{2}-b+1\right) }. \notag\end{aligned}$$
It is worth noting that (\[Identity\_Choi\]) can be proven using computer algebra.
$_{3}F_{2}$ recursive formulas\[Section: 3F2\]
==============================================
The generalized hypergeometric series $_{p}F_{q}$ is a natural generalization of the Gauss’s series $_{2}F_{1}$, and is defined as $$_{p}F_{q}\left( \left.
\begin{array}{c}
a_{1},\ldots ,a_{p} \\
b_{1},\ldots ,b_{q}\end{array}\right\vert z\right) =\sum_{m=0}^{\infty }\frac{\left( a_{1}\right)
_{m}\cdots \left( a_{p}\right) _{m}}{m!\left( b_{1}\right) _{m}\cdots \left(
b_{q}\right) _{m}}z^{m}. \label{pFq_def}$$
If any $a_{j}$, $j=1\ldots p$ is a negative integer, then the series ([pFq\_def]{}) terminates, thus it converges. If (\[pFq\_def\]) is not a terminating series, then it converges $\forall \left\vert z\right\vert
<\infty $, if $p\leq q$; and $\forall \left\vert z\right\vert <1$, if $p=q+1$. Also, (\[pFq\_def\]) diverges $\forall z\neq 0$, if $p>q+1$. If $p=q+1$ and $\left\vert z\right\vert =1$, then the series (\[pFq\_def\]) is absolutely convergent when $\left(
\sum_{j=1}^{q}b_{j}-\sum_{j=1}^{p}a_{j}\right) >0$; conditionally convergent when $z\neq 1$ and $-1<\,$$\left(
\sum_{j=1}^{q}b_{j}-\sum_{j=1}^{p}a_{j}\right) <0$; and divergent when $\,$$\left( \sum_{j=1}^{q}b_{j}-\sum_{j=1}^{p}a_{j}\right) \leq -1$.
If $k\in
\mathbb{N}
$, then $$\begin{aligned}
G_{k}\left( a,b,c,d\right) &=&G_{k-1}\left( a,b,c,d\right)
\label{Recursive_Miller} \\
&&+\frac{ab}{c\left( d+1\right) }G_{k-1}\left( a+1,b+1,c+1,d+1\right) ,
\notag\end{aligned}$$where$$G_{k}\left( a,b,c,d\right) =\,_{3}F_{2}\left( \left.
\begin{array}{c}
a,b,c+k+1 \\
d+1,c\end{array}\right\vert 1\right) , \label{G_k_Miller_def}$$and where, renaming parameters in [@Miller], we have$$G_{0}\left( a,b,c,d\right) =\frac{\Gamma \left( d+1\right) \Gamma \left(
d-a-b\right) }{c\,\Gamma \left( d-a+1\right) \Gamma \left( d-b+1\right) }\left[ a\left( b-c\right) +c\left( d-b\right) \right] . \label{G0_Miller}$$
Consider the contiguous relation [@Andrews Eqn. 3.7.9], exchanging the parameters $\alpha \longleftrightarrow \gamma $, $$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma +1 \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \label{Miller_1} \\
&=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) +\frac{\alpha \beta }{\delta \varepsilon }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) . \notag\end{aligned}$$and perform the substitutions $\alpha =a$, $\beta =b$, $\gamma =c+k+1$, $\delta =d+1$ and $\varepsilon =c$, to obtain the recursive formula ([Recursive\_Miller]{}).
Notice that $G_{0}\left( a,b,c,c\right) $ reduces to Gauss’s summation formula [@DLMF Eqn. 15.4.20].
From the above recursive equation (\[Recursive\_Miller\]), we obtain the following identity.
If $k=0,1,2\ldots $, then$$\begin{aligned}
&&\,_{3}F_{2}\left( \left.
\begin{array}{c}
a,b,c+k+1 \\
d+1,c\end{array}\right\vert 1\right) \label{Miller_relation} \\
&=&\frac{\left( -1\right) ^{k}\Gamma \left( d+1\right) \Gamma \left(
d-a-b-k\right) \left( a+b-d+1\right) _{k}}{c\,\Gamma \left( d-a+1\right)
\Gamma \left( d-b+1\right) } \notag \\
&&\times \left\{ \left[ a\left( b-c\right) +c\left( d-b\right) \right]
\,_{3}F_{2}\left( \left.
\begin{array}{c}
-k,a,b \\
c+1,a+b-d+1\end{array}\right\vert 1\right) \right. \notag \\
&&+\left. \frac{ab\left( c-d\right) k}{\left( c+1\right) \left(
a+b-d+1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
1-k,a+1,b+1 \\
c+2,a+b-d+2\end{array}\right\vert 1\right) \right\} , \notag\end{aligned}$$where notice that on the RHS of (\[Miller\_relation\]) the hypergeometric sums are finite sums.
With the aid of computer algebra, we can iterate the first terms of the recursive equation (\[Recursive\_Miller\]), starting from (\[G0\_Miller\]), obtaining:$$\begin{aligned}
&&G_{1}\left( a,b,c,d\right) \\
&=&-\frac{\Gamma \left( d+1\right) \Gamma \left( d-a-b-1\right) }{c\,\Gamma
\left( d-a+1\right) \Gamma \left( d-b+1\right) } \\
&&\left\{ \left( a+b-d-1\right) \left[ a\left( b-c\right) +c\left(
d-b\right) \right]
\begin{array}{c}
\displaystyle
\\
\displaystyle\end{array}\right. \\
&&\left.
\begin{array}{c}
\displaystyle
\\
\displaystyle\end{array}-\frac{ab\left[ a\left( b-c\right) -\left( b+1\right) c+\left( c+1\right) d\right] }{c+1}\right\} , \\
&&G_{2}\left( a,b,c,d\right) \\
&=&\frac{\Gamma \left( d+1\right) \Gamma \left( d-a-b-2\right) }{c\,\Gamma
\left( d-a+1\right) \Gamma \left( d-b+1\right) } \\
&&\left\{ \left( a+b-d-1\right) \left( a+b-d-2\right) \left[ a\left(
b-c\right) +c\left( d-b\right) \right]
\begin{array}{c}
\displaystyle
\\
\displaystyle\end{array}\right. \\
&&-\frac{2ab\left( a+b-d-2\right) \left[ a\left( b-c\right) -\left(
b+1\right) c+\left( c+1\right) d\right] }{c+1} \\
&&\left.
\begin{array}{c}
\displaystyle
\\
\displaystyle\end{array}+\frac{a\left( a+1\right) b\left( b+1\right) \left[ a\left( b-c\right)
-\left( b+2\right) c+\left( c+2\right) d\right] }{\left( c+1\right) \left(
c+2\right) }\right\} .\end{aligned}$$Therefore, we can conjecture the following general form,$$\begin{aligned}
&&G_{k}\left( a,b,c,d\right) \label{G_k_Sum} \\
&=&\frac{\left( -1\right) ^{k}\Gamma \left( d+1\right) \Gamma \left(
d-a-b-k\right) }{c\,\Gamma \left( d-a+1\right) \Gamma \left( d-b+1\right) }
\notag \\
&&\sum_{j=0}^{k}\frac{\left( -1\right) ^{j}\left( a+b-d+j+1\right)
_{k-j}\left( a\right) _{j}\left( b\right) _{j}}{\left( c+1\right) _{j}}\binom{k}{j} \notag \\
&&\quad \times \left[ a\left( b-c\right) -\left( b+j\right) c+\left(
c+j\right) d\right] , \notag\end{aligned}$$which can be proved by induction. Finally, split the sum given in ([G\_k\_Sum]{}) in two sums and recast them as hypergeometric sums, to obtain,$$\begin{aligned}
&&G_{k}\left( a,b,c,d\right) \label{G_k_resultado} \\
&=&\frac{\left( -1\right) ^{k}\Gamma \left( d+1\right) \Gamma \left(
d-a-b-k\right) \Gamma \left( a+b-d+k+1\right) }{c\,\Gamma \left(
d-a+1\right) \Gamma \left( d-b+1\right) \Gamma \left( a+b-d+1\right) }
\notag \\
&&\left\{ \left[ a\left( b-c\right) +c\left( d-c\right) \right]
\,_{3}F_{2}\left( \left.
\begin{array}{c}
-k,a,b \\
c+1,a+b-d+1\end{array}\right\vert 1\right) \right. \notag \\
&&+\left. \frac{ab\left( c-d\right) k}{\left( c+1\right) \left(
a+b-d+1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
1-k,a+1,b+1 \\
c+2,a+b-d+2\end{array}\right\vert 1\right) \right\} . \notag\end{aligned}$$Finally, match (\[G\_k\_resultado\]) to (\[G\_k\_Miller\_def\]), to obtain (\[Miller\_relation\]).
Next, we consider an extension of Pfaff-Saalschutz summation formula.
If $k,n\in
\mathbb{N}
$, then $$\begin{aligned}
&&G_{k}\left( n,a,b,c\right) =G_{k-1}\left( n,a+1,b,c+1\right)
\label{Recursive_Pfaff} \\
&&+\frac{nb}{\left( c+k\right) \left( a+b+1-n-c\right) }G_{k-1}\left(
n-1,a+1,b+1,c+2\right) , \notag\end{aligned}$$where$$G_{k}\left( n,a,b,c\right) =\,_{3}F_{2}\left( \left.
\begin{array}{c}
-n,a,b \\
c+k,a+b+1-n-c\end{array}\right\vert 1\right) ,$$and, according to Pfaff-Saalschutz balanced sum [@DLMF Eqn. 16.4.3],$$G_{0}\left( n,a,b,c\right) =\frac{\left( c-a\right) _{n}\left( c-b\right)
_{n}}{\left( c\right) _{n}\left( c-a-b\right) _{n}}. \label{Pfaff_sum}$$
Consider again the contiguous relation given in [@Andrews Eqn. 3.7.9], exchanging the parameters $\alpha \longleftrightarrow \beta $, thus $$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta +1,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \\
&=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) +\frac{\alpha \gamma }{\delta \varepsilon }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) .\end{aligned}$$Now, performing the substitutions $\alpha =-n$, $\beta =a$, $\gamma =b$, $\delta =c+k$, and $\varepsilon =a+b+1-n-c$, we arrive at the recursive equation (\[Recursive\_Pfaff\]).
Again, iterating the recursive equation (\[Recursive\_Pfaff\]), we can generalize Pfaff-Saalschutz summation formula as follows:
If $n\in
\mathbb{N}
$ and $k=0,1,2,\ldots $, then$$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
-n,a,b \\
c+k,a+b+1-n-c\end{array}\right\vert 1\right) \label{Identity_Pfaff} \\
&=&\frac{\left( c-a\right) _{n}\left( c-b+k\right) _{n}}{\left( c+k\right)
_{n}\left( c-a-b\right) _{n}}\,_{3}F_{2}\left( \left.
\begin{array}{c}
-k,-n,b \\
c-a,b-c-k-n+1\end{array}\right\vert 1\right) , \notag\end{aligned}$$where $k=0$ matches Pfaff-Saalschutz summation formula (\[Pfaff\_sum\]).
Computing the first iterations of (\[Recursive\_Pfaff\]), starting from (\[Pfaff\_sum\]), we can conjecture that$$\begin{aligned}
&&G_{k}\left( n,a,b,c\right) \notag \\
&=&\frac{\Gamma \left( n-a-c\right) }{\Gamma \left( c-b+k\right) \left(
c+k\right) _{n}\left( c-a-b\right) _{n}} \notag \\
&&\sum_{j=0}^{k}\left( -1\right) ^{j}\binom{k}{j}\frac{\left( b\right)
_{j}\left( n-j+1\right) _{j}\Gamma \left( c-b+n+k-j\right) }{\Gamma \left(
c-a+j\right) }, \label{G_K_Pfaff_sum}\end{aligned}$$which can be proven by induction. Finally, the sum given in ([G\_K\_Pfaff\_sum]{}) can be rewritten as a terminating hypergeometric sum, obtaining (\[Identity\_Pfaff\]).
If $k\in
\mathbb{N}
$, we have the recursive formula$$\begin{aligned}
&&G_{k+1}\left( a,b,c\right) =\frac{a\left[ a-1-2\left( b+c+k\right) \right]
}{k\left( b+c+k\right) }G_{k}\left( a+1,b+1,c+1\right)
\label{Recursive_Dixon} \\
&&-\frac{\left( a-b\right) \left( a-c\right) }{k\left( b+c+k\right) }G_{k}\left( a-1,b,c\right) , \notag\end{aligned}$$where$$G_{k}\left( a,b,c\right) =\,_{3}F_{2}\left( \left.
\begin{array}{c}
a,b+k,c+k \\
a-b+1,a-c+1\end{array}\right\vert 1\right) ,$$and where, taking $i=1$ and $j=0$ in [@Lavoie], we have$$\begin{aligned}
G_{1}\left( a,b,c\right) &=&\frac{\Gamma \left( 1+a-b\right) \Gamma \left(
1+a-c\right) }{2^{2c+1}bc\Gamma \left( a-2c\right) \Gamma \left(
a-b-c\right) } \\
&&\left[ \frac{\Gamma \left( \frac{a+1}{2}-c\right) \Gamma \left( \frac{a}{2}-b-c\right) }{\Gamma \left( \frac{a+1}{2}\right) \Gamma \left( \frac{a}{2}-b\right) }-\frac{\Gamma \left( \frac{a+1}{2}-b-c\right) \Gamma \left( \frac{a}{2}-c\right) }{\Gamma \left( \frac{a}{2}\right) \Gamma \left( \frac{a+1}{2}-b\right) }\right] .\end{aligned}$$
Consider the contiguous relation [@Andrews Eqn. 3.7.12], performing the substitution $\alpha \rightarrow \alpha +1$,$$\begin{aligned}
&&\delta \varepsilon \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \label{Dixon_1} \\
&=&\left( \alpha +1\right) \left( \delta +\varepsilon -\alpha -\beta -\gamma
-2\right) \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +2,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) \notag \\
&&+\left( \delta -\alpha -1\right) \left( \varepsilon -\alpha -1\right)
\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) , \notag\end{aligned}$$and also the contiguous relation [@Andrews Eqn. 3.7.9] $$\begin{aligned}
_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) &=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \label{Dixon_2} \\
&&+\frac{\beta \gamma }{\delta \varepsilon }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) . \notag\end{aligned}$$Thereby, eliminating the hypergeometric sum of the RHS of (\[Dixon\_1\])and (\[Dixon\_2\]), we arrive at$$\begin{aligned}
&&\delta \varepsilon \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \label{Dixon_3} \\
&&-\left( \alpha +1\right) \left( \delta +\varepsilon -\alpha -\beta -\gamma
-2\right) \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +2,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) \notag \\
&=&\left[ \left( \delta -\alpha -1\right) \left( \varepsilon -\alpha
-1\right) -\beta \gamma \right] \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) . \notag\end{aligned}$$Finally, substituting in (\[Dixon\_3\]) $\alpha =a$, $\beta =b+k$, $\gamma
=c+k$, $\delta =a-b+1$ and $\varepsilon =a-c+1$, we arrive at the recursive equation (\[Recursive\_Dixon\]).
Notice that $\forall k=0$, $G_{0}\left( a,b,c\right) $ reduces to Dixon’s theorem [@DLMF Eqn. 16.4.4], but in this case the recursive equation (\[Recursive\_Dixon\]) collapses.
The next two recursive formulas extend Watson’s sum [@DLMF Eqn. 16.4.6]. In [@LavoieWatson], we can find two other extensions of Watson’s sum by using contiguous relations. Next, we extend one of the results given in [@LavoieWatson], being the latter the particular case $k=1$ of the next theorem.
If $k\in
\mathbb{N}
$, then the following recursive equation holds true:$$\begin{aligned}
&&G_{k}\left( a,b,c\right) \label{Recursive_Watson_Lavoie} \\
&=&G_{k-1}\left( a,b,c\right) -\frac{2abc\ G_{k-1}\left( a+1,b+1,c+1\right)
}{\left( a+b+1\right) \left( 2c+k-1\right) \left( 2c+k\right) }, \notag\end{aligned}$$where$$G_{k}\left( a,b,c\right) =\,_{3}F_{2}\left( \left.
\begin{array}{c}
a,b,c \\
\frac{a+b+1}{2},2c+k\end{array}\right\vert 1\right) ,$$and, according to Watson’s sum [@DLMF Eqn. 16.4.6], we have$$G_{0}\left( a,b,c\right) =\frac{\sqrt{\pi }\Gamma \left( c+\frac{1}{2}\right) \Gamma \left( \frac{a+b+1}{2}\right) \Gamma \left( \frac{1-a-b}{2}+c\right) }{\Gamma \left( \frac{a+1}{2}\right) \Gamma \left( \frac{b+1}{2}\right) \Gamma \left( \frac{1-a}{2}+c\right) \Gamma \left( \frac{1-b}{2}+c\right) }. \label{Watson_sum}$$
Exchanging $\beta \longleftrightarrow \gamma $ and $\delta
\longleftrightarrow \varepsilon $ in the recursive relation given in [@Andrews Sect. 3.7], we have$$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma +1 \\
\delta ,\varepsilon +1\end{array}\right\vert 1\right) \\
&=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) +\frac{\alpha \beta \left( \varepsilon -\gamma \right)
}{\delta \varepsilon \left( \varepsilon +1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +2\end{array}\right\vert 1\right) .\end{aligned}$$Also, performing the change $\varepsilon \rightarrow \varepsilon +1$ in ([Miller\_1]{}), $$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma +1 \\
\delta ,\varepsilon +1\end{array}\right\vert 1\right) \\
&=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon +1\end{array}\right\vert 1\right) +\frac{\alpha \beta }{\delta \left( \varepsilon
+1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +2\end{array}\right\vert 1\right) .\end{aligned}$$Therefore, equating the above equations, we arrive at $$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon +1\end{array}\right\vert 1\right) \\
&=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) -\frac{\alpha \beta \gamma }{\delta \left( \varepsilon
+1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +2\end{array}\right\vert 1\right) .\end{aligned}$$Now, substituting $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =\frac{a+b+1}{2}$, and $\varepsilon =2c+k$, we arrive at the recursive equation ([Recursive\_Watson\_Lavoie]{}).
If $k\in
\mathbb{N}
$, then we have the following recursive equation:$$\begin{aligned}
&&G_{k}\left( a,b,c\right) =G_{k-1}\left( a,b,c\right) +\frac{b}{a+b+1}
\label{Recursive_Watson} \\
&&\left[ \frac{\left( a+k\right) \left( b+1\right) }{\left( 2c+1\right)
\left( a+b+3\right) }G_{k-1}\left( a+2,b+2,c+1\right) +G_{k-1}\left(
a+1,b+1,c\right) \right] , \notag\end{aligned}$$where$$G_{k}\left( a,b,c\right) =\,_{3}F_{2}\left( \left.
\begin{array}{c}
a+k,b,c \\
\frac{a+b+1}{2},2c\end{array}\right\vert 1\right) ,$$and $G_{0}\left( a,b,c\right) $ is given by Watson’s sum (\[Watson\_sum\]).
Exchanging $\alpha \longleftrightarrow \beta $ and setting $\beta
\rightarrow \beta +1$, $\gamma \rightarrow \gamma +1$, and $\varepsilon
\rightarrow \varepsilon +1$ in the recursive relation given in [@Andrews Sect. 3.7], we have$$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) \label{Watson_1} \\
&=&\frac{\left( \beta +1\right) \left( \delta -\alpha \right) \left( \gamma
+1\right) }{\delta \left( \delta +1\right) \left( \varepsilon +1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +2,\gamma +2 \\
\delta +2,\varepsilon +2\end{array}\right\vert 1\right) \notag \\
&&+\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta +1,\gamma +1 \\
\delta ,\varepsilon +1\end{array}\right\vert 1\right) . \notag\end{aligned}$$Now, substituting the RHS of (\[Watson\_1\]) in (\[Miller\_1\]), we arrive at$$\begin{aligned}
&&_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta +1,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \,=\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) \\
&&\quad +\frac{\delta \varepsilon }{\alpha \gamma }\left[ \frac{\left( \beta
+1\right) \left( \delta -\alpha \right) \left( \gamma +1\right) }{\delta
\left( \delta +1\right) \left( \varepsilon +1\right) }\,_{3}F_{2}\left(
\left.
\begin{array}{c}
\alpha +1,\beta +2,\gamma +2 \\
\delta +2,\varepsilon +2\end{array}\right\vert 1\right) \right. \\
&&\quad +\left. \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta +1,\gamma +1 \\
\delta ,\varepsilon +1\end{array}\right\vert 1\right) \right] \,.\end{aligned}$$Finally, taking $\alpha =c$, $\beta =a+k$, $\gamma =b$, $\delta =2c$, and $\varepsilon =\frac{a+b+1}{2}$, we arrive at the recursive equation ([Recursive\_Watson]{}).
Finally, we provide and extension of a summation formula given by Bailey. By induction from the recursive formula found, a simple closed-form expression is derived.
If $k=0,1,2,\ldots $, then the following recursive equation is satisfied:$$\begin{aligned}
&&G_{k+1}\left( a,b,c\right) =\frac{\left( 2c-b+k\right) \left(
2c-b+k+1\right) }{\left( a-1\right) \left( 2c-2b+k+1\right) \left(
c-b+k\right) } \label{Recursive_Bailey} \\
&&\left[ \left( c-1\right) G_{k}\left( a-1,b-1,c-1\right) -\left( c-a\right)
G_{k}\left( a-1,b,c\right) \right] , \notag\end{aligned}$$where$$\begin{aligned}
&&G_{k}\left( a,b,c\right) \notag \\
&=&\,_{3}F_{2}\left( \left.
\begin{array}{c}
a,b,c+1 \\
1+2c-b+k,c\end{array}\right\vert 1\right) \notag \\
&=&\frac{\left[ \left( a-2c\right) \left( b-c\right) +kc\right] \Gamma
\left( 2c-b+k+1\right) \Gamma \left( 2c-a-2b+k\right) }{c\ \Gamma \left(
2c-2b+k+1\right) \Gamma \left( 2c-a-b+k+1\right) }.
\label{Gk_Bailey_resultado}\end{aligned}$$
Considering the contiguous relation [@Andrews Eqn. 3.7.14], we have$$\begin{aligned}
&&\varepsilon \,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha ,\beta ,\gamma \\
\delta ,\varepsilon\end{array}\right\vert 1\right) -\left( \varepsilon -\alpha \right) \,_{3}F_{2}\left(
\left.
\begin{array}{c}
\alpha ,\beta +1,\gamma +1 \\
\delta +1,\varepsilon +1\end{array}\right\vert 1\right) \\
&=&\frac{\alpha \left( \delta -\beta \right) \left( \delta -\gamma \right) }{\delta \left( \delta +1\right) }\,_{3}F_{2}\left( \left.
\begin{array}{c}
\alpha +1,\beta +1,\gamma +1 \\
\delta +2,\varepsilon +1\end{array}\right\vert 1\right) ,\end{aligned}$$thus, setting $\alpha =a-1$, $\beta =b-1$, $\gamma =c$, $\delta =2c-b+k$, and $\varepsilon =c-1$, we arrive at the recursive equation ([Recursive\_Bailey]{}). Also, according to [@Bailey Eqn. 6.4(2)], after renaming the parameters, we have$$G_{0}\left( a,b,c\right) =\left( 1-\frac{a}{2c}\right) \frac{\Gamma \left(
2c-b+1\right) \Gamma \left( 2c-a-2b\right) }{\Gamma \left( 2c-2b\right)
\Gamma \left( 2c-a-b+1\right) }, \label{G0_Bailey}$$hence, recursive substitution of (\[G0\_Bailey\]) in ([Recursive\_Bailey]{}), after simplification, yields$$G_{1}\left( a,b,c\right) =\frac{\left[ \left( a-2c\right) \left( b-c\right)
+c\right] \Gamma \left( 2c-b+2\right) \Gamma \left( 2c-a-2b+1\right) }{c\
\Gamma \left( 2c-2b+2\right) \Gamma \left( 2c-a-b+2\right) },$$and$$G_{2}\left( a,b,c\right) =\frac{\left[ \left( a-2c\right) \left( b-c\right)
+2c\right] \Gamma \left( 2c-b+3\right) \Gamma \left( 2c-a-2b+2\right) }{c\
\Gamma \left( 2c-2b+3\right) \Gamma \left( 2c-a-b+3\right) }.$$Therefore, we may conjecture the general form given in ([Gk\_Bailey\_resultado]{}), which can be proved easily by induction using the recursive equation (\[Recursive\_Bailey\]).
Conclusions\[Section: Conclusions\]
===================================
We have obtained some recursive formulas to extend some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations. These recursive equations are quite suitable for symbolic and numerical evaluation by means of computer algebra. Moreover, in some cases, namely ([Identity\_2F1]{}), (\[Miller\_relation\]), (\[Identity\_Pfaff\]), and ([Gk\_Bailey\_resultado]{}), we have derived closed-form expressions. Also, as by-product, we have obtained an interesting identity in (\[Identity\_Choi\]). It is expected that the method used to obtain the different recursive equations can be applied to extend other hypergeometric summation formulas given in the literature.
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Moch, S., Uwer, P., Weinzierl, S.: Nested sums, expansion of transcendental functions, and multiscale multiloop integrals. J. Math. Phys. 43(6), 3363-3386 (2002)
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Koornwinder, T.H.: Identities of nonterminating series by Zeilberger’s algorithm. J. Comput. Appl. Math. 99(1-2), 449-461 (1998)
Lavoie, J.L., Grondin, F., Rathie A.K.: Generalizations of Whipple’s theorem on the sum of a 3F2. J. Comput. Appl. Math. 72(2), 293-300 (1996)
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|
---
abstract: 'This paper investigates the effects of data size and frequency range on distributional semantic models. We compare the performance of a number of representative models for several test settings over data of varying sizes, and over test items of various frequency. Our results show that neural network-based models underperform when the data is small, and that the most reliable model over data of varying sizes and frequency ranges is the inverted factorized model.'
author:
- |
Magnus Sahlgren\
Gavagai and SICS\
Slussplan 9, Box 1263\
111 30 Stockholm, 164 29 Kista\
Sweden\
[mange@\[gavagai|sics\].se]{}\
Alessandro Lenci\
University of Pisa\
via Santa Maria 36\
56126 Pisa\
Italy\
[[email protected]]{}\
bibliography:
- 'main.bib'
title: |
The Effects of Data Size and Frequency Range\
on Distributional Semantic Models
---
Introduction
============
Distributional Semantic Models (DSMs) have become a staple in natural language processing. The various parameters of DSMs — e.g. size of context windows, weighting schemes, dimensionality reduction techniques, and similarity measures — have been thoroughly studied [@Weeds:2004; @Sahlgren:2006; @Riordan:2011; @Bullinaria:2012; @Levy:2015], and are now well understood. The impact of various processing models — matrix-based models, neural networks, and hashing methods — have also enjoyed considerable attention lately, with at times conflicting conclusions [@Baroni:2014; @Levy:2015; @Schnabel:2015; @Osterlund:2015; @Sahlgren:2016]. The consensus interpretation of such experiments seems to be that the choice of processing model is less important than the parameterization of the models, since the various processing models all result in more or less equivalent DSMs (provided that the parameterization is comparable).
One of the least researched aspects of DSMs is the effect on the various models of data size and frequency range of the target items. The only previous work in this direction that we are aware of is Asr et al. , who report that on small data (the CHILDES corpus), simple matrix-based models outperform neural network-based ones. Unfortunately, Asr et al. do not include any experiments using the same models applied to bigger data, making it difficult to compare their results with previous studies, since implementational details and parameterization will be different.
There is thus still a need for a consistent and fair comparison of the performance of various DSMs when applied to data of varying sizes. In this paper, we seek an answer to the question: [**which DSM should we opt for if we only have access to limited amounts of data?**]{} We are also interested in the related question: [**which DSM should we opt for if our target items are infrequent?**]{} The latter question is particularly crucial, since one of the major assets of DSMs is their applicability to create semantic representations for ever-expanding vocabularies from text feeds, in which new words may continuously appear in the low-frequency ranges.
In the next section, we introduce the contending DSMs and the general experiment setup, before turning to the experiments and our interpretation of the results. We conclude with some general advice.
Distributional Semantic Models
==============================
One could classify DSMs in many different ways, such as the type of context and the method to build distributional vectors. Since our main goal here is to gain an understanding of the effect of data size and frequency range on the various models, we focus primarily on the differences in processing models, hence the following typology of DSMs.
### Explicit matrix models {#explicit-matrix-models .unnumbered}
We here include what could be referred to as *explicit* models, in which each vector dimension corresponds to a specific context [@Levy:Goldberg:2014]. The baseline model is a simple co-occurrence matrix $F$ (in the following referred to as <span style="font-variant:small-caps;">co</span> for Co-Occurrence). We also include the model that results from applying Positive Pointwise Mutual Information (<span style="font-variant:small-caps;">ppmi</span>) to the co-occurrence matrix. <span style="font-variant:small-caps;">ppmi</span> is defined as simply discarding any negative values of the PMI, computed as:
$$\label{eq:ppmi}
\textrm{PMI}(a,b) = \log \frac{f_{ab} \times T}{f_a f_b}$$
where $f_{ab}$ is the co-occurrence count of word $a$ and word $b$, $f_a$ and $f_b$ are the individual frequencies of the words, and $T$ is the number of tokens in the data.[^1]
### Factorized matrix models {#factorized-matrix-models .unnumbered}
This type of model applies an additional factorization of the weighted co-occurrence counts. We here include two variants of applying Singular Value Decomposition (SVD) to the <span style="font-variant:small-caps;">ppmi</span>-weighting co-occurrence matrix; one version that discards all but the first couple of hundred latent dimensions (<span style="font-variant:small-caps;">tsvd</span> for [*truncated*]{} SVD), and one version that instead [*removes*]{} the first couple of hundred latent dimensions (<span style="font-variant:small-caps;">isvd</span> for [*inverted*]{} SVD). SVD is defined in the standard way:
$$\label{eq:svd}
F = U\Sigma V^{T}$$
where $U$ holds the eigenvectors of $F$, $\Sigma$ holds the eigenvalues, and $V \in U(w)$ is a unitary matrix mapping the original basis of $F$ into its eigenbasis. Since $V$ is redundant due to invariance under unitary transformations, we can represent the factorization of $\hat{F}$ in its most compact form $\hat{F} \equiv U\Sigma$.
### Hashing models {#hashing-models .unnumbered}
A different approach to reduce the dimensionality of DSMs is to use a hashing method such as Random Indexing (<span style="font-variant:small-caps;">ri</span>) [@Kanerva:2000], which accumulates distributional vectors $\vec{d}(a)$ in an online fashion:
$$\label{eq:ri}
\vec{d}(a) \leftarrow \vec{d}(a_{i}) + \sum_{j=-c,j\neq 0}^{c} w(x^{(i+j)})\pi^{j}\vec{r}(x^{(i+j)})$$
where $c$ is the extension of the context window, $w(b)$ is a weight that quantifies the importance of context term $b$,[^2] $\vec{r}_{d}(b)$ is a [*sparse random index vector*]{} that acts as a fingerprint of context term $b$, and $\pi^{j}$ is a permutation that rotates the random index vectors one step to the left or right, depending on the position of the context items within the context windows, thus enabling the model to take word order into account [@Sahlgren:2008].
### Neural network models {#neural-network-models .unnumbered}
There are many variations of DSMs that use neural networks as processing model, ranging from simple recurrent networks [@Elman:1990] to more complex deep architectures [@Collobert:2008]. The incomparably most popular neural network model is the one implemented in the `word2vec` library, which uses the softmax for predicting $b$ given $a$ [@Mikolov:2013]:
$$\label{eq:sgns}
p(b|a) = \frac{\textrm{exp}(\vec{b}\cdot\vec{a})}{\sum_{b'\in C} \textrm{exp}(\vec{b'}\cdot\vec{a})}$$
where $C$ is the set of context words, and $\vec{b}$ and $\vec{a}$ are the vector representations for the context and target words, respectively. We include two versions of this general model; Continuous Bag of Words (<span style="font-variant:small-caps;">cbow</span>) that predicts a word based on the context, and SkipGram Negative Sampling (<span style="font-variant:small-caps;">sgns</span>) that predicts the context based on the current word.
Experiment setup
================
Since our main focus in this paper is the performance of the above-mentioned DSMs on data of varying sizes, we use one big corpus as starting point, and split the data into bins of varying sizes. We opt for the ukWaC corpus [@Ferraresi:2008], which comprises some 1.6 billion words after tokenization and lemmatization. We produce sub-corpora by taking the first 1 million, 10 million, 100 million, and 1 billion words.
Since the co-occurrence matrix built from the 1 billion-word ukWaC sample is very big (more than 4,000,000 $\times$ 4,000,000), we prune the co-occurrence matrix to 50,000 dimensions before the factorization step by simply removing infrequent context items.[^3] As comparison, we use 200 dimensions for <span style="font-variant:small-caps;">tsvd</span>, 2,800 (3,000-200) dimensions for <span style="font-variant:small-caps;">isvd</span>, 2,000 dimensions for <span style="font-variant:small-caps;">ri</span>, and 200 dimensions for <span style="font-variant:small-caps;">cbow</span> and <span style="font-variant:small-caps;">sgns</span>. These dimensionalities have been reported to perform well for the respective models [@Landauer:1997; @Sahlgren:2008; @Mikolov:2013; @Osterlund:2015]. All DSMs use the same parameters as far as possible with a narrow context window of $\pm 2$ words, which has been shown to produce good results in semantic tasks [@Sahlgren:2006; @Bullinaria:2012].
We use five standard benchmark tests in these experiments; two multiple-choice vocabulary tests (the TOEFL synonyms and the ESL synonyms), and three similarity/relatedness rating benchmarks (SimLex-999 (SL) [@Hill:2015], MEN [@Bruni:2014], and Stanford Rare Words (RW) [@Luong:2013]). The vocabulary tests measure the synonym relation, while the similarity rating tests measure a broader notion of semantic similarity (SL and RW) or relatedness (MEN).[^4] The results for the vocabulary tests are given in accuracy (i.e., percentage of correct answers), while the results for the similarity tests are given in Spearman rank correlation.
[|l|rr|rrr|]{} & [**TOEFL**]{} & [**ESL**]{} & [**SL**]{} & [**MEN**]{} & [**RW**]{}\
\
<span style="font-variant:small-caps;">co</span> & 17.50 & [**20.00**]{} & $-$1.64 & 10.72 & $-$3.96\
<span style="font-variant:small-caps;">ppmi</span> & 26.25 & 18.00 & 8.28 & 21.49 & $-$2.57\
<span style="font-variant:small-caps;">tsvd</span> & [**27.50**]{} & [**20.00**]{} & 4.43 & [**22.15**]{} & $-$1.56\
<span style="font-variant:small-caps;">isvd</span> & 22.50 & 14.00 & [**14.33**]{} & 19.74 & [**5.31**]{}\
<span style="font-variant:small-caps;">ri</span> & 20.00 & 16.00 & 5.65 & 17.94 & 1.92\
<span style="font-variant:small-caps;">sgns</span> & 15.00 & 8.00 & 3.64 & 12.34 & 1.46\
<span style="font-variant:small-caps;">cbow</span> & 15.00 & 10.00 & $-$0.16 & 11.59 & 1.39\
\
<span style="font-variant:small-caps;">co</span> & 40.00 & 22.00 & 4.77 & 15.20 & 0.95\
<span style="font-variant:small-caps;">ppmi</span> & 52.50 & 38.00 & 26.44 & 39.83 & 4.00\
<span style="font-variant:small-caps;">tsvd</span> & 38.75 & 30.00 & 19.27 & 34.33 & 5.53\
<span style="font-variant:small-caps;">isvd</span> & 45.00 & [**44.00**]{} & [**30.19**]{} & [**44.21**]{} & [**9.88**]{}\
<span style="font-variant:small-caps;">ri</span> & [**47.50**]{} & 24.00 & 20.44 & 34.56 & 3.32\
<span style="font-variant:small-caps;">sgns</span> & 43.75 & 42.00 & 28.30 & 26.59 & 2.38\
<span style="font-variant:small-caps;">cbow</span> & 40.00 & 30.00 & 22.22 & 28.33 & 3.04\
\
<span style="font-variant:small-caps;">co</span> & 45.00 & 30.00 & 10.00 & 19.36 & 3.12\
<span style="font-variant:small-caps;">ppmi</span> & [**66.25**]{} & 54.00 & 33.75 & 46.74 & 15.05\
<span style="font-variant:small-caps;">tsvd</span> & 46.25 & 34.00 & 25.11 & 42.49 & 13.00\
<span style="font-variant:small-caps;">isvd</span> & 66.25 & [**66.00**]{} & [**40.98**]{} & [**54.55**]{} & [**21.27**]{}\
<span style="font-variant:small-caps;">ri</span> & 55.00 & 48.00 & 32.31 & 45.71 & 10.15\
<span style="font-variant:small-caps;">sgns</span> & 65.00 & 58.00 & 40.75 & 52.83 & 11.73\
<span style="font-variant:small-caps;">cbow</span> & 61.25 & 46.00 & 36.15 & 48.30 & 15.62\
\
<span style="font-variant:small-caps;">co</span> & 55.00 & 40.00 & 11.85 & 21.83 & 6.82\
<span style="font-variant:small-caps;">ppmi</span> & 71.25 & 54.00 & 35.69 & 52.95 & 24.29\
<span style="font-variant:small-caps;">tsvd</span> & 56.25 & 46.00 & 31.36 & 52.05 & 13.35\
<span style="font-variant:small-caps;">isvd</span> & 71.25 & [**66.00**]{} & [**44.77**]{} & 60.11 & [**28.46**]{}\
<span style="font-variant:small-caps;">ri</span> & 61.25 & 50.00 & 35.35 & 50.51 & 18.58\
<span style="font-variant:small-caps;">sgns</span> & [**76.25**]{} & [**66.00**]{} & 41.94 & [**67.03**]{} & 24.50\
<span style="font-variant:small-caps;">cbow</span> & 75.00 & 56.00 & 38.31 & 59.84 & 22.80\
Comparison by data size
=======================
Table \[tab:size\] summarizes the results over the different test settings. The most notable aspect of these results is that the neural networks models do not produce competitive results for the smaller data, which corroborates the results by Asr et al. . The best results for the smallest data are produced by the factorized models, with both <span style="font-variant:small-caps;">tsvd</span> and <span style="font-variant:small-caps;">isvd</span> producing top scores in different test settings. It should be noted, however, that even the top scores for the smallest data set are substandard; only two models (<span style="font-variant:small-caps;">ppmi</span> and <span style="font-variant:small-caps;">tsvd</span>) manage to beat the random baseline of 25% for the TOEFL tests, and none of the models manage to beat the random baseline for the ESL test.
The <span style="font-variant:small-caps;">isvd</span> model produces consistently good results; it yields the best overall results for the 10 million and 100 million-word data, and is competitive with <span style="font-variant:small-caps;">sgns</span> on the 1 billion word data. Figure \[fig:sizes\] shows the average results and their standard deviations over all test settings.[^5] It is obvious that there are no huge differences between the various models, with the exception of the baseline <span style="font-variant:small-caps;">co</span> model, which consistently underperforms. The <span style="font-variant:small-caps;">tsvd</span> and <span style="font-variant:small-caps;">ri</span> models have comparable performance across the different data sizes, which is systematically lower than the <span style="font-variant:small-caps;">ppmi</span> model. The <span style="font-variant:small-caps;">isvd</span> model is the most consistently good model, with the neural network-based models steadily improving as data becomes bigger.
Looking at the different datasets, SL and RW are the hardest ones for all the models. In the case of SL, this confirms the results in [@Hill:2015], and might be due to the general bias of DSMs towards semantic relatedness, rather than genuine semantic similarity, as represented in SL. The substandard performance on RW might instead be due to the low frequency of the target items. It is interesting to note that these are benchmark tests in which neural models perform the worst even when trained on the largest data.
Comparison by frequency range
=============================
In order to investigate how each model handles different frequency ranges, we split the test items into three different classes that contain about a third of the frequency mass of the test items each. This split was produced by collecting all test items into a common vocabulary, and then sorting this vocabulary by its frequency in the ukWaC 1 billion-word corpus. We split the vocabulary into 3 equally large parts; the <span style="font-variant:small-caps;">HIGH</span> range with frequencies ranging from 3,515,086 (“do") to 16,830 (“organism"), the <span style="font-variant:small-caps;">MEDIUM</span> range with frequencies ranging between 16,795 (“desirable") and 729 (“prickly"), and the <span style="font-variant:small-caps;">LOW</span> range with frequencies ranging between 728 (“boardwalk") to hapax legomenon. We then split each individual test into these three ranges, depending on the frequencies of the test items. Test pairs were included in a given frequency class if and only if both the target and its relatum occur in the frequency range for that class. For the constituent words in the test item that belong to different frequency ranges, which is the most common case, we use a separate <span style="font-variant:small-caps;">MIXED</span> class. The resulting four classes contain 1,387 items for the <span style="font-variant:small-caps;">HIGH</span> range, 656 items for the <span style="font-variant:small-caps;">MEDIUM</span> range, 350 items for the <span style="font-variant:small-caps;">LOW</span> range, and 3,458 items for the <span style="font-variant:small-caps;">MIXED</span> range.[^6]
[**DSM**]{} [**<span style="font-variant:small-caps;">HIGH</span>**]{} [**<span style="font-variant:small-caps;">MEDIUM</span>**]{} [**<span style="font-variant:small-caps;">LOW</span>**]{} [**<span style="font-variant:small-caps;">MIXED</span>**]{}
---------------------------------------------------- ------------------------------------------------------------ -------------------------------------------------------------- ----------------------------------------------------------- -------------------------------------------------------------
<span style="font-variant:small-caps;">co</span> 32.61 ($\uparrow$62.5,$\downarrow$04.6) 35.77 ($\uparrow$66.6,$\downarrow$21.2) 12.57 ($\uparrow$35.7,$\downarrow$00.0) 27.14 ($\uparrow$56.6,$\downarrow$07.9)
<span style="font-variant:small-caps;">ppmi</span> 55.51 ($\uparrow$75.3,$\downarrow$28.0) 57.83 ($\uparrow$88.8,$\downarrow$18.7) 25.84 ($\uparrow$50.0,$\downarrow$00.0) 47.73 ($\uparrow$83.3,$\downarrow$27.1)
<span style="font-variant:small-caps;">tsvd</span> 50.52 ($\uparrow$70.9,$\downarrow$23.2) 54.75 ($\uparrow$77.9,$\downarrow$24.1) 17.85 ($\uparrow$50.0,$\downarrow$00.0) 41.08 ($\uparrow$56.6,$\downarrow$19.6)
<span style="font-variant:small-caps;">isvd</span> 63.31 ($\uparrow$87.5,$\downarrow$36.5) [**69.25**]{} ($\uparrow$88.8,$\downarrow$46.3) 10.94 ($\uparrow$16.0,$\downarrow$00.0) [**57.24**]{} ($\uparrow$83.3,$\downarrow$33.0)
<span style="font-variant:small-caps;">ri</span> 53.11 ($\uparrow$62.5,$\downarrow$30.1) 48.02 ($\uparrow$72.2,$\downarrow$20.4) 23.29 ($\uparrow$39.0,$\downarrow$00.0) 46.39 ($\uparrow$66.6,$\downarrow$21.0)
<span style="font-variant:small-caps;">sgns</span> [**68.81**]{} ($\uparrow$87.5,$\downarrow$36.4) 62.00 ($\uparrow$83.3,$\downarrow$27.4) 18.76 ($\uparrow$42.8,$\downarrow$00.0) 56.93 ($\uparrow$83.3,$\downarrow$30.2)
<span style="font-variant:small-caps;">cbow</span> 62.73 ($\uparrow$81.2,$\downarrow$31.9) 59.50 ($\uparrow$83.3,$\downarrow$32.4) [**27.13**]{} ($\uparrow$78.5,$\downarrow$00.0) 52.21 ($\uparrow$76.6,$\downarrow$25.9)
Table \[tab:freqs\] (next side) shows the average results over the different frequency ranges for the various DSMs trained on the 1 billion-word ukWaC data. We also include the highest and lowest individual test scores (signified by $\uparrow$ and $\downarrow$), in order to get an idea about the consistency of the results. As can be seen in the table, the most consistent model is <span style="font-variant:small-caps;">isvd</span>, which produces the best results in both the <span style="font-variant:small-caps;">MEDIUM</span> and <span style="font-variant:small-caps;">MIXED</span> frequency ranges. The neural network models <span style="font-variant:small-caps;">sgns</span> and <span style="font-variant:small-caps;">cbow</span> produce the best results in the <span style="font-variant:small-caps;">HIGH</span> and <span style="font-variant:small-caps;">LOW</span> range, respectively, with <span style="font-variant:small-caps;">cbow</span> clearly outperforming <span style="font-variant:small-caps;">sgns</span> in the latter case. The major difference between these models is that <span style="font-variant:small-caps;">cbow</span> predicts a word based on a context, while <span style="font-variant:small-caps;">sgns</span> predicts a context based on a word. Clearly, the former approach is more beneficial for low-frequent items.
The <span style="font-variant:small-caps;">ppmi</span>, <span style="font-variant:small-caps;">tsvd</span> and <span style="font-variant:small-caps;">ri</span> models perform similarly across the frequency ranges, with <span style="font-variant:small-caps;">ri</span> producing somewhat lower results in the <span style="font-variant:small-caps;">MEDIUM</span> range, and <span style="font-variant:small-caps;">tsvd</span> producing somewhat lower results in the <span style="font-variant:small-caps;">LOW</span> range. The <span style="font-variant:small-caps;">co</span> model underperforms in all frequency ranges. Worth noting is the fact that all models that are based on an explicit matrix (i.e. <span style="font-variant:small-caps;">co</span>, <span style="font-variant:small-caps;">ppmi</span>, <span style="font-variant:small-caps;">tsvd</span> and <span style="font-variant:small-caps;">isvd</span>) produce better results in the <span style="font-variant:small-caps;">MEDIUM</span> range than in the <span style="font-variant:small-caps;">HIGH</span> range.
The arguably most interesting results are in the <span style="font-variant:small-caps;">LOW</span> range. Unsurprisingly, there is a general and significant drop in performance for low frequency items, but with interesting differences among the various models. As already mentioned, the <span style="font-variant:small-caps;">cbow</span> model produces the best results, closely followed by <span style="font-variant:small-caps;">ppmi</span> and <span style="font-variant:small-caps;">ri</span>. It is noteworthy that the low-dimensional embeddings of the <span style="font-variant:small-caps;">cbow</span> model only gives a modest improvement over the high-dimensional explicit vectors of <span style="font-variant:small-caps;">ppmi</span>. The worst results are produced by the <span style="font-variant:small-caps;">isvd</span> model, which scores even lower than the baseline <span style="font-variant:small-caps;">co</span> model. This might be explained by the fact that <span style="font-variant:small-caps;">isvd</span> removes the latent dimensions with largest variance, which are arguably the most important dimensions for very low-frequent items. Increasing the number of latent dimensions with high variance in the <span style="font-variant:small-caps;">isvd</span> model improves the results in the <span style="font-variant:small-caps;">LOW</span> range (16.59 when removing only the top 100 dimensions).
Conclusion
==========
Our experiments confirm the results of Asr et al. , who show that neural network-based models are suboptimal to use for smaller amounts of data. On the other hand, our results also show that none of the standard DSMs work well in situations with small data. It might be an interesting novel research direction to investigate how to design DSMs that are applicable to small-data scenarios.
Our results demonstrate that the inverted factorized model (<span style="font-variant:small-caps;">isvd</span>) produces the most robust results over data of varying sizes, and across several different test settings. We interpret this finding as further corroborating the results of Bullinaria and Levy , and Österlund et al. , with the conclusion that the inverted factorized model is a robust competitive alternative to the widely used <span style="font-variant:small-caps;">sgns</span> and <span style="font-variant:small-caps;">cbow</span> neural network-based models.
We have also investigated the performance of the various models on test items in different frequency ranges, and our results in these experiments demonstrate that all tested models perform optimally in the medium-to-high frequency ranges. Interestingly, all models based on explicit count matrices (<span style="font-variant:small-caps;">co</span>, <span style="font-variant:small-caps;">ppmi</span>, <span style="font-variant:small-caps;">tsvd</span> and <span style="font-variant:small-caps;">isvd</span>) produce somewhat better results for items of medium frequency than for items of high frequency. The neural network-based models and <span style="font-variant:small-caps;">isvd</span>, on the other hand, produce the best results for high-frequent items.
None of the tested models perform optimally for low-frequent items. The best results for low-frequent test items in our experiments were produced using the <span style="font-variant:small-caps;">cbow</span> model, the <span style="font-variant:small-caps;">ppmi</span> model and the <span style="font-variant:small-caps;">ri</span> model, all of which uses weighted context items without any explicit factorization. By contrast, the <span style="font-variant:small-caps;">isvd</span> model underperforms significantly for the low-frequent items, which we suggest is an effect of removing latent dimensions with high variance.
This interpretation suggests that it might be interesting to investigate [*hybrid models*]{} that use different processing models — or at least different parameterizations — for different frequency ranges, and for different data sizes. We leave this as a suggestion for future research.
Acknowledgements
================
This research was supported by the Swedish Research Council under contract 2014-28199.
[^1]: We also experimented with [*smoothed*]{} <span style="font-variant:small-caps;">ppmi</span>, which raises the context counts to the power of $\alpha$ and normalizes them [@Levy:2015], thereby countering the tendency of mutual information to favor infrequent events: $f(b) = \frac{\#(b)^\alpha}{\sum_b \#(b)^\alpha}$, but it did not lead to any consistent improvements compared to PPMI.
[^2]: We use $w(b) = e^{-{\lambda \cdot \frac{f(b)}{V}}}$ where $f(b)$ is the frequency of context item $b$, $V$ is the total number of unique context items seen thus far (i.e. the current size of the growing vocabulary), and $\lambda$ is a constant that we set to 60 [@Sahlgren:2016].
[^3]: Such drastic reduction has a negative effect on the performance of the factorized methods for the 1 billion word data, but unfortunately is necessary for computational reasons.
[^4]: It is likely that the results on the similarity tests could be improved by using a wider context window, but such improvement would probably be consistent across all models, and is thus outside the scope of this paper.
[^5]: Although rank correlation is not directly comparable with accuracy, they are both bounded between zero and one, which means we can take the average to get an idea about overall performance.
[^6]: 233 test terms did not occur in the 1 billion-word corpus.
|
---
abstract: 'Due to Klein tunneling, electrostatic confinement of electrons in graphene is not possible. This hinders the use of graphene for quantum dot applications. Only through quasi-bound states with finite lifetime has one achieved to confine charge carriers. Here we propose that bilayer graphene with a local region of decoupled graphene layers is able to generate bound states under the application of an electrostatic gate. The discrete energy levels in such a quantum blister correspond to localized electron and hole states in the top and bottom layers. We find that this layer localization and the energy spectrum itself are tunable by a global electrostatic gate and that the latter also coincides with the electronic modes in a graphene disk. Curiously, states with energy close to the continuum exist primarily in the classically forbidden region outside the domain defining the blister. The results are robust against variations in size and shape of the blister which shows that it is a versatile system to achieve tunable electrostatic confinement in graphene.'
author:
- 'H. M. Abdullah'
- 'M. Van der Donck'
- 'H. Bahlouli'
- 'F. M. Peeters'
- 'B. Van Duppen'
title: 'Graphene quantum blisters: a tunable system to confine charge carriers'
---
Ever since the discovery of graphene, researchers have tried to confine electrons in graphene-based quantum dots (QDs)[@Guettinger2012] because of the vast range of new applications for QDs in for instance electronic circuitry[@Gueclue2013], photovoltaics[@Bacon2013], qubits[@Trauzettel2007], and gas sensing[@Sun2013]. Graphene as a basis for these QDs could enable fast and flexible devices. On a more fundamental level, the ultra-relativistic nature of graphene charge carriers made researchers wonder how they would respond to confinement[@Rozhkov_2011]. It is, however, exactly this peculiar property that prohibits the use of traditional QD fabrication techniques such as local electrostatic gating to confine carriers. The Klein tunnelling effect[@Katsnelson2006] allows electrons to use hole states in the gated region to escape the QD. The graphene quantum blister (GQB), proposed in this Article, overcomes this limitation and acts as a tunable graphene quantum dot that still harnesses the peculiar electronic properties of graphene.
The quest to confine Dirac Fermions in graphene QDs has resulted in many propositions. For instance, one has tried using magnetic fields[@Espinosa-Ortega2013; @Martino2007], cutting the flake into small nanostructures[@Mirzakhani2016; @Zebrowski2013] or using the substrate to induce a band gap[@Gutierrez-Rubio2015; @Recher2009]. However, magnetic fields bring along many difficulties in nano-sized systems [@Liu2017], QDs made from nanostructures are highly sensitive to the precise shape of the edge, which is hard to control [@Espinosa-Ortega2013], and also the band gap produced in graphene by a substrate is very difficult to control[@Decker2011]. Due to these difficulties experimental realization of graphene QDs is limited and this hinders applicability.
This has, however, not withheld researchers from trying to apply extreme external conditions. Under high magnetic field[@Jung2011] or supercritical charges[@Mao2016] confinement was realized, but only quasi-bound states with a relatively short life time[@Matulis2008] were observed. Recently, a few experiments[@Zhao2015; @Ghahari2017; @Gutierrez2016; @Lee2016; @Freitag2016] were conducted to detect short-lived quasi-bound states in single layer graphene by using advanced substrate engineering and the incorporation of an electrostatic potential induced by the tip of the scanning tunneling electron microscope (STM). There is only one recent experiment[@Qiao2017] that realized bound states with a longer lifetime in a QD in a graphene sheet through a strong coupling between the graphene sheet and the substrate. However, the bound states are only tunable through careful controlling of the distance between the sample and the STM tip.
![(Color online) (a) Schematic representation of a circular GQB with radius R. The inter-layer distance is shown in red. The black line corresponds to a local band gap for a global bias $\delta=0.12$ eV. The approximate band gap profile with an abrupt change at $\rho=R$ is shown in dashed blue. (b) Schematic representation of a cross section of the GQB depicting the position of the different atoms. The black lines are the $\pi$-orbitals, the vertical green lines represent the inter-layer coupling. For illustrative reasons, only a small number of atoms is shown. The GQBs in this study typically have radii of several hundreds of atoms. (c) Energy spectrum inside (left) and outside (right) the GQB. Red and blue bands correspond to top and bottom layers while the horizontal black lines in the left figure depict the discrete energy levels occuring due to confinement. These states are only allowed in the range $E<\left\vert \delta_{G} \right\vert$ as delimited by the yellow region in-between the solid black curves that correspond to the edge of the continuum outside the GQB.[]{data-label="fig-GQB"}](GQB_Sch_N2.pdf "fig:"){width="2.8"}\
![(Color online) (a) Schematic representation of a circular GQB with radius R. The inter-layer distance is shown in red. The black line corresponds to a local band gap for a global bias $\delta=0.12$ eV. The approximate band gap profile with an abrupt change at $\rho=R$ is shown in dashed blue. (b) Schematic representation of a cross section of the GQB depicting the position of the different atoms. The black lines are the $\pi$-orbitals, the vertical green lines represent the inter-layer coupling. For illustrative reasons, only a small number of atoms is shown. The GQBs in this study typically have radii of several hundreds of atoms. (c) Energy spectrum inside (left) and outside (right) the GQB. Red and blue bands correspond to top and bottom layers while the horizontal black lines in the left figure depict the discrete energy levels occuring due to confinement. These states are only allowed in the range $E<\left\vert \delta_{G} \right\vert$ as delimited by the yellow region in-between the solid black curves that correspond to the edge of the continuum outside the GQB.[]{data-label="fig-GQB"}](cross_section_GQB.pdf "fig:"){width="2.8"}\
![(Color online) (a) Schematic representation of a circular GQB with radius R. The inter-layer distance is shown in red. The black line corresponds to a local band gap for a global bias $\delta=0.12$ eV. The approximate band gap profile with an abrupt change at $\rho=R$ is shown in dashed blue. (b) Schematic representation of a cross section of the GQB depicting the position of the different atoms. The black lines are the $\pi$-orbitals, the vertical green lines represent the inter-layer coupling. For illustrative reasons, only a small number of atoms is shown. The GQBs in this study typically have radii of several hundreds of atoms. (c) Energy spectrum inside (left) and outside (right) the GQB. Red and blue bands correspond to top and bottom layers while the horizontal black lines in the left figure depict the discrete energy levels occuring due to confinement. These states are only allowed in the range $E<\left\vert \delta_{G} \right\vert$ as delimited by the yellow region in-between the solid black curves that correspond to the edge of the continuum outside the GQB.[]{data-label="fig-GQB"}](Bands.pdf "fig:"){width="\linewidth"}
Recently, delaminated bilayer graphene (BLG) attracted attention because of its possibility for layer selective transport [@Abdullah2017; @Abdullah_2016; @Lane2018; @Jaskolski2018]. These structures have also been experimentally observed in mechanically exfoliated graphene samples[@Yin2016]. GQBs are based on delaminated BLG, but here the delamination is concentrated in a circular region. By application of a global bias gate, states are trapped in this region but retain the interesting graphene-like characteristics. The proposed GQB supports bound states and overcomes the above mentioned limitations. It is free of magnetic fields, relatively possible to manufacture without losing graphene’s quality. Finally GQBs also allow external tunability of the electronic spectrum by application of a simple global gate[@Ohta_2006] and one can even control the layer localization of the confined states themselves.
A GQB correstponds to Bernal bilayer graphene where locally the upper layer is deformed, hence creating a blister in the top layer as shown in Fig. \[fig-GQB\]. Its electronic spectrum can be probed using STM[@Morgenstern2017], but in contrast to other experiments the electric field of the STM tip is not necessary to confine electrons[@Freitag2016; @Zhao2015]. As a result of the deformation, the inter-layer coupling strength $\gamma_{1}$ is strongly reduced and practically zero inside the blister. Therefore, the charge carriers have a degenerate linear energy spectrum inside the blister as they belong to independent layers. Outside the blister, however, the two layers are coupled in a Bernal bilayer structure and have the characteristics of a parabolic energy spectrum. By applying a global gate that induces a potential difference between top and bottom layer, a gap can be opened outside the GQB but inside the blister the linear energy spectrum of the two separate layers is only shifted up and down in energy, allowing states for energies in the bilayer gap. These states are bound in the GQB as shown in Fig. \[fig-GQB\](c). Since they cannot exist anywhere except in the GQB, the life time of the state diverges and we, therefore, have bound states.
In order to create the blister structure described in the previous paragraph, one could follow several routes. This first one uses the local separation of two graphene layers that is found in several samples [@Yan2016; @Schmitz2017a; @Clark_2014]. By applying a global gate to these of nanostructures, states will confine inside the blister. A second route follows a deliberate introduction atoms in-between two graphene layers with, for example, intercalation techniques[@P1988; @Kim2011a; @Wang2018]. A final route could consist of using graphene samples that are decorated with nanoclusters as a basis material during growth of bilayer graphene. It was shown that current techniques can precisely control over the size and content of these clusters[@Scheerder2017]. It is, therefore, expected that by following this technique, we could also precisely control the radius of the GQBs that are made in this way. Creating GQBs as such remains an open quest, but a major advantage of using nanoclusters is that the material type also influences the electronic properties of the confined modes. For example if the nanoclusters are metallic, an dipole will be induced in the nanocluster, which in its turn influences the electric potential felt by the states in a specific and material-dependent way. In this article, to investigate this effect we consider two extreme examples; one in which the nanocluster does not influence the local potential and one in which the local inter-layer potential has been flipped. These results show that whether confinement occurs does not depend on the precise contents of the nanocluster. A detailed modelling of the influence of the nanocluster’s material properties on the confined states is, however, beyond the scope of the current article.
To model the system we use the following Hamiltonian in the continuum limit around the $K$-point [@Snyman_2007]: $$\label{starting_Hamiltonian}
\hat{H}({\ensuremath{\boldsymbol{r}}})=\left(
\begin{array}{cccc}
\delta({\ensuremath{\boldsymbol{r}}}) & v_{\rm F}\hat{\pi}_{+} & \gamma_{1}({\ensuremath{\boldsymbol{r}}}) & 0 \\
v_{\rm F}\hat{\pi}_{-} & \delta({\ensuremath{\boldsymbol{r}}}) & 0 & 0\\
\gamma_{1}({\ensuremath{\boldsymbol{r}}}) & 0& - \delta({\ensuremath{\boldsymbol{r}}}) & v_{\rm F}\hat{\pi}_{-} \\
0 & 0& v_{\rm F}\hat{\pi}_{+} & - \delta({\ensuremath{\boldsymbol{r}}}) \\
\end{array}\right)~,$$ in the basis of orbital eigenstates of the four atoms in the BLG unit cell $\Psi = (\Phi_{A1},\Phi_{B1},\Phi_{B2},\Phi_{A2})^{\dag}$. In Eq. , $\hat{\pi}_{\pm} = \hat{p}_{x} \pm i \hat{p}_{y}$ is the canonical momentum and the quantity $\delta({\ensuremath{\boldsymbol{r}}})$ denotes the potential bias between the two layers induced by the global electrostatic gate. Notice that the latter quantity depends on the position as the structure of the GQB affects the local potential experienced on both layers. $\gamma_{1}({\ensuremath{\boldsymbol{r}}})$ is the inter-layer coupling between the $A1$ and $B2$ atoms and, together with $\delta({\ensuremath{\boldsymbol{r}}})$, determines the local band gap. A cross section of the atomic configuration is shown in Fig. \[fig-GQB\](b). The energy spectrum obtained from Eq. shows a band gap [@Abdullah_2017] $$\delta_{\rm G}({\ensuremath{\boldsymbol{r}}}) = \delta({\ensuremath{\boldsymbol{r}}}) \left(1+4\frac{\delta^{2}({\ensuremath{\boldsymbol{r}}})}{\gamma_{1}^{2}({\ensuremath{\boldsymbol{r}}})}\right)^{-1/2}~.
\label{eq_energy_range}$$ The strength of the inter-layer coupling $\gamma_{1}({\ensuremath{\boldsymbol{r}}})$ is determined by the distance between both layers. Since the coupling is related to the overlap between the orbital eigenstates of the two carbon atoms right above each other, it decreases exponentially with inter-layer distance. The inter-layer coupling can be written as [@Donck2016] $$\gamma_{1}({\ensuremath{\boldsymbol{r}}}) = \gamma^{0}_{1} \exp\left(-\beta \frac{c({\ensuremath{\boldsymbol{r}}})-c_{0}}{c_{0}}\right)~,
\label{gamma1_coupling}$$ Here, $\gamma_{1}^{0} = 0.38~{\rm eV}$, $\beta = 13.3$, and $c_{0} \approx 0.3~{\rm nm}$ is the equilibrium inter-layer distance[@Lobato_2011; @Donck2016].
Fig. \[fig-GQB\](a) shows the band gap in the GQB as a function of the distance to the center of a gaussian GQB. The result shows that the gap vanishes inside the blister and then increases very sharply at the edge of the GQB. The eigenstates and energy levels of confined states inside the blister can, therefore, be determined by assuming a sharp step in the band gap at $\rho = R$ by matching the different components of the wave functions of the two graphene layers inside with those outside the GQB[@Xavier2010].
![(Color online) Energy levels of a GQB and corresponding layer occupation indicated by the color for angular quantum number $m=0$. The solid curves in (a) and (b) correspond to a blister with homogenous bias $\delta({\ensuremath{\boldsymbol{r}}})$ everywhere or an opposite bias inside the GQB respectively. Yellow horizonal lines delimit the energy range for confinement, i.e. $E=\pm\delta_{G} $. In both graphs $\delta=95$ meV. Dashed and dotted-dashed curves represent the first energy levels of pure holes and electrons confined states inside the blister. (c) Energy levels of GQB as a function of the global homogeneous bias for $m=0$ and $R=34.6$ nm. Black dashed and yellow solid curves correspond to $E=\pm \delta$ and $E= \pm \delta_G$, respectively. []{data-label="Energy_levels_with_Layer_Occupation"}](Final_color_levels_Real_units2.pdf){width="\linewidth"}
The obtained eigenvalues are purely real and thus correspond to bound states with diverging lifetime. This is a manifestation of the fact that the bias-induced band gap only exists outside the blister. This contrasts with quasi-bound states in the presence of an electrostatic potential in single layer graphene QDs. For these structures, the eigenvalues are complex and thus the states have a finite lifetime[@Matulis2008; @Hewageegana2008].
The resulting energy levels for angular quantum number $m=0$ are shown in Fig. \[Energy\_levels\_with\_Layer\_Occupation\](a). It consists of discrete energy states that have an oscillatory dependence on the size $R$ of the GQB. For very small radii two in-gap states exist with energy near the edge of the conduction and valence bands. As the size increases, the energy levels approach each other and show avoided crossings. For non-zero angular quantum number similar results are obtained.
To understand the origin of the energy bands and their anti-crossings, in Fig. \[Energy\_levels\_with\_Layer\_Occupation\](a) we have color coded the spectrum indicating the layer to which the corresponding eigenstate belongs. From this it is clear that for small radii the states with positive energy belong to the layer, while the negative energy states are positioned at the . Because the inter-layer bias is applied to the entire sample, also inside the GQB the electronic states are shifted by $-\delta$ or $\delta$ for states on the top or bottom layer, respectively, as shown in the left panel of Fig. \[fig-GQB\](c). Therefore, the bottom layer is effectively hole-doped while the top layer is electron-doped due to the bias gate. This is reflected in the behavior of the confined states; indeed the electron state on the top layer decreases in energy as the GQB increases in size, while the hole state at the top layer increases.
As the two energy levels approach each other, the levels show an anti-crossing at the radius for which the particles are equally distributed over both layers. This happens every time a hole state from the layer crosses an electron state from the layer. The level repulsion is consistent with the Wigner-von Neumann theorem and occurs because the wave functions of both states share the same symmetry [@Greiner1985].
As a further proof of the origin of the different energy levels, in Fig. \[Energy\_levels\_with\_Layer\_Occupation\](a), we also show the energy levels of a pure hole (dashed) and electron (dot-dashed) doped GQB as a function of the radius of the GQB. This model system is formed by assuming that both layers inside the dot are at the same potential, while outside the blister an inter-layer bias still opens a gap. In Fig. \[Energy\_levels\_with\_Layer\_Occupation\](a) the energy levels correspond very closely to the numerically calculated results in the GQB. This shows that the anti-crossings result from wave function overlap in the connected bilayer graphene region, i.e. outside the blister. For large GQBs, we find that the energy levels, apart from anti-crossings, can be considered as stemming from two disconnected graphene nanodisks.
To gain a first insight into how the electrostatic properties of the GQBs nanoclusters could affect the properties of the confined modes, we consider a non-homogenous bias, which is directly visible in the energy levels of the GQB. In Fig. \[Energy\_levels\_with\_Layer\_Occupation\](b) we show these levels when the bias inside the GQB is exactly opposite to that in the rest of the sample. Since now the top layer is electron doped and the bottom layer is hole doped, the layer occupation is reversed with respect to the previous case with a homogeneous bias. We still observe anti-crossings when electron and hole states become degenerate but they occur at larger radii. Fig. \[Energy\_levels\_with\_Layer\_Occupation\](b) shows that the electron states now belong to the bottom layer and the holes belong to the top layer. Furthermore, the energy levels now correspond to the second branch of the pure electron and hole doped systems as indicated by the dashed and dot-dashed curves. A detailed description of the effect of specific material’s clusters on the confined states is beyond the scope of the present study, but the results shown in Fig. \[Energy\_levels\_with\_Layer\_Occupation\](b) do show that even for an extreme case of abrupt opposite bias, the confinement is maintained.
The tunability of GQBs is shown in Fig. \[Energy\_levels\_with\_Layer\_Occupation\](c) where the energy levels of a GQB of fixed size are shown as a function of inter-layer bias. The result shows that the number of confined energy levels can be tuned over a wide range by simply changing the applied bias gate. These results can be directly verified by local scanning tunnelling microscopy measurements [@Lee2016].
Since the inter-layer coupling is active only outside the blister, therefore one expects that the eigenstates are mainly localized inside the radius of the GQB. Peculiarly, however, we find that this is not true for all spinor components of the eigenstate. In Fig. \[Real\_Part\_Wave\] we show the real part of the wave function for each component of a blister that supports two positive energy levels. While for the low-energy state (bottom row) the wave function is almost completely localized inside the blister, we see that the high-energy state (top row) has a significant portion outside the blister’s radius on the $B1$ component of the bottom layer.
![Real part of the different components of the wavefunction for states with energy $E= 76$ meV and $E= 17$ meV for top and bottom rows, respectively. The blister has a radius $R=17.3$ nm and bias $\delta=95$ meV. The radius of the blister is indicated by a white dashed circle. []{data-label="Real_Part_Wave"}](Angular_Plot21.pdf){width="\linewidth"}
. (a) Shows a variation of the height of the dome by blue (1.5 $c_0$) and red (10 $c_0$) respectively. In (b) the dashed-orange curve shows the the effect of varying the inter-layer coupling strength through accounting for capacitive effects. []{data-label="GQb_Levels_Num"}](GQB_Levels_Num1.pdf){width="\linewidth"}
Up to now, the analysis was performed by modelling the edge of the blister as an abrupt interface in the band gap describing electronic states. This assumption is justified because of the very sharp transition between gapless and gapped states as shown in Fig. \[fig-GQB\](a), which is a consequence of the exponential dependence of the inter-layer coupling strength on the inter-layer distance expressed through Eq. . Because the band gap changes sharply at the edge of the system, the morphological details of the blister are obscured and many different shapes of circular blisters effectively have the same energy levels. However, when the height variation of the blister becomes much smaller, this argument might not hold any more. Therefore, in Fig. \[GQb\_Levels\_Num\](a) we show the energy levels for Gaussian GQBs with varying height. In contrast to the previous analysis for these results we have resorted to numerical calculation of the energy levels using a finite element package. The results show that even small blisters support localized eigenstates with a similar energy spectrum. Notice that the morphology of the blister only affects the strength of the anti-crossings but that already for GQB with a height of twice the equilibrium inter-layer distance, the energy levels are very close to the completely decoupled case discussed above.
We also investigate the effect of a change in inter-layer bias due to capacitive effects. Indeed, since the bias arises due to electrostatic gates, the top layer will be influenced differently when closer to the top gate than the bottom layer. In Fig. \[GQb\_Levels\_Num\](b) we show numerical results (dashed-orange) for a locally changing inter-layer bias. While also here the confined states result in a robust discretized energy spectrum, the wavelength of the oscillations due to anti-crossings is strongly reduced. This is because in the latter case the cones inside the GQB are shifted more strongly in energy and, therefore, the confined states have a shorter wavelength.
Finally, note that the deformation of the top layer to form a GQB is in principle associated with a local triaxial strain. Therefore, the inter-atomic distance in the top layer can be slightly larger than the equilibrium distance. This can affect the Fermi velocity $v_{\rm F}$ of the states in the top layer, however, as discussed by Neek-Amal et al[@Neek-Amal2013], triaxial strain will only introduce pseudo-magentic fields near the edge of a finite size graphene flake and in the center it is zero. In our case, the size of the GQB is much smaller than the total size of the bilayer graphene sheet and, therefore, strain has a negligible effect on the results obtained in this study.
In conclusion, graphene quantum blisters are unique electrostatic tunable graphene-based quantum dots. They support bound states with diverging lifetime that can be elegantly realized by means of only electrostatic gating and are robust against changes in the GQBs morphology. A big advantage of GQBs is the tunability through gate variations. Also, we pointed out that by changing the contents of the blisters, one could access another degree of freedom to establish quantum dot systems with specific energy levels as required for different applications. Therefore, we expect that the GQBs can form the basis of a new subfield in graphene physics where the graphene sheet structure is used together with electric fields to achieve tunable quantum systems.
Acknowledgments {#acknowledgments .unnumbered}
===============
HMA and HB acknowledge the Saudi Center for Theoretical Physics (SCTP) for their generous support and the support of KFUPM under physics research group projects RG1502-1 and RG1502-2. This work is supported by the Flemish Science Foundation (FWO-Vl) by a post-doctoral fellowship (BVD) and a doctoral fellowship (MVdD).
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|
---
abstract: |
In this paper we give a proof of the Lefschetz fixed point formula of Freed$^{\rm [1]}$ for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced in \[2\] and then we generalize this formula under the noncommutative geometry framework.\
[**Keywords:**]{} Clifford asymptotics; Even spectral triple; Chern-Connes character\
[**2000 MR Subject Classification**]{}58j20
author:
- |
Yong Wang\
[*Nankai Institute of Mathematics Tianjin 300071, P. R. China*]{}\
[* email: [email protected]*]{}
title: '[**Chern-Connes Character for the Invariant Dirac Operator in Odd Dimensions**]{} [^1]'
---
=14.2cm =21.3cm =-0.30in =-0.30in
\[section\] \[section\] \[section\] \[section\] \[section\]
Introduction
=============
In \[2\], Lafferty, Yu and Zhang presented a simple and direct geometric proof of the Lefschetz fixed point formula for an orientation-preserving isometry on an even dimensional spin manifold by Clifford asymptotics of heat kernel. Chern and Hu \[3\] used the method in \[2\] to compute the equivariant Chern-Connes character for the invariant Dirac operator on an even dimensional spin manifold. In \[4\] an alternate approach to certain technical estimates in \[3\] was given.\
In parallel, Freed$^{[1]}$ considered the case of an orientation-reversing involution acting on an odd dimensional spin manifold and gave the associated Lefschetz formulas by K-theretical way$^{\rm [5]}$. The heat kernel method$^{ [6]}$ may be used to prove this odd Lefschetz formula as claimed in \[1\].\
Inspired by \[2\] and \[3\], in this paper we give a direct geometric proof of the Freed’s odd Lefschetz formula. We also construct an even spectral triple (see Section 3) by the Dirac operator and the orientation-reversing involution, then compute the Chern-Connes character for this spectral triple.\
The paper is organized as follows: In Section 2.1, we present some notations and discuss the standard setup. Evaluating the Clifford asymptotics of the local Lefschetz index is given in Section 2.2 and Section 2.3. In Section 3, we construct an even spectral triple and then compute its Chern-Connes character.
A direct geometric proof of the Freed’s odd Lefschetz formula
===============================================================
[**2.1 Preliminaries**]{}\
Firstly we give the standard setup (also see Section 1 in \[1\]). Let $M$ be a closed, connected and oriented Riemannian manifold of odd dimension $n$ with a fixed spin structure ${\rm Spin}(M)$, and $S$ be the bundle of spinors on $M$. Denote by $D$ the associated Dirac operator on $\Gamma(M;S)$, the space of smooth sections of the bundle $S$. Let $\tau:~M\rightarrow M$ be an orientation-reversing isometric involution. Assume there exists a self-adjoint lift $\widetilde{\tau}:~\Gamma(M;S)\rightarrow \Gamma(M;S)$ of $\tau$ satisfying $$\widetilde{\tau}^2=1;~~D\widetilde{\tau}=-\widetilde{\tau}D.\eqno(2.1)$$ When $\tau$ preserves Pin structure, such a lift $\widetilde{\tau}$ always exists. Now the $+1$ and $-1$ eigenspaces of $\widetilde{\tau}$ give a splitting of the spinor fields $$\Gamma(M;S)\cong \Gamma^+(M;S){\small \oplus} \Gamma^{-}(M;S) \eqno(2.2)$$ and the Dirac operator interchanges $\Gamma^+(M;S)$ and $\Gamma^-(M;S)$. We denote by $D^+$ the restriction of $D$ on $\Gamma^+(M;S)$. The purpose of this section is to compute $${\rm index}[D^+:~\Gamma^+(M;S)\rightarrow \Gamma^-(M;S)]\eqno(2.3)$$
In the following we give an explicit construction of $\widetilde{\tau}$. The tangent map of $\tau$ gives a map $d\tau:~O(M)\rightarrow O(M)$. Let the associated bundle ${\rm
Pin}(M)={\rm Spin}(M)\otimes_R{\rm Pin}(n)$ be the induced Pin structure on $M$ where $R:~{\rm Spin}(n)\times {\rm
Pin}(n)\rightarrow {\rm Pin}(n)$ is the Clifford multiplication. Assume $\tau$ preserves this Pin structure, i.e. $d\tau$ has a lift $\overline{d\tau}$ such that the diagram $$\begin{array}{cc}
\ {\rm Pin}(M) \\
\ \pi\downarrow\\
\ O(M)
\end{array}
\begin{array}{cc}
\ \begin{array}{cc} \ \overline{d\tau}\\
\ \longrightarrow \end{array}
\\
\ \\
\ \longrightarrow\\
\ \begin{array}{cc} \ {d\tau} \end{array}
\end{array}
\begin{array}{cc}
\ {\rm Pin}(M) \\
\ \downarrow\pi\\
\ O(M)
\end{array}.$$ is commutative where $\pi:{\rm Pin}(M)\rightarrow O(M)$ is the double covering and $\overline{d\tau}$ commutes with the ${\rm
Pin}(n)$-action. We recall the odd dimensional Spin($n$) representation$^{\rm [7]}$. Let ${\rm Cl}^+(n+1)$ be the even part of the Clifford algebra generated by $e_1,\cdots,e_{n+1}$ and [ $\cal {I_+}$]{} be the associated positive irreducible representation. Let $\rho_1:{\rm Pin}(n)\rightarrow {\rm
Cl}^+(n+1);~e_i\rightarrow e_ie_{n+1}$ for $1\leq i\leq n$ be an algebra homomorphism and $\rho_2:{\rm Cl}^+(n+1)\rightarrow {\rm
End}(\cal I_+)$ be a representation of ${\rm Cl}^+(n+1)^{[7]}$, then $\rho=\rho_2\rho_1$ is a Pin($n$) representation. Note that Spin$(n)$ is a subgroup of Pin$(n)$, so we have $$S={\rm
Spin}(M)\times_{\rho}{\cal I_+}={\rm Pin}(M)\times_{\rho}{\cal
I_+}.\eqno(2.4)$$ A linear map $\widetilde{\tau_0}$ is defined as follows. Suppose that $\phi\in\Gamma(S)$ is expressed locally over an open set $U_{\tau x}$ by $\phi=[(\sigma,f)]$ for $x\in M$ and a neighborhood $U_{\tau x}$ of $\tau x$, where $\sigma:~U_{\tau
x}\rightarrow {\rm Spin(M)}$ is a local Spin frame field and $f:~
U_{\tau x}\rightarrow {\cal {I_+}}$ is a spinor-valued function, and $[(\sigma,f)]$ denotes the equivalence class of $(\sigma,f)$ in $S= {\rm Spin(M)}\times_{\rho}{\cal {I_+}}.$ Let $$\widetilde{\tau_0}:~S_{\tau x}\rightarrow S_x;~(\widetilde{\tau_0}\phi)(x)
=[(((\overline{d\tau})^{-1}\sigma)(x),f(\tau x))] \eqno(2.5)$$ where the right of (2.5) denotes the equivalence class in $S={\rm
Pin}(M)\times_{\rho}{\cal {I_+}}$. Then $\widetilde{\tau_0}D=-D\widetilde{\tau_0}$. By Lemma 1.5 of \[1\], then $\widetilde{\tau_0}^2$ is a constant multiple of the identity. Let $F_1,\cdots,F_r$ be components of the fixed point set of $\tau$ and ${\rm codim}F_q=2m_q+1~ (1\leq q \leq r)$ and $m_i\geq m_j$ for $i<j$, then $\widetilde{\tau_0}^2=(-1)^{m_1+1}$ over the neighborhood of $F_1$ (see Section 2.2). So we define $$\widetilde{\tau}=(\sqrt{-1})^{m_1+1}\widetilde{\tau_0},\eqno(2.6)$$ then $\widetilde{\tau}$ satisfies the condition (2.1). Note that since $\tau$ preserves the Pin structure, ${\rm
codim}F_i\equiv{\rm codim}F_j~{\rm mod}~ 4$ (similar to Proposition 8.46 in \[8\]). So (2.6) up to a sign is independent of the choice of components. We take the Pin($n$)-invariant Hermitian inner product on ${\cal {I_+}}$, then by (2.5) and (2.6), we have $\widetilde{\tau}{\widetilde{\tau}}_a=1$ where ${\widetilde{\tau}}_a$ is the adjoint operator of $\widetilde{\tau}$. Considering $\widetilde{\tau}^2=1$ then $\widetilde{\tau}={\widetilde{\tau}}_a$.\
By Mckean-Singer formula, we have $${\rm Ind}D^+={\rm Tr}(\widetilde{\tau}e^{-tD^2}).\eqno(2.7)$$ Let $P_t(x,y):S_y\rightarrow S_x$ be the fundamental solutions for the heat operator $\partial/\partial t+D^2.$ The standard heat equation argument yields $${\rm Tr}(\widetilde{\tau}e^{-tD^2})=\int_M{\rm
Tr}[\widetilde{\tau}P_t(\tau x,x)]dx. \eqno(2.8)$$ We shall use the abbreviation ${\cal{L}}(t,x)={\rm Tr}[\widetilde{\tau}P_t(\tau
x,x)]$. Let $\nu$ be the normal bundle of the fixed point set and ${\nu}(\varepsilon)=\{ x\in{\nu}|~||x||<\varepsilon\}$ for $\varepsilon>0$. Similar to the discussions in \[2\], we get
$${\rm Ind}D^+=\sum_{q=1}^r\int_{F_q}{\cal{L}}_{\rm loc}(\tau)(\xi)d\xi \eqno(2.9)$$ [*where*]{}$${\cal{L}}_{\rm loc}(\tau)(\xi)={\rm
lim}_{t\rightarrow 0}\int_{\nu_{\xi}(\varepsilon)}{\cal{L}}(t,{\rm
exp}c)dc\eqno(2.10)$$ [*exists and is independent of*]{} $\varepsilon$.
Since $\tau$ preserves the Pin structure, each $F_q$ has a natural orientation (similar to Proposition 6.14 in \[6\]). Let ${\rm dim}F_q=2n'$ and $\xi\in F_q$, then as in \[2\] there exists an oriented orthonormal frame field $E=(E_1,\cdots,E_n)$ in a neighborhood $U$ of $\xi$ such that:\
(a) for $\zeta\in U\bigcap F_q$, $(E_1(\zeta),\cdots,E_{2n'}(\zeta))$ is an oriented orthonormal basis of $T_{\zeta}F_q$ while the vector fields $E_{2n'+1}(\zeta),\cdots,E_{n}(\zeta)$ are normal to $T_{\zeta}F_q$.\
(b) $E$ is parallel along the geodesics normal to $F_q$.\
With respect to $(E_1,\cdots,E_n)$, $d\tau$ is expressed as a matrix-valued function ${\cal T}$ for $x\in U$ $$d\tau E(x)=E(\tau x){\cal T}(x).$$ Moreover there is a neighborhood $V$ of $\xi$ in $F_q$ such that $E$ is defined on $U={\rm exp}(\nu|_V\bigcap \nu(\varepsilon))$ for sufficient small $\varepsilon$. If $B_0(\varepsilon)$ is the ball of radius $\varepsilon$ in ${\bf R}^{2m_q+1}$; we define the homeomorphism $\Phi:~V\times B_0(\varepsilon)\rightarrow U$ by setting $$\Phi(x';c_1,\cdots,c_{2m_q+1})=x={\rm exp}_{x'}(\sum^{2m_q+1}_{\alpha=1}c_{\alpha}E_{2n'+\alpha}(x')).\eqno(2.11)$$ Denote by $(x';c)$ the orthogonal coordinates of $x$ with respect to $E=(E_1,\cdots,E_n)$ at $\xi$. Then we have that\
(a) ${\cal T}(x';c)={\cal T}(x')$; and\
(b) the isometry $\tau$ has the form $\tau(x';c)=(x';-c)$ and for $\forall x\in U$ $${\cal T}(x)=\left[\begin{array}{lcr}
\ I & 0 \\
\ 0 & -I
\end{array}\right];$$ (c) Let $E^{\tau x}$ be an oriented frame field defined over the patch $U$ by requiring that $E^{\tau x}(\tau x)=E(\tau x)$ and that $E^{\tau x}$ be parallel along geodesic through $\tau x$. Define the coordinates $\{ y_i\}$ of $x$ as $$(x';c)=x={\rm exp}_{\tau x}(\sum^n_{i=1}y_iE^{\tau x}_i(\tau
x))$$ then $E^{\tau x}(x)=E(x)$ and $y_i=0$ for $1\leq i \leq
2n';$ $y_{2n'+\alpha}=2c_\alpha$ for $1\leq \alpha \leq 2m_q+1.$ Note that (b) comes from $\tau^2=$id, i.e. ${\cal T}(x)^2=$id. Since $x=(x',c),~x'$ and $(x',-c)=\tau x$ belong to the same geodesic normal to $F_q$, (c) is correct.
[**2.2 The Clifford asymptotics**]{}
Choose a Spin frame field $\sigma:U\rightarrow {\rm Spin}(M)$ such that $\pi'\sigma=(E^{\tau x}_1,\cdots,E^{\tau x}_n)$ where $\pi':
{\rm Spin}(M)\rightarrow SO(M)$ is the double covering. For $x\in U$, let $\overline{P_t}(x)$, $\widetilde{\tau}^*(x)\in {\rm
Hom}({\cal {I_+}},{\cal {I_+}})$ be defined through the equivalence relations for $u,v\in {\cal {I_+}}$ $$P_t(\tau x,x)[(\sigma(x),v)]=[(\sigma(\tau x),\overline{P_t}(x)v)]\eqno(2.12)$$ and $$\widetilde{\tau}[(\sigma(\tau x),u)]=[(\sigma(x),\widetilde{\tau}^*(x)u].\eqno(2.13)$$ Similar to Lemma 4.1 in \[10\], we have:
[*For $x$ in a sufficient small neighborhood of $F_q$ and $t>0$, the integrand ${\cal L}(t,x)$ is evaluated by*]{} $${\cal L}(t,x)={\rm
Tr}(\widetilde{\tau}^*(x)\overline{P_t}(x)).\eqno(2.14)$$ As in \[2\] and \[9\], in the normal coordinates $y_1,\cdots,y_n$ at $\tau x$ with respect to the frame field $E^{\tau x}=(E^{\tau
x}_1,\cdots,E^{\tau x}_n)$, the operator $${\chi}(y^{\alpha}D^{\beta}_ye^{\gamma})=|\beta|-|\alpha|+|\gamma|\eqno(2.15)$$ for multi-indices $\alpha,~\beta,$ and $\gamma$, with $y^{\alpha}=y^{\alpha_1}_1\cdots y^{\alpha_n}_n,$ $D^{\beta}_y=(\partial/\partial_1)^{\beta_1}\cdots
(\partial/\partial_n)^{\beta_n}$ and $e^{\gamma}=e^{\gamma_1}_1\cdots e^{\gamma_n}_n$ for $\gamma_i\in\{ 0,1 \}.$
In the coordinates $(x_0;c)$ with respect to the frame field $E=(E_1,\cdots,E_n)$, set $c=\sqrt{t}b$ and define operator $\overline{\chi}$ on the monomials $\phi(t)e_{i_1}\cdots e_{i_s}$ by $$\overline{\chi}(\phi(t)e_{i_1}\cdots e_{i_s})=s-{\rm sup}\{
l\in {\bf Z}{\bf |}~{\rm lim}_{t\rightarrow
0}\frac{|\phi(t)|}{t^{l/2}}<\infty \},\eqno(2.16)$$ where $\phi(t)\in {\bf R}.$ We denote by $P=Q+(\overline{\chi}<m)$ the congruence of $P$ and $Q$ modulo the space generated by monomials with $\overline{\chi}<m$.
By Section 2.1, then $$d\tau E^{\tau x}(x)=d\tau E(x)=E(\tau x){\cal T}(x)=E^{\tau x}(\tau x)\left[\begin{array}{lcr}
\ I & 0 \\
\ 0 & -I
\end{array}\right].\eqno(2.17)$$ Thus $$\overline{d\tau}\sigma(x)=\sigma(\tau
x)c(e_{2n'+1})\cdots c(e_n).\eqno(2.18).$$ By (2.6), then $$\widetilde{\tau}^*=(\sqrt{-1})^{m_1+1}c(e_{2n'+1})\cdots
c(e_n)=(\sqrt{-1})^{m_1-m_q}((\sqrt{-1})^{m_q+1}c(e_{2n'+1})\cdots
c(e_n)).\eqno(2.19)$$ Let $x'$ be a point near $x=(\xi;c)$ and let $y=(y'_1,\cdots,y'_n)$ be the normal coordinate of $x'$ at point $\tau x$ with respect to the orthonormal frame field $E^{\tau x}=(E^{\tau x}_1,\cdots,E^{\tau
x}_n)$ defined in Section 2.1. As in \[2\], let $\widetilde{A}$ be the $n\times n$ matrix defined by $$\widetilde{A_{ij}}=-\frac{1}{2}\sum^n_{k,l=1}R^{\tau x}_{ijkl}(\tau
x)c(e_k)c(e_l),\eqno(2.20)$$ where $R^{\tau x}_{ijkl}(\tau
x)$ are the coefficients of the Riemannian curvature tensor under the frame field $E^{\tau x}=(E^{\tau x}_1,\cdots,E^{\tau
x}_n)$ at point $\tau x$. We define $\widetilde{A}^l(y')$ as $${\widetilde{A}}^l(y')=\sum^n_{i,j=1}y'_iy'_j{\widetilde{A}}^l_{ij}\eqno(2.21)$$ for $l=1,2,\cdots.$
Similarly to \[9\], in the odd dimensional case, there is a function $P(t;z_1,z_2,\cdots;w_1,w_2,\cdots)$, which is a power series in $t$ with coefficient polynomials in $z_i$ and $w_i$ such that\
$~~\overline{P_t(x)}=(1/4\pi t)^{\frac{n}{2}}{\rm exp}(-d^2(x,\tau
x)/4t)$ $$\times[P(t;{\rm Tr}\widetilde{A}^2,\cdots,{\rm Tr}\widetilde{A}^{2k},\cdots;
\widetilde{A}^2(y)\cdots,\widetilde{A}^{2l}(y),\cdots)+\sum_{m\geq
0 }t^m(\overline{\chi}<2m)],\eqno(2.22)$$ where in the diagonal form we have, by solving harmonic oscillator-type equations, $$P(t;((-1)^k2(x^{2k}_1+\cdots+x^{2k}_{n'+m_q}));((-1)^l\sum_{\alpha=1}^{n'+m_q}(y_{2\alpha-1}^2
+y_{2\alpha}^2)x_{\alpha}^{2l}))=(4\pi t)^{n'+m_q}$$ $$\times {\rm
exp}(\sum^{n-1}_{\alpha=1}y_{\alpha}^2/4t)\prod_{s=1}^{n'+m_q}\left[
\frac{\sqrt{-1}x_s}{8\pi {\rm sinh}\frac{\sqrt{-1}x_st}{2}}
{\rm exp}(-\frac{\sqrt{-1}x_s}{8}{\rm coth}\frac{\sqrt{-1}x_st}{2}(y_{2s-1}^2+y_{2s}^2))
\right].\eqno(2.23)$$ As Lemma 4.3 in \[2\], we have:
[*Let $\widetilde{A_0}$ be the matrix defined by $$\widetilde{A_0}_{ij}=-\frac{1}{2}\sum^{2n'}_{k,l=1}R_{ijkl}(\xi)c(e_k)c(e_l),~~~1\leq i,j\leq n ,\eqno(2.24)$$ and define the tangential component $A^{\top}$ and the normal component $A^{\bot}$ by $${A^{\top}}_{ij}=-\frac{1}{2}\sum^{2n'}_{k,l=1}R_{ijkl}(\xi)c(e_k)c(e_l),~~~1\leq i,j\leq 2n', \eqno(2.25)$$ $${A^{\bot}}_{ij}=-\frac{1}{2}\sum^{2n'}_{k,l=1}R_{ijkl}(\xi)c(e_k)c(e_l),~~~2n'+1\leq i,j\leq n .\eqno(2.26)$$ Then $$\widetilde{A_0}=\left(\begin{array}{lcr}
\ A^{\top} & 0 \\
~~0 & A^{\bot}
\end{array}\right).\eqno(2.27)$$ Further, the relations $${\rm Tr}\widetilde{A}^{2k}={\rm Tr}(A^{\top})^{2k}+{\rm
Tr}(A^{\bot})^{2k}+\sum_{\alpha=1}^{n-2n'}c(e_{2n'+\alpha})(\overline{\chi}<4k)+(\overline{\chi}<4k)\eqno(2.28)$$ and $$\widetilde{A}^{2k}(y/\sqrt{t})=4(A^{\bot})^{2k}(b)+
\sum_{\alpha=1}^{n-2n'}c(e_{2n'+\alpha})(\overline{\chi}<4k)+(\overline{\chi}<4k)\eqno(2.29)$$ hold.*]{}
Combining (2.19),(2.22) and Lemma 2.3, we get
$$\widetilde{\tau}^*(x)\overline{P_t}(x)=e^{-||b||^2}\left[\frac{(\sqrt{-1})^{m_1+1}}{(4\pi t)^{n/2}}
P(t;({\rm Tr}(A^{\top})^{2k}+{\rm
Tr}(A^{\bot})^{2k});(4t(A^{\bot})^{2k}(b)))\right.$$ $$\left.\times c(e_{2n'+1})\cdots
c(e_n)+(\overline{\chi}<2n-2n')_b\right]\eqno(2.30)$$ [*where $c=\sqrt{t}b$ and $(\overline{\chi}<2n-2n')_b$ denotes the space spanned by which are polynomials in $b$ and satisfy $\overline{\chi}<2n-2n'$.*]{}
[**2.3 Evaluation of the local index**]{}
(\[2\]) $${\rm lim}_{t\rightarrow 0}\int _{\nu_{\xi}(\varepsilon)}e^{-||b||^2}{\rm
Tr}(\phi) dc=0\eqno(2.31)$$ [*where*]{} $\phi\in (\overline{\chi}<2n-2n')_b.$\
To compute the trace it suffices to compute the coefficient of the $c(e_1)\cdots c(e_n)$ term in Lemma 2.4. Note that $A^{\bot}$ and $A^{\top}$ are of order $\overline{\chi}\leq
2$, containing terms $c(e_i)c(e_j)$ with $1\leq i,j \leq 2n'$ and ${\rm Tr}(c(e_1)\cdots
c(e_n))=(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}$. Since $c(e_i)c(e_j)=-c(e_j)c(e_i)+(\overline{\chi}<1)$, if we formally replace $c(e_i)$ by $\omega_i$ where $\omega=(\omega_1,\cdots,\omega_n)$ is the frame dual to $E$, and then substitute $\Omega^{\top}$ and $\Omega^{\bot}$ for $A^{\top}$ and $A^{\bot}$, where $$\Omega^{\top}=-\frac{1}{2}\sum^{n}_{k,l=1}R_{ijkl}(\xi)\omega_k\wedge\omega_l,~~~1\leq i,j\leq 2n' ;\eqno(2.32)$$ $${\Omega^{\bot}}=-\frac{1}{2}\sum^{n}_{k,l=1}R_{ijkl}(\xi)\omega_k\wedge\omega_l,~~~2n'+1\leq i,j\leq n. \eqno(2.33)$$ To compute the trace, we only need to compute the top form (of order $2n'$) on $F_q$, then we multiply it by $(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}$. In order to compute this differential form, we need the odd dimensional case of the Chern root algorithm (see \[9\]).
Let $$\Omega=\left[\begin{array}{lcr}
\ \Omega^{\top} & 0 \\
~~0 & \Omega^{\bot}
\end{array}\right]$$ be given formally as $$\Omega^{\top}=\left[\begin{array}{lllll}
0 & u_1 & & & \\
-u_1 & 0 & & & \\
& &\ddots & & \\
& & & 0 & u_{n'} \\
& & & -u_{n'} & 0 \\
\end{array}\right],~~~~~~
\Omega^{\bot}=\left[\begin{array}{llllll}
0 & v_1 & & & & \\
-v_1 & 0 & & & & \\
& &\ddots & & & \\
& & & 0 & v_{m_q} & \\
& & & -v_{m_q} & 0 & \\
& & & & & 0 \\
\end{array}\right],$$ where $u_i$ and $v_i$ are indeterminants. Then $$4t(\Omega^{\bot})^{2k}(b)=(-1)^k4t\sum_{\alpha=1}^{m_q}v_{\alpha}^{2k}(b^2_{2\alpha-1}+b^2_{2\alpha});\eqno(2.34)$$ $${\rm Tr}\Omega^{2k}=2(-1)^k(\sum_{\alpha=1}^{n'}u^{2k}_{\alpha}+\sum_{\beta=1}^{m_q}v_{\beta}^{2k}).\eqno(2.35)$$ By (2.23),(2.30),(2.34) and (2.35), we have: $$\begin{aligned}
{\cal L}_{\rm loc}(\tau)&=&{\rm lim}_{t\rightarrow 0}\int _{{\bf R
}^{n-2n'}}(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}t^{\frac{n-2n'}{2}}e^{-||b||^2}
\frac{(\sqrt{-1})^{m_1+1}}{(4\pi t)^{n/2}}\\
& ~ &\times
P(t;(2(-1)^k(\sum_{\alpha=1}^{n'}u^{2k}_{\alpha}+\sum_{\beta=1}^{m_q}v_{\beta}^{2k}));
((-1)^k4t\sum_{\alpha=1}^{m_q}v_{\alpha}^{2k}(b^2_{2\alpha-1}+b^2_{2\alpha})))db\\
& = & {\rm lim}_{t\rightarrow 0}\int _{{\bf R
}^{n-2n'}}(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}t^{\frac{n-2n'}{2}}e^{-b_n^2}
\frac{(\sqrt{-1})^{m_1+1}}{(4\pi t)^{n/2}}\\
& ~ &
\times\prod^{n'}_{\alpha=1}\frac{\sqrt{-1}tu_{\alpha}/2}{{\rm
sinh}\sqrt{-1}tu_{\alpha}/2}
\prod^{m_q}_{\beta=1}\frac{\sqrt{-1}tv_{\beta}/2}{{\rm
sinh}\sqrt{-1}tv_{\beta}/2}\\
& ~ & \times {\rm exp}(-\sum
^{m_q}_{s=1}\frac{\sqrt{-1}v_st}{2}{\rm
coth}\frac{\sqrt{-1}v_st}{2} (b_{2s-1}^2+b_{2s}^2))db.\end{aligned}$$\
Note that $\int_{\bf R}e^{-b_n^2}db_n=\sqrt{\pi}$. In the final calculation after integrating out $b$, we will take the form of order $2n'$ on $F_q$, and hence the factor of $t$ cancels. So $$\begin{aligned}
{\cal L}_{\rm loc}(\tau)&=&\sqrt{\pi} \int _{{\bf R
}^{2m_q}}(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}
\frac{(\sqrt{-1})^{m_1+1}}{(4\pi)^{n/2}}\\
&~&\times\prod^{n'}_{\alpha=1}\frac{\sqrt{-1}u_{\alpha}/2}{{\rm
sinh}\sqrt{-1}u_{\alpha}/2}
\prod^{m_q}_{\beta=1}\frac{\sqrt{-1}v_{\beta}/2}{{\rm
sinh}\sqrt{-1}v_{\beta}/2}\\
& ~ & \times {\rm exp}(-\sum
^{m_q}_{s=1}\frac{\sqrt{-1}v_s}{2}{\rm coth}\frac{\sqrt{-1}v_s}{2}
(b_{2s-1}^2+b_{2s}^2))db_1\cdots db_{2m_q}\\
&=&(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}\sqrt{\pi}
\frac{(\sqrt{-1})^{m_1+1}}{(4\pi)^{n/2}}\int _{{\bf R }^{2m_q}}
\prod^{n'}_{\alpha=1}\frac{\sqrt{-1}u_{\alpha}/2}{{\rm
sinh}\sqrt{-1}u_{\alpha}/2}
\prod^{m_q}_{\beta=1}\frac{\sqrt{-1}v_{\beta}/2}{{\rm
sinh}\sqrt{-1}v_{\beta}/2}\\
&~&\times {\rm exp}(-\frac{1}{2}\sum_s\frac{v_s}{{\rm
sin}v_s/2}{\rm sin}(\frac{\pi+v_s}{2})(b_{2s-1}^2+b_{2s}^2))db_1\cdots db_{2m_q}\\
&=&\frac{(-\sqrt{-1})^{[\frac{n}{2}]+1}(\sqrt{-1})^{m_1+1}}
{\pi^{n'}2^{[\frac{n}{2}]+1}}
\prod^{n'}_{\alpha=1}\frac{\sqrt{-1}u_{\alpha}/2}{{\rm
sinh}\sqrt{-1}u_{\alpha}/2} \prod^{m_q}_{\beta=1}[{\rm
sin}(\frac{v_\beta+\pi}{2})]^{-1}.\end{aligned}$$ Let $\frac{u_{\alpha}}{2\pi}=u_{\alpha}^*,~\frac{v_{\beta}}{2\pi}=v_{\beta}^*$ be the Chern roots, then $$\begin{aligned}
{\cal L}_{\rm loc}(\tau)^{(2n')}&=&\left[
\frac{(-\sqrt{-1})^{[\frac{n}{2}]+1}(\sqrt{-1})^{m_1+1}}{\pi^{n'}2^{[\frac{n}{2}]+1}}(\sqrt{-1})^{n'}(2\pi)^{n'}\right.\\
& &\times\left.\prod^{n'}_{\alpha=1}\frac{u_{\alpha}/4\pi}{{\rm
sinh}u_{\alpha}/4\pi} \prod^{m_q}_{\beta=1}[{\rm
sin}(\frac{v_\beta}{4\pi\sqrt{-1}}+\frac{\pi}{2})]^{-1}\right]^{(2n')}\\
&=&\left[\frac{(\sqrt{-1})^{m_1}}{2}
\prod^{n'}_{\alpha=1}\frac{u_{\alpha}^*/2}{{\rm
sinh}u_{\alpha}^*/2} \prod^{m_q}_{\beta=1}[2{\rm
sinh}(\frac{v_\beta}{4\pi}+\frac{\sqrt{-1}\pi}{2})]^{-1}\right]^{(2n')}\\
&=&\frac{(\sqrt{-1})^{m_1-m_q}}{2}\left[
\prod^{n'}_{\alpha=1}\frac{u_{\alpha}^*/2}{{\rm
sinh}u_{\alpha}^*/2}
\prod^{m_q}_{\beta=1}(e^{\frac{v^*_{\beta}}{2}}+e^{-\frac{v^*_{\beta}}{2}})^{-1}\right]^{(2n')}.\end{aligned}$$ As in \[5\], we write the characteristic class $${\rm ch}\triangle(N_q)=\prod^{m_q}_{\beta=1}(e^{\frac{v^*_{\beta}}{2}}+e^{-\frac{v^*_{\beta}}{2}})\eqno(2.36)$$ where $N_q$ denotes the normal bundle of $F_q$. We thus obtain the following theorem.\
(\[1\]) [*Let $M$ be an odd dimensional compact oriented Spin manifold and $\tau:M\rightarrow M$ be an orientation-reversing isometric involution which preserves Pin structure. Suppose that $F_1,\cdots,F_r$ are components of the fixed point set, then*]{} $${\rm ind}D^+=\frac{1}{2}\sum_{q=1}^r
\int_{F_q}(\sqrt{-1})^{m_1-m_q}\widehat{A}(TF_q)[{\rm
ch}\triangle(N_q)]^{-1}.\eqno(2.37)$$
By (2.6), the grading operator $\widetilde{\tau}$ depends on $m_1$. If we choose another component $F_i$, then $\widetilde{\tau}$ is up to $(\sqrt{-1})^{m_i-m_1}$, but we note by the change of $\widetilde{\tau}$ and (2.2),(2.3), then ${\rm ind}D^+$ also change $(\sqrt{-1})^{m_i-m_1}.$
The Chern-Connes character of even spectral triple $(C^{\infty}_{\tau}(M),L^2(M,S),D,\widetilde{\tau})$
========================================================================================================
Let $M,\tau$ and $\widetilde{\tau}$ be given as in Section 2.1, let $$C^{\infty}_{\tau}(M)=\{ a\in
C^{\infty}(M)|a\tau(x)=a(x);~\forall x\in M \}\subset
C^{\infty}(M),$$ then $(C^{\infty}_{\tau}(M),L^2(M,S),D,\widetilde{\tau})$ is an $\theta$-summable even spectral triple (for definition see \[10\] or \[11\] ). In the following we will compute its Chern-Connes character. Firstly let us review the definition of the Chern-Connes character represented by the JLO cocycle in the entire cyclic cohomology .
(\[12\]) Let $(A,H,D,\gamma)$ be an even $\theta$-summable spectral triple associated to a Banach algebra $A$ with identity, then its Chern character ${\rm
ch}_{*}(A,H,D,\gamma)=\{ {\rm ch}_k(D)|~k\geq 0 ~{\rm and~ even}
\}$ in the entire cyclic cohomology is defined by $${\bf {\rm
ch}}_k(D)(a^0,\cdots ,a^k)=\int_{\triangle_k}{\rm str}
(a^0e^{-s_1D^2}[D,a^1]e^{-(s_2-s_1)D^2}[D,a^2]$$ $$\cdots
e^{-(s_k-s_{k-1})D^2}[D,a^k] e^{-(1-s_k)D^2})ds,\eqno(3.1)$$ where $a^i\in A$ and $\triangle_k=\{(s_1,\cdots,s_k)|~0\leq
s_1\leq\cdots\leq s_k\leq 1\}$. For $t>0$, considering the deformed Chern-Connes character ${\rm ch}_{*}(\sqrt{t}D)=\{ {\rm
ch}_k(\sqrt{t}D)|~k\geq 0 ~{\rm and~ even}\}$ is expressed by $${\bf {\rm
ch}}_k(\sqrt{t}D)(a^0,\cdots
,a^k)=t^{\frac{k}{2}}\int_{\triangle_k}{\rm str}
(a^0e^{-s_1tD^2}[D,a^1]e^{-(s_2-s_1)tD^2}[D,a^2]$$ $$\cdots
e^{-(s_k-s_{k-1})tD^2}[D,a^k] e^{-(1-s_k)tD^2})ds.\eqno(3.2)$$ We write $$\lambda(p)=(\lambda_1,\cdots,\lambda_p);~|\lambda(p)|=\lambda_1+\cdots+\lambda_p;
~\lambda(p)!=\lambda_1!\cdots\lambda_p!$$ $$\widetilde{\lambda}(p)!=(\lambda_1+1)(\lambda_1+\lambda_2+2)\cdots(\lambda_1+\cdots+\lambda_{p}+p).$$ For an operator $B$ and any positive integer $l$, write $B^{[l]}=[D^2,B^{[l-1]}],~B^{[0]}=B$. We use the notation $$D^{\lambda(p)}=f^0[c(df^{1})]^{[\lambda _{1}]}
[c(df^2)]^{[\lambda _2]}\cdots [c(df^p)]^{[\lambda
_p]};~~D^{\lambda(p)}_t=t^{\frac{p}{2}+|\lambda(p)|}D^{\lambda(p)}\eqno(3.3)$$ where $f^j\in C^{\infty}_{\tau}(M)$ for $0\leq j\leq p$. By $0\leq
\lambda(p)\leq n-k$, we mean $0\leq \lambda_j\leq n-k$ for $1\leq
j\leq p$. Recall a result in \[3\] or \[4\].
(\[3\],\[4\]) (i) [*When $k\leq n$ and $t\rightarrow 0^+$, we have: $${\bf {\rm ch}}_k(\sqrt
{t}D)(f^0,\cdots ,f^k)=\sum_{0\leq \lambda(k)\leq
n-k}\frac{(-1)^{|\lambda(k)|}t^{|\lambda(k)|+\frac{k}{2}}}{\lambda(k)!\widetilde{\lambda(k)}!}
{\rm str}\{D^{\lambda(k)}e^{-tD^2}\}+O(\sqrt{t})¡£\eqno(3.4)$$*]{} (ii) [*If $k>n$, then when $t\rightarrow 0^+$, we have:*]{} $${\rm lim}_{t\rightarrow 0}{\bf {\rm ch}}_k(\sqrt {t}D)(f^0,\cdots
,f^k)=0.\eqno(3.5)$$ In the following, we’ll compute ${\rm lim}_{t\rightarrow 0}{\rm
str}\{D^{\lambda(k)}_te^{-tD^2}\}$ by using the method in Section 2. We consider the coordinates systems in Section 2.1, 2.2. Similar to Theorem 2.1 and Lemma 2.2, we have
$${\rm lim}_{t\rightarrow 0}{\rm
Tr}(\widetilde{\tau}D^{\lambda(k)}_te^{-tD^2})=\sum^r_{q=1}\int_{F_q}
({\rm
lim}_{t\rightarrow 0}\int_{\nu_{\xi}(\varepsilon)}{\rm
Tr}(\widetilde{\tau}^*(x)\overline{P_t^{\lambda}}(x))dx)d\xi,\eqno(3.6)$$ [*where*]{} $$D^{\lambda(k)}_tP_t(\tau x,x)[(\sigma(x),v)]=[(\sigma(\tau x),\overline{P_t^{\lambda}}(x)v)].\eqno(3.7)$$
(\[3\]) [*For $f^0,\cdots,f^k\in
C^{\infty}_{\tau}(M)$ and $\lambda\neq 0$, then*]{} $\overline{\chi}(D^{\lambda(k)}_t)<0$.
Let $\xi\in F_q$, we will compute ${\rm lim}_{t\rightarrow
0}\int_{\nu_{\xi}(\varepsilon)}{\rm
Tr}(\widetilde{\tau}^*(x)\overline{P_t^{\lambda}}(x))dx.$
[*If $\lambda=(\lambda_1,\cdots,\lambda_k)\neq 0$, then we have:*]{} $$\widetilde{\tau}^*(x)\overline{P_t^{\lambda}}(x)=e^{-||b||^2}(\overline{\chi}<2n-2n')_b,\eqno(3.8)$$ [*and*]{} ${\rm lim}_{t\rightarrow
0}\int_{\nu_{\xi}(\varepsilon)}{\rm
Tr}(\widetilde{\tau}^*(x)\overline{P_t^{\lambda}}(x))dx=0$ [*i.e.*]{} ${\rm lim}_{t\rightarrow 0}{\rm
Tr}(\widetilde{\tau}D^{\lambda(k)}_te^{-tD^2})=0$.\
[*Proof.*]{} This theorem comes from Lemma 2.4, Lemma 3.4 and Lemma 2.5. $\Box$\
As in \[3\], for any $g\in
C^{\infty}_{\tau}(M),~x=(\xi;c)=(\xi;\sqrt{t}b)$, then $g(x)=g(\xi)+(\overline{\chi}<0)$ and $(dg)(x)=(dg)(\xi)+(\overline{\chi}<1).$ So we have $$t^{\frac{k}{2}}f^0c(df^1)\cdots c(df^k)=t^{\frac{k}{2}}f^0(\xi)c(df^1)(\xi)\cdots
c(df^k)(\xi)+(\overline{\chi}<0).\eqno(3.9)$$ By Lemma 2.4 and (3.9), we obtain: $$\widetilde{\tau}^*(x)t^{\frac{k}{2}}f^0c(df^1)\cdots c(df^k)\overline{P_t}(x)
=(\sqrt{-1})^{m_1+1}t^{\frac{k}{2}}f^0(\xi)c(d(f^1))(\xi)\cdots
c(d(f^k))(\xi)$$ $$\times e^{-||b||^2}\frac{1}{(4\pi t)^{n/2}}
P(t;(\cdots,({\rm Tr}(A^{\top})^{2l}+{\rm
Tr}(A^{\bot})^{2l}),\cdots);(\cdots,(4t(A^{\bot})^{2l}(b)),\cdots))$$ $$\times c(e_{2n'+1})\cdots
c(e_n)+ e^{-||b||^2}(\overline{\chi}<2n-2n')_b.\eqno(3.10)$$ As in Section 2.3, using $\Omega^{\top}$ and $\Omega^{\bot}$ instead of $A^{\top}$ and $A^{\bot}$ and multiplying the constant $(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}$, then we get $$\begin{aligned}
& & {\rm lim}_{t\rightarrow 0}\int_{\nu_{\xi}(\varepsilon)}{\rm
Tr}[\widetilde{\tau}^*(x)t^{\frac{k}{2}}f^0c(df^1)\cdots
c(df^k)\overline{P_t}(x)]dx\\
&=&{\rm lim}_{t\rightarrow
0}\int_{{\bf R
}^{n-2n'}}(-\sqrt{-1})^{[\frac{n}{2}]+1}2^{[\frac{n}{2}]}
t^{\frac{n-2n'}{2}}t^{\frac{k}{2}}\\
& ~ & \times f^0(\xi)c(df^1)(\xi)\cdots c(df^k)(\xi)e^{-||b||^2}
\frac{(\sqrt{-1})^{m_1+1}}{(4\pi t)^{n/2}}\\
& ~ &\times
P(t;(\cdots,(2(-1)^l(\sum_{\alpha=1}^{n'}u^{2l}_{\alpha}+\sum_{\beta=1}^{m_q}v_{\beta}^{2l})),\cdots);
(\cdots,((-1)^l4t\sum_{\alpha=1}^{m_q}v_{\alpha}^{2l}(b^2_{2\alpha-1}+b^2_{2\alpha})),\cdots))db.\end{aligned}$$ Similar to the computation in Section 2.3, we obtain:
[*For $f^0,\cdots,f^k\in
C^{\infty}_{\tau}(M)$ and $k$ even,*]{}\
$~~~{\rm lim}_{t\rightarrow 0}{\bf {\rm ch}}_k(\sqrt
{t}D)(f^0,\cdots ,f^k)$ $$=\frac{1}{k!(2\pi\sqrt{-1})^{\frac{k}{2}}}
\sum_{q=1}^r \frac{(\sqrt{-1})^{m_1-m_q}}{2}\int_{F_q}f^0\wedge
df^1\wedge\cdots\wedge df^k\wedge \widehat{A}(TF_q)[{\rm
ch}\triangle(N_q)]^{-1}.\eqno(3.11)$$ [*where $f^j$ is considered as $f^j|_{F_q}$ for $0\leq j\leq k.$* ]{}
Since the computing of the Chern-Connes character does not require the condition $f^j\in
C^{\infty}_{\tau}(M)$, so (3.11) is correct for any $f^j\in
C^{\infty}(M)$. When $k=0$ and $f^0=1$, we get the theorem 2.6.
[ The author is indebted to Professor Weiping Zhang for his guidance and very helpful discussions. He also thanks Professor Huitao Feng for his generous help and referees for their careful reading and helpful comments.]{}
[20]{}
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Berline N., Getzler E., Vergne M., Heat Kernals and Dirac Operator, Springer-Verlag, Berlin 1992.
Lawson B., Michelson M. L., Spin Geometry, Princeton Univ. Press, 1993.
Atiyah M. F., Bott R., The Lefschetz fixed point theorem for elliptic complexes II: Applications, Ann. of Math., 1968, 88: 451-491.
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[^1]: partially supported by MOEC and the 973 project.
|
---
abstract: 'The counterfactuality of the recently proposed protocols for direct quantum communication is analyzed. It is argued that the protocols can be counterfactual only for one value of the transmitted bit. The protocols achieve a reduced probability of detection of the particle in the transmission channel by increasing the number of paths in the channel. However, this probability is not lower than the probability of detecting a particle actually passing through such a multi-path channel, which was found to be surprisingly small. The relation between security and counterfactuality of the protocols is discussed. An analysis of counterfactuality of the protocols in the framework of the Bohmian interpretation is performed.'
author:
- 'L. Vaidman'
title: 'Counterfactuality of “counterfactual” communication '
---
Introduction
============
Penrose [@Penrose] coined the term “counterfactuals” for describing quantum interaction-free measurements (IFM) [@IFM].
> Counterfactuals are things that might have happened, although they did not in fact happen.
He argued that in the IFM, an object is found because it might have absorbed a photon, although actually it did not. The idea of the IFM has been applied to “counterfactual computation” [@Joz], a setup in which one particular outcome of a computation becomes known in spite of the fact that the computer did not run the algorithm. Noh [@Noh] created counterfactual cryptography, a method for secret key distribution using events in which the particle was not present in the transmission channel. Noh used a random choice of orthogonal input states (like in [@GV]) in contrast with the non-orthogonal states of BB84 cryptographic protocol [@BB84].
It was argued [@Ho06], that a modification of the counterfactual computation proposed above, which includes quantum Zeno effect, can achieve counterfactuality for [*all*]{} outcomes of the computation. Recently, this idea has been used for performing “counterfactual communication” [@Salih], which supposedly allowed to send information from Bob to Alice without transferring any particle between them. The transmission happens in a counterfactual way: the mere possibility of transmitting the particle allows transmitting the value of the bit.
I find all these results very paradoxical since they contradict physical intuition of causality: information is usually transmitted continuously in space. I argued [@PSA] that to resolve the paradoxical feature of the IFM one has to adopt the many-worlds interpretation (MWI) of quantum mechanics [@Eve; @myMWI], in which the particle touches the object in a parallel world restoring causality at least within the complete physical universe which includes all the worlds. However, I believe that a protocol which can transmit both values of a bit without any particle present in the transmission channel is impossible, irrespectively of the interpretation of quantum mechanics one adopts. I have expressed this opinion already [@Va07; @MyCom], but more protocols were suggested [@Li] and the controversy remains open [@LiCom; @LiComRep; @SalihReply]. The clarification of these conceptual issues is particularly important due to the recent increasing interest in the applications of counterfactual protocols [@CFInt1; @CFInt2; @CFInt3; @CFInt4; @CFInt6; @CFInt7; @CFInt8; @CFInt9; @CFInt10]. Here I discuss this issue in more detail and try to resolve the controversy.
The plan of the paper is as follows. In Section II I introduce the general setup of quantum communication protocols. In Sections III and IV I describe two recent protocols which are claimed to be counterfactual. In Section V I analyze various possible definitions of counterfactuality and define my preferred criterion which is based on the magnitude of the trace left in the transmission channel. In Section VI I calculate the trace left by a single particle present in the channel, i.e. the trace of a non-counterfactual communication protocol. In Sections VII-IX I show that the trace in the protocols claimed to be “counterfactual” is not less than the trace in a non-counterfactual protocol. In Section X I analyze the security of “counterfactual” protocols against an eavesdropper. Section XI is devoted to counterfactuality in the framework of the Bohmian interpretation. I summarize the results in Section XII.
Communication with quantum particles {#ComQP}
====================================
There is a surprisingly low bound on the number of bits which can be sent using 1 qubit: The Holevo bound of 1, when the qubit is not entangled [@Hol], and 2, when entanglement is allowed [@supcod]. This is when we transmit one particle with an internal structure of a qubit such as a polarization state of a photon. In this paper I analyze protocols in which the particle does not have an internal structure: the information is encoded in the presence or absence of the particle.
Let Alice and Bob be on the two separate sides of a region, see Fig. 1. Bob has a mirror on his side which causes particles sent by Alice to be bounced back to her. For a bit value of 1, Bob places a shutter which absorbs Alice’s particles, while for 0, the shutter is absent.
![ Simple communication with a quantum particle. Alice sends a particle to Bob and knows the bit chosen by Bob through observation whether or not the particle comes back to her. []{data-label="fig:1"}](1.png){width="7.2cm"}
Quantum mechanics, via IFM, allows, at least sometimes, to transmit the bit 1 in a counterfactual manner, i.e., without having any particle in the transmission channel, see Fig. 2. Alice arranges a Mach-Zehnder interferometer (MZI) tuned to destructive interference in one of the ports, say $D_1$, such that one arm of the interferometer crosses the place where Bob’s shutter might be. Detection of the particle in the dark port of the interferometer tells her with certainty that the bit is equal to 1 (the shutter is present).
![ A single bit value communication with IFM. The interferometer is tuned in such a way that detector $D_1$ never clicks if the paths are free. Alice knows that bit 1 has been chosen (Bob blocked the path) when she observes the click in $D_1$. []{data-label="fig:2"}](2.png){width="7.2cm"}
The simplest argument that in this case the particle was not present in the transmission channel is: “If the particle were in the transmission channel, it could not be detected by Alice”. In my view, this argument cannot be used for claims about quantum particles [@past]. I, instead, suggest to rely on the fact that the particle does not leave any trace in the transmission channel.
Note that counterfactual transmission of just one bit value can be achieved using a classical particle [@Gisin]. Alice and Bob agree in advance that at a particular time, for a particular value of a bit, Bob sends the particle to Alice, while for the other bit value, he sends nothing. Note, however, that this classical protocol cannot achieve the task of the quantum IFM. In the IFM, Alice learns about the shutter in Bob’s place without prior agreement with Bob and without Bob knowing that she acquired this information.
In the IFM shown in Fig. 2, Alice does not obtain a definite information about the classical bit 0. Without the shutter, the click in the bright port happens with certainty, but it might happen (with probability 25%) with the shutter too. This protocol is also not the most efficient method for communication of the bit 1. When the particle is detected by a bright port (probability 25%) we get no decisive information, and in half of the cases the particle is lost (then we get the information that the bit is 1 but not in a counterfactual manner).
The quantum method can be modified to be a reliable transmission of both values of the bit. To this end, instead of the shutter, Bob inserts a half-wave plate (HWP), see Fig. 3. This transforms the dark port to bright port and vice versa. However, half of the wave always passes through the communication channel, so one cannot argue that this is a counterfactual communication.
![ Communication with MZI and HWP. The interferometer is tuned to destructive interference towards $D_1$. Bob communicates the bit to Alice by changing the destructive interference to detector $D_2$ by inserting the HWP in the right path of the particle. []{data-label="fig:3"}](3.png){width="8.2cm"}
![ Efficient communication using IFM and quantum Zeno effect. a). The chain of interferometers wih highly reflective beam splitters is tuned such that the particle has destructive interference towards $D_1$. b). Bob blocks the interferometers and then, due to Zeno effect, the particle reaches detector $D_1$ with probability close to 1. []{data-label="fig:4"}](4.png){width="7.2cm"}
Consider next what happens when we combine the quantum Zeno effect with the IFM [@ZenoIFM]. It allows to perform a counterfactual transmission of bit 1 with probability arbitrary close to 1. The device consists of a chain of $N$ interferometers with identical beam splitters with small transmittance ${T_1=\sin^2 \alpha}$, see Fig. 4a. Each one of the beam splitters in the chain performs the following unitary evolution of the state of the particle: $$\begin{aligned}
\label{BS} \nonumber
|L\rangle &\rightarrow & \cos \alpha |L\rangle + \sin \alpha |R\rangle,\\
|R\rangle &\rightarrow & -\sin \alpha |L\rangle + \cos \alpha
|R\rangle .\end{aligned}$$ A straightforward calculation shows that $n$ beam splitters perform the following evolution of the wave packets of the particle entering the chain: $$\begin{aligned}
\label{n-steps}
\label{BS} \nonumber
|L\rangle &\rightarrow & \cos n\alpha ~|L\rangle + \sin n\alpha ~|R\rangle,\\
~|R\rangle &\rightarrow & -\sin n\alpha |L\rangle + \cos n\alpha
~|R\rangle .
\end{aligned}$$ For the particular choice of the transmittance parameter, $\alpha = \frac {\pi}{2N}$, after passing all $N$ beam splitters, the wave packet of the particle moves from one side to the other: $|L\rangle \rightarrow |R\rangle$, ${|R\rangle \rightarrow -|L\rangle}$.
The Zeno effect takes place when the right arms of the interferometers are blocked, Fig. 4b. The state remains $|L\rangle$ with probability close to 1 when $N$ is large, $ \cos^{2N} \frac{\pi}{2N} \simeq 1-\frac {\pi^2}{4N} $.
In summary, for bit 0 Bob does nothing and Alice gets the click with certainty at the right port in the detector $D_2$. For bit 1 Bob blocks the interferometers and Alice gets the click with a very high probability in the detector $D_1$.
“Direct Counterfactual Quantum Communication" {#DirectCF}
=============================================
![ Communication with nested MZIs. a). The inner interferometer is tuned such that the particle cannot pass through the right arm of the external interferometer. b). There is a destructive interference towards $D_2$ when the right path of the inner interferometer is blocked. []{data-label="fig:5"}](5a.png "fig:"){width="6.2cm"} ![ Communication with nested MZIs. a). The inner interferometer is tuned such that the particle cannot pass through the right arm of the external interferometer. b). There is a destructive interference towards $D_2$ when the right path of the inner interferometer is blocked. []{data-label="fig:5"}](5b.png "fig:"){width="6.2cm"}
In this section I describe the recent protocol by Salih et al. [@Salih] which followed the idea of counterfactual computation [@Ho06; @Va07].
Let us first consider a MZI nested in another MZI, see Fig. 5. The inner interferometer is tuned for destructive interference toward the second beam splitter of the external interferometer, see Fig. 5a. The external interferometer is tuned for destructive interference towards $D_2$ when the lower path of the inner interferometer is blocked, see Fig. 5b. This configuration provides (sometimes) definite information about value 0 of the bit, namely the absence of the shutter. Indeed, click in $D_2$ is possible only if the shutter is not present. One can naively argue that Alice obtains this information in a counterfactual way, since the particle cannot pass through Bob’s site and reach $D_2$. However, as detailed in Section \[Criteria\], the presence of a weak trace inside the inner interferometer contradicts it.
{height="15.2cm"} {height="15.2cm"}
Salih et al. [@Salih] further argued that a scheme with numerous nested interferometers leads to a protocol for transmitting both values of the bit in a counterfactual way with an efficiency which is arbitrarily close to 100%. In the protocol, $M-1$ chains of the $N-1$ interferometers described in Fig. 4 are parts of another chain of interferometers with $M$ beam splitters having transmittance, $T_1=\sin^2 \frac {\pi}{2M}$. To simplify the analysis, I modify Salih et al. protocol making it slightly less efficient, but still good according to their line of argument. I replace the mirrors of the external chain of interferometers by highly reflective beam splitters. Transmitted waves are lost, but the modification balances the losses in the inner chains when shutters are introduced, such that the states of particles which are not lost are still described by the same equation (\[n-steps\]).
The setup is described in Fig. 6. The external chain of interferometers has $M$ beam splitters with transmittance $T_2=\sin^2 \frac{\pi}{2M}$ and the transmittance of the side beam splitters serving as mirrors is $T_3=1-\cos^{2N} \frac{\pi}{2N}$.
All right mirrors of the internal chains are in Bob’s territory. He knows that Alice sends a particle at a particular time at the top of the external chain in the state $|L\rangle $. For communicating bit 1, Bob blocks all inner interferometers, see Fig. 6a. Then, after $m$ beam splitters of the external interferometer, the [*normalized*]{} quantum state is $$\label{j-step}
|\Psi^{(1)}_m\rangle = \cos^{(m-1)N} \frac{\pi}{2N}\left ( \cos \frac{m\pi}{2M} |L\rangle + \sin \frac{m\pi}{2M} |R\rangle \right )+...~,$$ and after all $M $ beam splitters the state is $$\label{M-step}
|\Psi^{(1)}_M\rangle = \cos^{(M-1)N} \frac{\pi}{2N} |R\rangle +...~.$$ In both equations “...” signify states orthogonal to the term which is shown. If $1 \ll M \ll N$, then the norm of the leading term in (\[M-step\]) is close to 1: $$\label{M-stepNORM}
\cos^{2(M-1)N} \frac{\pi}{2N} \simeq 1- \frac{\pi^2 M}{4N}.$$ Thus, in the limit of large $N$, Bob’s choice of bit 1 leads to a click of Alice’s detector $D_2$. Since the state $|\Psi^{(1)}_M\rangle $ is orthogonal to the state $|L\rangle$ at the output port of the interferometer, there is zero probability for a click of $D_1$. The particle can be lost in Alice’s or Bob’s territories, and then none of the detectors click, but the probability of such a case vanishes for large $N$.
If Bob wants to communicate the bit 0, instead, he does nothing, Fig. 6b. Then, every wave packet entering any of the inner chains of the interferometers follows evolution (\[n-steps\]) inside this chain and does not come back to the external interferometer. At the output of the interferometer, the normalized quantum state is $$\label{M0-step}
|\Psi^{(0)}_M\rangle = \cos^{(M-1)N} \frac{\pi}{2N} \cos^{M} \frac{\pi}{2M} |L\rangle +...$$ Under the condition $1\ll M \ll N$, the norm of the leading term in (\[M0-step\]) is also close to 1: $$\label{M-stepNORM}
\cos^{2(M-1)N} \frac{\pi}{2N} \cos^{2M} \frac{\pi}{2M} \simeq 1- \frac{\pi^2}{4}\left(\frac{M}{ N} +\frac{1}{M}\right).$$ Therefore, the detection of the particle in the left port by $D_1$ tells Alice that Bob sent bit 1.
Note, that the probability for a failure might become large if the condition $1\ll M \ll N$ is not fulfilled. The particle might be lost or detected by $D_2$. However, if the condition holds, the probability for a failure is vanishingly small. The probability for loosing the particle and getting no result is of order $ \frac{\pi^2 M}{4N}$ for bit 1 and $\frac{\pi^2}{4}\left(\frac{M}{ N} +\frac{1}{M}\right)$ for bit 0.
The click of $D_2$ tells Alice with certainty that the bit is 1, and the click of $D_1$ tells her that the bit is 0 with only a very small probability for an error: about $\frac{\pi^2}{4M^2}$. This is a good direct communication protocol.
“Direct quantum communication with almost invisible photons” {#DirectnonZeno}
============================================================
As in the simple example in Sec. \[ComQP\] (Fig. 3), using HWPs instead of absorbing shutters leads to a communication protocol which is theoretically free of errors. Li et al. [@Li] suggest such a protocol and argue that it has “arbitrarily small probability of the existence of the particle in the transmission channel”.
The configuration is similar to the experiment of Salih et al. [@Salih]: a chain of $M-1$ interferometers with inner chains of $N-1$ interferometers (but now $M,N$ have to be even numbers). Without absorbers, the evolution is unitary and the Zeno effect is not used in this protocol. There is no need to modify the protocol by replacing mirrors with beam splitters, because there are no losses to compensate.
The new protocol is different also in the transmittance of the beam splitters in the inner chains. The parameter $\alpha$ is bigger by a factor of 2: $\alpha = \frac {\pi}{N}$. As a result, the chain (without the HWPs) works as two consecutive inner chains of the protocol discussed in the previous section. The first half of the chain moves the wave packet to the right side and the second brings it back to the left. From (\[n-steps\]) we obtain the transformation of the wave packet in the inner chain of the interferometers $|L\rangle \rightarrow -|L\rangle$, see Fig. 7a.
![ The chain of interferometers with highly reflective beam splitters manipulated by the HWPs. a). The wave packet of the particle moves from left to right and then to left again but obtains the phase $\pi$. b). HWPs “undo” the transformation on every second interferometer and the wave packet ends up in the original state on the left without additional phase. []{data-label="fig:7"}](7.png){width="6.2cm"}
When the HWPs are inserted in every interferometer of the inner chain, see Fig. 7b, they cause a $\pi$ phase change of every state $|R\rangle$ and, consequently, every second beam splitter reverses the operation of the previous one: $$\label{2step}
|L\rangle \rightarrow \cos \alpha |L\rangle + \sin \alpha |R\rangle \rightarrow \cos \alpha |L\rangle - \sin \alpha |R\rangle \rightarrow |L\rangle .$$ Since every chain has an even number of beam splitters, the transformation of the wave packet in the inner chain is $|L\rangle \rightarrow |L\rangle$.
{height="15.2cm"} {height="15.2cm"}
The setup for sending bit 0 is described on Fig. 8a. Bob leaves the inner interferometers untouched. Then, each inner chain of the interferometers changes the phase of the quantum state of the particle: $|L\rangle \rightarrow - |L\rangle$. A state $|L\rangle$ of the inner interferometer is a state $|R\rangle$ of the external interferometer. Thus, the operation of the first external interferometer is $$\label{2step}
|L\rangle \rightarrow \cos \alpha |L\rangle + \sin \alpha |R\rangle \rightarrow \cos \alpha |L\rangle - \sin \alpha |R\rangle \rightarrow - |L\rangle .$$ Since the number of beam splitters in the external chain is even, at the end of the process the particle is on the left side and it is detected by detector $D_1$ with certainty.
If Bob wants to communicate the bit 1, instead, he inserts HWPs in all the interferometers of the inner chains Fig. 8b. Now, after every two beam splitters of the inner chain, the wave packet comes back to the left side without acquiring additional phase. Thus, every inner chain works as a mirror and the external chain of the interferometers moves the particle from left to right, to be detected with certainty by detector $D_2$. Alice knows with certainty the choice of Bob by observing which detector clicks. This is an ideal direct communication protocol: theoretically, when there are no losses, there is zero probability for an error.
Criteria for Counterfactuality {#Criteria}
===============================
The question I want to answer in this paper is: Can the protocols of Sections \[DirectCF\] and \[DirectnonZeno\] be considered as [*counterfactual*]{} communication protocols? Let us consider the following three statements which try to capture the meaning of counterfactuality.
1\) The probability of finding the particle in the transmission channel is zero or can be made arbitrarily small.
2\) The particle did not pass through the transmission channel.
3\) The particle was not present in the transmission channel (or the probability of its existence in the transmission channel can be made arbitrarily small).
In the papers on counterfactual communication [@Salih; @Li] all these statements were considered to be interchangeable, i.e. all are true and each one of them represents counterfactuality. I argue that the situation is more subtle and clarification is needed.
Statement (1). A non-demolition measurement of the presence of the particle in the transmission channel disturbs completely the operation of the communication protocols which are considered. When such a measurement is added to the protocol, Bob does not transmit information to Alice by his actions. So, the truth or falsehood of this statement is not a decisive indication of the counterfactuality of the protocols.
However, since there is a separate controversy about the validity of this statement for the two protocols under discussion, it should be clarified too. The papers on counterfactual communication claim that this is a correct statement while I [@MyCom] claim that in these protocols the probability of finding the particle in the transmission channel is 1.
The source of this controversy is our different assumptions. The communication protocols involve preparation and detection of the particle. I consider the probability of finding the particle in the transmission channel under the condition of the same final detection as in the protocol without intermediate measurement. In this case, the probability of finding the particle is exactly 1, since had it it not been found, the result of the final detection could not be that of the undisturbed protocol. On the other hand, without the condition on the result of the final detection, the probability to detect the particle in the transmission channel is vanishingly small. These are two correct statements about the probability of finding the particle in the intermediate measurement: the probability is 1 when both the proper preparation and the proper final detection are done, and it is vanishingly small when only the preparation of the particle is assumed. None of these statements shed much light on the issue of counterfactuality of the protocols [*without*]{} intermediate non-demolition measurements.
Statement (2). In contrast to such a claim for a classical particle, the meaning of this statement for a quantum particle is not well defined. For a classical particle, the operational meaning of (2) is (1), but as discussed above, statement (1) is not helpful in the quantum case. In a double slit experiment with a screen, there is no good answer through which slit the particle passed and through which it did not pass. However, if the detector which finds the particle is placed after one of the slits, and the wave packet passing through the other slit does not reach the detector, then it is frequently declared, following Wheeler [@Whe], that the particle did not pass through the second slit. (Note that this contradicts the textbook picture, attributed to von Neumann, according to which the wave passes through both slits and then collapses to the location of the detector.) If we adopt Wheeler’s definition, then statement (2) is correct for the protocol of Section \[DirectCF\]. The wave packets of the particle passing through the transmission channel do not reach the detector which detects the particle in this protocol. I, however, argued that we should not adopt Wheeler’s definition for discussing the past of a quantum particle [@past].
The concept of a quantum particle passing through a channel has no clear meaning in the standard quantum mechanics in which particles do not have trajectories. It is rigorously defined in the framework of Bohmian mechanics [@Bohm52]. However, since the authors of papers on counterfactual communication never mentioned Bohmian mechanics, it is not particularly relevant to the current controversy. Still, due to to its conceptual importance I, following the advice of a referee, will provide the analysis in the framework of the Bohmian interpretation, but only at the end of the paper, in Section \[Bohm\].
Statement (3). Apart from Bohmian mechanics, quantum mechanics does not provide a rigorous meaning also for statement (3). Without a clear ontological definition I suggest to introduce an operational meaning. We cannot rely on an operational definition based on statement (1), since strong, even nondemolition, measurements change the setup we want to analyze. So, my proposal is to look at the [*weak*]{} trace the particle leaves.
All particles interact locally with the environment. If the particle is present in a particular place, it leaves some trace there, and it does not leave any trace where it was not present. Therefore, we can run the protocol we want to analyze, and then look at the trace left in the environment. If in the transmission channel there is a non-zero trace, we will say that the protocol is not counterfactual. The counterpart of (3), a definition of counterfactual as a process without a local trace, is less robust, because, although very unlikely, it is possible that the particle changed the local environment via some local interaction, but then changed it back to its original state.
Since we are all along analyzing interference experiments, the trace left by the particles has to be small, as otherwise the interference is destroyed. When the trace is small, one may argue that it can be neglected. I, however, claim that it can be neglected only if it is small compare to the trace which a single particle with the same coupling being at the same place would leave. Hence, the remaining task is the comparison between the trace left in the transmission channel in the “counterfactual” communication protocols [@Salih; @Li] and the trace left by a single particle passing through the channel. In the next section I will analyze the minimal trace left by a single particle being in a transmission channel and the comparison will be made in the following sections.
The trace left by a single particle {#presence}
====================================
In the framework of standard quantum mechanics there is no rigorous way to decide if statement (3) holds, that is: whether or not the particle was present in the transmission channel. In a two-slit experiment it is not clear whether the particle was in a particular slit. If a particle starts on one side of a plate with two slits and it is found later on the other side, we do not know if the particle was in the two slits together, or in one of them. However, we firmly believe that it cannot be that the particle was not present in both.
![ A single particle in a single localized wave packet passes from Alice to Bob in the transmission channel. Some trace is invariably left in the channel. We model it as a shift of a local degree of freedom of the channel (the pointer) described by (\[a\]). []{data-label="fig:9"}](9.png){width="7.2cm"}
Consider first a single-path transmission channel with a single particle in the form of a single localized wave packet. The wave packet passes from Alice to Bob, see Fig. 9. Let us model the interaction of the particle with the transmission channel as von Neumann measurement with a Gaussian probe. The initial wave function of the pointer is $$\label{psi0}
\langle x|\Phi_0\rangle = \frac{1}{\sqrt{\Delta \sqrt{\pi}}} e^{-\frac{x^2}{2\Delta^2}}.$$ When a particle is present in the transmission channel, the interaction shifts $x$ by $\delta$. Thus, the state of the measuring device after the interaction is $$\label{a}
|\Phi\rangle =\sqrt{1-\epsilon^2} |\Phi_0\rangle+ \epsilon |\Phi_\perp\rangle,$$ where $|\Phi_\perp\rangle$ is orthogonal to the initial pointer state and $ \epsilon=\sqrt{1-e^{-\frac{\delta^2}{\Delta^2}}}$.
How to quantify the strength of the trace? One option is to consider the probability of detecting the change in the state of the channel in an idealized experiment. The probability equals $\epsilon^2$. Another option is just to use the shift of the pointer via the parameter $\left|\frac{\delta}{\Delta}\right|$.
For a strong trace, the probability criterion does not represent the trace well: it remains almost 1 for $\left|\frac{\delta}{\Delta}\right|=10$ and also for $\left|\frac{\delta}{\Delta}\right|=1000$. In practice, however, it is plausible that in a realistic experiment, when in addition to the quantum uncertainty of the pointer there is an uncertainty of the grid on which we read the pointer, only very large $\left|\frac{\delta}{\Delta}\right|$ can be observed.
For a small value of $\left|\frac{\delta}{\Delta}\right|$, the probability of detection is proportional to the square of this parameter. The transmission of two particles doubles the shift, but increases the probability of detection by a factor of 4. The linear response seems to be a better representation of the trace.
![ A single particle in a superposition of several localized wave packets passes from Alice to Bob. We assume that the beam splitters are arranged in such a way that all packets have equal amplitudes and the beam splitters on Bob’s side are tuned to interfere constructively toward the detector. []{data-label="fig:10"}](10.png){width="7.2cm"}
Consider now a single particle passing through the transmission channel which consists of $N$ identical paths as above. The quantum state of the particle is an equal weight superposition of wave packets in all paths, $|\Psi_{in}\rangle=\frac{1}{\sqrt N}\sum_{i=1}^N |i\rangle$, where $|i\rangle$ signifies the wave packet of the particle inside path $i$ in the transmission channel, see Fig. 10. After the particle passes the transmission channel, the state of the particle and the pointers representing the transmission channel becomes $$\label{1-Nchan}
\frac{1}{\sqrt N}\sum_{i=1}^N
\prod_{j\neq i}| \Phi_0\rangle_j (\sqrt{1-\epsilon^2} |\Phi_0\rangle_i+ \epsilon |\Phi_\perp\rangle_i)|i\rangle.$$ The probability to detect the particle in the transmission channel is the probability to find one of the states $|\Phi_\perp\rangle_i$. It is $\epsilon^2$ as in the case of the single-path channel. The sum of the expectation values of the shifts of the $x_i$s is also the same, $ \sum \langle x_i \rangle = \delta$.
It is important to consider the post-selection measurement of the state of the particle made by Bob. Let Bob select the undisturbed state $|\Psi_{fin}\rangle=\frac{1}{\sqrt N}\sum_{i=1}^N |i\rangle$. This corresponds to detection of the particle by Bob’s detector in Fig. 10. For a good, low noise channel, the probability to find this state is very close to 1. The state of the transmission channel then becomes $$\label{1-Nchan}
\frac{N\sqrt{1-\epsilon^2}\prod_{j=1}^N | \Phi_0\rangle_j +\epsilon\sum_{i=1}^N |\Phi_\perp\rangle_i \prod_{j\neq i} |\Phi_0\rangle_j}{\sqrt{ N^2(1-\epsilon^2)+N \epsilon^2}}
.$$ At this stage, the probability to find one of the states $|\Phi_\perp\rangle_i$ is reduced dramatically. This is because the failure of post-selection by Bob implies that the probability to find one of the states $|\Phi_\perp\rangle_i$ is 1. For small $\epsilon$, the probability to detect the particle in the transmission channel after the successful post-selection is approximately $\frac{\epsilon^2}{N}$.
It is interesting that the post-selection of the particle state does not change the expectation value of the sum of the pointer variables $\langle \sum x_i \rangle = \delta$. One way to see this is to note that $\langle \sum x_i \rangle = \delta$ is proportional to the weak value [@AV90] of the sum of the projections on all parts of the channel which equals 1 because the initial state is the eigenstate of the sum of the projections with the eigenvalue 1 [@AV91]. For describing the magnitude of the trace, the directions of the shifts are not important. So the relevant parameter is $\sum_i|\langle x_i \rangle| $. We have found a lower bound, $\sum_i|\langle x_i \rangle|\geq \delta $.
The weak trace {#weakpresence}
===============
I analyse next the trace left in the transmission channel in communication protocols discussed above. All protocols are based on interference, therefore, when they work properly, only a small trace can be left. I use the same model: in every path of the transmission channel, the presence of the particle shifts the Gaussian pointer, see (\[a\]). I assume that the coupling is weak: $\delta \ll \Delta$, and consequently, $\epsilon \ll 1$. For simplicity, I consider the trace created by particles moving from Alice to Bob and disregard the trace created on the way from Bob to Alice. Then, the example described in Fig. 1, is identical to that of Fig. 9. and the trace in the communication channel is the shift $\delta$ of the pointer and the probability to discover the presence of the particle by observing the trace, is $\epsilon^2$. In the communication of the bit 0 using IFM, Fig. 2, and in the protocol with HWPs, Fig. 3 the trace is of the same order of magnitude.
In the IFM communication of the bit 1, see Fig. 2, after the interaction with the transmission channel, the state of the particle and the pointer is $$\label{1-IFM}
\frac{1}{\sqrt 2}\left [
\Phi_0\rangle |L\rangle + (\sqrt{1-\epsilon^2} |\Phi_0\rangle+ \epsilon |\Phi_\perp\rangle)|absorbed\rangle\right].$$ If the particle is absorbed by Bob, the trace in the channel is exactly as in the single-path channel of Section \[presence\]: shift by $\delta$ and the probability to find the trace is $\epsilon^2$. If the particle is detected by Alice, then there is no trace in the transmission channel. The wave packet “tagged” by an orthogonal state of the channel, $|\Phi_\perp\rangle$ cannot reach Alice.
Now I consider the IFM experiment with the chain of the interferometers, Fig. 4, starting with the communication of bit 0, the case in which Bob leaves the interferometers undisturbed. The exact expressions are complicated, but for weak coupling, only the first order contibution in $\epsilon$ is significant. Neglecting the coupling to the channel, we obtain from (\[n-steps\]) the state of the particle in the $n$th interferometer $$\label{n steps}
\cos \frac{n\pi}{2N} |L\rangle + \sin \frac{n\pi}{2N} |R\rangle .$$ The wave packet $|R\rangle$, “tagged” at the $n$th interferometer by the state $|\Phi_\perp\rangle_n$ in the transmission channel, interferes only with itself and reaches detectors in the state $$\label{N-n steps}
\cos \frac{(N-n)\pi}{2N} |R\rangle - \sin \frac{(N-n)\pi}{2N} |L\rangle .$$ In this experiment, the particle ends up in detector $D_2$ (state $|R\rangle$) with the probability close to 1. The state of the transmission channel then is $$\label{afterNchanABC}
{\cal N} \left ( \prod_{n=1}^{N-1} | \Phi_0\rangle_n + \epsilon \sum_{n=1}^{N-1} \sin^2 \frac{n\pi}{2N}\prod_{j\neq n} |\Phi_\perp\rangle_n \right ),$$ with normalization $|{\cal N}|$ close to 1. Therefore, the probability to detect the particle in the transmission channel, i.e., to find at least one of the states $|\Phi_\perp\rangle_n$, $$\label{afterNchanProb}
\epsilon^2 \sum_{n=1}^{N-1} \sin^4\frac{n\pi}{2N} \sim \epsilon^2 N\frac{2}{\pi}\int_0^{\frac{\pi}{2}} \sin^4x dx = \frac{3\epsilon^2 N}{8},$$ is much larger than the minimal probability to find a single particle present in this multiple-path channel, which can be as low as $\frac{\epsilon^2}{N}$. Thus, the case of the bit 0 is definitely not a “counterfactual” communication. This can also be seen by calculating the pointers shifts. These shifts are proportional to the expectation value of the projection on the paths of the transmission channel. The shift in path $n$ is $\delta \sin^2 \frac{n\pi}{2N}$ and the sum of all shifts, $\sim\delta \frac{N}{2}$, is much larger than $\delta$, the standard for the presence of a single particle in the channel.
The situation is different for communication of bit 1, when Bob blocks the paths of the interferometers, Fig. 4b. Due to Zeno effect, the probability of absorbtion by Bob is negligible. Detector $D_1$ clicks with probability close to 1 telling Alice that the bit value is 1. In this case, there is no trace in the communication channel. It is, therefore, a counterfactual communication for bit value 1.
Let us turn now to the case of nested interferometers, Fig. 5. The case which is particularly interesting is described in Fig. 5a. Bob does not put the shutter in, and detector $D_1$ clicks. Alice knows that the bit is 0, and naively, it seems to be a counterfactual communication since the particle “could not pass through the transmission channel”. Indeed, the wave packet entering the nested interferometer does not reach detector $D_1$.
The interferometer was defined only by demanding destructive interferences in particular situations. To make quantitative predictions, we have to specify the beam splitters of the interferometer. In a possible implementation of the interferometer [@Danan], the first two beam splitters transform the localized wave packet entering the interferometer into a superposition: $$\label{ABC}
|\Psi_{in}\rangle \rightarrow \frac{1}{\sqrt 3}(|A\rangle+|B\rangle+|C\rangle),$$ and the other two beam splitters transform each of the states as: $$\begin{aligned}
\label{ABC2}
\nonumber
|A\rangle &\rightarrow& \frac{1}{\sqrt 3} |1\rangle-\frac{1}{\sqrt 6}|2\rangle +\frac{1}{\sqrt 2}|3\rangle,\\
|B\rangle &\rightarrow& -\frac{1}{\sqrt 3} |1\rangle+\frac{1}{\sqrt 6}|2\rangle +\frac{1}{\sqrt 2}|3\rangle, \\
\nonumber
|C\rangle &\rightarrow& \frac{1}{\sqrt 3} |1\rangle+ \sqrt{\frac{2}{3}}|2\rangle,
\end{aligned}$$ where states $|i\rangle$ signifies a wave packet entering detector $D_i$, ($i$=1,2,3). It is is easy to see that these rules ensure the required destructive interferences.
After the interaction, the joint state of the particle and the channel is $$\label{aABC}
\frac{1}{\sqrt 3} [|A\rangle (\sqrt{1-\epsilon^2} |\Phi_0\rangle+ \epsilon |\Phi_\perp\rangle)+(|B\rangle+|C\rangle)|\Phi_0\rangle].$$ From (\[ABC2\]) we see that the detection of the particle in detector $D_1$ post-selects the state $ \frac{1}{\sqrt 3}(|A\rangle-|B\rangle+|C\rangle).$ Therefore, the state of the channel after the post-selection is $$\label{a1}
\sqrt{1-\epsilon^2}|\Phi_0\rangle+ \epsilon |\Phi_\perp\rangle.$$ This is exactly the same state of the channel as in the case that a single particle passed through it, see (\[a\]). Thus, contrary to the naive expectation, the scheme with nested interferometers does not provide counterfactual communication [@Va07].
The weak trace in “Direct counterfactual quantum communication” {#weakpresenceCount}
===============================================================
Now we are ready to analyze the trace left in the “Direct counterfactual quantum communication". The case of bit 1, Fig. 6a, is simple. The trace in the communication channel is correlated with the final location of the particle. If it is absorbed by Alice, which happens with a probability close to 1 and corresponds to the proper operation of the protocol, the trace is zero. The wave packets “tagged” by orthogonal states of the channel, $|\Phi_\perp\rangle_{m,n}$, cannot reach Alice. The trace is present only if the particle is absorbed by one of the Bob’s shutters which happens with vanishing probability. This is a counterfactual communication protocol for bit 1.
The more interesting case is that of bit 0, when Bob leaves the interferometer undisturbed, Fig. 6b. We assume that the interaction with the channel is small, $\epsilon \ll \frac{1}{M}$, $\epsilon \ll \frac{1}{N}$, so only the first order in $\epsilon$ should be considered.
The amplitude of the wave packet of the particle in the $n$th path of $m$th chain of the inner interferometers $(m,n)$, is $$\label{AMPnm}
\cos^{(m-1)N} \frac{\pi}{2N} \cos^{(m-1)} \frac{\pi}{2M} \sin \frac{\pi}{2M} \sin \frac{n\pi}{2N}.$$ A particle present in the path $(m,n)$ changes the state of the corresponding pointer according to (\[a\]): $$\label{M0-step1}
|\Phi_0\rangle_{m,n} |m,n\rangle \rightarrow \left( \sqrt{1-\epsilon^2} |\Phi_0\rangle_{m,n}+ \epsilon |\Phi_\perp\rangle_{m,n}\right)|m,n\rangle.$$ The wave packet $|m,n\rangle$ “tagged” by the orthogonal state $|\Phi_\perp\rangle_{m,n}$ interferes only with itself and leaves the inner chain in the state, see (\[N-n steps\]): $$\label{N-n stepsSa}
|m,n\rangle \rightarrow \sin \frac{n\pi}{2N} |R\rangle - \cos \frac{n\pi}{2N} |L\rangle .$$ State $|R\rangle$ is lost and the state $|L\rangle$ of the last inner interferometer of the chain $m$ enters from the right the remaining $M-m-1$ large interferometers. The transformation of this state (which is named $|R\rangle$ in the following equation) in the remaining interferometers is: $$\begin{aligned}
\nonumber \label{endlarge}
|R\rangle&\rightarrow& \sin \frac{\pi}{2M} \cos^{(M-m-1)N} \frac{ \pi}{2N} \cos^{(M-m-2)} \frac{\pi}{2M} \times \\
&\times& \left ( \cos \frac{\pi}{2M} |L\rangle + \sin \frac{ \pi}{2M} |R\rangle \right )+ ...~~,\end{aligned}$$ where “...” signifies the wave packets which do not reach the detectors.
In the protocol, the particle is found with probability close to 1 by detector $D_1$. After the detection of the particle, the amplitude of the term $|\Phi_\perp\rangle_{m,n}$ corresponding to detection of the particle in the path $(m,n)$ can be found by collecting the factors in (23-26). It is $$\label{AMPnmPerp}
\frac{\epsilon}{2} \cos^{(M-2)N} \frac{\pi}{2N} \sin^2 \frac{\pi}{2M} \sin \frac{n\pi}{N} \cos^{(M-2)} \frac{\pi}{2M}.$$ As explained in Section \[DirectCF\], the protocol works properly if $1<<M<<N$, so that $$\label{AMPnmappr}
\cos^{(M-2)N} \frac{\pi}{2N} \sim 1,~~~ \cos^{(M-2)} \frac{\pi}{2M}\sim 1,$$ and the probability that one of the orthogonal states $|\Phi_\perp\rangle_{m,n}$ will be found in the transmission channel is, approximately, $$\label{AMPnmapprox}
\sum_{m,n} \frac{\epsilon^2}{4} \sin^4 \frac{\pi}{2M} \sin^2 \frac{n\pi}{N} \sim \frac{\epsilon^2 \pi^ 4 N}{2^7 M^3}.$$
The number of paths in the channel is $\sim MN$. We have seen that a single particle present in such a channel can be found with probability as low as $ \frac{\epsilon^2 }{MN}$ which is smaller than the probability of detection of the particle in the protocol by a factor of approximately $ \frac{N^2 }{M^2}$. Since the protocol works well only when $N\gg M$, the trace in the protocol is larger than the trace of a single particle passing through the channel.
Another criterion of counterfactuality is the sum of displacements of the pointers in all paths of the channel, the standard for which is $\delta$. It can be found by calculating the absolute values of weak values of all projections: $\delta \sum_{m,n}|({\bf P}_{m,n})_w|$ with pre- and post-selection specified by the protocol. The scalar product in the denominator of the weak value is close to 1 since the probability of the post-selection is close to 1. The amplitude of the forward evolving state at path $(m,n)$ is given by (\[AMPnm\]) and, similarly, the amplitude of the backward evolving state is $$\label{AMPnm2}
\cos^{(M-m-1)N} \frac{\pi}{2N} \cos^{(M-m-1)} \frac{\pi}{2M} \sin \frac{\pi}{2M} \sin \frac{(N-n)\pi}{2N}.$$ Thus, the weak value of the projection on the path $(m,n)$ can be approximated as $$\label{WVProj}
{(\bf P}_{m,n})_w\sim \frac{\pi^2}{8M^2}\sin \frac{n\pi}{N},$$ and the sum of all shifts of pointers is $$\label{WVProj}
\delta \sum_{m,n}|({\bf P}_{m,n})_w| \sim \delta \frac{\pi^2 N}{16M}.$$ Since $N\gg M$, the trace in the protocol is much larger than the standard of the trace of a single particle present in such multiple-path channel.
The weak trace in “Direct quantum communication with almost invisible photons” {#weakpresence}
===============================================================================
Let us turn to the “Direct quantum communication with almost invisible photons” protocol [@Li]. When Bob transmits 0, i.e. does nothing, Fig. 8a, the amplitude in the path $(m,n)$ is $$\begin{aligned}
\label{Inv1}
\nonumber
\sin \frac{\pi}{2M} \sin \frac{n\pi}{ N} &{\rm for}& ~m~~ {\rm odd}, \\
0~~~~~~~~ &{\rm for}& ~ m~~ {\rm even}.
\end{aligned}$$ The wave packet $|m,n\rangle$, “tagged” by the orthogonal state $|\Phi_\perp\rangle_{m,n}$, interferes only with itself and it leaves the inner chain in the state $$\label{Inv2}
|m,n\rangle \rightarrow \sin \frac{n\pi}{N} |R\rangle - \cos \frac{n\pi}{N} |L\rangle .$$ The state $|R\rangle$ is lost and the state $|L\rangle$ of the last inner interferometer of the chain $m$ enters from the right the remaining $M-m-1$ large interferometers. Using the fact that “tagging” takes place only for odd $m$, and that the number of the large interferometers is odd ($M$, the number of beam splitters is even), and that after every second beam splitter the wave function repeats itself, we can conclude that the wave packet leaves the last beam splitter of the last inner chain with the same amplitude. The wave packet entering the last beam splitter of the large interferometers transforms into $$\label{Inv3}
\cos \frac{\pi}{2M} |R\rangle - \sin \frac{ \pi}{2M} |L\rangle.$$
In the protocol, the particle is detected with probability close to 1 by detector $D_1$ which detects the state $|L\rangle$. Thus, after detection of the particle, the amplitude of the term $|\Phi_\perp\rangle_{m,n}$ corresponding to the detection of the particle in the path $(m,n)$ can be found by collecting factors from (\[Inv1\]-\[Inv3\]) and a factor $\epsilon$ due to the interaction: $$\label{AMPnmPerpInv}
\frac{\epsilon}{2} \sin \frac{2n\pi}{ N} \sin^2 \frac{ \pi}{2M}.$$ This expression holds only for odd $m$, the amplitude in the paths with even $m$ vanishes. Summing the probabilities of finding the record the particle leaves in all the paths, i.e. summing on odd $m$s up to $M-1$ and on integers $n$ up to $N-1$, we obtain the probability of detection of the particle: $$\label{AMPnmapprox}
\sum_{m,n} \frac{\epsilon^2}{4} \sin^4 \frac{\pi}{2M} \sin^2 \frac{2n\pi}{N} \sim \frac{\epsilon^2 \pi^4N}{2^8 M^3}.$$
A single particle passing through this transmission channel can be found with probability of order $\frac{\epsilon^2}{MN}$. Depending on the ratio $\frac{N}{M}$, it can be smaller or larger than (\[AMPnmapprox\]), so we cannot yet decide on the counterfactuality of the protocol.
Compare now the sum of the pointer shifts in the channel. It can be found by calculating the absolute values of weak values of all projections: $\delta \sum_{m,n}|({\bf P}_{m,n})_w|$ with the pre- and post-selection specified by the protocol. In this case, the pre- and post-selected states are identical, so the weak values are expectation values: $$\label{Inv1111}
({\bf P}_{m,n})_w= \left \{ \begin{array} {cl} \sin^2 \frac{\pi}{2M} \sin^2 \frac{n\pi}{ N} &{\rm for} ~m~~ {\rm odd}, \\
0 &{\rm for} ~ m~~ {\rm even}. \end{array} \right.$$ and $$\label{WVProj0}
\delta \sum_{m,n}|({\bf P}_{m,n})_w| \sim \delta \frac{\pi^2 N}{16M}.$$
The sum of the pointer shifts when there is one particle in the transmission channel equals $\delta$, so the ratio $\frac{N}{M}$ tells us when the sum of displacements in the protocol (\[WVProj0\]) is smaller or larger than that of a single particle present in the channel. We can see that the criterion of the pointer shifts is in agreement with the criterion of the probability of detection.
Let us now repeat the analysis for the case of bit 1, when Bob puts HWPs in every inner interferometer. Now the amplitude in the path $(m,n)$ is $$\begin{aligned}
\label{Inv4}
\nonumber
\sin \frac{m\pi}{2M} \sin \frac{\pi}{ N} &{\rm for}& ~n~~ {\rm odd}, \\
0~~~~~~~~ &{\rm for}& ~ n~~ {\rm even}.
\end{aligned}$$
The wave packet $|m,n\rangle$, can be “tagged” by the orthogonal state $|\Phi_\perp\rangle_{m,n}$ only for odd $n$. Thus, due to the presence of the HWPs, the wave packet comes back unchanged every second beam splitter. It leaves the chain of the inner interferometers in the state $$\label{Inv5}
|m,n\rangle \rightarrow \cos \frac{ \pi}{N} |R\rangle - \sin \frac{\pi}{N} |L\rangle .$$ State $|R\rangle$ is lost, and the state $|L\rangle$ of the last inner interferometer of the chain $m$ enters from the right the remaining $M-m-1$ large interferometers. In the chain of the large interferometers it performs usual evolution (\[n-steps\]) and ends up in the state $$\label{Inv6}
\sin \frac{m\pi}{2M} |R\rangle -\cos \frac{m\pi}{2m} |L\rangle .$$
In the protocol, the particle is detected with probability close to 1 by detector $D_2$ which detects the state $|R\rangle$. Thus, after the detection of the particle, the amplitude of the term $|\Phi_\perp\rangle_{m,n}$ corresponding to the detection of the particle in the path $(m,n)$ can be found by collecting factors from (\[Inv4\]-\[Inv6\]) and the factor $\epsilon$ due to the interaction: $$\label{AMPnmPerpInv}
\frac{\epsilon}{2} \sin^2 \frac{ \pi}{ N} \sin^2 \frac{ m\pi}{2M}.$$ This expression holds only for odd $n$, the amplitude in the paths with even $n$, vanishes. Summing the probabilities of finding the record the particle leaves in all the paths, we obtain the probability of detection of the particle: $$\label{AMPnmapprox2}
\sum_{m,n} \frac{\epsilon^2}{4} \sin^4 \frac{m\pi}{2M} \sin^4 \frac{\pi}{N} \sim \frac{\epsilon^2 3\pi^4 M}{2^6 N^3}.$$
Again, as in the case of bit 0, the ratio $\frac{N}{M}$ tells us if it is larger or smaller than the minimal probability of detection in case of a single particle present in the transmission channel. However, the dependence is opposite: if for bit 0 the probability of detection in the protocol was smaller than single-particle standard, for bit 1 it will be larger, and vice versa.
The pointer shifts criterion of counterfactuality is in agreement with these results. The sum of pointers shifts in all paths of the channel is proportional to the sum of the weak values of all projections: $\delta \sum_{m,n}|({\bf P}_{m,n})_w|$. Also in this case, the pre- and post-selected states are identical and the weak values are expectation values: $$\label{Inv1112}
({\bf P}_{m,n})_w= \left \{ \begin{array} {cl} \sin^2 \frac{m\pi}{2M} \sin^2 \frac{\pi}{ N} &{\rm for} ~n~~ {\rm odd}, \\
0 &{\rm for} ~ n~~ {\rm even}, \end{array} \right.$$ and $$\label{WVProj1}
\delta \sum_{m,n}|({\bf P}_{m,n})_w| \sim \delta \frac{\pi^2 M}{4N}.$$ The ratio $\frac{N}{M}$ tells us when the sum of displacements in the protocol (\[WVProj1\]) is smaller or larger than that of a single particle. If it is smaller for bit 1, it is larger for bit 0, and vice versa.
Security of Counterfactual Protocols {#secur}
====================================
One of the motivations for “counterfactual” protocols, in which no particles are present in the transmission channel, is that it is secure against Eve who is trying to eavesdrop the communication: she has no particles to look at [@CFseq]. The obvious cryptographic weakness of the protocols with shutters which are 100% counterfactual is that Eve can use an active attack detecting the presence of the shutters. Moreover, the shutter can be detected by Eve in a counterfactual way [@CFattack], and recently there has been a claim of a very efficient attack of this kind [@CFInt5].
If Eve uses an active attack, probing Bob’s bit by sending her particles, the counterfactuality property does not help. So, for the analysis of counterfactuality we should only consider passive attacks in which Eve “eavesdrops”, i.e. measures the presence of the particle in the transmission channel. Under this condition, the counterfactual protocols with shutters are secure.
Consider the following counterfactual key distribution protocol. There are two identical chains of interferometers as in Fig. 4. One of the chains is defined as bit 0, and the other as bit 1. On each run of the protocol, Alice randomly sends a single particle through one of the interferometers and Bob randomly chooses one of the interferometers and places the shutters in all of its paths. Every time Alice detects the particle in detector $D_1$ of one of the interferometers, she makes a public announcement. Detector $D_1$ can click only if Alice and Bob chose the same interferometer, i.e. they chose the same bit. This creates a common key.
The multiple shutters Bob placed represent the weak point of the protocol due to the reason above, although we can improve it using detectors instead of shutters and telling Alice to send particles not every time, but only at some random times (using ideas of [@GV]). Anyway, we made a postulate here that Eve only performs some (weak) nondemolition measurements of the presence of the particle running in the interferometer. Let us see if Eve can get some information about the key in this way.
If Eve detects the particle in one of the interferometers, it cannot be the one which generates a correct bit in the key. For a correct bit generation event, Alice and Bob have to choose the same interferometer. Eve can detect the particle only if it is present, i.e., it has to be chosen by Alice. If Bob also chooses this interferometer, then after detection by Eve, the particle has to be absorbed by Bob, so it will not reach Alice.
Only if the interferometer is not chosen by Bob, the particle seen by Eve in the nondemolition measurement can reach Alice, and there is a nonzero probability that the detector $D_1$ will click and thus Alice will declare generation of a bit in the key. But this will be an error bit, since Bob has chosen the other interferometer. Eve introduces errors, and the only information she gets is about these error bits.
Let us turn to the “Direct counterfactual quantum communication” protocol. When Bob places the shutters, Fig. 6a, Eve cannot get information about the correct bit because Eve’s detection causes the loss of the particle. (It is not surprising, since in this case the protocol is counterfactual.)
If the bit is 0, and the interferometer is free, Fig. 6b, Eve’s detection will not necessarily lead to the loss of the particle. Detection of the particle by Eve in the last chain of inner interferometers will lead to part of its wave packet to enter the last beam splitter from the right, so most probably it will create an error: $D_2$ clicks, but Bob’s bit is 0. However, Eve’s detection of the particle in any of the first $M-2$ chains of inner interferometers will lead to part of the particle wave packet to enter the last beam splitter from the left side, so most probably $D_1$ will click, corresponding to a successful transmission of the correct bit. Thus, sometimes, Eve gets a reliable information about the transmitted bit.
Eve, who eavesdrops only by observing the original particle of the communication protocol, cannot learn any correct bit of real counterfactual protocols with shutters which transmit bit 1, but she does learn some correct bits in “counterfactual” communication protocols of bit 0. This provides another argument why such protocols should not be named counterfactual.
Counterfactuality according to the Bohmian Mechanics {#Bohm}
=====================================================
It seems that Bohmian mechanics [@Bohm52], which postulates that every particle has a trajectory, should provide the ultimate answer regarding the counterfactuality of a protocol. There are unambiguous answers to criteria (2) and (3) of Sec. \[Criteria\]: either the Bohmian trajectory passes through the transmission channel, or it does not. However, the fact that the Bohmian trajectory does not pass through the transmission channel does not tell us that Eve, who has an access to this channel, cannot get some information about this communication.
For completeness of the counterfactuality analysis, I will consider the following technical question. Is the Bohmian trajectory present in the transmission channel of the protocols discussed in this paper?
Bohmian position of a particle cannot be in a place where the (forward evolving) wave function vanishes. Therefore, the successful IFMs of the presence of an opaque object, described in Fig. 2 and Fig. 4b, are counterfactual according to the Bohmian trajectory criterion. Moreover, the IFM of the absence of an opaque object, Fig. 5a, is counterfactual too. Direct counterfactual communication [@Salih], as its predecessor [@Ho06] and their variation presented in Fig. 6, are counterfactual. In all these protocols there is no continuous path with non-vanishing wave function between the source and the detector the particle reached.
![ The modification of the experiment shown on Fig. 5a which is not counterfactual according to the Bohmian criterion. The interferometer is tuned such that there is a destructive interference toward $D_2$ when the shutter is present. Thus, if detector $D_2$ clicks, we know that path $A$ is free. Although naively, the particle reaching $D_2$ cannot pass through the inner interfereometer, the Bohmian trajectory of the particle (solid line) does pass through $A$. []{data-label="fig:11"}](11.png){width="7.2cm"}
In the “Direct quantum communication with almost invisible photons” [@Li], see Fig. 8, the probability that the Bohmian particle will pass through the transmission channel can be found from the maximal amplitude in the paths passing through the channel. For bit 0, the amplitude is given by (\[Inv1\]), and therefore, the maximum probability is approximately $\frac{\pi^2}{4M^2}$. For bit 1, the amplitude is given by (\[Inv4\]), and the maximum probability is approximately $\frac{\pi^2}{N^2}$. Thus, for large $M$ and $N$, the probability that the communication is counterfactual is close to 1 for both bit values.
However, it should be mentioned, that in most cases when the communication is not counterfactual, the Bohmian particle crosses the transmission channel, not once, but many times. Given equal probability for bit values, the expectation value of the number of crosses of the transmission channel by the Bohmian particle can be obtained from the sum of the probabilities on all paths. The amplitudes are given by (\[Inv1\]) and (\[Inv4\]). Consequently, the expectation value of the number of crosses is, approximately, $\frac{\pi^2}{4}\left (\frac{M}{N}+\frac{N}{4M}\right)$, and it cannot be much less than 1 for any choice of $M$ and $N$.
Note that in the “Direct counterfactual communication protocol” [*experiment*]{} [@Salih], the expectation value of the number of crosses of the transmission channel is also of order 1, but all these events happen when the particle is lost, and these cases are discarded according to the protocol.
![ Simple protocol which is counterfactual according to Bohmian criterion. Bob places the mirror in the position 1 or 2. This makes clicks of Alice’s corresponding detectors $D_1$ or $D_2$ possible. Only empty wave passes through the transmission channel when Alice’s detector clicks. []{data-label="fig:11"}](12.png){width="7.2cm"}
I mentioned above that the protocol presented in Fig. 5a, which demonstrates the absence of an object in the apparently interaction-free manner, is counterfactual according to Bohm. It is interesting that a slight modification of this setup, similar to the experiment actually performed [@Danan], is [*not*]{} counterfactual. Consider a setup presented in Fig. 11. A straightforward calculations of Bohmian trajectories (similar to that of Bell [@Bell], or in a simplified way [@gili]) show that [*all*]{} particles detected by $D_2$ passed through arm $A$.
In fact, the experiment [@Danan] indicated the presence of the weak trace in $A$, but it had nothing to do with the presence of the Bohmian trajectory there. The weak trace appeared also in $B$ and $C$ where the Bohmian particle was not present.
Empty waves cause observable difference when they “catch” Bohmian particles. This allows a very simple communication protocol which is counterfactual according to Bohmian criterion, see Fig. 12. Particles are sent through a MZI without a second beam splitter at particular times. One of the mirrors is in Bob’s place and he has two options for placing it, such that the wave bouncing off the mirror will end up in detector $D_1$ or $D_2$ on Alice’s side. The wave packet going through another arm ends at detector $D_3$ on Bob’s side. If the particle does not reach $D_3$, Bob knows that it reaches Alice and that [ it he]{} has chosen the detector which will detect the particle. Every time $D_1$ or $D_2$ clicks the Bohmian particle does not pass through Bob’s place. Only an empty wave was directed by Bob’s mirror. Indeed, the wave packets overlap at point $O_1$ or $O_2$ and the Bohmian particle must “change hands” in the overlap, so Bohmian particles reaching Alice’s detectors never cross communication channel.
Probably the majority would not consider the communication protocol described in Fig. 12 as counterfactual. According to Wheeler’s common sense argument [@Whe], the particle reaching Alice could come only through Bob’s site. This example, and Englert et al. setup [@SUR], in which a strong trace (observed at a later time) is left in a place where the Bohmian particle was not present, explain why the Bohmian criterion of counterfactuality does not agree with the intuition of most physicists.
Conclusions {#conc}
============
The standard quantum formalism, in contrast to Bohmian mechanics, does not specify the position of a quantum particle. Thus, it does not provide an unambiguous answer to the question: Is a particular communication protocol counterfactual? I.e.: Was the particle present in the transmission channel? In this paper I analyzed an approach to answering this question based on the weak trace the particle leaves in the channel. I compared the trace left in the channel in recently proposed protocols claimed to be counterfactual with the trace in the protocol constructed to transmit a single particle in the same channel.
In the analysis, I considered two criteria for comparing the traces. First, the probability of finding a conclusive evidence for the presence of the particle, and second, the expectation value of the sum of total shifts of some variables of the channel.
The question of counterfactuality of the protocols is considered in cases the protocols work properly, i.e. when the particle is detected by the right detector. It means that the particle is pre- and post-selected. The protocols were compared with the transmission of a single pre- and post-selected particle. In all these cases the probability of post-selection was closed to 1.
The analyses using the two criteria of the trace led to the same conclusion. [*It is possible to communicate only one value of a bit in a counterfactual way.*]{}
The protocol “Direct counterfactual quantum communication" [@Salih] is fully counterfactual for bit value 1. The trace is identically 0. Nothing changes in the transmission channel and therefore there is zero probability to detect the particle in the transmission channel. Passive eavesdropping provides no information about the transmitted bit.
However, the protocol is not counterfactual for the bit value 0. It is true that by increasing the number of paths in the channel, the probability of finding a conclusive evidence of the presence of the particle reduces, but increasing the number of paths also reduces the probability to find a single quantum particle when it passes the channel. The probability to find the presence of the particle in the transmission channel in the event of successful operation of the protocol is larger than the probability to detect a particle successfully passing through this channel. Eve, using passive attack, obtains some information about the transmitted bit.
The criterion of the shifts of variables of the channel tells us the same. The expectation value of the sum of the shifts is zero for bit 1, but for bit value 0 it is larger than the sum of the shifts when a single particle passes the channel.
The protocol “Direct quantum communication with almost invisible photons” is not fully counterfactual for any bit value. Some trace is always left in the transmission channel. However, if we are ready to consider a protocol as counterfactual when it leaves a trace which is much smaller than the trace of a single particle passing through this channel, then we can arrange that it will be counterfactual for one of the bit values. By playing with the numbers $N$ and $M$ of the inner and the external interferometers respectively, the protocol can be made counterfactual for value 0 or for value 1 of the bit. It [*cannot*]{} be made counterfactual for both.
The analysis of the trace left by a particle passing through a $N$-path channel of Section VI showed a surprising result: the probability of detection in the channel of the successfully transmitted particle is reduced by the factor of $\frac{1}{N}$. It helped me to analyse the counterfactuality of the protocols, but it might also open new avenues for useful quantum communication applications.
I also hope that this study will lead to a deeper understanding of the question: “Where are particles passing through interferometers?” [@past; @morepast; @Bart; @Jordan; @Poto].
I thank Yakir Aharonov, Eliahu Cohen and Shmuel Nussinov for helpful discussions. This work has been supported in part by the Israel Science Foundation Grant No. 1311/14 and the German-Israeli Foundation Grant No. I-1275-303.14.
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abstract: 'This talk focuses on the role of light scalars in cosmology, both Nambu Goldstone bosons and pseudo moduli. The former include QCD axions, which might constitute the dark matter, and more general axions, which, under certain conditions, might play the role of inflatons, implementing [*natural inflation*]{}. The latter are the actors in (generalized) hybrid inflation. They rather naturally yield large field inflation, even mimicking chaotic inflation for suitable ranges of parameters.'
bibliography:
- 'nambu\_talk.bib'
---
[SCIPP 15/04\
UTTG-08-15\
]{}
1.2cm
[**Light Scalars and the Cosmos: Nambu-Goldstone and Otherwise**]{}
Nambu Memorial Symposium, University of Chicago, 2016
1.4cm
[Michael Dine]{}\
0.4cm [*$^{(a)}$Santa Cruz Institute for Particle Physics and\
Department of Physics, University of California at Santa Cruz\
Santa Cruz CA 95064* ]{}\
1.5cm
Introduction: Homage to Nambu
=============================
From my graduate student days, when I first encountered his work on symmetry breaking in the strong interactions, Nambu has been one of my intellectual heroes. This was reinforced during my years at City College, where I regularly heard stories of Nambu from Bunji Sakita, who himself was an admirer. While he was somewhat younger than Nambu, Sakita often regaled me with stories of the War years in Japan, involving both his experiences and Nambu’s. I finally got to know Nambu in the 1980’s, and all of my interactions with him were intellectually stimulating and enhanced by his charm and wit. I remember many conversations at Chicago, but remarks he made at the 1984 Argonne meeting on String Theory, which were thoughtful but cautionary, particularly stand out. My final interactions came shortly after his Nobel Prize. Like many, I sent him a congratulatory note, only to receive a “mailbox is full” reply. About a year later, though, I received the most thoughtful note.
In my own work, Nambu’s influence is perhaps heaviest in the areas of string theory and in the appearance of light scalars in the case of continuous global symmetry breaking. His work with Jona-Lasinio has always been instructive for me in that it takes a model which in detail cannot be taken too seriously, but extracts important, universal features. Some of what I say in this talk I hope can be viewed as a modest effort in this style. While much of my discussion will center on Nambu-Goldstone bosons, I will also consider some questions in strong dynamics, where fermionic condensates will play important roles.
In his famous work on symmetry breaking, the light scalars were the pions of the strong interactions. Today, particularly important roles for light scalars arise in cosmology. Examples include axions as dark matter, but also candidates for the inflaton of slow roll inflation. Indeed, the following statements are often made about inflation:
1. The Planck satellite ruled out hybrid inflation
2. If tensor fluctuations are observed (requiring Planck scale variation of the inflaton) then inflation is necessarily “natural", or “chaotic".
3. In the case of “natural" inflation, this must be understood as “monodromy" inflation or “aligned axions".
Today I want to push back (gently) on these statements.
Small vs. Large Field Inflation
================================
We often distinguish two ategories of Inflationary models:
1. Large Field Inflation: $\phi \gg M_p$. In this rubric, most often one speaks of chaotic inflation or natural inflation and its variant, monodromy inflation. This framework predicts potentially observable gravity waves.
2. Small Field Inflation: $\phi \ll M_p$. This is often associated with “hybrid inflation". In this case, there are no observable gravity waves. For small field hybrid inflation, the challenge is to understand $n_s$ as reported by Planck ($n_s = 0.9603 \pm 0.0073.$) Indeed, the Planck paper[@planck] asserts that hybrid inflation is ruled out.
Theoretical Challenges For Large Field Inflation
------------------------------------------------
Both the small and large field scenarios for inflation face significant theoretical challenges. For the two principle implementations of large field:
1. Chaotic inflation[@chaotic] is typically modeled with monomial potentials; arguably this is the [*definition*]{} of chaotic inflation. One requires very small coefficients for the monomial, and this is typically not enforced by any conventional symmetry (often approximate, continuous shift symmetries for non-compact fields are invoked, but we have little understanding of how such symmetries would arise, say, in string theory). The dominance of a particular monomial requires even greater suppression of a host of terms of the form $\phi^n \over M_p^{n-4}$ for many $n$. Again, why should this be?
2. Natural inflation: natural inflation invokes Nambu-Goldstone bosons as inflatons[@naturalinflation]. This is, indeed, the sense in which natural inflation is strictly natural according to the criterion of ’t Hooft[@thooftnaturalness]: such models contain small parameters protected by approximate symmetries. Axions typically have a periodicity (discrete shift symmetry), a 2 f\_a where $a$ is a canonically normalized scalar field. So $a$ is compact. Correspondingly, the potential for $a$ is of the form: V(a) = \^4 ([a f\_a]{} + ). Successful inflation requires that $a$ vary over a range much larger than the Planck mass, i.e. $f_a \gg M_p$. This is in the realm of graviational physics. The one framework in which we can assess the plausibility of such large $f_a$ is string theory, where this doesn’t seem to be realized in known constructions[@dineetalnatural]. This observation has been elevated to a principle, the [*Weak Gravity Conjecture*]{}, which makes natural inflation, as originally proposed, seem unlikely[@weakgravityconjecture].
In recent years, a plausible generalization of natural inflation has emerged: monodromy inflation[@silversteinwestphal]. Theories of monodromy inflation mimic a large $f_a$ by allowing the axion to vary over a much larger range. This can be understood in terms of a potential: V(a) = \^4 ([[a f\_a]{} + 2 k N]{}). The $2 \pi$ periodicity survives if $k \rightarrow k-1$ when $a$ shifts. This phenomenon was argued to be realized in string theory in [@silversteinwestphal]; here we’ll see some simple (and in fact familiar) realizations in field theory. We should note that there are related proposals, such as “multi-natural inflation", which we will not consider here[@choinatural].
Theoretical Challenges for Small field Inflation
------------------------------------------------
Here we will focus mainly on hybrid inflation. Hybrid inflation is often understood in terms of rather specific (usually supersymmetric) models, but it can be understood in general terms as inflation on a pseudomoduli space. In its favor, as we explain, it is somewhat natural (arguably much moreso than chaotic inflation). But Planck scale corrections are still important and must be suppressed. There is no obvious symmetry explanation for this suppression, so there appears an irreducible tuning at the level of $10^{-2}$ or so. One also requires very tiny couplings to account for $\delta \rho \over \rho$. This smallness may be technically natural, but it is disturbing nevertheless.
More generally, there is a challenge in either the small or large field frameworks: to what extent can one make predictions which would tie to a detailed microscopic picture. We will propose an alternative viewpoint on modeling inflation in this talk, but we won’t give a completely satisfying answer to this question.
Nambu-Goldstone Bosons as the Actors in Inflation: [*Natural Inflation*]{} and its Variants
===========================================================================================
With V(a) = \^4 (/f\_a). The constraints on $f_a$ arise from satisfying the slow roll conditions, for example: =M\_p\^2 [V\^ V]{} 1 require $f_a \gg M_p$. As we have mentioned, this is difficult to realize in string theory. Silverstein and Westphal[@silversteinwestphal] suggested an alternative. They noted that in string models in the presence of branes, axions can “wind". They have, as a result, an approximate periodicity greater than $2\pi$. Silverstein and Westphal found that the potentials for these fields were monomial, yielding a form of chaotic inflation.
The string constructions are somewhat complicated. A readily understood class of models of monodromy inflation can be exhibited in field theory[@dinedrapermonodromyinflation]. Consider an SU(N) supersymmetric gauge theory without chiral matter. In such a theory, there is a gaugino condensate. The idea of non-zero fermion bilinears goes back, of course, to Nambu. In the strong interactions, these have been studied phenomenologically since that time, and more quantitatively in lattice gauge theory. In the last few decades, supersymmetry has provided a context in which such condensates can sometimes be computed analytically. In particular, in the $SU(N)$ supersymmetric gauge theory[@seibergholomorphy], = \^3 e\^[2 i k N]{} e\^[i /N]{}. The phase reflects the spontaneous breaking of a $Z_N$ symmetry.
If we perturb the theory with a susy-breaking gaugino mass term, $m_\lambda \lambda \lambda$, then V() = m\_\^3 ([N]{} + [2 k N]{}). \[thetapotential\] So, as we anticipated above, the naive periodicity $\theta \rightarrow \theta + 2 \pi$ is compensated by changing the branch, $k$. If we elevate $\theta$ to a (pseudo) Nambu-Goldstone boson, $\theta = a(x)/f_a$, then if the axion moves slowly, it simply crosses to the other branches. The tunneling rate scales as e\^[-N\^4 ([m\_]{} )\^3]{}. so the tunneling rate for changes of $k$ is extremely small[@shifmantunneling; @dinedrapermonodromyinflation]. Further observations on the tunneling rate will appear in [@dineetaltoappear].
For sufficiently large $N$ ($N \sim 50-100$) and $m_\lambda$ small (not too small, as will be clear from [@dineetaltoappear], we seem to have what we require for successful natural inflation. One might wonder whether this a particularly plausible story (for example. For example, is such large $N$ in the [*Swampland*]{}?[@swampland]). Similar questions can be raised about the ingredients of brane constructions, which one might think roughly dual to these field theories.
Other Field Theory Realizations
--------------------------------
Witten, long ago, put forward a particularly interesting proposal for monodromy in QCD[@witteninstantonsnot; @wittenetaprime; @wittenetaprimetwo; @witten1998]. He advocated considering QCD from the viewpoint of the Large N expansion. In large $N$, he argued that instanton effects should be exponentially suppressed. Instead, he proposed that quantities like correlation functions of $F \tilde F$, even at zero momentum, should exhibit the same behavior with $N$ as in perturbation theory. For example, in pure gauge theory, a correlator (d\^4 x F F )\^n N\^[2-n]{}. To account for the $2\pi$ periodicity of $\theta$, he suggested that pure QCD, for example, should have $N$ branches, with the $\theta$ dependence of the vacuum energy, for example behaving as E() = [min]{}\_k (+ 2 k)\^2.
It is interesting to revisit these questions in light of our understanding of instantons and $\theta$ in supersymmetric theories, where we have a great deal of theoretical control. This control extends to the inclusion of small soft breakings; of course, we need to pass to large soft breakings if we are to understand real QCD. In any case, already examining eqn. \[thetapotential\], we see exactly the sort of structure anticipated by Witten. In fact, exploring these theories, one can also reproduce Witten’s modification of the non-linear lagrangian of the pseudogoldstone bosons to include the $\eta^\prime$. The origin of the branches in each of these cases is clear: they are associated with the breaking of an approximate $Z_N$ symmetry.
But said this way, it is natural to ask what happens as one increases, say, $m_\lambda$, so that the discrete symmetry is badly broken – and ordinary QCD recovered. One might speculate that the $N$ branches would collapse into a small number, and a more conventional periodicity in $\theta$ would be recovered. The problem with this idea is the presumed suppression of instanton effects.
So it is is interesting to revisit these as well. This will be done in [@dineetaltoappear], but the main point is simple, and not surprising given our experience with SUSY theories. In cases where instanton calculations are reliable, they are not suppressed with $N$. In fact, they reproduce the counting expected from perturbation theory. So on the one hand, the instanton argument for branches does not hold; on the other hand, where one has control, Witten’s conjectures are realized. In [@dineetaltoappear], possible behaviors are enumerated, and a set of possible lattice tests will be enumerated.
To summarize this section, we have considered the two popular realizations of large field inflation:
1. Chaotic inflation with monimials: Here one has the longstanding puzzle of accounting for the suppression of an infinite set of operators – polynomials in fields – at large fields. These are non-compact fields and it is hard to see how this arises from conventional symmetries.
2. Monodromy inflation has simple realizations in field theory, but these require very large gauge groups. It is not clear how plausible this is. Whether similar plausibility issues exist for the string constructions would seem a question worthy of investigation.
So it is interesting to consider other possibilities for large field inflation. Indeed, string theory suggests a simple alternative picture.
Non-Compact String Moduli as an Arena for Inflation
===================================================
String theory possesses non-compact fields which, at least in certain regions of the field space, would seem likely to develop very flat potentials. These are the [*moduli*]{} of the theory. The idea that such fields are candidate inflatons has been around a long time (e.g. [@banksmoduliinflation]). Here we will add some new elements.
To justify the existence of a pseudomoduli space, we assume approximate supersymmetry. The non-compact moduli of interest are typically scalar partners of compact moduli (axions). Together these form the lowest components of superfields, $\Phi = r + i a$. The discrete shift symmetry of the axion, as well as the requirement of sensible behavior in asymptotic regions of the moduli space (e.g. where the theory is typically weakly coupled) allow one to write, for example, the superpotential as a series of terms of the form $e^{-N \Phi}$. This connection between axions and moduli will provide us with a conceptual peg for our discussion. We will see that the combination of large and small field inflation has close parallels to large and small field solutions of strong CP problem. In this section we will:
1. Review (small field) hybrid inflation extracting some general lessons.
2. Discuss large and small field solutions of the strong CP problem
3. Consider Inflation with non-compact moduli
4. Enumerate the Ingredients for successful modular inflation
5. Discuss large field excursions in the moduli space
Hybrid Inflation: Small Field
------------------------------
Hybrid inflation is often defined in terms of fields and potentials with rather detailed, special features, e.g. a so-called waterfall field[@hybrid1; @hybrid2; @hybrid3; @hybrid4]. But hybrid inflation can be characterized in a more conceptual way. Inflation occurs in all such models on a pseudomoduli space, in a region where supersymmetry is badly broken (possibly by a larger amount than in the present universe) and the potential is slowly varying[@dinehybrid].
Essentially all hybrid models in the literature are small field models; this allows quite explicit constructions using rules of conventional effective field theory, but it is not clear that small field inflation is selected by any deeper principle. The simplest (supersymmetric) hybrid model involves two fields, $I$ and $\phi$, with superpotential: W = I (\^2 - \^2). $\phi$ is known as the waterfall field.
Classically, for large $I$, the potential is independent of $I$; $$V_{cl} = \mu^4 ~(\phi=0).$$ The quantum mechanical corrections control the dynamics of the inflaton: V(I) =\^4(1 + [\^2 16 \^2]{} (I \^2/\^2)). $\kappa$ is constrained to be extremely small in order that the fluctuation spectrum be of the correct size; $\kappa$ is proportional, in fact, to $V_I$, the energy during inflation. The quantum corrections determine the slow roll parameters. One has: V\_I = 2.5 10\^[-8]{} \^2 M\_p\^4 = 0.17 ([10\^[15]{} [GeV]{}]{} )\^2 = 7.1 10\^5 ([M\_P]{} )\^2.
### Corrections to the Simplest Model
In addition to the quantum corrections we have described, higher dimension operators of various types, even if Planck suppressed, can have dramatic effects. These include:
1. Kahler Potential Corrections: One expects corrections to the Kahler potential; we will assume here that they are Planck suppressed; the constraints are more severe if they are suppressed by some smaller scale. We organize the effective field theory in powers of $I$[@dinehybrid]. The quartic term in $K$, K = [M\_p\^2 ]{}I\^I I\^I gives too large an $\eta$ unless $\alpha \sim 10^{-2}$. This appears to be one irreducible source of fine tuning in this framework.
2. Superpotential Corrections: thesel are potentially very problematic. For example, one can write: W = [I\^n M\_p\^[n-3]{}]{} At least the low $n$ terms must be suppressed. This might occur as a result of discrete symmetries.The leading power of $I$ in the superpotential controls the scale of inflation. For example, $N=4$, gives $\mu \approx 10^{11}$ GeV and $\kappa \approx 10^{-10}$. With $N=5$, one obtains $\mu \approx 10^{13}$ GeV, and $\kappa \approx 10^{-5}$ The scale $\mu$ grows slowly with $N$, reaching $10^{14}$ GeV at $N=7$ and $10^{15}$ GeV for $N=12$.
This result is interesting from the perspective of understanding (predicting?) the scale of inflation. It is hard to understand a high scale of inflation in this framework without a rather absurd sort of discrete symmetry. This might be taken as an argument for a low scale of inflation. On the other hand, pointing in the opposite direction is $\kappa$, which gets smaller rapidly with $V_0$, In addition, achieving $n_s < 1$, consistent with Planck, required a balancing of Kahler and superpotential corrections. Indeed, the abstract of the Planck theory paper[@planck] includes the assertion: “the simplest hybrid inflationary models, and monomial potential models of degree $n>2$ do not provide a good fit to the data."
Because of the requirement of small parameters, and the inevitably significant amount of fine tuning, the theoretical arguments for small field models over large field models for inflation are hardly so persuasive. Even at low scales it is necessary to have control over Planck scale corrections, and tuning of parameters (at least at the part in $10^{-2}$ level) is required. One also needs a very small dimensionless parameter, progressively smaller as the scale of inflation becomes smaller. Quite likely, any successful model requires significant discrete symmetries (or even more severe tunings).
Generalizing hybrid inflation to large fields: moduli inflation
---------------------------------------------------------------
So it is clearly interesting to explore the possibility of inflation on (non-compact) moduli spaces with fields undergoing variations of order Planck scale or larger. Such moduli spaces are quite familiar from string theory. First it is instructive to consider another situation where such a small field/large field dichotomy arises: the axion solution to the strong CP problem.
### Small Field and Large Field Solutions to the Strong CP Problem
To [*solve*]{} the strong CP problem one must account for an accidental global symmetry which is of extremely high [*quality*]{} . Most models designed to obtain a Peccei-Quinn symmetry can be described as small field models; they are constructed with small axion decay constant, $f_a \ll M_p$, with $f_a = \langle\phi\rangle$ In this case, one can organize the effective field theory in powers of $\phi/M_p$ (again, if higher dimension operators are suppressed by a scale $M \ll M_p$, the objections discussed here to the Peccei-Quinn solution are even more severe).
We can define a notion of Axion [*Quality*]{}[@dineetalquality]. We require $$Q_a \equiv {1 \over f_a m_a^2}{\partial V \over \partial a} = 10^{4} {f_a} {\partial V \over \partial a} < 10^{-11}.$$
In small field models, if the axion is the phase of $\phi$, the PQ symmetry is the transformation $\phi \rightarrow e^{i \alpha} \phi$. This symmetry must be [*extremely*]{} good[@kamionkowski]. One needs to suppress $\phi^N \over M_p^{N-3}$ up to very high $N$. E.g. $Z_N$, with $N> 11$ or more, depending on $f_a$. This is not terribly plausible. It involves models of a high degree of complexity, designed to solve a problem of essentially no consequence (small $\theta$ is not, by itself, singled out by anthropic or similar considerations[@dineetalquality; @ubalditheta].
### Large field solutions of the Strong CP Problem
String theory has long suggested a large field perspective on the axion problem[@wittenso10]. String theory, as we have stressed, frequently possesses axions. These axions exhibit continuous shift symmetries in some approximation (e.g. perturbatively in the string coupling). Non-perturbatively these symmetries are broken, but usually one has a discrete shift symmetry left which is exact: a a + 2 .We have normalized $a$ to be dimensionless; $f_a$ depends on the precise form of the axion kinetic term. The (non-compact) moduli which accompany these axions typically have Planck scale vev’s. Calling the full chiral axion superfield ${\cal A} = s + ia + \dots$, this periodicity implies that, for large $s$, in the superpotential the axion appears as $e^{-\cal A}$. Solving the strong CP problem then requires suppressing only a small number of possible terms[@bobkovraby; @dinefestucciawu].
As always, in string theory, one has to understand stabilization of moduli. More honestly (thinking of Nambu’s cautionary remarks) in the current state of our knowledge, we can at best conjecture that moduli are stabilized. If string theory is to produce an axion which can solve the strong CP problem, typically several moduli must be stabilized. Whatever the mechanism, the axion multiplet is special. If the superpotential plays a significant role in stabilization of the [*saxion*]{}, it is difficult to understand why the axion should be light. $e^{-{\cal A}}$ would badly break the PQ symmetry if responsible for saxion stabilization. So the stabilization must result from Kahler potential effects (presumably connected with supersymmetry breaking). In perturbative string models the Kahler potential is often a function of ${\cal A} + {\cal A}^\dagger$. There is no guarantee that would-be corrections to $K$ which stabilize ${\cal A}$ do not violate this symmetry substantially, but will take as a hypothesis.
For example, as a model, suppose there is some other modulus, $T = t + ib$, appearing in the superpotential as $e^{-T}$, where $e^{-T}$ might set the scale for supersymmetry breaking. W(T) = A e\^[-T/b]{} + W\_0 with small $W_0$, leading to T b (W\_0). The potential for $s$ would arise from terms in the supergravity potential: V\_s = e\^[K]{} W \^2 g\^[[A]{} [A]{}\^\*]{} + …For suitable $K({\cal A},{\cal A}^*)$, $V$ might exhibit a minimum as a function of $s$. If $s$ is, say, twice $t$ at the minimum, $e^{-{\cal A}}$ is severely suppressed, as is the potential for the (QCD) axion, $a$.
### A Remark on Distances in the Modulus Geometry
Typical metrics for non-compact moduli fall off as powers of the field for large field. Defining $s$ to be dimension one, g\_[[A]{},[A]{}\^\*]{}=C\^2 M\_p\^2/s\^2 for some constant, $C$. So large $s$ is far away (a distance of order $a~M_p \log(s/M_p)$) in field space. If, for example, the smallness of $e^{-(s + ia)}$ is to account for an axion mass small enough to solve the strong CP problem, we might require $s \sim 110~M_p$ , corresponding to a distance of order $8 M_p$ from $s = M_p$ if $C = \sqrt{3}$.
Non-Compact Moduli as Inflatons
--------------------------------
So the strong CP problem points to Planck scale regions of field space as the arena for phenomenology. This has parallels in inflation. Moduli of the sort required for the axion solution might also play a role as inflatons. There are some plausible ingredients for moduli as the players in inflation:
1. In the present epoch, one or more moduli which are responsible for hierarchical supersymmetry breaking.
2. In the present epoch, a modulus whose superpotential is highly suppressed, and whose compact component is the QCD axion. This is not necessary for inflation, but is the essence of a modular (large field) solution to the strong CP problem.
3. At an earlier epoch, a stationary point in the effective action with higher scale supersymmetry breaking then at present and a positive cosmological constant.
4. At an earlier epoch, a field with a particularly flat potential which is a candidate for slow roll inflation.
Fields need not play the same role in the inflationary era that they do now. The Peccei-Quinn symmetry might be badly broken during inflation. Then the axion will be heavy during this period and isocurvature fluctuations may not be an issue[@dineanisimov]. In such a case the initial axion misalignment angle, $\theta_0$, would be fixed rather than being a random variable. We know that the scale of inflation is well below $M_p^4$. So it is plausible that even during inflation moduli have large vev’s, $e^{-{\cal A}},~e^{-T} \ll 1$, though much smaller than at present. For example, suppose that there exist a pair of moduli, ${\cal A}, T$ responsible for supersymmetry breaking, and an additional field, $I$, which will play the role of the inflaton. During inflation, H\_I \~W \~e\^[-t]{} For typical Kahler potentials, the curvature of the $t$ and $i$ potentials will be of order $H_I$ (for $i$, this is the usual “$\eta$ problem"). We will exhibit a model with lower curvature below.
A successful model requires a complicated interplay between effects due to the Kahler potential and superpotential[@dinelargefield].
- The [*potential*]{} must possess local, supersymmetry breaking minima in ${\cal A}$ and $T$, one of higher, one of lower, energy. The former is the setting for the inflationary phase; the latter for the current, nearly Minkowski, universe.
- In the inflationary domain, the potential for $I$, must be very flat over some range.
- In the inflationary domain, the imaginary parts of ${\cal A}$ and $T$ should have masses comparable to $H_I$ (or slightly larger), if the system is to avoid difficulties with isocurvature fluctuations. This would arise if $e^{-s} \approx e^{-t}$.
- In the present universe, the imaginary part of ${\cal A}$ should be quite light and that of $I$ much lighter.
- There are additional constraints from the requirement that inflation ends. For some value of Re $I$, the inflationary minimum for $T$ and ${\cal A}$ must be destabilized (presumably due to Kahler potential couplings of $I$ to ${\cal A}$ and $T$). At this point, the system must transit to another local minimum of the potential, with nearly vanishing cosmological constant.
- The process of transiting from the inflationary region of the moduli space to the present day one is subject to serious constraints. Even assuming that there is a path from the inflationary regime to the present one, the system is subject to the well-known concerns about moduli in the early universe[@bankskaplannelson; @ibanez]. If they are sufficiently massive (as might be expected given current constraints on supersymmetric particles), they may reheat the universe to nucleosynthesis temperatures, avoiding the standard cosmological moduli problem. $T$ and ${\cal A}$ are vulnerable to the moduli overshoot problem[@brusteinsteinhardt], for which various solutions have been proposed.
### Inflationary Models: Large $r$
Here we describe a simple model which yields large $r$ and satisfies some of the conditions enumerated above (because it involves only a single field and we specify the potential only in a limited range it cannot satisfy all)[@dinelargefield] For the Kahler potential we take: K = -[N]{}\^2 (I + I\^\*). With I = e\^[/[N]{}]{} the kinetic term for $\phi$ is simply $\vert\partial \phi \vert^2$. V() = e\^[-[N]{} ]{} V\_0, $V_0$ being the minimum of the $S$, $T$ potential.
The slow roll parameters are: = [1 2]{} [N]{}\^2; = [N]{}\^2 = 2 . Note n-1 = -2 . If $r = 0.2=16 \epsilon$,n\_s -1 = 0.025 on the high end of the range favored by the Planck measurement.
This model is discussed in the Planck theory paper which rules it out based on their measurement of $n_s$. Suitable modifications are discussed in [@dinelargefield]. A model with similar features (with $\cosh$ rather than exponential potential) has been discussed in[@nojiri].
### Connection to Chaotic Inflation
Chaotic inflation has, for decades, provided a simple model for slow roll inflation, and its prediction of transplanckian field motion and observable gravitational radiation is compatible with our discussion of large field modular inflation. As we look at the moduli inflation model of the previous section (and more generally moduli models of large field inflation), we see, in fact, a realization of the ideas of chaotic inflation[@dinelargefield]. Again, the potential behaves as V \~H\_I\^2 M\_p\^2 e\^[[N]{} ]{}
The exponent changes, during inflation, by a factor of about $3/2$. So we can make a crude approximation, expanding the exponent and keeping only a few terms. If we focus on each monomial in the expansion, the coefficient of $\phi^p$, in Planck units, is: \^p = [10\^[-8]{} [N]{}\^p p!]{}. where $N$ is the humber of $e$-folds.
We can compare this with the required coefficients of chaotic inflation driven by a monomial potential, $\phi^p$. In this case, \_p = [3 10\^[-7]{} (2Np)\^[[p 2]{} - 1]{}]{} These coefficients are not so different. For example, for $p=1$, the moduli coefficient is about $2 \times 10^{-9}$, while for the chaotic case it is about four times smaller; the discrepancy is about a factor of two larger for $p=2$. So we see that these numbers, which would one hardly expect to be identical, are in a similar ballpark.
So moduli inflation provides a rationale for the effective field theories of chaotic inflation. The typical potential is not a monomial, but one has motion on a non-compact field space, over distances of several $M_p$, with a scale, in Planck units, roughly that expected for chaotic inflation. The structure is enforced by supersymmetry and discrete shift symmetries.
Summary
========
Physicists are likely a long way from writing down [*the*]{} microscopic model which describes inflation. For the time being the most sensible approach is to consider classes of models, the constraints coming from observations, and possible general features and predictions. Here we have discussed some features of several classes of models:
1. Chaotic inflation: we have reviewed why it is puzzling as usually formulated, and have seen that something like it may arise in frameworks which are more natural.
2. Natural inflation in its simplest formulation unlikely.
3. Monodromy inflation: has simple field theory realizations, with large amounts of inflation requiring [*very*]{} large gauge groups; we have speculated about the implications for the plausibility of the mechanism.
4. Hybrid inflation: we have stressed that hybrid inflation should be thought of as inflation on a pseudomoduli space. from this vantage point, large field seems plausible (the simplest forms of small field are ruled out, and even these are highly tuned). Large field hybrid inflation favors higher scales for inflation, and has features which can mimic chaotic inflation.
In all cases, detailed implementations are challenging; one wants to ask whether there are any generic features one can extract ($r$, non-gaussianity,...) and compare with data.
Explaining inflation from an underlying microscopic theory is an extremely challenging problem, quite possibly inaccessible to our current theoretical technologies. As we have reviewed, even in so-called small field inflation, it requires control over Planck scale phenomena. Within string theory, this requires understanding of supersymmetry breaking (whether large or small) and fixing of moduli in the present universe as well as at much earlier times. It requires an understanding of cosmological singularities, and almost certainly of something like a landscape.
We have stressed a parallel between small/large field inflation and small/large field solutions to the strong CP problem. The existence of moduli in string models is strongly suggestive of the large field solutions to both problems. The proposal we have put forward here is similar to the large field solutions of the strong CP problem.
Several moduli likely play a role in inflation in order to achieve the needed degree of supersymmetry breaking and slow roll. We have noted that small $r$ is more tuned than large $r$, giving some weight to the former possibility. We have noted the contrast with small field inflation, where extreme tuning to achieve low scale inflation is replaced by the requirement of an extremely small dimensionless coupling.
Returning to the strong CP problem, any would-be Peccei-Quinn symmetry is an accident, and the accident which holds in the current configuration of the universe need not hold during inflation; this would resolve the axion isocurvature problem. It would imply that $\theta_0$ is not a random variable.
The inflationary paradigm is highly successful; the question is whether we can provide some compelling microscopic framework and whether it is testable. In the present proposal, one does not attempt (at least for now) a detailed microscopic understanding, but considers a class of theories. Within those considered here:
1. Higher scales of inflation are preferred
2. High scale axions are likely, and the idea of an [*axiverse*]{} gains additional plausability[@axiverse].
In a more detailed picture, one might hope to connect some lower energy phenomenon, such as supersymmetry breaking, with inflation.
.2cm [**Acknowledgements:**]{} I thank my collaborators Patrick Draper, Guido Festuccia, Laurel Stephenson-Haskins, and Lorenzo Ubaldi for the many insights they have shared with me. This work was supported in part by the U.S. Department of Energy grant number DE-FG02-04ER41286.
|
---
abstract: 'In a recent paper we discovered that a fermionic condensate is formed from gravitational interactions due to the covariant coupling of fermions in the presence of a torsion-fermion contact interaction. The condensate gap gives a negative contribution to the bare cosmological constant. In this letter, we show that the cosmological constant problem can be solved without fine tuning of the bare cosmological constant. We demonstrate how a universe with a large initial cosmological constant undergoes inflation, during which time the energy gap grows as the volume of the universe. Eventually the gap becomes large enough to cancel out the bare cosmological term, inflation ends and we end up in a universe with an almost vanishing cosmological term. We provide a detailed numerical analysis of the system of equations governing the self regulating relaxation of the cosmological constant.'
author:
- Stephon Alexander
- Deepak Vaid
bibliography:
- 'chiral\_condensate5.bib'
title: A fine tuning free resolution of the cosmological constant problem
---
Introduction
============
There are many faces to the cosmological constant/dark energy problem. First, the naive perturbative theoretical evaluation of the vacuum energy of all particles in the standard model gives a result that disagrees with observations by 120 orders of magnitude [@Carroll-2001]. Second, a confluence of cosmological and astrophysical observations, such as the WMAP satellite [@spergel-2006] and Type Ia supernovae [@astier-2006], indicate that the cosmological constant or something very similar to it, currently dominates the universe.
Perhaps the most striking aspect of the cosmological constant problem is seen in the details of the inflationary paradigm[@Brandenberger:2002wm]. Inflation is driven by a constant part of the Energy-Momentum tensor of a scalar field, which is indistinguishable from a pure cosmological constant. Therefore, any mechanism which relaxes the cosmological constant would also prevent inflation from happening. One way out of this possible conundrum is to do away with fundamental scalar fields, allow inflation to occur with a large cosmological constant and investigate any self consistent mechanism which negates the cosmological constant to almost zero. Such a mechanism would solve all three cosmological constant problems:
- The cosmological constant would be dynamically relaxed due to the non-trivial dynamics of inflation itself; it would be self regulatory.
- Dark energy and the coincidence problem would be explained if a residual amount of cosmological constant would be left over by the end of inflation.
- Since inflation is not driven by a fundamental scalar fields, fine tuning of the cosmological constant is no longer needed.
Attempts at tackling the this problem via cosmological condensates include [@brandenberger-1997; @brout-2003; @arkanihamed-2004; @alexander-2004; @alexander-2005]. More recently, Prokopec proposed a mechanism involving a Yukawa coupling between a scalar field and fermions [@prokopec-2006].
A simple way to obtain inflation in the absence of matter is due to the presence of a non-zero, positive cosmological term on the right hand side of Einstein’s equation:
$$\begin{aligned}
G_{ab} = 8\pi G \Lambda_0 g_{ab}\nonumber \\
\Rightarrow a(t) = a_0 (t) e^{\sqrt{\frac{\Lambda_0}{3}}t}\end{aligned}$$
where we have used the FRW metric ansatz to obtain our solution. $a(t)$ is the scale factor. From the solution it is clear that the Hubble rate $H = \sqrt{\frac{\Lambda_0}{3}}$.
While this simple model gives us an inflating universe it is clearly not in line with reality because it does not predict an end to inflation. A way to get around this obstacle is to introduce matter, traditionally scalar fields, into the picture. Then the first Friedmann equation becomes:
$$3\left( \frac{\dot a}{a} \right)^2 = \Lambda_0 + \frac{1}{2}
\dot \phi^2 + V(\phi)$$
where $\phi$ is the scalar field.
We also have the E.O.M for the scalar field:
$$\ddot \phi + 3 \frac{\dot a}{a} \dot \phi + \frac{dV}{d\phi} = 0$$
where $V(\phi)$ is the scalar field potential. Such models typically require special initial conditions for the scalar field called the “slow-roll” conditions. The scalar field must start off at a large initial value and then start rolling slowly down an almost flat potential. This results in an inflationary universe. After a sufficient number of e-foldings, the scalar fields reaches the steeper part of the potential where it decays via parametric resonance leading to reheating and particle production after inflation has ended.
Unfortunately, such models have several shortcomings:
- The shape of the potential is arbitrary and we have no physical way of choosing the one that would correspond to our universe from an almost infinitely large family.
- We require the scalar field to start off at a large initial value. What mechanism would cause the scalar field to be “pumped up” to this value initially?
- The mass of the scalar field is an arbitrary parameter. It can be fixed once we fix the potential, but it remains a source of vagueness.
- Most importantly, from whence did this scalar field come . Perhaps if one tries to tackle this fundamental question head on the others might also be amenable to a solution.
In this letter we propose a dynamical solution to the CC problem assuming only the Standard Model and General Relativity. There are no fundamental scalar fields to tune. Therefore the universe will be dominated by a large cosmological constant, which naturally generates inflation. The non-trivial observation here is that the dynamics of inflation itself holds the key to relaxing the cosmological constant without fine tuning. How is this possible? The exponential time dependent behavior of de Sitter space counterintuitively enhances correlations between fermion pairs. These correlations become so strong that these fermions form a Cooper pair.
In a recent paper [@alexander-2006], we showed how the presence of torsion and fermionic matter in gravity naturally leads to the formation of a fermionic condensate with a gap which depends on the 3-volume. In this letter we will analyze explicitly the dynamics of the universe with a cosmological constant in the presence of this gap. Numerical calculations then show that with a large initial cosmological term and generic initial conditions for the scalar field and its momenta, we obtain a universe which undergoes an inflationary phase during which the gap grows as a function of $a^3$, causing the effective cosmological term to diminish to a small positive value.
In Section 2 we discuss the E.O.M for our system. In Section 3 we present the numerical results and finally we conclude with some discussion or our results and what they imply for our understanding of inflation and the cosmological term.
Friedmann and scalar equations
==============================
We briefly summarize the steps that were taken in [@alexander-2006]. We started with the Holst action for gravity with fermions:
$$S_{H+D} = \frac{1}{2\kappa}\int d^{4}x\,e(\,e^{\mu}_{I}e^{\nu}_{J}R^{IJ}_{\mu\nu} - \frac{2}{3}\Lambda_0) -
\frac{1}{2\kappa\gamma}\int d^{4}x\,e\,e^{\mu}_{I}e^{\nu}_{J}\star R^{IJ}_{\mu\nu} + \frac{i}{2}\int d^{4}x\,e\,(\bar{\Psi}\gamma^{I}e^{\mu}_{I}D_{\mu}\Psi -
\overline{D_{\mu}\Psi}\gamma^{I}e^{\mu}_{I}\Psi)$$
$e^\mu_I$ is the tetrad field. $R_{\mu\nu}^{IJ}$ is the curvature tensor. The second term in the above equation is analogous to the $\Theta$ term in Yang-Mills theory and is required if we want to work with arbitrary values of the Immirzi parameter ($\gamma$). After varying the action w.r.t the connection $A^\mu_{IJ}$ and solving the Gauss constraint we which that $A^\mu_{IJ} =
\omega^\mu_{IJ} + C^\mu_{IJ}$[^1], where $\omega^\mu_{IJ}$ is the tetrad compatible connection and $C^\mu_{IJ}$ can be expressed in terms of the axial vector current:
$$C_\mu^{IJ} = \frac{\kappa}{4}\frac{\gamma^2}{\gamma^2 + 1}j^M_a\left \{ \epsilon_{MK}{}^{IJ}e^K_\mu
- \frac{1}{2\gamma}\delta^{[J}_M e^{I]}_\mu \right\}$$
where $ j^M_a = \bar\Psi\gamma_5\gamma^M\Psi$. Inserting the torsion into the first order action we find the resulting second order action which now contains a four-fermi interaction and the tetrad is the only independent variable, the connection having already been solved for in the previous step.
$$S[e,\Psi] = S_{H+D}[\omega(e)] -\frac{3}{2}\pi G \frac{\gamma^{2}}{\gamma^{2}+1} \int d^{4}x\,e (j_a^I)^2$$
Then we did the 3+1 decomposition of the action to find the Hamiltonian, which after making the ansatz of a FRW metric becomes:
$$\begin{aligned}
\label{ham}
{\cal H} &=& -\frac{3}{\kappa}a^3H^2 +
a^3\Lambda_0 + \frac{i}{a}\big(\psi_L^\dag\sigma^a\partial_a\psi_L - \psi_R^\dag\sigma^a\partial_a\psi_R\big) \nonumber \\
&+& \frac{3\kappa}{32a^3}\frac{\gamma^2}{\gamma^2+1}\left[\psi_L^\dag\psi_L - \psi_R^\dag\psi_R\right]^2 = 0\end{aligned}$$
We see that the right hand side is the sum of the gravitational, Dirac and interaction terms. $\psi_L (\psi_R)$ is the spinor for left (right) handed fermions. $\gamma$ is the Immirzi parameter. $H
= \frac{\dot a}{a}$ is the Hubble rate.
The key ingredient that dynamically cancels the cosmological constant arises from the four-fermion interaction in the r.h.s of (\[ham\]). This effect arises from an interplay between general covariance and non-perturbative quantum mechanics. General covariance guarantees the four-fermion interaction. What about the non-perturbative quantum mechanics? We see that the effective coupling of the four-fermion interaction becomes large for small values of the scale factor (ie. at early times). The form of this Hamiltonian maps directly into the BCS Hamiltonian of superconductivity, except it is the gravitational field that is playing the role of the phonons. As a result, just like in the BCS theory (see eg. [@Fetter_Walecka]), an energy gap $\Delta$ opens up which reflects the instability of the ground state associated with the bare cosmological constant. An effective cosmological constant with a lower energy is generated from the formation of the gap. To obtain the gap, we diagonalize the fermionic part of this Hamiltonian by expanding the fermions in normal modes and using a Boguliubov transformation. The resulting Hamiltonian is:
$$\begin{aligned}
\mathcal{H} &=& -\frac{3}{\kappa}H^2 + \frac{1}{\kappa}(\Lambda_0 -
\Lambda_{corr})
+ \nonumber \\
&& \int \frac{d^3k}{(2\pi)^3} \sqrt{E_k^2 + \Delta^2}(m_k + \bar m_k + n_{-k} + \bar
n_{-k})\end{aligned}$$
where the non-perturbative correction to the bare cosmological constant is[^2]:
$$\begin{aligned}
\label{gap1}
\Lambda_{corr} & = & 2 \Delta^2 \nonumber \\
\Delta & = & \frac{2\hbar\omega_D \exp^{\frac{\nu}{2}}}{\exp^{\nu} - 1},
\qquad \left(\nu = \frac{2}{\kappa\, a^3 k_f^2}\frac{\gamma^2 + 1}{\gamma^2} \right)\end{aligned}$$
$k_f$ is the fermi energy, $E_k$ is the energy of the $k^{th}$ mode of the condensate and $\gamma$ is the Immirzi parameter. $m_k (n_k)$ and $\bar m_k (\bar n_k)$ are the creation (annihilation) operators for the condensate of the left and right-handed fermions respectively. The $a^3$ factor in $\Delta$ comes from the fact that the density of states in an expanding universe scales as the 3-volume. We see that the last term in the Hamiltonian constraint corresponds to the quantized expression for a scalar field condensate, $\phi_{c}$, with mass $\Delta$ [^3]. Replacing this with the classical expression for a scalar field we get:
$$\mathcal{H} = -\frac{3}{\kappa}H^2 + \frac{1}{\kappa}(\Lambda_0 - 2\Delta(a)^2)
+ \frac{1}{2}\dot\phi_c^2 + \frac{1}{2}\Delta(a)^2 \phi_c^2 = 0$$
It is important to keep in mind that $\phi_c$ is not a fundamental scalar field. Its annihilation and creation operators ($m_k$ and $n_k$) correspond to excitations of the condensate. This leads to the first Friedmann equation with a time-dependent correction to the cosmological constant and a scalar field as our matter:
$$\label{friedmann_eom}
3\left(\frac{\dot a}{a}\right)^2 = \Lambda_0 - 2\Delta(a)^2
+ \frac{1}{2}\dot\phi_c^2 + \frac{1}{2}\Delta(a)^2 \phi_c^2$$
after setting $\kappa = 1$.
The equation of motion for a scalar in a FRW background is:
$$\label{scalar_eom}
\ddot \phi_c + 3\frac{\dot a}{a}\dot \phi_c + \Delta(a) \phi_c^2 = 0$$
We see that the energy gap (\[gap1\]) increases monotonically with $a$. From this we can guess the qualitative behavior of the scale factor. As long as the initial value of the scale factor is such that $2 \Delta^2 < \Lambda_0$, then from (\[friedmann\_eom\]) we see that the right hand side will be positive definite resulting in an inflating universe. The Hubble rate plays the role of friction for the scalar field. As time develops the friction will drive $\dot
\phi_c$ to reach zero. From then until inflation ends, $\phi_c$ will be a constant. Eventually the scale factor becomes large enough and the right side of (\[friedmann\_eom\]) will start to decrease. $H$ will then decrease and reach its minimum when:
$$\label{critical_gap}
\Lambda_0 = 2 \Delta^2 - \frac{1}{2}\Delta^{2}\phi_{c}^{2} -
\frac{1}{2}\dot \phi_c^2$$
$\phi_c$ will then start rolling down the potential hill again, which is becoming steeper because $a(t)$ and hence $\Delta$ is still increasing. This presence of the scalar condensate coupling in the r.h.s of (\[critical\_gap\]) means that when the system dynamically relaxes to $\Lambda_{eff} = 0$, it is in a state in which the energy density of the effective cosmological constant $\Lambda_{eff}=\Lambda_{0}-2 \Delta^2 $ traces the energy density of matter. This condition is similar to the relaxation mechanism due to backreaction of IR gravitational waves in which the backreaction effects ceases to negate the cosmological constant and one reaches a scaling solution where the energy density of matter and radiation traces the effective cosmological constant[@brandenberger-2002; @abramo-1999]. We will see in the next section that once the cosmological constant is canceled the tracking solution is dynamically reached without any fine tuning and the cosmological constant will remain vanishingly small.
We can see that the kinetic energy of the scalar field will dissipate eventually, due to a small but non-zero $H$. $H = 0$ is the late-time attractor for this system. As $a(t)$ increases, the R.H.S. of (\[friedmann\_eom\])will decrease and eventually reach zero. The solution is stable with respect to perturbations around this point because of the presence of the gap.
The expression (\[critical\_gap\]) allows us to calculate the value of the scale factor at the end of inflation. For large $a$, $\Delta
\sim a^3 M_{pl}^2 k_f^2 $. Then from (\[critical\_gap\]) we have:
$$\label{final_a}
a_f = \left ( \frac{\Lambda_0 M_{pl}^2}{2 E_D^2 k_f^4} \right)^\frac{1}{6}$$
where $E_D = \hbar \omega_D$. Then for the number of e-foldings we find:
$$\label{efoldings}
N = ln\left( \frac{a_f}{a_i} \right) \sim - \frac{1}{6} ln(E_D^2 k_f^4)$$
where we have set the scale factor at the beginning of inflation $a_i = 1$. If we assume that $E_D \sim M_{pl}$ and $N = 60$ then this implies that $k_f \sim e^{-90}$.
Numerical solution and results
==============================
For our numerical calculation we work in Planck units ($\kappa \sim
M_{pl}^2 = 1$). We set $E_D$ and $k_f$ to be $M_{pl} \sim 1$. We must emphasize that the qualitative behavior is completely independent of the values of these parameters. In particular, if we set $k_f = e^{-90}$ we would get 60 e-foldings. It is reasonable to assume that the bare cosmological constant cannot exceed $M_{pl}^4$ and thus we set $\Lambda_0 = M_{pl}^4 \sim 1$. Then the expression for the gap becomes:
$$\label{gap}
\Delta = 2\frac{\exp^{\frac{1}{a(t)^3}}}{\exp^{\frac{2}{a(t)^3}} - 1}$$
We solved equations (\[scalar\_eom\]) and (\[friedmann\_eom\]) numerically. An analytic solution is not possible because of non-analytic form of the gap (\[gap\]). Fig. \[fig:gap\] shows the behavior of the scale factor and the hubble rate as a function of time.
![\[fig:gap\] Scale factor, hubble rate and condensate gap as a function of time](simplot.eps)
We find that initially the universe undergoes inflationary expansion (indicated by the constant value of the hubble rate). When the gap becomes large enough to cancel out the bare cosmological constant, inflation ceases.
The behavior of the scalar field and momentum is in accord with the expectations outlined in the previous section. The scalar field increases or decreases initially depending on the sign of the initial value of the scalar momentum. It quickly levels off to a constant value for the rest of the inflationary period, as the momentum is driven towards zero by a positive $H$ and stays there until inflation ends. This behavior is independent of the initial values (which ranged from $0.5$ to $-0.5$ in various runs) and shows that during inflation the Hubble rate during inflation is always $\sqrt{\Lambda_0}/3$. In fact, the scalar field plays no role in the relaxation of the bare cosmological constant. A numerical calculation setting $\phi_c = \dot \phi_c = 0$ confirms this. Fig. \[fig:scalar\] shows the scalar field evolution for one set of initial values.
\[scalar\_figure\] ![\[fig:scalar\] Scalar field and momentum](scalar.eps "fig:")
Discussion
==========
In a universe filled with fermions and with a positive cosmological constants is unstable. Their exists an interaction between fermions propagated by torsion at the level of the effective field theory. This interaction leads to the formation of Cooper pairs and a condensate forms whose free energy is lower than that of the deSitter background. Consequently the bare cosmological constant, which we identify to be the free energy of the deSitter background, is lowered by an amount proportional to the square of the condensate gap. We have cosmic expansion because initially the gap does not cancel out the bare cosmological constant completely. The size of the gap depends on the 3-volume. Hence as the expansion occurs the effective cosmological “constant” becomes smaller, until eventually after a period of inflation we emerge from the deSitter vacuum into flat Minkowski space, where $H \sim 0$.
The number of e-foldings during inflation is given by (\[efoldings\]) and can be tuned by adjusting $E_D$ and $k_f$. The behavior of the scale factor is also independent of the scalar field evolution.
There are three free parameters in our model. The bare cosmological constant $\Lambda_0$, the fermi energy $k_f$ and the Debye energy $E_D$. In a condensate $E_D$ is the cutoff frequency and is determined by the lattice size. In a cosmological context therefore we can speculate that it should be $\sim M_{pl}$. $k_f$ can constrained according to (\[efoldings\]) to be $\sim e^{-90}$. $\Lambda_0$ determines the Hubble rate during inflation. From the WMAP data [@spergel-2006], the upper limit on $H/M_{pl}^2$ is constrained to be $10^{-4}$. From this we can deduce that the bare cosmological constant, needs to be fixed by hand to be $\Lambda \sim
H^2 \sim 10^{-8} M_{pl}^4$ in order to conform to observations.
We have presented here a non-perturbative mechanism which relaxes the bare cosmological constant to zero. As a bonus we find that the relaxation is accompanied by an inflationary period. The duration of inflation is determined by two parameters ($E_D$ and $k_f$) whose precise determination requires physics beyond the standard model. The lack of fine-tuning is demonstrated by the fact that the solution has an attractor with $H=0$ independent of the values of the free parameters.
The scalar field discussed here is an emergent degree of freedom. After inflation, oscillations of this field can lead to reheating. However, to what extent this would be a viable description of the post-inflationary period remains to be seen in future work.
We would like to thank Robert Brandenberger, Michael Peskin and Tirtho Biswas for many helpful discussions and suggestions.
[^1]: Which implies that the torsion is non-zero
[^2]: $\Delta$ is obtained by solving the gap equation obtained in [@alexander-2006] in a self-consistent manner
[^3]: To be precise we note that there are *two* scalar fields, corresponding to the two pairs of annihilation and creation operators. However in the following we use only one scalar field for simplicity. Noting that the left handed massless fermions are the antiparticles of the right handed ones, we can conjecture that this expression is the quantized form of a *complex* scalar field, which would imply that we are dealing with an axion
|
---
abstract: |
New results of NA61/SHINE on determination of charged hadron yields in proton-carbon interactions are presented. They aim to improve predictions of the neutrino flux in the T2K experiment. The data were recorded using a secondary-proton beam of 31 GeV/$c$ momentum from CERN SPS which impinges on a graphite target. To determine the inclusive production cross section for charged pions, kaons and protons the thin ($0.04\, \lambda_I$) target was exploited. Results of this measurement are used in the T2K beam simulation program to reweight hadron yields in the interaction vertex. At the same time, NA61/SHINE results obtained with the T2K replica target ($1.9\, \lambda_I$) allow to constrain hadron yields at the surface of the target. It would correspond to the constraint up to 90% of the neutrino flux, thus reducing significantly a model dependence of the neutrino beam prediction. All measured spectra are compared to predictions of hadron production models.
In addition a status of the analysis of data collected by NA61/SHINE for the NuMI target (Fermilab) is reviewed. These data will be used further in neutrino beam calculations for the MINERvA, MINOS(+) and LBNE experiments.
title: 'Hadron production measurement from NA61/SHINE'
---
[Hadron production measurement from NA61/SHINE]{}
[A.Korzenev[^1], on behalf of the NA61/SHINE collaboration]{}
[DPNC, University of Geneva, Switzerland]{}
[[email protected]]{}
A precise prediction of the expected neutrino flux is required for the T2K experiment [@review; @T2K_flux_paper]. It is used to calculate a neutrino cross section at the near detector, while at the far detector it provides an estimate of the expected signal for the study of neutrino oscillations. Prediction of the neutrino flux is constrained by using dedicated hadron production measurements. These measurements were therefore performed in the NA61/SHINE experiment [@proposal_NA61] at CERN. We refer the reader to [@NA49] for the description of main detector components, software, calibration and analysis methods which were basically inherited from the NA49 experiment.
The cross section measurement using a secondary-proton beam of 31 GeV/$c$ momentum from CERNSPS scattered off a thin graphite target (0.04 $\lambda_{I}$) have been performed by NA61 in years 2007 and 2009. The first NA61 physics papers were devoted to the production of $\pi^\pm$ and K$^+$ [@Abgrall:2011ae; @Abgrall:2011ts]. For this analysis data collected in 2007 have been used. By now these results were integrated to the T2K beam simulation program to constrain the production of hadrons in the primary interaction of the beam protons in the target [@Abe:2012gx; @Abe:2011sj].
Although pilot data 2007 covered a significant part of the relevant hadron production phase space of T2K [@T2K_flux_paper] the statistical uncertainty is quite large. In the year 2008 important changes have been introduced to the experimental setup of NA61: new trigger logic, TPC read-out and DAQ upgrade, additional sections of ToF wall, new beam-telescope detectors. As a consequence of these upgrades the number of events recorded in 2009 and 2010 for about a same period of time have been increased by an order of magnitude as compared to the 2007 data. This larger sample allows simultaneous extraction of yields of $\pi^{\pm}$, K$^{\pm}$, K$^0_s$ and protons. Furthermore the phase space of NA61 has been increased (see Fig.\[fig:na61\_coverage\]). Additional sections of ToF improve the coverage at high $\theta$. In the forward direction one profits from the use of the Gap TPC detector. It plays a key role in the analysis of forward produced particles. The coverage of this kinematic domain is important for the muon monitor measurements in T2K [@Abe:2011ks]. Statistics collected in 2007 and 2009 with the thin target is $0.7$ and $5.4$ millions of triggers, respectively.
![ The phase space of $\pi^+$, $\pi^-$, K$^+$, K$^-$, K$^0_L$ and protons contributing to the predicted neutrino flux at SK in the “positive” focusing configuration [@T2K_flux_paper], and the regions covered by new 2009 data ([*dashed line*]{}) and by previously published NA61/SHINE measurements ([*solid line*]{}) [@Abgrall:2011ae; @Abgrall:2011ts].[]{data-label="fig:na61_coverage"}](pi_pl "fig:"){width="30.00000%"} ![ The phase space of $\pi^+$, $\pi^-$, K$^+$, K$^-$, K$^0_L$ and protons contributing to the predicted neutrino flux at SK in the “positive” focusing configuration [@T2K_flux_paper], and the regions covered by new 2009 data ([*dashed line*]{}) and by previously published NA61/SHINE measurements ([*solid line*]{}) [@Abgrall:2011ae; @Abgrall:2011ts].[]{data-label="fig:na61_coverage"}](pi_mi "fig:"){width="30.00000%"} ![ The phase space of $\pi^+$, $\pi^-$, K$^+$, K$^-$, K$^0_L$ and protons contributing to the predicted neutrino flux at SK in the “positive” focusing configuration [@T2K_flux_paper], and the regions covered by new 2009 data ([*dashed line*]{}) and by previously published NA61/SHINE measurements ([*solid line*]{}) [@Abgrall:2011ae; @Abgrall:2011ts].[]{data-label="fig:na61_coverage"}](k_pl "fig:"){width="30.00000%"} ![ The phase space of $\pi^+$, $\pi^-$, K$^+$, K$^-$, K$^0_L$ and protons contributing to the predicted neutrino flux at SK in the “positive” focusing configuration [@T2K_flux_paper], and the regions covered by new 2009 data ([*dashed line*]{}) and by previously published NA61/SHINE measurements ([*solid line*]{}) [@Abgrall:2011ae; @Abgrall:2011ts].[]{data-label="fig:na61_coverage"}](k_mi "fig:"){width="30.00000%"} ![ The phase space of $\pi^+$, $\pi^-$, K$^+$, K$^-$, K$^0_L$ and protons contributing to the predicted neutrino flux at SK in the “positive” focusing configuration [@T2K_flux_paper], and the regions covered by new 2009 data ([*dashed line*]{}) and by previously published NA61/SHINE measurements ([*solid line*]{}) [@Abgrall:2011ae; @Abgrall:2011ts].[]{data-label="fig:na61_coverage"}](kaons_neutral_fluka "fig:"){width="30.00000%" height="24.20000%"} ![ The phase space of $\pi^+$, $\pi^-$, K$^+$, K$^-$, K$^0_L$ and protons contributing to the predicted neutrino flux at SK in the “positive” focusing configuration [@T2K_flux_paper], and the regions covered by new 2009 data ([*dashed line*]{}) and by previously published NA61/SHINE measurements ([*solid line*]{}) [@Abgrall:2011ae; @Abgrall:2011ts].[]{data-label="fig:na61_coverage"}](p_pl "fig:"){width="30.00000%"}
An essential fraction of the neutrino flux arises from secondary re-interactions as long targets are used in T2K [@T2K_flux_paper; @Abgrall:2012pp]. The lack of direct data, hence the use of sparse data sets, to cover these contributions limits the achievable precision on the flux prediction. Therefore measurements with a full-size replica of the T2K target (1.9 $\lambda_{I}$) have been performed by NA61 in years 2007, 2009 and 2010. A total of $0.2$, $4$ and $10$ millions of triggers have been recorded on tape, respectively. Analysis of these data will allow to reduce the systematic uncertainties of neutrino flux due to the treatment of secondary re-interactions in the target.
### Normalization and production cross section {#normalization-and-production-cross-section .unnumbered}
For the normalization of hadron spectra and the calculation of the production cross section we use a procedure described in [@Abgrall:2011ae; @NA49]. The idea is to measure an interaction probability for cases when the graphite target was inserted and removed. Using these values one calculates the so-called “trigger” cross section which, in turn, is an input for the analysis of “physics” cross sections. Due to upgrade of the spectrometer and higher beam intensity in 2009 the procedure was slightly modified as compared to the analysis of the 2007 data.
[r]{}[0.46]{} {width="45.00000%"}
In the 2009 campaign the beam trigger ran simultaneously with physics triggers. The ability to apply the event-by-event selection improved the systematics significantly. To account for the high rate the beam trigger was prescaled by a factor of 100.
As a result of the analysis of the 2009 data we obtain the inelastic cross section $$\begin{aligned}
\sigma_{inel} = 261.3 \pm 2.8(\mbox{stat}) \pm 2.2 (\mbox{model}) \pm 1.0 (\mbox{trig})~{\rm mb}
\nonumber \end{aligned}$$ which comprises all processes due to strong interactions excluding the coherent nuclear elastic scattering. By subtracting from $\sigma_{inel}$ the cross section of quasielastic interactions, which amounts to $27.8 \pm 2.2$ (stat) mb, one determines the production cross section $$\begin{aligned}
\sigma_{prod} = 233.5 \pm 2.8(\mbox{stat}) \pm 4.2(\mbox{model}) \pm 1.0 (\mbox{trig})~{\rm mb}.\nonumber\end{aligned}$$ It is used further to normalize hadron cross sections to be able to compare to MC models. All model-dependent corrections were estimated with GEANT4.9.5 [@GEANT4; @GEANT4bis] using FTF\_BIC physics list. A comparison to the previously published results is presented in Fig.\[Davide\].
The total uncertainty of $\sigma_{prod}$ is 5.1 mb, almost a factor of two smaller than the one obtained with the 2007 data. The dominant contribution to the uncertainty comes from the physics model used to recalculate the production cross section from the “trigger” cross section.
### Measurement of charged hadron cross sections for T2K {#measurement-of-charged-hadron-cross-sections-for-t2k .unnumbered}
Depending on the momentum interval and the particle type, different analysis techniques were tested for the analysis of data 2007 [@Abgrall:2011ae]. More than 90% of primary negatively charged particles produced at this energy are $\pi^-$. Thus the analysis of $\pi^-$ spectra can be done without additional particle identification (so called $h^-$ approach). For other species of particles the identification is mandatory. In particular for the region of momenta $p>1$ GeV/$c$, which is the most crucial for T2K neutrino kinematics, a combined analysis of time-of-flight (ToF) measurements and energy loss, $dE/dx$, measurements in TPC was used.
{width="45.00000%"} {width="45.00000%"}
Raw particle yields have been corrected step-by-step using the NA61 Monte Carlo simulation program with VENUS [@VENUS] for primary interactions and a GEANT3-based part for tracking of secondary particles through the detector. The following effects have been accounted for: geometrical acceptance of the spectrometer; efficiency of the reconstruction chain; decays before reaching the ToF wall; ToF detection efficiency; pions coming from $\Lambda$ and K$^0_s$ decays (feed-down correction).
Spectra of $\pi^{+}$ and $\pi^{-}$ obtained with data 2007 and 2009 are presented in Fig.\[NA61\_pions\]. For data 2007 three analysis techniques were applied: $dE/dx$ method for momenta smaller than 1 GeV/c; combined ToF-$dE/dx$ method for momenta larger than 1 GeV/c; $h^-$ method for $\pi^-$ cross section. Spectra were compared in overlapping regions to check their consistency. Complementary domains were combined to reach maximum acceptance. Due to large statistics in 2009 it was decided to use only the ToF-$dE/dx$ method since it provides high selection purity of the sample and low dependence on a MC model.
The analysis of K$^\pm$ is more complicated due to their small fraction in the overall sample [@Abgrall:2011ts]. For instance the K$^+$ signal vanishes over the predominant pion one at the low momentum range while at higher momenta protons dominate. Analysis technique is basically similar to the ToF–$dE/dx$ one used for pions. In general statistical error of the K$^+$ spectra with data 2007 is by a factor of 3 larger than the systematic one [@Abgrall:2011ts] and only two intervals in $\theta$ were considered. Data collected in 2009 improved the precision strongly. In Fig.\[NA61\_kaons\] the differential cross section of kaons normalized to the production cross section is shown. Larger statistics of 2009 allowed to split the phase space into 9 intervals in $\theta$. Graphs are overlaid with predictions from several recommended GEANT4 physics lists [@GEANT4; @GEANT4bis].
Distribution of proton multiplicities obtained with data 2009 as a function of momentum in different intervals of $\theta$ is shown in Fig.\[NA61\_protons\]. Comparison to GEANT4 models shows that none of them can satisfactory describe the data.
{width="45.00000%"} {width="45.00000%"}
{width="99.00000%"}
{width="99.00000%"}
### Cross section of neutral particles $K_S^0$ and $\Lambda$ {#cross-section-of-neutral-particles-k_s0-and-lambda .unnumbered}
Understanding of the neutral strange particle production in NA61/SHINE is of great interest for two reasons. First, it allows to decrease the systematic uncertainty associated with the charged particle production, namely pions [@Abgrall:2011ae] and protons. Second, the measurement of the $K^0_S$ production will improve our knowledge of the $\nu_e$ flux at the T2K experiment coming from the three body decay of the $K^0_L \to \pi^0 e^\pm \nu_e (\overline \nu_e)$ [@T2K_flux_paper]. The technique used for the analysis of the 2009 data have been already tested on the 2007 data [@K0_2007]. We reconstruct $K_S^0$ and $\Lambda$ in so-called $V^0$ mode: decay into two charged particle of opposite signs $K^0_S \to \pi^+ + \pi^-$ and $\Lambda \to p \phantom{^+} + \pi^-$. Particle yields have been extracted in the analysis of invariant mass spectra applying corresponding mass hypotheses for daughter tracks. The main background sources were associated with converted photons and the combinatorial background which is mainly due to particles produced in the primary interaction. Selection cuts have been applied on the following variables: the number of clusters for daughter tracks, the impact parameter of the $V^0$ track in the primary vertex, the distance of closest approach between daughter tracks, the distance between primary and decay vertices, the cosine of the angle between the trajectory of $V^0$ and daughter particle in the center-of-mass system of $V^0$, the energy loss in TPC by daughter tracks, and finally $V^0$ satisfying the hypothesis of $K_S^0$ were removed from the analysis of $\Lambda$ and vice versa. Multiplicities of $K_S^0$ are shown in Fig.\[NA61\_K0S\] together with a prediction of several models. Analysis of $\Lambda$ is in progress.
Being able to measure simultaneously K$^+$, K$^-$ and K$^0_S$ we can provide a test for several hypotheses predicting a relative yield of charged and neutral kaons. These hypotheses derived from the isospin symmetry and/or some basic assumption on parton distributions in nucleon [@BMPT]. The comparison of the measured differential multiplicity of K$^0_S$ to the prediction obtained with the charged kaons is shown in Fig.\[NA61\_K0S\]. A reasonable agreement is observed for prediction of ’isospin’ and ’quark-counting’ hypotheses presented in the figure. However more certain statement would require a higher statistical precision for K$^0_S$.
### Analysis of the T2K replica-target data {#analysis-of-the-t2k-replica-target-data .unnumbered}
First physics results from the analysis of the replica-target data taken in 2007 have been published [@Abgrall:2012pp]. A dedicated reconstruction method has been developed to provide results in a form that is of direct interest for T2K. Yields of positively charged pions are reconstructed at the surface of the T2K replica target in bins of the laboratory momentum and polar angle as a function of the longitudinal position along the target. By parametrizing hadron yields on a surface of the target one predicts up to 90% of the flux for both $\nu_{\mu}$ and $\nu_e$ components while only 60% of neutrinos are coming from decay of particles produced in the primary interaction. Two methods (constraint of hadroproduction data at primary interaction and on a target surface) are consistent within their uncertainties achieved on statistics of pilot run 2007. The ultimate precision will come from the analysis of the replica-target data 2009 and 2010.
### Data taking for the Fermilab neutrino beam {#data-taking-for-the-fermilab-neutrino-beam .unnumbered}
Following discussion initiated at NUFACT2011 a group from 10 US institutions expressed their interest in possibility of collecting data relevant for NuMI experiments (MINERvA, NOvA, MINOS+), Booster experiments (MiniBooNE, MicroBooNE) and LBNE [@Schmitz:2012]. An important pilot run with a proton beam of 120 GeV and a thin graphite target took place in July 2012. It resulted to 3.5 millions of recoded events. An experience with these pilot data gives a basis to estimate an amount of efforts needed to complete the Fermilab’s neutrino program in NA61. A complete data taking is expected after the 2013/2014 shutdown of the CERN accelerator complex.
[00]{}
M.Bonesini and A.Guglielmi, Phys. Rep. 433 (2006) 65 K. Abe [*et al.*]{}, Phys.Rev. D87 (2013) 012001 N. Antoniou [*et al.*]{}, CERN-SPSC-2006-034, proposal of NA61 C.Alt [*et al.*]{}, Eur.Phys.J. C45 (2006) 343 N. Abgrall [*et al.*]{}, Phys.Rev. C84 (2011) 034604
N. Abgrall [*et al.*]{}, Phys.Rev. C85 (2012) 035210 K. Abe [*et al.*]{}, Phys.Rev. D85 (2012) 031103 K. Abe [*et al.*]{}, Phys. Rev. Lett. 107 (2011) 041801 K. Abe [*et al.*]{}, Nucl.Instrum.Meth. A659 (2011) 106 N. Abgrall [*et al.*]{}, Nucl.Instrum.Meth. A701 (2013) 99 G. Battistoni [*et al.*]{}, AIP Conf. Proc. 896 (2007) 31
S.Agostinelli [*et al.*]{}, Nucl. Instrum. Meth. A506 (2003) 250
J.Allison [*et al.*]{}, IEEE Transactions on Nuclear Science, 53(1), 270 (2006) N. Abgrall [*et al.*]{}, CERN-PH-EP-2013-160, arXiv:1309.1997 \[physics.acc-ph\]
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D. Schmitz [*et al.*]{}, Letter of Intent, SPSC, May 2012
[^1]: Presented at EPS-HEP2013 in Stockholm on July 19, 2013.
|
---
abstract: 'This article studies the poset of simple permutations with respect to the pattern involvement. We specify results on critically indecomposable posets obtained by Schmerl and Trotter in [@SchmerlTrotter1993] to simple permutations and prove that if $\sigma, \pi$ are two simple permutations such that $\pi < \sigma$ then there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)}, \ldots, \sigma^{(k)}=\pi$ such that $|\sigma^{(i)}| - |\sigma^{(i+1)}| = 1$ - or $2$ when permutations are exceptional- and $\sigma^{(i+1)} < \sigma^{(i)}$. This characterization induces an algorithm polynomial in the size of the output to compute the simple permutations in a wreath-closed permutation class.'
address:
- 'LIAFA, UMR 7089, Université Paris Diderot, Paris, France'
- 'LIX, UMR 7161, Ecole Polytechnique, Palaiseau, France'
author:
- Adeline Pierrot
- Dominique Rossin
bibliography:
- 'biblio.bib'
title: Simple permutations poset
---
Introduction
============
Simple permutations are the permutations which map no proper non-singleton interval onto an interval. Those permutations play a key role in the study of permutation classes, that is closed sets of permutations. More precisely, they are core objects in the substitution decomposition of permutations. For example, if a class contains a finite number of simple permutations then it is finitely based, meaning that the class can be expressed as the set of permutations that do not contain as pattern any permutation of a finite set $B$. Moreover, the generating function of the permutation class is algebraic. On second hand, even if the pattern involvement problem is NP-complete in general, there exists a FPT algorithm [@BR06] where the parameter is the length of the largest simple permutation that appears as a pattern of the involved permutations.
In this article, we study the set of simple permutations with respect to the pattern containment relation. In [@SchmerlTrotter1993], Schmerl and Trotter study critically indecomposable partially ordered sets of integers and prove many structural results. They notice that their results still holds for all relational structures, in our case, permutations.
In this article, we focus on simple permutations and show that the general results on integers can be refined in our case. More precisely, if $\sigma, \pi$ are two simple permutations such that $\pi$ is pattern of $\sigma$ -$\pi < \sigma$- [@SchmerlTrotter1993] proves that there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)}, \ldots, \sigma^{(k)}=\pi$ such that $|\sigma^{(i)}| - |\sigma^{(i+1)}| = 1$ or $2$ and $\sigma^{(i+1)} \prec \sigma^{(i)}$. Using the structure of simple permutations, we strengthen the result and show that in the case of permutations, we can find a chain with all size differences of $1$ except when $\sigma$ is exceptional. In the latter there exists a chain with all size differences of $2$.
This structural result on permutations and pattern involvment has many consequences. First, it allows us to compute the average number of points in a simple permutations that can be removed -one at each time- in order to obtain another simple permutation. On another hand, it gives rise to a polynomial algorithm to generate simple permutations of a wreath-closed class of permutations. This algorithm roughly starts by looking to simple permutations of size $4$ and iterates over the size of permutations. The characterization of the preceding chain translates into a polynomial time algorithm for finding simple permutations of size $n+1$ in $Av(B)$ knowing only simple permutations of size $n$ and $n-1$ in this class. Note that our algorithm requires no pattern involvement test. Although we give this algorithm in the general framework of wreath-closed permutation classes, we apply it for classes containing only a finite number of simple permutations. It can be also used to generate the simple permutations of any wreath-closed class up to a given size, even if the number of simple permutations is infinite.
Together with preceding results of Albert, Atkinson, Brignall [@AA05; @Bri06; @Bri07; @BHV06b] and our results [@BBR09; @BBPR09], this algorithm allows to compute the generating function of a wreath-closed class of permutations containing a finite number of simple permutations. The overall complexity of this algorithm is polynomial in the number of simple permutations. Some statistical results are given in the last section.
Definitions
===========
A permutation $\sigma$ is a bijective map from $\{1\ldots n\}$ onto $\{ 1 \ldots n\}$ with $n = |\sigma|$. We either represent a permutation by a word $\sigma = \sigma_1\sigma_2\ldots \sigma_n$ where $\sigma_i = \sigma(i)$ or its graphical representation in a grid (see Figure \[fig:exceptional\] for examples).
Let $\pi$ and $\sigma$ be two permutations. We say that $\pi = \pi_1 \ldots \pi_k$ is a [*pattern*]{} of $\sigma$ and we write $\pi \preceq \sigma$ if and only if there exist $i_1 < i_2 < \ldots < i_k$ such that $\pi$ is order-isomorphic to $\sigma_{i_1} \sigma_{i_2} \ldots \sigma_{i_k}$. A permutation class is a downward-closed set of permutations under pattern relation. We can also define permutation classes by the set of minimal (for $\preceq$) permutations not in the class which is an antichain for the pattern relation. This set is called the [*basis*]{} of the class. We denote by $Av(B)$ the permutation class which is the set of permutations that do not contain any of the permutations $\pi \in B$ as a pattern. For example $Av(231)$ is the class of one-stack sortable permutations. When the basis $B$ contains only simple permutations the permutation class $Av(B)$ is said to be [*wreath-closed*]{}. Wreath-closed classes are defined in [@AA05] in a different way but the authors prove that this definition is equivalent.
An interval in a permutation is a consecutive set of elements $\sigma_i \sigma_{i+1}\ldots \sigma_j$ such that the set of values $\{ \sigma_i, \sigma_{i+1},\ldots \sigma_j \}$ is an interval. A permutation is said to be simple if and only if its intervals are trivial -the singletons and the whole permutation-. As example 1, 12, 21, 2413 and 3142 are the simple permutations of size $\leq 4$. A subset of simple permutations, called exceptional ones plays a key role in this article.
Exceptional permutations are permutations defined below for every $m \geq 2$ (see Figure \[fig:exceptional\]):
- $2\ 4\ 6\ 8\ldots (2m)\ 1\ 3\ 5\ldots (2m-1)$ — type 1
- $(2m-1)\ (2m-3)\ldots 1\ (2m)\ (2m-2)\ldots 2$ — type 2
- $(m+1)\ 1\ (m+2)\ 2\ldots (2m)\ m$ — type 3
- $m\ (2m)\ (m-1)\ (2m-1)\ldots 1\ (m+1)$ — type 4
; iin [2,4,6,8,10,1,3,5,7,9]{}
;
(1,1) grid (,);
;
iin [ 2,4,6,8,10,1,3,5,7,9 ]{}
(+.5,i+.5) \[fill\] circle (.2);
;
;
; iin [9,7,5,3,1,10,8,6,4,2]{}
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iin [ 9,7,5,3,1,10,8,6,4,2 ]{}
(+.5,i+.5) \[fill\] circle (.2);
;
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; iin [6,1,7,2,8,3,9,4,10,5]{}
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iin [ 6,1,7,2,8,3,9,4,10,5 ]{}
(+.5,i+.5) \[fill\] circle (.2);
;
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iin [ 5,10,4,9,3,8,2,7,1,6 ]{}
(+.5,i+.5) \[fill\] circle (.2);
;
;
Notice that exceptional permutations are simple. Notice also that, if we remove the symbols $2m-1$ and $2m$ from the first two types, we obtain an other exceptional permutation of the same type; and likewise if we remove the symbols in the last two positions from the types 3 and 4 and renormalize the result.
\[prop:TypeExceptional\] Let $\sigma, \sigma'$ be two exceptional permutations with $|\sigma| \leq |\sigma'|$. Then $\sigma \preceq \sigma'$ if and only if $\sigma$ and $\sigma'$ are exceptional permutations of the same type.
; iin [1,3,5,7,9,11,2,4,6,8,10]{}
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iin [ 1,3,5,7,9,11,2,4,6,8,10 ]{}
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A more general kind of permutations containing exceptionnal permutations will appear naturally in this article. An [*alternation*]{} is a permutation in which every odd entry lies to the left of every even entry, or any symmetry of such a permutation. A [*parallel*]{} alternation is one in which these two sets of entries form monotone subsequences, either both increasing or both decreasing. A [*wedge*]{} alternation is one in which the two sets of entries form monotone subsequences pointing in opposite directions. See Figures \[fig:para\] and \[fig:alternations\] for examples. Among parallel or wedge alternations, only exceptional permutations are simple.
Pattern containment on simple permutations {#sec:MotifsSimples}
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Simple patterns of simple permutations
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In this section, we study the poset of simple permutations with respect to the pattern relation. More precisely, we specify and extend results from [@SchmerlTrotter1993] to permutations. To each permutation $\sigma$ is associated a poset $P( \sigma )=([1..n],\prec)$ where $i \prec j \Leftrightarrow (i < j \text{ and } \sigma_i < \sigma_j)$. A poset on $[1..n]$ is called [*indecomposable*]{} [@SchmerlTrotter1993] if it does not contain any non-trivial interval with respect to the relations , $\prec$ in our case. It is called [*critically indecomposable*]{} if furthermore whenever an element is removed, the resulting poset is not indecomposable. In the specific case of permutations those poset characteristics can be translated as stated in the following propositions:
The poset $P(\sigma)$ is indecomposable if and only if $\sigma$ is simple.
\[prop:critic\] $P(\sigma)$ is critically indecomposable if and only if $\sigma$ is exceptional.
This is a consequence of Corollary $5.8 (2)$ of [@SchmerlTrotter1993].
The standard poset isomorphism can be transposed to integer posets.
Let $A,B \subset {\ensuremath{\mathbb N}}$ posets, then $A \equiv B$ if $A \sim B$ as posets and the isomorphism keeps the natural integer ordering on ${\ensuremath{\mathbb N}}$.
Let $\sigma$ and $\pi$ be permutations. Then $\pi \preceq \sigma$ if and only if there exists $A \subset P(\sigma)$ such that $A \equiv P(\pi)$.
The next four propositions are mere translations of results from [@SchmerlTrotter1993] on permutations.
\[prop:size4\] Let $\sigma$ be a simple permutation with $|\sigma| \geq 3$, then either $2\,4\,1\,3$ or $3\,1\,4\,2$ is a simple pattern of $\sigma$.
This is the mere translation of Theorem $2.1$ of [@SchmerlTrotter1993], noticing that there are no simple permutations of size $3$ and there are only two simple permutations of size $4$.
\[prop:proposition0.2\] Let $\pi, \sigma$ be two simple permutations with $\pi \preceq \sigma$. If $3 \leq |\pi| \leq |\sigma|-2$, then there exists a simple permutation $\tau$ such that $\pi \preceq \tau \preceq \sigma$ and $|\tau| = |\pi|+2$.
\[prop:exceptional\] Let $\sigma$ be an exceptional permutation. If $3 \leq m \leq |\sigma|$, then $\sigma$ has a simple pattern of size $m$ if and only if $m$ is even.
This is a direct consequence of Corollary $3.1$ of [@SchmerlTrotter1993] as every exceptional permutation is of even size.
\[prop:simplePattern\] Let $\sigma$ be a non exceptional simple permutation. If $4 \leq m \leq |\sigma|$ then $\sigma$ has a simple pattern of size $m$.
The preceding result holds for non exceptional permutations. For exceptional ones, the following proposition concludes:
\[prop:except\] If $\sigma$ is an exceptional permutation, then for every $m$ such that $3 \leq m \leq |\sigma|$:
- If $m$ is odd, then $\sigma$ has no simple pattern of size $m$.
- Otherwise $m$ is even and $\sigma$ has exactly one simple pattern of size $m$ which is the exceptional permutation of the same type as $\sigma$.
The first item -$m$ is odd- is a direct consequence of Proposition \[prop:exceptional\]. For the second point, $\sigma$ has at least one simple pattern $\pi$ of size $m$ by Proposition \[prop:exceptional\]. Suppose now that $\pi$ is not exceptional, then $m \geq 5$ as simple permutations of size $4$ are exceptional. Then, using Proposition \[prop:simplePattern\], $\pi$ has a simple pattern $\tau$ of size $5$, thus $\tau$ is a pattern of $\sigma$ but of odd size which is forbidden by Proposition \[prop:exceptional\]. So $\pi$ is exceptional and of the same type as $\sigma$ from Proposition \[prop:TypeExceptional\].
A direct consequence of the preceding proposition is that all simple patterns of exceptional permutations are exceptional. For non exceptional ones the following proposition describes the pattern containment relation:
\[prop:size-1\] Let $\sigma$ be a simple permutation of size $\geq 5$. Then $\sigma$ has a simple pattern of size $|\sigma|-1$ if and only if $\sigma$ is not exceptional.
Consequence of Proposition \[prop:exceptional\] and Proposition \[prop:simplePattern\] above.
Simple pattern containing a given simple permutation
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The results obtained in the preceding section describe how a simple permutation can give other simple permutations by deleting elements. In the sequel, we add another constraint on patterns, that is we want to delete elements in a simple permutation $\sigma$ containing a simple permutation $\pi$ as a pattern only by deleting one or two elements to obtain another simple permutation $\sigma'$ such that $\pi \preceq \sigma'$.
Theorem \[thm:theorem0.3\] deals with the case $|\pi| = |\sigma|-2$ and relies on the following intermediate result:
\[prop:intervalleOuCoin\] Let $\tau$ be a non simple permutation such that $\tau \setminus \{\tau_i\}$ is simple. Then $\tau_i$ belongs to an interval of size $2$ of $\tau$ or is in a corner of the graphical representation of $\tau$.
As $\tau$ is not simple, $\tau$ contains at least one non-trivial interval $I$. As $I$ is an interval of $\tau$, $I \setminus \{\tau_i\}$ is an interval of $\tau \setminus \{\tau_i\}$. But $\tau \setminus \{\tau_i\}$ is simple thus $I \setminus \{\tau_i\}$ is a trivial interval of $\tau \setminus \{\tau_i\}$, hence is a singleton $\{\tau_k\}$ or the whole permutation $\tau \setminus \{\tau_i\}$. But $I$ is non-trivial so in the first case $I = \{\tau_i, \tau_k\}$ and $\tau_i$ belongs to an interval of size $2$ of $\tau$, and in the second case $I = \tau \setminus \{\tau_i\}$ and $\tau_i$ is in a corner of the graphical representation of $\tau$.
\[thm:theorem0.3\] Let $\sigma = \sigma_1 \sigma_2 \ldots \sigma_n$ be a non exceptional simple permutation of size $n \geq 4$ and $\pi$ a simple permutation of size $n-2$ such that $\pi \preceq \sigma$. Then there exists a simple permutation $\tau$ of size $n-1$ such that $\pi \preceq \tau \preceq \sigma$.
Supppose that such a permutation $\tau$ does not exist. We prove that this leads to a contradiction. Let $i,j$ such that $\pi = \sigma \setminus \{\sigma_i, \sigma_j\}$. If $\sigma \setminus \{ \sigma_i \}$ is simple then $\tau = \sigma \setminus \{ \sigma_i \}$ would contradict our hypothesis. Thus $\sigma \setminus \{ \sigma_i \}$ is not simple, but $\pi = \sigma \setminus \{\sigma_i, \sigma_j\}$ is simple. From Proposition \[prop:intervalleOuCoin\] $\sigma_j$ belongs to an interval of size $2$ of $\sigma \setminus \{ \sigma_i \}$ or is in a corner of the bounding box of the graphical representation of $\sigma \setminus \{ \sigma_i \}$ thanks to $\pi$. By symmetry between $i$ and $j$ the same results holds when exchanging these two indices. So there are $3$ different cases:
- $\sigma_i$ and $\sigma_j$ are both in a corner thanks to $\pi$. In that case $\pi$ is a non trivial interval of $\sigma$, which contradicts the fact that $\sigma$ is simple.
- $\sigma_i$ belongs to an interval $I$ of size $2$ of $\sigma \setminus \{ \sigma_j \}$ and $\sigma_j$ is in a corner of $\sigma \setminus \{ \sigma_i \}$ thanks to $\pi$ (the same proof holds when exchanging $i$ and $j$). $\sigma$ is simple thus $\sigma_j$ is not in a corner of $\sigma$, but is in a corner of $\sigma \setminus \{ \sigma_i \}$ thus $\sigma_i$ is the only point separating $\sigma_j$ from a corner (see Figure \[fig:cas2\] for an example). Let $i_1$ such that $I = \{i,i_1\}$, then $\sigma_j$ is the only point separating $\sigma_{i_1}$ from $\sigma_i$, so $\pi = \sigma \setminus \{ \sigma_{i_1},\sigma_j \}$. If $\sigma \setminus \{ \sigma_{i_1} \}$ is simple then $\tau = \sigma \setminus \{ \sigma_{i_1} \}$ would answer the theorem, contradicting our hypothesis. Thus $\sigma \setminus \{ \sigma_{i_1} \}$ is not simple but $\pi = \sigma \setminus \{\sigma_{i_1}, \sigma_j\}$ is simple, hence from Proposition \[prop:intervalleOuCoin\] $\sigma_j$ belongs to an interval $J$ of size $2$ of $\sigma \setminus \{ \sigma_{i_1} \}$ or is in a corner of $\sigma \setminus \{ \sigma_{i_1} \}$ which is impossible as $\sigma_i$ separate it from one corner and $|\sigma| \geq 4$. Let $j_1$ such that $J = \{ j,j_1 \}$, then $\pi = \sigma \setminus \{ \sigma_{i_1}, \sigma_{j_1} \}$. If $\sigma \setminus \{ \sigma_{j_1} \}$ is simple then $\tau = \sigma \setminus \{ \sigma_{i_1} \}$ answer the theorem, contradiction. Let $i_0 = i$ and $j_0 = j$, we recursively build $i_0,j_0,i_1,j_1,\ldots$ such that $\forall k, \pi = \sigma \setminus \{ \sigma_{i_k},\sigma_{j_k} \} = \sigma \setminus \{ \sigma_{j_k}, \sigma_{i_{k+1}} \}$ and $\sigma \setminus \sigma_{i_k}$ and $\sigma \setminus \sigma_{j_k}$ are not simple, until reaching all points of $\sigma$. $\sigma_i$ and $\sigma_{i_1}$ are in increasing -or decreasing- order. For each case, -see Figure \[fig:cas2\]-, $\sigma_{i_1}$ has a determined position. Then positions of $\sigma_{i_k}$ and $\sigma_{j_k}$ are fixed for all $k$ as $\sigma_{i_k}$ does not separate $\sigma_{i_{k-1}}$ from $\sigma_{i_{k-2}}$. Depending of the position of $\sigma_{i_1}$, $\sigma$ is either a parallel alternation or a wedge alternation thus is exceptional or not simple, a contradiction.
(-1,-1) rectangle (10,8); (1.5,0.5) \[fill\] circle (0.2); (0.5,4.5) \[fill\] circle (0.2); at (1.5,-1) [$\sigma_j$]{}; at (-1,4.5) [$\sigma_i$]{}; (2,1) rectangle node [$\pi$]{} +(6.3,7); (0,0) rectangle (8.3,8); (1,0) – (1,8);
(-1,-1) rectangle (10,8); (1,0) grid (4,2); (0,4) grid (3,6); (1.5,0.5) \[fill\] circle (0.2); (3.5,1.5) \[fill\] circle (0.2); (0.5,4.5) \[fill\] circle (0.2); (2.5,5.5) circle (0.2); at (1.5,-1) [$\sigma_j$]{}; at (5,1.7) [$\sigma_{j_1}$]{}; at (-1,4) [$\sigma_i$]{}; at (4,5.6) [$\sigma_{i_1}$]{}; (2,1) rectangle node [$\pi$]{} +(6.3,7); (0,0) rectangle (8.3,8);
(-1,-1) rectangle (10,8); (1,0) grid (4,2); (0,4) grid (3,6); (1.5,0.5) \[fill\] circle (0.2); (3.5,1.5) \[fill\] circle (0.2); (0.5,5.5) \[fill\] circle (0.2); (2.5,4.5) circle (0.2); at (1.5,-1) [$\sigma_j$]{}; at (5,1.7) [$\sigma_{j_1}$]{}; at (-1,5.5) [$\sigma_i$]{}; at (4,3.9) [$\sigma_{i_1}$]{}; (2,1) rectangle node [$\pi$]{} +(6.3,7); (0,0) rectangle (8.3,8);
- $\sigma_i$ belongs to an interval $I$ of size $2$ of $\sigma \setminus \{\sigma_{j}\}$ and $\sigma_j$ belongs to an interval $J$ of size $2$ of $\sigma \setminus \{\sigma_{i}\}$. Let $i_1$ such that $I = \{ i,i_1 \}$, $I$ is an interval of $\sigma \setminus \{\sigma_{j}\}$ but $\sigma$ is simple thus $\sigma_j$ is the only point separating $\sigma_i$ from $\sigma_{i_1}$. Let $j_1$ such that $J = \{ j,j_1 \}$, $J$ is an interval of $\sigma \setminus \{\sigma_{i}\}$ but $\sigma$ is simple thus $\sigma_i$ is the only point separating $\sigma_j$ from $\sigma_{j_1}$. This indeed is one of the two cases depicted in Figure \[fig:cas3\] (up to symmetry). But $\pi = \sigma \setminus \{ \sigma_i, \sigma_{j} \} = \sigma \setminus \{ \sigma_{i_1}, \sigma_{j} \}$. If $\sigma \setminus \{ \sigma_{i_1} \}$ is simple then $\tau = \sigma \setminus \{ \sigma_{i_1} \}$ answer the theorem, a contradiction. Thus $\sigma \setminus \{ \sigma_{i_1} \}$ is not simple but $\pi = \sigma \setminus \{ \sigma_{i_1}, \sigma_{j} \}$ is simple thus from Proposition \[prop:intervalleOuCoin\] $\sigma_j$ belongs to an interval $J'$ of size $2$ of $\sigma \setminus \{ \sigma_{i_1} \}$ or lies in a corner of $\sigma \setminus \{ \sigma_{i_1} \}$ which is impossible if $i_1 \neq n$ (up to symmetry). Let $J' = \{ j, j'\}$ then $\pi = \sigma \setminus \{ \sigma_{i_1},\sigma_j \} = \sigma \setminus \{ \sigma_{i_1},\sigma_{j'} \}$. If $\sigma \setminus \{ \sigma_{j'} \}$ is simple then $\tau = \sigma \setminus \{ \sigma_{i_1}\}$ answer the theorem, contradiction. Moreover $\pi = \sigma \setminus \{ \sigma_{i},\sigma_j \} = \sigma \setminus \{ \sigma_{i},\sigma_{j_1} \}$. If $\sigma \setminus \{ \sigma_{j_1} \}$ is simple then $\tau = \sigma \setminus \{ \sigma_{j_1}\}$ fulfil the theorem, contradiction. Thus $\sigma \setminus \{ \sigma_{j_1}\}$ is not simple but $\pi = \sigma \setminus \{ \sigma_{j_1},\sigma_i \}$ is simple so that $\sigma_i$ belongs to an interval $I'$ of size $2$ of $\sigma \setminus \{ \sigma_{j_1} \}$ or lies in a corner of $\sigma \setminus \{ \sigma_{j_1}\}$, which is impossible if $j_1 \neq 1$ (up to symmetry). Let $i'$ such that $I' = \{ i,i'\}$, then $\pi = \sigma \setminus \{ \sigma_{i},\sigma_{j_1} \} = \sigma \setminus \{ \sigma_{i'},\sigma_{j_1} \}$. If $\sigma \setminus \{ \sigma_{i'} \}$ is simple then $\tau = \sigma \setminus \{ \sigma_{i'}\}$ fulfil our theorem, contradiction. Similarly to the preceding case, if $i_0 = i, j_0 = j$ then we can prove by induction until reaching all points of $\sigma$ that either $\sigma$ is a parallel alternation or a wedge permutation so that $\sigma$ is exceptional or not simple which leads to a contradiction.
(0,0) rectangle (9,9); (2,1) grid (5,3); (3,5) grid (6,7); (2.5,1.5) \[fill\] circle (0.2); (4.5,2.5) \[fill\] circle (0.2); (3.5,5.5) \[fill\] circle (0.2); (5.5,6.5) circle (0.2); at (1.4,0.6) [$\sigma_{j_1}$]{}; at (6,2.5) [$\sigma_j$]{}; at (2,5) [$\sigma_i$]{}; at (7,6.5) [$\sigma_{i_1}$]{}; (0,0) rectangle (8,8);
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(0,0) rectangle (9,9); (2,1) grid (5,3); (3,5) grid (6,7); (4,2) grid (7,4); (1,4) grid (4,6); (6.5,3.5) \[fill\] circle (0.2); (5.5,6.5) \[fill\] circle (0.2); (2.5,1.5) \[fill\] circle (0.2); (4.5,2.5) \[fill\] circle (0.2); (1.5,4.5) \[fill\] circle (0.2); (3.5,5.5) \[fill\] circle (0.2); at (1.4,0.6) [$\sigma_{j_1}$]{}; at (5.8,1.2) [$\sigma_j$]{}; at (8,3.2) [$\sigma_{j'}$]{}; at (2.2,6.5) [$\sigma_i$]{}; at (0.7,3.6) [$\sigma_{i'}$]{}; at (7,6.5) [$\sigma_{i_1}$]{}; (0,0) rectangle (8,8);
(0,0) rectangle (9,9); (2,1) grid (5,3); (3,5) grid (6,7); (1,4) grid (4,6); (4,0) grid (7,2); (1.5,4.5) \[fill\] circle (0.2); (2.5,2.5) \[fill\] circle (0.2); (4.5,1.5) \[fill\] circle (0.2); (3.5,5.5) \[fill\] circle (0.2); (5.5,6.5) circle (0.2); (6.5,0.5) \[fill\] circle (0.2); at (0.8,2.5) [$\sigma_{j_1}$]{}; at (5.8,2.5) [$\sigma_j$]{}; at (8,0.2) [$\sigma_{j'}$]{}; at (0.7,3.7) [$\sigma_{i'}$]{}; at (2.2,6.5) [$\sigma_i$]{}; at (7,6.5) [$\sigma_{i_1}$]{}; (0,0) rectangle (8,8);
Thanks to Theorem \[thm:theorem0.3\] and a simple induction, we are able to state our main result on pattern involvement.
\[thm:main\] Let $\sigma \not= \pi$ be two simple permutations, $\sigma$ non exceptional. If $\pi \prec \sigma$ and $|\pi| \geq 3$ then there exists a simple permutation $\tau$ such that $\pi \preceq \tau \prec \sigma$ and $|\tau| = |\sigma|-1$.
We prove this result by induction on $|\sigma|-|\pi|$ using Proposition \[prop:proposition0.2\]. If $|\sigma|-|\pi|$ is odd, using recursively Proposition \[prop:proposition0.2\] we find a simple permutation $\tau$ such that $\pi \preceq \tau \preceq \sigma$ and $|\tau| = |\sigma|-1$. If $|\sigma|-|\pi|$ is even, we find a simple permutation $\tau'$ such that $\pi \preceq \tau' \preceq \sigma$ and $|\tau'| = |\sigma|-2$ and we apply Theorem \[thm:theorem0.3\] which ensure the existence of a simple permutation $\tau$ such that $\pi \preceq \tau' \preceq \tau \preceq \sigma$ and $|\tau| = |\sigma|-1$.
Simple permutations poset {#sec:poset}
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We study the poset of simple permutations of size $\geq 4$ with respect to the pattern containment relation. We can represent this poset by an oriented graph $G$, whose vertices are the simple permutations and there is an edge from a simple permutation $\sigma$ to a simple permutation $\pi$ if and only if $\pi \prec \sigma$ and there is no simple permutation $\tau$ such that $\pi \prec \tau \prec \sigma$. Then $\pi \prec \sigma$ if and only if there is a path from $\sigma$ to $\pi$ in $G$. From Theorem \[thm:main\], if $\sigma$ is not exceptional there is an edge from $\sigma$ to $\pi$ if and only if we can obtain $\pi$ from $\sigma$ by deleting one point, and from Proposition \[prop:except\] if $\sigma$ is exceptional, there is an edge from $\sigma$ to $\pi$ if and only if $\pi$ is exceptional of the same type of $\sigma$ and $|\sigma| = |\pi|+2$. In this section we study other properties of $G$.
Paths in the simple permutations poset
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In the next theorem, we prove that if a simple permutation $\sigma$ has a simple pattern $\pi$, then there is a path in $G$ from $\sigma$ to $\pi$ in the graph whose first part consists of non exceptional simple permutations of consecutive sizes and second part of exceptional permutations (one of the parts can be empty). From Proposition \[prop:except\] it is obvious that reciprocally, all paths from $\sigma$ to $\pi$ are of this form. Then we extend this result to prove that whenever $\sigma$ is not exceptionnal, there is such a path such that the second part of the path is empty, that is we can reach $\pi$ from $\sigma$ by deleting one element at a time and all involved permutations are simple.
\[thm:chain\] Let $\pi \neq \sigma$ be simple permutations. If $\pi \preceq \sigma$ and $|\pi| \geq 3$, then there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)}, \ldots, \sigma^{(k-1)}, \sigma^{(k)}=\pi$ and $m \in \{0 \dots k\}$ such that $\sigma^{(i)} \preceq \sigma^{(i-1)}$ and:
- $|\sigma^{(i-1)}| - |\sigma^{(i)}| = 1$ if $1 \leq i \leq m$,
- $|\sigma^{(i-1)}| - |\sigma^{(i)}| = 2$ if $m+1 \leq i \leq k$
- if $m < k$ then $\sigma^{(i)}$ is exceptional for $m \leq i \leq k$.
If $\sigma$ is exceptional, then $\pi$ is exceptional of the same type as $\sigma$ (Proposition \[prop:except\]). Then we set $m = 0$ and $k = (|\sigma|-|\pi|)/2$, and $\sigma^{(i)}$ are exceptional permutations of the same type as $\sigma$ and size between $|\pi|$ and $|\sigma|$.
If $\sigma$ is not exceptional, we set $\sigma^{(0)} = \sigma$, and we construct $\sigma^{(i)}$ by induction while $\sigma^{(i-1)}$ is not exceptional and $\pi \neq \sigma^{(i-1)}$: from Theorem \[thm:main\], there exists a simple permutation $\sigma^{(i)}$ such that $\pi \preceq \sigma^{(i)} \preceq \sigma^{(i-1)}$ and $|\sigma^{(i)}|=|\sigma^{(i-1)}|-1$. We iterate until $\sigma^{(j)} = \pi$, then $m = k = |\sigma|-|\pi|$ and we have the result, or until $\sigma^{(j)}$ is exceptional. Then $\pi$ is exceptional of the same type as $\sigma^{(j)}$. Then we set $m = j$ and $k = m+(|\sigma^{(j)}|-|\pi|)/2$, and $\sigma^{(i)}$ for $j \leq i \leq k$ are exceptional permutations of the same type as $\pi$ and size between $|\pi|$ et $|\sigma^{(j)}|$.
Note that the paths in $G$ between two simple permutations can be of different length. As example with $\sigma = 5263714$ and $\pi = 3142$, we have a path of length 3 (by $526314$ and $42613$, non exceptional, see Figure \[fig:path3\]) and a path of length 2 (by $415263$, exceptional, see Figure \[fig:path2\]).
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But if $\sigma$ and $\pi$ are non exceptional, all path from $\sigma$ to $\pi$ have a length $|\sigma|-|\pi|$, and if $\sigma$ and $\pi$ are exceptional, all path from $\sigma$ to $\pi$ have a length $(|\sigma|-|\pi|)/2$.
The last case is $\sigma$ not exceptional and $\pi$ exceptional. Theorem \[thm:1to1\] prove that we can always choose a path with only one exceptional permutation: $\pi$.
\[prop:motif-excep\] Let $\pi$ be an exceptional permutation of size $2(n+1)$ where $n \geq 2$, $P$ be a set of $n$ points of $\pi$ and $\pi'$ the exceptional permutation of size $2n$ of the same type as $\pi$. Then there exists a pattern $\pi'$ in $\pi$ which contain all points of $P$.
We have only to prove the result for exceptional permutations of type $3$, the result for the other types follows by symmetry. Suppose that $\pi$ is of type $3$. Then by deleting two points of $\pi$ of consecutive indices, we obtain a pattern $\pi'$. So we have only to prove that there exists two points of consecutive indices which are not in $P$. If it doesn’t exist two points of consecutive indices which are not in $P$, then $P$ contains at least half of points of $\pi$ i.e. $n+1$ points, a contradiction.
\[propF\] Let $\pi' \prec \pi$ be exceptional permutations with $|\pi'|=|\pi|-2$ and $\sigma$ a non exceptional simple permutation such that $\pi \prec \sigma$ and $|\sigma| = |\pi|+1$. Then there exists a simple permutation $\tau$ such that $\pi' \prec \tau \prec \sigma$ and $|\tau| = |\pi'|+1$.
Let $n = |\sigma| = |\pi|+1$, as $\pi \prec \sigma$ there exists an index $k \in \{1 \dots n\}$ such that we obtain $\pi$ from $\sigma$ by deleting $\sigma_k$. As $\sigma$ is simple, we can’t have ($k \in \{1, n\}$ and $\sigma_k \in \{1, n\}$). So there exist $i$ and $j$ in $\{1 \dots n\}$ such that $i = k-1$ and $j = k+1$, or $\sigma_i = \sigma_k-1$ and $\sigma_j = \sigma_k+1$. As $\sigma$ is simple, There exists a point $\sigma_{i'}$ which separates $\sigma_i$ from $\sigma_k$ and a point $\sigma_{j'}$ which separates $\sigma_j$ from $\sigma_k$.
If $|\pi| \geq 10$, from Proposition \[prop:motif-excep\] there exists a pattern $\pi'$ in $\pi$ (thus in $\sigma$) which contains $\sigma_i$, $\sigma_j$, $\sigma_{i'}$ and $\sigma_{j'}$. Let $\tau$ be the permutation obtained from this pattern $\pi'$ in $\sigma$ and point $\sigma_k$. Then $\pi' \prec \tau \prec \sigma$ and $|\tau| = |\pi'|+1$. Moreover if $\tau$ were not simple, as $\pi' = \tau \setminus \{\sigma_k\}$ is simple, from Proposition \[prop:intervalleOuCoin\] $\sigma_k$ belongs to an interval $I$ of size $2$ of $\tau$ ($\sigma_k$ is not in a corner of the graphical representation of $\tau$ because $i < k < j$ or $\sigma_i < \sigma_k < \sigma_j$). Set $I = \{\sigma_{\ell}, \sigma_k\}$, then $\ell =i$ or $\ell =j$, excluded because $\sigma_{i'}$ separates $\sigma_i$ from $\sigma_k$ and $\sigma_{j'}$ separate $\sigma_j$ from $\sigma_k$. Thus $\tau$ is simple and we have the result expected.
As $\pi'$ is exceptional, $|\pi'| \geq 4$ so it remains only to prove the result when $|\pi|=6$ or $|\pi|=8$. We can prove by exhaustive verification that in this case there also exists a set $P$ of points of $\pi$ building a pattern $\pi'$ such that $\tau = P \cup \sigma_k$ fulfil our proposition.
\[thm:1to1\] Let $\sigma \not= \pi$ be two simple permutations, $\sigma$ non exceptional and $\ell = |\sigma|-|\pi|$. If $\pi \preceq \sigma$ and $|\pi| \geq 3$ then there exists a chain of simple permutations $\sigma^{(0)} = \sigma, \sigma^{(1)}, \ldots, \sigma^{(\ell-1)}, \sigma^{(\ell)}=\pi$ such that $\forall i$, $\sigma^{(i)} \preceq \sigma^{(i-1)}$ and $|\sigma^{(i-1)}| - |\sigma^{(i)}| = 1$.
Let $(\sigma^{(i)})_{0\leq i\leq k}$ be a chain of simple permutations given by Theorem \[thm:chain\] with $m$ maximum. If $m < k$ then $\sigma^{(m)}$ and $\sigma^{(m+1)}$ are exceptional, and $\sigma^{(m-1)}$ is not exceptional ($m > 0$ because $\sigma$ is not exceptional). We can apply Proposition \[propF\], and we have a simple permutation $\tau$ such that $\sigma^{(m+1)} \preceq \tau \preceq \sigma^{(m-1)}$ and $|\tau| = |\sigma^{(m+1)}|+1$. By Theorem \[thm:theorem0.3\], we have a simple permutation $\rho$ such that $\tau \preceq \rho \preceq \sigma^{(m-1)}$. Then we set $\pi^{(i)} = \sigma^{(i)}$ if $1 \leq i \leq m-1$, $\pi^{(m)} = \rho$, $\pi^{(m+1)} = \tau$ and $\pi^{(i)} = \sigma^{(i-1)}$ if $m+2 \leq i \leq k+1$. Then we have $|\pi^{(i-1)}| - |\pi^{(i)}| = 1$ if $1 \leq i \leq m+1$, which give us a chain of simple permutations verifying conditions of Theorem \[thm:chain\] so $m$ is not maximum, a contradiction. Thus $m = k$ and we have the result expected.
=\[draw,fill=white,rectangle,rounded corners=2,drop shadow,line width=0.5pt\]; =\[fill=red!20, drop shadow\]; (0,8) node (2 7 4 8 1 6 3 5 ) [$2\,7\,4\,8\,1\,6\,3\,5\,$]{}; (-2.0,7) node (2 4 7 1 6 3 5 ) [$2\,4\,7\,1\,6\,3\,5\,$]{}; (2 4 7 1 6 3 5 .north) – (2 7 4 8 1 6 3 5 .south); (-1.0,7) node (2 6 4 7 1 3 5 ) [$2\,6\,4\,7\,1\,3\,5\,$]{}; (2 6 4 7 1 3 5 .north) – (2 7 4 8 1 6 3 5 .south); (0.0,7) node (2 6 4 7 1 5 3 ) [$2\,6\,4\,7\,1\,5\,3\,$]{}; (2 6 4 7 1 5 3 .north) – (2 7 4 8 1 6 3 5 .south); (1.0,7) node (2 7 4 1 6 3 5 ) [$2\,7\,4\,1\,6\,3\,5\,$]{}; (2 7 4 1 6 3 5 .north) – (2 7 4 8 1 6 3 5 .south); (2.0,7) node (6 3 7 1 5 2 4 ) [$6\,3\,7\,1\,5\,2\,4\,$]{}; (6 3 7 1 5 2 4 .north) – (2 7 4 8 1 6 3 5 .south); (-4.0,6) node (2 4 1 6 3 5 ) [$2\,4\,1\,6\,3\,5\,$]{}; (2 4 1 6 3 5 .north) – (2 4 7 1 6 3 5 .south); (2 4 1 6 3 5 .north) – (2 7 4 1 6 3 5 .south); (-3.0,6) node\[exceptional\] (2 4 6 1 3 5 ) [$2\,4\,6\,1\,3\,5\,$]{}; (2 4 6 1 3 5 .north) – (2 4 7 1 6 3 5 .south); (2 4 6 1 3 5 .north) – (2 6 4 7 1 3 5 .south); (-2.0,6) node (2 4 6 1 5 3 ) [$2\,4\,6\,1\,5\,3\,$]{}; (2 4 6 1 5 3 .north) – (2 4 7 1 6 3 5 .south); (2 4 6 1 5 3 .north) – (2 6 4 7 1 5 3 .south); (-1.0,6) node (2 5 3 6 1 4 ) [$2\,5\,3\,6\,1\,4\,$]{}; (2 5 3 6 1 4 .north) – (2 6 4 7 1 3 5 .south); (2 5 3 6 1 4 .north) – (2 6 4 7 1 5 3 .south); (0.0,6) node (2 6 4 1 3 5 ) [$2\,6\,4\,1\,3\,5\,$]{}; (2 6 4 1 3 5 .north) – (2 6 4 7 1 3 5 .south); (2 6 4 1 3 5 .north) – (2 7 4 1 6 3 5 .south); (1.0,6) node (2 6 4 1 5 3 ) [$2\,6\,4\,1\,5\,3\,$]{}; (2 6 4 1 5 3 .north) – (2 6 4 7 1 5 3 .south); (2 6 4 1 5 3 .north) – (2 7 4 1 6 3 5 .south); (2.0,6) node (3 6 1 5 2 4 ) [$3\,6\,1\,5\,2\,4\,$]{}; (3 6 1 5 2 4 .north) – (2 4 7 1 6 3 5 .south); (3 6 1 5 2 4 .north) – (6 3 7 1 5 2 4 .south); (3.0,6) node (5 2 6 4 1 3 ) [$5\,2\,6\,4\,1\,3\,$]{}; (5 2 6 4 1 3 .north) – (6 3 7 1 5 2 4 .south); (4.0,6) node (5 3 6 1 4 2 ) [$5\,3\,6\,1\,4\,2\,$]{}; (5 3 6 1 4 2 .north) – (2 6 4 7 1 5 3 .south); (5 3 6 1 4 2 .north) – (6 3 7 1 5 2 4 .south); (-2.0,5) node (2 4 1 5 3 ) [$2\,4\,1\,5\,3\,$]{}; (2 4 1 5 3 .north) – (2 4 1 6 3 5 .south); (2 4 1 5 3 .north) – (2 4 6 1 5 3 .south); (2 4 1 5 3 .north) – (2 6 4 1 5 3 .south); (-1.0,5) node (2 5 3 1 4 ) [$2\,5\,3\,1\,4\,$]{}; (2 5 3 1 4 .north) – (2 5 3 6 1 4 .south); (2 5 3 1 4 .north) – (2 6 4 1 3 5 .south); (2 5 3 1 4 .north) – (2 6 4 1 5 3 .south); (0.0,5) node (3 1 5 2 4 ) [$3\,1\,5\,2\,4\,$]{}; (3 1 5 2 4 .north) – (2 4 1 6 3 5 .south); (3 1 5 2 4 .north) – (3 6 1 5 2 4 .south); (1.0,5) node (3 5 1 4 2 ) [$3\,5\,1\,4\,2\,$]{}; (3 5 1 4 2 .north) – (2 4 6 1 5 3 .south); (3 5 1 4 2 .north) – (3 6 1 5 2 4 .south); (3 5 1 4 2 .north) – (5 3 6 1 4 2 .south); (2.0,5) node (4 2 5 1 3 ) [$4\,2\,5\,1\,3\,$]{}; (4 2 5 1 3 .north) – (2 5 3 6 1 4 .south); (4 2 5 1 3 .north) – (5 2 6 4 1 3 .south); (4 2 5 1 3 .north) – (5 3 6 1 4 2 .south); (-0.5,4) node\[exceptional\] (2 4 1 3 ) [$2\,4\,1\,3\,$]{}; (2 4 1 3 .north) – (2 4 1 5 3 .south); (2 4 1 3 .north) – (2 5 3 1 4 .south); (2 4 1 3 .north) – (3 1 5 2 4 .south); (2 4 1 3 .north) – (3 5 1 4 2 .south); (2 4 1 3 .north) – (4 2 5 1 3 .south); (0.5,4) node\[exceptional\] (3 1 4 2 ) [$3\,1\,4\,2\,$]{}; (3 1 4 2 .north) – (2 4 1 5 3 .south); (3 1 4 2 .north) – (3 1 5 2 4 .south); (3 1 4 2 .north) – (3 5 1 4 2 .south); (3 1 4 2 .north) – (4 2 5 1 3 .south); (2 4 6 1 3 5 .south) .. controls +(0,-1) and +(-1,0) .. (2 4 1 3 .west);
Degree of vertices in the poset
-------------------------------
The preceding section proves that the poset is ranked if we omit exceptionnal permutations meaning that each level of the poset corresponds to simple permutations of given size and there exist only edges between permutations of contiguous ranks. In this section, we study the possible edges between two contiguous levels which provide statistics on simple permutations. More precisely, Proposition \[prop:size-1\] proves that if $\sigma$ is a non-exceptional simple permutation, then it exists a simple permutation $\sigma'$ of size $|\sigma|-1$ such that $\sigma' \prec \sigma$. In other words, there exists a point $\sigma_i$ of $\sigma$ such that the permutation obtained when deleting $\sigma_i$ and renormalizing is simple. But how many points of $\sigma$ have this property? To answer this question, we look at the -possibly multi-graph where vertices are simple permutations and as many edges between $\sigma$ and $\sigma'$ as the number of possible ways to insert an element in $\sigma'$ to obtain $\sigma$. Proving that this indeed is a graph shows that our question is equivalent to counting the number of edges between two consecutive levels in the original poset.
\[simple->simple\] Let $\sigma = \sigma_{1}\sigma_{2}\dots \sigma_{n}$ be a simple permutation and $\tau$ be a simple permutation of size $n+1$ such that $\sigma \preceq \tau$. Then there exists only one way to obtain $\tau$ by adding a point in $\sigma$.
Suppose that there exist at least 2 ways to do it. Then there are integers $a \neq b$ and $i<k$ such that:\
$\tau = \sigma'_{1} \dots \sigma'_{i-1}\ a\ \sigma'_{i} \dots \sigma'_{k-1}\ \sigma'_{k} \dots \sigma'_{n}$ and\
$\tau = \sigma''_{1}\dots\sigma''_{i-1}\ \sigma''_{i}\dots \sigma''_{k-1}\ b\ \sigma''_{k}\dots\sigma''_{n}$\
where $\sigma'_{j}=\left\{
\begin{array}{ll}
\sigma_{j} & \mathrm{if}\ \sigma_{j} < a\\
\sigma_{j}+1 & \mathrm{otherwise} \\
\end{array}
\right.$ and $\sigma''_{j}=\left\{
\begin{array}{ll}
\sigma_{j} & \mathrm{if}\ \sigma_{j} < b\\
\sigma_{j}+1 & \mathrm{otherwise.} \\
\end{array}
\right.$
In particular the equality between these two ways to write $\tau$ implies that if $i < k-1$, then $\sigma'_{i} = \tau_{i+1} = \sigma''_{i+1}$, which is impossible because $\sigma$ is simple so $|\sigma_{i}-\sigma_{i+1}| \geq 2$. Thus $i = k-1$, but then the equality implies that $a = \tau_i =\sigma''_i$ and $b = \tau_{i+1} =\sigma'_i$, so $\{a, b\} = \{\sigma_i, \sigma_i+1\}$, which is impossible because $\tau$ is simple. Consequently there is only one way to write $\tau$ from $\sigma$.
Recalling that $G$ is the graph representing the poset of simple permutations defined at the beginning of Section \[sec:poset\], we consider the graph $G_1$ obtained from $G$ by deleting edges between two exceptional permutations: note that there is an edge in $G_1$ from a simple permutation $\sigma$ to a simple permutation $\pi$ if and only if we can obtain $\pi$ from $\sigma$ by deleting one point.
Let $\pi$ be a simple permutation. We define the set of parents $S_{\pi+}$ -resp. children $S_{\pi-}$ - of $\pi$ in $G_1$ by:\
$S_{\pi+} = \{\sigma\ |\ \sigma$ is simple, $\pi \preceq \sigma$ and $|\sigma| = |\pi|+1\}$ and\
$S_{\pi-} = \{\sigma\ |\ \sigma$ is simple, $\sigma \preceq \pi$ and $|\sigma| = |\pi|-1\}$
\[indegree\] Let $\pi$ be a simple permutation of size $n$.\
Then ${|S_{\pi+}| = (n+1)(n-3)}$.
Permutations of $S_{\pi+}$ are simple permutations obtained from $\pi$ by adding one point. There are $(n+1)^2$ ways to insert a point in $\pi$ (giving permutations not necessarily different): if we consider the graphical representation of $\pi$ in a grid, adding one point to $\pi$ corresponds to choosing one point in the grid, which is of size $(n+1)^2$. But we want only simple permutations, that exclude $4(n+1)$ points in the grid: for one fixed point $\pi_i$ of $\pi$, we can’t take one of the $4$ corners of the cell where it is, that exclude $4n$ points which are all different because $\pi$ is simple so there is no points in contiguous cells. And we can’t take one of the $4$ corners of the grid, and these $4$ points have not been excluded yet because $\pi$ is simple so there is no point in a corner. There are $4(n+1)$ points excluded among $(n+1)^2$ possibilities and Proposition \[prop:intervalleOuCoin\] ensure that they are the only points to exclude, so we have $(n+1)(n-3)$ points left which give simple permutations. We have now to ensure that we can’t have the same simple permutations from two different points, which is given by Proposition \[simple->simple\].
So $|S_{\pi+}|$, which is the indegree of $\pi$ in the graph $G_1$, is independant of $\pi$. We are now interested in $|S_{\pi-}|$, the outdegree of $\pi$ in $G_1$. We know that it depends on $\pi$, and especially that $|S_{\pi-}| = 0$ if and only if $\pi$ is exceptional. We know also that $|S_{\pi-}| \leq |\pi|$. We consider the average outdegree in $G_1$.
file [ficDistrib\_5.txt]{}; file [ficDistrib\_10.txt]{}; file [ficDistrib\_15.txt]{}; file [ficDistrib\_20.txt]{}; file [ficDistrib\_25.txt]{}; file [ficDistrib\_30.txt]{}; file [ficDistrib\_35.txt]{}; file [ficDistrib\_40.txt]{}; file [ficDistrib\_45.txt]{}; file [ficDistrib\_50.txt]{};
\[prop:n4\] Let $D_n$ be the average outdegree of simple permutations of size $n$ in $G_1$. Then $D_n = n-4-\frac{4}{n} + O(\frac{1}{n^2})$.
In $G_1$, there is an edge from a simple permutation $\sigma$ to a simple permutation $\pi$ if and only if we can obtain $\pi$ from $\sigma$ by deleting one point. So edges that come from permutations of size $n$ are those which go to permutations of size $n-1$. Let $s_n$ be the number of simple permutations of size $n$. From Proposition \[indegree\] we know that there are $s_{n-1} \times n(n-4)$ such edges. So $D_n = \dfrac{s_{n-1} \times n(n-4)}{s_n}$. But from Theorem 5 of [@AAK03] we know that $s_n = \dfrac{n!}{e^2}\Big(1-\dfrac{4}{n}+\dfrac{2}{n(n-1)}+O(n^{-3})\Big)$ and a straightforward computation allows us to conclude.
We are now interested in the number $S_n^k$ of simple permutations of size $n$ and of outdegree $k$ fixed. As example we know that $S_n^0 = 4$ for every even $n$ and $S_n^0 = 0$ for every odd $n$ (number of exceptional permutations). Figure \[fig:outdegree\] show the percentage of simple permutations which have outdegree $k$. Each plot shows this distribution for a given size of permutations as indicated in the caption. Notice that these plots illustrate the result given in Proposition \[prop:n4\].
Let $S_n^k = |\{\pi \ |\ |S_{\pi-}| = k\}|$ be the number of simple permutations of size $n$ and of outdegree $k$ in $G_1$. Then for every fixed $k$, the proportion $\dfrac{S_n^k}{s_n}$ of simple permutations of outdegree $k$ among simple permutations of size $n$ tends to zero when $n$ tends to infinity.
By definition $s_n \times D_n = \sum_{i=0}^{n} i\times S_n^i$. Suppose that there exists $k$ such that $\dfrac{S_n^k}{s_n}$ does not tend to zero, then there exists $\epsilon > 0$ such that $\forall n_0$, $\exists n \geq n_0$ such that $\dfrac{S_n^k}{s_n} > \epsilon$. But then $$D_n = \sum_{i=0}^{n} i \times \dfrac{S_n^i}{s_n}
= k \times \dfrac{S_n^k}{s_n} + \sum_{i\neq k, i=0}^{n} i \times \dfrac{S_n^i}{s_n}
\leq k + \sum_{i\neq k, i=0}^{n} n \times \dfrac{S_n^i}{s_n}
= k + n\Big(1-\dfrac{S_n^k}{s_n}\Big)
\leq k + n(1-\epsilon)$$ but from Proposition \[prop:n4\], for $n_0$ large enough $D_n \geq n-5$, a contradiction.
An algorithm to generate simple permutations in a wreath-closed permutation class {#sec:algo}
=================================================================================
Theorem \[thm:main\] characterizes the pattern relation between non exceptional simple permutations. This theorem ensures that if $\sigma$ is a non exceptional simple permutation and $Av(B)$ a wreath-closed class of permutations, then $\sigma$ does not belong to $Av(B)$
- if and only if it contains a permutation of $B$ as a pattern.
- if and only if it is equal to a permutation of $B$ or contains as a pattern a simple permutation of size $|\sigma|-1$ which does not belong to $Av(B)$.
This recursive test leads to Algorithm \[alg:algo1\] (see p.17).
$Si_1 \leftarrow \{1\}, Si_2 \leftarrow \{ 12,21\}, Si_3 \leftarrow \emptyset, Si_4\leftarrow \{ 2413,3142\} \setminus B$ $n \leftarrow 5$
Its validity is proved in the next Theorem based on results from Section \[sec:MotifsSimples\]. Note that in order to avoid trivial cases, we assume that $B$ does not contain $12$ or $21$.
\[th:algo\] The set $Si_n$ computed by Algorithm \[alg:algo1\] is the set of simple permutations of size $n$ contained in $Av(B)$.
The preceding theorem holds for $n \leq 4$. For $n \geq 5$, we show it by induction.
We have to prove that every simple permutation $\sigma$ of size $n$ in $Av(B)$ belongs to $Si_n$. Let $\sigma$ be a simple permutation of size $n$ in $Av(B)$. If $\sigma$ is not exceptional, there exists $\pi$ simple such that $\pi \preceq \sigma$ and $|\pi| = |\sigma|-1$ (Proposition \[prop:size-1\]). By induction hypothesis, $\pi \in Si_{n-1}$ so that $\sigma$ is considered at line $6$ of our algorithm. As $\sigma \in Av(B)$, $\sigma \not\in B$ and every simple pattern $\tau$ of $\sigma$ of length $n-1$ is in $Av(B)$ and by induction hypothesis lies in $Si_{n-1}$. Thus line $18$ is reached and $\sigma$ is added to $Si_n$. If $\sigma$ is exceptional, $\sigma$ is considered at line $24$ of our algorithm and is added to $Si_n$ by induction hypothesis.
Reciprocally, let us prove that every permutation $\sigma \in Si_n$ is a simple permutation of size $n$ of $Av(B)$. If $\sigma \in Si_n$ notice first that $\sigma$ is simple and of size $n$. Suppose now that $\sigma \not\in Av(B)$ then it exists $\pi \in B$ ($\pi$ simple) such that $\pi \preceq \sigma$. We have $\sigma \neq \pi$ otherwise $\sigma \in B$ but there is no permutation of $B$ in $Si_n$ (because of lines $7$ and $24$ of the algorithm). If $\sigma$ is not exceptional, using Theorem \[thm:main\], we can find $\tau$ simple of size $n-1$ such that $\pi \preceq \tau \preceq \sigma$, so $\tau \not\in Av(B)$ and by induction hypothesis $\tau \not\in Si_{n-1}$. But our algorithm tests every pattern of $\sigma$ of size $n-1$ in line $9$ so $\sigma$ is not added to $Si_n$. If $\sigma$ is exceptional, then $|\pi|$ is even (Proposition \[prop:except\]) so $\pi \preceq \sigma'$ where $\sigma'$ is the exceptional permutation of the same type as $\sigma$ of size $|\sigma| - 2$. By induction hypothesis $\sigma' \notin Si_{n-2}$ so $\sigma$ is not added to $Si_n$ and we have the result.
Algorithm \[alg:algo1\] terminates if and only if $Av(B)$ contains only a finite number of simple permutations. In this case it gives all simple permutations in $Av(B)$.
If Algorithm \[alg:algo1\] terminates, there exists $n \geq 5$ such that $Av(B)$ contains no simple permutation of size $n-1$ or $n-2$. Suppose that $Av(B)$ contains a simple permutation $\sigma$ of size $k \geq n$, then from Proposition \[prop:exceptional\] and Proposition \[prop:simplePattern\] $\sigma$ has a simple pattern of size $n-1$ or $n-2$ in $Av(B)$, a contradiction. So $Av(B)$ contains no simple permutation of size greater than $n-2$ and Theorem \[th:algo\] ensures that the algorithm gives all simple permutations in $Av(B)$.
Conversely if $Av(B)$ contains only a finite number of simple permutations, let $k$ be the size of the greater simple permutation in $Av(B)$. From Theorem \[th:algo\], the algorithm computes $Si_{k+1}=Si_{k+2}=\emptyset$ and the algorithm terminates.
Before running Algorithm \[alg:algo1\] we can test whether $Av(B)$ contains a finite number of simple permutations in time ${\mathcal O}(n \log n)$ where $n=\sum_{\pi \in B} |\pi|$ thanks to the algorithm given in [@BBPR09]. If $Av(B)$ contains a finite number of simple permutations, Algorithm \[alg:algo1\] give all simple permutations in $Av(B)$. If $Av(B)$ contains an infinite number of simple permutations, we can use a modified version of the algorithm to obtain all simple permutations in $Av(B)$ of size less than a fixed integer $k$: it is sufficient to replace in the algorithm “while $Si_{n-1} \not= \emptyset$ or $Si_{n-2} \not= \emptyset$” by "for $n \leq k$”.
Let us now evaluate the complexity of our algorithm.
The complexity of Algorithm \[alg:algo1\] is ${\mathcal O}\big(\sum_{n=5}^{k+1} n^{4} |Si_{n-1}|\big)$ where $k$ is the size of the longest simple permutation in $Av(B)$.
First, we encode every set of permutations as tries, allowing a linear algorithm to check if a permutation is in the set. The [*while*]{} loop beginning at line $3$ is done for $n$ from $5$ to $k+2$. The inner loop beginning at line $5$ is repeated $|Si_{n-1}|$ times (with $|Si_{k+1}|=0$). The loop of line $6$ is repeated $n(n-4)$ times (see Proposition \[indegree\]). This loop performs the following tests:
- Compute $\sigma$ : ${\mathcal O}(n)$
- Test whether $\sigma$ is in $B$ : ${\mathcal O}(n)$ using tries.
- Loop at line $9$ is performed $n$ times and perform each time the following operations:
- Compute $\tau$ : ${\mathcal O}(n)$
- Test whether $\tau$ is simple : ${\mathcal O}(n)$
- Test whether $\tau \in S_{n-1}$ : ${\mathcal O}(n)$
- Add if necessary $\sigma$ into $Si_n$ : ${\mathcal O}(n)$ as we use tries.
Thus the inner part of loop in line $5$ has complexity ${\mathcal O}(n(n-4)(n+n+n(n+n+n)+n))$ leading to the claimed result.
Indeed, the exceptional case is easy to implement in ${\mathcal O}(n)$ time as there are at most $4$ exceptional permutations of a given size.
Concluding remarks
==================
Theorem \[thm:1to1\] gives a structural result on simple permutations poset. It has many implications, two of which are explicited in this article: the first one being the average number of points that can be removed in a simple permutation and remain simple, the second one leading to a polynomial time algorithm for computing the set of simple permutations in a wreath-closed class. For the latter problem, we restrict ourselves to wreath-closed class of permutations and unfortunately cannot apply it directly to general classes. Indeed, to adapt our algorithm to the general case, the part of the algorithm from line $7$ to line $20$ can be replaced by testing if $\sigma \in Av(B)$ and in that case adding it to $Si_n$. Unfortunately, there is no efficient algorithm to test if a permutation is in a class $Av(B)$. The only known algorithm is to test if $\sigma$ avoids every permutations in $B$. Thus we have the following proposition:
For every class $Av(B)$ containing a finite number of simple permutations, we can compute the simple permutations in $Av(B)$ in time $\sum_{n=5}^{k+1} |Si_{n-1}|n^2f_{Av(B)}(n)$ where $k$ is the size of the longest simple permutation in $Av(B)$ and $f_{Av(B)}(n)$ the complexity of testing if a permutation of size $n$ belongs to $Av(B)$.
Note that in the preceding proposition function $f_{Av(B)}$ is bounded by $\sum_{\tau \in B} n^{|\tau|}$. Indeed a naive algorithm consists in testing the pattern condition for each permutation $\tau$ in the basis. In some cases, this test can be improved, see for example [@AAAH01; @BRV07]. In general case, this give a complexity of order ${\mathcal O} \big(|B|.|Si_B|.k^{p+2}\big)$ for computing the set $Si_B$ of simple permutations in $Av(B)$, with $p = \max \{|\tau| : \tau \in B\}$ and $k = \max \{|\pi| : \pi \in Si_B\}$. For wreath-closed classes, Algorithm \[alg:algo1\] has a complexity of order ${\mathcal O}\big(|Si_B|.k^{4}\big)$
An open question is whether there exists a more efficient algorithm in the general case and more precisely for testing if a permutation belongs to a given class.
|
---
abstract: 'Galactic ultra compact binaries are expected to be the dominant source of gravitational waves in the milli-Hertz frequency band. Of the tens of millions of galactic binaries with periods shorter than an hour, it is estimated that a few tens of thousand will be resolved by the future Laser Interferometer Space Antenna (LISA). The unresolved remainder will be the main source of “noise” between 1-3 milli-Hertz. Typical galactic binaries are millions of years from merger, and consequently their signals will persist for the the duration of the LISA mission. Extracting tens of thousands of overlapping galactic signals and characterizing the unresolved component is a central challenge in LISA data analysis, and a key contribution to arriving at a global solution that simultaneously fits for all signals in the band. Here we present an end-to-end analysis pipeline for galactic binaries that uses trans-dimensional Bayesian inference to develop a time-evolving catalog of sources as data arrive from the LISA constellation.'
author:
- 'Tyson B. Littenberg'
- 'Neil J. Cornish'
- Kristen Lackeos
- Travis Robson
bibliography:
- 'refs.bib'
title: Global Analysis of the Gravitational Wave Signal from Galactic Binaries
---
Introduction
============
The most prolific source of gravitational waves (GWs) in the mHz band are galactic ultra compact binaries (UCBs), primarily comprised of two white dwarf stars. Ref. [@Korol:2017qcx] describes a contemporary prediction for the population of UCBs detectable by the Laser Interferometer Space Antenna (LISA) [@LISA]. GWs from UCBs are continuous sources for LISA, several thousands of which will be individually resolvable. The remaining binaries blend together to form a confusion-limited foreground that is expected to be the dominant “noise” contribution to the LISA data stream at frequencies below ${\sim}3$ mHz, the extent of which depending on the population of binaries and the observing time of LISA [@Cornish:2017vip].
Of the thousands of resolvable binaries, the best-measured systems will serve as laboratories for studying the dynamical evolution of the binaries. Encoded within the orbital dynamics are relativistic effects, the internal structure of WD stars, and effects of mass transfer [@Taam1980; @Savonije1986; @Willems2008; @Nelemans2010; @Littenberg_2019; @Piro_2019]. The observable population of UCBs will depend on astrophysical processes undergone by binary stars that are currently not well understood, including the formation of the compact objects themselves, binary evolution, and the end result for such binaries [@Webbink1984]. UCBs are detectable anywhere in the galaxy because the GW signals are unobscured by intervening material in the Galactic plane, providing an unbiased sample to infer large scale structure of the Milky Way [@Adams:2012qw; @Korol2018]. While LISA will dramatically increase our understanding of UCBs in the galaxy, there is an ever-increasing number of systems discovered by electromagnetic (EM) observations that will be easily detectable by LISA [@Kupfer_2018; @Burdge_2019; @Burdge_2019b; @Brown_2020]. Thus UCBs are guaranteed multimessneger sources and the joint EM+GW observations provide physical constraints on masses, radii, and orbital dynamics far beyond what independent EM or GW observations can achieve alone [@Shah2014a; @Littenberg_2019b].
The optimal detection, characterization, and removal of UCBs from the data stream has been long recognized as a fundamentally important and challenging aspect of the broader LISA analysis. Over-fitting the galaxy will result in a large contamination fraction in the catalog of detected sources, while under-fitting the UCB population will degrade the analyses of extragalactic sources in the data due to the excess residual.
In this paper we describe a modern implementation of a UCB analysis pipeline which is a direct descendent of the trailblazing algorithms designed in response to the original Mock LISA Data Challenges (MLDCs) [@Babak_2008; @Babak_2010], and similar methods developed for astrophysical transients and non-Gaussian detector noise currently in use for ground-based GW observations [@Cornish:2014kda; @Littenberg:2015].
{width="45.00000%"} {width="45.00000%"}
Previous work
=============
Compared to other GW sources, UCBs are simple to model. When in the LISA band, the binary is widely separated and the stars’ velocities are small compared to the speed of light $c$. Therefore the waveforms are well predicted using only leading order terms for the orbital dynamics of the binary [@Peters_1963] and appear as nearly monochromatic (constant frequency) sources. Accurate template waveforms are computed at low computational cost using a fast/slow decomposition of the waveform convolved with the instrument response [@Cornish:2007if].
The UCB population is nevertheless a challenging source for LISA analysis due to the sheer number of sources expected to be in the measurement band, rather than the complication of detecting and characterizing individual systems. Each source is well-modeled by $\mathcal{O}(10)$ parameters and over $10^4$ sources are expected to be individually resolvable by LISA, resulting in a ${\sim}10^5$ parameter model and thus ruling out any brute-force grid-based method. Compounding the challenge is the fact that the GW signals, though narrow-band, are densely packed within the LISA measurement band to the extent that sources are overlapping. As a consequence, a hierarchichal/iterative scheme where bright sources are removed and the data is reanalyzed produces biased parameter estimation and poorer detection efficiency: Each iteration leaves behind some residual due to imperfect subtraction, and enough iterations are required for the residuals to build up to the point where they limit further analysis [@gClean]. It was determined in the early 2000s that stochastic sampling algorithms performing a global fit to the resolvable binaries, while simultaneously fitting a model for the residual confusion or instrument noise and using Bayesian model selection to optimize the number of detectable sources, provided an effective approach.
The first full-scale demonstration of a galactic binary analysis was put forward by Crowder and Cornish [@Cornish:2005qw; @Crowder:2006eu] with the Blocked Annealed Metropolis (BAM) Algorithm. The BAM Algorithm started from the full multi-year data set provided by the Mock LISA Data Challenges (MLDCs) [@Babak_2008]. Because the sources are narrow-band compared to the full measurement band of the detector, the search was conducted independently on sub-regions in frequency. The analysis region in each segment was buffered by additional frequency bins that overlapped with neighboring segments. The noise spectrum was artificially increased over the buffer frequencies to suppress signal power from sources in neighboring bands which spread into the analysis window.
The template waveforms were computed in the time domain, Fourier transformed, and tested against the frequency domain data. In accordance to the MLDC simulations, the waveform model did not include the intrinsic frequency evolution of the binaries, and the frequency-dependent detector noise level was assumed to be known *a priori*. The BAM analysis was a quasi-Bayesian approach, using a generalized multi-source $\mathcal{F}$ statistic likelihood that maximized, rather than marginalized, over four of the extrinsic parameters of each waveform. Model parameters used flat priors except for the sky location which was derived from an analytic model for the spatial distribution of binaries in the galaxy, projected onto the sky as viewed by LISA. To improve the convergence of the algorithm, particularly for high-${\rm{SNR}}$ signals, the sampler used simulated annealing [@Kirkpatrick671] during the burn-in phase.
To sample from the likelihood function, BAM employed a custom Markov Chain Monte Carlo (MCMC) algorithm with a mixture of proposal distributions including uniform draws from the prior, jumps along eigenvectors of the Fisher information matrix for a given source, and localized uniform jumps over a range scaled by the estimated parameter errors. The BAM Algorithm made use of domain knowledge by explicitly proposing jumps by the modulation frequency $f \rightarrow f \pm 1/{\rm yr}$ to explore sidebands of the signal imparted by LISA’s orbital motion.
To determine the number of detectable sources, BAM employed an approximate Bayesian model selection criteria, where models of increasing dimension (i.e., number of detectable sources) were hierarchically evaluated, starting with a single source in each analysis segment and progressively adding additional sources to the fit. The different dimension models were ranked using the Laplace approximation to the Bayesian evidence [@Jeffreys61]. The stopping criteria for the model exploration was met when the approximated model evidence reached a maximum.
In response to the next generation of MLDCs, Littenberg [@Littenberg:2011zg] extended the BAM Algorithm in several key ways, but maintained the original concept of analyzing independent segments with attention paid to the segment boundaries to avoid edge effects. The primary advancement of this generation of the search pipeline was the use of replica exchange between chains of different temperatures (parallel tempering) [@PhysRevLett.57.2607] and marginalizing over the number of sources in the data (as opposed to hierarchically stepping through models) using a Reversible Jump MCMC (RJMCMC) [@doi:10.1093/biomet/82.4.711] to identify the range of plausible models. To guard against potentially poor mixing of the RJMCMC a dedicated fixed-dimension follow-up analysis with Bayesian evidence computed via thermodynamic integration [@Goggans_2004] was used for the final model selection determination. The algorithm continued using the $\mathcal{F}$ statistic likelihood and simulated annealing during burn in (the “search phase”) but switched to the full likelihood, sampling over all model parameters, during the parameter estimation and model selection phase of the analysis. The algorithm additionally made use of the burn-in by building proposal distributions from the biased samples derived during the non-Markovian search phase using a naive binning of the model parameters. The algorithm included a parameterized noise model, by fitting coefficients to the expected noise power spectral density (proportional to the variance of the noise). The waveform model included frequency evolution, and was computed directly in the Fourier domain using the fast-slow decomposition described in [@CornLitt07].
Experience gained from the noise modeling and trans-dimensional algorithms originally applied to the LISA galactic binary problem permeated into analyses of ground-based GW data from the LIGO-Virgo detectors. For spectral estimation the [`BayesLine`]{}algorithm uses a two-component phenomenological model to measure the frequency-dependent variance of the detector noise [@Littenberg:2015], while the [`BayesWave`]{}algorithm uses a linear combination of wavelets to fit short-duration non-Gaussian features in the data [@Cornish:2014kda]. The wavelet model in each detector is independent when fitting noise transients, and is coherent across the network when fitting GW models. The Bayes factor between the coherent and incoherent models is used as a detection statistic as part of a hierarchichal search pipeline [@PhysRevD.93.022002]. The number of wavelets, and components to the noise model, are all determined with an RJMCMC algorithm. The large volume of data, number of event candidates, and thorough measurement of search backgrounds motivated development of global proposal distributions to improve convergence times of the samplers. The [`BayesWave`]{}and [`BayesLine`]{}models were both inspired by the previous work on the galactic binary problem, with the wavelets substituting the UCB waveforms and the [`BayesLine`]{}model replacing the confusion noise fits. Completing the feedback loop, lessons learned from the development and deployment of the methods on the LIGO-Virgo data have formed part of the foundation in this work, particularly through global proposal distributions, numerical methods for reducing computational time of likelihood evaluations, and infrastructure for deploying the pipeline on distributed computing resources.
A New Hope
==========
The new UCB algorithm incorporates many of the features from the earlier efforts, but improves on them in several ways. The biggest change is the adoption of a time evolving strategy, which reflects the reality of the data collection. Analyzing the data as it is acquired also eliminates dedicated algorithm tuning choices for dealing with very loud sources. When new data are acquired the analysis starts on the residual after the bright sources identified previously are removed from the data. In each analysis segment, the removed sources are added back into the data before the RJMCMC begins sampling. This eliminates the problem of having power leakage between analysis segments, and the resultant noise model manipulation to suppress the model from being biased by edge effects in each segment. The time-evolving analysis is also naturally “annealed” as the ${\rm{SNR}}$ of sources builds slowly over time.
Other significant changes include improvements to the RJMCMC implementation with the addition of global proposal distributions which eliminate the need for a separate, non-Markovain, search phase or the fixed-dimension follow-up analysis for evidence calculation–the model selection is now robustly handled by the RJMCMC itself as originally intended.
For the first time in the context of our UCB work, we have also considered how to distill the unwieldy output from the RJMCMC to more readily useable, higher-level, data products which is how the majority of the research community will interact with the LISA observations.
The code described in this work is open source and available under the GNU/GPL v2 license [@littenberg_tyson_2020_3756199].
Model and Implementation
========================
Bayesian inference requires the specification of a likelihood function and prior probability distributions for the model components. The implementation of the analysis employs stochastic sampling techniques, in our case the trans-dimensional Reversible Jump Markov Chain Monte Carlo (RJMCMC) [@doi:10.1093/biomet/82.4.711] algorithm with replica exchange [@PhysRevLett.57.2607], to approximate the high dimensional integrals that define the marginalized posterior distributions. As with all MCMC algorithms, the choice of proposal distributions is critical to the performance. Here we detail the model and the implementation, hopefully in sufficient detail for the analysis to be repeated by others.
Likelihood function
-------------------
The LISA science analysis can be carried out using any complete collection of Time Delay Interferometry (TDI) channels [@Prince:2002hp; @Adams:2010vc]. For example, we could use the set of Michelson-type channels $I=\{X,Y,Z\}$, or any linear combination thereof. Schematically we can write ${{\bf d}}_I = {\bf h}_I + {\bf n}_I$, where ${\bf}h_I$ is the response of the $I^{\rm th}$ channel to all the gravitational wave signals in the Universe, and ${\bf n}_I$ is the combination of all the noise sources impacting that channel. Here the “noise” will include gravitational wave signals that are individually too quiet to extract from the data. The goal of the analysis is to reconstruct the detectable gravitational wave signal using a signal model ${\bf h}_I$ such that the residual ${\bf r}_I = {{\bf d}}_I - {\bf h}_I$ is consistent with the noise model. For Gaussian noise the likelihood is written as: $$p({{\bf d}}| {\bf h}) = \frac{1}{(2\pi \, \det{\bf C})^{1/2}} \, e^{- \frac{1}{2}(d_{Ik} - h_{Ik}) C^{-1}_{(I k)(J m)} (d_{Jm} - h_{Jm})}\, ,$$ where ${\bf C}$ is the noise correlation matrix, and the implicit sum over indicies spans the TDI channels $I=\{X,Y,Z\}$ and the data samples $k$. If the data are stationary, then the noise correlation matrix is partially diagonalized by moving to the frequency domain: $C_{(I k)(J m)} = S_{IJ}(f_k) \delta_{km}$, where $S_{IJ}(f)$ is the cross-power spectral density between channels $I,J$ [@Adams:2010vc]. The cross-spectral density matrix is diagonalized by performing a linear transformation in the space of TDI variables. If the noise levels are are equal on each spacecraft, this leads to the $I'=\{A,E,T\}$ variables [@Prince:2002hp] via the mapping $$\left[ \begin{array}{c} A \\ \\ E \\ \\ T \end{array} \right] = \begin{bmatrix} \frac{2}{3} & -\frac{1}{3} & -\frac{1}{3} \\ && \\ 0 & - \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{3}} \\ && \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{bmatrix} \left[ \begin{array}{c} X \\ \\Y \\ \\Z \end{array} \right]$$ In practice, the noise levels in each spacecraft will not be equal, and the $\{A,E,T\}$ variables will not diagonalize the noise correlation matrix [@Adams:2010vc]. However, $\{A,E,T\}$ serve another purpose as they diagonalize the gravitational wave polarization response of the detector for signals with frequencies $f < f_* = 1/(2 \pi L) \simeq 19.1 \; {\rm mHz}$, such that $A \sim h_+$, $E \sim h_\times$ and $T \sim h_\odot$. Since the breathing mode $h_\odot$ vanishes in general relativity, the gravitational wave response of the $T$ channel is highly suppressed for $f < f_*$, making the $T$ channel particularly valuable for noise characterization and the detection of stochastic backgrounds [@Tinto:2001ii; @Hogan:2001jn] and un-modeled signals [@travis4].
Full expressions for the instrument noise contributions to the cross spectra $S_{IJ}(f)$ are given in Ref. [@Adams:2010vc]. Added to these expressions will be contributions from the “confusion noise” from the millions of signals that are too quiet to detect individually. The confusion noise will add to the overall noise as well as introduce off-diagonal terms in the frequency domain noise correlation matrix ${\bf C}$, as the confusion noise is inherently non-stationary with periodic amplitude modulations imparted by LISA’s orbital motion [@PhysRevD.69.123005].
For now we have made a number of simplifying assumptions that will be relaxed in future work: We ignore the non-stationarity of the noise and assume that the noise correlation matrix is diagonal in the frequency domain; In addition, since we are mostly interested in signals with frequencies well below the transfer frequency $f_* \simeq 19.1 \; {\rm mHz}$, we only use the $A$ and $E$ data combinations in the analysis, and we assume that the noise in these channels is uncorrelated; Rather than working with a component level model for the noise, as was done in Ref. [@Adams:2010vc], we break the analysis up into narrow frequency bands $[f_i, f_i+\Delta f]$ and approximate the noise in each band as an undetermined constant $S_i$. The noise level in each band becomes a parameter to be explored by the RJMCMC algorithm, resulting in a piecewise fit to the instrument noise over the full analysis band.
The signal model ${{\mathbf h}}({\mathbf\Lambda})$ is the superposition of each individual UCB in the model parameterized by ${\mathbf\lambda}$: $${{\mathbf h}}_I({\mathbf\Lambda}) = \sum_{a=0}^{{N_{\rm GW}}} {{\boldsymbol h}}_I({\mathbf\lambda}_a)$$ where ${{\boldsymbol h}}_I({\mathbf\lambda}_a)$ denotes the detector response of the $I^{\rm th}$ data channel to the signal from a galactic binary with parameters ${\mathbf\lambda}_a$. Note that the number of detectable systems, ${N_{\rm GW}}$, is [*a priori*]{} unknown, and has to be determined from the analysis. Indeed, we will arrive at a probability distribution for ${N_{\rm GW}}$, which implies that there will be no single definitive source catalog. The individual binary systems are modeled as isolated point masses on slowly evolving quasi-circular orbits neglecting the possibility of orbital eccentricity [@Seto:2001pg], tides [@2012MNRAS.421..426F] or third bodies [@Robson:2018svj]. The signals are modeled using leading order post-Newtonian waveforms. The instrument response includes finite arm-length effects of the LISA constellation and arbitrary spacecraft orbits, but the TDI prescription currently implemented makes the simplifying assumption that the arm lengths are equal and unchanging with time. Adopting more realistic instrument response functions increases the computational cost but does not change the complexity of the analysis.
To compute the waveforms, a fast/slow decomposition is employed that allows the waveforms to be modeled efficiently in the frequency domain [@Cornish:2007if]. The basic idea is to use trigonometric identities to re-write the detector response to the signal in the form $h(t) = a(t) \cos(2 \pi f_k t)$ where $f_k = n_k/T_{\rm obs}$, $n_k = {\rm int}[ {f_0}T_{\rm obs}]$, and ${f_0}$ is the gravitational wave frequency of the signal (twice the orbital frequency) at some fiducial reference time. The Fourier transform of $h(t)$ is then $\tilde h(f) =\frac{1}{2} ( \tilde a(f-f_k) + \tilde a(f+f_k))$. Since $a(t)$,which includes the orbital evolution and time-varying detector response, varies much more slowly than the carrier signal $\tilde h(f) =\frac{1}{2} \tilde a(f-f_k)$, the Fourier transform of $a(t)$ is computed numerically using a lower sample cadence than needed to cover the carrier. A sample cadence of days is usually sufficient. Note that in the original implementation [@Cornish:2007if] the signal was written as $h(t) = a(t) \cos(2 \pi {f_0}t)$, which was less efficient as it required the convolution $\tilde{h} * \tilde{a}$. By mapping the carrier frequency to a multiple of the inverse observation time the Fourier transform of the carrier becomes a pair of delta functions and the convolution becomes the sum of just two terms, one of which effectively vanishes.
Each binary is parameterized by ${N_{\rm P}}$ parameters. ${N_{\rm P}}$ is typically eight, with ${\mathbf\lambda}\rightarrow ({\mathcal{A}}, {f_0}, {\dot{f}}, {\varphi_0}, \iota, \psi, \theta, \phi)$, where ${\mathcal{A}}$ is the amplitude, ${f_0}$ is the initial frequency, ${\dot{f}}$ is the (constant) time derivative of the, ${\varphi_0}$ is the initial phase, $\iota$ is the inclination of the orbit, $\psi$ the polarization angle and $\theta,\phi$ are the sky location in an ecliptic coordinate system. If the evolution of the binary were purely driven by gravitational wave emission we could replace the parameters $\left\{{\mathcal{A}}, {\dot{f}}\right\}$ by the chirp mass ${\cal M}$ and luminosity distance $D_L$ via the mapping $$\begin{aligned}
\label{MD}
{\dot{f}}&=& \frac{96}{5} \pi^{8/3} {\cal M}^{5/3} {f_0}^{11/3} \nonumber \\
{\mathcal{A}}&=& \frac{2 {\cal M}^{5/3} \pi^{2/3} {f_0}^{2/3}}{D_L} \, .\end{aligned}$$ We prefer the $\left\{{\mathcal{A}},{\dot{f}}\right\}$ parameterization as it is flexible enough to fit systems with non-GW contributions to the orbital dynamics, e.g. mass transferring systems, and it is better suited to modeling systems where ${\dot{f}}$ is poorly constrained (it is better to have just one parameter filling its prior range than two). For binaries with unambiguously positive ${\dot{f}}$, and assuming GW-dominated evolution of the orbit, we resample the posteriors to $\cal M$ and $\D_L$ in post-processing [@Littenberg_2019].
We also have optional settings to increase ${N_{\rm P}}$ by including the second derivative of the frequency [@Littenberg_2019] in which case the frequency derivative is no longer constant, so the parameter ${\dot{f}}\rightarrow{\dot{f}}_0$ is fixed at the same fiducial time as ${f_0}$ and ${\varphi_0}$. Additional, optional changes to the source parameterization includes holding an arbitrary number of parameters fixed at input values determined, for example, by EM observations [@Littenberg_2019b], or to include parameters which use the UCBs as phase/amplitude standards for self-calibration of the data [@Littenberg_2018].
Prior distributions
-------------------
The model parameters are given by the $N_n$ noise levels for each frequency band $S_i$ and the collection of ${N_{\rm GW}}\times{N_{\rm P}}$ signal parameters $\Lambda$. The number of noise parameters $N_n$ is fixed by our choice of bandwidth $\Delta f$ and the frequency range we wish to cover in the analysis. In the current configuration of the pipeline we use analysis windows with $\Delta f \sim\mathcal{O}(\mu\rm{Hz})$ in width resulting in $N_n=\mathcal{O}(10^4)$ noise parameters to cover the full measurement band of the mission. We use a uniform prior range $S_I \in [10^{-1} S_I(f_i), 10^{2} S_I(f_i)]$ where $S_I(f_i)$ is the theoretical value for the noise level of data channel $I$ used to generate the data. In practice the prior ranges on the noise will be set using information from the commissioning phase of the mission.
The total number of detectable signals ${N_{\rm GW}}$ per frequency band are unknown. We use a uniform prior covering the range ${N_{\rm GW}}\in U[0,30]$. For the individual source parameters we used uniform priors on the initial phase ${\varphi_0}\in [0,2\pi]$ and polarization angle $\psi \in [0,\pi]$, and a uniform prior on the cosine of the inclination $\cos \iota \in [-1,1]$. In each analysis window the initial frequency ${f_0}$ was taken to have a uniform prior covering the range $[f_i, f_i+\Delta f]$.
The allowed range of the frequency derivative is informed by population synthesis models which provide information on the mass and frequency distribution of galactic binaries [@Toonen_2012]. While the expression for the frequency derivative is only valid for isolated point masses, the balancing of accretion torques and gravitational wave emission in mass-transferring AM CVn type systems is thought to lead to a similar magnitude for the frequency derivative, but with the sign reversed [@Nelemans2010]. Using these considerations as input, we adopt a uniform prior on ${\dot{f}}$ in each frequency band that covers the range ${\dot{f}}=[ - 5\times 10^{-6} f_i^{13/3}, 8\times 10^{8} f_i^{11/3}]$.
![\[fig:skyprior\] The sky prior plotted in ecliptic coordinates. The color scale is logarithmic prior density $\ln p(\theta,\phi)$.](figures/galaxy_prior.pdf){width="50.00000%"}
For RJMCMC algorithms with scale parameters–in our case the amplitude–the choice of prior influences both the recovery of those parameters as well as on the model posterior. For example, a simple uniform prior between $U[0,{\mathcal{A}}_{\rm max}]$ will support including low-amplitude sources in the model. Adding a source to the model with ${\rm{SNR}}\sim0$ will not degrade the likelihood, and the remaining model parameters will sample their prior such that the so-called “Occam penalty” from including extra (constrained) parameters is small. The need to derive an amplitude prior that results in model posteriors as we intuitively expect–namely that templates are included in the model predominantly when there is a detectable source for them to fit–and does not bias the recovery of the amplitude parameter was addressed in the [`BayesWave`]{}algorithm [@Cornish:2014kda]. There the prior on the amplitudes had to be considered to suppress large numbers of low-amplitude wavelets saturating the model prior. The solution was to evaluate the prior not on the amplitudes themselves, but on the [[[SNR]{}]{}]{} of the wavelet. The prior was tuned to go to 0 at low [[SNR]{}]{}, peak in the regime where most wavelets were expected to appear in the model (near the “detection” threshold), and taper off at high [[SNR]{}]{}. We adopt that approach for the UCB model as follows.
Up to geometrical factors of order unity, the ${\rm{SNR}}$ of a galactic binary $\rho$ is related to the amplitude via the linear mapping $$\rho = \frac{{\mathcal{A}}}{2} \left( \frac{T_{\rm obs}\sin^2({f_0}/f_*)}{S_A({f_0})}\right)^{1/2} \, .$$ The prior on the amplitude is then mapped from a prior on $\rho$ of the form $$p(\rho) = \frac{3 \rho}{4 {\rho}_*^2 (1 + {\rho}/(4{\rho}_*))^5}$$ which peaks at $\rho=\rho_*$ and falls off as $\rho^{-4}$ for large $\rho$. Because most detections will be close to the detection threshold we set $\rho_* = 10$. For bright sources the likelihood, which scales as $e^{\rho^2}$, overwhelms the prior, and there is little influence in the the recovered amplitudes from our choice of prior.
For the sky location the pipeline has support for two options: Either the model can use a uniform priors on the sky or a prior weighted towards the sources being distributed in the galaxy according to an analytic model for its overall shape. As currently implemented we use a simple bulge-plus-disk model for the stellar distribution of the form $$\varrho = \varrho_0 \left[ \alpha e^{-r^2/R_b^2} + (1-\alpha) e^{-u/Rd} \rm{sech}^2\left(\frac{z}{Z_d}\right)\right].$$ Here $r^2 = x^2+y^2 +z^2$ and $u^2=x^2+y^2$, and $x,y,z$ are a set of Cartesian coordinates with origin at the center of the galaxy and the $z$ axis orthogonal to the galactic plane. The parameters are the overall density scaling $\varrho_0$, bulge fraction $\alpha$, bulge radius $R_b$, disk radius $R_d$ and disk scale height $Z_d$. Ideally we would make these quantities hyper-parameters in a hierarchical Bayesian scheme [@Adams:2012qw], but for now we have fixed them to the fiducial values $\alpha=0.25$, $R_b=0.8$ kpc, $R_d=2.5$ kpc, and $Z_b=0.4$ kpc and $\varrho_0$ determined by numerically normalizing the distribution, . LISA views the galaxy from a location that is offset from the galactic center by an amount $R_G$ in the $x$-direction, and use ecliptic coordinates to define the sky locations. This necessitates that we apply a translation and rotation to the original galactic coordinates. We then compute the density $\varrho(\theta,\phi)$ in the new coordinate system and normalize the density on the sky to unity for use as a prior. In order to ensure full sky coverage we rescale the normalized density by a factor of $(1-\beta)$ and add to it a uniform sky distribution that has total probability $\beta$. Figure \[fig:skyprior\] shows the sky prior for the choice $\beta = 0.1$.
Trans-dimensional MCMC
----------------------
Trans-dimension modeling is a powerful technique that simultaneously explores the range of plausible models for the data as well as the parameters of each candidate model. The trans-dimensional approach is particularly valuable in situations where it is unclear how many components should be included in the model and there is a danger of either over- or under-fitting the data. Trans-dimensional modeling allows us to explore a wide class of models in keeping with our motto “model everything and let the data sort it out” [@Cornish:2014kda]. While fixed dimension (signal model) sampling techniques have thus far proven sufficient for LIGO-Virgo analyses of isolated events, we see no alternative to using trans-dimensional algorithms for the multi-source fitting required for LISA data analysis.
Trans-dimensional MCMC algorithms are really no different from ordinary MCMC algorithms. They simply operate on an extended parameter space that is written in terms of a model indicator parameter $k$ and the associated parameter vector $\vec{\theta}_k$. It is worth noting that the number of models can be vast. For example, suppose we were addressing the full LISA data analysis problem using a model that included up to $N_{\rm UCB}\sim 10^5$ galactic binaries, $N_{\rm BH} \sim 10^3$ supermassive black holes, $N_{\rm EMRI}\sim 10^3$ extreme mass ratio inspirals and $N_{\rm n}\sim 10^3$ parameters in the noise model. Since the number of parameters for each model component are not fixed, the total number of possible models is the [*product*]{}, not the sum, of the number of possible sub-components, resulting in $\sim 10^{14}$ possible models in this instance. The advantage of the RJMCMC method is that it is not necessary to enumerate or sample from all possible models but, rather, to have the [*possibility*]{} of visiting the complete range of models. This is in contrast to the product space approach [@10.2307/2346151], which requires that all models be enumerated and explored while most of the computing effort is spent exploring models that have little or no support. Just as an ordinary MCMC spends the majority of its time exploring the regions of parameter space with high posterior density, the RJMCMC algorithm spends most of the time exploring the most favorable models.
Our goal is to compute the joint posterior of model $k$ and parameters ${\mathbf\theta}_k$ $$p(k, {\mathbf\theta}_k | {{\bf d}}) = \frac{ p({{\bf d}}| k, {\mathbf\theta}_k) p(k, {\mathbf\theta}_k)}{ p({{\bf d}})}$$ which is factored as $$p(k, {\mathbf\theta}_k | {{\bf d}}) = p(k | {{\bf d}}) p({\mathbf\theta}_k | k, {{\bf d}}) \, ,$$ where $p(k|{{\bf d}})$ is the posterior on the model probabilities and $p({\mathbf\theta}_k | k, {{\bf d}})$ is the usual parameter posterior distribution for model $k$. The quantity $O_{ij} = p(i | {{\bf d}})/p(j | {{\bf d}})$ is the odds ratio between models $i,j$. The RJMCMC algorithm generates samples from the joint posterior distribution $p(k, \vec{\theta}_k | {{\bf d}})$ by developing a Markov Chain via proposing transitions from state $\{k, {\mathbf\theta}_k\}$ to state $\{l,{\mathbf\theta}_l\}$ using a proposal distribution $q(\{k, {\mathbf\theta}_k\}, \{l,{\mathbf\theta}_l\})$. Transitions are accepted with probability $\alpha = \min \left\{ 1, H_{l\rightarrow k} \right\}$ with the Hastings Ratio $$\label{rjmcmc}
H_{l\rightarrow k} = \frac{p({{\bf d}}| k, {\mathbf\theta}_k)}{p({{\bf d}}| l, {\mathbf\theta}_l)} \, \frac{p(k, {\mathbf\theta}_k)}{p(l, {\mathbf\theta}_l)} \, \frac{q(\{k, {\mathbf\theta}_k\}, \{l,{\mathbf\theta}_l\})}{q(\{l,{\mathbf\theta}_l\}, \{k, {\mathbf\theta}_k\} )}.$$ Proposals are usually separated into within-model moves, where $k=l$ and only the model parameters ${\mathbf\theta}_k$ are updated, and between-model moves where both the model indicator $l$ and the model parameters ${\mathbf\theta}_l$ are updated. Written in the form of Eq. \[rjmcmc\] the RJMCMC algorithm is no different than the usual Metropolis-Hastings algorithm. In practice the implementation is complicated by the need to match dimensions between the model states, which introduces a Jacobian determinant of the mapping function [@doi:10.1093/biomet/82.4.711]. This can all become very confusing, and may explain the slow adoption of trans-dimensional modeling in the gravitational wave community. Thankfully the models we consider are [*nested*]{}, such that the transition from state $k$ to $l$ involves the addition or removal of a model component. In the case of nested models the mapping function is a linear addition or subtraction of parameters, and the Jacobian is simply the ratio of the prior volumes [@doi:10.1111/j.1365-246X.2006.03155.x]. For example, the Hasting ratio for adding a single UCB source with parameters ${\mathbf\lambda}_{k+1}$ to the current state of the model already using $k$ templates (with joint parameters ${\mathbf\Lambda}_k$) is $$H_{k\rightarrow k+1} = \frac{p({{\bf d}}|{\mathbf\Lambda}_k, {\mathbf\lambda}_{k+1}) p({\mathbf\lambda}_{k+1}) } {p({{\bf d}}|{\mathbf\Lambda}_k) q({\mathbf\lambda}_{k+1})}$$ where $q({\mathbf\lambda}_{k+1})$ is the proposal distribution that generated the new source parameters, and we assume for the reverse move ($k+1\rightarrow k$) that existing sources are selected for removal with uniform probability.
The efficiency of any MCMC algorithm depends critically on the choice of proposal distributions. The necessity for finding good proposal distributions is even more acute for the trans-dimensional moves of a RJMCMC algorithm. In the UCB pipeline, an increase in dimension comes about when a new waveform template is added to the solution. For such a move to be accepted the parameters for the new source must land sufficiently close to the true parameters of some signal for the transition to be accepted. Arbitrarily choosing the ${N_{\rm P}}$ parameters that define a signal has low probability of improving the likelihood enough for the transition to be accepted. The strategy we have adopted to improve the efficiency, which is explicitly detailed in the following section, is to identify promising regions of parameter space in pre-processing, in effect producing coarse global maps of the likelihood function, and using these maps as proposal distributions. The global proposals are also effective at promoting exploration of the multiple posterior modes that are a common feature of GW parameter spaces for single sources.
To further aid in mixing we use replica exchange (also know as parallel tempering). Parallel tempering uses a collection of chains to explore models with the modified likelihood $p({{\bf d}}| {\mathbf\Lambda}, \beta) = p({{\bf d}}| {\mathbf\Lambda})^{\beta}$, where $\beta\in[0,1]$ is an inverse “temperature”. Chains with high temperatures (low $\beta$) explore a flattened likelihood landscape and move more easily between posterior modes, while chains with lower temperature sample the likelihood around candidate sources and map out the peaks in more detail. Only those chains with $\beta=1$ provide samples from the target posterior. A collection of chains at different temperatures are run in parallel, and information is passed up and down the temperature ladder by proposing parameter swaps, which are accepted with probability $\alpha = \min\left\{ 1,H_{i\leftrightarrow j}\right\}$ and $$\label{ptmcmc}
H_{i\leftrightarrow j} = \frac{ p({{\bf d}}| i, {\mathbf\Lambda}_i,\beta_i) \, p({{\bf d}}| j, {\mathbf\Lambda}_j,\beta_j)}{ p({{\bf d}}| i, {\mathbf\Lambda}_i,\beta_j) \, p({{\bf d}}| j, {\mathbf\Lambda}_j,\beta_i) } \, .$$ Here we are proposing to swap the parameters of the model $\{i, {\mathbf\Lambda}_i\}$ at inverse temperature $\beta_i$ with the model $\{j, {\mathbf\Lambda}_j\}$ at inverse temperature $\beta_j$. Note that if $\beta_i=\beta_j$ the swap is always accepted. Models with higher temperatures typically have lower likelihoods. If the likelihoods of the two models are very different the Hastings Ratio $H_{i\leftrightarrow j}$ will be small. We only propose exchanges between chains that are near one another in temperature.
Choosing the temperature ladder so that chain swaps are readily accepted is a challenge. The situation we need to avoid is a break in the chain, where a collection of hotter chains decouples from the colder chains such that no transitions occur between the two groups. When that happens the effort spent evolving the hot chains is wasted as their findings are never communicated down the temperate ladder to the $\beta=1$ chain(s) that accumulate the posterior samples. It is generally more effective to run a large number of chains that are closely spaced in temperature for few iterations than it is to run with fewer chains for longer. We adopt the scheme described in Ref. [@Vousden_2015] where the temperature spacing between chains is adjusted based on acceptance rates of chain swaps, and the degree to which the temperatures adjust based on the acceptance rates, asymptotically approaches zero as the number of chain iterations increases. Thus the temperature spacing is dynamically adjusting to the rapidly changing model when the sampler is “burning in” but settles into a steady-state when the sampler is exploring the posterior.
Proposal Distributions
----------------------
As mentioned previously, the efficiency of a MCMC algorithm is heavily dependent on the design of the proposal distributions. This “tuning” requirement for an efficient MCMC has led to the development of samplers designed to be more agnostic to the parameter space such as ensemble samplers (e.g. [@Foreman_Mackey_2013]), Hamiltonian Monte Carlo [@betancourt2017conceptual], etc. However, there has been less development of alternatives to sampling transdimensional posteriors and the scale of the LISA UCB problem may be prohibitive to brute-force evaluation of many competing models. It is our view that continued innovation in development of custom proposal distributions that leverage the hard-earned domain knowledge is worth the investment. To that end, we observe that the posterior is the ideal proposal distribution–setting $q(\{i,{\mathbf\Lambda}_i\}, \{j, {\mathbf\Lambda}_j\} ) = p(i, {\mathbf\Lambda}_i | {{\bf d}})$ we have $H_{i\rightarrow j}=1$, so every proposed move is accepted and the correlation between successive samples can be made arbitrarily small. Of course, if we could produce independent samples from the posterior in advance there would be no need to perform the MCMC, but this observation provides guidance in the design of effective proposal distributions–we seek distributions that are computationally efficient approximations to the posterior distribution, which usually amounts to finding good approximations to the likelihood function. Consider the log likelihood for model $k$ describing $N_k$ galactic binaries, which is written as $$\begin{aligned}
\label{ll}
\ln p({{\bf d}}| k, {\mathbf\Lambda}_k ) &=& \sum_{i=1}^{N_k} \ln p({{\bf d}}| {\mathbf\lambda}_i) + \frac{1}{2}(N_k-1){\langle d | d \rangle} \nonumber \\
&+&\sum_{i>j} {\langle {\mathbf\lambda}_i | {\mathbf\lambda}_j \rangle} \, ,\end{aligned}$$ where $${\langle a | b \rangle} \equiv a_{Im} C^{-1}_{(I m)(J n)}(\vec{\kappa}) b_{Jn}$$ and we are neglecting terms from the noise parameters. The first term in the expression for the log likelihood in Eq.\[ll\] is the sum of the individual likelihoods for each source, while the final term describes the correlations between the sources. While accounting for these correlations is crucial to the global analysis, the correlation between any pair of sources is typically quite small, and we ignore them in the interest of finding a computationally efficient approximation to the likelihood to use as a proposal. Figure \[fig:overlap\] shows the maximum match between pairs of sources with ${\rm{SNR}}> 7$, using a simulated galactic population and assuming 1, 2, and 4 year observation periods. Here the match, or overlap, is defined as: $$\label{eq:match}
M_{ij} \equiv \frac{{\langle {{\boldsymbol h}}({\mathbf\lambda}_i) | {{\boldsymbol h}}({\mathbf\lambda}_j) \rangle}}{\sqrt{ {\langle {{\boldsymbol h}}({\mathbf\lambda}_i) | {{\boldsymbol h}}({\mathbf\lambda}_i) \rangle} {\langle {{\boldsymbol h}}({\mathbf\lambda}_j) | {{\boldsymbol h}}({\mathbf\lambda}_j) \rangle} }}\, ,$$ and we are using the $A,E$ TDI data channels. Less than 1% of sources have overlaps greater than 50%, and the fraction diminishes with increased observing time. Thus we will develop proposals for individual sources and propose updates to their parameters independently of other sources in the model. The MCMC still marginalizes over the broader parameter space, including the rare but non-zero case of non-negligible covariances between sources, in effect executing a blocked Gibbs sampler where the blocks are individual source’s parameters.
![\[fig:overlap\] Survival function of the maximum match between any pair of detectable sources computed using a simulated galactic population of UCBs. For 1 year of observing (green) $\lesssim1\%$ of sources have overlaps greater than 50%. That fraction is reduced to $0.1\%$ after 2 years (orange), and $0$ after 4 years (purple) as the resolving power of LISA increases.](figures/match.pdf){width="50.00000%"}
### $\mathcal{F}$ statistic Proposal
We construct a global proposal density using the single source $\mathcal{F}$ statistic to compute the individual likelihoods $\ln p({{\bf d}}| {\mathbf\lambda}_i)$ maximized over the extrinsic parameters ${\mathcal{A}}, {\varphi_0}, \iota, \psi$. Up to constants that depend on the noise parameters, the maximized log likelihood is equal to $${\cal F}({f_0},\theta, \phi) = \frac{1}{2} {\langle {\bf g}_i | {\bf g}_j \rangle}^{-1} {\langle {{\bf d}}| {\bf g}_i \rangle} {\langle {{\bf d}}| {\bf g}_j \rangle}$$ where the four filters ${\bf g}_i$ are found by computing waveforms with parameters ${f_0}, {\dot{f}}=0, \theta, \phi$, ${\mathcal{A}}=2$ and $$\begin{aligned}
\label{filters}
&& {\bf g}_1 = {{\boldsymbol h}}\left({\varphi_0}=0, \iota=\frac{\pi}{2},\psi=0\right) \\
&& {\bf g}_2 = {{\boldsymbol h}}\left({\varphi_0}=\pi, \iota=\frac{\pi}{2},\psi=\frac{\pi}{4}\right) \\
&& {\bf g}_3 = {{\boldsymbol h}}\left({\varphi_0}=\frac{3\pi}{2}, \iota=\frac{\pi}{2},\psi=0\right) \\
&& {\bf g}_4 = {{\boldsymbol h}}\left({\varphi_0}=\frac{\pi}{2}, \iota=\frac{\pi}{2},\psi=\frac{\pi}{4}\right) \, .\end{aligned}$$ The $\mathcal{F}$ statistic proposal is the three dimensional histogram precomputed from the data using a grid in ${f_0}, \theta, \phi$. We use a fixed grid spacing governed by what is needed for the best resolved sources which are found in the ecliptic plane (which maximises the doppler modulations imparted by LISAs orbital motion) and at the highest frequencies covered by the analysis. The probability density of a cell $(a,b,c)$ of the three-dimensional histogram is ${\cal F}_{a,b,c}$ normalized by the sum of ${\cal F}$ over all cells, and the parameter volume of the cell.
The optimal spacing of the grid can be estimated from the reduced Fisher information matrix $\gamma_{ij}$, which is found by projecting out the parameters ${\mathcal{A}}, {\varphi_0}, \iota, \psi$ from the full Fisher information matrix $\Gamma_{ij} = {\langle \partial{{\boldsymbol h}}/\partial{\mathbf\lambda}_i | \partial{{\boldsymbol h}}/\partial{\mathbf\lambda}_j \rangle}$ [@Cornish:2005qw]. The reduced Fisher matrix is not constant across the parameter space and will naturally reduce the grid size as ${f_0}$ gets larger, and for sky locations near the ecliptic equator compared to those near the poles. The grid spacing will also become finer as the observation time grows. These modifications, as well as extending to a 4D grid including ${\dot{f}}$, will further improve the efficiency of the proposal and are left for future development.
![\[fig:fstat-proposal\] Frequency slices of the multidimensional $\mathcal{F}$ statistic proposal for the same segment of data shown in Fig. \[fig:money\_plot\]. The color scale is linear in the proposal density, and each panel is on the same scale. The proposal promotes frequencies and sky locations consistent with the signal in the data (top right and bottom left panels) and returns a low-density and diffuse distribution at frequencies consistent with random noise (top left and bottom right panels).](figures/fstat_sky.pdf){width="50.00000%"}
### Multi-modal Proposals
Due to the parameterization of the gravitational wave signals, and the instrument response to those signals, there are known exact or near degeneracies which appear as distinct modes in the likelihood/posterior distribution. While MCMC algorithms are not generically efficient at sampling from multimodal distributions, we have developed dedicated proposal distributions to exploit the predictable multi-modality and improve the chain convergence time.
Due to the annual orbital motion of the LISA constellation, continuous monochromatic sources will have non-zero sidebands at the modulation frequency $f_m = 1/{\rm year}$. Sources that are detectable at low ${\rm{SNR}}$ after several years of observation can have likelihood support at multiple modes separated by $f_m$, while for high ${\rm{SNR}}$ sources the secondary modes are subdominant local maxima, challenging to generic MCMC sampling algorithms. We have adopted a dedicated proposal that updates the UCB initial frequency by ${f_0}\rightarrow {f_0}+ n f_m$ where $n$ is drawn from $N[0,1]$ and mapped to the nearest integer. The sky location of the source correlates with the frequency through the doppler modulations imparted by the detector’s orbital motion, so the proposal alternates between updates to the extrinsic parameters using the Fisher matrix proposal, F-statistic proposal, and draws from the prior. A similar proposal was deployed and demonstrated in Refs. [@Crowder:2006eu; @Littenberg:2011zg].
We also take advantage of a linear correlation between the gravitational wave phase ${\varphi_0}$ and polarization angle $\psi$, and a perfectly degenerate pair of modes over the prior $\psi \in [0,\pi]$ and ${\varphi_0}\in [0,2\pi]$ by proposing $\{\psi,{\varphi_0}\}\rightarrow \{\psi \pm \delta/2,{\varphi_0}\pm \delta\}$ where $\delta \in U[0,2\pi]$ and the sign of the shift in the parameters is random, as the sign of the $\psi/{\varphi_0}$ correlation depends on the sign of $\cos{\iota}$, i.e. if the stars are orbiting clockwise or counterclockwise as viewed by the observer.
### Posterior-Based Proposals
The UCBs are continuous sources for LISA and will be detectable from the beginning of operations throughout the lifetime of the mission. Our knowledge of the gravitational wave signal from the galaxy will therefore build gradually over time. We have designed a proposal distribution to leverage this steady accumulation of information about the galaxy by analyzing the data as it is acquired, and building proposal functions for the MCMC from the archived posterior distributions inferred at each epoch of the analysis.
For a particular narrow-band segment of data, the full posterior is a complicated distribution due to the probabilistically determined number of sources in the data, and their potentially complicated, multimodal structure. The posterior is known to us only through the discrete set of samples returned by the MCMC but for use as a proposal must be a continuous function over all of parameter space (as we must be able to evaluate the proposal anywhere in order to maintain detailed balance in the Markov chain). Therefore some simplifications must be made to convert the discrete samples of the chain into a continuous function.
In the release of the pipeline accompanying this paper, we select chain samples from the maximum marginalized likelihood (i.e. highest evidence) model at the current epoch to build the proposals used in the subsequent analysis when more data are available.. We post-process the chain samples to cluster those that are fitting discretely identified sources, and to filter out samples from the prior or from weaker candidate sources that don’t meet our threshold for inclusion in the source catalog. The post-production analysis is described in Sec. \[sec:catalog\].
Each source $i$ identified in the post-production step will have at least two modes, because of the degeneracy in the $\psi-{\varphi_0}$ plane. For each mode $n$, we compute the vector of parameter means $\bar{\mathbf\lambda}_{i,n}$ from the one-dimensional marginalized posteriors, the full ${N_{\rm P}}\times{N_{\rm P}}$ covariance matrix $\mathbf{C}_{i,n}$ from the chain samples, and the relative weighting $\alpha_{i,n}$ which is the number of samples in the mode normalized by the total number of samples used to build the proposal.
The proposal is evaluated for arbitrary parameters ${\mathbf\lambda}$ as $$p({\mathbf\lambda}) = \sum_{i=0}^{i<I} \sum_{n=0}^{i<2} \alpha_{i,n} \frac{ e^{-\frac{1}{2} ({\mathbf\lambda}-\bar{\mathbf\lambda}_{i,n}) \mathbf{C}_{i,n}^{-1} ({\mathbf\lambda}-\bar{\mathbf\lambda}_{i,n})} }{\left((2\pi)^{{N_{\rm P}}} \det\mathbf{C}_{i,n}\right)^{1/2}} .$$
To draw new samples from this distribution, we first select which mode by rejection sampling on $\alpha_{i,n}$, and then draw new parameters ${\mathbf\lambda}$ via: $${\mathbf\lambda}= \bar{\mathbf\lambda}_{i,n} + \mathbf{L}_{i,n}\mathbf{n}$$ where $\mathbf{n}$ is an ${N_{\rm P}}$-dimension vector of draws from a zero-mean unit-variance Gaussian, and $\mathbf{L}_{i,n}$ is the LU decomposition of $\mathbf{C}_{i,n}$.
Fig. \[fig:cov-proposal\] shows the 1 and 2$\sigma$ contours of the set of covariance matrices computed from a 6 month observation of simulated LISA data around 4 mHz in two projections of the full posterior: the ${f_0}-{\mathcal{A}}$ plane (top) and sky location (bottom). Shown in gray is the scatter plot of all chain samples before being filtered by the catalog production step described in the next section. The color scheme is consistent between the two panels. Note that for well localized (e.g. high amplitude) sources the covariance matrix is a good representation of the posterior, as should be the case since the posterior should trend towards a Gaussian distribution with increased ${\rm{SNR}}$, and will therefore serve as an efficient proposal when new data are acquired.
![\[fig:cov-proposal\] Two-dimensional projections of the multi-source covariance matrix proposal produced after analyzing 6 months of simulated data round 4 mHz. The gray scatter plots show all of the chain samples from the analysis which are then filtered and clustered into discrete sources by the catalog production step. The mean parameter values and covariance matrix for each discrete source are computed from the chain samples and used a proposal for the next step of the analysis after more data are acquired. Parameter combinations shown are the frequency-amplitude plane (top panel) and sky location (bottom panel). Ellipses enclose the 1 and $2\sigma$ contours of the covariance matrices, and sources are colored consistently in the top and bottom panels.](figures/covariance_f-A.pdf "fig:"){width="50.00000%"} ![\[fig:cov-proposal\] Two-dimensional projections of the multi-source covariance matrix proposal produced after analyzing 6 months of simulated data round 4 mHz. The gray scatter plots show all of the chain samples from the analysis which are then filtered and clustered into discrete sources by the catalog production step. The mean parameter values and covariance matrix for each discrete source are computed from the chain samples and used a proposal for the next step of the analysis after more data are acquired. Parameter combinations shown are the frequency-amplitude plane (top panel) and sky location (bottom panel). Ellipses enclose the 1 and $2\sigma$ contours of the covariance matrices, and sources are colored consistently in the top and bottom panels.](figures/covariance_sky.pdf "fig:"){width="50.00000%"}
Fig. \[fig:logL\] shows the log-likelihood of the model as a function of chain step for observations of increasing duration $T$ with (teal) and without (orange) using the covariance matrix proposal built from each intermediate analysis. This demonstration was on the same data from Fig. \[fig:money\_plot\] containing the type of high-${f_0}$ and high-${\rm{SNR}}$ source that proved challenging for the previous RJMCMC algorithm [@Littenberg:2011zg]. With the covariance matrix proposal the chain convergence time is orders of magnitude shorter than using the naive sampler, to the point where the $T=24$ month run failed to converge in the number of samples it took the analysis with the covariance matrix proposal to finish.
![\[fig:logL\] Log-likelihood chains from analyses of the same data as shown in Fig. \[fig:money\_plot\] run with (teal) and without (orange) the covariance matrix proposal. As the observing time increases, the chain sampling efficiency gained by including the proposal built from previous analyses becomes more significant.](figures/highf_logL.png){width="50.00000%"}
Using the customized proposals described in this section allows the sampler to robustly mix over model space and explore the parameters of each model supported by the data. The pipeline dependably converges without the need for non-Markovian maximization steps as were used in the “burn in” phase of our previously published UCB pipelines, and reliably produces results for model selection and parameter estimation analyses simultaneously.
Data selection
--------------
While the UCB pipeline is pursuing a global analysis of the data, we leverage the narrow-band nature of the sources to parallelize the processing. Sources separated by more than their bandwidth–typically less than a few hundred frequency bins–are uncorrelated and can therefore be analyzed independently of one another.
As was done in previous UCB algorithms [@Crowder:2006eu; @Littenberg:2011zg], we divide the full Fourier domain data stream into adjacent segments and process each in parallel, without any exchange of information during the analysis between segments. To prevent edge effects from templates trying to fit sources outside of the analysis window, each segment is padded on either side in frequency with data amounting to the typical bandwidth of a source, thus overlapping the neighboring segments. The MCMC is free to explore the data in the padded region, but during post-production only samples fitting sources in the original analysis window are kept, preventing the same source from being included in the catalog twice. Meanwhile, sources within the target analysis region but close to the boundary will not have part of their signal cut off in the likelihood integral.
Unlike in Refs. [@Crowder:2006eu; @Littenberg:2011zg], there is no manipulation of the likelihood or noise model to prevent loud sources outside of analysis region from corrupting the fit. Instead, we leverage the time-evolving analysis by ingesting the list of detections from previous epochs of the catalog, forward modeling the sources as they would appear in the current data set and subtracting them from the data. This will be an imperfect subtraction but is adequate to suppress the signal power in the tails of the source which extend into the adjacent segments and, due to the padding, does not alter the data in the target analysis region. In the event that an imperfect subtraction leaves a detectable residual, it will not corrupt the final catalog of detected sources because templates fitting that residual will be in the padded region of the segment and removed in post-processing. The downside is merely in the computational efficiency, as poorly-subtracted loud signal with central frequency out of band for the analysis will require several templates co-adding to mitigate the excess power, wasting computing cycles and increasing the burden on the MCMC to produce converged samples. The effectiveness of the subtraction will improve as the duration of observing time between analyses decreases, and is an area to explore when optimizing the overall cost of the multi-year analysis.
The strategy for mitigating edge effects is prone to failure if the posterior distribution of a source straddles the boundary. The frequency is precisely constrained for any UCB detection so having a source so precariously located is unlikely but nonetheless needs to be guarded against. While not yet implemented, we envision checks for sources near the boundaries in post-production to see if posterior samples from different windows should be combined, and/or adaptively choosing where to place the segment boundaries based on the current understanding of source locations from previous epochs of the analysis. There is no requirement on the size or number of analysis windows except that they are much larger than the typical source bandwidth, and the segment boundaries do not need to remain consistent between iterations of the analysis as more data are added.
![\[fig:padding\] Demonstration of data selection and padding procedure. The top panel shows the power spectrum of an example analysis segment in black and the reconstructed waveforms from the analysis in various colors. The vertical dashed lines mark the region of the analysis region where sources will be selected for the catalog. Gray reconstructions are from the analyses of the adjacent segments. The bottom panel shows the same frequency interval in the $\{f_0,{\mathcal{A}}\}$ plane with injected signals marked as gray circles and a scatter plot of the MCMC samples in green. Note that the chain samples extend into the padded region and fit sources there, but those waveforms are not included in the top-panel’s reconstructions](figures/padding.pdf){width="50.00000%"}
Fig. \[fig:padding\] demonstrates the data selection and padding procedure by displaying results from the center analysis region of three adjacent windows processed with the time-evolving RJMCMC algorithm. The top and bottom panels show the reconstructed waveforms and posterior samples, respectively. The posterior samples extend outside of the analysis region (marked by vertical dashed lines) to fit loud signals in neighboring frequency bins, but are rejected during the catalog production step. The frequency padding ensures that the waveform templates of sources inside of the analysis region are not truncated at the boundary. Sources recovered from the neighboring analyses are marked in gray. Note that there is no conflict between the fit near the boundaries despite their being overlapping sources in this example at the upper frequency boundary.
Catalog Production {#sec:catalog}
==================
The output of the RJMCMC algorithm is thousands of random draws from the variable dimension posterior, with each sample containing an equally likely set of parameters *and* number of sources in the model. Going from the raw chain samples to inferences about individual detected sources is subtle, as a model using ${N_{\rm GW}}$ templates does not necessarily contain $N$ discrete sources. For example, the model may be mixing between states where the ${N_{\rm GW}}^{\rm th}$ template is fitting one (or several) weak sources, or sampling from the prior, and such a model could be on similar footing with the ${N_{\rm GW}}-1$ or ${N_{\rm GW}}+1$ template models purely on the grounds of the evidence calculation. How then to answer the questions “How many sources were detected?” or “What are the parameters of the detected sources?” in a way that is robust to the more nuanced cases where the data supports a broad set of models containing several ambiguous candidates?
Filtering and Clustering Posterior Samples
------------------------------------------
In Ref. [@Littenberg:2011zg], for the sake of responding to the Mock LISA Data Challenge, post-processing the chains went only as far as selecting the maximum likelihood chain sample from the maximum likelihood model. Condensing the rich information in the posterior samples down to single point estimate defeats the purpose of all the MCMC machinery in the first place. Furthermore, due to the large number of sources being fit simultaneously and the finite number of samples, the maximum likelihood sample within a particular dimension model does not necessarily correspond to the maximum likelihood parameters for each of the many sources in the analysis should they have been fit by the model in isolation.
It was therefore necessary that we begin to seriously consider how to post-process the raw chain samples into a more manageable data product for the sake of producing source catalogs that are easily ingested by end users of the LISA observations, but are not overly reduced to the point of being prohibitively incomplete or misleading. We originally explored using standard “off the shelf” clustering algorithms to take the ${N_{\rm GW}}\times{N_{\rm P}}$ samples from the chain and group them into the discrete sources being fit by the model. Although not an exhaustive effort, this proved challenging due to the large dimension of parameter space, different sources located close to one another in parameter space, and the multi-modal posteriors.
A more robust approach was to group the parameters of the model by using the match between the waveforms as defined in Eq. \[eq:match\] and applying a match threshold $M^*$ that must be exceeded for the parameter sets to be interpreted as fitting the same source. Seeing as it is the waveforms that are fundamentally what is being fit to the data, whereas the model parameters are just how we map from the template space to the data, clustering chain samples based on the waveform match, rather than the parameters, is naturally more effective.
The catalog production algorithm goes as follows: Beginning with the first sample of the chain, we compute the waveform from the parameters, produce a new *Entry* to the catalog (i.e., a new discrete detection candidate), and store the chain sample in that Entry. The parameters and corresponding waveform become the *Reference Sample* for the Entry. For each subsequent chain sample we again compute the waveform and check it against each catalog Entry. If the GW frequency of the chain sample is within 10 frequency bins of the Reference Sample we compute the match $M_{ij}$ and, if $M_{ij} > M^*$ the sample is appended to the Entry, effectively filtering all chain samples but those associated with the discrete feature in the data corresponding to the Entry. The check on how close the two samples are in frequency is to avoid wasteful inner-product calculations that will obviously result in $M_{ij}\sim0$. If a chain sample has been checked against all current Entries without exceeding the match threshold $M^*$ it becomes the reference sample for a new Entry in the catalog. Once the entire chain has been processed, the Catalog will contain many more candidate Entries than actual sources in the data (imagine a chain that has templates in the model occasionally sampling from the prior). However, the total number of chain samples in an Entry is proportional to the evidence $p({{\bf d}}) = \int p({\mathbf\lambda}|{{\bf d}})\, d{\mathbf\lambda}$ for that candidate source. Thus each Entry has an associated evidence that is used to further filter insignificant features. The default match threshold is $M^*=0.5$ but is easily adjustable by the user.
For each Entry, additional post-processing is then done to produce data products of varying degrees of detail depending on the needs of the end user. We select a point-estimate as the sample containing the median of the marginalized posterior on ${f_0}$, and store the ${\rm{SNR}}$, based on the reasoning that ${f_0}$ is by far the best constrained parameter and likely the most robust way of labeling/tracking the sources. We also compute the full multimodal ${N_{\rm P}}\times{N_{\rm P}}$ covariance matrix $C_{ij}$ as a condensed representation of the measurement uncertainty, and for use as a proposal when more data are acquired. From the ensemble of waveforms for each Sample in the Entry, we also compute the posterior on the reconstructed waveform. Finally, metadata about the Catalog is stored including the total number of above-threshold Entries, the full set of posterior samples, and the model evidence. A block diagram for the data products and how they are organized is shown in Fig. \[fig:catalog\].
![\[fig:catalog\] Proposed scheme for packaging chain output into higher level data products for publication in source catalogs. Raw chain output and evidences are available, as well as the posterior samples after having been filtered and clustered into discrete detected sources. Each discrete source candidate will have its own detection confidence (evidence), chain samples, point estimate, and covariance matrix error estimates so that the user can choose the most appropriate level of detail for their application of the catalog, along with metadata including the source name and history (for continuity over catalog releases), etc.](figures/CatalogBlockDiagram.pdf){width="50.00000%"}
Catalog Continuity
------------------
As the observing time grows the UCB catalog will evolve. New sources will become resolvable, marginal candidates may fade into the instrument noise, and overlapping binaries which may have been previously fit with a single template will be resolved as separate sources with similar orbital periods. Our scheme of identifying the binaries by their median value of $f_0$ will also evolve between releases of the catalog. While the association for a particular source from one catalog to the next is obvious upon inspection, the sheer number of sources requires an automated way of generating and storing the ancestry of a catalog entry in meta data.
To ensure continuity of the catalog between releases, we construct the “family tree” of sources in the catalog after each incremental analysis is performed. A source’s “parent” is determined by again using the waveform match criteria, now comparing the new entry to sources in the previous catalog computed using the previous run’s observing time. In other words, we are taking Entries found in the current step of the analysis and “backward modeling” the waveforms as they would have appeared during the production of the previous catalog. The waveforms are compared to the recovered waveforms from the previous epoch to identify which sources are associated across catalogs, tracing a source’s identification over the entire mission lifetime, and making it easy to quickly identify new sources at each release of the catalog.
Demonstration
=============
To demonstrate the algorithm performance we have selected two stress-tests using data simulated for the LISA Data Challenge *Radler* dataset [^1]. The first is a high-frequency, high-${\rm{SNR}}$ isolated source that challenges the convergence of the pipeline due to the many sub-dominant local maxima in the likelihood function. As shown in Figs. \[fig:money\_plot\] and \[fig:logL\], new features in the algorithm have the desired affect of improving the convergence time.
We have also tested the pipeline on data at lower frequencies where the number of detectable sources is high, focusing on a ${\sim}140\ \mu$Hz wide segment starting at 3.98 mHz. The segment is subdivided into three regions to test the performance at analysis boundaries, and processed after 1.5, 3, 6, 12, and 24 months of observing. For the 24 month analysis, the full bandwidth was further divided into six regions to complete the analysis more quickly. Fig. \[fig:4mHz-model\] shows a heat map of the posterior distribution function on the model dimension for the six adjacent frequency segments analyzed to cover the ${\sim}140\ \mu$Hz test segment. The maximum likelihood model is selected for post-processing to generate a resolved source catalog. In the event that multiple dimension models have equal likelihood the lower dimension model is selected.
Fig. \[fig:4mHz-waveforms\] shows the data, residual, and noise model (top panel) and the posterior distributions on the reconstructed waveforms which met the criteria for inclusion into the detected source catalog after 24 months of observing (bottom panel). The waveforms, residuals, and noise reconstructions are plotted with 50% and 90% credible intervals, though the constraints are sufficiently tight that the widths of the intervals are small on this scale. The reconstructed waveforms are shown over a narrower-band region than the full analysis segment, containing the middle two of the six adjacent analysis windows.
![\[fig:4mHz-model\] Heat map of posterior distribution function as a function of frequency segment and number of signals in the model.](figures/model_posterior.pdf){width="50.00000%"}
{width="100.00000%"}
The recovered source parameters are tested against the true values used in the data simulation and we find that our inferences about the data correspond to the simulated signals that we would expect to be detected. Fig. \[fig:4mHz-posteriors\] shows the 1- and 2-sigma contours of the marginalized 2D posteriors for the frequency-amplitude plane (top) and sky location (bottom) with gray circles marking the true parameter values. These results come from a single analysis window because the results from the full test region are overwhelming when all plotted together.
![\[fig:4mHz-posteriors\] Two-dimensional marginalized posteriors for a single analysis window of the full test segment of simulated data around 4 mHz after 12 months of observing time by LISA. The analysis was built up from 1.5, 3, and 6 month observations. Gray circles mark the parameter values of the injected sources. The top panel shows the frequency-amplitude plane, and the bottom panel shows the sky location in ecliptic coordinates. Contours enclose the 1 and $2\sigma$ posterior probability regions for each discrete source found in the catalog production, and the color scheme is consistent with Fig. \[fig:4mHz-waveforms\].](figures/4mHz_f-A.pdf "fig:"){width="50.00000%"}\
![\[fig:4mHz-posteriors\] Two-dimensional marginalized posteriors for a single analysis window of the full test segment of simulated data around 4 mHz after 12 months of observing time by LISA. The analysis was built up from 1.5, 3, and 6 month observations. Gray circles mark the parameter values of the injected sources. The top panel shows the frequency-amplitude plane, and the bottom panel shows the sky location in ecliptic coordinates. Contours enclose the 1 and $2\sigma$ posterior probability regions for each discrete source found in the catalog production, and the color scheme is consistent with Fig. \[fig:4mHz-waveforms\].](figures/4mHz_sky.pdf "fig:"){width="50.00000%"}\
{width="100.00000%"}
Fig. \[fig:continuity\] is a graphical representation of the family tree concept for tracking how the source catalog evolves over time. From this diagram one can trace the genealogy of a source in the current catalog through the previous releases. The diagram is color-coded such that new sources are displayed in green, sources unambiguously associated with an entry from the previous catalog in white, and sources that share a “parent” with another source are in blue. Based on the encouraging results of the narrow-band analysis shown here we will begin the analysis of the full data set. A thorough study of the pipeline’s detection efficiency, the robustness of the parameter estimation, and optimization of MCMC and post-production settings will be presented with the culmination of the full analysis.
Future Directions
=================
The algorithm presented here is a first step towards a fully functional prototype pipeline for LISA analysis. We envision continuous development as the LISA mission design becomes more detailed, and as our understanding of the source population, both within and beyond the galaxy, matures.
The main areas in need of further work are: (1) Combining the galactic binary analysis with analyses for other types of sources; (2) Better noise modeling, including non-stationarity on long and short timescales; (3) Handling gaps in the data; (4) More realistic instrument response modeling and TDI generation; (5) Further improvements to the convergence time of the pipeline.
![\[fig:global\] The UCB search as one component of a global fit. The residuals from each source analysis block are passed along to the next analysis in a sequence of Gibbs updates. New data is incorporated into the fit during the mission. The noise model and instrument models are updated on a regular basis.](figures/cycle.png){width="50.00000%"}
Figure \[fig:global\] shows one possible approach for incorporating the galactic analysis as part of a global fit. In this scheme, the analyses for each source type, such as super massive black hole binaries (SMBH), stellar origin (LIGO-Virgo) binaries (SOBH), un-modeled gravitational waves (UGW), extreme mass ratio inspirals (EMRI), and stochastic signals (Stochastic) are cycled through, which each analysis block passing updated residuals (i.e., the data minus the current global fit) along to the next analysis block. New data is added to the analysis as it arrives. The noise model and the instrument model (e.g., spacecraft orbital parameters, calibration parameters, etc.) are regularly updated. This blocked Gibbs scheme has the advantage of allowing compartmentalized development, and should be fairly efficient given that the overlap between different signal types is small.
A more revolutionary change to the algorithm is on the near horizon, where we will switch to computing the waveforms and the likelihood using a discrete time-frequency wavelet representation. A fast wavelet domain waveform and likelihood have already been developed [@Cornish:2020]. This change of basis allows us to properly model the non-stationary noise from the unresolved signals which are modulated by the LISA orbital motion, as well as any long-term non-stationarity in the instrument noise. Rectangular grids in the time-frequency plane are possible using wavelet wave packets [@Klimenko_2008] which make it easy to add new data as observations continue, instead of needing the new data samples to fit into a particular choice for the wavelet time-frequency scaling, e.g. being $2^n$ or a product of primes. Wavelets are also ideal for handling gaps in the data as they have built-in windowing that suppresses spectral leakage with minimal loss of information. The time-frequency likelihood [@Cornish:2020] also enable smooth modeling of the dynamic noise power spectrum $S(f , t)$ using [`BayesLine`]{}type methods extended to two dimensions.
Convergence of the sampler will be improved by including directed jumps in the extrinsic parameters when using the $\mathcal{F}$ statistic proposal (as opposed to the uniform draws that are currently used). The effectiveness of the posterior-based proposals can be improved by including inter-source correlations in the proposal distributions. This would be prohibitively expensive if applied to all parameters as the full correlation matrix is $D \times D$, where $D={N_{\rm GW}}\times{N_{\rm P}}\sim 10^4$. However, if the sources are ordered by frequency, the $D \times D$ correlation matrix of source parameters will be band diagonal. We can therefore focus only on parameters that are significantly correlated, and only between sources that are close together in parameter space, while explicitly setting to zero most of the off-diagonal elements of the full correlation matrix. There may also be some correlations with the noise model parameters, but we do not expect these to be significant.
Along a similar vein, we will include correlations between sources in the Fisher matrix proposals. This will only be necessary for sources with high overlaps [@Crowder:2004ca] which will be identified adaptively within the sampler. Then the Fisher matrix is computed using the parameters set ${\mathbf\lambda}= \{{\mathbf\lambda}_1, {\mathbf\lambda}_2\}$ and waveform model ${\bf h}({\mathbf\lambda}) = {\bf h}_1({\mathbf\lambda}_1) + {\bf h}_2({\mathbf\lambda}_2)$.
There is a large parameter space of analysis settings to explore when optimizing the computational cost of the full analysis, as well as the “wall” time for processing new data. The first round of tuning the deployment strategy for the pipeline will come from studying the optimal segmenting of the full measurement band, and the cadence for reprocessing the data as the observing time increases.
We will extend the waveform model to allow for more complicated signals including eccentric white dwarf binaries, hierarchical systems and stellar mass binary black holes which are the progenitors of the merging systems observed by ground-based interferometers [@Sesana_2016], and develop infrastructure to jointly analyze multimessenger sources simultaneously observable by both LISA and EM observatories [@Korol:2017qcx; @Kupfer_2018; @Burdge_2019; @Littenberg_2019b].
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Q. Baghi, J. Baker, C. Cutler, J. Slutsky, and J. I. Thorpe for insightful discussions during the development of this pipeline, particularly related to the catalog development. We also thank the LISA Consortium’s LDC working group for curating and supporting the simulated data used in this study. TL acknowledges the support of NASA grant NNH15ZDA001N-APRA and the NASA LISA Study Office. TR and NJC appreciate the support of the NASA LISA Preparatory Science grant 80NSSC19K0320. KL’s research was supported by an appointment to the NASA Postdoctoral Program at the NASA Marshall Space Flight Center, administered by Universities Space Research Association under contract with NASA. NJC expresses his appreciation to Nelson Christensen, Direct of Artemis at the Observatoire de la Côte d’Azur, for being a wonderful sabattical host.
[^1]: <https://lisa-ldc.lal.in2p3.fr/ldc>
|
---
abstract: 'We investigate two-photon entangled states using two important degrees of freedom of the electromagnetic field, namely orbital angular momentum (OAM) and spin angular momentum. For photons propagating in the same direction we apply the idea of *entanglement duality* and develop schemes to do *entanglement sorting* based either on OAM or polarization. In each case the entanglement is tested using appropriate witnesses. We finally present generalizations of these ideas to three- and four-photon entangled states.'
author:
- 'D. Bhatti'
- 'J. von Zanthier'
- 'G. S. Agarwal'
bibliography:
- 'literatur2014oct.bib'
title: Entanglement of Polarization and Orbital Angular Momentum
---
Introduction
============
It is known that for identical particles one can construct entangled states by using different degrees of freedom, e.g., linear momentum, polarization or even orbital angular momentum (OAM). For instance, for two photons traveling in different directions ${\mathbf{k}}_{1}$ and ${\mathbf{k}}_{2}$ one can consider an entangled state involving linear momentum and polarization degrees of freedom $$\ket{\Psi_{Pol}}=\frac{1}{\sqrt{2}}\left(\ket{H,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{2}}+\ket{V,{\mathbf{k}}_{1}}\ket{H,{\mathbf{k}}_{2}}\right) \ .
\label{eq:dualHVpath1}$$ Such states are now routinely produced by type II parametric down conversion [@Shih(1995)] and have been extensively studied in the literature [@Zeilinger(2012)]. More recently it was realized that orbital angular momentum is another degree of freedom of the radiation field which can be fruitfully employed for entanglement generation. This led among others to the production of entangled states of the form $$\ket{\Psi_{OAM}}=\frac{1}{\sqrt{2}}\left(\ket{l,{\mathbf{k}}_{1}}\ket{-l,{\mathbf{k}}_{2}}+\ket{-l,{\mathbf{k}}_{1}}\ket{l,{\mathbf{k}}_{2}}\right) \ ,
\label{eq:dualOAMpath1}$$ which have been used in tests of nonlocality as well as for applications in quantum communication and cryptography [@Mair(2001); @Franke-Arnold(2008); @Okamoto(2008); @Leach(2009); @Karimi(2010); @Leach(2010)].\
A more interesting possibility would be to consider two photons with the same linear momentum but with different polarizations and OAM degrees of freedom $$\ket{\Psi}=\frac{1}{\sqrt{2}}\left(\ket{H,l}\ket{V,-l}+\ket{H,-l}\ket{V,l}\right) \ .
\label{eq:dualPolOAM1}$$ In this case we have entanglement between two types of angular momenta, namely spin angular momentum and orbital angular momentum. Entanglement of these two degrees of freedom has not been studied extensively in the literature so far.\
In this paper we focus our attention precisely on entangled states of the form of Eq. (\[eq:dualPolOAM1\]). We show how to produce these entangled states and use the recently formulated *duality* of identical particle entanglement [@Bose(2013)] to perform *entanglement sorting*. Note that the state given in Eq. (\[eq:dualPolOAM1\]) has the interesting property that one can detect the entangled character by studying either polarization variables or OAM variables. These studies would yield identical information if the particles are indistinguishable.\
The paper is organized as follows. In Sec. \[sec:2\] we describe the production of single path two-photon states displaying polarization-OAM entanglement. To this aim we use different types of polarization to OAM transferrers to generate polarization-entangled photon pairs in separate path modes whereupon a polarizing beam splitter (PBS) is employed to project both photons into a single path. Using entanglement duality we then describe entanglement sorting in sections \[sec:3\] and \[sec:4\]. More specifically, we describe in Sec. \[sec:3\] the detection of entanglement in OAM variables, whereas in Sec. \[sec:4\] we register the entanglement in the polarization degrees of freedom. In both cases entanglement witnesses are constructed. In Sec. \[sec:5\] we generalize the idea of entanglement sorting to three- and four-photon entangled states, and in Sec. \[sec:6\] we finally conclude.
Entangled States of Polarization and Orbital Angular Momentum {#sec:2}
=============================================================
To create polarization-OAM entanglement of two indistinguishable photons travelling in the same direction (cf. Eq. (\[eq:dualPolOAM1\])) a two-photon source has to be employed, e.g., using parametric down conversion. The produced down converted polarization-path entangled photon pairs are then described by the quantum state given in Eq. (\[eq:dualHVpath1\]). Since in our case we want to replace the spatial degree of freedom by OAM variables, two different polarization to OAM transferrers ($\pi\rightarrow l$) are applied thereafter, where $\pi$ ($l$) denotes the polarization (OAM quantum number) of the particle, followed by a PBS for spatial mode mixing.\
A possible $\pi\rightarrow l$ transferrer described in [@Nagali(2009)] consists of a quarter wave plate (QWP) changing $H$ ($V$) polarization to right (left) circular polarization, a q-plate and a PBS. A q-plate transforms left (right) circular polarization to right (left) circular polarization while at the same time producing OAM of $l$ ($-l$) [@Marrucci(2006)]. Recently, q-plates producing OAM with values of up to $|l|=100$ have been reported [@DAmbrosio(2013)]. Measuring only the horizontal output mode of the q-plate by use of the PBS, every polarization state $1/\sqrt{2}(\ket{H}+\ket{V})$ with vanishing OAM ($\ket{l=0}$) will be transformed into the following superposition state of OAM $\pm l$ [@Nagali(2009)] $$\frac{1}{\sqrt{2}}\left(\ket{H}+\ket{V}\right)\ket{l=0} \ \stackrel{\text{q-plate}}{\longrightarrow} \ \frac{1}{\sqrt{2}}\left(\ket{l}+\ket{-l}\right)\ket{H} \ .
\label{eq:transferrer1}$$ In principle the same result can be achieved by using a second type of $\pi\rightarrow l$ transferrer as described in [@Fickler(2012)]. Hereby, spatial light modulators (SLM) are employed which can produce OAM with values of up to $|l|=300$. By separating the photons according to their polarization and sending them on different SLM, OAM of $l$ ($-l$) can be transferred independently to $H$ ($V$) polarized photons. The subsequent recombination of the distinct photon modes and deleting the polarization degree of freedom by use of a linear polarizer of diagonal polarization $D_{+}$ ($\ket{D_{+}}=1/\sqrt{2}(\ket{H}+\ket{V}$)) completes the entanglement transfer [@Fickler(2012)] $$\begin{aligned}
\frac{1}{\sqrt{2}}\left(\ket{H}+\ket{V}\right)\ket{l=0} \ & \stackrel{\text{SLM}}{\longrightarrow} \ \frac{1}{\sqrt{2}}\left(\ket{H,l}+\ket{V,-l}\right) \\
& \stackrel{D_{+}}{\longrightarrow} \ \frac{1}{\sqrt{2}}\left(\ket{l}+\ket{-l}\right)\ket{D_{+}} \ .
\label{eq:transferrer2} \\
\end{aligned}$$ The output states of the two $\pi\rightarrow l$ transferrers only differ in their polarization (cf. Eq. (\[eq:transferrer1\]) and (\[eq:transferrer2\])). Half wave plates (HWP) of angle $\theta_{\text{HWP}1}=22.5^{\circ}$ can then be used to rotate the polarization $D_{+}$ to $H$. Another possibility would be to rotate the coordinate system around an angle of $\theta_{sys}=45^{\circ}$; however, for a rotated coordinate system all subsequent devices have to be rotated around the same angle $\theta_{sys}$ as well.\
In both cases of $\pi\rightarrow l$ transferrers the polarization has to be changed in one of the two spatial modes (e.g., ${\mathbf{k}}_{2}$). This is done by employing a HWP of angle $\theta_{\text{HWP}2}=45^{\circ}$, which finally produces a polarization-OAM entangled state but still in two spatial path modes (see Fig. \[fig:source2\]) $$\begin{aligned}
\ket{\Psi'}= \frac{1}{\sqrt{2}}\left(\ket{H,l,{\mathbf{k}}_{1}} \ket{V,-l,{\mathbf{k}}_{2}}+\ket{H,-l,{\mathbf{k}}_{1}}\ket{V,l,{\mathbf{k}}_{2}}\right) \ .
\label{eq:statePolOAMPath}
\end{aligned}$$
\
Since the state given in Eq. (\[eq:statePolOAMPath\]) displays two-photon entanglement for either variables, OAM and path, or polarization and OAM, several tests of entanglement are possible. For example, OAM-path entanglement could be tested in an analogous way as the path-polarization experiments presented in [@Bose(2013)]. However, due to the simultaneous entanglement of OAM, polarization and path, the two photons could not interfere when using polarization sensitive photon-detectors since in this case the photons become distinguishable. On the other hand, projecting all photons onto diagonal polarized states would project the complete state onto a maximally OAM-path entangled state. In this case interference becomes possible even for polarization sensitive detectors. Another experiment could be to test polarization-OAM entanglement of the two photons still being in distinct spatial modes. This measurement would require entanglement sorting via polarization and OAM, as being suggested in the following for two photons travelling along the same direction.\
In what follows we thus concentrate on the two-photon state prepared in only one single path. To this aim the additional spatial degree of freedom has to be removed. Since photons in mode ${\mathbf{k}}_{1}$ are always horizontally polarized and photons in mode ${\mathbf{k}}_{2}$ are always vertically polarized, all photons can be projected onto a single path mode by mixing modes ${\mathbf{k}}_{1}$ and ${\mathbf{k}}_{2}$ via a PBS (see Fig. \[fig:source2\]). This eliminates the spatial degree of freedom and creates entanglement solely between polarization and OAM variables of the form (cf. Eq. (\[eq:dualPolOAM1\])) $$\begin{aligned}
\ket{\Psi} & = \frac{1}{\sqrt{2}}\left(\ket{H,l}\right. \left. \ket{V,-l}+\ket{V,l}\ket{H,-l}\right) \\
& \equiv \frac{1}{\sqrt{2}}\left(\ket{l,H} \ket{-l,V}+\ket{-l,H}\ket{l,V}\right) \ .
\label{eq:statePolOAM}
\end{aligned}$$ Starting from Eq. (\[eq:statePolOAM\]) we next make use of the properties of entanglement duality [@Bose(2013)]. As it is shown in Eq. (\[eq:statePolOAM\]) the polarization-OAM entanglement can be written in two equivalent ways, by interchanging the variables ($H,V$) and ($l,-l$). This has been interpreted as two distinct possibilities of labeling the entanglement of two identical particles [@Bose(2013)]. Both entanglement labelings display two-particle entanglement. Now, the produced dual state can be separated into two spatial modes depending on its labeling variables, i.e., either via the polarization or the OAM degrees of freedom. This so-called entanglement sorting allows for a direct measurement of the corresponding variables’ entanglement (see Fig. \[fig:OAMmeasure\] and \[fig:Polmeasure\]) as in both cases experimentally implementable witnesses can be set up to prove the entanglement. Hereby, an observable $\hat{W}$ is called entanglement witness when indicating entanglement in the following way [@Guhne(2009)] $$\begin{aligned}
\text{Tr}[\hat{W}\rho_{s}] \geq 0 \ , \ \ \ \ \
\text{Tr}[\hat{W}\rho_{e}] < 0 \ ,
\end{aligned}$$ with $\rho_{s}$ representing all separable states and $\rho_{e}$ representing at least one entangled state.
Entanglement Sorting via Polarization {#sec:3}
=====================================
Sorting the state via polarization by use of a PBS divides the given state $\ket{\Psi} $ of Eq. (\[eq:statePolOAM\]) into $H$ and $V$ polarized photons (see Fig. \[fig:OAMmeasure\]). Then flipping polarization in the vertical mode from $V$ to $H$ employing a HWP of $\theta=45^{\circ}$ eliminates the polarization degree of freedom and produces the OAM-path entangled state $\ket{\Psi_{OAM}}$ of Eq. (\[eq:dualOAMpath1\]). This allows to measure OAM-path entanglement. As shown in [@Fickler(2012)] a subsequent combination of a radial symmetric slit mask ($2l$ slits) and a bucket detector in each spatial output mode $1$ and $2$ allows for measuring the OAM-entanglement using an appropriate entanglement witness $\hat{W}$ and considering coincident counts only, even for very high quantum numbers $l$. In the following the experimental idea from [@Fickler(2012)] will be explained, but a different entanglement witness will be employed. The derivation of this entanglement witness can be found in [@Guhne(2002); @Guhne(2009)].
\
For the experimental approach one can make use of the fact that any state of the form [@Fickler(2012)] $$\ket{\Psi_{j}}= \frac{1}{\sqrt{2}}\ (\ket{l,{\mathbf{k}}_{j}}+e^{i\varphi_{j}}\ket{-l,{\mathbf{k}}_{j}}) \ ,
\label{eq:State2lInt}$$ displays a radial intensity profile with $2l$ intensity maxima arranged in a circle, where $j=1,2$ denotes the spatial path mode. Therefore, a rotatable slit mask of the same symmetry can be used to measure every possible superposition as a function of the mask’s rotation angle $\gamma'_{j}$. Obviously, the state measured by the mask then reads [@Fickler(2012)] $$\begin{aligned}
\ket{\chi_{j}(\phi_{j})}=\frac{1}{\sqrt{2}}\ (\ket{l,{\mathbf{k}}_{j}}+e^{i\phi_{j}}\ket{-l,{\mathbf{k}}_{j}}) \ ,
\label{eq:statemasktext}
\end{aligned}$$ where the phase factor $\phi_{j}$ of the projected state is determined by the angular position of the slit mask [@Fickler(2012)] $$\gamma'_{j}=\frac{\phi_{j}\cdot 360^{\circ}}{2l\cdot 2\pi} \Leftrightarrow \phi_{j}=\frac{\gamma'_{j} \cdot 2l \cdot 2\pi}{360^{\circ}} \ .$$ Coincidence measurements of both modes can then be performed as a function of the angular positions $\gamma'_{1}$ and $\gamma'_{2}$ of the slit masks in the two output modes.\
The entanglement witness can be formulated in an experimentally implementable way consisting of different mutually unbiased measures [@Guhne(2002)] $$\begin{aligned}
\hat{W}= & \frac{1}{2}\mathbb{1} -\ket{\Psi_{OAM}}\bra{\Psi_{OAM}} \\
= & \frac{1}{2} \left( \ket{\,l_{1},l_{2}}\bra{l_{1},l_{2}\,} + \ket{-l_{1},-l_{2}}\bra{-l_{1},-l_{2}} \right. \\
& - \ket{{d_{+}}_{1},{d_{+}}_{2}}\bra{{d_{+}}_{1},{d_{+}}_{2}} - \ket{{d_{-}}_{1},{d_{-}}_{2}}\bra{{d_{-}}_{1},{d_{-}}_{2}} \\
& \left. + \ket{\mathcal{L}_{1},\mathcal{R}_{2}}\bra{\mathcal{L}_{1},\mathcal{R}_{2}} + \ket{\mathcal{R}_{1},\mathcal{L}_{2}}\bra{\mathcal{R}_{1},\mathcal{L}_{2}} \right) \ ,
\label{eq:Witness}
\end{aligned}$$ where $\ket{A_{1},B_{2}}\equiv \ket{A,{\mathbf{k}}_{1}}\ket{B,{\mathbf{k}}_{2}}$. All parts of the witness can be expressed by projected states of the slit masks (cf. Eq. (\[eq:statemasktext\])). This can be accomplished by simply adjusting the slit masks to appropriate angular positions. From this follow the identities $\ket{{d_{+}}_{j}}=\ket{\chi_{j}(0)}$, $\ket{{d_{-}}_{j}}=\ket{\chi_{j}(\pi)}$ denoting diagonal OAM states, and $\ket{\mathcal{L}_{j}}=\ket{\chi_{j}(\pi/2)}$ and $\ket{\mathcal{R}_{j}}=\ket{\chi_{j}(-\pi/2)}$ denoting circular OAM states in analogy to polarization states. The complete witness in terms of the mask projections now reads (cf. Eq. (\[eq:Witness\])) $$\begin{aligned}
\hat{W}= & \frac{1}{2}\mathbb{1} -\ket{\Psi_{OAM}}\bra{\Psi_{OAM}} \\
= & \frac{1}{2} \big(\ \ket{\,l_{1},l_{2}}\bra{l_{1},l_{2}\,} + \ket{-l_{1},-l_{2}}\bra{-l_{1},-l_{2}} \\
&\hspace{4mm} \ - \ket{\chi_{1}(0),\chi_{2}(0)}\bra{\chi_{1}(0),\chi_{2}(0)} \\
&\hspace{4mm} \ - \ket{\chi_{1}(\pi),\chi_{2}(\pi)}\bra{\chi_{1}(\pi),\chi_{2}(\pi)} \\
&\hspace{4mm} \ + \ket{\chi_{1}(\pi/2),\chi_{2}(-\pi/2)}\bra{\chi_{1}(\pi/2),\chi_{2}(-\pi/2)} \\
&\hspace{4mm} \ + \ket{\chi_{1}(-\pi/2),\chi_{2}(\pi/2)}\bra{\chi_{1}(-\pi/2),\chi_{2}(\pi/2)}\ \big) \ .
\label{eq:WitnessProjections}
\end{aligned}$$ Since in the experiment one is measuring intensities, i.e., coincidence counts, the measurements have to be normalized by the sum of the intensities of the basis states $\ket{l_{1},l_{2}}$, $\ket{l_{1},-l_{2}}$, $\ket{-l_{1},l_{2}}$ and $\ket{-l_{1},-l_{2}}$ [@Barbieri(2003)].\
Calculating the theoretical expectation value of the entanglement witness (Eq. (\[eq:Witness\])) and the maximally entangled state (Eq. (\[eq:dualOAMpath1\])) gives a value of $$\bra{\Psi_{OAM}}\hat{W}\ket{\Psi_{OAM}} = - \frac{1}{2} \ .$$ Similar to [@Fickler(2012)] we now want to show that a general separable state (cf. Eq. (\[eq:State2lInt\])) $$\begin{aligned}
\ket{\Psi_{s}}=&\ket{\Psi_{1}',\Psi_{2}'} \\
=&\left(a\ket{l,{\mathbf{k}}_{1}}\vphantom{e^{i\varphi_{1}}}+b e^{i\varphi_{1}}\ket{-l,{\mathbf{k}}_{1}}\right) \\
& \hspace{15mm} \otimes\left(c\ket{l,{\mathbf{k}}_{2}} + d e^{i\varphi_{2}}\ket{-l,{\mathbf{k}}_{2}}\right) ,
\label{eq:statePsi12}
\end{aligned}$$ does *not* violate the entanglement witness, where $a,b,c,d\in \mathbb{R}$ and $a^{2}+b^{2}=c^{2}+d^{2}=1$. For this to prove we have to determine the expectation value of the witness $\bra{\Psi_{s}}\hat{W}\ket{\Psi_{s}}$ and show that $\bra{\Psi_{s}}\hat{W}\ket{\Psi_{s}} \geq 0$. For the state $\ket{\Psi_{s}}$ from Eq. (\[eq:statePsi12\]) the components of the entanglement witness (cf. Eq. (\[eq:Witness\])) take the following form $$\begin{aligned}
\braket{\Psi_{s}|\;l_{1},l_{2}\,|\Psi_{s}} &= a^{2}c^{2} \ , \\
\braket{\Psi_{s}|-l_{1},-l_{2}|\Psi_{s}} &= b^{2}d^{2} \ , \\
\braket{\Psi_{s}|{d_{+}}_{1},{d_{+}}_{2}|\Psi_{s}} &= | \frac{1}{2} (a+be^{i\varphi_{1}})(c+de^{i\varphi_{2}}) |^{2} \ , \\
\braket{\Psi_{s}|{d_{-}}_{1},{d_{-}}_{2}|\Psi_{s}} &= | \frac{1}{2} (a-be^{i\varphi_{1}})(c-de^{i\varphi_{2}}) |^{2} \ , \\
\braket{\Psi_{s}|\mathcal{L}_{1},\mathcal{R}_{2}|\Psi_{s}} &= | \frac{1}{2} (a-ibe^{i\varphi_{1}})(c+ide^{i\varphi_{2}}) |^{2} \ , \\
\braket{\Psi_{s}|\mathcal{R}_{1},\mathcal{L}_{2}|\Psi_{s}} &= | \frac{1}{2} (a+ibe^{i\varphi_{1}})(c-ide^{i\varphi_{2}}) |^{2} \ ,
\label{eq:WitnessOAM}
\end{aligned}$$ where $\braket{\Psi_{s}|A_{1},B_{2}|\Psi_{s}} = \braket{\Psi_{s}|A_{1},B_{2}}\braket{A_{1},B_{2}|\Psi_{s}} $. Solving the absolute squares $$\begin{aligned}
\braket{\Psi_{s}|{d_{+}}_{1},{d_{+}}_{2}|\Psi_{s}} = & \frac{1}{4} [1 + 2ab \cos(\varphi_{1}) + 2cd \cos(\varphi_{2}) \\
& \hspace{14mm} + 4abcd \cos(\varphi_{1}) \cos(\varphi_{2})] \ , \\
\braket{\Psi_{s}|{d_{-}}_{1},{d_{-}}_{2}|\Psi_{s}} = & \frac{1}{4} [1 - 2ab \cos(\varphi_{1}) - 2cd \cos(\varphi_{2}) \\
& \hspace{14mm} + 4abcd \cos(\varphi_{1}) \cos(\varphi_{2})] \ , \\
\braket{\Psi_{s}|\mathcal{L}_{1},\mathcal{R}_{2}|\Psi_{s}} = & \frac{1}{4} [1 + 2ab \sin(\varphi_{1}) - 2cd \sin(\varphi_{2}) \\
& \hspace{14mm} - 4abcd \sin(\varphi_{1}) \sin(\varphi_{2})] \ , \\
\braket{\Psi_{s}|\mathcal{R}_{1},\mathcal{L}_{2}|\Psi_{s}} = & \frac{1}{4} [1 - 2ab \sin(\varphi_{1}) + 2cd \sin(\varphi_{2}) \\
& \hspace{14mm} - 4abcd \sin(\varphi_{1}) \sin(\varphi_{2})] \ , \\
\end{aligned}$$ leads to the following value for the entanglement witness (cf. Eq. (\[eq:Witness\])) $$\begin{aligned}
\bra{\Psi_{s}}\hat{W}\ket{\Psi_{s}} = & \; \frac{1}{2} \; [ a^{2}c^{2} + b^{2}d^{2} - 2abcd\cos(\varphi_{1})\cos(\varphi_{2}) \\
& \hspace{24mm} - 2abcd\sin(\varphi_{1})\sin(\varphi_{2}) ] \ .
\label{eq:WitnessSep}
\end{aligned}$$ Eq. (\[eq:WitnessSep\]) reaches a minimal value of $0$ (for $\varphi_{1}=\varphi_{2}$, $a=d$ and $b=c$) what confirms the validity of the witness and the OAM-path entanglement of the two-photon state $\ket{\Psi_{OAM}}$.\
Note that for very high quantum numbers $l$ the coincidence signal might have to be corrected by subtracting accidental coincident counts. However, experiments for at least $l=100$ still display very good results without correction [@Fickler(2012)].\
Another possibility for measuring the OAM-entanglement witness has been implemented consisting of SLM in the two spatial output modes [@Agnew(2012)]. In this case SLM take the role of the slit masks and project the OAM-modes onto the distinct superposition states what allows for measuring the witness.
Entanglement sorting via Angular Momentum {#sec:4}
=========================================
The second possibility of entanglement sorting requires at first to divide up the polarization-OAM entangled state of Eq. (\[eq:statePolOAM\]) according to the OAM numbers $\pm l$. Holographic fork masks of appropriate order $l$ can be used to achieve this separation [@Heckenberg(1992)], as they diffract incident photons while at the same time change their OAM. In the first diffraction order photons with OAM-change of $\Delta l=+ l$ can be found in one direction and photons with OAM-change $\Delta l= -l$ in the other direction. Coupling the photons into single mode fibers, transmitting photons with $l=0$ only, allows for separation and detection of photons with OAM $\pm l$ [@Nagali(2009)].
The state after the separation due to OAM components carries the polarization entanglement and corresponds to $\ket{\Psi_{Pol}}$ of Eq. (\[eq:dualHVpath1\]). Proving polarization-entanglement can now be accomplished by using again an appropriate entanglement witness. In [@Barbieri(2003)] this approach has been used to verify entanglement for a polarization singlet state. Here the witness for testing the given state (cf. Eq. (\[eq:dualHVpath1\])) is of the same form as Eq. (\[eq:Witness\]) but formulated in terms of polarization [@Guhne(2002)] $$\begin{aligned}
\hat{W}= & \frac{1}{2}\mathbb{1} -\ket{\Psi_{Pol}}\bra{\Psi_{Pol}} \\
= & \frac{1}{2} \left( \ket{H_{1},H_{2}}\bra{H_{1},H_{2}} + \ket{V_{1},V_{2}}\bra{V_{1},V_{2}} \right. \\
& -\! \ket{{D_{+}}_{1},{D_{+}}_{2}}\bra{{D_{+}}_{1},{D_{+}}_{2}}\! -\! \ket{{D_{-}}_{1},{D_{-}}_{2}}\bra{{D_{-}}_{1},{D_{-}}_{2}} \\
& \left. + \ket{L_{1},R_{2}}\bra{L_{1},R_{2}} + \ket{R_{1},L_{2}}\bra{R_{1},L_{2}} \right) \ ,
\label{eq:WitnessPol}
\end{aligned}$$ where again $\ket{A_{1},B_{2}}\equiv \ket{A,{\mathbf{k}}_{1}}\ket{B,{\mathbf{k}}_{2}}$ and the diagonal polarization states $\ket{D_{\pm}}=(\ket{H}\pm\ket{V})/\sqrt{2}$ and the circular polarization states left $\ket{L}=(\ket{H}+i\ket{V})/\sqrt{2}$ and right $\ket{R}=(\ket{H}-i\ket{V})/\sqrt{2}$ have been inserted.\
The different combinations of polarizations in the two distinct output modes can be measured by applying QWP, HWP and a PBS sequentially [@Barbieri(2003)] where again coincident counts have to be taken into account. Additionally, the measurements have to be normalized by the sum of the coincident rates in the basis $\ket{H_{1},H_{2}}$, $\ket{H_{1},V_{2}}$, $\ket{V_{1},H_{2}}$ and $\ket{V_{1},V_{2}}$ in order to obtain probabilities [@Barbieri(2003)].
Generalization to three- and four-photon entanglement {#sec:5}
=====================================================
The combination of OAM and polarization degree of freedom also allows for implementing entanglement sorting with more than two identical photons. In [@Bouwmeester(1999)] it was shown that three-photon polarization-entanglement can be produced in the form of a Greenberger-Horne-Zeilinger (GHZ) state $$\begin{aligned}
&\ket{\Psi_{GHZ}}=\frac{1}{\sqrt{2}}\left(\ket{H,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{2}}\ket{V,{\mathbf{k}}_{3}} \right. \\
& \hspace{30mm} \left. +\ket{V,{\mathbf{k}}_{1}}\ket{H,{\mathbf{k}}_{2}}\ket{H,{\mathbf{k}}_{3}}\right) \ ,
\label{eq:GHZ}
\end{aligned}$$ where the photons are propagating in three different spatial modes. Employing $\pi\rightarrow l$ transferrers (cf. Eq. (\[eq:transferrer1\]) or (\[eq:transferrer2\])) and a polarization flipper leads to a three-photon polarization-OAM-path entanglement. All photons can now be combined into one single path by use of a collective lens, which gives rise to the following state $$\begin{aligned}
&\ket{\Psi_{GHZ}'}\\
&\hspace{5mm}=\frac{1}{\sqrt{2}}\left(\ket{H,l}\ket{V,-l}\ket{V,-l}+\ket{V,l}\ket{V,l}\ket{H,-l}\right) \\
&\hspace{5mm}\equiv\frac{1}{\sqrt{2}}\left(\ket{l,H}\ket{-l,V}\ket{-l,V}+ \ket{-l,H}\ket{l,V}\ket{l,V}\right) .
\label{eq:threephotonstate1}
\end{aligned}$$ Here, entanglement sorting via polarization produces the OAM-path entangled state $$\begin{aligned}
&\ket{\Psi_{OAM,3}}=\frac{1}{\sqrt{2}} \left( \ket{l,{\mathbf{k}}_{1}}\ket{-l,{\mathbf{k}}_{2}}\ket{-l,{\mathbf{k}}_{2}} \right. \\
& \hspace{30mm} + \left. \ket{-l,{\mathbf{k}}_{1}}\ket{l,{\mathbf{k}}_{2}}\ket{l,{\mathbf{k}}_{2}} \right) \ ,
\end{aligned}$$ whereas entanglement sorting via OAM produces polarization-path entanglement of the form $$\begin{aligned}
&\ket{\Psi_{Pol,3}}=\frac{1}{\sqrt{2}} \left( \ket{H,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{2}}\ket{V,{\mathbf{k}}_{2}} \right. \\
& \hspace{30mm} + \left. \ket{V,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{1}}\ket{H,{\mathbf{k}}_{2}} \right) \ .
\end{aligned}$$ Obviously, entanglement sorting for three-photon entanglement can only be accomplished in a modified way since the asymmetric photon distribution causes the sorted states to be of asymmetric form.\
In case of four-photon entanglement which could arise from type-II parametric down-conversion four photons are created in two distinct directions [@Weinfurter(2001); @Weinfurter(2003)] $$\begin{aligned}
\ket{\Psi_{Pol,4}} = \frac{1}{\sqrt{3}} \left( \vphantom{\ket{H,{\mathbf{k}}_{1}}} \right.\!\! & \left. \ket{H,{\mathbf{k}}_{1}}\ket{H,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{2}}\ket{V,{\mathbf{k}}_{2}} \right. \\
+ & \left. \ket{V,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{1}}\ket{H,{\mathbf{k}}_{2}}\ket{H,{\mathbf{k}}_{2}} \right. \\
\vphantom{\frac{1}{\sqrt{3}} } + & \left. \ket{H,{\mathbf{k}}_{1}}\ket{V,{\mathbf{k}}_{1}}\ket{H,{\mathbf{k}}_{2}}\ket{V,{\mathbf{k}}_{2}} \right) \ .
\label{eq:Polfour}
\end{aligned}$$ By using SLM to induce OAM of $l$ and $-l$ in spatial modes ${\mathbf{k}}_{1}$ and ${\mathbf{k}}_{2}$, respectively, and collecting all photons in a single path mode with an optical lens, the following four-photon polarization-OAM entangled state is generated $$\begin{aligned}
\ket{\Psi_{4}} = \frac{1}{\sqrt{3}} \left( \vphantom{\ket{H,l}} \right. \!\! & \left. \ket{H,l}\ket{H,l}\ket{V,-l}\ket{V,-l} \right. \\
+ & \left. \ket{V,l}\ket{V,l}\ket{H,-l}\ket{H,-l} \right. \\
\vphantom{\frac{1}{\sqrt{3}} } + & \left.\ket{H,l}\ket{V,l}\ket{H,-l}\ket{V,-l} \right) \ .
\label{eq:PolOAMfour}
\end{aligned}$$ Due to the symmetry of the four-photon state dual entanglement sorting can be accomplished in the same way as in the case of two-photon polarization-OAM entanglement. For example, applying entanglement sorting via polarization by use of a PBS leads to OAM-path entanglement (cf. Eq. (\[eq:Polfour\])) $$\begin{aligned}
\ket{\Psi_{OAM,4}} = \frac{1}{\sqrt{3}} \left( \vphantom{\ket{l,{\mathbf{k}}_{1}}} \right. \!\! & \left. \ket{l,{\mathbf{k}}_{1}}\ket{l,{\mathbf{k}}_{1}}\ket{-l,{\mathbf{k}}_{2}}\ket{-l,{\mathbf{k}}_{2}} \right. \\
+ & \left. \ket{-l,{\mathbf{k}}_{1}}\ket{-l,{\mathbf{k}}_{1}}\ket{l,{\mathbf{k}}_{2}}\ket{l,{\mathbf{k}}_{2}} \right. \\
\vphantom{\frac{1}{\sqrt{3}} } + & \left. \ket{l,{\mathbf{k}}_{1}}\ket{-l,{\mathbf{k}}_{1}}\ket{l,{\mathbf{k}}_{2}}\ket{-l,{\mathbf{k}}_{2}} \right) \ ,
\label{eq:OAMfour}
\end{aligned}$$ whereas entanglement sorting via OAM by use of a fork hologram projects the state $\ket{\Psi_{4}}$ (cf. Eq. (\[eq:PolOAMfour\])) onto the polarization-path entangled state $\ket{\Psi_{Pol,4}}$ (cf. Eq. (\[eq:Polfour\])).
Conclusion {#sec:6}
==========
In conclusion we presented in Sec. \[sec:2\] an experimental setup able to produce polarization-OAM entangled two-photon states where both photons propagate in the same direction. For these states we demonstrated in sections \[sec:3\] and \[sec:4\] that due to the duality of the identical photons entanglement sorting can be accomplished depending on their angular momenta, i.e., via their polarization or OAM degrees of freedom.\
In Sec. \[sec:3\] we showed that entanglement sorting via polarization can be achieved using a PBS while in Sec. \[sec:4\] a holographic fork mask was employed to achieve entanglement sorting via OAM. Hence, we could show that simple experimental setups can be used to perform entanglement sorting and, thereby, prove the dual entanglement of the two angular momenta degrees of freedom.\
In Sec. \[sec:5\] we finally pointed out that entanglement sorting can be generalized to higher photon numbers, discussing in particular the cases $N=3$ and $N=4$. From these discussions it became evident that in case of even photon numbers entanglement duality occurs and symmetric entanglement sorting is possible – meaning that the two entanglement sorted states have symmetric path-entangled structure – whereas in case of odd photon numbers the entanglement sorted states are of asymmetric form.
|
---
author:
- Fahad Mahmood
- Dipanjan Chaudhuri
- Sarang Gopalakrishnan
- Rahul Nandkishore
- 'N. P. Armitage'
- 'Fahad Mahmood$^{1,2,3}$, Dipanjan Chaudhuri$^1$, Sarang Gopalakrishnan$^{4,5}$, Rahul Nandkishore$^6$, and N. P. Armitage$^1$\'
title: Observation of a marginal Fermi glass using THz 2D coherent spectroscopy
---
**A longstanding open problem in condensed matter physics is whether or not a strongly disordered interacting insulator can be mapped to a system of effectively non-interacting localized excitations. We investigate this issue on the insulating side of the 3D metal-insulator transition (MIT) in phosphorus doped silicon using the new technique of terahertz two dimensional coherent spectroscopy. Despite the intrinsically disordered nature of these materials, we observe coherent excitations and strong photon echoes that provide us with a powerful method for the study of their decay processes. We extract the first measurements of energy relaxation ($T_1$) and decoherence ($T_2$) times close to the MIT in this classic system. We observe that (i) both relaxation rates are linear in excitation frequency with a slope close to unity, (ii) the energy relaxation timescale $T_1$ counterintuitively *increases* with increasing temperature and (iii) the coherence relaxation timescale $T_2$ has little temperature dependence between $\SI{5}{K}$ and $\SI{25}{K}$, but counterintuitively *increases* as the material is doped towards the MIT. We argue that these features imply that (a) the system behaves as a well isolated electronic system on the timescales of interest, and (b) relaxation is controlled by electron-electron interactions. We discuss the potential relaxation channels that may explain the behavior. Our observations constitute a qualitatively new phenomenology, driven by the interplay of strong disorder and strong electron-electron interactions, which we dub the marginal Fermi glass.**
Understanding systems with strong disorder and strong interactions is a central open issue in condensed matter physics. It is a remarkable fact that many [*metals*]{} can be understood in terms of weakly interacting fermionic quasiparticles near the Fermi energy ($E_F$) despite the fact that the bare Coulomb interaction is not particularly small or short-ranged. This has been canonized in terms of the Landau Fermi liquid theory [@nozierespines], where the effects of interactions renormalize quasiparticle parameters like the effective mass, but do not change the underlying effective structure of the theory from that of free-electrons. Scattering rates of the quasiparticles in a Landau Fermi liquid go like $(E-E_F)^2$; thus quasiparticles are arbitrarily well-defined near $E_F$. These effects arise as a consequence of both the Pauli exclusion principle, which reduces the phase space for scattering, and screening that renders the bare Coulomb interaction effectively short-range.
The tendency of strong disorder is to localize particles. Anderson showed that in the absence of interactions sufficiently strong disorder could localize wavefunctions with a sharp boundary in energy between localized and extended states [@pwa1958]. This is a generic wave phenomenon that applies equally to acoustic, electromagnetic, or neutral matter waves [@abrahams201050]. Although such “Anderson localization” is frequently invoked in the study of disordered electronic insulators, it is unclear to what extent this phenomenon actually applies to real materials.
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In this regard, in 1970 Anderson proposed the notion – in analogy with the Fermi liquid – of a “Fermi glass" as a localized disordered state of matter adiabatically connected to the noninteracting Anderson insulator, whose universal properties arose through Pauli exclusion alone [@Anderson70a]. Anderson conjectured that via the protection afforded by the Fermi energy, such a state of matter would also have well-defined single-particle-like excitations at low energy. It was later understood that the localized nature of such systems and lack of metallic screening made these considerations more subtle [@Efros75a; @fleishman1980interactions; @freedman1977theory]. Recently, it was realized that insulators might feature an even stronger notion of adiabatic continuity than metals. Refs. [@gmp; @basko2006metal] argued that a disordered system with [*short-range interactions*]{} could be “many-body-localized" and thus have infinitely sharp excitations even at nonzero temperatures and far from $E_F$. This has been a large area of current inquiry—see [@mblarcmp; @mblrmp] for reviews. However the effects of [*long-range Coulomb interactions*]{} are still not fully understood. It has been shown [@burin1998; @burin; @yao2014many; @gutman] that long-range interactions invalidate perturbative arguments for localization. While non-perturbative methods have been applied in certain settings [@lrmbl], it remains unknown whether a “Fermi glass” exists, i.e., whether a frequency or temperature window exists where the excitations of an interacting insulator are renormalized, weakly interacting, electron-like quasiparticles.
In this work we use the new technique of terahertz 2D coherent spectroscopy (THz 2DCS)[@woerner2013ultrafast; @lu2017; @wan2019resolving] to shed light on this fundamental problem. We investigate the canonical disordered material phosphorus doped silicon (Si:P) on the insulating side of the metal-insulator transition [@rosenbaum1983metal; @paalanen1983critical]. Among other aspects, THz 2DCS allows us to measure both $T_1$ and $T_2$ times of inhomogeneously broadened spectra in the THz range. At low temperature we find a temperature independent regime governed by electron-electron interactions. We find that relaxation rates of the optical excitations are linear in frequency with a proportionality constant of order one. This establishes that in our frequency range, the low energy excitations are not well defined. This is consistent with a picture in which localized electronic systems are [*not*]{} adiabatically connected to the Anderson insulator. We call this state of matter the [*marginal Fermi glass*]{}. This electronic relaxation is consistent with the existence of an electronic continuum that arises through long-range Coulomb interactions which could destabilize the localized state at non-zero temperatures.
In Fig. \[LinearTimeTraces\]a, we show the real part of the linear response THz range conductivity of a representative 39$\%$ sample. In previous work [@Helgren04a; @Helgren02a] it was shown that the optical response of Si:P near the MIT was in accord with the theory of Mott-Efros-Shklovskii [@Lee01a; @shklovskii1981phononless]. Here one models the excited states of the system as an ensemble of resonant pairs that can be mapped to a random ensemble of two-level systems (i.e., the “pair approximation") that gives a conductivity $ \sigma_{1}(\omega) = \alpha e^{2}N_{0}^{2}\xi^{5}\omega[ln(2I_{0}/\hbar\omega)]^{4}[\hbar\omega + U(r_{w})]$ e.g., an almost linear conductivity is found at low frequencies and a quadratic one at higher frequencies. These power laws come from phase space considerations (see Supplementary Information (SI)). They crossover at an energy scale $U(r_{\omega}) = e^{2}/\varepsilon_{1} r_{\omega}$ which is the attraction between an electron and hole in a dipolar excitation at a distance $r_\omega = \xi[ln(2I_{0}/\hbar\omega)]$. Here $\alpha$ is a constant close to one, and $I_{0}$ is the pre-factor of the overlap integral (commonly taken to be the Bohr energy of the dopant). One aspect not considered in the usual treatment is that each of the excitations that contributes to $\sigma(\omega)$ has a finite lifetime. The functional form of $\sigma(\omega)$ is insensitive to moderate level broadening, so it is uninformative about excitation lifetimes. Quantifying homogeneous broadening due to quasiparticle decay in the face of overtly inhomogeneously broadened spectra is the principal difficulty in characterizing interactions in these systems.
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2D coherent spectroscopy is a nonlinear 4-wave mixing technique that can, among other aspects, directly reveal couplings between excitations and separate homogeneous from inhomogeneous broadening [@Mukamel1995; @hamm2011concepts; @aue1976two; @cundiff2013optical]. It has been incredibly powerful in its radio and infrared frequency incarnations for the study of chemical systems. It has been extended recently to the THz range to study graphene and quantum wells [@kuehn2011two; @woerner2013ultrafast], molecular rotations [@lu2016nonlinear], and spin waves in conventional magnets [@lu2017]. It has also been proposed to give unique information on fractionalized spin phases [@wan2019resolving; @choi2020theory].
As discussed in the method section, two THz pulses (A and B) are incident on a sample. The transmitted electric field is recorded as a function of the separation between them ($\tau$) and the time from pulse B ($t$). The nonlinear signal is defined $E_{NL}(\tau,t) = E_{AB}(\tau,t) - E_{A}(\tau,t) - E_{B}(t)$. Here $E_{AB}$ is the transmitted signal when both THz pulses are present while $E_{A}$ and $E_{B}$ are the transmitted signals with each pulse A and B present individually. Fig. 1c-f shows $E_A$, $E_B$, $E_{AB}$, and $E_{NL}$ as a function of $t$ and $\tau$ for the 39% sample. Fig. 2 shows the resulting 2D THz spectra $E_{NL}(\nu_\tau,\nu_t)$ from the Fourier transforms with respect to $\tau$ and $t$ for each doping studied at a temperature of . We note that typical P-P spacings in these samples are of order [@Helgren04a], and hence the associated Coulomb energy is of order , which corresponds to a timescale of $\approx \SI{0.3}{ps}$. Thus, the experimental timescales of a few ps of these measurements are easily long enough for the effect of interactions to be important.
{width="17cm"}
In an inversion symmetric system like Si:P the leading nonlinear response is $\chi^{(3)}$ electric dipole reradiation. Therefore with two pulses, there are contributions to $E_{NL}$ in which pulse A interacts twice and pulse B once with the sample and other contributions where pulse A interacts once and pulse B twice. Moreover, the $\chi^{(3)}$ response can be separated into non-rephasing (NR) and rephasing (R) contributions [@woerner2013ultrafast; @lu2017]. The R signal arises due to a reverse phase accumulation during time $t$ as compared to $\tau$ and thus occurs at negative frequencies of either $\nu_t$ or $\nu\tau$ when compared with the NR signal. The different non-linear signals in each quadrant in the 2D frequency plan can be understood in terms of ‘frequency vectors’ as outlined in [@woerner2013ultrafast; @wan2019resolving] and the SI. For the case where pulse A precedes B and pulse B has two sample interactions (AB scheme), the $E_{NL}$ signal in the 4th quadrant is the ‘photon echo’ R contribution. Within a picture where excitations are resonant pairs, its anti-diagonal widths are a measure of the decoherence rates ($\Gamma_2 = 1/T_2$) [@Mukamel1995; @hamm2011concepts; @TomakoffNotes]. The strong signal along the diagonal in the 1st quadrant is a pump-probe (PP) contribution from pulse B interacting twice from the sample and arriving before A (BA scheme). It is sensitive to decay of the excited state populations and its anti-diagonal width is, within the pair approximation, a measure of the energy relaxation rate ($\Gamma_1 = 1/T_1$) at the excitation frequency of the projection onto either axis. Please see the SI for a detailed description of the full 2DCS response of a generic two-level system subject to finite longitudinal and transverse relaxation rates $1/T_1$ and $1/T_2$.
{width="12cm"}
As can be seen in Fig. 2, the signals shift towards lower frequencies on approaching the MIT. The echo signal is most apparent for the least doped (39%) sample (Fig. 2a). Similarly, the PP streak along the 1st quadrant diagonal narrows (decreasing $\Gamma_{1}$) with increasing doping. To quantify the relaxation rates $\Gamma_{1,2}$, we take cuts along the anti-diagonal in both quadrants. In so doing, we can get the relaxation rates as a function of energy. Fig. \[frequencydependence\]a shows representative cuts of the 39$\%$ sample taken along the green and purple dashed lines in Fig. 2a. The cuts in each case can be well fit to a single Lorentzian to extract the FWHM as a measure of the relaxation rates. We plot the frequency dependencies of the relaxation rates in Fig. \[frequencydependence\]b and the doping dependencies in Fig. \[frequencydependence\]c. The relative widths of $\Gamma_{1,2}$ easily satisfy the fundamental relation for “magnetic" resonance with $2/T_2 \geq 1/ T_1$. The frequency dependence of $\Gamma_{1}$ is shown in Fig. \[frequencydependence\]d at different doping levels. Note that the x-axis in Fig. 3b and Fig. 3d is the frequency $\nu_t$ at which the anti-diagonal cuts peak in Fig. 2. Due to low signal it was challenging to extract $\Gamma_{2}$ over the full doping range for all samples.
One can see in Fig. \[frequencydependence\]b and d that the relaxation rates are roughly linear in excitation frequency in the sub-THz regime and are consistent with an extrapolation to zero in the limit of zero frequency. Qualitatively, this frequency dependence is reminiscent of the behavior of the relaxation rate as a function of energy for a metal in the sense that they go to zero as $\omega \rightarrow 0$ (e.g., as $E \rightarrow E_F$) as the phase space for electronic relaxation collapses. However, the quantitative dependence is different as the relaxation rates are linear in frequency with a slope close to unity. We also find that the doping dependence (Fig. \[frequencydependence\]c) is such that the relaxation rates [*decrease*]{} as we approach the transition to the metallic phase. Fig. \[tempdependence\] shows the temperature dependence of the relaxation rates. Over a range of temperatures from $5K$ to $25K$, $1/T_2$ does not change at all to within experimental uncertainty, while the $1/T_1$ relaxation rate actually [*decreases*]{} with increasing temperature.
What can be inferred from these results? The temperature dependence in Fig. \[tempdependence\] and the frequency dependence in Fig. \[frequencydependence\] rule out phonons as a dominant relaxation channel. Relaxation from phonons is known (see e.g. [@spectraldiffusion]), to lead to relaxation rates that are increasing functions of temperature and with frequency dependence (see SI) that goes as $\omega^3$ at low $\omega$. Thus, the electronic system can be considered well isolated on the timescales of interest with coupling to phonons unimportant for this relaxation. We further note that the temperature dependence of the energy relaxation rate $1/T_1$ (Fig.\[tempdependence\]a) has the opposite sign from what one might naively expect - the relaxation rate [*decreases*]{} as we increase temperature. This also rules out explanations based on spectral diffusion (see SI), but can be naturally explained if we postulate that $T_1$ comes from the interaction mediated coherent tunneling of electron-hole excitations (see SI). Raising the temperature increases screening and suppresses coherent tunneling [@fisherzwerger].
The doping dependence of the relaxation rate (Fig. \[frequencydependence\]c) provides further insight. This behavior is counterintuitive: relaxation slows as we approach the metallic phase. We argue that in fact, it is evidence that electron-electron interactions dominate relaxation. This follows from essentially dimensional considerations. Microscopically, the system consists of randomly placed P atoms; electrons hop between these atoms, and repel each other via the Coulomb interaction. The system is doped toward the MIT by increasing the density of P atoms. Since the hopping is exponentially suppressed in the P-P distance and the Coulomb interactions are only algebraically suppressed, increasing the density *decreases* the ratio of interactions to hopping, and thus makes the system effectively more weakly interacting, causing the quasiparticle lifetimes to increase.
The linear in frequency relaxation rate with a slope close to unity is a dependence reminiscent of strongly correlated metals that exhibit the marginal Fermi liquid phenomenology [@varma2002singular; @varma1989phenomenology] and here demonstrates something similar; particle-hole excitations are only marginally well-defined in the relevant frequency range. That the relaxation rates appear to extrapolate to zero in the zero frequency limit, despite the fact that the system is at finite temperature, likely reflects the fact that we are probing energy scales larger than the thermal scale; the smallest frequency probed ($\approx \SI{5}{THz}$) corresponds to a temperature of $\approx \SI{24}{K}$ and the data in Fig.\[frequencydependence\](a) is taken at $\SI{5}{K}$. It is likely that the relaxation rates saturate to a non-zero value at a frequency lower than we can probe due to the finite temperature of the experiment.
We now sketch a mechanism that can give the linear in $\omega$ dependence (see SI for details). The low-frequency excitations above the localized state are resonant particle-hole excitations, in which a particle is moved between two nearby localized orbitals. These excitations are local electric dipoles, and thus naturally interact via $1/R^3$ dipole-dipole interactions [@burin1994low; @gutman]. As discussed by Levitov [@levitov1989absence], dipolar interactions are known to cause delocalization in three dimensions. Coulomb interactions parametrically enhance the density of dipoles at low frequencies [@shklovskii1981phononless], through a blockade effect; even if two nearby sites have on-site energies below $E_F$, occupying one of them may push the other site above $E_F$. These local anticorrelations among occupation numbers give a phase space of particle hole excitations that is $\omega$-independent at low frequencies (cf. Fermi liquids, where this phase space scales as $\omega$). A dipolar excitation can coherently hop on this network, at a rate one can calculate (see SI) to be $\sim \omega$. This mechanism also has temperature and doping dependence consistent with the earlier dimensional analysis. It is important to point out that our experiment is not in the regime far from the MIT where the Shklovski-Efros-Levitov (S.E.L.) calculation is well controlled (see SI). Nevertheless it remains possible that the experimental results are quantitatively explicable via some nontrivial extension of its central ideas to systems near the MIT. Regardless of the precise mechanism, the relaxation comes from the interplay of strong electron electron interactions with strong disorder, in a regime where controlled analytic calculation does not appear feasible. It should be noted that in the absence of the Efros-Shklovskii ground state reconstruction [@shklovskii1981phononless], the Levitov argument [@levitov1989absence] would predict relaxation rates that scale as $\sim \omega^2$, implying sharply defined low energy excitations (see SI). One needs both it and the long-range dipolar interaction to get the linear in $\omega$ relaxation.
Finally, we note that while discussions of Fermi liquid theory are usually couched in terms of the lifetime of single electrons ‘injected’ above a filled Fermi sea, the lifetimes we are measuring here are those of elementary ‘dipoles’ (particle-hole excitations). In a Fermi liquid these relaxation rates scale the same way; in disordered systems they generally do not (see SI). Nevertheless, in the microcanonical ensemble (relevant for optical experiments, where we do inject particles), particle hole excitations are the low-energy excitations of the system, and the marginality of their lifetime is the key diagnostic of the marginal Fermi glass[^1].
To conclude, we have examined the energy relaxation rate and photon echo decay rates in P doped Si using THz 2DCS. We have discovered a host of surprising features, including relaxation rates that are linear functions of frequency with slope close to unity; echo decay rates that are temperature independent within experimental precision; energy relaxation rates that are [*decreasing*]{} functions of temperature over the range probed; and a doping dependence such that the relaxation rates [*decrease*]{} as we approach the transition to the metallic phase. We have argued that these results indicate that (i) the system is behaving as a well isolated electronic system, with coupling to phonons negligible on experimental timescales (ii) the relaxation is dominated by [*electron-electron interactions*]{} and arises through the interplay of strong interactions and strong disorder. By analogy with the ‘marginal Fermi liquid’ behavior observed in strongly correlated metals, we dub the phase characterized by our experiments a ‘marginal Fermi glass.’
[**Methods.**]{} Experiments were performed on nominally uncompensated phosphorous-doped silicon (Si:P) samples, which were cut from a Czochralski grown boule to a specification of in diameter with a P-dopant gradient along the axis. This boule was subsequently sliced and then polished down to wafers. Samples from this boule were previously used for studies of the THz-range conductivity in the phononless regime [@Helgren04a; @Helgren02a] and optical pump-THz probe measurements [@thorsmolle2010ultrafast]. We measured samples from 39-85$\%$ of the critical concentration of the 3D metal-insulator transition (MIT) in a regime where the localization length was of order or longer than the inter-dopant spacing. Note that these concentrations are far higher than used in THz free electron laser (FEL) studies demonstrating photon echo at the $1s \rightarrow 2p_0$ transition ($\sim \SI{8.29}{THz}$) [@greenland2010coherent; @lynch2010first]. P concentrations were calibrated with the room temperature resistivity using the Thurber scale [@Thurber80a].
To perform 2D non-linear THz spectroscopy, two intense THz pulses (A and B) generated by the tilted pulse front technique and separated by a time-delay $\tau$ (Fig. 1b) are focused onto each sample in a collinear geometry (see SI for details of the experimental setup) [@woerner2013ultrafast; @lu2017]. The transmitted THz fields were detected by standard electro-optic (EO) sampling using a , pulse that is delayed by time $t$ relative to pulse B. Displayed data was taken with a maximum electric field of for each pulse. A differential chopping scheme is used to extract the non-linear signal ($E_{NL}(\tau,t) = E_{AB}(\tau,t) - E_{A}(\tau,t) - E_{B}(t)$) resulting from the interaction of the two THz pulses with the sample. Here $E_{AB}$ is the transmitted signal when both THz pulses are present while $E_{A}$ and $E_{B}$ are the transmitted signals with each pulse A and B present individually. A 2D Fourier transform of $E_{NL}$ with respect to $\tau$ and $t$ gives the complex 2D spectra as a function of the frequency variables $\nu_\tau$ and $\nu_t$. Fits to Lorentzians were restricted to the central part of peaks due to the phase twisting present in these spectra as discussed in the SI.
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{#section .unnumbered}
We would like to thank Y. Galperin, Y.-B. Kim, A. Legros, I. Martin, A. Millis, V. Oganesyan, B. Shklovskii, and Y. Yuan for helpful discussions. This project was supported by a now canceled DARPA DRINQS program grant. S.G. acknowledges support from NSF Grant No. DMR-1653271.
{#section-1 .unnumbered}
FM and DC built the 2D THz setup and did experiments and analysis. SG and RN gave theoretical support. NPA directed the project. All authors contributed to the writing and editing of the manuscript.
{#section-2 .unnumbered}
**Competing financial interests:** The authors declare no competing financial interests.\
**Data availability:** All relevant data are available on reasonable request from NPA.
****\
Experimental setup
==================
To implement THz 2D coherent spectroscopy (2DCS), intense THz pulses were generated by the tilted pulse front technique by optical rectification in LiNbO$_3$ crystals [@hirori2011single; @woerner2013ultrafast; @Lu2017]. 800 nm laser pulses from an Astrella one-box Ti:Sapphire amplifier system (1 kHz, 30 fs, 7 mJ/pulse) were separated via beam splitters (BS) into two roughly identical beams, A and B. A time-delay $\tau$ was introduced between the two using a mechanical linear translation stage. Each of the laser pulses were then directed onto a diffraction grating to generate the requisite pulse front tilt for optimal phase matching. The output pulse from the grating was imaged onto the 0.6$\%$ MgO doped LiNbO$_3$ crystals to generate THz pulses. The two beams were combined using a wire-grid polarizer and then focused onto the samples in a collinear geometry with four parabolic mirrors in an 8f configuration as shown in Supplementary Fig. \[Schematic\]. The transmitted THz fields were detected by standard electro-optic (EO) sampling in ZnTe using a third 30-fs 800 nm pulse that is delayed by time $t$ relative to pulse A. Peak electric fields on the sample are estimated to be $\sim$ 50 kV/cm for each pulse.
A differential chopping scheme is used to extract the non-linear signal defined as $E_{NL}(t,\tau) = E_{AB}(t,\tau) - E_{A}(t,\tau) - E_{B}(t)$ that results from the interaction of the two THz pulses with the sample. Here $E_{AB}$ is the transmitted signal when both THz pulses are present while $E_{A}$ and $E_{B}$ are the transmitted signals with each pulse A and B present individually. Fig. 1c-f in the main text shows $E_A$, $E_B$, $E_{AB}$, and $E_{NL}$ as a function of $t$ and $\tau$ for the 39% sample. A 2D Fourier transform of $E_{NL}$ with respect to $t$ and $\tau$ gave the complex 2D spectra as a function of the frequency variables $\nu_A$ and $\nu_B$.
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2D nonlinear response for a single two level system
===================================================
As discussed in the main text and in Sec. \[dipoleham\], it is believed that one can understand the optical excitations of an electronic glass as an ensemble of random two-level systems. Although there may be aspects of the current data that are beyond this simple picture, it is instructive to work out the theoretical expectation for the THz 2DCS response for a single idealized two-level system with finite longitudinal and transverse relaxation rates $1/T_1$ and $1/T_2$. Related treatments exist in Refs. [@hamm2011concepts; @Mukamel1995; @hamm2005principles]. In addition to being directly relevant to the particular physical system we are considering, this exposition should be regarded as an elementary introduction to THz 2DCS in general.
In all optical spectroscopies, one measures an emitted field in response to a time-dependent perturbing electromagnetic field. The emitted field is caused by a time-dependent polarization and the sample response characterized by a frequency dependent complex susceptibility. The emitted field has a 90$^\circ$ phase lag from a sample’s macroscopic polarization. In linear response this 90$^\circ$ phase lag combines with the part of the polarizability (the imaginary part) that has a 90$^\circ$ phase lag to the driving field to give an emitted field that is 180$^\circ$ out of phase with the driving field and hence destructive interference. This gives the phenomena of absorption, which is why the imaginary part of the polarization is associated with dissipation. This is a natural way to analyze the linear response. However in the nonlinear regime, rather than considering a material as a system that absorbs photons, it is more illuminating to think of a time dependent polarization as a [*source*]{} of electromagnetic radiation. Therefore, our task in describing the nonlinear response is to analyze the time-dependent polarization. The time-dependent polarization is equal to the expectation value of the dipole operator e.g. $P(t) = \langle \mu(t) \rangle$. We use the “Mukamelian" or perturbative expansion of the density matrix [@Mukamel1995] to model the time dependent dipole operator and eventually the nonlinear response of a two-level system.
Preliminaries
-------------
The expectation value of any operator $\hat{A}$ can be expressed as trace of the density matrix multiplied by that operator. The density matrix $\rho = |\Psi \rangle \langle \Psi| $ expanded in terms of basis states (typically eigenstates of $ \hat{H}_0 $ below) e.g. $|\Psi \rangle = \sum_{n} c_n |n\rangle$ is
$$\rho = |\Psi \rangle \langle \Psi| = \sum_{nm} c_n c_m^* |n\rangle \langle m|.$$
Then for instance the expectation value for the dipole operator $ \langle \Psi(t)| \mu | \Psi(t)\rangle$ is
$$\begin{aligned}
\langle \mu \rangle & = \sum_m c^*_m \sum_n c_n \langle m| \hat{\mu} |n \rangle \\
& = \sum_{mn} c^*_m c_n \mu_{mn} \\
& = \sum_{mn} \rho_{nm} \mu_{mn} = \mathrm{Tr}(\hat{\rho} \hat{\mu}) \\
& = \langle \hat{\rho} \hat{\mu} \rangle.
\label{PolarExpect}\end{aligned}$$
Therefore the time dependent polarization is given by the time-dependence of the density matrix. The time evolution of the density matrix is given by its commutator with the system Hamiltonian e.g.
$$\frac{d \rho}{dt } = - \frac{i}{\hbar} [ \hat{H},\rho].
\label{Liouville}$$
This follows from the fact that the time evolution of the density matrix is
$$\frac{d \rho}{dt} = \frac{d}{dt} ( |\Psi \rangle \langle \Psi| ) = ( \frac{d }{dt} |\Psi \rangle) \langle \Psi| + |\Psi \rangle (\frac{d }{dt} \langle \Psi| ) ).
\label{timeevolve}$$
Using the Schroedinger equation that gives the time evolution of $|\Psi \rangle$ e.g. $ \frac{d }{dt} |\Psi \rangle = - \frac{i}{\hbar} \hat{H} |\Psi\rangle $ (and its complex congugate) and plugging into Eq. \[timeevolve\], one obtains the so-called Liouville-von Neumann equation that is written as
$$\begin{aligned}
\frac{d }{dt} \rho &= - \frac{i}{\hbar} \hat{H} |\Psi\rangle \langle \Psi | + \frac{i}{\hbar} |\Psi\rangle \langle \Psi | \hat{H} \\
&= - \frac{i}{\hbar} \hat{H} \rho + \frac{i}{\hbar} \rho \hat{H} \\
&= - \frac{i}{\hbar} [\hat{H},\rho],\end{aligned}$$
which is the same as Eq. \[Liouville\] above. The essential point is that the time dependence of $\rho$ follows from the Hamiltonian acting from both the right [*and*]{} left sides of the density matrix.
Linear Response
---------------
We will use this density matrix formalism in a number of different ways. First let us describe two-level system’s linear response. Consider a total Hamiltonian
$$\hat{H}(t) = \hat{H}_0 + \hat{W}(t)
\label{FullHamiltonian}$$
where $ \hat{H}_0$ is the time-independent part of the system Hamiltonian and $\hat{W}(t)$ is an interaction with a time dependent field. Here and below we assume that the set of basis states used for the density matrix $|n\rangle$ diagonalizes the system Hamiltonian $\hat{H}_0$ as
$$|\Psi \rangle = c_0 e^{-i \omega_0 t } | 0 \rangle + i c_1 e^{-i \omega_1 t } | 1 \rangle .$$
For reasons that will be useful below, we have chosen the $c_
n$ coefficients such that a factor of $i$ is included in the definition of the wavefunction with $c_1$. In the absence of dephasing and population relaxation the elements of the density matrix are
$$\rho = \left(\begin{array}{cc}\rho_{00} & \rho_{01} \\ \rho_{10} & \rho_{11}\end{array}\right) = \left(\begin{array}{cc}c_0^2 & - i c_0c_1e^{i\omega_{01}t} \\ i c_0c_1e^{-i\omega_{01}t} & c_1^2\end{array}\right),$$
where $\omega_{01} = \omega_{1} - \omega_{0} $. In the presence of dephasing and population relaxation standard Bloch sphere dynamics gives the time evolution of the density matrix as
$$\rho = \left(\begin{array}{cc}c_0^2 + c_1^2(1 - e^{-t/T_1} ) & - i c_0c_1e^{i\omega_{01}t} e^{-t/T_2} \\ i c_0c_1e^{-i\omega_{01}t}e^{-t/T_2} & c_1^2 e^{-t/T_1} \end{array}\right),
\label{TimeEvolve}$$
which describes a precessing Bloch vector. As always the homogeneous dephasing and population relaxation are related by $1/T_2 = 1/2T_1 + 1/T_2^*$ where $T_2^*$ is pure dephasing time that may be caused by fluctuations of the environment.
The interaction of the light pulse with the sample is accounted by the term $\hat{W}(t) = - \hat{\mu} E(t)$ in Eq. \[FullHamiltonian\]. $E(t)$ is a real valued field and the transition dipole operator is
$$\hat{\mu} = \left(\begin{array}{cc} 0 & \mu_{01} \\ \mu_{10} & 0 \end{array}\right).$$
We use a perturbative expansion of the Liouville-von Neumann equation to account for the time dependence of the density matrix under the influence of $\hat{W}(t)$. In this analysis we use the [*semi-impulsive*]{} limit, where the light pulses are assumed to be short compared with any time scale of the system but long compared to the oscillation period of the light field. Therefore we describe the light field by an expression where envelopes of the pulses are approximated by $\delta$-functions, but we retain oscillations e.g.
$$E(t) = | \varepsilon | \delta(t) \; \mathrm{cos}(\omega t) = \frac{| \varepsilon |}{2} \delta(t) (e^{i \omega t} + e^{ - i \omega t} )
\label{impulse}$$
where $|\varepsilon|\delta(t)$ is the electric field envelope function and $\omega$ is the carrier frequency. Substituting Eq. \[impulse\] into the $\hat{W}(t)$ part of the Hamilitonian and integrating Eq. \[Liouville\] with respect to time one gets the expression for the correction to the density matrix after one instantaneous interaction with the light pulse
$$\rho^{(1)} \approx \frac{i}{\hbar} \frac{ | \varepsilon|}{2} (\mu(0)\rho(-\infty) - \rho(-\infty) \mu(0) ).
\label{correction}$$
This represents a system with a density matrix representing the ground state $\rho(-\infty)$ interacting with a light pulse at time zero. Note that the dipole operator respresenting the light pulse operates from both the left and the right sides of the density matrix e.g. the ket and bra sides and hence that the two terms in Eq. \[correction\] are complex congugates of each other. Also note that in actuality [*four*]{} terms exist in the interaction of the light field with the ground state (the two complex conjugate terms representing the light field multiplied by the two terms of the Liouville-von Neumann equation) however only the two in Eq. \[correction\] are effectively non-zero as transitions can only be made to higher energies from the ground state. This means that the $\varepsilon $ term of the electric field makes transitions only only the ket side of the density matrix whereas the $\varepsilon^* $ term makes transitions on the bra side. Dropping the other two terms is known as the rotating wave approximation. After interaction with the light pulse, the full density matrix has the form
$$\rho(0^+) = \rho^{(0)} + \rho^{(1)} = \left(\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array}\right) + \frac{i}{\hbar} \frac{ | \varepsilon|}{2 } \left(\begin{array}{cc} 0 & - \mu_{01} \\ \mu_{01} & 0 \end{array}\right) .
\label{correction2}$$
The system’s density matrix then evolves freely in a manner given by Eq. \[TimeEvolve\], which describes an oscillating Bloch vector
$$\rho(t) = \rho^{(0)} + \rho^{(1)} = \left(\begin{array}{cc}1 & 0 \\ 0 & 0 \end{array}\right) + \frac{i}{\hbar} \frac{ | \varepsilon|}{2 } \left(\begin{array}{cc} 0 & - \mu_{01} e^{i\omega_{01}t}e^{-t/T_2} \\ \mu_{01} e^{-i\omega_{01}t}e^{-t/T_2} & 0 \end{array}\right) .
\label{correction3}$$
An oscillating Bloch vector will emit an electromagnetic field and so it is clear that only the time dependent part of Eq. \[correction3\] e.g. $\rho^{(1)}$ will give an emitted wave. The light emission is described by another interaction with the dipole operator at a time $t$ and via Eq. \[PolarExpect\] the emitted field is given by $\langle P^{(1)}(t) \rangle = \mathrm{Tr}(\rho(t)\mu)$. When evaluated, this expression gives
$$P^{(1)}(t) = \varepsilon \frac{ \mu_{01}^2}{\hbar}e^{-t/T_2} \; \mathrm{sin} (\omega_{01} t).
\label{FID}$$
This is the so-called “free induction decay" in the semi-impulsive limit. When the radiation from this time-dependent polarization is added to the original incoming signal $E(t)$, destructive interference takes place and less radiation will be transmitted at a frequency $\omega_{01}$ resulting in absorption. The Fourier transform of Eq. \[FID\] is
$$P^{(1)}(\omega) = \varepsilon \frac{ \mu_{01}^2}{\hbar} \frac{\omega_{01} }{ \omega_{01}^2 + 1/T_2^2 - \omega^2 - i 2 \omega /T_2 }.
\label{FIDomega}$$
As as a delta function impulse has a flat frequency profile with an electric field component $\varepsilon $ at each frequency, the Fourier transform of a response to a delta function impulse is equivalent to a response function. So therefore the linear response defined as $P(\omega) = \chi^{(1)} E(\omega)$ (with $\varepsilon = E(\omega) $) is
$$\chi^{(1)}(\omega) = \frac{ \mu_{01}^2}{\hbar} \frac{\omega_{01} }{ \omega_{01}^2 + 1/T_2^2 - \omega^2 - i 2 \omega /T_2 }.
\label{LinearResponse}$$
This is the well known functional form for a Drude-Lorentz oscillator.
Non-Linear Response
-------------------
The above perturbative expansion of density matrices is – frankly speaking – a tedious method of calculating linear response. However, it is a very powerful method to calculate the [*nonlinear*]{} response. To calculate the the nonlinear response we proceed in analogous fashion starting from the time-dependent density matrix, but extending the formalism to multiple pulses. For the third order response an analogous scheme is used e.g. semi-impulsive excitation of a ground state density matrix by the first pulse at $t=0$, free evolution of the density matrix for a time $t_1$ and then semi-impulsive excitation by the second light pulse, free evolution of the density matrix for a time $t_2$ and then semi-impulsive excitation by the third light pulse, and then emission that is described by interaction with a fourth dipole operator after an additional time $t_3$ (See Fig. \[Pulses\]). Again we describe the interaction of light at each step via the Liouville-von Neumman formalism using Eq. \[correction\], which expresses the fact that the dipole operator operates on both the ket and bra sides of the density matrix. The third order response can be expressed compactly in terms of nested commutators as
$$\langle P^{(3)} \rangle = - i \frac{| \varepsilon_2 \varepsilon_1 \varepsilon_0 |}{8 \hbar^3} \langle \mu(t_3 + t_2 + t_1) [\mu(t_2 + t_1),[\mu(t_1),[\mu(0), \rho(-\infty)]]] \rangle.
\label{NL}$$
Here our notation is such that $\mu(t)$ refers to action of the dipole operator at a time $t$. It does not explicitly refer to the time dependence of the operator e.g. we are still using the Schroedinger picture for time-evolution. The commutators can be expanded as $$\begin{aligned}
&\langle \mu_3[\mu_2,[\mu_1,[\mu_0,\rho(-\infty) ]]]\rangle = \nonumber \\
& \; \; \;\; \; \; \langle \mu_3\mu_1 \rho(-\infty) \mu_0 \mu_2 \rangle - \langle \mu_2\mu_0 \rho(-\infty) \mu_1 \mu_3 \rangle \; + &| \; \; \; R_1 + R_1^* \nonumber \\
& \; \; \; \; \; \; \langle \mu_3\mu_2 \rho(-\infty) \mu_0 \mu_1 \rangle - \langle \mu_1\mu_0 \rho(-\infty) \mu_2 \mu_3 \rangle \; + &| \; \; \; R_2 + R_2^* \nonumber \\
&\; \; \; \; \; \; \langle \mu_3\mu_0 \rho(-\infty) \mu_1 \mu_2 \rangle - \langle \mu_2\mu_1 \rho(-\infty) \mu_0 \mu_3 \rangle \; + &| \; \; \; R_4 + R_4^* \nonumber \\
& \; \; \; \; \; \; \langle \mu_3\mu_2 \mu_1 \mu_0 \rho(-\infty) \rangle - \langle \rho(-\infty) \mu_0\mu_1 \mu_2 \mu_3 \rangle . \; &| \; \; \; R_5 + R_5^*
\label{NLexpanded}\end{aligned}$$ The terms $R_1$, $R_1^*$, etc. denote 3rd order response functions to semi-impulsive driving fields and represent different Liouville pathways for excitation e.g. different unique sequences of manipulating the density matrices from ket and bra sides that can give emission[^2]. Here, we have used the notation of Ref. [@hamm2011concepts] with regards to the dipole operators and the response functions[^3].
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We now consider the evolution of the full density matrix with first order corrections at each step as it is affected by the light pulses. We illustrate this for $R_4$ and for notational simplicity here we ignore the changes to the density matrix during times $t_1$, $t_2$, and $t_3$. We let $\beta_n = \frac{i}{\hbar} \frac{\varepsilon_n}{2} \mu_{01}$ and get $$\begin{aligned}
R_4 \; | \; \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right) \xrightarrow{\mu_0 \rho} & \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right) + \left(\begin{array}{cc} 0 & 0 \\ \beta_0 & 0 \end{array}\right) \xrightarrow{\rho \mu_1} \left(\begin{array}{cc} 1 & 0 \\ \beta_0 & 0 \end{array}\right) + \left(\begin{array}{cc} 0 & \beta_1 \\ 0 & \beta_0 \beta_1 \end{array}\right) \xrightarrow{\rho \mu_2} \left(\begin{array}{cc} 1 & \beta_1 \\ \beta_0 & \beta_0 \beta_1 \end{array}\right) + \left(\begin{array}{cc} \beta_1 \beta_2 & \beta_2 \\ \beta_0 \beta_1 \beta_2 & \beta_0 \beta_2 \end{array}\right) \xrightarrow{\mu_3 \rho } \nonumber \\ &
\left(\begin{array}{cc} 1 + \beta_1 \beta_2 & \beta_1 + \beta_2 \\ \beta_0 + \beta_0 \beta_1 \beta_2 & \beta_0 \beta_1 + \beta_0 \beta_2 \end{array}\right) +
\left(\begin{array}{cc} \beta_0 \beta_3 + \beta_0 \beta_1 \beta_2 \beta_3 & \beta_0 \beta_1 \beta_3 + \beta_0 \beta_2 \beta_3 \\ \beta_3 + \beta_1 \beta_2 \beta_3 & \beta_1 \beta_3 + \beta_2 \beta_3 \end{array}\right).
\label{FullDM}\end{aligned}$$ Note that only the diagonal elements of the second term on the bottom line of Eq. \[FullDM\] contain dependencies on the radiation dipole operator $\beta_3$ that gives an external field. The trace of this matrix gives a quantity proportional to the reradiated field. If one applies the protocol considered in the text and discussed above where one analyzes only the [*nonlinear*]{} signal defined as $E_{NL} = E_{AB}$ - $E_{A}$ - $E_{B}$ this subtracts off the diagonal parts of the density matrix (e.g. $\beta_1 \beta_3$) that gives the linear response contributions to the reradiating field. The residual contains only effects of the $\chi^{(3)}$ nonlinear susceptibility. Therefore going forward, we can analyze the $\chi^{(3)}$ response by considering only the repeated operation of the dipole operator on the [*corrections*]{} to the density matrices after each field interaction.
Here we illustrate the evolution of the density matrix for the response function $R_1$, by now considering the full time dependencies, but at each step of interaction with the light pulse or time evolution retaining only the first order correction to the density matrix at each step. This gives
$$\begin{aligned}
R_1 \; | \; &\left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right) \xrightarrow{\rho(t) \mu_{0} } \frac{ |\varepsilon_0|}{2} \left(\begin{array}{cc} 0 & i \frac{ \mu_{10} }{\hbar} \\ 0 & 0 \end{array}\right) \xrightarrow{ t_1} \frac{ |\varepsilon_0|}{2} \left(\begin{array}{cc} 0 & i \frac{ \mu_{10} }{\hbar} e^{i \omega_{01} t_1 } e^{-t_1/T_2} \\ 0 & 0 \end{array}\right) \xrightarrow{\mu_1 \rho(t) } \frac{|\varepsilon_0 \varepsilon_1|}{4} \left(\begin{array}{cc} 0 & 0\\ 0 & - \frac{ \mu_{10}^2 }{\hbar^2} e^{i \omega_{01} t_1 } e^{-t_1/T_2} \end{array}\right) \xrightarrow{t_2}\nonumber \\ & \;\;\;\;\;\;\;\;\;\; \frac{|\varepsilon_0 \varepsilon_1|}{4} \left(\begin{array}{cc} 0 & 0\\ 0 & - \frac{ \mu_{10}^2 }{\hbar^2} e^{i \omega_{01} t_1 } e^{-t_1/T_2} e^{-t_2/T_1} \end{array}\right) \xrightarrow{\rho(t) \mu_2} \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{8} \left(\begin{array}{cc} 0 & 0\\ - i \frac{ \mu_{10}^3 }{\hbar^3} e^{i \omega_{01} t_1 } e^{-t_1/T_2} e^{-t_2/T_1} & 0 \end{array}\right) \xrightarrow{t_3} \nonumber \\ & \;\; \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{8} \left(\begin{array}{cc} 0 & 0\\ - i \frac{ \mu_{10}^3 }{\hbar^3} e^{i \omega_{01} (t_1 - t_3 ) } e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} & 0 \end{array}\right) \xrightarrow{\mu_3 \rho(t)} \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{8} \left(\begin{array}{cc} - i \frac{ \mu_{10}^4 }{\hbar^3} e^{i \omega_{01}( t_1 - t_3 ) } e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} & 0 \\ 0 & 0 \end{array}\right).\end{aligned}$$
Here $\mu_n$ represents the action of the electric field pulse at the time $t_n$. When combined with the complex conjugated term $R_1^*$ one gets the $R_1 + R_1^*$ contribution to the total response in the semi-impulsive limit as
$$P^{(3)}_{R_1 + R_1^*}(t_1,t_2,t_3) = \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} \mathrm{sin}\; [ \omega_{01}( t_1 - t_3 ) ].\label{R1}$$
The other terms in the expansion of the density matrix can be evaluated in a similar fashion by considering each term in Eq. \[NLexpanded\]. In so doing, one finds that $P^{(3)}_{R_1 + R_1^*} = P^{(3)}_{R_2 + R_2^*} $ and that $P^{(3)}_{R_4 + R_4^*} = P^{(3)}_{R_5 + R_5^*} $. The response for $R_4 + R_4^*$ is
$$P^{(3)}_{R_4 + R_4^*}(t_1,t_2,t_3) = - \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} \mathrm{sin}\; [ \omega_{01}( t_1 + t_3 ) ]. \label{R4}$$
Note the different time dependencies of the oscillating term on $t_1$ and $t_3$ in Eqs. \[R1\] and \[R4\]. The coherent time evolution of the state is in opposite directions for times $t_1$ and $t_3$ for $R_1 + R_1^*$, whereas they are in the same direction for $R_4 + R_4^*$. In the former case, this is known as rephasing and gives the phenomena of photon echo for response $R_1 + R_1^*$ (and $R_2 + R_2^*$) making it the most important spectroscopic contribution. $R_4 + R_4^*$ and $R_5 + R_5^*$ are known as non-rephasing contributions.
In an actual experiment one does not typically use three pulses. Instead two pulses (A and B) are used and then with the scheme of Fig. \[Pulses\] either $t_1 \rightarrow 0$ or $t_2 \rightarrow 0 $. In the below, we assume that pulse A arrives at a time zero, pulse B arrives at a later time $\tau$, and the electric fields are measured at time $t$ that is measured in reference to pulse B. We refer to this as the AB pulse sequence. One has then four separate contributions that corresponds rephasing and non-rephasing versions of the first pulse giving two field interactions or the second pulse giving two field interactions. They are
$$\begin{aligned}
P^{(3)}_{R_1 + R_1^*}(\tau,t:AB) =&\; \; \; \; \frac{| \varepsilon_A^2 \varepsilon_B |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-\tau/T_1}e^{-t/T_2} \mathrm{sin}\; [ \omega_{01}( - t ) ] &t_1 \rightarrow 0 & \; \; \; \; \; \; \mathrm{AB \; Pump-probe} \label{ExperimentalSignal1} \\
P^{(3)}_{R_4 + R_4^*}(\tau,t:AB) =& - \frac{| \varepsilon_A^2 \varepsilon_B |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-\tau/T_1}e^{-t/T_2} \mathrm{sin}\; [ \omega_{01}( t ) ] &t_1 \rightarrow 0 & \; \; \; \; \; \; \mathrm{AB \; Pump-probe} \label{ExperimentalSignal2}\\
P^{(3)}_{R_1 + R_1^*}(\tau,t:AB) =& \; \; \; \; \frac{| \varepsilon_A \varepsilon_B^2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-(\tau + t)/T_2} \mathrm{sin}\; [ \omega_{01}( \tau - t ) ] &t_2 \rightarrow 0 & \; \; \; \; \; \; \mathrm{AB \; Rephasing} \label{ExperimentalSignal3}\\
P^{(3)}_{R_4 + R_4^*}(\tau,t:AB) =& - \frac{| \varepsilon_A \varepsilon_B^2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-(\tau +t)/T_2} \mathrm{sin}\; [ \omega_{01}( \tau + t ) ] . &t_2 \rightarrow 0 & \; \; \; \; \; \; \mathrm{AB \; Non-rephasing}
\label{ExperimentalSignal4}\end{aligned}$$
Note $t \geq 0$ and $\tau \geq 0$ for the equations above and by causality, $P^{(3)} = 0$ for $t < 0$ or $\tau < 0$. Here AB refers to the pulse sequence where pulse A arrives before pulse B. One can see that for $t_1 \rightarrow 0$, the $R_1 + R_1^*$ and $R_4 + R_4^*$ contributions become identical. This contribution which depends only on $\omega_{01}$ and $1/T_1$ is referred to as the pump-probe (PP) signal. The pump-probe $R_1$ contribution corresponds to a bra side action (e.g. operation from the right) of $\varepsilon^*_A$ on the density matrix, then immediate action of $\varepsilon_A$ on the ket side, then time evolution for a time $\tau$, then the action of $\varepsilon_B$ on the bra side of $\rho$, and then emission at a time $t$. This gives a response proportional to $(\varepsilon^*_A \varepsilon_A) \varepsilon_B$. The pump-probe $R_1^*$ contribution gives a ket side interaction with $\varepsilon_A$, and then immediate action of $\varepsilon^*_A$ on the bra side, then time evolution for a time $\tau$, then the action of $\varepsilon^*$ on the ket side of $\rho$, and then emission at a time $t$. This gives a response proportional to ($\varepsilon_A \varepsilon^*_A) \varepsilon^*_B$. The pump-probe $R_4$ corresponds to a ket side interaction with $\varepsilon_A$, an immediate action of $\varepsilon^*_A$ on the bra side, the time evolution for a time $\tau$, then (in contrast to $R_1^*$) the action of $\varepsilon^*$ on the [*bra*]{} side of $\rho$, and then emission at a time $t$. This gives a response proportional to $(\varepsilon_A \varepsilon^*_A) \varepsilon_B$, which will be the same as $R_1$.
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For $t_2 \rightarrow 0$, there are two distinct contributions. These are known as the rephasing (R) (Eq. \[ExperimentalSignal3\]) and non-rephasing (NR) (Eq. \[ExperimentalSignal4\]) signals. The rephasing $R_1$ contribution corresponds to a bra side action of $\varepsilon^*_A$ on the density matrix, then time evolution for a time $\tau$, then action of $\varepsilon_B$ on the ket side, then the immediate action of $\varepsilon_B$ on the bra side of $\rho$, and then emission at a time $t$. This gives a response proportional to $\varepsilon^*_A (\varepsilon_B \varepsilon_B)$. The rephasing $R_1^*$ contribution corresponds to a ket side action of $\varepsilon_A$ on the density matrix, then time evolution for a time $\tau$, then action of $\varepsilon^*_B$ on the bra side, then the immediate action of $\varepsilon^*_B$ on the ket side of $\rho$, and then emission at a time $t$. This gives a response that depends on $\varepsilon_A (\varepsilon^*_B \varepsilon^*_B)$. The non-rephasing $R_4$ contribution corresponds to a ket side action of $\varepsilon_A$ on the density matrix, then time evolution for a time $\tau$, then action of $\varepsilon^*_B$ on the bra side, then the immediate action of $\varepsilon_B$ on the bra side of $\rho$, and then emission at a time $t$, which gives a response that depends on $\varepsilon_A (\varepsilon^*_B \varepsilon_B)$. The non-rephasing contribution is also known as the perturbed free-induction decay [@kuehn2011two] as it can be see as the perturbation by pumping of a free-induction decay. These rather complicated dependencies and different Liouville pathways can be represented compactly in terms of “Feynman diagrams". We refer the interested reader to the literature [@hamm2011concepts] for their use and interpretation.
We plot these function and their sum in Fig. \[TimeTraces2D\]. Note that the different contributions have different characteristics with regards to the directions that oscillation occurs and that the signal decays. This is the essential utility of 2DCS. For the AB sequence the pump probe signals oscillate and shows $T_2$ decay in the $\hat{t}$ directions, but $T_1$ decay in the $\hat{\tau}$ direction. The rephasing contribution shows oscillations in the $(\hat{\tau} - \hat{t})/\sqrt{2}$ direction, but $T_2$ decay in the orthogonal $(\hat{\tau} + \hat{t})/\sqrt{2}$ direction. The non-rephasing contribution shows both its oscillations and $T_2$ decay in the $(\hat{\tau} + \hat{t})/\sqrt{2}$ direction. These differences are essential when considering the role of inhomogeneous broadening on the experimental signal and limits the usefulness of the NR signal (see discussion below) when multiple closely spaced oscillators exist.
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In the analysis of experimental data one typically takes the Fourier transforms of the experimental quantities. The Fourier transforms of the full PP and R signals as a function of $\tau$ and $t$ in the AB pulse sequence are
$$\begin{aligned}
P^{(3)}_{PP}(\omega_\tau,\omega_t:AB) =&
\frac{i}{2} | \varepsilon_A^2 \varepsilon_B | \frac{ \mu_{10}^4 }{\hbar^3} \frac{ 1 }{ 1/T_1 + i \omega_\tau } \; \Big[ \frac{1}{1/T_2 + i (\omega_t - \omega_{01})} - \frac{1}{1/T_2 + i (\omega_t + \omega_{01})} \Big] \nonumber \\
=& - | \varepsilon_A^2 \varepsilon_B | \frac{ \mu_{10}^4 }{\hbar^3} \frac{ 1 }{ 1/T_1 + i \omega_\tau } \; \Big[ \frac{\omega_{01} }{ \omega_{01}^2 + 1/T_2^2 - \omega_t^2 - i 2 \omega_t /T_2 } \Big]
\label{ExperimentalSignalFT2},\\
P^{(3)}_{R}(\omega_\tau,\omega_t:AB) =& \frac{-i}{4} | \varepsilon_A \varepsilon_B^2 | \frac{ \mu_{10}^4 }{\hbar^3} \Big[ \frac{1}{1/T_2 + i (\omega_\tau + \omega_{01})}\frac{1}{1/T_2 + i (\omega_t - \omega_{01})} - \frac{1}{1/T_2 + i (\omega_\tau - \omega_{01})} \frac{1}{1/T_2 + i (\omega_t + \omega_{01})} \Big] .
\label{ExperimentalSignalFT4}\end{aligned}$$
We plot these functions in Fig. \[FT\_2D\].
In the response functions above peaks are found for the PP signal in the AB pulse sequence at ($\omega_\tau = 0, \omega_t = \pm \omega_t$) with a $\hat{\omega}_\tau$ direction width set by $1/T_1$ and a $\hat{\omega}_t$ direction width set by $1/T_2$. The R signal is found at ($\omega_\tau = \mp \omega_{01}, \omega_t = \pm \omega_{01}$) with a width that is set by $1/T_2$ in the $(\hat{\omega}_\tau + \hat{\omega}_t)/\sqrt{2}$ direction and is infinitesimally narrow in the orthogonal direction. The NR signal is found at ($\omega_\tau = \pm \omega_{01}, \omega_t = \pm \omega_{01}$) and again is broadened in the $(\hat{\omega}_\tau + \hat{\omega}_t)/\sqrt{2}$ direction by $1/T_2$ and is not broadened at all in the orthogonal direction. Note that an intrinsic feature of 2DCS spectra in general and that can be seen explicitly in Eqs. \[ExperimentalSignalFT2\] and \[ExperimentalSignalFT4\] is “phase twisting." Each frequency axis gives a complex contribution to the response and the overall 2D response is the product of these complex contributions. Therefore as shown in Fig. \[FT\_2D\], neither the real or imaginary parts of the $\chi^{(3)}$ response correspond to purely absorptive spectra characterized by simple peaked lineshapes. They show more complicated mixed absorptive and dispersive character, which complicates their analysis. A phasing procedure used in previous THz 2DCS experiments [@kuehn2011two] to get purely absorptive lineshapes is challenging to implement here, because of the extreme inhomogeneous broadening and overlapping contributions in 2D frequency space. For this reason we chose to analyze the magnitudes of the nonlinear response in the experimental data which can be approximated with a Lorentzian lineshape to extract the widths $1/T_1$ and $1/T_2$. This is discussed in detail below.
In the treatment above, pulse A precedes pulse B. However in our experiment we scan pulse A through pulse B, such that we can acquire data where pulse B precedes pulse A. One can get the contribution of these to the experiment by the substitutions into Eqs. \[ExperimentalSignal1\] - \[ExperimentalSignal4\] of $t \rightarrow t + \tau$ and $\tau \rightarrow - \tau$. These time dependencies (plotted in Fig. \[TimeTraces2D\_BA\]) are
$$\begin{aligned}
P^{(3)}_{R_4 + R_4^*}(-\tau,t+\tau:BA) =& - \frac{| \varepsilon_B^2 \varepsilon_A |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{\tau/T_1}e^{-(t + \tau)/T_2} \mathrm{sin}\; [ \omega_{01}( t + \tau ) ] & & \; \; \; \; \; \; \mathrm{BA\;Pump-probe} \label{ExperimentalSignal2BA}\\
P^{(3)}_{R_1 + R_1^*}(-\tau,t+\tau:BA) =& \; \; \; \; \frac{| \varepsilon_B \varepsilon_A^2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{- t/T_2} \mathrm{sin}\; [ \omega_{01}( -2\tau - t ) ] & & \; \; \; \; \; \; \mathrm{BA\;Rephasing} \label{ExperimentalSignal3BA}\\
P^{(3)}_{R_4 + R_4^*}(-\tau,t+\tau:BA) =& - \frac{| \varepsilon_B \varepsilon_A^2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-t/T_2} \mathrm{sin}\; [ \omega_{01}( t ) ] . & & \; \; \; \; \; \; \mathrm{BA \;Non-rephasing}
\label{ExperimentalSignal4BA}\end{aligned}$$
Note $t \geq 0$ and $\tau \leq 0$ for the equations above and by causality, $P^{(3)} = 0$ for $t+\tau < 0$. The magnitude of Fourier transform of these functions are plotted in Fig. \[FT\_2D\_BA\]. Of course there is nothing fundamentally different in the BA sequence than the AB sequence. The differences are only in the labels of the timing of arrival of peaks and measurement.
Identification of the various contributions to the nonlinear response is made easier by the realization that a scheme of “frequency vectors" can be used to identify the particular contribution to the response [@kuehn2011two; @woerner2013ultrafast]. Given a weakly non-linear oscillator with frequency $\omega_{01}$, the frequency vectors that correspond to pulses A and B are $\vec{\omega}_A = \omega_{01}(\hat{\omega}_\tau + \hat{\omega}_t)$ and $\vec{\omega}_B = \omega_{01} \hat{\omega}_t$. Conjugated electric field pulses are represented by a reversed vector. Thus the PP contribution in the AB sequence can be placed by $\vec{\omega}_{PP:AB} = \vec{\omega}_A - \vec{\omega}_A + \vec{\omega}_B$. The R signal with AB pulses is $\vec{\omega}_{R:AB} = - \vec{\omega}_A
+\vec{\omega}_B + \vec{\omega}_B$ and the NR signal with AB pulses is $\vec{\omega}_{NR:AB} = \vec{\omega}_A
-\vec{\omega}_B + \vec{\omega}_B$. The PP contribution in the BA sequence is $\vec{\omega}_{PP:BA} = \vec{\omega}_B - \vec{\omega}_B + \vec{\omega}_A$. The R contribution in the BA sequence is $\vec{\omega}_{R:BA} = -\vec{\omega}_B + \vec{\omega}_A + \vec{\omega}_A$. The NR contribution in the BA sequence is $\vec{\omega}_{NR:BA} = \vec{\omega}_B - \vec{\omega}_A + \vec{\omega}_A$. These contributions and the resulting frequency vector scheme can be seen in Fig. \[FreqVectors\].
{width="8cm"} {width="8cm"}
The special importance of the PP and R contributions can be seen when one considers the response of a physical system that has many closely spaced overlapping resonances. The present case of an electron glass is clear realization of such a scenario. If the different excitations are largely independent (e.g. non-coupled), then we can understand the total system response as a convolution of the response of a single two-level system with the density of states of two-level excitations. For the simple case of a normal distribution of two-level systems, one gets for the convolution of the impulse responses given in Eqs. \[R1\] and \[R4\] as
{width="11cm"}
$$\begin{aligned}
\overline{P^{(3)}_{R_1 + R_1^*} } &= \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} \int d\omega_{01} \Big( \frac{e ^{ - {(\omega_{01} -
\overline{ \omega}_{01} )^2} / 2 \sigma^2}}{\sigma \sqrt{2 \pi}} \Big) \; \mathrm{sin}\; [ \omega_{01}( t_1 - t_3 ) ] \nonumber \\ &= \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3}
e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} \mathrm{sin}\; [ \overline{ \omega}_{01} ( t_1 - t_3 ) ]\; e^{ -\sigma^2 (t_1 - t_3)^2/2 },
\label{R1convolve}\end{aligned}$$
$$\begin{aligned}
\overline{P^{(3)}_{R_4 + R_4^*} } &= \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3} e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} \int d\omega_{01} \Big( \frac{e ^{ - {(\omega_{01} -
\overline{ \omega}_{01} )^2} / 2 \sigma^2}}{\sigma \sqrt{2 \pi}} \Big) \; \mathrm{sin} [ \omega_{01}( t_1 + t_3 ) ] \nonumber \\ &= \frac{| \varepsilon_0 \varepsilon_1 \varepsilon_2 |}{4} \frac{ \mu_{10}^4 }{\hbar^3}
e^{-(t_1 + t_3)/T_2} e^{-t_2/T_1} \; \mathrm{sin} [\overline{ \omega}_{01} ( t_1 + t_3 ) ] \; e^{ -\sigma^2 (t_1 + t_3)^2/2 }.
\label{R4convolve}\end{aligned}$$
where $ \overline{ \omega}_{01} $ is the mean resonance frequency and $\sigma$ is the spread. We plot the simulated experimental signals in Fig. \[TimeTracesCV2D\] for both $t_1 \rightarrow 0$ and $t_2 \rightarrow 0$ (giving R, NR, and PP signals), using the same experimental values in Fig. \[TimeTraces2D\], but including the effects of convolution with $\sigma = 0.5$. One can see that due to the rapid dephasing of multiple detuned oscillators the experimental signals decays quickly in the modulation directions, however one can still measure $T_1$ by the PP signal’s decay in the $\hat{\tau}$ direction and $T_2$ by the R signal’s decay in the $(\hat{\tau} + \hat{t})/\sqrt2$ direction. For the non-rephasing contribution, additional decay of the experimental signal occurs as function of $t_1 + t_3$, which limits the NR signal’s utility. As can be see in Fig. \[TimeTracesCV2D\] (and Eq. \[R1convolve\]) in contrast the rephasing signal shows the phenomena of “photon echo" when $t_1 = t_3$ (e.g along the diagonal). The important distinction between the PP and the R contributions as compared to the NR response is that they show their decay from oscillator dephasing in the orthogonal direction from the decay that arises from lifetime effects. In the frequency domain this will lead to a diagonal (horizontal) streaking of the signal for the R (PP) signal due to the multiple oscillators. However the direction perpendicular to the streaking direction gives $1/T_1$ for the PP signal and $1/T_2$ for the R signal. This is the essential important feature of 2DCS.
Analysis
--------
Due to the “phase twisting" in the nonlinear response functions discussed above and their dependence on both absorptive and dissipative elements it is more complicated to extract out parameter values like relaxation times than in linear response. For example, as shown in Fig. \[FT\_2D\] (a) and (b), the real and imaginary parts of the Fourier transform of the R peak along the anti-diagonal both have both positive and negative values and do not display a simple absorptive (Lorentzian) lineshape. Also as mentioned above, the phasing procedure used in previous THz 2DCS experiments [@kuehn2011two] to get purely absorptive lineshapes is challenging to implement here, because of the extreme inhomogeneous broadening and overlapping contributions in 2D frequency space. For this reason we chose to analyze the magnitudes of the experimental data in the main text as near resonances, magnitudes are dominated by the absorptive part of the lineshape. One can show algebraically that the magnitude of the R peak for the AB pulse sequence along the anti-diagonal direction can be approximated by a single Lorentzian with width $1/T_2$. This is illustrated in Fig. \[ADcuts\](a) which shows a cut along the anti-diagonal direction in the magnitude of the calculated Fourier transform of the R signal i.e., $\vert P^{(3)}_{R} \vert$ shown in Fig. \[FT\_2D\] (c). We overlay this cut with a Lorentzian $ \propto \frac{1}{\frac{1}{T_2^2}+\left(\omega -\omega_{01}\right){}^2}$ (yellow line in Fig. \[ADcuts\](a)). The peak is centered at the resonance frequency $\omega_{01}$. Note that the frequency axis (which is the anti-diagonal cut) in Fig. \[ADcuts\] is scaled by the factor $1/\sqrt{2}$ to give it the same scaling as the $\hat{\omega}_t$ and $\hat{\omega}_\tau$ directions. The same process is followed in analyzing the anti-diagonal cuts in the main text. As can be seen in Fig. \[ADcuts\](a), the cut along the anti-diagonal is quite well described by a single Lorentzian with width $1/T_2$. Thus, to extract $1/T_2$ from the experimental data, we fit a single Lorentzian to the anti-diagonal cut across the rephasing peak over an approximately FWHM as shown in Fig. 3a of the main text.
{width="11cm"}
Similarly, it can be shown that the magnitude of the PP peak for the BA pulse sequence along it’s anti-diagonal direction can be approximated by a single Lorentizan with width $1/T_1$ if one only fits the region near the peak where the response is dominated by dissipative effects. Fig. \[ADcuts\](b) shows a cut along the anti-diagonal direction in the magnitude of the calculated Fourier transform of the PP signal i.e., $\vert P^{(3)}_{PP} \vert$ shown in Fig. \[FT\_2D\_BA\] (d). The yellow line in Fig. \[ADcuts\](b) is $ \propto \frac{1}{\frac{1}{T_1^2}+\left(\omega -\omega_{01}\right){}^2}$. Thus, to extract $1/T_1$ from the experimental data, we fit a single Lorentzian to the anti-diagonal cut across the PP signal for the BA pulse sequence as shown in Fig. 3a of the main text. The fits to the data in the main text were performed in a range that is approximately one FWHM so as to emphasize the dissipative character of the response.
Broader context of 2DCS
-----------------------
We have considered here only the simplest case of the 2DCS response of a two-level system. This is a very useful example for the present case of an electron glass. Of course most physical systems are much more complicated. The presence of multiple oscillators with additional nonlinearities and couplings between them will result in very rich spectra.
The interested reader may ask how the above treatment can be related to the picture put forth in Ref. [@wan2019resolving] where the 2DCS response of an Ising chain in transverse field was calculated. There the excited spinon pairs can be mapped to a two-level system where each pair of spinons with momenta $\pm k$ correspond to a two-level system in which the ground and excited states correspond to the absence or presence respectively of these pairs. As discussed in the Supplemental Material to Ref. [@wan2019resolving] an added level of complexity occurs in that case because the spinon basis does not necessarily correspond to the same basis in which a $y$ oriented THz B field causes pure spin-flip transitions e.g. “diagonal transitions" are also allowed. There the Bogoliubov (energy dependent) coherence factor $\theta_k$ sets the rotation of the spinon basis from the physical reference frame in which a $y$ axis THz B field acts. For $\theta_k = \pi/2$ (satisfied in the middle of the two spinon band), diagonal transitions are not allowed and the results of Ref. [@wan2019resolving] reduce to the expressions above.
We anticipate that THz 2DCS will become a powerful general tool for condensed matter physics. For some more complicated models systems that have been analyzed previously we refer interested readers to the physical chemistry literature [@hamm2011concepts; @Mukamel1995; @hamm2005principles]. Although there has been some attempts to work out the 2DCS response for interesting quantum magnets that exhibit fractionalization [@wan2019resolving; @choi2020theory], the 2D response for many materials that are at the forefront of modern condensed matter physics has not been considered at all. This is a wide open area both theoretically and experimentally.
Mechanisms for relaxation in electron glasses
=============================================
In this section, we discuss possible mechanisms for the relaxation of particle-hole excitations in the regime considered in the main text, i.e., that of an electron glass that is relatively near the metal-insulator transition. We will proceed as follows. In Sec. \[recap\], we summarize the experimental findings in the main text, and argue that they strongly constrain the plausible relaxation mechanisms. We find that electron-electron interactions seem to be the only mechanism potentially consistent with experiment. In Sec. \[dipoleham\] we derive an effective model of dipolar two-level systems (TLS’s), corresponding to soft particle-hole excitations. In Sec. \[relaxrates\] we show that dipolar interactions among the TLS’s can give rise to energy relaxation with a characteristic rate $\Gamma_1 \sim \omega$. Finally we discuss (Sec. \[discussion\]) to what extent this mechanism can account for the experimental findings.
Experimental constraints on possible explanations {#recap}
-------------------------------------------------
We recall the main experimental conclusions about the relaxation rates $\Gamma_1(\omega)$ and $\Gamma_2(\omega)$ for an excitation at frequency $\omega$:
1. Both decay rates obey $\Gamma \sim \omega$, with a proportionality constant of order unity in the frequency range we probed ($\sim 0.2-1$ THz).
2. $\Gamma_1$ decreases (weakly) as the temperature is raised (i.e., relaxation slows down as the system heats up!), whereas $\Gamma_2$ is roughly temperature-independent in the range of temperatures $5 \mathrm{K} \leq T \leq 25 \mathrm{K}$.
3. Both decay rates decrease as the system is doped toward the metal-insulator transition.
4. The features do not strongly on the intensity with which the system is driven.
We also add some observations about the regime in which the experiment is performed.
1. A frequency of $0.5$ THz corresponds to a temperature of around $24$ K. Thus, at the lowest temperature we are probing excitations for which $\omega > k_B T$, although we move into the range $\omega \sim k_B T$ at higher $T$.
2. The bare Coulomb interaction at the inter-phosphorus distance is approximately $3$ THz. At these temperatures (at least for the more localized samples) we are probing transitions that take place inside the Coulomb gap.
3. The samples are doped relatively close to the metal-insulator transition [@Helgren02a; @Helgren04a], so the single-particle localization length is not a small parameter.
Together, these observations imply that the experiment probes a parameter regime in which theory is not well-controlled. Nevertheless, some explanations seem implausible given the experimental results.
### Rearrangements of thermally excited quasiparticles
In short-range many-body localized systems at nonzero temperature, relaxation takes place through large-scale collective rearrangements [@gopalakrishnan2015low]. The phase space for such rearrangements scales as $\omega^{-\phi}$, where $\phi \propto s(T)$, and $s(T)$ is the entropy density at temperature $T$. Since $s(T) \sim T$ at low temperatures, this mechanism would imply a temperature-dependent *exponent* for the relaxation, with low-frequency relaxation being parametrically faster at higher temperatures. This mechanism is inconsistent with the data. Other mechanisms by which a TLS decays by coupling to thermal noise can be ruled out on similar grounds: noise is stronger at higher temperatures, so these mechanisms give a relaxation rate that increases with temperature, whereas we see the opposite.
### Phonons
Phonons are a natural relaxation mechanism: a TLS excited can relax by emitting or absorbing a phonon. However, phonon-based mechanisms do not naturally capture the frequency- and temperature-dependence seen in the experiment.
(i) *Acoustic phonons* give the wrong frequency- and temperature-dependence. The matrix element for a TLS at frequency $\omega$ to couple to phonons goes as $\sim \omega$ [@spectraldiffusion], so the relaxation rate from coupling to phonons is $\sim \omega g(\omega) (2N({\omega},T) + 1)$, where $N(\omega, T)$ is the occupation number for bosons at frequency $\omega$ and temperature $T$. Here a factor of $N(\omega, T)$ comes from phonon absorption, and a factor of $N(\omega, T)+1$ comes from phonon emission. Such effects cannot describe the experimental observations for two reasons. Firstly if we examine the temperature dependence we conclude that the phonon mediated relaxation rate depends on temperature according to $2N(\omega, T)+1$, which increases with temperature. However, the experimentally measured energy relaxation rate decreases with increasing temperature. Secondly, the frequency dependence of the relaxation rate, in the limit when frequency is large compared to temperature, is $\omega g(\omega)$. As the phonon density of states must vanish at $\omega \rightarrow 0$ (generally as $\omega^2$), so this will produce a relaxation rate that vanishes faster than linearly with frequency. At low frequencies, it should go as $\omega^3$, which is inconsistent with the experimental observations [@spectraldiffusion]. While this channel must exist, it seems to be subleading.
(ii) *Optical phonons* do not seem relevant as the optical phonon branch of silicon is at $15$ THz [@brockhouse; @beltukov], which is well above the frequencies we are probing. In any case, the decay rate due to such phonons again would increase with temperature, due to Bose enhancement, and this is at odds with observation.
### Overheating
Nonlinear current-voltage characteristics near the metal-insulator transition can exhibit bistability [@altshuler2009jumps], because phonons are ineffective at equilibrating electronic degrees of freedom to base temperature. If the electronic temperature were indeed to decouple in this way, the measured temperature-dependence would be unreliable. This scenario, however, predicts that stronger pumping should heat up the electronic degrees of freedom more. This is inconsistent with the observed insensitivity of the relaxation rates to THz intensity.
Effective dipolar Hamiltonian for electron glass {#dipoleham}
------------------------------------------------
Here we start with the microscopic Hamiltonian describing impurity-band electrons in Si:P and derive a low-energy effective Hamiltonian in terms of two-level systems that interact through dipolar interactions. To allow for a controlled theoretical analysis we will assume that the localization length of single electronic excitations is short; however, we will not make any assumptions about the Coulomb interaction strength. The derivation takes place in various stages that are laid out in the following sections.
### From substitutional to on-site disorder
We begin with a general tight-binding Hamiltonian for interacting electrons in the presence of positional randomness:
$$\label{hmic}
H = \sum_{i \neq j} t_{ij} (c^\dagger_i c_j + \mathrm{h.c.}) + \frac{\mathcal{V}}{|\mathbf{r}_i - \mathbf{r}_j|} n_i n_j.$$
Here, $\mathcal{V}$ is the characteristic coupling strength for the Coulomb interaction in this dielectric medium, $c_j$ is the annihilation operator for an electron on site $j$, and $n_j$ is the electron density on site $j$. The hopping amplitude $t_{ij} \sim \exp(-c|\mathbf{r}_i - \mathbf{r}_j|)$ falls off rapidly with the spatial separation between pairs of atoms. The positional randomness in Eq. gives rise to on-site randomness through two mechanisms. First, the average potential energy on site $i$ is $\sum_j \mathcal{V}/|\mathbf{r}_i - \mathbf{r}_j| \langle n_j \rangle$, which is spatially random. If we integrate out states far from the Fermi energy, freezing in their occupation numbers, the remaining states will experience a random potential due to the randomly positioned frozen electrons. Second, even in the absence of interactions, the hopping term renormalize the on-site energy. We can evaluate the renormalized on-site energy $E_i$ self-consistently in (Brillouin-Wigner) perturbation theory. At second order we get $E_i = \sum_j \frac{|t_{ij}|^2}{E_i}$, so $
E_i = \sqrt{\sum_j |t_{ij}|^2}$. In practice, the $t_{ij}$ are exponentially sensitive to bond lengths so we can approximate $E_i = \max(|t_{ij}|)$, where the maximum is taken over all $j \neq i$. Pairs of sites that are anomalously nearby hybridize strongly (and are shifted far from the Fermi energy); perturbative corrections from coupling to these randomly positioned tightly bound dimers generate on-site disorder for typical sites.
### From on-site disorder to resonant pairs
The above considerations let us replace the Hamiltonian with an effective Hamiltonian that describes particles on a site-diluted lattice, and contains the following terms: a random on-site potential $\epsilon_i$, drawn from a distribution of characteristic width $W$; a Coulomb interaction of strength $V_{ij} \equiv \mathcal{V}/|\mathbf{r}_i - \mathbf{r}_j|$ between any two sites; and a hopping term $t_{ij}$ of characteristic scale $t$ between neighboring pairs of sites:
$$\label{h2}
H = \sum_{i} \epsilon_i n_i + \sum_{i\neq j} V_{ij} n_i n_j + \sum_{\langle ij \rangle} t_{ij} c^{\dag}_i c_j$$
We proceed as follows. If we set all the $t_{ij} = 0$ the eigenstates of Eq. are product states in which each site is either occupied or unoccupied. The ground state of this system is the classical “electron glass”. At low temperatures, there are many metastable states, all with statistically similar properties that the system is in some mixture of. The ground state and metastable states are, by definition, stable against single-electron moves. This stability implies a pseudogap (Coulomb gap) in the density of states [@Efros75a].
Starting from this classical ground state, we would like to create a particle-hole excitation at a low frequency $\omega \ll W, t$, by moving an electron from a filled level $a$ to an empty level $b$. This dimer has four states, which we label $|00\rangle$ (neither filled), $|\downarrow\rangle$ (only $a$ filled, i.e., ground state), $|\uparrow\rangle$ (only $b$ filled) and $|11\rangle$ (both filled). We ignore the $|00\rangle$ state in what follows. The remaining three states have the following energies:
$$\label{SEcount}
E_{\downarrow} = \epsilon_a + \sum_{c \neq a, b} V_{ac}, \quad E_{\uparrow} = \epsilon_b + \sum_{c \neq a,b} V_{bc}, \quad E_{11} = E_{\downarrow} + E_{\uparrow} + V_{ab}.$$
A pump at frequency $\omega$ can induce transitions between pairs of levels with energies separated by $\omega$ (up to some resolution set by the properties of the pulse). The matrix element connecting states $|\downarrow\rangle$ and $ |\uparrow\rangle$ comes from their hybridization due to the hopping. In what follows it is crucial to understand how the hopping shifts these levels. The matrix element goes as $t_{ab} = W e^{-|r_{ab}|/\xi}$ where $\xi$ is the localization length. (Within the locator approximation, $t_{ab}$ is due to a sequence of $\sim |r_{ab}|$ non-degenerate perturbative hops, so $t_{ab} \sim W(t/W)^{|r_{ab}|}$, implying that $\xi \sim 1/\log(W/t)$. Once this hybridization is included, the splitting between the two levels is
$$\mathcal{E}_{ab} = \sqrt{t_{ab}^2 + (E_{\uparrow} - E_{\downarrow})^2}.$$
The light can excite a transition when its frequency $\omega$ matches the energy of the energy splitting. This condition can not hold unless $t_{ab} \alt \omega$. Therefore, there is a minimal distance $r_\omega \sim \xi \log(W/\omega)$ above which such resonances are possible [@Mott79a]. Moreover, if $r \gg r_\omega$, hopping is very unlikely (i.e., has probability $\sim e^{-r/r_\omega}$) to appreciably hybridize the levels, and the transition amplitude is strongly suppressed. Therefore, as first pointed out by Mott, resonant pairs at splitting $\omega$ form at an optimal distance $r_\omega$ [@Mott79a]. At this optimal scale, the states $|\uparrow\rangle, |\downarrow\rangle$ of the resonant pair are strongly hybridized, and form a TLS governed by the Hamiltonian
$$\label{htls}
H_{\mathrm{TLS}} = t_{ab} \sigma^x_{ab} + (E_{\downarrow} - E_{\uparrow}) \sigma^z_{ab} \equiv \mathcal{E}_{ab} \tau^z_{ab}.$$
In what follows, we call such pairs of levels resonant pairs. We will next estimate the density of such resonant pairs.
### Density of active resonant pairs
To contribute to low-frequency dynamics, a resonant pair must be “active” in the ground state, i.e., it should be in the states $|\downarrow\rangle$ or $|\uparrow\rangle$. We now estimate the density of pairs that meet this criterion. There are two regimes:
Mott regime
: When $\omega$ is relatively large, i.e., $\omega \agt \mathcal{V}/|r_\omega|$, the interaction correction to $E_{11}$ in Eq. is unimportant. Thus, for a resonant pair to be active, the lower eigenstate of the TLS must be within $\omega$ of the Fermi energy. The density of resonant pairs at frequency $\omega$ then scales as $\omega r_\omega^2$.
Shklovskii-Efros regime
: When $\omega \alt \mathcal{V}/|r_\omega|$, the interaction correction in Eq. crucially changes the counting, through a mechanism similar to Coulomb blockade. As long as *either* $E_0$ or $E_1$ is within $\mathcal{V}/|r_\omega|$ of the Fermi energy, the $|11\rangle$ state is energetically unfavorable, so the pair of sites is singly occupied in the ground state. Therefore, the phase space for the TLS goes as $\mathcal{V}|r_\omega|$, which is essentially constant at low frequencies [@shklovskii1981phononless].
The crossover between these two regimes is also captured by the low-frequency (phononless) conductivity, which goes as $\omega^2$ in the Mott regime and $\omega$ in the Shklovskii-Efros regime [@shklovskii1981phononless].
### Interactions between resonant pairs
We now argue that the Coulomb interaction induces effective dipolar ($1/R^3$) interactions between resonant pairs. The result is well known [@burin1994low; @burin; @yao2014many], but we rederive it for completeness. We consider two active resonant pairs $ab$ and $cd$, separated by a distance that is much larger than the size of the dipoles themselves i.e. $r_{ac} \gg \text{max}(r_{ab}, r_{cd})$. Since both pairs are assumed to be active, the two-pair Hamiltonian acts in a four-dimensional space consisting of the two TLS’s $\mathbf{\sigma}_{ab}$ and $\mathbf{\sigma}_{cd}$. The Coulomb interaction is diagonal in the $\sigma^z$ basis for both TLSs. For example, if the pair $ab$ is in its $|\downarrow\rangle$ state, the potential on site $c$ is given by $V_0 + V_{ac}$, where $V_0$ is the potential due to all spins other than the resonant pairs under consideration; similarly, the potential on site $d$ in this configuration is $V_0 + V_{ad}$. Writing this out, we see that the Coulomb interaction between the two resonant pairs takes the form
$$H_C = \frac{1}{4}(1 - \sigma^z_{ab}) (V_{ac} - V_{ad}) \sigma^z_{cd} + \frac{1}{4}(1 + \sigma^z_{ab}) (V_{bc} - V_{bd}) \sigma^z_{cd} = \frac{1}{2} (V_{bc} - V_{ac} + V_{ad} - V_{bd}) \sigma^z_{ab} \sigma^z_{cd} + \ldots$$
where the terms ignored in $\ldots$ are overall static shifts that are included in the full Hamiltonian. Assuming $r_{ac} \gg \mathrm{max}(r_{ab}, r_{cd})$, one can expand the effective coupling constant in a multipole expansion to get that $V_{ab, cd} \sim \mathcal{V} r_{ab} r_{cd} / r_{ac}^3$. (In this expression we have ignored some geometric factors that will not be important in what follows.) Thus, the Hamiltonian for two coupled resonant pairs is
$$\label{4lev}
H_{\mathrm{2-pair}} = \Delta_{ab} \sigma^z_{ab} + t_{ab} \sigma^x_{ab} + \Delta_{cd} \sigma^z_{cd} + t_{cd} \sigma^x_{cd} + \frac{c'\mathcal{V} r_{ab} r_{cd}}{r_{ac}^3} \sigma^z_{ab} \sigma^z_{cd}.$$
where $c'$ is a numerical prefactor. In the regime of interest, we can simplify this further. The Hamiltonian in Eq. describes a four-dimensional Hilbert space. However, when the Coulomb interaction between the two resonant pairs is weaker than (or comparable to) their splitting, we can make a secular approximation and eliminate the matrix elements of the Coulomb interaction that do not preserve the number of excitations. Doing so, and transforming to the eigenbasis of the isolated TLSs (denoted by Pauli matrices $\tau$), we get the equation for two interacting resonant pairs:
$$\label{secular}
H_{\mathrm{2-pair}} = \mathcal{E}_{ab} \tau^z_{ab} + \mathcal{E}_{cd} \tau^z_{cd} + \frac{\tilde c \mathcal{V} r_{ab} r_{cd}}{r_{ac}^3} (\tau^+_{ab} \tau^-_{cd} + \mathrm{h.c.})$$
where $\tilde c\neq c'$ is a prefactor that is not important for our argument. In what follows we will consider relaxation in an ensemble of resonant pairs interacting pairwise through Eq. .
Relaxation of coupled resonant pairs {#relaxrates}
------------------------------------
After eliminating the degrees of freedom that are not low-frequency resonant pairs, we finally reduce the Hamiltonian to a spin model for coupled TLS’s, of the form
$$\label{cpair}
H_{\mathrm{pair}} = \sum_\alpha \mathcal{E}_\alpha \tau^z_\alpha + \sum_{\alpha\neq \beta} \frac{ c|\mathcal{V}| p_\alpha p_\beta}{r_{\alpha\beta}^3} (\tau^+_\alpha \tau^-_\beta + \mathrm{h.c.}).$$
where $ c$ is an prefactor that is not important for the argument, $\alpha$ labels resonant pairs, and $p_\alpha \sim \xi\log(W/\mathcal{E}_\alpha)$ is the size of pair $\alpha$. In the dilute limit and when all pairs but one are in their ground state, Eq. describes a single particle hopping on a disordered lattice with dipolar interactions, a problem addressed in Ref. [@levitov1989absence]. The density of resonant pairs depends on which regime one considers:
$$P(\mathcal{E}_\alpha = \omega) \sim \frac{1}{W^2} \left\{
\begin{array}{lr}
\mathcal{V} \xi \log(W/\omega) & \text{Shklovskii-Efros} \\
& \\
\omega \xi^2 \log^2(W/\omega) & \text{Mott}
\end{array}\right.$$
We will deal with these cases separately.
### Shklovskii-Efros regime
The experiment flips a pseudospin at frequency $\omega$ and we are interested in how the pseudospin decays. We can eliminate all states in the Hilbert space of Eq. with energies greater than $c \omega$, where $c$ is some constant of order unity. (One can regard this as decoupling such states through a Schrieffer-Wolff transformation; the transformation will generate additional short-range interactions that do not affect the analysis.) This leaves behind a sparse network of effective spins, with characteristic bandwidth $c \omega$ and spatial separation $[c \omega \mathcal{V} \xi \log(W/\omega) / W^2]^{-1/3}$. The interaction between typical spins at this scale goes as $V_{\mathrm{eff}} \sim \omega \xi^3 \log^3(W/\omega) (\mathcal{V}/W)^2$. Thus, a dipole excited by a frequency $\omega$ typically has a nearest neighbor at a distance where the interaction is resonant.
Note that this occurs precisely because the dipolar interaction scales as inverse volume in three dimensions i.e. it is marginal [@levitov1989absence]. More generally, if interactions fall off as $1/R^\alpha$ we would get $V_{\mathrm{eff}} \sim \omega^{d/\alpha}$, so the disorder would become irrelevant (for $\alpha < d$) or the interactions would become irrelevant (for $d < \alpha$) giving rise to dipole localization.
Since the rate at which TLS’s hop on the effective lattice scales as $V_{\mathrm{eff}}$, this mechanism would predict
$$\label{lse}
\Gamma(\omega) \simeq \omega \xi^3 \log^3(W/\omega) (\mathcal{V}/W)^2.$$
This $\omega$-dependence is consistent with experimental observations, though it is not obvious why the approximations we have made to derive Eq. should be valid for the parameters in the experiment. In particular, our calculation has assumed that the localization length $\xi$ is small compared to the $P-P$ spacing, that the ‘dipoles’ are well separated (inter-dipole spacing much larger than dipole size), and that the frequency $\omega$ is much smaller than the characteristic disorder and interaction scales. These assumptions are suitable deep in the insulating phase, but none of these assumptions are safe in the experimental regime close to the MIT, where $\xi$ is large compared to $P-P$ spacing, and where $\omega$ is of the same order as the characteristic disorder and interaction scales (such that the inter-dipole separation is comparable to the dipole size). These issues are explored further in the discussion section below.
### Mott regime
Adapting the previous discussion to the Mott regime is straightforward. The spatial density of low-energy dipoles now scales as $\omega^2$ rather than $\omega$. Thus the spacing of low-energy pairs therefore scales as $\omega^{-2/3}$, and the interactions at that scale go as $\omega^2$. In this case, therefore, the interactions are parametrically weaker than the hopping of dipoles on the effective lattice. Naively one might estimate that $\Gamma(\omega) \sim \omega^2$. Resonant pairs below some frequency scale have no neighbors to inter-resonate with; they delocalize instead through the long-range part of the dipole interaction [@levitov1989absence], with relaxation rate that is therefore much slower. So unlike the previous case these dipoles do get arbitrarily sharp as $\omega \rightarrow 0$, which implies such a system would be a “Fermi glass”.
### Short-range interactions
Although not likely relevant to experiment, it is illustrative to apply the reasoning above to the case of short-range interactions. Once again, the density of active resonant pairs at frequency $\omega$ scales as a power of $\omega$ (which depends on whether $\omega$ is in the Shklovskii-Efros or Mott regimes). The typical spacing between dipoles is thus $\omega^{-\phi}$ for $\phi = 1/3$ (Shklovskii-Efros regime) or $2/3$ (Mott regime). However, the matrix element between dipoles at this scale is now exponentially suppressed, and scales as $\sim \exp(-1/\omega^\phi) \ll \omega$. A naive application of the Golden Rule might suggest that the lifetime should scale similarly, yielding arbitrarily sharp excitations near the Fermi energy. However, in this case we expect that a typical dipole has *no* resonant neighbors, and should therefore be strictly localized, below some critical frequency.
### An alternative derivation
The discussion above involved some heuristic steps, so it is helpful to check it using a more systematic approach. To this end we adopt the self-consistent theory of localization [@abou1973selfconsistent], as follows. We compute the lifetime $\Gamma(\omega)$ of a dipole due to hopping to other sites on the lattice, while assuming that these other sites also decay at rate $\Gamma(\omega)$. A self-consistent approach is necessary because otherwise each line in a localized system is infinitely sharp, and exact resonances cannot be found. To keep the model tractable we assume that the decay rate is only a function of $\omega$ (i.e., we neglect spatial heterogeneity that could in principle be important). This leads us to a self-consistent equation for the function $\Gamma(\omega)$:
$$\Gamma_\alpha = \sum_\beta V_{\alpha\beta}^2 \mathrm{Im} \left( \frac{1}{\omega_\alpha - \omega_\beta + i \Gamma(\omega_\beta)} \right).$$
Converting the sum to an integral and invoking spatial homogeneity we get
$$\Gamma(\omega) = \int_{R(\omega)}^\infty dr r^2 \frac{\mathcal{V}^2}{r^6} \int d\Omega \frac{\mathcal{V} \xi \log(W/\Omega)}{W^2} \frac{\Gamma(\Omega)}{(\omega - \Omega)^2 + \Gamma(\Omega)^2}.$$
Here we have used the crucial fact that resonant pairs at $\omega$ cannot be spaced much closer than $R(\omega) \sim (\mathcal{V}/\omega)^{1/3}$, as otherwise the dipolar interaction would hybridize and split them strongly. Outside this zone, the detunings and frequencies are uncorrelated, allowing us to simplify this expression (neglecting logarithmic factors) to be
$$\Gamma(\omega) = \omega \frac{\mathcal{V}^2}{W^2} \int d\Omega \frac{\Gamma(\Omega)}{(\omega - \Omega)^2 + \Gamma(\Omega)^2}.$$
One can check that the only power-law scaling that satisfies this self-consistent equation is $\Gamma \sim \omega (\mathcal{V}/W)^2$, confirming our previous estimate up to logarithmic factors that have been suppressed.
Discussion
----------
The analysis of the previous section shows how a combination of Coulomb blockade and dipolar hopping can give rise to the experimentally observed $\Gamma(\omega) \sim \omega$ for the relaxation of a TLS in an electron glass. The prefactor (ignoring logarithms) is $\xi^3 (\mathcal{V}/W)^2$, which is not necessarily small, if the typical microscopic hopping amplitude between adjacent P sites is comparable to the Coulomb interaction between them. This framework also naturally reproduces the sharpening of the TLS frequency as one increases the temperature up to $T \simeq \omega$: when delocalization takes place through coherent tunneling at zero temperature, the finite-temperature corrections usually suppress transport through decoherence (see, e.g., Ref. [@fisherzwerger]), and furthermore raising temperature increases screening and hence suppresses the Coulomb interaction.
As mentioned above, the validity of our analysis relied on the assumptions that $\xi$ is small compared to the $P-P$ spacing and $\omega$ is small compared to the typical hopping and interaction scale. These assumptions are not quantitatively accurate in the experimental regime (close to the metal insulator transition), where $\xi$ becomes large compared to the $P-P$ spacing, and $\omega$ is of the same order as the typical interaction scale between neighboring $P$ dopants. We can briefly comment on how some version of this mechanism might operate closer to the metal-insulator transition. There are two essential ingredients: (i) hopping of dipolar excitations among electron-hole pairs and (ii) the enhancement of phase space for active resonant pairs due to Coulomb blockade. As one approaches the metal-insulator transition, the picture of well-separated dipoles breaks down: rather, one has a much more densely connected network of overlapping electron-hole pairs. Nevertheless, an excitation at frequency $\omega$ can only hybridize with states that are at frequency $\simeq \omega$, and (by Mott’s argument) the matrix element for this cannot exceed $\omega$. Thus, the $\omega$-dependence of decay rates should persist. As $\xi$ increases past unity, however, the Coulomb blockade effect becomes less effective; naively, one should replace $\mathcal{V}$ in Eq. by the charging energy $\mathcal{V}/\xi^3$. Also, the bandwidth $W$ is increasingly dominated by kinetic energy, so $\mathcal{V}/W$ decreases as a consequence. These considerations are consistent with the dimensional argument in the main text. Thus, dipolar relaxation seems consistent with the observed trends near the metal-insulator transition, though we have not been able to perform a controlled calculation in that regime. Our discussion so far has been limited to estimating the lifetime of a single particle-hole excitation in an electron glass (i.e., the $T_1$ time). We now turn to two other issues: the relation between the dipole lifetime and the lifetime of the single-particle Green’s function, and the relation between $T_1$ and $T_2$ times.
### Electron lifetime vs. dipole lifetime
The calculation outlined above gives the lifetime of an elementary dipole in an electron glass. The question of whether a quasiparticle is sharp, however, is normally phrased in terms of the decay rate of the single-electron Green’s function, probed, for example, by tunneling an electron into the system. In translationally invariant Fermi liquids the two quantities are effectively the same; momentum and energy conservation forbid the particle from recombining with its hole, so they relax separately as if they were injected excitations. In the Fermi glass, by contrast, the lack of momentum conservation allows the particle and hole to recombine.
We now briefly comment on what happens if one injects an electron into the electron glass. In the Mott regime, this calculation is straightforward; the electron relaxes into two particles and a hole. Since all three final states must lie within $\omega$ of the Fermi level, and there are two free energies, the phase space scales as $\omega^2$. Therefore the lifetime of an injected particle is parametrically longer than that of a dipole: it scales as $\omega^{-3}$. In the Shklovskii-Efros regime this question is much more delicate, owing to the presence of the Coulomb gap: we do not address this question here, but remark that a similar separation of scales seems possible. As noted in the main text, we are treating the particle-hole lifetime rather than the injected-particle lifetime as the fundamental quantity, since this is the relevant low energy excitation for an isolated system.
### Thermal effects, spectral diffusion, and $T_1$ vs. $T_2$
Within our zero-temperature theory the processes responsible for $T_1$ and $T_2$ are the same, so we expect these quantities to be related by some simple scale factor; this is indeed what we observe. However, the trends with increasing temperature are different: $T_1$ grows with temperature, while $T_2$ remains roughly constant (so the ratio $T_1/T_2$ increases). A natural interpretation is that this approximate temperature-independence comes from a competition between the increasing $T_1$ time and the opening of thermal dephasing channels.
Another potential relaxation channel is spectral diffusion [@spectraldiffusion]. The essential idea behind spectral diffusion is that the splitting of a particular TLS is a time-varying quantity, undergoing stochastic fluctuations with “amplitude" $\Delta \omega$ and correlation time $\tau$. When the correlation time is sufficiently short, we expect that the energy splitting of a TLS fluctuates at a rate $\sim \Delta \omega^2 \tau$ [@hamm2011concepts]. Note that this rate decreases as $\tau$ gets shorter: this is analogous to “motional narrowing,” where rapid fluctuations in the environment get averaged out. If $\tau$ were to decrease with temperature, while $\Delta \omega$ stayed roughly constant, spectral diffusion may give rise to lifetimes that increased with temperature. As we will discuss below, the natural physical mechanisms for spectral diffusion have the frequency amplitude increasing as its correlation time decreases, so the predicted lifetime shortens as the system is heated up.
Two possible mechanisms for noise on a particular resonant pair are phonons and dipolar fluctuations of other resonant pairs. As we have already discussed, effects due to phonons should be strongly suppressed at low frequency and temperature, in a manner inconsistent with our observations. We now turn to interactions among resonant pairs. A given resonant pair is surrounded by thermally fluctuating pairs with typical energy (and lifetime) $\sim T$. When $\hbar \omega \alt k_B T$, each such fluctuation is large compared with the splitting of the TLS; therefore, in this regime one cannot think of a TLS as having a stable splitting $\mathcal{E}$. Rather one must work in the $z$ basis of Eq. , and consider incoherent transitions due to the hopping. In this regime, any specific level fluctuates over a range $\sim T$ on a timescale $\sim 1/T$, so the decay rate is not strongly frequency dependent but scales as $T$; this is inconsistent with our observations. Another way to see this is as follows: the lifetime of a dipole at temperature $T$ scales as $1/T$, but because the interaction scales as inverse volume, the strength of the noise generated by nearby dipoles at temperature $T$ also scales as $T$, so $\Delta \omega^2 \tau \sim T$.
In addition to energy decay, spectral diffusion potentially gives rise to dephasing. This is because the rate of phase accumulation depends on the energy of the excitation, and if the energy is fluctuating about some mean, then the rate of phase accumulation is also fluctuating. Very roughly, fluctuations in the energy of the excitation lead to a random walk in the phase, and by the central limit theorem, if the line broadening from spectral diffusion is $\sim k_B T$, then the rate of dephasing from spectral diffusion goes $\sim \sqrt{k_B T}$.
It is worth emphasizing that the line broadening and the dephasing rate arising from spectral diffusion are both [*increasing*]{} functions of temperature. In contrast, we measure that the energy relaxation rate is a decreasing function of temperature, and the dephasing rate is approximately temperature independent. Although it could be that more complicated models of spectral diffusion that incorporate the effects of temperature dependent screening do describe the data, simple models of spectral diffusion do not describe our data. We will also mention that despite these arguments, it could still be quite interesting to try to measure the spectral diffusion directly via the multi-pulse 2DCS experiments [@hamm2005principles]. Accepted protocols for measuring spectral diffusion exist in molecular systems that could be adapted to the THz 2DCS. These will be topics for further study.
Summary
-------
We end this discussion of relaxation rates with a brief summary. Many natural potential mechanisms—such as thermal noise, spectral diffusion, Joule heating, and electron-phonon interactions—do not seem to be relevant to the physics in the regimes we are probing. In some cases this can be seen a priori, in other cases because these mechanisms do not fit the data. The remaining candidate is the Coulomb interaction. We know that Coulomb interactions affect the response of the system in the frequency range being probed (this is evident, for example, in the Shklovskii-Efros dependence of conductivity in Fig. 2 of the main text). We argued that these interactions give rise to relaxation via a continuum of dipoles, made up of localized pairs of single-particle orbitals; the relaxation rate goes as $\Gamma_1 \sim \omega$, as observed in the experiment. This dipolar theory (developed for low frequency response in deeply localized systems) is not well controlled in the experimental parameter regime, which probes intermediate frequencies in systems near the metal-insulator transition. However, it is consistent with the experimental phenomenology at low temperatures. Extending this theory of dipolar relaxation near the metal-insulator transition and to finite temperatures is an interesting task for future work.
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[^1]: We remark, however, that the particle-hole lifetime does not distinguish between Fermi glasses and certain localized “non-Fermi glass” phases [@nfg].
[^2]: The terms that are missing in the notation of Ref. [@hamm2011concepts] $R_3$, $R_3^*$, and $R_6$, $R_6^*$ represent the double excitation of the ket and bra sides of the density matrix before de-excitation. However, such double excitation is not allowed in a two-level system and so we do not consider these effects here.
[^3]: Note that usual definitions of the third order response function differ from our definition of the third order response by a minus sign. Usually two factors of i that originate in the perturbative expansion of the density matrix are neglected in the definitions for third order response in 2DCS. We have not neglected them here.
|
---
abstract: 'The OPERA experiment is aiming at the first direct detection of neutrino oscillations in appearance mode through the study of the $\nu_\mu\rightarrow\nu_\tau$ channel. The OPERA detector is placed in the CNGS long baseline $\nu_\mu$ beam 730 km away from the neutrino source. The analysis of a sub-sample of the data taken in the 2008-2009 runs was completed. After a brief description of the beam and the experimental setup, we report on event analysis and on a first candidate event, its background estimation and statistical significance.'
address: |
ETH, Institute for Particle Physics,\
Zurich, Switzerland\
$^*$E-mail: [email protected]
author:
- 'L. S. Esposito$^*$, on behalf of the OPERA Collaboration'
title: 'Search for $\nu_\mu\rightarrow\nu_\tau$ oscillations in appearance mode in the OPERA experiment'
---
Introduction {#intro}
============
The phenomenon of neutrino oscillation, the transition from a neutrino flavour to another, was anticipated nearly 50 years ago[@pontecorvo]. Two types of experimental methods can be used to detect such oscillations: observe the appearance of a neutrino flavour initially absent in the beam or measure the disappearance rate of the initial flavour. The disappearance of muonic neutrinos have been convincingly observed in different experiments [@SK; @MINOS]. In the SNO [@SNO] solar neutrino experiment, the measured rate of neutral-current (NC) interactions was shown to be compatible with the rate expected from total solar neutrino flux, supporting the idea that flavour transitions occur among the three active flavours of the Standard Model. However, there is no direct evidence of neutrino oscillation by the appearance method where the new flavour is identified. The OPERA experiment is aiming at the first direct detection of neutrino oscillations in appearance mode by the identification of the $\tau$ lepton produced in $\nu_\tau$ CC interaction in an almost pure muon neutrino beam by an event-by-event measurement. In a different approach, a statistical analysis of the atmospheric neutrino in Super-Kamiokande shows that data are inconsistent with no $\tau$ appearance hypothesis due to $\nu_\mu\rightarrow\nu_\tau$ oscillations [@SK_tau].
Beam
====
OPERA is exposed to the long-baseline CNGS $\nu_\mu$ beam [@CNGS] from CERN in the Gran Sasso Laboratories (LNGS), 730 km from the CERN neutrino source. The beam is optimized for the observation of $\nu_\tau$ CC interactions. The average neutrino energy is $\sim$17 GeV. The $\bar{\nu}_\mu$ contamination is 2.1% in terms of interactions; the $\nu_e$ and $\bar{\nu}_e$ contaminations are lower than 1%, while the number of prompt $\nu_\tau$ is negligible. With a total CNGS beam intensity of $22.5 \times 10^{19}$ pot (protons on target), about 24300 neutrino events would be collected. The experiment should observe about 10 $\nu_\tau$ CC events for the present $\Delta m^2_{23}$ allowed region with a background of less than one event. During the physics runs in 2008, 2009 and 2010 the total achieved intensity was $9.34 \times 10^{19}$ pot.
The detector {#detector}
============
The OPERA detector installed in the underground laboratory of LNGS has three main components (see Fig. \[fig:det\]).
![The OPERA detector in the LNGS hall C, at a depth corresponding to 3,100 m water equivalent overburden. The CNGS neutrino beam comes from left.[]{data-label="fig:det"}](detector.pdf){width="0.7\linewidth"}
The first one is an active target, consisting of about 150k Emulsion Cloud Chamber (ECC) modules called ”bricks”. A brick is a sandwich structure of 57 nuclear emulsion films and 56 1 mm thick lead plates. The lead plates serve as neutrino interaction target and the emulsion films as 3-dimensional tracking detector providing track coordinates with a sub-micron accuracy and track angles with a few mrad accuracy. The material of a brick along the beam direction corresponds to about 10 radiation length and 0.33 interaction length. The brick size is 10 cm$\times$12.5 cm$\times$8 cm and it weighs about 8.3 kg. The total active target mass is thus 1.25 ktons. The target is subdivided in two identical units, each consisting of walls of bricks, containing about 2800 units each. The second main component of the detector is the target tracker (TT) system, a set of 62 scintillator planes in total, interleaved with the brick walls. Each TT plane consists of 256 2.6 cm wide plastic strips. The sandwich structure of the brick walls and the TT planes is illustrated in Fig. \[fig:ecc-tt\].
![Arrangement of the ECC brick walls and the TT scintillator planes.[]{data-label="fig:ecc-tt"}](ecc-cs-tt.pdf){width="0.7\linewidth"}
The third detector component consists in two muon spectrometers following the two target units. Equipped with RPC and high precision drift chambers, they allow determining the muon momentum with better than 20% accuracy up to 30 GeV/c.
Event analysis {#analysis}
==============
Neutrino events analysis starts with processing of signals from the electronic detectors, reconstructing tracks of particles outgoing from the interaction vertex and measuring muons momentum in the magnetic spectrometers. Tracking information is combined with reconstructed hadronic shower axis and the output of a Neural Network for the selection of the wall where the interaction occurred, providing a list of bricks with the associated probability that the interaction occurred therein. For events with a muon in the final state, a prediction for the slope of the muon and its impact on the brick is provided. For NC events a hadronic shower calculated with TT hits provides general direction to the vertex. This part of the event analysis is called Brick Finding (BF). The brick with the highest probability is extracted from the detector for analysis. No new bricks are inserted in the detector during the experiment, so the total target mass is gradually decreasing with time. Therefore, a high efficiency of the BF is important for minimization of the target mass loss and the reduction of emulsion processing load. After extraction of the brick predicted in the electronic detectors, its validation is performed by analysis of two interface emulsion films (called Changeable Sheets, CS) [@CS] that are inserted in between each ECC brick and TT scintillator strips. The CS doublet is analysed in the scanning facilities before a brick is disassembled. The information of the CS is then used for a precise prediction of the position of the tracks in the most downstream films of the brick, hence guiding the scan-back vertex finding procedure. If no tracks are found in the CS, the brick is returned back to the detector with another CS doublet attached; otherwise, after a brick has been validated, it is dismantled and its emulsion films are developed and dispatched to the various scanning laboratories, in Europe or in Japan After a brick has been validated, its emulsion films are developed and dispatched to the various scanning laboratories. All tracks measured in the CS are followed back until they are not found. The stopping point is considered as either a primary or a secondary vertex, then the vertex is confirmed by scanning a volume with 1 cm$^2$ in 15 films, 5 upstream and 10 downstream of the stopping point. A further analysis called decay search procedure is applied to located vertices to detect possible decay or interaction topologies. When secondary vertices are found in the event, a kinematical analysis is performed using track angles and momenta. Momenta of charged particles can be measured in ECC using the angular deviations of tracks by Multiple Coulomb Scattering (MCS) in lead [@MCS]. This method gives a momentum resolution better than 22% for particles with momenta lower than 6 GeV/c, passing through a brick. For higher momentum, the position deviations are used for the measurement. The resolution is better than 33% for particles with momenta lower than 12 GeV/c, passing through a brick. Momenta of muons reaching the spectrometer are measured with a resolution of 20% up to 30 GeV/c.
The analysis of a sub-sample of 1088 events of the neutrino data taken in the 2008-2009 runs was completed, corresponding to $1.89 \times 10^{19}$ pot. Charmed particles have similar lifetimes and decay topologies if charged, so the detection of charm decays is used to check the $\tau$ detection efficiency. In the sample of $\nu_\mu$ CC interactions, 20 charm decays have been observed that survived all the cuts, in agreement with expectations from a MC study, $16.0\pm 2.9$. Out of them 3 have a 1-prong topology where $0.8\pm0.2$ are expected. The background for the total charm sample is about 2 events. Several $\nu_e$-induced events have also been observed.
The first $\nu_\tau$ candidate event {#tau}
====================================
The decay search procedure yielded one candidate event satisfying the selection criteria for the $\nu_\tau$ interaction search [@OPERA_tau]. The cuts are the same as those defined in the experimental proposal [@OPERA]. The event is displayed in Fig. \[event\].
![Display of the $\tau^-$ candidate event. [*Top left:*]{} view transverse to the neutrino direction. [*Top right:*]{} same view zoomed on the vertices. [*Bottom:*]{} longitudinal view.[]{data-label="event"}](beam_m.pdf "fig:"){width="0.445\linewidth"} ![Display of the $\tau^-$ candidate event. [*Top left:*]{} view transverse to the neutrino direction. [*Top right:*]{} same view zoomed on the vertices. [*Bottom:*]{} longitudinal view.[]{data-label="event"}](beam_zoom_m.pdf "fig:"){width="0.445\linewidth"}\
![Display of the $\tau^-$ candidate event. [*Top left:*]{} view transverse to the neutrino direction. [*Top right:*]{} same view zoomed on the vertices. [*Bottom:*]{} longitudinal view.[]{data-label="event"}](viewer_m.pdf "fig:"){width="0.9\linewidth"}
The selection criteria are that there are no primary tracks compatible with a muon or an electron, and the secondary vertex and the primary vertex survive the cuts shown in Table \[tab:var\].
\[tab:var\]
The primary vertex consists of seven tracks of which one track has a kink. Two electromagnetic showers caused by $\gamma$ rays have been detected and they are compatible with pointing to the secondary vertex. The invariant mass of the two detected $\gamma$ rays yields a mass consistent with the $\pi^{0}$ mass, $(120\pm20~(stat) \pm 35~(syst))$ MeV/c$^{2}$. Then, the invariant mass of the $\pi^{-}\gamma\gamma$ system has a value compatible with that of the $\rho (770)$, $(640 \pm 125 ~(stat) \pm 100~(syst)$ MeV/c$^{2}$. The $\rho$ appears in about 25% of the $\tau$ decays.
Background estimation and statistical significance {#background}
==================================================
The secondary vertex is compatible with the decay of $\tau \rightarrow h^{-} (n \pi^{0}) \nu_\tau$. The main background sources to this channel are
- the decays of charmed particles produced in $\nu_\mu$ CC interactions where the primary muon is not identified.
- the 1-prong interactions of primary hadrons produced in $\nu_\mu$ CC interactions where the primary muon is not identified or in $\nu_\mu$ NC interactions.
The charm background produced in the $\nu_\mu$ interactions in the analyzed sample is $0.007\pm0.004$ events, that produced in the $\nu_e$ interactions is less than $10^{-3}$ events.
The background from hadron interactions has been evaluated with a FLUKA based MC code, updated with respect to the proposal simulations. The kink probability to occur in 2 mm lead integrated over the $\nu_\mu$ NC hadronic spectrum yield a background probability of $(1.9 \pm 0.1) \times 10^{-4} $/NC. This probability decreases to $(3.8 \pm 0.2) \times 10^{-5} $/NC taking into account the cuts on the event global kinematics. This leads to a total of $0.011\pm0.006$ events when misclassified CC events are included. Cross-checks of the 1-prong hadron background estimation have been performed. The tracks of hadrons from a sub-sample of neutrino interactions have been followed far from the primary vertex to search decay-like interactions. A total length of 9 m has been measured and no event has been found in the signal region. This corresponds to a probability over 2 mm lead smaller than $1.54 \times 10^{-3}$ at 90% C.L. Within the low statistics, the $P_t$ distribution agrees with the simulation.
In summary we observed 1 event in the 1-prong hadron $\tau$ decay channel, with a background expectation $0.018\pm0.007~(syst)$. The probability to observe 1 event due to a background fluctuation is 1.8% (2.36$\sigma$). As all $\tau$ decay modes were included in the search (1-prong $\mu$, 1-prong e, 1-prong hadron, 3-prong hadron), the total background becomes $0.045 \pm 0.023 ~(syst)$, the probability to observe 1 event due to a background fluctuation becomes 4.5% ($2.01 \sigma$).
Conclusion
==========
Data taking in the CNGS beam is going smoothly. The analysis of a sub-sample of the neutrino data taken in the 2008-2009 runs was completed, corresponding to out of $22.5 \times 10^{19}$ proposed pot. Decay topologies due to charmed particles have been observed in good agreement with expectations, as well as several events induced by $\nu_e$ present as a contamination in the $\nu_\mu$ beam. One muon-less event showing a $\tau$ to 1-prong hadronic decay topology has been detected. It is the first $\nu_\tau$ candidate event in OPERA, with a statistical significance of $2.36 \sigma$ (1-prong hadronic decay mode) and $2.01 \sigma$ (all decay modes). Analysis of the 2008+2009 full sample will be completed in two months. Analysis of 2010 events is being performed in parallel. Next CNGS runs will be in 2011 and 2012 (CERN stop in 2013).
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|
---
author:
- 'Yushan Gao, Ander Biguri, and Thomas Blumensath[^1]'
title: 'Block stochastic gradient descent for large-scale tomographic reconstruction in a parallel network'
---
u
Introduction {#sec:introduction}
============
transmission X-ray computed tomography (CT), when using non-standard scan trajectories or when operating with high noise levels, traditional analytical reconstruction techniques such as the filtered backprojection algorithm (FBP) [@sagara2010abdominal; @hoffman1979quantitation] and the Feldkamp Davis Kress (FDK) [@feldkamp1984practical; @rodet2004cone] method are no longer applicable. In these circumstances, less efficient, iterative reconstruction methods can provide significantly better reconstructions [@gervaise2012ct; @wang2008outlook; @deng2009parallelism; @willemink2013iterative]. These methods model the x-ray system as a linear system: $$\y=\A \x+\mathbf{e},
\label{yax}$$ where $\y=[y_1,\cdots,y_r]^T,\x=[x_1,\cdots,x_c]^T$ and $\e=[e_1,\cdots,e_r]^T$ are x-ray projection data, the unknown vectorised image and measurement noise respectively. The system matrix $\A\in \mathbb{R}^{r*c}$ has non-negative elements, which can be computed using Siddon’s method [@jacobs1998fast]. Image reconstruction can then be cast as an optimisation problem [@soleimani2015introduction; @guo2016convergence; @beister2012iterative]: $$\min_\x f(\x)=\min_\x \frac{1}{2}\|\y-\A\x\|_2^2,
\label{min}$$
In many applications, such as industrial CT scanning, the system matrix $\A$ can be enormous [@ni2006review]. Iterative methods apply matrices $\A$ and $\A^T$ to compute “forward projection”(FP) and “back projection”(BP) respectively, to iteratively find an approximate solution to minimize Eq.\[min\]. Note that in realistic applications, due to its size, the matrix $\A$ is never stored [@van2015astra], FP and BP are instead computed ‘on the fly’ using Graphical Processor Units (GPUs).
For our discussion, we classify iterative methods into column action methods and row action methods. Column action methods include iterative coordinate descent (ICD)[@yu2011fast; @benson2010block] and axial block coordinate descent (ABCD) [@fessler2011axial; @kim2012parallelizable]. They divide $\x$ into several blocks and update individual blocks in each iteration using the most recent estimates of all other blocks. Row action methods include Kaczmarz methods(ART) [@li2013adaptive], simultaneous iterative reconstruction technique(SIRT) [@gregor2008computational], and component averaging (CAV) [@censor2001component] and their ordered set variations [@censor2001bicav; @xu2010efficiency]. Unlike column action methods, row action methods divide the projection data $\y$ into several blocks and update all of $\x$ simultaneously using one or several blocks of $\y$. Despite the superior reconstructions achievable with iterative methods in many applications, the high computational complexity remains a significant bottleneck limiting their application in realistic settings. To overcome these issues, parallization is desirable. For example, the ICD algorithm can be run on multiple CPUs [@wang2016high] or on several graphics processing units(GPUs) [@sabne2017model; @wang2017massively]. Parallization of row action methods is straightforward: each node receives a copy of $\x$ and different blocks of $\y$. Each node independently updates $\x$ and message passing between nodes computes weighted sums of partial results [@bilbao2004performance; @flores2012fast]. Compared with column action methods, row action methods is more amenable to parallel processing in a multiple-node network because that different nodes do not have to update the same elements in the error vector $\y-\A\x$[@rui2012evaluation].
A range of row and column action methods have been specifically designed for tomographic reconstruction. Recently, advances in machine learning have also led to significant advances in stochastic optimization and many of these ideas are also applicable to tomographic reconstruction. Most methods here are row action methods. These include stochastic gradient descent [@ruder2016overview], stochastic variance reduced gradient(SVRG) [@johnson2013accelerating], incremental aggregated gradient (IAG) [@IAG] and stochastic average gradient (SAG)[@SAG]. These stochastic algorithms have often been parallelized to operate on large data sets [@recht2011hogwild; @zhao2016fast; @leblond2016asaga; @zhang2015fast].
The BSGD algorithm
==================
In this paper, we develop a parallel row action algorithm specifically for large scale tomographic reconstruction. Whilst previous work in parallel tomographic reconstruction has concentrated on standard tomography, where an object is rotated around a single axis, we are here particularly interested in a setting that allows more general trajectories such as those found in laminographic scanning [@Woodetal2018]. With “large scale” we here mean that both $\y$ and $\x$ are too large to be stored within one computing node. We thus divide $\y$, $\A$ and $\x$ into several blocks and design algorithms in which each node only has partial access to both $\y$ and $\x$ at each iteration. Let $\A$ be divided into $M$ row blocks and $N$ column blocks (possibly after row and column permutation). Let $\{I_i\}_{i=1}^M$ be a set of row indices and $\{J_j\}_{j=1}^N$ a set of column indices. $\A_I^J$ thus is a sub-matrix indexed by $I\in \{I_i\}_{i=1}^M$ and $J\in \{J_j\}_{j=1}^N$. The forward X-ray projection process can then be approximated as: $$\begin{bmatrix}
\y_{I_1}\\
\vdots\\
\y_{I_M}
\end{bmatrix}\approx
\begin{bmatrix}
\A_{I_1}^{J_1} & \cdots & \A_{I_1}^{J_N}\\
\vdots & \vdots & \vdots \\
\A_{I_M}^{J_1} & \cdots & \A_{I_M}^{J_N}
\end{bmatrix}
\begin{bmatrix}
\x_{J_1}\\
\vdots \\
\x_{J_N}
\end{bmatrix}\equiv \begin{bmatrix}
\A_{I_1}\\
\vdots \\
\A_{I_M}
\end{bmatrix}\x.
\label{e1}$$
With this partitioning, both column and row action methods can be inefficient in terms of communication between nodes since they all require full access to either all of $\y$ or all of $\x$. For example, in row action methods, the most time consuming operations are the FP and BP [@palenstijn2015distributed]. These projections are parallelizable [@rosen2013iterative]. For the FP $\A_I\x\equiv \sum_{j=1}^N \A_I^{J_j}\x_{J_j}$, each parallel node calculates a forward projection $\A_I^{J_j}\x_{J_j}$. The summation over $j$ is then calculated at a master node or using an ALLREDUCE procedure [@jones2006hybrid]. A similar parallel scheme is also applicable to the BP. If the number of computing nodes is smaller than the number of column or row blocks $N$ or $M$, then the parallel calculations require several communications between data storage and computation nodes, which can be time consuming [@bilbao2004performance; @deng2009parallelism].
To reduce the communication overhead and the algorithm’s dependency on access to all of $\y$ or $\x$, we previously proposed a parallel algorithm called Coordinate-Reduced Steepest Gradient Descent (CSGD) [@gao2018joint] to solve the block linear model Eq.\[e1\]. However, our previous method only converged to a weighted least squares solution. Here we propose an improved algorithm, which we call ‘Block Stochastic Gradient Descent’’ (BSGD). We empirically show that BSGD converges closer to the least squares solution than CSGD. The new method is similar to SAG and accumulates previously calculated direction to obtain the current update direction, but it differs from SAG in that we also incorporate a coordinate descent strategy. In the origin SAG algorithm, the gradient summands must be calculated by accessing all of $\x$ while in BSGD only part of $\x$ is required and updated. Furthermore, we exploit the sparsity of the system matrix $\A$ found in CT imaging and proposed an “importance sampling” strategy for BSGD (BSGD-IM). An automatic parameter tuning strategy is also adopted to tune the step length. Simulation results show that the convergence speed of BSGD-IM is faster than other row action methods such as SAG and SVRG, making BSGD an ideal candidate for distributed tomographic reconstruction.
We derive BSGD for large scale CT reconstruction where a parallel multiple-GPU network is available. The network uses a master-servant architecture and the servants/nodes(i.e. GPUs) in the network have limited access to both projection data $\y$ and reconstructed volume $\x$. To facilitate latter discussions, we define $\r=\y-\A\x$ and let $\r_I$ be the subset of $\r$ representing $\y_I-\A_I\x$.
To motivate our approach, let us consider iterative mini-batch stochastic gradient descent [@li2014efficient] with a decreasing step length $\mu$. At the $k^{th}$ iteration, this method computes: $$\begin{aligned}
& \r_I=\y_I-\A_I\x \\
& \g = 2\A_I^T\r_I \\
& \x^{k+1}=\x^k+\mu\g.\\
\label{SGDITE}
\end{aligned}$$ In our setting where a master or storage node assigns data to servant nodes for processing, if local memory at a servant node is restricted, then each node can only process a partial block $\x_J$ and calculate $\A_I^J\x_J$. To compute $\A_I\x$, we would thus need to repeatedly sent different blocks to the servant nodes before the result is combined in the master node to update $\r_I$. This process is shown in Fig.\[FigParalProtype\].
![In a master-servant network, in each iteration, each servant node receives one block $\x_{J}$ to calculate $\A_I^J\x_J$. The results are accumulated in the master node to update the corresponding residual $\r_I$. If the number of the servant nodes is less than the number of column blocks $N$, then we require repeated communication between servants and the master nodes.[]{data-label="FigParalProtype"}](gao1.pdf){width="55mm"}
Computation of $\A_I^T\r_I$ would need to follow a similar strategy. Instead of updating $\x$ only once the exact residual $\r$ has been computed, our innovation is to compute a stochastic approximation of the residual by only processing a subset of $\x$ in each iteration and using previously computed estimates of $\A_I^J\x_J$ for the blocks not used in this step. The hope is that the increase in uncertainty in the gradient estimate is compensated for by a reduction in computation and communication cost. BSGD thus does not compute all $\A_I^T\r_I$ and $\A_I\x_I$ in each iteration. For a fixed number of servant nodes, let $\alpha$ and $\gamma$ be the fraction of row and column blocks that can be used in parallel computations at any one time. During each iteration, we thus propose to only use $\alpha \gamma MN$ sub-matrices ($\A_I^J$) together with the corresponding data ($\y_I$) and volume($\x_J$) sub-vectors to compute updates. To gradually reduce the error between the stochastic gradient and the true gradient, BSGD adopts a gradient aggregation strategy that is similar to that of SAG. As we show below for our CT reconstruction problem, this accumulation strategy enables the algorithm to converge with a constant step length $\mu$.
The BSGD algorithm is shown in Algo.\[alg1\]. When $\gamma=1$, the method becomes SAG. In this paper we mainly focus on $\alpha$ and $\gamma<1$, which allows us to use a reduced number of servant nodes.
Initial: $\g=\mathbf{0},\{\hat{\g}^i\}_{i=1}^M=\{\mathbf{0}\},\{\z^j\}_{j=1}^N=\{\mathbf{0}\}, \r=\y$, $\mu=$ const. $\x_{est}=\mathbf{0}$. randomly select $\alpha M$ row blocks from $\{I_i\}_{i=1}^M$ and $\gamma N$ column blocks from $\{J_j\}_{j=1}^N$ $\z_{I_i}^j=\A_{I_i}^{J_j}{\x_{est}}_{J_j}$ $\r=\y-\sum_{j=1}^N\z^j$ ${\hat{\g}}_{J_j}^i=2(\A_{I_i}^{J_j})^T\r_{I_i}$ $\g=\sum_{i=1}^M\hat{\g}^i$ ${\x_{est}}_{J_j}={\x_{est}}_{J_j} + \mu\g_{J_j}$
Improving BSGD performance
--------------------------
There are two tricks to improve BSGD performance. The first trick is the use of an importance sampling strategy which we call “BSGD-IM”. In tomographic reconstruction, the matrix $\A$ is often sparse with different blocks $\A_I^J$ often varying widely in their sparsity. This sparsity can be enhanced further for 3D tomographic problems if we partition $\y$ such that each partition is made up of a selection of sub-blocks, where each sub-block corresponds to a partition of the X-ray detector at one projection angle as shown in Fig. \[Impor\].
![Example cone beam CT setting. The 3D volume is divided into 8 sub volumes. In the cone beam scanning geometry, the projection of one sub-block is mainly concentrated in a small area on the detector. If the detector is also divided into 4 sub-areas, then the currently selected sub-volume (bold red frame) is mainly projected onto the top-left and top-right sub-detector area, i.e. $\hat{I}_1$ and $\hat{I}_2$ area.[]{data-label="Impor"}](gao2.pdf){width="60mm"}
Inspired by this property, BSGD-IM breaks each single projection into several sub-projections (4 sub-projections in Fig.\[Impor\]) and only samples 1 sub-projections for the selected sub-volume $\x_J$. This sampling is not done uniformly but is based on a selection criteria that uses the relative sparsity of each matrix $\{\A_{\hat{I}_i}^J\}$, i.e. the denser a sub-matrix is, the higher the probability that it is selected. As we do not have access to matrix $\A$, to estimate the sparsity pattern of $\A$, BSGD-IM computes the fraction of a volume block’s projection area on each sub-detector. When the row block $I_i$ contains several projection angles, the importance sampling strategy is repeatedly applied to each projection angle contained in the row block. The advantage of BSGD-IM is that it provides each computation node more projection angles within one row block than BSGD when a nodes’ storage capacity is limited and thus reduces the row block number $M$ as well as the total storage requirement. For example, if we assume that one GPU can only process a single projection in the original BSGD, then BSGD-IM with the partition of Fig.\[Impor\] can use four projection angles by choosing one detector sub-areas for each projection angle.
BSGD-IM is shown in Algo.\[alg2\]. Whilst importance sampling speeds up initial convergence, it also introduces a bias is the stochastic gradient due to the inhomogeneous sampling. To overcome this, it is suggested that after initial fast convergence, the last few iterations should be run without importance sampling.
Initial: $\g=\mathbf{0},\{\hat{\g}^i\}_{i=1}^M=\{\mathbf{0}\},\{\z^j\}_{j=1}^N=\{\mathbf{0}\}, \r=\y$, $\mu=$ const. $\x_{est}=\mathbf{0}$. randomly select $\alpha M$ row blocks from $\{I_i\}_{i=1}^M$ and $\gamma N$ column blocks from $\{J_j\}_{j=1}^N$. For each column block $\x_{J_j}$, importance sample one sub-detector area from each single projection view. All indexes represented by those selected sub-detector areas form the row index set $\hat{I}_t$. $\z_{\hat{I}_t}^j=\A_{\hat{I}_t}^{J_j}{\x_{est}}_{J_j}$ $\r=\y-\sum_{j=1}^N\z^j$ ${\hat{\g}}_{J_j}^i=2(\A_{\hat{I}_t}^{J_j})^T\r_{\hat{I}_t}$ $\g=\sum_{i=1}^M\hat{\g}^i$ ${\x_{est}}_{J_j}={\x_{est}}_{J_j} + \mu\g_{J_j}$
The second trick is to use automatic parameter tuning. Broadly speaking, up to a limit, increasing $\mu$ increases convergence speed. However, in practice, it is difficult to determine the upper limit. As a result, in realistic large scale tomographic reconstruction, instead of using a fixed step-length $\mu$, we developed an automatic parameter tuning approach. Parameter tuning is not a new concept in machine learning and optimization. For example, the hypergradient descent[@baydin2017online] or the Barzilai-Borwein (BB) method [@tan2016barzilai] can be used for SGD or SVRG. However these methods are not directly applicable to BSGD, as they require updates to all of $\x$ in each iteration. Furthermore, BSGD uses dummy variables $\z$ to stores information about previous $\x$. Due to this, the stochastic gradient $\g$ of BSGD is much noisier than the estimate obtained by traditional stochastic gradient methods. We here proposed an automatic parameter tuning methods that is different from the BB method or hypergradient descent method. It only exploits the parameters generated during the iteration process: the residual $\r$ and iteration direction $\g$, as shown in Algo.\[cre1\].
$\epsilon$ and $\delta$ are positive constants. At each $k^{th}$iteration where $mod(k,M)==0$, sum up all $\g$ in the past $M$ epochs as an effective update direction (EUD) $\overline{\g}^{k/M}$. calculate the inner product between two consecutive $\overline{\g}$ as $\theta^{k/M}=\frac{(\overline{\g}^{k/M})^T\overline{\g}^{k/M-1}}{\|\overline{\g}^{k/M}\|\|\overline{\g}^{k/M-1}\|}$. $\mu=(1+\epsilon)*\mu$ $\mu=(1-\delta)\mu$
This automatic parameter tuning is applied after $M$ iterations. It tests whether $\r$ decreased during the past $2M$ epochs, in which case $\mu$ is increased by $1+\epsilon$. To reduce $\mu$, using a similar condition on $\r$ alone (i.e. line 8 in Algo.\[cre1\], named as “criteria 1”) was not found to be sufficient to ensure convergence. We thus use an additional criteria (line 9 in Algo.\[cre1\], named as “criteria 2”). The criteria 2 is motivated by the general parameter tuning methods that computes inner-products between adjacent gradients and determines to increase or decrease $\mu$ according to the positivity of the inner-product [@baydin2017online; @plakhov2004stochastic]. We thus compare the inner-products of two gradients. To do this, we accumulate several stochastic gradients during a period of several iterations ($M$ epochs in Algo.\[cre1\]) to compute an effective update direction (EUD) $\overline{\g}$ to reduce the stochastic error variance. We have observed that when BSGD converges with a properly chosen fixed $\mu$, then the change of two adjacent EUDs do not vary significantly. On the contrary, these two directions vary significantly when BSGD suffers from oscillatory behaviour or an increase in the norm of $\r$. This is due to our method using some old values $\z$ in the calculation of each update. If the change in $\x$ is not too large, then these old values for $\z$ are good approximations to the current values. As a result, the two EUDs, should also be similar to each other. If we assume that during $M$ epochs it is likely that all of $\x$ (and thus all of the $\z$’s) have been updated, then two EUDs can be computed from the previous $2M$ epochs. The inner product of the two normalized EUDs should always be close to 1. If not, it means that the step length is too big and the iteration is likely to diverge.
For both increasing and decreasing $\mu$ part, the frequency of parameter changing is $M$ epochs rather than 1 epoch. One reason is that the high stochastic noise effect in the gradient update can be reduced after a period of epochs. Another reason is that the calculation of $\|\r\|$ can be time consuming when the size of $\r$ is large, reducing the frequency of computation on $\|\r\|$ is beneficial to save the reconstruction time. We have experimentally validated that setting the test frequency to $M$ leads to a good compromise between increased computational demand and improved overall convergence speed.
Incorporating TV regularization into BSGD
-----------------------------------------
When we have few projections, a TV regularization term is often used to increase the reconstruction quality. Reconstruction with a TV regularization term often minimizes a quadratic objective function plus a non-smooth TV-regularization term: $$\underbrace{(\y-\A\x)^T(\y-\A\x)}_{f(\x)}
+\underbrace{2\lambda \text{TV}(\x)}_{g(\x)},
\label{TVequ}$$ where $\lambda$ is a relaxation parameter and $\text{TV}(\x)$ is the total variation (TV) of $\x$. For 2D images, the total variation penalty can be defined as: $$\text{TV}(\x)=\sum_{c,d}\sqrt{(x_{c,d}-x_{c-1,d})^2+(x_{c,d}-x_{c,d-1})^2},$$ where $x_{c,d}$ is the intensity of image pixel in row $c$ and column $d$.
Traditional methods, including ISTA[@combettes2005signal] and FISTA[@beck2009fast], minimize the TV regularized objective function with two steps: Each iteration starts with using the gradient of $f(\x)$ to reduce the data fidelity, i.e. to reduce $f(\x)$, followed by a TV-based de-noising procedure. We empirically show that BSGD is able to replace the gradient descent (GD) step. Since BSGD updates only some components of $\x$ with partial projection data in each iteration, the TV-based de-noising procedure is only performed after a period of time, enabling the computation load of BSGD is scalable to that in ISTA or FISTA. The algorithm is shown in Algo.\[BSGDTV\].
Initial: $\g=\mathbf{0},\{\hat{\g}^i\}_{i=1}^M=\{\mathbf{0}\},\{\z^j\}_{j=1}^N=\{\mathbf{0}\}, \r=\y$, $\mu=$ const. $\x_{est}=\mathbf{0}$. randomly select $\alpha M$ row blocks from $\{I_i\}_{i=1}^M$ and $\gamma N$ column blocks from $\{J_j\}_{j=1}^N$ $\z_{I_i}^j=\A_{I_i}^{J_j}{\x_{est}}_{J_j}$ $\r=\y-\sum_{j=1}^N\z^j$ ${\hat{\g}}_{J_j}^i=2(\A_{I_i}^{J_j})^T\r_{I_i}$ $\g=\sum_{i=1}^M\hat{\g}^i$ ${\x_{est}}_{J_j}={\x_{est}}_{J_j} + \mu\g_{J_j}$ $\x_{est}=\arg\min_{\t} \Vert\t-\x_{est}\Vert^2+2\mu\lambda \text{TV}(\t)$
Results
=======
We start the evaluation of our method using a 2D scanning setup (sections \[A\] to \[D\]) to explore convergence properties of BSGD. In the final section (\[E\]), we then look at a more representative 3D cone beam setting.
BSGD convergence {#A}
----------------
We here use the scanning geometry as shown in Fig.\[Scanning geometry\].
![A standard 2D scanning geometry with a Shepp-Logan phantom, where P is the x-ray source, O is the centre of the object and the rotation centre. D is the centre of the detector. Source and detector rotate around the centre and take measurements at different angles. The linear detector is evenly divided into to sub-areas DE and DF, which will be used in importance sampling discussed later. In this paper, unless particularly mentioned, the size of the image pixels (or voxels in 3D) and the detector pixel size are both 1. []{data-label="Scanning geometry"}](gao3.pdf){width="60mm"}
In our first experiments we set $K$ to 16, $OP$ and $OD$ is 50, the detector has 30 elements and the angular interval is $10^\circ$ so that $\A\in\mathbb{R}^{1080*256}$. This model is used from section \[A\] to section \[AutParaTun\].
We first examine if BSGD (without additional regularisation) converges to the least square solution of the linear model. We here set $M=4$ and $N=2$. $\alpha$ and $\gamma$ are initially set to 1. We add Gaussian noise to the data so that the Signal to Noise ratio (SNR) of $\y$ is 17.5 dB. The distance to the least square solution (**DS**) is defined as $$DS=\Vert \x_{rec}-\x_{lsq} \Vert,$$ where $\x_{rec}$ is the reconstructed image vector and $\x_{lsq}$ is the least square solution obtained here using the LSQR method.
We compared BSGD with other mature methods including SIRT and CAV as well as with our previous algorithm CSGD. The results are shown in Fig.\[ExamSimu38\], where we see the linear convergence of our method to the least squares solution.
![In contrast to BSGD, SIRT, CAV and CSGD do not achieve the least square solution. []{data-label="ExamSimu38"}](gao4.pdf){width="70mm"}
All parameters in the methods were well tuned to ensure the fastest convergence rate. We see that BSGD not only approaches the least square solution, it also shows a faster convergence rate compared to the other methods.
The initial simulations used $\alpha = \gamma =1$. We thus next study the more realistic setting where $\alpha$ and $\gamma$ are smaller than 1.The results, shown in Fig.\[Algo2toLsqb\], show that even in this scenario, BSGD still approaches the least square solution. We here divided $\A$ into 64 blocks, using different partitions that varied in the numbers of row and column blocks (2 row blocks and 31 columns blocks, 4 row blocks and 16 column blocks and 8 row blocks and 8 column blocks). We set $\alpha$ and $\gamma$ to $\frac{1}{M}$ and $\frac{1}{N}$ respectively. To measure convergence speed and communication costs between master node and computation node in a realistic parallel network, we plot the $DS$ as a function of the number of multiplications of a vector by $\A_I^J$(or $(\A_I^J)^T$), which is proportional to the number of forward/backwards projections as well as the corresponding communication time. We see from Fig.\[Algo2toLsqb\] that BSGD performs better in terms of reaching the least squares solution. We also see differences in the convergence speed depending on the way in which we partition the matrix.
![BSGD shows better converge compared to CSGD in terms of achieving the least squares solution. Different ways to partition the rows and columns lead to different convergence speeds.[]{data-label="Algo2toLsqb"}](gao5.pdf){width="70mm"}
Setting $\alpha,\gamma,M$ and $N$
---------------------------------
We nexrt study the influence of $\alpha$ and $\gamma$ for a fixed partition of $\A$. In Fig.\[Algo2toLsqb\] $\alpha$ and $\gamma$ were set to an extreme value where only one node is used. In realistic applications, several nodes might be available. Assume we have 4 nodes so that $MN\alpha\gamma =1$. Simulations, shown in Fig.\[fig1\], show that reducing $\gamma$ slows down convergence.
Based on these and similar results, we suggest to use the following selection criteria for $\alpha$,$\gamma$ $$\left\{
\begin{array}{lr}
\gamma=\min\{1,\frac{NodeNum}{N}\},\\
\alpha=\frac{NodeNum}{MN\gamma}, & \\
\end{array}
\right.
\label{Eq1}$$ where the $NodeNum$ is the number of separate computation nodes.
According to Eq.\[Eq1\], we divide $\A$ into different numbers of row and column blocks and compare convergence speed. The results shown in Fig.\[CompaDiffMN\] indicate that BSGD does not encourage the partition in the column direction.
![ BSGD does not encourage the partition in the column direction. When $N$ increases, convergence speed decreases.[]{data-label="CompaDiffMN"}](gao8.pdf){width="70mm"}
Whilst this indicates that we do not want to partition in the column direction, different partitions lead to different storage demand. This is shown in Fig.\[fig4\], where we show the amount of storage that is required in the compute nodes and the master node.
Note that in the previous simulations, we kept the product $MN$ fixed as in this case, computation speed per iteration remains constant. However, when storage demand is the limiting factor, then the $m+n$ become limited ($m,n$ are rows and columns of $\A_I^J$). To explore this, we fix $m+n\leq 140$. The results, shown in Fig.\[CompareFixSto\] for different values of $M$ and $N$ are now plotted against the number of times we compute multiplications by matrices $\A_I^J$, multiplied by $mn$ to normalise computation time differences.
We here assume that there are only two nodes in the parallel network and the selection criteria for $\alpha$ and $\gamma$ follows Eq.\[Eq1\]. The convergence speed and storage demand in the master node is shown in Fig.\[CompareFixSto\]. Note that when $M=135,N=2$ (green line), BSGD becomes SAG. For large $N$, storage demand in the master node increases significantly. Luckily, convergence is the fastest for moderate values of $N$.
Automatic parameter tuning {#AutParaTun}
--------------------------
We first demonstrate that criteria 1 ( line 8 in Algo \[cre1\]) on its own is not sufficient for parameter tuning. If $\delta$ is small, $\mu$ is not effectively reduced, whilst for large $\delta$, the $\mu$ tends to become too small and the iterations get ‘stuck’. Simulation results to prove this are shown in Fig.\[CompareFixSto2\].
![$\mu_0$ is the initial step length. Different color stands for different $\mu_0$. Only using criteria 1 does not overcome issues with parameter selection. The dashed lines use a large $\delta$ leading to the algorithm getting stuck, whilst solid lines use a small $\delta$ leading to oscillation and divergence.[]{data-label="CompareFixSto2"}](gao13.pdf){width="70mm"}
Automatic parameter tuning works if we combine criteria 1 with criteria 2 (line 9 in Algo \[cre1\] ). We here use $\delta =0.4$. The compute nodes is still set as 2. The results, shown in Fig.\[auto\], demonstrate that automatic tuning (dashed lines) allows faster convergence than the original method (dashed line) and the method using criteria 1 only (solid line).
Incorporating the TV constraint {#D}
-------------------------------
To explore Total Variation regularisation, in Fig.\[Scanning geometry\], the $K$ is increased to 64. $OP$ and $OD$ is 100 and the detector has 180 elements. The scanning angle increment is $1^{\circ}$ and the point source only rotates through $180^{\circ}$. In this section, the $\textbf{
Signal Noise Ratio}$ (SNR) of $\x$ is defined as $20\log_{10}\frac{\|\x_{true}\|}{\|\x_{dif}\|}$, where $\Vert\x_{dif}\Vert$ is the $\ell_2$ norm of the difference between reconstructed image vector and the original vector $\x_{true}$. The $\lambda=0.1$ and the projection $\y$ are influenced by Gaussian noise, with an SNR (similar definition holds) of 28.8dB.
The change of SNR is shown in Fig.\[Compare2\].
![Comparison between BSGD-TV, FISTA, ISTA and Gradient Descend (GD) (without TV constraint). The step length $\mu$ for ISTA, GD and BSGD-TV is 0.0004. BSGD-TV uses $M=20,N=4,\alpha=0.05 and \gamma=0.5$ with 2 nodes available in the network. $\lambda$ in Eq.\[TVequ\] is $0.1$. The low SNR of GD suggests the necessity of incorporating the TV norm here. It can be seen that the BSGD-TV converge faster than FISTA methods.[]{data-label="Compare2"}](gao16.pdf){width="70mm"}
We here plot SNR against effective epochs, where an effective epoch is a normalized iteration count that corrects for the fact that the stochastic version of our algorithm only updates a subset of elements at each iteration.Comparing BSGD-TV against FISTA and ISTA, BSGD-TV shows a faster convergence speed, even though it only update blocks of $\x$ in each iteration and only applies the TV-based de-noising after a period of iterations. A visual comparison after a fixed number of effective epochs is shown in Fig. \[figcompare2\].
Note that ADMM-TV [@boyd2011distributed] is another algorithm that can be distributed over several nodes, however, our previous work has demonstrated that ADMM-TV is significantly slower than TV constraint version of CSGD [@gao2018joint] and thus has not been included in the comparison here.
Applying BSGD in 3D CT reconstruction {#E}
-------------------------------------
In this section, a workstation containing two NVIDIA GEFORCE GTX 1080Ti GPUs is adopted to demonstrate BSGD’s performance on realistic data sizes. We used a 3D cone beam scanning geometry similar to the 2D simulation as defined in Fig.\[Scanning geometry\]. In simulation, $OP=1536$ mm, $OD=1000$ mm and the detector is a square panel with side length($EF$) of 400 mm. The reconstruction volume is a cube with side length of 256 mm. The point source and the centre of the square detector are located at the middle slice of the 3D volume. They rotate around the volume horizontally for a full circle with angular increments of $1^\circ$. We test BSGD for increasing data sizes (see Table \[Table\_Parameter\]) and compare it to other methods.
case1 case2 case3
------------------------- ------------------- ------------------- -------------------
Detector 400\*400 1000\*1000 2000\*2000
Volume 256\*256\*256 512\*512\*512 1024\*1024\*1024
Rows of $\mathbf{A}$ $5.76\times 10^7$ $3.6\times 10^8$ $1.44\times 10^9$
Columns of $\mathbf{A}$ $1.68\times 10^7$ $1.34\times 10^8$ $1.07\times 10^9$
: Three different reconstruction scales
\[Table\_Parameter\]
The simulations are performed in MATLAB R2016b together with a purpose built version of the TIGRE toolkit [@biguri2016tigre] that uses OpenMP to synchronize the two GPUs, performing two FPs or two BPs simultaneously. Simulation results show that BSGD with importance sampling and automatic parameter tuning can be applied in realistic settings and is faster than existing methods in a multi-GPU work station.
### Time required for FP, BP and other operations
We start by looking at the time required to compute FB/BP and contrast this time to the other computational overheads of the method. For one GPU, we process two data blocks one after the other whilst for two GPUs two blocks are computed in parallel. We here divided the image into 8 cubic blocks ($N=8$) and randomly partitioned the 360 projections into 5 groups ($M=5$). As we here look at computation speed of individual FP and BPs, no noise was added to the projections.
The overall time spent on FP and BP per iteration are shown in Fig.\[ProjectionCompare\]. The measurements here are averaged over 10 repeated runs.
We also measured the proportion of time each iteration of BSGD spent on FP and BP during reconstruction (See Fig.\[percentage\]), which shows that for increasing problem sizes, FP and BP become increasingly smaller fractions of overall cost. As data transfer and other operations are similar in the 1 and 2 GPU settings, this further demonstrates that multi-GPU reconstruction is beneficial to reduce the time spent on FP/BP.
### Quality of reconstruction
We next look at the quality of reconstruction for the three scenarios. Since there are two GPUs are available, we thus set $M=5,N=8,\alpha=\frac{1}{5}$ and $\gamma=\frac{2}{8}$. We here measure quality in terms of SNR. The results are shown in Fig.\[FigSnrLarge\].
Reconstructed slices are shown in Fig.\[figsubfigIma\].
![Slices from the 3D reconstruction for different problem dimensions, showing the effectiveness of BSGD under different cases.[]{data-label="figsubfigIma"}](gao27.pdf){width="80mm"}
### Comparison with other methods
In this section, we demonstrate the advantage of BSGD compared with other methods. A cubic skull skeleton provided by the TIGRE toolkit was used and reconstructed using different methods. The object to detector distance was 536 mm, source to object distance was 1000 mm and we again collected 360 equally spaced projections and reconstructed onto a 256 by 256 by 256 grid. The detector used $512 \times 512$ pixels. The side length of each voxel and detector pixel were 1 mm, so that $\A\in\R^{9.4\times 10^7* 1.7\times 10^7}$. The projection data $\y$ is deteriorated by white Gaussian noise with SNR of 28.1 dB. In our simulations, we assume that each computation node (GPU) can only process $\frac{1}{8}$ of the volume and 18 projections. Since in the previous simulation we have demonstrated that BSGD can outperform CAV and SIRT, we here compared BSGD-IM with SAG, SVRG, GD, GD-BB[@barzilai1988two], FISTA and ORBCDVD [@wang2014randomized]. Except for BSGD-IM, the other methods divide $\A$ into $20*8$ sub-matrices (i.e. $M=20,N=8$ to enable each row blocks contain 18 projections and each column blocks contain ${\frac{1}{8}}^{th} $volume. $\alpha=\frac{1}{20},\gamma=\frac{1}{4}$ is set according to Eq.\[Eq1\]) and consecutively process the sub-matrices on each GPU one at a time. BSGD-IM, sampling $\frac{1}{4}^{th}$ projection from each projection angle, allows us to divide $\A$ into $5*8$ sub-matrices with $\alpha$ and $\gamma$ set to 0.2 and 0.5. By this division, it is guaranteed that the computation amount for FP and BP of BSGD-IM are the same with the other methods. This is because that despite that the BSGD-IM process 72 projection angles each time, for each projection angle, only $\frac{1}{4}^{th}$ projection data are used for each projection angle. As a result, the actual projection data size is equivalent to $72*\frac{1}{4}=18$ full projection angles, which is the size for other methods.
In the large scale reconstruction case, calculating the least square solution itself can be time consuming, thus using the term $DS$ is inapplicable in realistic case. To reflect the speed of the iteration result approaching to the least square solution, we thus plot $GAP=\|\y-\A\x_{est}\|$ as a function of the number of usages of $A_I^J$ for each FP and BP. This is reasonable since the least square solution minimizes $GAP$ and a faster downward trend suggests a faster reconstruction speed. Convergence results are shown in Fig.\[Compare\].
![BSGD-RAN is similar to BSGD-IM, but it uniformly selects the sub-projections at each projection angle while BSGD-IM selects sub-projections based on sub-matrix sparsity. Both BSGD methods use automatic parameter tuning. The parameters in the other methods were optimised to ensure optimal performance.[]{data-label="Compare"}](gao28.pdf){width="70mm"}
It can be seen that BSGD-IM is faster than the other methods. Reconstruction results are shown in Fig.\[figcompare\], where we show a subsection of a 2D slice after 2000 forward and backward projections.
Fixed point analysis
====================
In this part we show that BSGD has a single fixed point at the least square solution of the optimisation problem. The system matrix $\A\in\mathbb{R}^{r*c}$, has $M$ row blocks and $N$ column blocks. We vectorise the sets $\{\z^j\}_{j=1}^N$ and $\{\g^i\}_{i=1}^M$ and put their elements into the vector $\overline\z\in\mathbb{R}^{Nr*1}$ and $\overline\g\in\mathbb{R}^{Mc*1}$. $\overline\A\in \mathbb{R}^{Nr*c}$ is a deformation of $\A$, defined as: $$\overline\A = \begin{bmatrix}
\A_{I_1}^{J_1} & \mathbf{0}& \hdots& \mathbf{0}\\
\vdots & \vdots & & \vdots \\
\A_{I_M}^{J_1} & \mathbf{0} & \hdots & \mathbf{0}\\
\mathbf{0} & \A_{I_1}^{J_2} & \hdots & \mathbf{0}\\
\vdots & \vdots & & \vdots \\
\mathbf{0} & \A_{I_M}^{J_2} & \hdots & \mathbf{0}\\
\mathbf{0} & \mathbf{0} & & \A_{I_1}^{J_N}\\
\vdots &\vdots & &\vdots\\
\mathbf{0} & \mathbf{0}&\hdots & \A_{I_M}^{J_N}
\end{bmatrix}$$ $\overline{\A^T}\in\mathbb{R}^{Mc*r}$ is a similar deformation of $\A^T$. Let us also introduce the matrix $\overline{\I}_{Nr}=[\I_r,\I_r,\hdots,\I_r]\in\mathbb{R}^{r*Nr}$, where we concatenate $N$ identity matrices $\I_r$ each of size $r*r$. With this notation we obtain: $$\begin{aligned}
& \overline{\I}_{Nr}\overline{\A}=\A,\\
& \overline{\I}_{Mc}\overline{\A^T}=\A^T,\\
& \sum_{j=1}^N\z^j=\overline{\I}_{Nr} \overline{\z},\\
& \g = \overline{\I}_{Mc} \overline{\g},\\
\end{aligned}
\label{overlineA}$$ where the definition of $\g$ and $\{\z^j\}$ can be found in the algorithm description. To encode the random updates over subsets of index pairs $I_i,J_j$, we introduce the random matrices $\bR_1\in \mathbb{R}^{Nr*Nr}$, $\bR_2\in \mathbb{R}^{Mc*Mc}$ and $\bR_3\in \mathbb{R}^{c*c}$ , which are diagonal matrices whose diagonal entries are either $0$ or $1$. With this notation, we can write the update of $\z$, $\g$ and $\x$ as: $$\overline{\z}^{k+1} = \overline{\z}^k + \bR_1 \left[ \overline\A\x^k-\overline{\z}^k \right],
\label{Updaz}$$ $$\overline{\g}^{k+1} = \overline{\g}^k + \bR_2 \left[ \overline{\A^T} \left(\y-\overline{\I}_{Nr}\overline{\z}^{k+1}\right)-\overline{\g}^k \right]
\label{Updag}$$ and $$\x^{k+1} = \x^k + \mu\bR_3 \overline{\I}_{Mc} \overline{\g}^{k+1},
\label{Updax}$$ where $k$ is the epoch number. Inserting Eq.\[Updaz\] into Eq.\[Updag\] and then Eq.\[Updag\] into Eq.\[Updax\], the recursion in Eq.\[EquMat\] is obtained: $$\begin{bmatrix}
\overline{\z}^{k+1} \\
\overline{\g}^{k+1} \\
\x^{k+1}
\end{bmatrix} = \mathbf{M}\begin{bmatrix}
\overline{\z}^k \\
\overline{\g}^k \\
\x^k
\end{bmatrix}+\begin{bmatrix}
\mathbf{0} \\
\bR_2\overline{\A^T}\y \\
\mu\bR_3\overline{\I}_{Mc}\bR_2\overline{\A^T}\y,
\end{bmatrix},
\label{EquMat}$$ where $\mathbf{M}$ is
In the fixed point analysis, note that for any fixed point $\x^{\star}$, the random updates of $\z$ mean that we require that $\z_{I_i}^j=\A_{I_i}^{J_j}\x_J^\star$ for all $i$ and $j$, so that the fixed $\z^{\star}$ must be of the form $\z^{\star}=\overline{\A}\x^\star$. So we need: $$%\begin{align}
\begin{bmatrix}
\overline{\A}\x^{\star} \\
\overline{\g}^{\star} \\
\x^{\star}
\end{bmatrix} = \mathbf{M}\begin{bmatrix}
\overline{\A}\x^{\star} \\
\overline{\g}^{\star} \\
\x^\star
\end{bmatrix}+\begin{bmatrix}
\mathbf{0} \\
\bR_2\overline{\A^T}\y \\
\mu\bR_3\overline{\I}_{Mc}\bR_2\overline{\A^T}\y
\end{bmatrix}.
\label{EquMat2}
%\end{align}$$ Since the first line of Eq.\[EquMat2\] is an identity, we only focus on the second and third line. The second line can be expressed as: $$\bR_2 \left( \overline{\A^T} (\y- \A\x^*) - \overline{\g}^*\right)=\mathbf{0}.$$ As $\bR_2$ is a random diagonal matrix, this implies that $ \g^*=\overline{\A^T} (\y- \A\x^*) $. The third line, after the deformation, can be expressed as: $$\bR_3\overline{\I}_{Mc}\bR_2\left( \overline{\A^T}\left( \y-\A\x^{\star}\right)-\overline{\g}^\star \right)+\bR_3\overline{\I}_{Mc}\overline{\g}^\star=\mathbf{0}$$ which suggest that $\g^\star\equiv\overline{\I}_{Mc}\overline{\g}^\star\equiv2\A^T(\y-\A\x^\star)=\mathbf{0}$. This proves that the fixed point is the least square solution. Our empirical results show convergence to the fixed point. A theoretical analysis and formal convergence proof is in preparation.
Conclusion
==========
BSGD can be viewed as an improvement of our previous CSGD algorithm. It mainly focus on the case where a distributed network is adopted to reconstruct a large scale CT image/volume in parallel and the nodes in the network have limited access to both projection data and volume. When noise is Gaussian type, which is a common case in CT reconstruction area, iteration results obtained by BSGD approaches closer to the least square solution than other mature CT reconstruction algorithms such as SIRT, CAV and our previously CSGD method. Compared with the other optimization algorithm proposed in machine learning area, such as SVRG, ORBCDVD, the BSGD has higher computation efficiency. It also has the ability to address the sparse view CT reconstruction by combining itself with TV regularization. Simulations prove that both BSGD and BSGD-TV have the ability to be applied on a distributed network.
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[^1]: Manuscript received March 27, 2019.
|
---
abstract: 'Let $\mathcal{A}$ be a laterally complete commutative regular algebra and $X$ be a laterally complete $\mathcal{A}$-module. In this paper we introduce a notion of passport $\Gamma(X)$ for $X$, which consist of uniquely defined partition of unity in the Boolean algebra of idempotents in $\mathcal{A}$ and the set of pairwise different cardinal numbers. It is proved that $\mathcal{A}$-modules $X$ and $Y$ are isomorphic if and only if $\Gamma(X) = \Gamma(Y)$.'
author:
- 'Vladimir I. Chilin, Jasurbek A. Karimov'
title: Classification of modules over laterally complete regular algebras
---
*Key words:* Commutative regular algebra, Homogeneous module, Finite dimensional module
*Mathematics Subject Classification (2000):* MSC 13C05, MSC 16D70
Introduction {#intro}
============
J. Kaplansky [@lit8] introduced a class of $AW^\ast$-algebras to describe $C^\ast$-algebras, which is close to von Neumann algebras by their algebraic and order structure. The class of $AW^\ast$-algebras became a subject of many researches in the operator theory (see review in [@lit7]). One of the important results in this direction is the realization of an arbitrary $AW^\ast$-algebra $M$ of type $I$ as a $\ast$-algebra of all linear bounded operators, which act in a special Banach module over the center $Z(M)$ of the algebra $M$ [@lit9]. The Banach $Z(M)$-valued norm in this module is generated by the scalar product with values in the commutative $AW^\ast$-algebra $Z(M)$. Later, these modules were called Kaplansky-Hilbert modules (KHM). Detailed exposition of many useful properties of KHM is given, for example, in ([@lit5], 7.4). One of the important properties is a representation of an arbitrary Kaplansky-Hilbert module as a direct sum of homogeneous KHM ([@lit10], [@lit5], 7.4.7).
Development of the noncommutative integration theory stimulated an interest to the different classes of algebras of unbounded operators, in particular, to the $\ast$-algebras $LS(M)$ of locally measurable operators, affiliated with von Neumann algebras or $AW^\ast$-algebras $M$. If $M$ is a von Neumann algebra, then the center $Z(LS(M))$ in the algebra $LS(M)$ identifies with the algebra $L^0(\Omega, \Sigma, \mu)$ of all classes of equal almost everywhere measurable complex functions, defined on some measurable space $(\Omega, \Sigma, \mu)$ with a complete locally finite measure $\mu$ ([@lit6], 2.1, 2.2). If $M$ is an $AW^\ast$-algebra, then $Z(LS(M))$ is an extended $f$-algebra $C_\infty(Q)$, where $Q$ is the Stone compact coresponding to the Boolean algebra of central projectors in $M$ [@lit7]. The problem (like the one in the work of J. Kaplansky [@lit9] for $AW^\ast$-algebras) on possibility of realization of $\ast$-algebras $LS(M)$, in the case, when $M$ has the type $I$, as $\ast$-algebras of linear $L^0(\Omega, \Sigma, \mu)$-bounded (respectively, $C_\infty(Q)$-bounded) operators, which act in corresponding KHM over the $L^0(\Omega, \Sigma, \mu)$ or over the $C_\infty(Q)$ naturally arises. In order to solve this problem it is necessary to construct corresponding theory of KHM over the algebras $L^0(\Omega, \Sigma, \mu)$ and $C_\infty(Q)$. In a particular case of KHM over the algebras $L^0(\Omega, \Sigma, \mu)$ this problem is solved in [@lit3], where the decomposition of KHM over $L^0(\Omega, \Sigma, \mu)$ as a direct sum of homogeneous KHM is given. Similar decomposition as a direct sum of strictly $\gamma$-homogeneous modules is given in the paper [@lit7a] for arbitrary regular laterally complete modules over the algebra $C_\infty(Q)$ (the definitions see in the Section 3 below).
The algebra $C_\infty(Q)$ is an example of a commutative unital regular algebra over the field of real numbers. In this algebra the following property of lateral completeness holds: for any set $\{a_i\}_{i \in I}$ of pairwise disjoint elements in $C_\infty(Q)$ there exists an element $a \in C_\infty(Q)$ such that $as(a_i) = a_i$ for all $i \in I$, where $s(a_i)$ is a support of the element $a_i$ (the definitions see in the Section 2 below). This property of $C_\infty(Q)$ plays a crucial role in classification of regular laterally complete $C_\infty(Q)$-mpdules [@lit7a]. Thereby, it is natural to consider the class of laterally complete commutative unital regular algebras $\mathcal{A}$ over arbitrary fields and to obtain variants of structure theorems for modules over such algebras. Current work is devoted to solving this problem. For every faithful regular laterally complete $\mathcal{A}$-module $X$ the concept of passport $\Gamma(X)$, which consist of the uniquely defined partition of unity in the Boolean algebra of idempotents in $\mathcal{A}$ and the set of pairwise different cardinal numbers is constructed. It is proved, that the equality of passports $\Gamma(X)$ and $\Gamma(Y)$ is necessary and sufficient condition for isomorphism of $\mathcal{A}$-modules $X$ and $Y$.
Laterally complete commutative regular algebras {#sec:1}
===============================================
Let $\mathcal{A}$ be a commutative algebra over the field $K$ with the unity $\mathbf{1}$ and $\nabla=\{e\in\mathcal{A}: e^2=e\}$ be a set of all idempotents in $\mathcal{A}$. For all $e, f \in\nabla$ we write $e \leq f$ if $ef=e$. It is well known (see, for example [@lit11 Prop. 1.6]) that this binary relation is partial order in $\nabla$ and $\nabla$ is a Boolean algebra with respect to this order. Moreover, we have the following equalities: $e \vee f = e + f - ef$, $e \wedge f = e f$, $Ce = \mathbf{1} - e$ with respect to the lattice operations and the complement $Ce$ in $\nabla$.
The commutative unital algebra $\mathcal{A}$ is called regular if the following equivalent conditions hold [@lit12 §2, item 4]:
1\. For any $a\in\mathcal{A}$ there exists $b\in\mathcal{A}$ such that $a=a^2b$;
2\. For any $a\in\mathcal{A}$ there exists $e\in\nabla$ such that $a\mathcal{A}=e\mathcal{A}$.
A regular algebra $\mathcal{A}$ is a regular semigroup with respect to the multiplication operation [@lit13 Ch. I, §1.9]. In this case all idempotents in $\mathcal{A}$ commute pairwisely. Therefore, $\mathcal{A}$ is a commutative inverse semigroup, i.e. for any $a\in\mathcal{A}$ there exists an unique element $i(a)\in\mathcal{A}$, which is an unique solution of the system: $a^2x=a$, $ax^2=x$ [@lit13 Ch. I, §1.9]. The element $i(a)$ is called an inversion of the element $a$. Obviously, $ai(a)\in\nabla$ for any $a\in\mathcal{A}$. In this case the map $i:\mathcal{A}\rightarrow\mathcal{A}$ is a bijection and an automorphism (by multiplication) in semigroup $\mathcal{A}$. Moreover, $i(i(a))=a$ and $i(g)=g$ for all $a\in\mathcal{A}$, $g\in\nabla$.
Let $\mathcal{A}$ be a commutative unital regular algebra and $\nabla$ be a Boolean algebra of all idempotents in $\mathcal{A}$. Idempotent $s(a)\in\nabla$ is called the support of an element $a\in\mathcal{A}$ if $s(a)a=a$ and $ga=a$, $g\in\nabla$ imply $s(a) \leq g$. It is clear that $s(a)=ai(a)=s(i(a))$. In particular, $s(e)=ei(e)=e$ for any $e\in\nabla$.
It is easy to show that supports of elements in a commutative regular unital algebra $\mathcal{A}$ satisfy the following properties:
\[art9\_utv\_2\_2\] Let $a,b\in \mathcal{A}$, then
(i). $s(ab) = s(a)s(b)$, in particular, $ab = 0 \Leftrightarrow s(a)s(b) = 0$;
(ii). If $ab = 0$, then $i(a+b) = i(a) + i(b)$ and $s(a+b) = s(a) + s(b)$.
Two elements $a$ and $b$ in a commutative unital regular algebra $\mathcal{A}$ are called *disjoint elements*, if $ab = 0$, which equivalent to the equality $s(a)s(b) = 0$ (see Proposition \[art9\_utv\_2\_2\] (i)). If the Boolean algebra $\nabla$ of all idempotents in $\mathcal{A}$ is complete, $a\in\mathcal{A}$ and $r(a)=\sup\{e\in\nabla: ae = 0\}$, then $$\begin{gathered}
s(a)r(a) = s(a) \wedge r(a) = s(a) \wedge (\sup\{e: ae = 0\})=\\
= \sup\{s(a) \wedge e: ae = 0\} = \sup\{s(a)e: ae = 0\} = 0.\end{gathered}$$ Hence $s(a) \leq \mathbf{1} - r(a)$. If $q = (\mathbf{1} - r(a) - s(a))$, then $aq = as(a)q = 0$, thus $q \leq r(a)$. This yields that $q = 0$, i.e. $s(a) = \mathbf{1} - r(a)$. This implies the following
\[art9\_utv\_2\_25\] Let $\mathcal{A}$ be a commutative unital regular algebra and let $\nabla$ be complete Boolean algebra of idempotents in $\mathcal{A}$. If $\{e_i\}_{i \in I}$ is a partition of unity in $\nabla$, $a,b \in \mathcal{A}$ and $ae_i=be_i$ for all $i \in I$, then $a=b$.
Since $(a-b)e_i=0$ for any $i \in I$, then $\mathbf{1}=\sup\limits_{i \in I}e_i \leq r(a-b)$, i.e. $r(a-b)=\mathbf{1}$. Hence, $s(a-b)=0$, i.e. $a=b$.
Commutative unital regular algebra $\mathcal{A}$ is called *laterally complete* (*$l$-complete*) if the Boolean algebra of its idempotents is complete and for any set $\{a_i\}_{i \in I}$ of pairwise disjoint elements in $\mathcal{A}$ there exists an element $a\in\mathcal{A}$ such that $as(a_i) = a_i$ for all $i \in I$. The element $a\in\mathcal{A}$ such that $as(a_i) = a_i$, $i \in I$, in general, is not uniquely determined. However, by Proposition \[art9\_utv\_2\_25\], it follows that the element $a$ is unique in the case, when $\sup\limits_{i \in I}s(a_i) = \mathbf{1}$. In general case, due to the equality $as(a_i) = a_i = bs(a_i)$ for all $i \in I$ and $a,b\in\mathcal{A}$, it follows that $a\sup_{i \in I}s(a_i) = b\sup_{i \in I}s(a_i)$.
Let us give examples of $l$-complete and not $l$-complete commutative regular algebras. Let $\Delta$ be an arbitrary set and $K^\Delta$ be a Cartesian product of $\Delta$ copies of the field $K$, i.e. the set of all $K$-valued functions on $\Delta$. The set $K^\Delta$ is a commutative unital regular algebra with respect to pointwise algebraic operations, moreover, the Boolean algebra $\nabla$ of all idempotents in $K^\Delta$ is an isomorphic atomic Boolean algebra of all subsets in $\Delta$. In particular $\nabla$ is complete Boolean algebra. If $\{a_j=(\alpha^{(j)}_q)_{q\in\Delta}, j \in J\}$ is a family of pairwise disjoint elements in $K^\Delta$, then setting $\Delta_j = \{q\in\Delta: \alpha^{(j)}_q \neq 0\}$, $j \in J$ and $a=(\alpha_q)_{q\in\Delta} \in K^\Delta$, where $\alpha_q = \alpha^{(j)}_q$ for any $q\in\Delta_j$, $j \in J$, and $\alpha_q = 0$ for $q\in\Delta\setminus\bigcup\limits_{j \in J}\Delta_j$, we obtain that $as(a_j) = a_j$ for all $j \in J$. Hence, $K^\Delta$ is a $l$-complete algebra.
Now let $\mathcal{A}$ be an arbitrary commutative unital regular algebra over the field $K$ and $\nabla$ be a Boolean algebra of all idempotents in $\mathcal{A}$. An element $a\in\mathcal{A}$ is called a *step element* in $\mathcal{A}$ if it has the following form $a = \sum_{k=1}^n\lambda_k e_k$, here $\lambda_k \in K$, $e_k\in\nabla$, $k=1,\dots,n$. The set $K(\nabla)$ of all step elements is the smallest subalgebra in $\mathcal{A}$, which contains $\nabla$. Any nonzero element $a = \sum_{k=1}^n \lambda_k e_k$ in $K(\nabla)$ can be represented as $a = \sum_{l=1}^m\alpha_l g_l$, here $g_l\in\nabla$, $g_l g_k = 0$ when $l \ne k$, $0 \ne \alpha_k \in K$, $l,k=1,\dots,m$. Setting $b = \sum_{l=1}^m\alpha_l^{-1} g_l \in K(\nabla)$, we obtain $a^2b=a$. Hence, $K(\nabla)$ is a regular subalgebra in $\mathcal{A}$. Since $\nabla \subset K(\nabla)$, the Boolean algebra of idempotents in $K(\nabla)$ coincides with $\nabla$. Assume that $\mathrm{card}\,(K) = \infty$ and $\mathrm{card}\,(\nabla) = \infty$. We choose a countable set $K_0 = \{\lambda_n\}_{n=1}^\infty$ of pairwise different nonzero elements in $K$ and a countable set $\{e_n\}_{n=1}^\infty$ of nonzero pairwise disjoint elements in $\nabla$. Let us consider a set $\{\lambda_n e_n\}_{n=1}^\infty$ of pairwise disjoint elements in $K(\nabla)$. Assume that there exists $b=\sum_{l=1}^m \alpha_l g_l\in K(\nabla)$, $0\ne\alpha_l \in K$, $g_l\in\nabla$, $g_l g_k = 0$ and $l \ne k$, $l,k=1,\dots,m$, such that $be_n=bs(\lambda_n e_n)=\lambda_n e_n$. In this case for any positive integer $n$ there exists natural number $l(n)$, such that $\alpha_{l(n)}g_{l(n)}e_n=\lambda_ng_{l(n)}e_n\neq 0$, i.e. $\alpha_{l(n)}=\lambda_n$. This implies that the set $\{\lambda_n\}_{n=1}^\infty$ is finite, which is not true. Hence, the commutative unital regular algebra $K(\nabla)$ is not $l$-complete.
Let $\nabla$ be complete Boolean algebra and let $Q(\nabla)$ be a Stone compact corresponding to $\nabla$. An algebra $C_\infty(Q(\nabla))$ of all continuous functions $a: Q(\nabla) \rightarrow [-\infty,+\infty]$, taking the values $\pm\infty$ only on nowhere dense sets in $Q(\nabla)$ [@lit5 1.4.2], is an important example of a $l$-complete commutative regular algebra.
An element $e\in C_\infty(Q(\nabla))$ is an idempotent if and only if $e(t) = \chi_V(t)$, $t\in Q(\nabla)$, for some clopen set $V \subset Q(\nabla)$, where $$\chi_V(t) = \left\{
\begin{array}{ll}
1, & t \in V\textrm{;}\\
0, & t \notin V\textrm{,}
\end{array} \right.$$ i.e. $\chi_V(t)$ is a characteristic function of the set $V$. In particular, the Boolean algebra $\nabla$ can be identified with the Boolean algebra of all idempotents in algebra $C_\infty(Q(\nabla))$.
If $a\in C_\infty(Q(\nabla))$, then $G(a) = \{t \in Q(\nabla): 0<|a(t)|<+\infty\}$ is open set in the Stone compact set $Q(\nabla)$. Hence, the closure $V(a) = \overline{G(a)}$ in $Q(\nabla)$ of the set $G(a)$ is an clopen set, i.e. $\chi_{V(a)}$ is an idempotent in the algebra $C_\infty(Q(\nabla))$. We consider a continuous function $b(t)$, given on the dense open set $G(a) \cup (Q(\nabla) \setminus V(a))$ and defines by the following equation $$b(t) = \left\{
\begin{array}{ll}
\frac{1}{a(t)}, & t \in G(a)\textrm{,}\\
0, & t \in Q(\nabla) \setminus V(a)\textrm{.}
\end{array} \right.$$ This function uniquely extends to a continuous function defined on $Q(\nabla)$ with values in $[-\infty,+\infty]$ [@lit14 Ch.5, §2] (we also denote this extension by $b(t)$). Since $ab=\chi_{V(a)}$, then $a^2b = a$ and $s(a) = \chi_{V(a)}$. Hence, $C_\infty(Q(\nabla))$ is a commutative unital regular algebra over the field of real numbers $\mathbf{R}$. In this case, the Boolean algebra of all idempotents in $C_\infty(Q(\nabla))$ is complete.
It is known that (see, for example [@lit5 1.4.2]) $C_\infty(Q(\nabla))$ is an extended complete vector lattice. In particular, for any set $\{a_j\}_{j\in J}$ of pairwise disjoint positive elements in $C_\infty(Q(\nabla))$ there exists the least upper bound $a=\sup_{j\in J}a_j$ and $as(a_j) = a_j$ for all $j \in J$. It follows that the commutative regular algebra $C_\infty(Q(\nabla))$ is laterally complete.
In the case, when $\nabla$ is a complete atomic Boolean algebra and $\Delta$ is the set of all atoms in $\nabla$, then $C_\infty(Q(\nabla))$ is isomorphic to the algebra $\mathbf{R}^\Delta$.
The following examples of laterally complete commutative regular algebras are variants of algebras $C_\infty(Q(\nabla))$ for any topological fields, in particular, for the field $\mathbf{Q}_p$ of $p$-adic numbers.
Let $K$ be an arbitrary field and $t$ be the Hausdorff topology on $K$. If operations $\alpha\rightarrow(-\alpha)$, $\alpha\rightarrow\alpha^{-1}$ and operations $(\alpha,\beta)\rightarrow\alpha+\beta$, $(\alpha,\beta)\rightarrow\alpha\beta$, $\alpha,\beta \in K$, are continuous with respect to this topology, we say that $(K,t)$ is a *topological field* (see, for example, [@lit15 Ch.20, §165]).
Let $(K,t)$ be a topological field, $(X,\tau)$ be any topological space and $\nabla(X)$ be a Boolean algebra of all clopen subsets in $(X,\tau)$. A map $\varphi:(X,\tau)\rightarrow(K,t)$ is called *almost continuous* if there exists a dense open set $U$ in $(X,\tau)$ such that the restriction $\varphi|_U: U \rightarrow (K,t)$ of the map $\varphi$ on the subset $U$ is continuous in $U$. The set of all almost continuous maps from $(X,\tau)$ to $(K,t)$ we denote by $AC(X,K)$.
We define pointwise algebraic operations in $AC(X,K)$ by $$(\varphi+\psi)(t) = \varphi(t)+\psi(t);$$ $$(\alpha\varphi)(t) = \alpha\varphi(t);$$ $$(\varphi\cdot\psi)(t) = \varphi(t)\psi(t)$$ for all $\varphi,\psi \in AC(X,K)$, $\alpha\in K$, $t\in X$.
Since an intersection of two dense open sets in $(X,\tau)$ is a dense open set in $(X,\tau)$, then $\varphi+\psi$, $\varphi\cdot\psi \in AC(X,K)$ for any $\varphi, \psi \in AC(X,K)$. Obviously, $\alpha\varphi \in AC(X,K)$ for all $\varphi \in AC(X,K)$, $\alpha \in K$. It can be easily checked that $AC(X,K)$ is a commutative algebra over $K$ with the unit element $\mathbf{1}(t) = 1_K$ for all $t \in X$, where $1_K$ is the unit element of $K$. In this case, the algebra $C(X,K)$ of all continuous maps from $(X,\tau)$ to $(K,t)$ is a subalgebra in $AC(X,K)$.
In the algebra $AC(X,K)$ consider the following ideal $$I_0(X,K) = \{\varphi \in AC(X,K):\ \text{interior of preimage}\ \varphi^{-1}(0)\ \text{is dense in}\ (X,\tau)\}.$$ By $C_\infty(X,K)$ denote the quotient algebra $AC(X,K)/I_0(X,K)$ and by $$\pi: AC(X,K) \rightarrow AC(X,K)/I_0(X,K)$$ denote the corresponding canonical homomorphism.
\[art9\_teor\_2\_26\] The quotient algebra $C_\infty(X,K)$ is a commutative unital regular algebra over the field $K$. Moreover, if $(X,\tau)$ is a Stone compact set, then algebra $C_\infty(X,K)$ is laterally complete, and the Boolean algebra $\nabla$ of all its idempotents is isomorphic to the Boolean algebra $\nabla(X)$.
Since $AC(X,K)$ is a commutative unital algebra over $K$, then $C_\infty(X,K)$ is also a commutative unital algebra over $K$ with unit element $\pi(\mathbf{1})$. Now we show that $C_\infty(X,K)$ is a regular algebra, i.e. for any $\varphi \in AC(X,K)$ there exists $\psi \in AC(X,K)$, such that $\pi^2(\varphi)\pi(\psi)=\pi(\varphi)$.
We fix an element $\varphi \in AC(X,K)$ and choose a dense open set $U\in\tau$, such that the restriction $\varphi|_U: U \rightarrow (K,t)$ is continuous. Since $K\setminus\{0\}$ is an open set in $(K,t)$, then the set $V = U\cap\varphi^{-1}(K\setminus\{0\})$ is open in $(X,\tau)$. Clearly, the set $W = X \setminus \overline{V}^\tau$ is also open in $(X,\tau)$, in this case $V \cup W$ is a dense open set in $(X,\tau)$.
We define a map $\psi: X \rightarrow K$, as follow: $\psi(x) = (\varphi(x))^{-1}$ if $x \in V$, and $\psi(x) = 0$ if $x \in X \setminus V$. It is clear that $\psi \in AC(X,K)$ and $\varphi^2\psi-\varphi \in I_0(X,K)$, i.e. $\pi^2(\varphi)\pi(\psi)=\pi(\varphi)$. Hence, the algebra $C_\infty(X,K)$ is regular.
For any clopen set $U\in\nabla(X)$ its characteristic function $\chi_U$ belongs to $AC(X,K)$, in this case, $\pi(\chi_U)^2 = \pi(\chi_U^2) = \pi(\chi_U)$, i.e. $\pi(\chi_U)$ is an idempotent in the algebra $C_\infty(X,K)$.
Assume that $(X,\tau)$ is a Stone compact and we show that for any idempotent $e \in C_\infty(X,K)$ there exists $U \in \nabla(X)$ such that $e = \pi(\chi_U)$.
If $e\in\nabla$, then $e = \pi(\varphi)$ for some $\varphi \in AC(X,K)$ and $$\pi(\varphi) = e^2 = \pi(\varphi^2),$$ i.e. $(\varphi^2-\varphi) \in I_0(X,K)$. Hence, there exists a dense open set $V$ in $X$ such that $\varphi^2(t)-\varphi(t) = 0$ for all $t \in V$. Denote by $U$ a dense open set in $X$ such that the restriction $\varphi|_U: U \rightarrow K$ is continuous. Put $U_0 = \varphi^{-1}(\{0\})\cap(U \cap V)$, $U_1=\varphi^{-1}(\{1_K\})\cap(U \cap V)$. Since $U_0 \cap U_1 = \emptyset, U_0 \cup U_1 = U \cap V \in \tau$ and the sets $U_0$, $U_1$ are closed in $U \cap V$ with respect to the topology induced from $(X, \tau)$, it follows that $U_0, U_1 \in \tau$. Hence, the set $U_\varphi = \overline{U_1}$ belongs to the Boolean algebra $\nabla(X)$, besides, $U_\varphi \cap U_0 = \emptyset$.
Since $U_0\cup U_1=U\cap V$ is a dense open set in $(X, \tau)$ and $\varphi(t)=\chi_{U_\varphi}(t)$ for all $t \in U_0\cup U_1$, it follows that $e=\pi(\varphi)=\pi(\chi_{U_\varphi})$. Thus, the mapping $\Phi:\nabla(X)\rightarrow\nabla$ defined by the equality $\Phi(U)=\pi(\chi_U)$, $U \in\nabla(X)$, is a surjection.
Moreover, for $U,V\in\nabla(X)$ the following equalities hold $$\Phi(U \cap V) = \pi(\chi_{U \cap V}) = \pi(\chi_U\chi_V) = \pi(\chi_U)\pi(\chi_V) = \Phi(U)\Phi(V),$$ $$\Phi(X \setminus U) = \pi(\chi_{X \setminus U}) = \pi(\mathbf{1} - \chi_U) = \Phi(X)-\Phi(U).$$ Furthermore, the equality $\Phi(U)=\Phi(V)$ implies that the continuous mappings $\chi_U$ and $\chi_V$ coincide on a dense set in $X$. Therefore $\chi_U=\chi_V$, that is $U=V$.
Hence, $\Phi$ is an isomorphism from the Boolean algebra $\nabla(X)$ onto the Boolean algebra $\nabla$ of all idempotents from $C_\infty(X,K)$, in particular, $\nabla$ is a complete Boolean algebra.
Finally, to prove $l$-completeness of the algebra $C_\infty(X,K)$ we show that for any family $\{\pi(\varphi_i): \varphi\in AC(X,K)\}_{i\in I}$ of nonzero pairwise disjoint elements in $C_\infty(X,K)$ there exists $\varphi \in AC(X,K)$ such that $\pi(\varphi)s(\pi(\varphi_i)) = \pi(\varphi_i)$ for all $i \in I$. For any $i \in I$ we choose a dense open set $U_i$ such that the restriction $\varphi_i|_{U_i}$ is continuous and put $V_i = U_i\cap\varphi_i^{-1}(K\setminus\{0\})$, $i \in I$. It is not hard to prove that $s(\pi(\varphi_i)) = \Phi(\overline{V_i})$. In particular, $V_i \cap V_j = \emptyset$ when $i \ne j$, $i,j \in I$. Define the mapping $\varphi: X \rightarrow K$, as follows $\varphi(t) = \varphi_i(t)$ if $t \in V_i$ and $\varphi(t) = 0$ if $t \in X\setminus\left(\bigcup\limits_{i \in I}V_i\right)$. Clearly, $\varphi \in AC(X,K)$ and $\pi(\varphi)s(\pi(\varphi_i)) = \pi(\varphi\chi_{\overline{V_i}}) = \pi(\varphi_i\chi_{\overline{V_i}}) = \pi(\varphi_i)$ for all $i \in I$.
Laterally complete regular modules {#sec:2}
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Let $\mathcal{A}$ be a laterally complete commutative regular algebra and let $\nabla$ be a Boolean algebra of all idempotents in $\mathcal{A}$. Let $X$ be a left $\mathcal{A}$-module with algebraic operations $x + y$ and $a x$, $x,y \in X$, $a\in\mathcal{A}$. Since the algebra $\mathcal{A}$ is commutative, then a left $\mathcal{A}$-module $X$ becomes a right $\mathcal{A}$-module, if we put $x a := a x$, $x \in X$, $a \in \mathcal{A}$. Hence, we can assume, that $X$ is a bimodule over $\mathcal{A}$, where the following equality $a x = x a$ holds for any $x \in X$, $a\in\mathcal{A}$. Next, an $\mathcal{A}$-bimodule $X$ we shall call an $\mathcal{A}$-module.
An $\mathcal{A}$-module $X$ is called faithful, if for any nonzero $e \in \nabla$ there exists $x \in X$ such that $ex \ne 0$. Clearly, for a faithful $\mathcal{A}$-module $X$ the set $X_e := eX$ is a faithful $\mathcal{A}_e$-module for any $0 \ne e \in \nabla$, where $\mathcal{A}_e := e\mathcal{A}$.
An $\mathcal{A}$-module $X$ is said to be a regular module, if for any $x\in\mathcal{A}$ the condition $ex = 0$ for all $e \in L \subset \nabla$ implies $(\sup L)x = 0$. In this case, for $x \in X$ the idempotent $$s(x) = \mathbf{1} - \sup\{e \in \nabla: ex = 0\}$$ is called the support of an element $x$. In case, when $X = \mathcal{A}$, the notions of support of an element in an $\mathcal{A}$-module $X$ and of support of an element in $\mathcal{A}$ coincide. If $X$ is a regular $\mathcal{A}$-module, then $X_e$ is also a regular $\mathcal{A}_e$-module for any nonzero $e\in\nabla$.
We need the following properties of supports of elements in a regular $\mathcal{A}$-module $X$.
\[art9\_utv\_2\_3\] Let $X$ be a regular $\mathcal{A}$-module, $x,y \in X$, $a \in \mathcal{A}$. Then
(i). $s(x)x = x$;
(ii). if $e\in\nabla$ and $ex = x$, then $e \geq s(x)$;
(iii). $s(a x) = s(a)s(x)$.
$(i)$. If $r(x) = \sup\{e\in\nabla: ex = 0\}$, then $s(x) = \mathbf{1} - r(x)$ and $r(x)x = 0$. Hence, $x = (s(x) + r(x))x = s(x)x$.
$(ii)$. As $ex = x$, then $(\mathbf{1} - e)x = 0$, and therefore $\mathbf{1} - e \leq r(x)$. Thus $e \geq \mathbf{1} - r(x) = s(x)$.
$(iii)$. Since $(s(a)s(x)) \cdot (ax) = (s(a)a) \cdot (s(x)x) = ax$, then by $(ii)$ we have $s(a x) \leq s(a)s(x)$. If $g = s(a)s(x) - s(a x) \ne 0$, then $ga \ne 0$, $g \leq s(a)$ and $gs(ax)=0$. Hence $gax = 0$ and $0 = i(ga)(gax) = (i(g)i(a)ga)x = (gi(a)a)x = gs(a)x = gx \ne 0$. This contradiction implies $g = 0$, i.e. $s(a x) = s(a)s(x)$.
We say that a regular $\mathcal{A}$-module $X$ is laterally complete ($l$-complete), if for any set $\{x_i\}_{i \in I} \subset X$ and for any partition $\{e_i\}_{i \in I}$ of unity of the Boolean algebra $\nabla$ there exists $x \in X$ such that $e_i x = e_i x_i$ for all $i \in I$. In this case, the element $x$ is called mixing of the set $\{x_i\}_{i \in I}$ with respect to the partition of unity $\{e_i\}_{i \in I}$ and denote by $\underset{i \in I}{\mathrm{mix}}(e_i x_i)$. Mixing $\underset{i \in I}{\mathrm{mix}}(e_i x_i)$ is defined uniquely, whereas the equalities $e_i x = e_i x_i = e_i y$, $x, y \in X$, $i \in I$, implies $e_i (x - y) = 0$ for all $i \in I$, and, by regularity of the $\mathcal{A}$-module $X$, we obtain $x = y$.
Let $\{x_i\}_{i \in I} \subset E \subset X$ and let $\{e_i\}_{i \in I}$ be a partition of unity in $\nabla$. The set of all mixings $\underset{i \in I}{\mathrm{mix}}(e_i x_i)$ is called a cyclic hull of the set $E$ in $X$ and denotes by $\mathrm{mix}(E)$. Obviously, the inclusion $E \subset \mathrm{mix}(E)$ is always true. If $E = \mathrm{mix}(E)$, then $E$ is called a cyclic set in $X$ (compare with [@lit4], 1.1.2).
Thus, a regular $\mathcal{A}$-module $X$ is a $l$-complete $\mathcal{A}$-module if and only if $X$ is a cyclic set. In particular, in any $l$-complete $\mathcal{A}$-module $X$ its submodule $X_e$ is also a $l$-complete $\mathcal{A}_e$-module for any nonzero idempotent $e$ in $\mathcal{A}$.
We need the following properties of cyclic hulls of sets.
\[art9\_utv\_2\_6\] Let $X$ be a $l$-complete $\mathcal{A}$-module and let $E$ be a nonempty subset in $X$, $a \in \mathcal{A}$. Then
(i). $\mathrm{mix}(\mathrm{mix}(E)) = \mathrm{mix}(E)$;
(ii). $\mathrm{mix}(a E) = a \mathrm{mix}(E)$;
(iii). If $Y$ is an $\mathcal{A}$-submodule in $X$, then $\mathrm{mix}(Y)$ is a $l$-complete $\mathcal{A}$-submodule in $X$;
(iv). If $U$ is an isomorphism from $\mathcal{A}$-module $X$ onto $\mathcal{A}$-module $Z$, then $Z$ is a $l$-complete $\mathcal{A}$-module and $\mathrm{mix}(U(E)) = U(\mathrm{mix}(E))$.
$(i)$. It is sufficient to show that $\mathrm{mix}(\mathrm{mix}(E)) \subset \mathrm{mix}(E)$. If $x\in\mathrm{mix}(\mathrm{mix}(E))$, then $x = \underset{i \in I}{\mathrm{mix}}(e_i x_i)$, where $x_i \in \mathrm{mix}(E)$, $i \in I$. Since $x_i\in\mathrm{mix}(E)$, then $x_i = \underset{j \in J(i)}{\mathrm{mix}}(e_j^{(i)} x_j^{(i)})$, where $x_j^{(i)} \in E$, $j \in J(i)$ and $\{e_j^{(i)}\}_{j \in J(i)}$ is a partition of unity in the Boolean algebra $\nabla$ for all $i \in I$. Fix $i \in I$ and put $g_j^{(i)} := e_i e_j^{(i)}$. It is clear that $\{g_j^{(i)}\}_{j \in J(i)}$ is a partition of the idempotent $e_i$. Hence, $\{g_j^{(i)}\}_{j \in J(i), i \in I}$ is a partition of unity $\mathbf{1}$. Besides, $$g_j^{(i)}x = g_j^{(i)} e_i x = g_j^{(i)} e_i x_i = e_i e_j^{(i)} x_i = e_i e_j^{(i)} x_j^{(i)} = g_j^{(i)} x_j^{(i)}.$$ This yields that $x = \underset{j \in J(i), i \in I}{\mathrm{mix}}(g_j^{(i)}x_j^{(i)}) \in \mathrm{mix}(E)$.
$(ii)$. If $x\in\mathrm{mix}(a E)$, then $x = \underset{i \in I}{\mathrm{mix}}(e_i ay_i)$, where $y_i \in E, i \in I$. Since $X$ is a $l$-complete $\mathcal{A}$-module, then there exists $y = \underset{i \in I}{\mathrm{mix}}(e_i y_i) \in \mathrm{mix}(E)$ and $e_i x = a e_i y_i = e_i (a y)$ for all $i \in I$. Hence, $e_i(x - a y) = 0$, and regularity of the $\mathcal{A}$-module $X$ implies the equality $x = a y$. Thus, $\mathrm{mix}(a E)\subset a\mathrm{mix}(E)$.
Conversely, if $x\in a \mathrm{mix}(E)$, then $x = a z$, where $z = \underset{i \in I}{\mathrm{mix}}(e_i z_i), z_i \in E, i \in I$. Since $a z_i \in a E$ and $e_i x = e_i (a z) = e_i a e_i z = e_i (a z_i)$ for all $i \in I$, we have that $x = \underset{i \in I}{\mathrm{mix}}(e_i(a z_i)) \in \mathrm{mix}(a E)$. Hence, $a \mathrm{mix}(E) \subset \mathrm{mix}(a E)$.
$(iii)$. Let $x, y \in \mathrm{mix}(Y)$, $x = \underset{i \in I}{\mathrm{mix}}(e_i x_i)$, $y = \underset{j \in J}{\mathrm{mix}}(g_j y_j)$, where $x_i, y_j \in Y$, $i \in I$, $j \in J$, $\{e_i\}_{i \in I}$, $\{g_j\}_{j \in J}$ are partitions of unity in $\nabla$. Clearly, that $p_{ij} = e_i g_j$, $i \in I$, $j \in J$, is also a partition of unity in $\nabla$ and $p_{ij}(x+y)=p_{ij}(x_i+y_j)$, where $x_i + y_j \in Y$ for all $i \in I$, $j \in J$. This means that $(x + y)\in\mathrm{mix}(Y)$.
Since $a Y \subset Y$, then by $(ii)$ we have that $a x \in a \mathrm{mix}(Y) = \mathrm{mix}(a Y) \subset \mathrm{mix}(Y)$. Hence, $\mathrm{mix}(Y)$ is an $\mathcal{A}$-submodule in $X$, and by regularity of the $\mathcal{A}$-module $X$, it is a regular $\mathcal{A}$-module. The equality $\mathrm{mix}(Y) = \mathrm{mix}(\mathrm{mix}Y)$ (see $(i)$) implies that $\mathrm{mix}Y$ is a $l$-complete $\mathcal{A}$-module.
$(iv)$. If $U(x) = y \in Z$, $x \in X$, $\emptyset \ne L \subset \nabla$ and $ey = 0$ for all $e \in L$, then $U(ex) = e U(x) = ey = 0$. Since $U$ is a bijection, then $ex = 0$ for any $e \in L$. By regularity of the $\mathcal{A}$-module $X$, we have that $(\sup L)x = 0$, and, therefore, $(\sup L)y = U((\sup L)x) = 0$. Hence, $Z$ is a regular $\mathcal{A}$-module. In the same way we show that $Z$ is a $l$-complete $\mathcal{A}$-module and the equality $\mathrm{mix}(U(E)) = U(\mathrm{mix}(E))$ holds.
Let $\nabla$ be an arbitrary complete Boolean algebra. For any nonzero element $e\in\nabla$ we put $\nabla_e = \{q\in\nabla: q \leq e\}$. The set $\nabla_e$ is a Boolean algebra with the unity $e$ with respect to partial order, induced from $\nabla$.
We say that a set $B$ in $\nabla$ is a minorant subset for nonempty set $E\subset\nabla$, if for any nonzero $e \in E$ there exists nonzero $q \in B$ such that $q \leq e$. We need the following property of complete Boolean algebras.
[([@lit5], 1.1.6)]{} \[art9\_teor\_2\_1\] If $\nabla$ is a complete Boolean algebra, $e$ is a nonzero element in $\nabla$ and $B$ is a minorant subset for $\nabla_e$, then there exists a disjoint subset $L \subset B$ such that $\sup L = e$.
We say that a Boolean algebra $\nabla$ has *a countable type* or is *$\sigma$-finite*, if any nonfinite family of nonzero pairwise disjoint elements in $\nabla$ is a countable set. A complete Boolean algebra $\nabla$ is called *multi-$\sigma$-finite*, if for any nonzero element $g\in\nabla$ there exists $0 \ne e\in\nabla$ such that $e \leq g$ and the Boolean algebra $\nabla_e$ has a countable type. By theorem \[art9\_teor\_2\_1\], a multi-$\sigma$-finite Boolean algebra $\nabla$ always has a partition $\{e_i\}_{i \in I}$ of unity $\mathbf{1}$ such that the Boolean algebra $\nabla_{e_i}$ has a countable type for all $i \in I$.
By theorem \[art9\_teor\_2\_1\] we set the following useful properties of $l$-complete $\mathcal{A}$-modules.
\[art9\_utv\_2\_7\] Let $X$ be an arbitrary $l$-complete $\mathcal{A}$-module and $\nabla$ be a complete Boolean algebra of all idempotents in $\mathcal{A}$. Then
(i). If $X$ is a faithful $\mathcal{A}$-module, then there exists an element $x \in X$ such that $s(x) = \mathbf{1}$;
(ii). If $Y$ is a $l$-complete $\mathcal{A}$-submodule in a regular $\mathcal{A}$-module $X$ and for any nonzero $e \in \nabla$ there exists a nonzero $g_e \in \nabla$ such that $g_e \leq e$ and $g_e Y = g_e X$, then $Y = X$.
*Proof* is in the same way as the proof of Proposition 2.4 in [@lit7a].
We need a representation of a faithful $l$-complete $\mathcal{A}$-module $X$ as the Cartesian product of a faithful $l$-complete $\mathcal{A}_{e_i}$-modules family, where $\{e_i\}_{i \in I}$ is a partition of unity in the Boolean algebra $\nabla$ of all idempotents in $\mathcal{A}$. In the Cartesian product $$\prod_{i \in I}e_i X = \{\{y_i\}_{i \in I}: y_i \in e_i X\}$$ of $\mathcal{A}$-submodules $e_i X$ we consider coordinate-wise algebraic operations. It is clear that $\prod\limits_{i \in I}e_i X$ is a faithful $l$-complete $\mathcal{A}$-module. We define a map $U: X \rightarrow \prod\limits_{i \in I}e_i X$ given by $U(x) = \{e_i x\}_{i \in I}$. Obviously, $U$ is a homomorphizm from $X$ onto $\prod\limits_{i \in I}e_iX$. If $U(x) = U(y)$, then $e_i x = e_i y$ for all $i \in I$, and by regularity of the $\mathcal{A}$-module $X$, it follows that $x = y$.
If $z = \{x_i\}_{i \in I} \in \prod\limits_{i \in I}e_i X$, where $x_i \in e_i X \subset X$, $i \in I$, then $l$-completeness of the $\mathcal{A}$-module $X$ implies that there exists an element $x \in X$ such that $e_i x = e_i x_i = x_i$ for all $i \in I$. Hence, $U(x) = z$, i.e. $U$ is a surjection.
Thus, the following proposition holds.
\[art9\_utv\_2\_8\] If $X$ is a faithful $l$-complete $\mathcal{A}$-module, $\{e_i\}_{i \in I}$ is a partition of unity of the Boolean algebra $\nabla$ of all idempotents in $\mathcal{A}$, then $\prod\limits_{i \in I}e_i X$ is also a faithful $l$-complete $\mathcal{A}$-module and $U$ is an isomorphism from $X$ onto $\prod\limits_{i \in I}e_i X$.
Homogenous $\mathcal{A}$-modules {#sec:3}
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Let $\mathcal{A}$ be a laterally complete commutative regular algebra, let $\nabla$ be a complete Boolean algebra of all idempotents in $\mathcal{A}$, let $X$ be a faithful $\mathcal{A}$-module. The following $\mathcal{A}$-submodule in $X$ is called $\mathcal{A}$-linear hull of a nonempty subset $Y \subset \mathcal{A}$ $$\mathrm{Lin}(Y, \mathcal{A}) = \left\{ \sum_{i=1}^na_i y_i: a_i \in \mathcal{A}, y_i \in Y, i=1,\ldots,n, n \in\mathcal{N} \right\},$$ where $\mathcal{N}$ is the set of all natural numbers. If $X$ is a $l$-complete $\mathcal{A}$-module, then by proposition \[art9\_utv\_2\_6\] $(iii)$, $\mathrm{mix}(\mathrm{Lin}(Y, \mathcal{A}))$ is also a $l$-complete $\mathcal{A}$-submodule in $X$.
A set $\{x_i\}_{i \in I}$ in an $\mathcal{A}$-module $X$ is called $\mathcal{A}$-*linearly independent*, if for any $a_1, \ldots, a_n \in \mathcal{A}$, $x_{i_1}, \ldots, x_{i_n} \in \{x_i\}_{i \in I}$, $n\in \mathcal{N}$, the equality $\sum\limits_{k=1}^n a_k x_{i_k} = 0$ implies equalities $a_1 = \ldots = a_n = 0$.
\[art9\_utv\_3\_3\] If $Y = \{x_1, \ldots, x_k\}$ is a finite $\mathcal{A}$-linearly independent subset in a $l$-complete $\mathcal{A}$-module $X$, then $\mathrm{mix}(\mathrm{Lin}(Y, \mathcal{A})) = \mathrm{Lin}(Y, \mathcal{A})$.
It is sufficient to show the following inclusion $\mathrm{mix}(\mathrm{Lin}(Y, \mathcal{A})) \subset \mathrm{Lin}(Y, \mathcal{A})$. Let $x \in \mathrm{mix} (\mathrm{Lin}(Y, \mathcal{A}))$, $\{e_i\}_{i \in I}$ be a partition of unity in the Boolean algebra $\nabla$ and let $\{y_i\}_{i \in I} \subset \mathrm{Lin} (Y, \mathcal{A})$ be such that $e_i x = e_i y_i$ for all $i \in I$. Since $e_i x = e_i y_i \in \mathrm{Lin}(Y, \mathcal{A})$, then $e_i x = \sum\limits_{j=1}^k a_j^{(i)} x_j$ for some $a_j^{(i)} \in \mathcal{A}$, $j=1, \ldots, k$. Hence, $e_i x = e_i(e_i x) = \sum\limits_{j=1}^k e_i a_j^{(i)} x_j$. Since $\mathcal{A}$ is a $l$-complete commutative regular algebra and $\{e_i\}_{i \in I}$ is a partition of unity in $\nabla$, then there exists a unique element $\beta_j \in \mathcal{A}$ such that $e_i \beta_j = e_i a_j^{(i)}$ for all $i\in I$, where $j \in \{1, \ldots, k\}$. Thus, $e_i x = \sum\limits_{j=1}^k e_i \beta_j x_j = e_i \left(\sum\limits_{j=1}^k \beta_j x_j\right)$ for any $i \in I$, and this implies the equality $x = \sum\limits_{j=1}^k \beta_j x_j \in \mathrm{Lin}(Y, \mathcal{A})$.
We say that an $\mathcal{A}$-linearly independent system $\{x_i\}_{i \in I}$ from a $l$-complete $\mathcal{A}$-module $X$ is $\mathcal{A}$-*Hamel basis*, if $$\mathrm{mix}(\mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A})) = X.$$
In the case when an $\mathcal{A}$-Hamel basis is a finite set, we say that it is an $\mathcal{A}$-basis in $X$.
\[art9\_teor\_4\_2a\] If $\{x_i\}_{i=1}^n$, $\{y_j\}_{j=1}^k$ are $\mathcal{A}$-basises in an $\mathcal{A}$-module $X$, then $n=k$.
First we shall show the following $\mathcal{A}$-variant of one known fact from the linear algebra.
\[art9\_lemma\_3\_3\] Let $\{z_i\}_{i=1}^n \subset X, \{y_j\}_{j=1}^k \subset X, \{ey_j\}_{j=1}^k \subset \mathrm{Lin}(\{ez_i\}_{i=1}^n, \mathcal{A}_e)$ for nonzero $e \in \nabla$. If the set $\{e y_1, \ldots, ey_k\}$ is $\mathcal{A}_e$-linearly independent, then $k \leq n$.
We use the mathematical induction. Let us suppose that for $n=1$, $k > 1$ the equalities $e y_1 = a_1 e z_1, \ldots, e y_k = a_k e z_1$ hold, where $a_i \in \mathcal{A}_e$, $i = 1, \ldots, k$. Since $a_2 e y_1 + (-a_1) e y_2 = 0$, then $ea_1 = ea_2 = 0$, i.e. $e y_1 = e y_2 = 0$, this contradicts to $\mathcal{A}_e$-linear independence of the elements $e y_1$ and $e y_2$. Hence, $k=1$.
Now assume that the lemma holds for $n=l-1$. Let $\{z_i\}_{i=1}^l \subset X$ and the following equalities hold $$\label{lemma_sfdm2_8_eq_1}
ey_j = \sum_{i=1}^l a_{ji}ez_i, a_{ji} \in \mathcal{A}_e, \quad j = 1, \ldots, k, i= 1, \ldots, l.$$
Let $a_{j_0 l}e x_l \ne 0$ for some $j_0 \in \{1, \ldots, k\}$. By reindexing $\{y_j\}_{j=1}^k$, we can assume that $a_{kl}ex_l \ne 0$, in particular $p = s(a_{kl}e) \ne 0$, wherein $p \leq e$. Since the set $\{e y_j\}_{j=1}^k$ is $\mathcal{A}_e$-linearly independent, then the set $\{p y_j\}_{j=1}^k$ is $\mathcal{A}_p$-linearly independent, wherein, by , we have $$\label{lemma_sfdm2_8_eq_2}
p y_j = \sum_{i=1}^l a_{ji}p z_i, \quad j = 1, \ldots, k.$$
Since $\mathcal{A}$ is a regular algebra, then for the inversion $h = i(a_{kl}) \in \mathcal{A}$ the equality $ha_{kl} = s(a_{kl})$ holds. Therefore the following equality $$p y_k = \sum_{i=1}^{l-1} a_{ki}pz_i + a_{kl}pz_l$$ implies $$\label{lemma_sfdm2_8_eq_3}
pz_l = h p y_k - \sum_{i=1}^{l-1} a_{ki} h p z_i.$$ Substitute $pz_l$ from in the first $(k-1)$ equalities from and collect similar terms, we obtain $$p y_j - ha_{jl}py_k = \sum_{i=1}^{l-1}\beta_{ji}pz_i \in \mathrm{Lin}(\{pz_i\}_{i=1}^{l-1}, \mathcal{A}_p)$$ for some $\beta_{ji}\in\mathcal{A}_p$, $i=1,\ldots,l-1$, $j=1,\ldots,k-1$.
Let us show that the elements $u_j = py_j - ha_{jl}py_k$, $j = 1, \ldots, k-1$ are $\mathcal{A}_p$-linearly independent. Let $$\sum_{j=1}^{k-1}\gamma_jpy_j - \left(\sum_{j=1}^{k-1}\gamma_j h a_{jl}\right)py_k = \sum\limits_{j=1}^{k-1}\gamma_j u_j = 0,$$ where $\gamma_j\in\mathcal{A}_p$, $j=1,\ldots,k-1$. Since $\{py_j\}_{j=1}^k$ is $\mathcal{A}_p$-linearly independent, then $p\gamma_1 = p\gamma_2 = \ldots = p\gamma_{k-1} = 0$, i.e. the set $\{u_j\}_{j=1}^{k-1}$ is $\mathcal{A}_p$-linearly independent in $pX$. By the assumption of the mathematical induction we have that $k-1 \leq l-1$, and thus $k \leq l$. The Lemma \[art9\_lemma\_3\_3\] is proved.
Return to the proof of Theorem \[art9\_teor\_4\_2a\]. As $\{x_i\}_{i=1}^n$ is an $\mathcal{A}$-basis in $X$, then by Proposition \[art9\_utv\_3\_3\] we obtain that $X = \mathrm{Lin}(\{x_i\}_{i=1}^n, \mathcal{A})$. On the other hand, $\{y_j\}_{j=1}^k \subset X$ and $\{y_j\}_{j=1}^k$ is an $\mathcal{A}$-linearly independent set. Therefore, by Lemma \[art9\_lemma\_3\_3\] it follows that $k \leq n$.
Similarly, we show that $n \leq k$, and thus $n=k$.
Next we need the following characterization of $\mathcal{A}$-Hamel basises.
\[art9\_utv\_3\_2\] For an $\mathcal{A}$-linearly independent set $\{x_i\}_{i \in I}$ in a $l$-complete $\mathcal{A}$-module $X$ the following conditions are equivalent:
(i). $\{x_i\}_{i \in I}$ is an $\mathcal{A}$-Hamel basis;
(ii). For any $x \in X$ and any nonzero idempotent $e\in\mathcal{A}$ there exists a nonzero idempotent $g \leq e$, such that $gx \in g \mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A})$.
$(i) \Rightarrow (ii)$. If $X = \mathrm{mix}(\mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A}))$, then for $x \in X$ there exists a partition $\{e_j\}_{j \in J}$ of unity, such that $e_j x \in e_j \mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A})$. Since $\sup\limits_{j \in J}e_j = \mathbf{1}$, then for $0 \ne e \in \nabla$ there exists an element $j_0 \in J$ such that $g = e_{j_0}e \ne 0$, wherein $gx \in g \mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A})$.
$(ii) \Rightarrow (i)$. Fix $0 \ne x \in X$ and for any nonzero idempotent $e\in\nabla$ choose a nonzero idempotent $g(e,x) \leq e$ such that $g(e,x) x \in g(e, x) \mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A})$. By Theorem \[art9\_teor\_2\_1\], there exists a set $\{q_j\}_{j \in J}$ of pairwise disjoint idempotents in $\mathcal{A}$ such that $\sup\limits_{j \in J}q_j = \mathbf{1}$ and $q_j x \in q_j \mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A})$ for all $j \in J$. This means that $x \in \mathrm{mix}(\mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A}))$, which implies the equality $X = \mathrm{mix}(\mathrm{Lin}(\{x_i\}_{i \in I}, \mathcal{A}))$.
Fix some cardinal number $\gamma$. A faithful $l$-complete $\mathcal{A}$-module $X$ is called *$\gamma$-homogeneous*, if there exists an $\mathcal{A}$-Hamel basis $\{x_i\}_{i \in I}$ in $X$ with $\mathrm{card}\, I = \gamma$. We say that $\mathcal{A}$-module $X$ *homogeneous*, if it is a $\gamma$-homogeneous $\mathcal{A}$-module for some cardinal number $\gamma$.
If $X$ is a $\gamma$-homogeneous $\mathcal{A}$-module, then obviously, $eX$ is also $\gamma$-homogeneous $\mathcal{A}_e$-module for any nonzero idempotent $e\in\mathcal{A}$. Besides, by Proposition \[art9\_utv\_2\_6\] ($iv$) it follows that, if $\mathcal{A}$-module $Y$ is isomorphic to a $\gamma$-homogeneous $\mathcal{A}$-module $X$, then $Y$ is also a $\gamma$-homogeneous module.
By repeating the proof of Theorem 3.8 from [@lit7a], we establish the following proposition on isomorphisms of $\gamma$-homogeneous $\mathcal{A}$-modules.
\[art9\_teor\_3\_13a\] If $X$ and $Y$ are $\gamma$-homogeneous $\mathcal{A}$-modules, then $X$ and $Y$ are isomorphic.
Let us give examples of $\gamma$-homogeneous $\mathcal{A}$-modules for an arbitrary cardinal number $\gamma$ and for any $l$-complete commutative regular untaly algebra $\mathcal{A}$. Consider an arbitrary set of indexes $I$ with $\mathrm{card}\,I = \gamma$. Since the algebra $\mathcal{A}$ is $l$-complete, then the Cartesian product $$Y = \prod\limits_{i \in I} \mathcal{A} = \{\hat{\alpha} = \{\alpha_i\}_{i \in I}: \alpha_i \in \mathcal{A}, i \in I\}$$ is a $l$-complete $\mathcal{A}$-module with coordinate-wise algebraic operations.
For any $j \in I$ consider an element $\hat{g_j} = \{g_i^{(j)}\}_{i \in I}$ from $Y$, where $g_i^{(j)} = 0$, $i \ne j$ and $g_i^{(i)} = \mathbf{1}$, $i \in I$. Clearly, that the set $\{\hat{g_j}\}_{j \in I}$ is $\mathcal{A}$-linearly independent, and, therefore, the $\mathcal{A}$-submodule $X = \mathrm{mix}\,(\mathrm{Lin}\,(\{\hat{g}_j\}_{j \in I}, \mathcal{A}))$ in $Y$ is a $\gamma$-homogeneous $\mathcal{A}$-module.
If $\gamma$ is a positive integer $n$, then for the faithful $l$-complete $\mathcal{A}$-module $Y = \prod\limits_{i=1}^n \mathcal{A} = \mathcal{A}^n$ and for $\hat{g_j} = \{g_i^{(j)}\}_{i=1}^n$, $j=1, \ldots, n$ we have that $\mathrm{Lin}\,(\{\hat{g}_j\}_{j=1}^n, \mathcal{A}) = Y$, i.e. the set $\{\hat{g}_j\}_{j=1}^n$ is an $\mathcal{A}$-Hamel basis in $Y$. Thus, Proposition \[art9\_teor\_3\_13a\] implies the following
\[art9\_teor\_3\_14\] For any positive integer $n$ there exists a unique, up to isomorphism, $n$-homogeneous $\mathcal{A}$-module, which is isomorphic to $\mathcal{A}^n$.
Let $X$ be a faithful $l$-complete $\mathcal{A}$-module, which is $\gamma$-homogeneous and $\lambda$-homogeneous simultaneously. There is a natural question, whether in this case the equality $\gamma = \lambda$ holds. Similar question was studied in classification of Kaplansky-Hilbert modules (KHM) $X$ over a commutative $AW^\ast$-algebra $\mathcal{A}$ with the Boolean algebra of projections $\nabla$ (see [@lit10]). In the case, when $\nabla$ is a multi-$\sigma$-finite Boolean algebra in [@lit10] it is proved that for a KHM $X$ the equality $\lambda = \gamma$ is always true. However, for an arbitrary complete Boolean algebra $\nabla$ this equality cannot be established. Thereby, in ([@lit5], 7.4.6) the notion of *strictly $\gamma$-homogeneous* KHM $X$ is defined, and this gave an opportunity to classify KHM $X$ over an arbitrary commutative $AW^\ast$-algebra $\mathcal{A}$. For the same reason, below we introduce the notion of strictly $\gamma$-homogeneous faithful $l$-complete modules over laterally complete algebras $\mathcal{A}$. With this notion we obtain necessary and sufficient conditions for $l$-complete $\mathcal{A}$-modules to be isomorphic.
Let $X$ be a faithful $l$-complete $\mathcal{A}$-module, $0 \ne e \in \nabla$. By $\varkappa(e) = \varkappa_X(e)$ we denote the smallest cardinal number $\gamma$ such that the $\mathcal{A}_e$-module $X_e$ is $\gamma$-homogeneous. If the $\mathcal{A}$-module $X$ is homogeneous, then the cardinal number $\varkappa(e)$ is defined for all nonzero $e\in\nabla$. Further, by ([@lit5], 7.4.7), we assume that $\varkappa(0) = 0$.
We say that an $\mathcal{A}$-module $X$ is *strictly $\gamma$-homogeneous* (compare with [@lit5], 7.4.6), if $X$ is $\gamma$-homogeneous and $\gamma = \varkappa(e)$ for all nonzero $e\in\nabla$. If an $\mathcal{A}$-module $X$ is strictly $\gamma$-homogeneous for some cardinal number $\gamma$, then such $\mathcal{A}$-module $X$ is called *strictly homogeneous*.
Clearly, any strictly $\gamma$-homogeneous $\mathcal{A}$-module is a $\gamma$-homogeneous $\mathcal{A}$-module. By Lemma \[art9\_lemma\_3\_3\] it follows that every $n$-homogeneous $\mathcal{A}$-module $X$ is a strictly $n$-homogeneous module. By Proposition \[art9\_utv\_2\_6\] ($iv$) every $\mathcal{A}$-module $Y$, which is isomorphic to a strictly $\gamma$-homogeneous $\mathcal{A}$-module $X$, is also strictly $\gamma$-homogeneous.
The following theorem holds.
\[art9\_teor\_3\_9\] Let $\lambda$ and $\gamma$ be infinite cardinal numbers and let the Boolean algebra $\nabla$ of all idempotents in a $l$-complete commutative regular algebra $\mathcal{A}$ has countable type. If a faithful $l$-complete $\mathcal{A}$-module $X$ is $\lambda$-homogeneous and $\gamma$-homogeneous simultaneously, then $\gamma = \lambda$.
*Proof* of Theorem \[art9\_teor\_3\_9\] is similar to that of Theorem 3.4 in [@lit7a].
Using Theorem \[art9\_teor\_3\_9\] to the $\mathcal{A}_e$-module $X_e$, we have, that Theorem \[art9\_teor\_3\_9\] holds in the case, when in the Boolean algebra $\nabla$ of idempotents in $\mathcal{A}$ there exists nonzero element $e$, which has a countable type. Thus, repeating the proof of Corollary 3.7 in [@lit7a], we obtain the following necessary and sufficient conditions for coincidence of strictly $\gamma$-homogeneous and $\gamma$-homogeneous notions for $\mathcal{A}$-modules.
\[art9\_utv\_4\_8a\] Let a Boolean algebra $\nabla$ of all idempotents on a $l$-complete commutative regular algebra $\mathcal{A}$ be multi-$\sigma$-finite. If $\gamma$ is an infinite cardinal number and $X$ is a $\gamma$-homogeneous $\mathcal{A}$-module, then the module $X$ is strictly $\gamma$-homogeneous.
The following proposition enables to “glue” $\gamma$-homogeneous (strictly $\gamma$-homogeneous) $\mathcal{A}$-modules.
\[art9\_utv\_4\_9a\] Let $\mathcal{A}$ be a $l$-complete commutative regular algebra, let $X$ be a $l$-complete $\mathcal{A}$-module and let $\{e_i\}_{i \in I}$ be a set of pairwise disjoint nonzero idempotents in $\mathcal{A}$ and $e=\sup\limits_{i \in I}e_i$. If $ X_{e_i}$ is a $\gamma$-homogeneous (respectively, strictly $\gamma$-homogeneous) $\mathcal{A}_{e_i}$-module for all $i \in I$, then the $\mathcal{A}_e$-module $X_e$ is also $\gamma$-homogeneous (respectively, strictly $\gamma$-homogeneous).
*Proof* is similar to that of Proposition 3.10 in [@lit7a].
Classification of faithful $l$-complete $\mathcal{A}$-modules {#sec:4}
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In this section it is proved that every faithful laterally complete $\mathcal{A}$-module is isomorphic to a Cartesian product of strictly homogeneous $\mathcal{A}$-modules. The important step in obtaining such an isomorphism is the following theorem.
\[art9\_teor\_5\_1\] Let $\mathcal{A}$ be a $l$-complete commutative regular algebra, let $\nabla$ be a Boolean algebra of all idempotents in $\mathcal{A}$ and let $X$ be a faithful $l$-complete $\mathcal{A}$-module. Then there exists a nonzero idempotent $p\in\nabla$ such that $X_p$ is a strictly homogeneous $\mathcal{A}_p$-module.
Using Proposition \[art9\_utv\_2\_7\] ($i$), we choose $x_0 \in X$ such that $s(x_0) = \mathbf{1}$. If $X = \mathrm{Lin}(x_0, \mathcal{A})$, then $X$ is a strictly $1$-homogeneous module and Theorem \[art9\_teor\_5\_1\] is proved.
Assume that $X \ne \mathrm{mix}\,(\{x_0\})$. We consider in $X$ the following nonempty family of subsets $$\mathscr{E} = \{B \subset X: x_0 \in B, B - \mathcal{A}\text{-linearly independent set}\}.$$ We introduce in $\mathscr{E}$ a partial order by $B \leq C \Leftrightarrow B \subset C$. By Zorn’s lemma there exists maximal element $D$ in $\mathscr{E}$. If $D$ is an $\mathcal{A}$-Hamel basis in $X$, then $X$ is $(\mathrm{card}\, D)$-homogeneous $\mathcal{A}$-module.
Assume that $X \ne \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$. If for any nonzero $e\in\nabla$ there exists $0 \ne q_e \in \nabla$ such that $q_e\, \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A})) = q_e X$, then from Proposition \[art9\_utv\_2\_6\] ($iii$) and Proposition \[art9\_utv\_2\_7\] ($ii$) it follows that $X = \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$, which contradicts our assumption. Hence, there exists nonzero $e\in\nabla$ such that the following condition holds: $$g\, \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A})) \ne gX \ \hbox{for all non zero} \ g \in \nabla_e. \tag{1}$$
Denote by $\mathscr{L}$ a set of all nonzero $e\in\nabla$ with property $(1)$. Put $e_0 = \sup\mathscr{L}$ and show that the equality $e_0 = \mathbf{1}$ fails.
Assume that $e_0 = \mathbf{1}$. In this case for every nonzero $q \in \nabla$ there exists $e\in\mathscr{L}$ such that $g = qe \ne 0$. Hence, $gX \ne g\mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$ (see (1)), which implies $$q X \ne q\, \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A})). \tag{2}$$
Show that for any nonzero $q\in\nabla$ there exists a nonzero idempotent $r \leq q$ such that for any $0 \ne g \in \nabla_r$ the following property holds: $$\hbox{There exists } x_g \in gX \hbox{ such that } s(x_g)=g \hbox{ and } \ lx_g \not\in \mathrm{Lin}\,(D, \mathcal{A}) \hbox{ for all } 0 \ne l \in \nabla_g. \tag{3}$$
If this is not true, then there exists a nonzero $q \in\nabla$ such that for every $0 \ne r \in \nabla_q$ there exists a nonzero idempotent $g_r \in \nabla_r$ without property $(3)$, i.e. for any $x \in g_r X$ with $s(x) = g_r$ there exists a nonzero idempotent $e(x_g,r) \leq g_r \leq q$ such that $$e(x_g,r) x \in e(x_g,r) \mathrm{Lin}\,(D, \mathcal{A}) \subset \mathrm{Lin}\,(D, \mathcal{A}).$$
Show that, in this case, $g_q X = g_q \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$. Let $x$ be a nonzero element in $g_q X$, in particular, $0 \ne s(x) \leq g_q$. For any nonzero idempotent $a \leq s(x)$ there exists a nonzero idempotent $e(ax,a) \leq a$ such that $e(ax,a) x \in \mathrm{Lin}\,(D,\mathcal{A})$. By Theorem \[art9\_teor\_2\_1\], there exists a partition $\{e_i\}_{i \in I}$ of support $s(x)$ such that $e_i x \in s(x)\mathrm{Lin}\,(D, \mathcal{A})$ for all $i \in I$. This means that $x \in \mathrm{mix}\,(s(x)\mathrm{Lin}\,(D, \mathcal{A})) = s(x)\mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$ (see Proposition \[art9\_utv\_2\_6\] $(ii)$). Since $s(x) \leq g_q$, we have that $x \in g_q \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$, which implies the inclusion $g_q X \subset g_q \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$. On the other hand, by $l$-completeness of an $\mathcal{A}_{g_q}$-module $g_q X$ we have that $$g_q \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A})) \subset g_q \mathrm{mix}\,(X) = \mathrm{mix}\,(g_q X) = g_q X.$$ Hence, $g_q X = g_q \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A}))$, which contradicts to (2).
Thus, for every nonzero $q\in\nabla$ there exists a nonzero idempotent $r \leq q$ such that for any $0 \ne g \in \nabla_r$ property $(3)$ holds.
Again by Theorem \[art9\_teor\_2\_1\], we choose a partition $\{g_j\}_{j \in J}$ of the idempotent $r$ and a set $\{x_{g_j}\}_{j \in J}$ in $rX$, such that $s(x_{g_j}) = g_j$ and $l x_{g_j} \not\in \mathrm{Lin}\,(D, \mathcal{A})$ for all $0 \ne l \in \nabla_{g_i}$.
Since $rX$ is a $l$-complete $\mathcal{A}_r$-module, then there exists $x \in rX$ such that $g_j x = x_{g_j}$. In particular, $s(x) = r$, wherein $lx \not\in \mathrm{Lin}\,(D, \mathcal{A})$ for all $0 \ne l \in \nabla_r$.
Again by Theorem \[art9\_teor\_2\_1\] we choose a partition $\{r_k\}_{k \in K}$ of the unity $\mathbf{1}$ in the Boolean algebra $\nabla$ and a set $\{x_k\}_{k \in K}$ in $X$, such that $s(x_k) = r_k$ and $l x_k \not\in \mathrm{Lin}\,(D, \mathcal{A})$ for any $0 \ne l \in \nabla_{r_k}$. By $l$-completeness of the $\mathcal{A}$-module $X$ there exists $\hat{x} \in X$ such that $r_k \hat{x} = x_k$ for all $k \in K$. In this case $s(\hat{x}) = \mathbf{1}$ and $l \hat{x} \not\in \mathrm{Lin}\,(D, \mathcal{A})$ for any $0 \ne l \in \nabla$.
Show that the set $D \cup \{\hat{x}\}$ is $\mathcal{A}$-linearly independent. Let $a_0 \hat{x} + \sum\limits_{i=1}^n a_i x_i = 0$, where $a_0, a_i \in \mathcal{A}$, $x_i \in D$, $i=1, \ldots, n$. If $a_0 = 0$, then $\sum\limits_{i=1}^n a_i x_i = 0$ and by $\mathcal{A}$-linear independence of the set $D$ it follows that $a_i = 0$ for all $i=1, \ldots, n$. If $a_0 \ne 0$, then $s(a_0) \ne 0$ and for $i(a_0)=h \in \mathcal{A}$ we have that $h a_0 = s(a_0)$ and $s(a_0)\hat{x} = - \sum\limits_{i=1}^n a_i h x_i \in \mathrm{Lin}\,(D, \mathcal{A})$, which is not true. Hence, the set $D \cup \{\hat{x}\}$ is $\mathcal{A}$-linearly independent in $X$, which contradicts to maximality of the set $D$.
Thus the equality $e_0 = \mathbf{1}$ is impossible. This means that $e = \mathbf{1} - e_0 \ne 0$. By construction of the idempotent $e_0$, every nonzero idempotent $r \leq e$ does not have property $(1)$. Hence, for any $0 \ne r \in \nabla_e$ there exists a nonzero idempotent $p_r \leq r$ such that $$p_r X = p_r \mathrm{mix}\,(\mathrm{Lin}\,(D, \mathcal{A})) = \mathrm{mix}\,(\mathrm{Lin}\,(p_r D, \mathcal{A}_{p_r }))= p_r \mathrm{mix}\,(\mathrm{Lin}\,(e D, \mathcal{A}_{e})).$$ From Propositions \[art9\_utv\_2\_6\] $(iii)$ and \[art9\_utv\_2\_7\] $(ii)$ it follows that $$eX = \mathrm{mix}\,(\mathrm{Lin}\,(eD, \mathcal{A}_e)).$$ Since $eD$ is an $\mathcal{A}_e$-linearly independent subset in the $\mathcal{A}_e$-module $eX$, then $eD$ is an $\mathcal{A}_e$-basis in $eX$, i.e. $eX$ is a $\gamma$-homogeneous $\mathcal{A}_e$-module, where $\gamma = \mathrm{card}\,(eD)$. In particular, a cardinal number $\varkappa(p)$ is defined for all nonzero $p \in \nabla_e$. Let $\gamma_e$ be the smallest cardinal number in the set of cardinal numbers $\{\varkappa(p): 0 \ne p \leq e\}$, i.e. $\gamma_e = \varkappa(p)$ for some nonzero $p \leq e$. By the choice of the idempotent $p$ it follows that $\gamma_e = \varkappa(p) = \varkappa(q)$ for all $0 \ne q \in \nabla_p$. This means that the $\mathcal{A}_p$-module $X_p$ is strictly homogeneous.
Now everything is ready to obtain the isomorphism from the faithful laterally complete $\mathcal{A}$-module to the Cartesian product of strictly homogeneous $\mathcal{A}$-modules.
\[art9\_teor\_5\_2\] Let $\mathcal{A}$ be a $l$-complete commutative regular algebra, let $\nabla$ be a Boolean algebra of all idempotents in $\mathcal{A}$ and let $X$ be a faithful $l$-complete $\mathcal{A}$-module. Then there exist a uniquely defined set of pairwise disjoint nonzero idempotents $\{e_i\}_{i \in I} \subset \nabla$ and a set of pairwise different cardinal numbers $\{\gamma_i\}_{i \in I}$ such that $\sup\limits_{i \in I}e_i = \mathbf{1}$ and $X_{e_i}$ is a strictly $\gamma_i$-homogeneous $\mathcal{A}_{e_i}$-module for all $i \in I$. In this case, the $\mathcal{A}$-modules $X$ and $\prod\limits_{i \in I} X_{e_i}$ are isomorphic.
By Theorem \[art9\_teor\_5\_1\] for every nonzero idempotent $e\in\mathcal{A}$ there exists a nonzero idempotent $g \leq e$ such that $X_g$ is a strictly homogeneous $\mathcal{A}_g$-module. By Theorem \[art9\_teor\_2\_1\], choose a set of pairwise disjoint nonzero idempotents $\{q_j\}_{j \in J}$ such that $\sup\limits_{j \in J}q_j = \mathbf{1}$ and $q_j X$ is a strictly $\lambda_j$-homogeneous $\mathcal{A}_{q_j}$-module for all $j \in J$. We decompose the set of cardinal numbers $A = \{\lambda_j\}_{j \in J}$ as a union of disjoint subsets $A_i$ in such a way that every $A_i$ consists of equal cardinal numbers from $A$. By $\gamma_i$ denote an element in $A_i$. By Proposition \[art9\_utv\_4\_9a\], for $e_i = \sup\{q_j: \lambda_j \in A_i\}$ we have that the $\mathcal{A}_{e_i}$-module $X_{e_i}$ is strictly $\gamma_i$-homogeneous. Moreover, by Proposition \[art9\_utv\_2\_8\], the $\mathcal{A}$-module $X$ and $\prod\limits_{i \in I}e_i X$ are isomorphic.
Assume, that there exist other sets of pairwise disjoint nonzero idempotents $\{g_j\}_{j \in J}$ and pairwise different cardinal numbers $\{\mu_j\}_{j \in J}$, such that $\sup\limits_{j \in J}g_j = \mathbf{1}$ and $X_{g_j}$ is a strictly $\mu_j$-homogeneous $\mathcal{A}_{g_j}$-module for all $j \in J$. For any fixed $j \in J$, by the equality $\sup\limits_{i \in I}e_i = \mathbf{1}$, we have that $g_j = \sup\limits_{i \in I}e_i g_j$. If there exist two different indexes $i_1, i_2 \in I$ such that $e_{i_1} g_j \ne 0$ and $e_{i_2} p_j \ne 0$, then $$\mu_j = \varkappa(g_j) = \varkappa(e_{i_1}g_j) = \varkappa(e_{i_1}) = \gamma_{i_1} \ne \gamma_{i_2} = \varkappa(e_{i_2}) = \varkappa(e_{i_2}g_j) = \mu_j.$$ By this contradiction, it follows that $e_i g_j = 0$ for all $i \in I$ except one index, which we denote by $i(j)$. Since $e_{i(j)} g_j \neq 0$, we have that $$\mu_j = \varkappa(g_j) = \varkappa(e_{i(j)} g_j) = \varkappa(e_{i(j)}) = \gamma_{i(j)}.$$ If $g_j \ne e_{i(j)}$, then by the equality $\sup\limits_{j \in J}g_j = \mathbf{1}$, there exists index $j_1 \in J$, $j_1 \ne j$ such that $e_{i(j)}g_{j_1} \ne 0$. Hence, $$\mu_j = \gamma_{i(j)} = \varkappa(e_{i(j)}) = \varkappa(e_{i(j)} g_{j_1}) = \varkappa(g_{j_1}) = \mu_{j_1},$$ which is not true. Thus, $g_j = e_{i(j)}$ and $\mu_j = \gamma_{i(j)}$.
For the same reason, for any $i \in I$ there exists the unique index $j(i)$ such that $e_i = g_{j(i)}$ and $\gamma_i = \mu_{j(i)}$.
The partition $\{e_i\}_{i \in I}$ of unity in a Boolean algebra of idempotents in $\mathcal{A}$ and the set of cardinal numbers $\{\gamma_i\}_{i \in I}$ in Theorem \[art9\_teor\_5\_2\] are called a passport for a faithful laterally complete $\mathcal{A}$-module $X$ and denoted by $\Gamma(X) = \{(e_i(X), \gamma_i(X))\}_{i \in I(X)}$.
Thus, a passport $\Gamma(X) = \{(e_i(X), \gamma_i(X))\}_{i \in I(X)}$ for a faithful $l$-complete $\mathcal{A}$-module $X$ means that $X = \prod\limits_{i \in I(X)}e_i(X) X$ (up to an isomorphism), where $e_i(X) X$ is a strictly $\gamma_i(X)$-homogeneous $\mathcal{A}_{e_i}$-module for all $i \in I(X)$, $e_i(X) \ne 0$, $e_i(X) e_j(X) = 0$, $\gamma_i(X) \ne \gamma_j(X)$, $i \ne j$, $i,j \in I(X)$, $\sup\limits_{i \in I(X)}e_i(X) = \mathbf{1}$.
The following theorem gives a criterion for isomorphism between faithful $l$-complete $\mathcal{A}$-modules, by using the notion of passport for these $\mathcal{A}$-modules.
\[art9\_teor\_5\_3\] Let $\mathcal{A}$ be a $l$-complete commutative regular algebra, $X$ and $Y$ be a faithful $l$-complete $\mathcal{A}$-modules. The following conditions are equivalent:
\(i) $\Gamma(X) = \Gamma(Y)$;
\(ii) $\mathcal{A}$-modules $X$ and $Y$ are isomorphic.
$(i) \Rightarrow (ii)$. Let $\{(e_i(X), \gamma_i(X))\}_{i \in I(X)} = \Gamma(X) = \Gamma(Y) = \{(e_i(Y), \gamma_i(Y))\}_{i \in I(Y)}$, i.e. $I(X) = I(Y) := I$, $e_i(X) = e_i(Y) := e_i$ and $\gamma_i(X) = \gamma_i(Y) := \gamma_i$ for all $i \in I$. By Theorem \[art9\_teor\_5\_2\], there exists an isomorphism $U$ from $\mathcal{A}$-module $X$ onto $\mathcal{A}$-module $\prod\limits_{i \in I}e_i X$ (respectively an isomorphism $V$ from $\mathcal{A}$-module $Y$ onto $\mathcal{A}$-module $\prod\limits_{i \in I}e_i Y$), where $U(x) = \{e_i x\}_{i \in I}$ (respectively, $V(y) = \{e_i y\}_{i \in I}$) for every $x \in X$ (respectively, for every $y \in Y$).
Since $e_i X$ (respectively, $e_i Y$) is a strictly $\gamma_i$-homogeneous $\mathcal{A}_{e_i}$-module, then by Proposition \[art9\_teor\_3\_13a\], for all $i \in I$ there exists an isomorphism $U_i$ from the $\mathcal{A}_{e_i}$-module $e_i X$ onto the $\mathcal{A}_{e_i}$-module $e_i Y$. It is clear that a map $\Phi: X \rightarrow Y$, defined by the equality $$\Phi(x) = V^{-1}(\{U_i(e_i x)\}_{i \in I}).$$ is an isomorphism from the $\mathcal{A}$-module $X$ onto the $\mathcal{A}$-module $Y$.
$(ii) \Rightarrow (i)$. Let $\Psi$ be an isomorphism from $X$ onto $Y$ and $\Gamma(X) = \{(e_i(X), \gamma_i(X))\}_{i \in I(X)}$ be a passport for a faithful $l$-complete $\mathcal{A}$-module $X$. By Proposition \[art9\_utv\_2\_6\] $(iv)$, the following $\mathcal{A}_{e_i(X)}$-module $$Y_i = \Psi(e_i(X)X) = e_i(X) \Psi(X) = e_i(X) Y$$ is strictly $\gamma_i(X)$-homogeneous. This means that $\{(e_i(X), \gamma_i(X))\}_{i \in I}$ is a passport for the faithful $l$-complete $\mathcal{A}$-module $Y$, i.e. $\Gamma(X) = \Gamma(Y)$.
Let $\mathcal{A}$ be a $l$-complete commutative regular algebra, let $\nabla$ be a Boolean algebra of all idempotents in $\mathcal{A}$. A faithful $l$-complete $\mathcal{A}$-module $X$ is called finitely-dimensional, if there exist a finite partition $\{e_i\}_{i=1}^k$ of unity in the Boolean algebra $\nabla$ ($e_i \neq 0, i=1, \ldots, k)$ and a finite set $\{n_i\}_{i=1}^k$ of natural numbers ($n_1 < n_2 < \ldots < n_k$) such that $X_{e_i}$ is an $n_i$-homogeneous $\mathcal{A}_{e_i}$-module for all $i = 1, \ldots, k$.
This means that any finitely-dimensional $\mathcal{A}$-module $X$ has a passport of the following form $$\Gamma(X) = \{(e_i(X), n_i(X))\}_{i=1}^k,$$ where $$e_1(X) + \ldots + e_k(X) = \mathbf{1}, n_1(X) < \ldots < n_k(X) < \infty.$$
\[art9\_teor\_5\_4\] For a faithful $l$-complete $\mathcal{A}$-module $X$ the following conditions are equivalent:
(i). $X$ is a finitely-dimensional module;
(ii). $X$ is a finitely-generated module, i.e. there exists a finite set $\{x_i\}_{i=1}^m$ of elements in $X$ such that $X = \mathrm{Lin}(\{x_i\}_{i=1}^m, \mathcal{A})$;
(iii). There exists a positive integer $m$ such that for any nonzero idempotent $e\in\mathcal{A}$ any $\mathcal{A}_e$-linearly independent set in $X_e$ consists of not more than $m$ elements.
$(i)\Rightarrow(ii)$. Let $\Gamma(X) = \{(e_i(X), n_i(X))\}_{i=1}^k$ be a passport for the $\mathcal{A}$-module $X$. For every $i = 1, \ldots, k$ we choose the $\mathcal{A}_{e_i}$-basis $\{x_j^{(i)}\}_{j=1}^{n_i}$ in $X_{e_i}$. If $x \in X$, then $e_i x = \sum\limits_{j=1}^{n_i} a_j^{(i)}x_j^{(i)}$, where $a_j^{(i)} \in \mathcal{A}_{e_i}$. Hence, $$x = \sum\limits_{i=1}^k e_i x = \sum\limits_{i=1}^k\sum\limits_{j=1}^{n_i} a_j^{(i)}g_j^{(i)} \in \mathrm{Lin}\ (\{x_j^{(i)}\}_{j=\overline{1,n_i}, i=\overline{1,k}}, \mathcal{A}).$$ This means that $\mathcal{A}$-module $X$ is finitely-generated.
$(ii)\Rightarrow(iii)$. If $X = \mathrm{Lin}(\{x_i\}_{i=1}^m, \mathcal{A})$, $e$ is a nonzero idempotent in $\mathcal{A}$ and $\{y_j\}_{j=1}^l$ is an $\mathcal{A}_e$-linearly independent set in $X_e$, then by Lemma \[art9\_lemma\_3\_3\], it follows that $l \leq m$.
$(iii)\Rightarrow(i)$. By Theorem \[art9\_teor\_5\_2\], there exist a set of pairwise disjoint nonzero idempotents $\{e_i\}_{i \in I}$ and a set of pairwise different cardinal numbers $\{\gamma_i\}_{i \in I}$ such that $\sup\limits_{i \in I}e_i = \mathbf{1}$ and $X_{e_i}$ is a strictly $\gamma_i$-homogeneous $\mathcal{A}_{e_i}$-module for all $i \in I$. If $\gamma_i > m$, then in $X_{e_i}$ there exists a finite set $\{x_i\}_{i=1}^l$, which consist of $\mathcal{A}_e$-linearly independent elements, and besides $l > m$, which contradicts to condition $(iii)$. Hence, $\gamma_i \leq m$ for all $i \in I$. Since natural numbers $\{\gamma_i\}_{i \in I}$ are pairwise different, then $I$ is a finite set, i.e. $\{\gamma_i\}_{i \in I} = \{n_i\}_{i=1}^k$, where $n_1 < n_2 < \ldots n_k$. Hence, the $\mathcal{A}$-module $X$ is finitely-dimensional.
The following description of finitely-dimensional $\mathcal{A}$-modules follows directly from Theorem \[art9\_teor\_5\_2\] and Corollary \[art9\_teor\_3\_14\].
\[art9\_teor\_4\_5\] If $X$ is a finitely-dimensional $\mathcal{A}$-module, then there exist an uniquely defined finite partition $\{e_i\}_{i=1}^k$ of unity in the Boolean algebra of all idempotents in $\mathcal{A}$ and a finite set of positive integers $n_1 < \ldots < n_k$ such that the $\mathcal{A}$-module $X$ is isomorphic to the $\mathcal{A}$-module $\prod\limits_{i=1}^k \mathcal{A}^{n_i}_{e_i}$ (here $e_i \ne 0$ for all $i = 1, \ldots, k$).
A faithful $l$-complete $\mathcal{A}$-module $X$ is called $\sigma$-finitely-dimensional, if there exist a countable partition $\{e_i\}_{i=1}^\infty$ of unity in the Boolean algebra of all idempotents in $\mathcal{A}$ ($e_i \neq 0, i=1,2,\ldots$) and a countable set $\{n_i\}_{i=1}^\infty$ of positive integers ($n_1 < n_2 < \ldots $) such that $X_{e_i}$ is an $n_i$-homogeneous $\mathcal{A}_{e_i}$-module for all $i = 1, 2, \ldots$
By Theorem \[art9\_teor\_5\_2\] and Corollary \[art9\_teor\_3\_14\] we obtain the following description of $\sigma$-finitely-dimensional $\mathcal{A}$-modules.
\[art9\_teor\_4\_6\] If $X$ is a $\sigma$-finitely-dimensional $\mathcal{A}$-module, then there exist a uniquely defined countable partition $\{e_i\}_{i=1}^\infty$ of unity in the Boolean algebra of all idempotents in $\mathcal{A}$ and a countable set of positive integers $n_1 < n_2 < \ldots $ such that the $\mathcal{A}$-module $X$ is isomorphic to the $\mathcal{A}$-module $\prod\limits_{i=1}^\infty \mathcal{A}^{n_i}_{e_i}$ (here $e_i \ne 0$ for all $i = 1, 2, \ldots$).
Chilin V.I. Partially ordered Baire’s involutive algebras. Modern problems of Mathematics: The latest trends, VINITI, Moscow, **27** (1985), 99-128. (in Russian) Chilin V.I., Karimov J.A. Laterally complete $C_{\infty}(Q)$-modules. Vladikavkaz math J., **16(2)** (2014), 69-78. (in Russian) Clifford A.N., Preston G.B. The algebraic theory of semigroup. Amer. Math. Soc., Mathematical Surveys, Number 7, Vol.I (1964). Kaplansky J. Projections in Banach algebras. Ann. Math, **53** (1951), 235-249. Kaplansky J. Algebras of type I. Ann. Math, **56** (1952), 450-472. Kaplansky J. Modules over operator algebras. Amer. J. Math, **75(4)** (1953), 839-858. Karimov, J.A. Kaplansky-Hilbert modules over the algebra of measurable functions. Uzbek math. J., 4 (2010), 74-81. (in Russian) Kusraev, A.G. Vector duality and its applications. Nauka, Novosibirsk (1985). (in Russian) Kusraev, A.G. Dominated operators. Springer, Netherlands (2000). Maeda F. Kontinuierliche Geometrien. Berlin (1958). Muratov M.A., Chilin V.I. Algebra of measurable and locally measurbale operators. Proceedings of Mathematics institute of NAS Ukraine, **69** (2007). (in Russian) Skornyakov L.A. Dedekind’s lattices with complements and regular rings. Fizmatgiz, Moscow (1961). (in Russian) Van Der Waerden B.L. Algebra. Volueme II. Springer-Verlag New York (1991). Vulikh B.Z. Introduction to the theory of partially ordered spaces. Wolters-Noordhoff Sci. Publ., Groningen (1967).
National University of Uzbekistan, 100174, Vuzgorodok, Tashkent, Uzbekistan;
e-mail: [email protected], [email protected];
[email protected]
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address: |
$^1$Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\
$^2$Theory Group, Deutsches Elektronen-Synchrotron (DESY), D-22607 Hamburg, Germany\
$^3$ITFA, University of Amsterdam, Science Park 904, 1018 XE, Amsterdam, The Netherlands\
$^4$Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands
author:
- 'Ian Moult,$^1$ Iain W. Stewart,$^1$ Frank J. Tackmann,$^2$ Wouter J. Waalewijn,$^{3,4}$ [^1]'
title: EMPLOYING HELICITY AMPLITUDES FOR RESUMMATION IN SCET
---
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DESY 16-087
MIT–CTP 4805
NIKHEF 2016-022
-----------------
\
Introduction
============
Precise predictions for Standard Model backgrounds are important to uncover new physics at the LHC. We focus on processes with hadronic jets, which receive large QCD corrections. There has been tremendous progress in calculating these corrections in fixed-order perturbation theory, using the spinor helicity formalism, color ordering techniques and unitarity based methods. Currently, NLO predictions are available for processes with a large number of jets and their computation has been largely automatized [@NLO]. Jet measurements often introduce a sensitivity to QCD effects at a scale $p$ well below the partonic center-of-mass energy $Q$. Here $p$ corresponds to e.g. the typical jet invariant mass or a veto on additional jets. The hierachy between $p$ and $Q$ leads to large logarithms $\alpha_s^n\ln^m(p/Q)$ ($m \leq 2n$) in the cross section, that require resummation.
Soft-Collinear Effective Theory (SCET) [@SCET] is an effective theory of QCD that enables resummation. It treats collinear and soft radiation (see [Fig. \[fig:hard\]]{}) as dynamical degrees of freedom, $${{\mathcal L}}_{\rm SCET} = \sum_n {{\mathcal L}}_n + {{\mathcal L}}_{\rm soft} + {{\mathcal L}}_{\rm hard}
\,,$$ with ${{\mathcal L}}_n$ the Lagrangian for collinear radiation in the light-like $n$ direction. The hard scattering is integrated out, due to the large virtuality of the momentum exchange, giving rise to ${{\mathcal L}}_{\rm hard}= \sum_i C_i O_i$. Describing the spin content of operators $O_i$ with Dirac structures becomes cumbersome for complicated final states. We discuss a helicity operator basis which makes it easy to construct a complete basis and facilitates the matching from QCD onto SCET [@Moult:2015aoa].
In SCET resummation is achieved by decoupling the collinear and soft degrees of freedom in the Lagrangian [@Bauer:2001yt], leading to the following (schematic) factorized cross section $$\label{eq:sigma}
{\mathrm{d}}\sigma =
\int\!{\mathrm{d}}\Phi(\{q_i\})\, M(\{q_i\})
\sum_{\kappa,\lambda} {\vec{C}}_{{\lambda}_1 \cdot\cdot(\cdot\cdot{\lambda}_n)}^\dagger (\{q_i\}) {\widehat{S}}_\kappa {\vec{C}}_{{\lambda}_1\cdot\cdot(\cdot\cdot{\lambda}_n)}(\{q_i\}) \otimes
\Bigl[ B_{\kappa_a} B_{\kappa_b} \prod_J J_{\kappa_J} \Bigr]
\,,$$ where the underlying Born process is $\kappa_a (q_a)\, \kappa_b(q_b) \to \kappa_1(q_1) \kappa_2(q_2) \cdots$. ${\mathrm{d}}\Phi$ denotes the phase space integral and $M$ encodes the measurement on the hard kinematics. The restriction on collinear and soft radiation is encoded by the beam functions $B$, jet functions $J$ and soft function $S$. The matching coefficient ${\vec{C}}_{{\lambda}_1\cdot\cdot(\cdot\cdot{\lambda}_n)}$ depends on the helicities ${\lambda}_i$ of the colliding partons and is a vector in color space. It cannot be combined with its conjugate, because the soft function sitting between them is a color matrix. As we will see in [Eq. \[eq:hard\_match\]]{}, for our operator basis these Wilson coefficients are directly given in terms of color-ordered helicity amplitudes.
![Schematic LHC collision. The collinear (green and blue) and soft radiation (orange) are described dynamically in SCET. The hard scattering (zoomed in on the right) is encoded in matching coefficients.[]{data-label="fig:hard"}](waalewijn_fig){width="70.00000%"}
Helicity operators
==================
We start by constructing quarks and gluon fields with definite helicity and then use this to construct our helicity operator basis. We will need the (conjugate) spinor with helicity $\pm$ $$\label{eq:braket_def}
{|p\pm\rangle} = \frac{1 \pm {\gamma}_5}{2}\, u(p)
\,, \qquad
{\langlep\pm|} = \mathrm{sgn}(p^0)\, \bar{u}(p)\,\frac{1 \mp {\gamma}_5}{2}
\,,$$ and the polarization vector for an (outgoing) gluon with momentum $p$ (with reference vector $k$) $${\varepsilon}_+^\mu(p,k) = \frac{{\langlep\!+\!|{\gamma}^\mu|k+\rangle}}{\sqrt{2} \langle k\!-\!|p+\rangle}
\,,\quad
{\varepsilon}_-^\mu(p,k) = - \frac{{\langlep\!-\!|{\gamma}^\mu|k-\rangle}}{\sqrt{2} \langle k\!+\!|p-\rangle}
\,.$$ The smallest building blocks of operators are the quark and gluon fields $\chi_{n,{\omega}}$ and ${{\mathcal B}}_{n,{\omega}\perp}^\mu$, where $n = (1, \hat n)$ denotes the collinear direction and ${\omega}= (1, -\hat n) \cdot p$ is the large component of its momentum $p$. These fields are invariant under collinear gauge transformations through the inclusion of Wilson lines. We define a gluon field of definite helicity by $$\label{eq:cBpm_def}
{{\mathcal B}}^a_{i\pm} = -{\varepsilon}_{\mp\mu}(n_i, {\bar{n}}_i)\,{{\mathcal B}}^{a\mu}_{n_i,{\omega}_i\perp_i}
\,.$$ This definition is chosen such that that at tree level, $$\label{eq:B_tree}
{\bigl\langleg_{\lambda}^a(p)\bigr|{{\mathcal B}}_{i{\lambda}'}^{a'}\bigr|0\bigr\rangle} = \delta_{{\lambda},{\lambda}'}\, \delta^{a a'}\, {\tilde \delta}({\tilde p}_i - p)
\,,\qquad
{\bigl\langle0\bigr|{{\mathcal B}}_{i{\lambda}'}^{a'}\bigr|g_{\lambda}^a(p)\bigr\rangle} = (1 - \delta_{{\lambda},{\lambda}'})\, \delta^{aa'}\, {\tilde \delta}({\tilde p}_i + p)
\,,$$ where the delta function ${\tilde \delta}$ only fixes the large momentum component ${\tilde p}_i = {\omega}_i n_i/2$. Exploiting that fermions come in pairs, we define fermion vectors currents of definite helicity $$\label{eq:jpm_def}
J_{ij+}^{{{\bar \alpha}}\beta}
= \frac{\sqrt{2}\, {\varepsilon}_-^\mu(n_i, n_j)}{\sqrt{\phantom{2}\!\!\omega_i \, \omega_j }}\, \frac{\bar{\chi}^{{\bar \alpha}}_{i+}\, \gamma_\mu \chi^\beta_{j+}}{\langle n_i n_j\rangle}\,,
\qquad
J_{ij-}^{{{\bar \alpha}}\beta}
= -\, \frac{ \sqrt{2}\, {\varepsilon}_+^\mu(n_i, n_j)}{\sqrt{\phantom{2}\!\! \omega_i \, \omega_j }}\, \frac{\bar{\chi}^{{\bar \alpha}}_{i-}\, \gamma_\mu \chi^\beta_{j-}}{[n_i n_j]}
\,,$$ which have similarly simple tree-level matrix elements.
It is now straightforward to write down the basis for a specific process. For example, for $gg q\bar q H$ the helicity basis consists of a total of six independent operators, $$\label{eq:ggqqH_basis}
O_{++(\pm)}^{ab\, {{\bar \alpha}}\beta}
= \frac{1}{2}\, {{\mathcal B}}_{1+}^a\, {{\mathcal B}}_{2+}^b\, J_{34\pm}^{{{\bar \alpha}}\beta}\, H_5
\,, \quad
O_{+-(\pm)}^{ab\, {{\bar \alpha}}\beta}
= {{\mathcal B}}_{1+}^a\, {{\mathcal B}}_{2-}^b\, J_{34\pm}^{{{\bar \alpha}}\beta}\, H_5
\,, \quad
O_{--(\pm)}^{ab\, {{\bar \alpha}}\beta}
= \frac{1}{2} {{\mathcal B}}_{1-}^a\, {{\mathcal B}}_{2-}^b\, J_{34\pm}^{{{\bar \alpha}}\beta}\, H_5
\,.$$ The symmetry factors in front of the operators account for identical fields. They ensure the validity of [Eq. \[eq:Leff\_me\]]{}, leading to a simple matching equation.
For specific processes, it is convenient to decompose the color structure of the Wilson coefficients using a color basis $T_k^{a_1\cdots\alpha_n}$, where $k$ runs over the allowed color structures. This yields $$\label{eq:Cpm_color}
C_{+\cdot\cdot(\cdot\cdot-)}^{a_1\cdots\alpha_n}
= \sum_k C_{+\cdot\cdot(\cdot\cdot-)}^k T_k^{a_1\cdots\alpha_n}
\equiv {\bar{T}}^{ a_1\cdots\alpha_n} {\vec{C}}_{+\cdot\cdot(\cdot\cdot-)}
\,.$$ For the $gg q\bar q H$ process a suitable color basis is given by $$\label{eq:ggqqH_color}
{\bar{T}}^{ ab \alpha{{\bar \beta}}}
= \Bigl(
(T^a T^b)_{\alpha{{\bar \beta}}}\,,\, (T^b T^a)_{\alpha{{\bar \beta}}} \,,\, {\textrm{tr}}[T^a T^b]\, \delta_{\alpha{{\bar \beta}}}
\Bigr)
\,.$$
Matching
========
For our helicity operator basis, the tree-level matrix element of ${{\mathcal L}}_{\mathrm{hard}}$ is equal to the Wilson coefficient for the corresponding configuration of external particles, $$\label{eq:Leff_me}
{\bigl\langleg_1g_2\cdots q_{n-1}\bar{q}_n\bigr|{{\mathcal L}}_{\mathrm{hard}}\bigr|0\bigr\rangle}^{{\mathrm{tree}}}
= C_{+\cdot\cdot(\cdot\cdot-)}^{a_1 a_2\cdots\alpha_{n-1}{{\bar \alpha}}_n}({\tilde p}_1,{\tilde p}_2,\ldots,{\tilde p}_{n-1},{\tilde p}_n)
\,,$$ where $g_i \equiv g_{\lambda_i}^{a_i}(p_i)$ stands for a gluon with helicity $\lambda_i$, momentum $p_i$, color $a_i$, and analogously for (anti)quarks. This implies the tree-level matching equation $$\label{eq:matching_LO}
C_{+\cdot\cdot(\cdot\cdot-)}^{a_1\cdots{{\bar \alpha}}_n}({\tilde p}_1,\ldots,{\tilde p}_n)
= -{\mathrm{i}}{{\mathcal A}}^{\mathrm{tree}}(g_1 \cdots \bar{q}_n)
\,,$$ where ${{\mathcal A}}^{\mathrm{tree}}$ is the tree-level QCD helicity amplitude.
In dimensional regularization, all loop corrections to the matrix element in [Eq. \[eq:Leff\_me\]]{} are scaleless and vanish. These corrections consist of UV poles, which get renormalized, and IR poles, which cancel in the matching because SCET is an effective theory of QCD. This implies, $$\label{eq:hard_match}
C_{+\cdot\cdot(\cdot\cdot-)}^{a_1\cdots{{\bar \alpha}}_n}({\tilde p}_1,\ldots,{\tilde p}_n)
= -{\mathrm{i}}{{\mathcal A}}_{\mathrm{fin}}(g_1\cdots \bar{q}_n)
\equiv \frac{-{\mathrm{i}}\,
{\bar{T}}^{ a_1\cdots\bar\alpha_n} \widehat Z_C^{-1} \vec {\cal A}_{\rm ren}(g_1\cdots \bar{q}_n)}{ Z_\xi^{n_q/2} Z_A^{n_g/2} }
\,.$$ Here $Z_\xi$, and $Z_A$ are the wave function renormalization of the quark and gluon field. $\widehat Z_C$ is the renormalization factor of the Wilson coefficient, which is a matrix in color space. At one-loop order ${{\mathcal A}}_{\mathrm{fin}}$ is simply the IR-finite part of the renormalized QCD helicity amplitude.
As an explicit example, we consider $gg q\bar q H$, for which the helicity operator basis was given in [Eq. \[eq:ggqqH\_basis\]]{}. The color decomposition of the QCD helicity amplitudes into partial amplitudes is $$\label{eq:ggqqH_QCD}
{{\mathcal A}}\bigl(g_1 g_2\, q_{3} {{\bar{q}}}_{4} H_5 \bigr)
= {\mathrm{i}}\!\! \sum_{\sigma\in S_2} \bigl[T^{a_{\sigma(1)}} T^{a_{\sigma(2)}}\bigr]_{\alpha_3{{\bar \alpha}}_4}
A(\sigma(1),\sigma(2); 3_q, 4_{{\bar{q}}}; 5_H)
\!+{\mathrm{i}}\, {\textrm{tr}}[T^{a_1} T^{a_2}]\,\delta_{\alpha_3{{\bar \alpha}}_4} B(1,2; 3_q, 4_{{\bar{q}}}; 5_H)
.$$ Using the color basis in [Eq. \[eq:ggqqH\_color\]]{}, we can read off the Wilson coefficients. E.g. $$\label{eq:ggqqH_coeffs}
{\vec{C}}_{+-(+)}({\tilde p}_1,{\tilde p}_2;{\tilde p}_3,{\tilde p}_4;{\tilde p}_5) =
\left(\!\!\!\begin{tabular}{c}
$A_{\mathrm{fin}}(1^+,2^-;3_q^+,4_{{\bar{q}}}^-; 5_H)$ \\
$A_{\mathrm{fin}}(2^-,1^+;3_q^+,4_{{\bar{q}}}^-; 5_H)$ \\
$B_{\mathrm{fin}}(1^+,2^-;3_q^+,4_{{\bar{q}}}^-; 5_H)$ \\
\end{tabular}\!\!\!\right)
\,.$$ Charge conjugation invariance halves the number of independent Wilson coefficients.
Properties
==========
Our operator basis is automatically crossing symmetric, because the gluon fields ${{\mathcal B}}_{i\pm}$ can absorb or emit a gluon, and the quark current $J_{ij\pm}$ can destroy or produce a quark-antiquark pair, or destroy and create a quark or antiquark.
The helicity operator basis has simple behavior under discrete symmetries. For example, $$\label{eq:Cfield}
{\mathrm{C}}\, {{\mathcal B}}^a_{i\pm}\, T^a_{\alpha{{\bar \beta}}}\,{\mathrm{C}}= - {{\mathcal B}}^a_{i \pm} T^a_{\beta{{\bar \alpha}}}
\,, \quad
{\mathrm{C}}\, J^{{{\bar \alpha}}\beta}_{ij\pm}\,{\mathrm{C}}= -J^{{{\bar \beta}}\alpha}_{ji\mp}
\,,$$ Charge conjugation and parity invariance reduce the number of independent Wilson coefficients.
Since the polarizations of gluons can be treated in $d$ rather than 4 dimensions, it is natural to ask whether our helicity operator basis is complete. Operators with ${\epsilon}$-dimensional polarizations do arise in the matching for states with physical polarizations. They are also not generated by the renormalization group evolution: The only communication between collinear sectors is through soft radiation, which does not carry spin and therefore cannot change helicity.
Conclusions and outlook
=======================
We have described a helicity operator basis, that makes it straightforward to write down the complete basis for a hard scattering process. It also facilitates the matching from fixed-order calculations onto SCET, since the matching coefficients are directly given in terms of the color-ordered helicity amplitudes. We demonstrated its ease by obtaining the Wilson coefficients for $pp\to H + 0,1,2$ jets, $pp\to W/Z/\gamma + 0,1,2$ jets, and $pp\to 2,3$ jets at (next-to-)leading order [@Moult:2015aoa].
The spin of the operators does not play a crucial role at leading power, as the helicities are simply summed over in [Eq. \[eq:sigma\]]{}. This is not true for color, since soft gluons can exchange color. However, at subleading power also the spin structure is essential, since soft gluons can then also transfer spin. Our helicity approach was key in constructing a basis of subleading operators [@Kolodrubetz:2016uim].
Spin information also needs to be kept track of when matching between different SCET theories. For example, to describe two nearby hard jets one matches through an intermediate SCET where the two nearby jets are not separately resolved [@Bauer:2011uc]. To keep track of this spin information in the matching, helicity fields proved particularly useful [@Pietrulewicz:2016nwo].
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in part by the Office of Nuclear Physics of the U.S. Department of Energy under Contract No. DE-SC0011090, the DFG Emmy-Noether Grant No. TA 867/1-1, the Marie Curie International Incoming Fellowship PIIF-GA-2012-328913, the Simons Foundation Investigator Grant No. 327942, NSERC of Canada, and the D-ITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
References {#references .unnumbered}
==========
[10]{}
Z. Bern, et al. , 88:014025, 2013. R. K. Ellis, et al. , 01:012, 2009. S. Badger, et al , 89:034019, 2014. V. Hirschi, et al. , 05:044, 2011. G. Cullen, et al. , C72:1889, 2012. F. Cascioli, et al. , 108:111601, 2012.
C. W. Bauer, et al. , 63:014006, 2000. C. W. Bauer, et al. , 63:114020, 2001. C. W. Bauer and I. W. Stewart. , 516:134–142, 2001. C. W. Bauer, et al. , 65:054022, 2002.
I. Moult, I. W. Stewart, F. J. Tackmann, and W. J. Waalewijn. , 93:094003, 2016.
Daniel W. Kolodrubetz, Ian Moult, and Iain W. Stewart. arXiv:1601.02607.
C. W. Bauer, et al. , 85:074006, 2012.
Piotr Pietrulewicz, Frank J. Tackmann, and Wouter J. Waalewijn. arXiv:1601.05088.
[^1]: Speaker
|
---
abstract: |
We present a comprehensive series of simulations to study the secular evolution of disk galaxies expected in a LCDM universe. Our simulations are organized in a hierarchy of increasing complexity, ranging from rigid-halo collisionless simulations to fully live simulations with gas and star formation.
Our goal is to examine which structural properties of disk galaxies may result from secular evolution rather than from hierarchical assembly. In the vertical direction, we find that various mechanisms lead to heating, the strongest of which is the buckling instability of a bar, which leads to peanut-shaped bulges; these can be recognized face-on even in the presence of gas. We find that bars are robust structures that survive buckling and require a large ($\sim 20\%$ of the total mass of the disk) central mass concentration to be destroyed. This can occur in dissipative simulations, where bars induce strong gas inflows, but requires that radiative cooling overcome heating. We show how angular momentum redistribution leads to increasing central densities and disk scale lengths and to profile breaks at large radii. The breaks in these simulations are in excellent agreement with observations, even when the evolution is collisionless. Disk scale-lengths increase even when the total disk angular momentum is conserved; thus mapping halo angular momenta to scale-lengths is non-trivial. A decomposition of the resulting profile into a bulge$+$disk gives structural parameters in reasonable agreement with observations although kinematics betray their bar nature.
These findings have important implications for galaxy formation models, which have so far ignored or introduced in a very simplified way the effects of non-axisymmetric instabilities on the morphological evolution of disk galaxies.
author:
- 'Victor P. Debattista'
- Lucio Mayer
- 'C. Marcella Carollo'
- Ben Moore
- James Wadsley
- Thomas Quinn
title: The Secular Evolution of Disk Structural Parameters
---
Introduction
============
In the current paradigm, galaxy formation is hierarchical ( White & Rees 1978; Steinmetz & Navarro 2002). Indeed, evidence can be found of continued accretion onto both the Milky Way (Ibata 1994; Helmi 1999) and M31 (Ferguson 2002). Although the standard framework of disk formation in such cosmogonies was formulated some time ago (White & Rees 1978; Fall & Efstathiou 1980), disk galaxy formation remains a challenging problem for both simulations ( Navarro & Steinmetz 2000; Abadi 2003) and semi-analytical models ( Somerville & Primack 1999; Hatton 2003; van den Bosch 1998, 2000, 2001, 2002; Firmani & Avila-Reese 2000; Mo 1998; Dalcanton 1997; Benson 2003; Cole 1994, 2000; Baugh 1996; Kauffmann 1993). The latest generation of simulations produces disks having roughly the correct sizes and structural properties (Governato 2004; Abadi 2003; Sommer-Larsen 2003). This is due partly to the increased resolution that reduces the artificial loss of angular momentum of the baryonic component. At the same time, it has been realized that only halos with a quiet merging history after $z\sim 2$ can host disk galaxies today (but see Springel & Hernquist 2005). Therefore in the current picture most of the mass of the disk is assembled from the smooth accretion of gas cooling inside the dark halo (combined with the accretion along cold filaments for lower mass objects; Katz & Gunn 1991; Katz 2003; Birnboim & Dekel 2003; Kereš 2005) following an intense phase of merger activity that can give rise to the stellar halo and a massive old bulge.
During disk assembly, secular evolution must have played a role in shaping the structure of disk galaxies as we see them at $z=0$. Non-axisymmetric instabilities, particularly bars, drive a substantial redistribution of mass and angular momentum in the disk. A possible product of bar-driven evolution is the formation of a bulge-like component. Such a component will have a stellar population similar to that of the disk and is thus younger than the old spheroid formed by the last major merger. Low-mass bulges may result from such a mechanism; observed low-mass bulges have disk-like, almost-exponential stellar density (Andredakis & Sanders 1994; Courteau 1996; de Jong 1996; Carollo 2002; Carollo 2001; Carollo 1999; Carollo 1998; Carollo 1997; MacArthur 2003) and in some cases disk-like, cold kinematics (Kormendy 1993; Kormendy 2002). Comparison between bulge and disk parameters shows a correlation between the scale-lengths of bulges and disks (de Jong 1996; MacArthur 2003) and, on average, similar colors in bulges and inner disks (Terndrup 1994; Peletier & Balcells 1996; Courteau 1996; Carollo 2001). The disk-like properties of bulges and the links between bulge and disk properties have been suggested to indicate that bulges may form through the evolution of disk dynamical instabilities such as bars, which are present in about $70\%$ of disk galaxies (Knapen 1999; Eskridge 2000). Further evidence for secular evolution comes from edge-on galaxies, where bulges are often found to be box- or peanut-shaped (Lütticke 2000), a shape associated with evolution driven by the presence of a bar (Combes & Sanders 1981; Pfenniger 1984; Combes 1990; Pfenniger & Friedli 1991; Raha 1991; Kuijken & Merrifield 1995; Bureau & Freeman 1999; Bureau & Athanassoula 1999; Athanassoula & Bureau 1999).
Secular evolution must also be considered for comparisons of predictions based on semi-analytic models of disk formation and observations to be meaningful. In the standard picture baryons cooling in dark halos to form a disk have the same specific angular momentum of the dark matter, and this is conserved during collapse. Then the distribution of disk scale-lengths can be computed from the known distribution of halo angular momenta ( Dalcanton 1997; Mo 1998; van den Bosch 1998). However, de Jong & Lacey (2000) found that the width of the observed disk scale-length distribution at fixed luminosity is smaller than that predicted by such simple models. This suggests that the mapping between initial halo angular momenta and disk scale-lengths cannot be so trivial. In addition to complications from the cosmological side, including that the initial specific angular momentum distribution of the baryons need not be like that of the dark halo ( van den Bosch et al. 2002) and angular momentum distributions favoring disks more centrally concentrated than exponential (Firmani & Avila-Reese 2000; van den Bosch 2001; Bullock 2001), secular evolution will also change disk structure. In the simplest prescriptions this is not considered, while it is known since Hohl (1971) that bars can drive substantial evolution of disk profiles.
Which structural properties of present-day disk galaxies are primordial and which are the result of internal evolution? Already by $z\sim 1$ a population of disk galaxies with scale-lengths similar to those of a local population is observed (Lilly 1998), suggesting that the structural properties of disk galaxies have not changed substantially since then. If the quiescent phase of disk assembly starts early, as current cosmological simulations suggest, secular evolution might have already been operating by $z=1$. Verification that the secular evolutionary timescale can be sufficiently short is thus required. With the present algorithms and computing power, simulations of individual isolated galaxies are best able to achieve sufficient force and mass resolution to address such issues. Simulations of individual galaxies decoupled from the hierarchical growth have the additional advantage of exploring directly the role that internal secular evolution plays in shaping the currently observed galaxy population.
In this paper we report on a series of such simulations exploring how disks evolve in the presence of a bar. An important goal of this paper is to demonstrate the wealth of structural properties possible in cosmologically-motivated disk galaxies and to identify what properties of the mass distribution of disk galaxies may result from internal evolution, rather than arising directly from hierarchical assembly. We identify several structural properties we would like to test secular evolution for: vertical thickening, inner profile steepening, profile breaks, and bar destruction. We explore simulations in which the full disk-halo interaction is self-consistent in cosmologically-motivated dark matter halos with and without gas and star formation. We use various prescriptions for gas physics, to understand how this impacts the evolution. We also use rigid-halo simulations as in Debattista (2004; hereafter Paper I) which allow us to study a variety of phenomena at high spatial and mass resolution. These simulations also help us to isolate physical mechanisms by which secular evolution occurs. Together these simulations allow us to assess the impact of secular evolution on disk galaxies.
Methods {#sec:methods}
=======
Live-halo models {#ssec:livehalos}
----------------
----- ------ ------ ----- ------ ----- ----- ---------
NC1 4.92 3.55 0.7 0.05 0.0 - -
NC2 4.92 3.55 1.0 0.05 0.0 - -
NC3 4.92 3.55 1.7 0.05 0.0 - -
NG1 5.45 3.44 1.7 0.05 0.1 4 RC
NG2 5.45 3.44 1.7 0.05 0.1 4 A
NG3 5.45 3.44 1.7 0.05 0.1 4 RC + SF
NG4 10.9 2.35 1.7 0.11 0.5 0.4 A
NG5 5.45 3.44 1.7 0.05 0.5 0.8 RC
----- ------ ------ ----- ------ ----- ----- ---------
$M_{disk}$, $R_{\rm d}$, $Q_{star}$, $Q_{gas}$ are the disk mass, initial disk scale-length, and the minimum Toomre-$Q$ parameter of, respectively, the stellar and the gaseous disk. $f_d$ is the fraction of disk (stars+gas) to dark matter mass and $f_{\rm gas}$ is the fraction of the baryonic mass which is in the gaseous state in the initial conditions, while “Gas physics” lists the physics of the gas used: ’RC’ refers to ’radiative cooling’, ’A’ refers to ’adiabatic’ and ’SF’ refers to ’star formation’. \[tab:live\]
Live-halo models are built using the technique developed by Hernquist (1993; see also Springel & White 1999). The structural properties of halos and disks are tied together by the scaling relations expected in the currently favored structure formation model, $\Lambda$CDM. We start by choosing the value of the circular velocity of the halo at the virial radius, $V_{vir}$, which, for an assumed cosmology (hereafter $\Omega_0=0.3$, $\Lambda=0.7$, $H_0=65$ ), automatically determines the virial mass, $M_{vir}$, and virial radius, $R_{vir}$, of the halo (Mo, Mao & White 1998). Halos are isotropic and have angular momentum that is specified by the spin parameter, $\lambda = [J^2 |E|/ (G^2 M_{vir}^5)]^{1/2}$, where $J$ and $E$ are, respectively, the total angular momentum and total energy of the halo and $G$ is the gravitational constant. We use $\lambda=0.045$ throughout, close to the mean value measured in cosmological simulations ( Gardner 2001). The halo density profile is an NFW (Navarro, Frenk & White 1996) with a given value of the concentration $c$, where $c=R_{vir}/r_s$, $r_s$ being the halo scale radius. The higher the concentration, the higher the halo density near the center at a given value of $M_{vir}$ (and therefore the more steeply rising is its inner rotation curve). Adiabatic contraction of the halo due to the presence of the disk is taken into account by assuming that the spherical symmetry of the halo is retained and that the angular momentum of individual dark matter orbits is conserved (see Springel & White 1999). The disk mass fraction relative to the halo virial mass, $f_d=M_d/M_{vir}$ is $0.05$, consistent with estimates for galaxies in the local Universe (e.g. Jimenez, Verde & Oh, 2003) and is conservatively lower than the estimate of the universal baryonic mass fraction yielded by [*WMAP*]{} (Spergel et al. 2003). Our models implicitly assume that the disk forms out of collapsed gas that started with the same specific angular momentum as the halo and that such angular momentum was conserved during infall (Mo, Mao & White 1998). The disk has an exponential surface density profile with scale length $R_d$ that is determined by the value of $\lambda$ (which sets the degree of available centrifugal support) and by the values of $c$, $f_d$, and $M_{vir}$ (which together set the depth of the potential well). The setup of the stellar disk is complete once the Toomre parameter, $Q(R)$, is assigned (Toomre 1964). This corresponds to fixing the local radial velocity dispersion $\sigma_R$, since $Q(R)=\sigma_R \kappa/3.36G \Sigma_s$, where $\kappa$ is the local epicyclic frequency, $G$ is the gravitational constant, and $\Sigma_s$ is the disk surface density. The velocity field of the disk is calculated as in Springel & White (1999; see also Hernquist 1993); in particular, the radial and vertical velocity dispersions are assumed to be equal, and the azimuthal velocity dispersion is determined from the radial dispersion using the epicyclic approximation. Dark matter halos are sampled using $10^6$ particles and stellar disks by $2\times
10^5$ particles. The gravitational softening of both components is equal to $300$ pc. We reran a few simulations with higher resolution ($5 \times 10^6$ halo particles and $5\times 10^5$ disk particles) and smaller softenings ($50$ pc; see Paper I) to check for resolution effects. We found the analysis presented in this paper to be fairly insensitive to the resolution adopted, and in the remainder we always show results for runs with the standard resolution. The live-halo models used in this paper have structural parameters in line with the expectations of $\Lambda$CDM models for a Milky-Way sized system. These models are similar to the mass models of the Milky Way presented by Klypin, Zhao & Somerville (2002). The rotation curve of the collisionless models used in the paper is shown in Figure \[fig:rc1\]. All collisionless models have the same rotation curve since they differ only in terms of their Toomre parameter. These simulations were carried out with the parallel multistepping tree code PKDGRAV (Stadel 2001).
SPH simulations
---------------
Gasdynamical simulations were carried out with GASOLINE, an extension of PKDGRAV (Stadel 2001) that uses smoothed particle hydrodynamics (SPH) to solve the hydrodynamical equations (Wadsley, Stadel & Quinn 2004). The gas is ideal with equation of state $P=(\gamma -1)\rho u$, where $P$ is the pressure, $\rho$ is the density, $u$ is the specific thermal energy, and $\gamma = 5/3$ is the ratio of the specific heats (adiabatic index). We are assuming that the gaseous disk represents the partially ionized hydrogen component of the galaxy. In its general form the code solves an internal energy equation that includes an artificial viscosity term to model irreversible heating from shocks. The code adopts the standard Monaghan artificial viscosity and the Balsara criterion to reduce unwanted shear viscosity (Balsara 1995). In the adiabatic runs the thermal energy can rise as a result of compressional and shock heating and can drop because of expansion. In runs including radiative cooling energy can be released also through radiation. We use a standard cooling function for a primordial gas composition (helium and atomic hydrogen).
Dissipational galaxy models are built following the same prescription described in Section \[ssec:livehalos\] for live-halo models but include also a gaseous disk represented by $10^5$ SPH particles each with a gravitational softening of $300$ pc. The basic properties of the runs performed are shown in Table \[tab:live\]. The disk mass fraction is $f_d=0.05$ in all runs except run NG4, which has $f_d=0.12$. The gaseous disk has a temperature of $10^4$ K, consistent with the gas velocity dispersions derived in observations (Martin & Kennicutt 2001). The gaseous disk has an exponential surface density profile with the same scale-length as the stellar disk (see Mayer & Wadsley 2004), and its thickness is determined by local hydrostatic equilibrium. In a gaseous disk the Toomre parameter is defined as $Q(R)= c_s \kappa/\pi G \Sigma_g$, where $c_s$ is the sound speed and $\Sigma_g$ is the surface density of the gas. The global stability of the disk will be determined by the combined stability properties of the stellar and gaseous disks (Jog & Solomon 1991). Gravitational instabilities can be more vigorous in a cold gaseous disk and might affect the development of non-axisymmetry even in the stellar disk (Rafikov 2001). In particular, in models having $50\%$ gas, the gaseous disks have $Q<2$ over most of the radial extent of the galaxy, which should make the system unstable to non-axisymmetric perturbations irrespective of the stellar $Q$ (see Rafikov 2001 for the case in which the sound speed $c_s$ is $\sim 0.3$ of the radial stellar velocity dispersion as in our models). In the models with $10\%$ gas instead, the gaseous disk has a high $Q$, making them stable to axisymmetric perturbations although they still can be unstable to non-axisymmetric perturbations since $Q < 2$ for the stars (Binney & Tremaine 1987). The $Q$ profiles of gas and stars are shown in Figure \[fig:Qprofs\].
Finally, we also include star formation. The star formation algorithm follows that of Katz (1992), where stars form from cold, Jeans-unstable gas particles in regions of convergent flows (see also Governato 2004; Stinson 2006). The star formation efficiency parameter $c_* = 0.15$, but with the adopted scheme its value has only a minor effect on the star formation rate (Katz 1992). No supernova feedback is included in our simulations. The rotation curve of the models with gas can be seen in Figure \[fig:rc2\].
Rigid-halo models
-----------------
These simulations consist of a live disk inside a rigid halo, which permit large numbers of particles and thus allow high spatial resolution to be reached. High resolution is particularly useful for studying the vertical evolution of disks. Rigid-halo simulations are better suited to systems in which the disk is dominant in the inner regions because the interaction with the halo is intrinsically weaker (Debattista & Sellwood 2000).
The rigid halos were represented by either a logarithmic potential with a core $$\Phi_L(r) = \frac{v_{\rm h}^2}{2}~ \ln(r^2 + r_{\rm h}^2),$$ or a cuspy Hernquist model $$\Phi_H(r) = -\frac{M_{\rm h}}{r+r_{\rm h}}.$$
The initially axisymmetric disks were all Sérsic (1968) type, $$\rho_{\rm d}(R,z) = \frac{M_{\rm d}}{2 \pi R_{\rm d}^2} e^{-(R/R_{\rm
d})^{(1/n)}} \frac{1}{\sqrt{2 \pi} z_{\rm d}}e^{-\frac{1}{2}(z/z_{\rm
d})^2}$$ with scale-length $R_{\rm d}$, mass $M_{\rm d}$, and Gaussian thickness $z_{\rm d}$, truncated at a radius $R_t$ and a Sérsic index $n$. Disk kinematic setup used the epicyclic approximation to give constant Toomre-$Q$ and the vertical Jeans equation to set vertical motions appropriate for a constant thickness. The disks were represented by $4-7.5\times 10^6$ equal-mass particles. In units where $R_{\rm d} = M_{\rm d} = G = 1$, which gives a unit of time $(R_{\rm d}^3/GM_{\rm d})^{1/2}$, the values for the disk$+$halo parameters such that our rotation curves were always approximately flat to large radii are given in Table \[tab:rigid\]. One possible scaling to real units has $R_{\rm d} = 2.5$ and $V_c = 200$ , so that $M_{\rm d} = 2.3 \times 10^{10} M_\odot$ and the unit of time is 12.4 Myr. We adopt this time scaling throughout but present masses, lengths, and velocities in natural units.
These simulations were run on a three-dimensional cylindrical polar grid code (described in Sellwood & Valluri 1997) with $N_R\times
N_\phi \times N_z = 60 \times 64 \times 243$. We also ran tests with finer grids to verify that our results are not sensitive to the grid used. The radial spacing of grid cells increases logarithmically from the center, with the grid reaching to $\sim 2 R_t$ in most cases; except where noted, $R_t = 5 \rd$. For all of the simulations in Table \[tab:rigid\], the vertical spacing of the grid planes, $\delta z$, was set to $0.0125 \rd$ (except in run L1 where we reduced this to $0.0083 \rd$), but we confirmed that our results do no change with smaller $\delta z$. We used Fourier terms up to $m=8$ in the potential,[^1] which was softened with the standard Plummer kernel, of softening length $\epsilon = 0.017 \rd$, although we also tested smaller $\epsilon$ and larger maximum $m$. Time integration was performed with a leapfrog integrator with a fixed time-step, $\delta t = 0.01
(\equiv 1.24 \times 10^5 \mathrm{yr})$ for all runs with $n=1$; otherwise, we use $\delta t = 0.0025 (\equiv 3.1 \times 10^4
\mathrm{yr})$. With these values, a circular orbit at $R_{\rm d}/10$ typically is resolved into 600 steps. For the logarithmic halos, we set $(r_{\rm h},v_{\rm h}) = (3.3,0.68)$, while the Hernquist halos had $(r_{\rm h},M_{\rm h}) = (20.8,43.4)$.
---- ------- ----- ----- ------- ------ ------- ------- ------ ----- ------ -----
L1 0.025 1.6 1.0 Log. 3.3 -1.54 -3.80 0.44 1.1 0.32 1.7
L2 0.05 1.2 1.0 Log. 3.3 -1.15 -3.45 0.52 1.3 0.17 2.1
L3 0.05 1.6 1.0 Log. 3.3 -1.36 -3.77 0.56 0.9 0.47 1.7
L4 0.10 1.2 1.0 Log. 3.3 -1.77 -8.07 0.36 0.8 0.34 1.7
L5 0.10 1.6 1.0 Log. 3.3 -1.36 -7.19 0.69 0.8 0.68 1.5
L6 0.20 1.2 1.0 Log. 3.3 -1.11 -8.07 0.53 0.8 0.44 1.8
S1 0.05 1.0 1.5 Log. 3.3 -0.81 -3.54 0.61 1.8 0.16 2.3
S2 0.05 1.0 2.0 Log. 3.3 -1.20 -3.76 0.76 3.1 0.14 1.5
S3 0.05 1.0 2.5 Log. 3.3 -1.27 -5.91 0.63 1.7 0.12 1.5
H1 0.05 1.2 1.0 Hern. 20.8 -1.39 -5.68 0.46 1.1 0.22 2.4
H2 0.05 1.6 1.0 Hern. 20.8 -1.54 -6.83 0.37 1.1 0.51 1.4
H3 0.05 2.0 1.0 Hern. 20.8 -1.94 -5.22 0.06 1.2 0.70 1.0
---- ------- ----- ----- ------- ------ ------- ------- ------ ----- ------ -----
$z_{\rm d}$, $Q$ and $r_{\rm h}$ are the disk Gaussian scale-height, Toomre-$Q$ and halo scale-length, respectively. $n$ is the index of the initial Sérsic disk ($n=1$ is an exponential disk). In column “Halo” we describe the halo type: a logarithmic or Hernquist potential. $\ln A_\phi$ and $\ln A_z$ are the maximum amplitudes of the bar and of buckling. Strong buckling corresponds to $\ln A_z
\gtsim -4$. The quantities $B/D$, $n_b$, $R_{b,eff}/R_{d,f}$ and $R_{d,f}$ are all parameters of the bulge$+$disk decomposition at the end of the simulation. Here $R_{d,f}$ is the final value of the scale-length. The following simulations were presented also in Paper II: L2 (as R1), S3 (as R4), L6 (as R6) and H2 (as R7). \[tab:rigid\]
Tracking structural evolution
-----------------------------
Our models host disks that are massive enough to form bars in a few dynamical times. The formation of the bar is just one of the mechanisms that drive the morphological evolution of the disks in our simulations. Spiral structure and vertical instabilities, like the buckling instability, also lead to secular evolution of disk structural parameters, from stellar density profiles and disk scale-lengths to the bulge-to-disk ratio. In order to track the evolution of our models, we measured the amplitude of the bar, $A_\phi$, as the normalized amplitude of the $m=2$ density distribution: $$A_\phi = \frac{1}{N} \left| \sum_j e^{2 i \phi_j}\right|$$ where $\phi_j$ is the two-dimensional cylindrical polar angle coordinate of particle $j$. The sum extends over stellar particles only. We measured the $m=2$ bending amplitude, $A_z$, similarly: $$A_z = \frac{1}{N} \left| \sum_j z_j e^{2 i \phi_j}\right|.$$ These quantities allowed us to determine when a bar formed and whether it buckled.
Vertical evolution
==================
The vertical direction is best resolved in the rigid-halo simulations, so we begin by considering those. By far the fastest secular evolution in the vertical direction is driven by the buckling instability. This bending instability, which is caused by anisotropy, is very efficient at heating the disk vertically.
Raha (1992) described the distortion of a bar during buckling. As that work is not widely available, we present a description of buckling in run L2 before exploring its effects on stellar disks. In the animation accompanying this paper (see also Figure \[fig:bending\]) we show the evolution of this run between $t=1.0$ and $t=2.2$ Gyr. At $t=1.12$ Gyr, the system is largely symmetric about the midplane but develops a small bend by $t=1.18$ Gyr, which displaces the center toward positive $z$ and the outer parts of the bar toward negative $z$. In the outer parts of the bar, where it has its largest vertical excursion, the bend develops on the leading side of the bar where it persists for some time, eventually evolving into a trailing spiral. As it passes the major axis of the bar, it grows substantially. After this bend has dissipated, the process repeats another two times (see also Martinez-Valpuesta 2006), with small bends on the leading side of the bar, developing into stronger bends on crossing the bar’s major axis. At smaller radii within the bar, the peak bending amplitude occurs close to the minor axis (at $t=1.43$ Gyr). The region outside the bar also bends ( at $t=1.55$ Gyr) but generally with smaller amplitude. Small-scale bending persists to late times and is still ongoing as late as $t=2.60$ Gyr. At $t\simeq 1.42$ Gyr, the buckling produces the largest mean vertical displacement, $\overline{z} \simeq 0.157 \rd$, which is more than 3 times the initial disk thickness for this simulation, $z_{\rm d} = 0.05 \rd$.
Buckling leads to a significant vertical heating: at the center $\sigma_w/\sigma_u$ increases from $\sim 0.4$ to $\sim 0.85$, and $\sigma_w/\sigma_u$ averaged inside $R=1.2$ increases by a similar factor, where $\sigma_w$ is the vertical velocity dispersion and $\sigma_u$ is the radial velocity dispersion. The three phases of strong bending (which can be seen in the accompanying animation) can be identified with three phases of strong vertical heating.
Peanuts from buckling {#ssec:peanuts}
---------------------
After buckling, model L2 becomes distinctly peanut-shaped when viewed edge-on. The disk scale-height $h_z$ has increased by factors of $2-6$ depending on where it is measured. It is larger on the minor axis of the bar than on the major, and is smallest at the center, properties that are typical of all of the rigid-halo simulations but are more pronounced for the buckled bars (see also Sotnikova & Rodionov 2003). The peanut results in a negative double minimum in $d_4$, the fourth-order Gauss-Hermite moment (Gerhard 1993; van der Marel & Franx 1993) of the density distribution ( the peanut produces a flat-topped density distribution) [*within the bar*]{}. The peanut is also manifest in the face-on kinematics as a pronounced negative minimum in the Gauss-Hermite kinematic moment, $s_4$.[^2] No similar signature of a peanut is evident before buckling. In Debattista (2005; Paper II) we developed this into a diagnostic signature of peanuts seen nearly face-on.
Buckling need not always result in a peanut. Both models L1 and L2 formed strong bars and had roughly equal buckling that vertically heated both disks substantially. However, whereas L2 formed a strong peanut, L1 formed only a weak one.
Vertical heating with live-halos and gas
----------------------------------------
The vertical heating of disks in our simulations with live halos is dominated by the buckling instability when gas is absent. When gas is present, we found that it may suppress buckling, in agreement with Berentzen (1998). (We tested that this result does not depend on force resolution by repeating runs with a softening 6 times smaller, 50 pc.) But this depends on how readily gas dissipates its thermal energy (see Figure \[fig:gasamps\]), a point not appreciated by Berentzen (1998) who performed only isothermal simulations. When the gas can cool (NG1), buckling is suppressed. In this case the bar amplitude decreases significantly because of the central gas concentration produced in the inflow driven by the bar. The reduced bar strength implies a reduced radial anisotropy in the system, which then is less prone to buckling (Berentzen 1998). However, the bars in the $10\%$ gas case are not destroyed (see Section \[sec:barsurvival\]), which suggests that the complete suppression of buckling does not simply reflect the decrease in bar strength. This is evident especially in run NG3, which has gas cooling and star formation, in which a fairly strong bar is present and yet buckling did not occur. In Berentzen (1998) weak buckling was always associated with weak bars.
In axisymmetric systems, central concentrations suppress bending modes (Sotnikova & Radionov 2005). Demonstrating a similar result in the barred case in the presence of gas is non-trivial. Indeed, whether gas suppresses buckling directly because it can dissipate bending energy or because it leads to central mass concentrations (Berentzen 1998) proved difficult to determine because any experiment we could conceive of also led to different bars. For example, when we replaced the central (inner 600 pc) gas blob formed in run NG1 after $\sim 2$ Gyr with a point mass having equal mass and softening equal to its half-mass radius and evolved the system as purely collisionless, we found that the bar buckled, but in the meantime it also grew stronger. While we were not able to design a clear test for these two hypotheses, our results do exhibit a correlation between buckling amplitude and central mass concentration (Figure \[fig:bucklbar\]).
Vertical heating in the presence of gas occurs even without buckling, as shown in Figure \[fig:gasheating\]. In NG1 and NG3, in which no buckling occurs, we still see an increase in the vertical stellar velocity dispersion. This heating is gentle, with $\sigma_z/\sigma_R$ increasing nearly linearly with time and never falling below the critical threshold of $\sim 0.4$. The cause of this vertical heating appears to be scattering by spirals in the gas disk, which remains significantly thinner than the stellar disk in this simulation. In contrast, heating by buckling, as in run NG2 (in which the gas disk quickly became thicker than the stellar disk) is abrupt. The gas thickness in this simulation results from a steadily increasing temperature as a result of shock heating.
Peanuts without buckling
------------------------
The live-halo simulations show an alternative way in which peanuts can form. The gas-free live-halo simulations all buckled and in the process formed peanuts no different from those described above. In some simulations with gas ( NG3) we found peanuts forming without buckling. It is possible that these formed by direct resonant trapping of orbits in the growing bar potential (Quillen 2002). Indeed, in run NG3 $A_\phi$ increased by a factor of about 2 over a period of $\sim 2$ Gyr (Figure \[fig:gasamps\]).
Peanuts in the presence of gas
------------------------------
We showed in Paper II that peanuts produce prominent minima in the kinematic Gauss-Hermite moment, $s_4$, when viewed face-on. In Figure \[fig:gaspeanut\] we show that a peanut can still be recognized by a prominent negative minimum in $s_4$ in model NG2, despite the presence of gas. The reason for this is that the gas sinks to smaller radii than the peanut.
Bar Destruction {#sec:barsurvival}
===============
Both the buckling instability and the growth of massive central objects have been suggested to destroy bars. We examine each of these in this section.
The collisionless case
----------------------
After Raha (1991) showed that buckling weakens bars, it has often been assumed that bars are destroyed by buckling. To our knowledge, that buckling destroys bars has never been demonstrated; indeed, none of the bars in our simulations were destroyed by buckling. Merritt & Sellwood (1994) noted that buckling grew stronger when force resolution was increased because then particles have larger vertical oscillatory frequencies, destabilizing the bar. Thus, the most damaging buckling occurs in the rigid-halo simulations, where the vertical structure is best resolved. We verified that the survival of bars after buckling is not due to insufficient resolution by running an extensive series of numerical tests at higher resolution using rigid halos. For these tests, we used model L2, one of the most strongly buckling simulations. The bar amplitude $A_\phi$ after buckling in these tests turned out to not be strongly dependent on any numerical parameter. Thus, that buckling does not destroy bars is not an artifact of insufficient resolution. This conclusion is also supported by a higher resolution version of the live-halo simulation of Paper I, where we increased the number of particles ($N_{halo} = 4\times 10^6$, $N_{disk} = 2\times 10^5$) and decreased the softening ($\epsilon =
50$ pc for all particles). Again, although buckling was strong here as well, the bar was not destroyed.
We also checked that increasing spatial resolution does not lead to stronger buckling in weakly buckling simulations. We re-ran simulations L5 and H1 at higher resolution ($m=32$, $\delta z = 0.005$ and $\epsilon = 0.0083$) and found that $A_\phi$ is barely affected, demonstrating that vertical frequencies were well resolved and confirming that the weak bucklings are intrinsic.
The effect of bar slowdown
--------------------------
Although our grid code simulations have high resolutions, they have rigid halos; thus bar slowdown (Weinberg 1985; Debattista & Sellwood 1998, 2000; Sellwood & Debattista 2006) is not included. Araki (1985) showed that stability to bending modes in the infinite, uniform, non-rotating sheet required that $\sigma_w \geq 0.293
\sigma_u$. As a bar slows, $\sigma_u$ is likely to increase, which may drive a stable bar to instability. To test whether this happens, we slowed down some of our bars by introducing a retarding quadrupole moment $$\Phi_{ret} = \Phi_0~ f(R)~ g(s)~ e^{ -2 i (\phi_{\rm bar} - \phi_r)}$$ trailing behind a bar. Here $s = (t-t_0)/(t_1-t_0)$ and $g(s) = -16
s^2 (1 - s)^2$, so that the perturbation is gently switched on at $t_0$ and off at $t_1$. The phase of the bar, $\phi_{\rm bar}$, was computed at each time-step by computing the phase of the $m=2$ Fourier moment of all of the particles; since the disk also has spirals, there is a typical uncertainty of order $\pm 15\degrees$ in the bar angle. We therefore set $\phi_r = 30\degrees$ to be certain that the retarding potential always trails the bar. We chose the radial dependence of the retarding potential to be $f(R) = R/(1+R^2)^2$, which ensured that it peaks inside the bar radius.
We performed these experiments on runs L2, L4 and H2; a list of all of the experiments is given in Table \[tab:slowruns\]. For run L2, we switched on the quadrupole shortly after the bar formed and switched it off before it buckled. In this case we found that the buckling is then stronger, which leaves the bar $\sim 20\%$ weaker but still does not destroy it.
------- ----- ------ ------
L2.s1 16. 0.74 1.24
L2.s2 1.6 0.74 1.24
L4.s1 1.6 2.60 3.10
L4.s2 4.8 2.60 3.10
H2.s1 4.8 2.60 3.10
H2.s2 1.6 2.60 3.10
H2.s3 0.8 2.60 3.10
H2.s4 0.8 2.60 3.72
------- ----- ------ ------
: The series of simulations to test the effect of bar-slowdown on the buckling instability.
$\Phi_0$ measures the relative amplitude of the retarding perturbation, while $t_0$ and $t_1$ give the time when the perturbation is switched on and off. \[tab:slowruns\]
The other two systems on which we tried such experiments had not buckled when undisturbed. In these cases, we turned on the retarding quadrupole after the bar had settled and turned it off not less than 1.5 bar rotations later. Slowing down these bars resulted in very strong buckling, stronger even than in run L2. But even in these somewhat extreme cases the bar survives; we illustrate this in Figure \[fig:slowdown\], where we show the various bar slowdown experiments in model H2. While these bucklings do not destroy bars, which we determine simply by visual inspection, in a few cases they leave a much weaker bar, which would be better described as an SAB than an SB. In Figure \[fig:SAB\] we present the most extreme example of H2.s4, in which the final bar axis ratio was $b/a \simeq 0.85$; although weak, this can still clearly be recognized by visual inspection. The edge-on view of a slice taken around the bar’s major axis reveals a peanut, which can be recognzed by the double minimum in the $s_4$ diagnostic. These slowed bars probably represent an upper limit to the damage buckling can inflict on strong bars.
In the live-halo simulations, where the bar can interact with the halo and slow down, we continue to find that buckling does not destroy bars. Therefore we conclude that the bar buckling instability does not destroy bars (see also Martinez-Valpuesta & Shlosman 2004).
Central gas mass growth
-----------------------
We now consider bar destruction via the growth of a massive central gaseous object. The bottom panel of Figure \[fig:gasamps\] shows the evolution of the bar amplitude for models NG1-NG3. The bar amplitude depends very strongly on the gas physics: when the gas is adiabatic (NG2), it does not become very centrally concentrated and the bar amplitude is not very strongly affected by the gas. If the gas can cool (NG1), it sinks quickly to the center and remains there. Thus, the bar forms already much weaker. Continued infall at later times further weakens the bar. The main difference between the cooling and adiabatic simulations is in the amount of gas that sinks into the center of the disk. In NG1, gas accounts for $\sim 60\%$ of the mass within 500 pc (Figure \[fig:fracgas\]); with such a high fraction, it is unsurprising that the bar is weaker. The softening in these simulations is 300 pc; therefore, the central gas mass is not well resolved. At these scales the hydrodynamical resolution is higher than the gravitational force resolution (the gravitational softening volume contains many times the SPH smoothing volume) with the result that the collapse of gas toward the center is inhibited (see Bate & Burkert 1997). Therefore, the same amount of gas would probably have collapsed to an even smaller radius had we had greater force resolution, weakening the bar further. In NG2 gas cannot radiate away the intense compressional heating it experiences, and only $10\%$ by mass is found within 500 pc. Thus, its bar is stronger than in NG1. When star formation is allowed (NG3), the gas accumulated into the center drives a starburst. This converts most of the highly concentrated gas into stars, which can now support the bar. A strong bar again forms. Since our simulations do not include feedback, none of the mass that falls in flows back out; up to $t\sim
2.5$ Gyr, there is very little difference in the (azimuthally-averaged) density profile inside 3 kpc between runs NG1 and NG3. Later profile differences are most likely caused by the difference in the bars, which, being stronger in run NG3, leads to further infall to the center.
Not much changes at higher gas mass fraction if the gas is adiabatic (NG4). Then the gas layer is quite thick, producing a bar that, although somewhat weaker than in the purely stellar case (NC3), is still quite strong. When the gas can cool (NG5), the gaseous disk becomes violently gravitationally unstable and a new phenomenon appears, namely, the fragmentation of gas into clumps that sink to the center, dragging an associated stellar clump. Such clump instabilities have been found in previous simulations (Noguchi 1999; Immeli 2004) and have been shown to build central bulge-like objects directly. Immeli (2004) showed that the rate at which clouds dissipate their energy is the main parameter that determines whether the clump instability occurs. As a result of the central mass, only a weak bar forms, and this is eventually destroyed by continued gas inflow.
In our fully self-consistent simulations with cosmologically motivated halos, we found that the fraction of the total disk mass needed to destroy the bar ($\sim 20\%$; Figure \[fig:fracgas\]) is in very good agreement with that recently found by Shen & Sellwood (2004), as is the gradual decay of the bar amplitude (Figure \[fig:gasamps\]). This result is different from that of Bournaud (2005); we note that our model NG5 differs from theirs in two important ways. NG5 has a live halo versus their rigid halos, and the dark matter halo is strongly concentrated at the center. Both properties of our halos allow angular momentum to be transferred from bar to halo efficiently (Weinberg 1985; Debattista & Sellwood 1998), which may perhaps account for the difference in these results.
Inner Profile Evolution {#sec:innerprofs}
=======================
We explore the evolution of density profiles at large radii in §\[sec:truncations\], and in §\[sec:b+d\] we discuss the bulge$+$disk decompositions that result from profile evolution. In this section we explore this evolution qualitatively.
Buckling and central densities
------------------------------
Bar formation leads to a change in density profiles. As was already noted by Hohl (1971), the central density generally increases while the outer disk becomes shallower. In Figure \[fig:profevol\] we show that bar formation leads to an increase in the central density in run L2 (here we define central density from particle counts inside $0.1 R_{\rm d} \simeq 5 \epsilon$). Moreover, as already noted by Raha (1991), buckling may also increase the central density of a disk. We demonstrated [*directly*]{} that buckling is responsible for an increase in central density by re-simulating this system with an imposed symmetry about the mid-plane, which prevents buckling. Figure \[fig:profevol\] shows that when buckling is absent, no further increase in central density occurs. The buckled system is 2.0 times denser at the center than when it is prevented from buckling.
Disk scale-lengths and angular momentum
---------------------------------------
In Paper I we argued that the fact that the evolution of the scale-length of the outer disk changes under the influence of the bar, even when the total baryonic angular momentum is conserved, implies that the distribution of disk scale-lengths does not follow automatically from that of halo angular momenta. The increase in disk scale-length is due to transfer of angular momentum from the bar to the outer disk (Hohl 1971). It is remarkable that the disk outside the bar remains exponential out to the point where a break occurs, in both collisionless and dissipative systems. Hohl (1971) was the first to notice that exponential disks are naturally obtained after bar formation (outside the bar) even when the initial disk did not have an exponential profile. Since bars are ubiquitous in galaxies at low and high redshift (Jogee et al. 2004), it follows that the effect of secular evolution on disk sizes has to be included in any realistic galaxy formation model.
Not only do bars change disk scale-lengths, but the amount by which they change varies dramatically depending on the $Q$ profile, even for (nearly) identical initial angular momentum. Consider models H1, H2, and H3. These all have the same initial conditions other than Toomre-$Q$, leading to $\ltsim 10\%$ difference in the total baryon angular momentum. Nevertheless, the final values of $R_{\rm d}$ range from $1.0$ to $2.4$ (Table \[tab:rigid\]). We conclude that the direct mapping of halo spins into a distribution of disk scale-lengths (e.g. Mo, Mao & White 1998) will not yield correct predictions.
Outer Disk Breaks {#sec:truncations}
=================
Disk densities do not always exhibit a single exponential profile. More typically a sharp break between an inner and outer profile is evident. These breaks are often referred to as truncations following the apparently sharp drop-offs first discovered by van der Kruit (1979). Subsequently, van der Kruit & Searle (1981a,b) fitted sharp truncations to light profiles at large radii. However, de Grijs (2001) found that truncations occur over a relatively large region, rather than sharply. The larger sample of Pohlen (2002) confirmed this result; he found that truncations are better described by a double-exponential profile with a break radius. For his sample of mostly late-type systems, Pohlen (2002) estimates that the fraction of disk galaxies with breaks is $\gtsim 79\%$. Similar disk breaks have been found up to redshifts of $z\simeq 1$ (Pérez 2004; Trujillo & Pohlen 2005). What is the origin of these features in the light distribution of disks? In Section \[sec:innerprofs\] we showed that the angular momentum re-distribution caused by the bar leads to an increased central density and a shallower density profile outside this. This angular momentum redistribution cannot be efficient to arbitrarily large radii; thus, we may ask whether secular evolution can give rise to breaks in density profiles.
Studying breaks in $N$-body simulations is numerically challenging because they generally occur at large radii, where the density of particles is low. We therefore ran several high-resolution rigid-halo simulations with disks extended as far as $R_t = 12\rd$; with $7.5M$-particles the initial conditions still had $\sim 24K$ particles at $R \geq 8 \rd$ and $>3K$ at $R \geq 10 \rd$, sufficient to properly measure the density profile out to the large radii required. We chose $R_t$ this large in order to ensure that edge-modes (Toomre 1981) do not interfere with other secular effects. The series of simulations we used in this study is listed in Table \[tab:truncruns\].
In order to compare our simulations with observations, we used the double-exponential fitting form of Pohlen (2002). We only fit profiles at $2 \rd \leq R \leq 8 \rd$; the lower limit is needed to avoid the central bulge-like component. At very large radii, the surface density barely evolves because of the low self-gravity (although all of our models were evolved for at least three rotations at the outermost radius). Clearly the profile at larger radii reflects only our initial conditions. A reasonable transition radius between the initial profile and the secularly evolved profile occurs at $R\simeq 8 \rd$, which we use as our upper limit on the double-exponential fits. (For the initial pure exponential profile this is equivalent to $\sim 7$ mag fainter than the center.)
These fits give three dimensionless quantities: $R_{br}/R_{in}$, the ratio of break radius to inner scale-length, $R_{out}/R_{in}$, the ratio of outer scale-length to inner scale-length and $\mu_{0,in}-\mu_{0,out}$, the difference between the central surface-brightnesses of the two exponential fits, which we compare with the data of Pohlen (2002) and Pohlen & Trujillo (2006).
-------- ---- -----
L2.t8 8 1.2
L2.t12 12 1.2
L4.t12 12 1.2
L5.t12 12 1.6
H1.t12 12 1.2
H2.t12 12 1.6
H3.t12 12 2.0
T1.t12 12 1.2
-------- ---- -----
: The simulations testing disk breaks.
The first two characters in the name of each simulation reflects the model from Table \[tab:rigid\] on which that simulation is based by extending $R_t$ from $5\rd$ to either $8\rd$ or $12\rd$ (number after “t” in the name of each run). Model T1.t12 is not based on any in Table \[tab:rigid\]. It was produced by adding to run L2 a central Gaussian: $\Sigma(R) = \Sigma_0 (e^{-R/R_{\rm d}} + 4 e^{-1/2
(R/0.2)^2})$. \[tab:truncruns\]
The face-on view
----------------
Several of our simulations produced clear breaks of the double-exponential type. We present one example in Figure \[fig:breakevol\], where we show the initial and final profiles in run L2.t8. The profile very quickly develops from a single exponential to a double-exponential. In Figure \[fig:breakrad\] we plot the evolution of the parameters of the double-exponential profile for this run and for runs L2 and L2.t12. The formation of the break in these three simulations is obviously a discrete event. Their bars formed at $t\simeq 620$ Myr; $R_{br}$ does not evolve substantially after $t\simeq 990$ Myr. Because of this near coincidence in time, we conclude that, in these simulations, the process of bar [*formation*]{} can directly or indirectly somehow lead to the formation of broken profiles.
Figure \[fig:breakrad\] also investigates the difference between $R_t = 8\rd$ ([*black lines*]{}), $R_t = 12\rd$ ([*thick gray lines*]{}), and $R_t = 5\rd$ ([*thin gray lines*]{}) (models L2.t8, L2.t12, and L2, respectively). The similarity of the break parameters in the three simulations shows that the breaks do not result from edge-modes (Toomre 1981). In run L2, the initial disk did not extend as far as the final break radius; thus, the break is not an artifact of our initial disk extending to very large radii. Therefore, all parameters of the double-exponential profile that develops can be considered robust.
In Figure \[fig:pohlen\_faceon\] we compare the break parameters of the simulations with the observations of Pohlen (2002) and Pohlen & Trujillo (2006) for a combined sample of 31 face-on galaxies. Our simulations are in reasonable agreement with the observations, although they span a smaller part of the parameter space. The agreement in the narrow distributions in the $(R_{out}/R_{in}$, $\mu_{0,in}-\mu_{0,out})$-plane is quite striking.
We explore the angular momentum redistribution that leads to the breaks in Figure \[fig:angmom\]. This plots the distribution of angular momenta in the initial conditions and at the end of the simulation for runs H1.t12 (with $R_{br} \simeq 5\rd$) and H3.t12 (which did not form a break). In the left panel we see that very little angular momentum redistribution occurred in run H3.t12. On the other hand, in run H1.t12, bar formation leads to an excess of low angular momentum particles. At the same time, a second smaller peak forms at $2.1 \ltsim j_z \ltsim 2.6$ (see also Pfenniger & Friedli 1991). The right panel plots the location of particles in this angular momentum range. We find that the bulk of these particles occur inside the break radius, supporting the interpretation that breaks occur because of angular momentum redistribution.
The edge-on view
----------------
Observationally, breaks have often been sought in edge-on systems, since this orientation leads to higher surface brightnesses. Comparing to such data is complicated by the fact that these projections integrate along the entire line-of-sight. At very large radii, the density profile does not evolve and reflects initial conditions. In order to avoid being biased by these effects, we again limit our double-exponential fits to [*projected*]{} radii $R^\prime <
8$; however, we integrate along the entire line-of-sight since to do otherwise would require an arbitrary cutoff. In order to increase the signal-to-noise ratio of our measurements, we use all particles regardless of their height above or below the disk mid-plane. Moreover, in the edge-on case, the break parameters depend on the bar viewing orientation. Therefore, to compare with simulations, we consider all orientations of the bar between $0 \leq \phi_{bar} \leq
90 \degrees$. The right panel of Figure \[fig:breakevol\] shows an example of one of our fitted edge-on breaks.
The largest observational sample of disk breaks consists of 37 edge-on galaxies studied by Pohlen (2002). We compared our simulations to these data; the results are shown in Figure \[fig:pohlen\_angle\]. Variations in $\phi_{bar}$ lead to large variations in the parameters of the double exponential fit. Nonetheless, these fall within the range of observed systems. This is particularly striking in the $(\mu_{0,in} - \mu_{0,out},R_{out}/R_{in})$ plane, where the observational data span a narrow part of the space. Thus, we conclude that simple secular evolution suffices to produce realistic disk breaks.
Figure \[fig:pohlen\_angle\] shows the effect of the line-of-sight integration: runs L2.t8 ([*gray asterisks*]{}) and L2.t12 ([*cyan asterisks*]{}), which have very similar intrinsic ( face-on) breaks (Figure \[fig:breakrad\]), have very different breaks in the edge-on view. Nevertheless, in both cases the resulting parameters are in good agreement with those in real galaxies.
Unbroken profiles
-----------------
Not all disk galaxies exhibit breaks (Weiner 2001; Pohlen & Trujillo 2006). A successful theory of break formation must also be able to explain such unbroken profiles, which may present difficulties for a star-formation threshold explanation of disk breaks (Schaye 2004; Elmegreen & Hunter 2005). In our simulations, the formation of a bar is not always accompanied by the formation of a break in the disk density profile. For example, run H3.t12 formed a bar, which was not weak, but the profile remained largely exponential, as can be seen in Figure \[fig:breakQ\]. Similarly, model L5.t12, with $Q = 1.6$, failed to produce a break.
We investigate the dependence of break parameters on the disk temperature by considering a set of models in which only $Q$ of the initial conditions varies (H1.t12, H2.t12, and H3.t12). We plot the density profiles in Figure \[fig:breakQ\]. At $Q=1.2$, a prominent break develops in the density profile. At $Q=1.6$, a break is still evident, although weaker. By $Q=2.0$ no break forms in the density profile.
Interpretation: bar-spiral coupling
-----------------------------------
The breaks that develop in our simulations are associated with the angular momentum redistribution induced by the bar. In all cases the breaks are well outside the bar semi-major axis (always $\ltsim
3\rd$). The radius at which the breaks develop instead appears to be set by spirals resonantly coupled to the bar. Resonant couplings between bars and spirals have been described before ( Tagger 1987; Sygnet 1988; Masset & Tagger 1997; Rautiainen & Salo 1999).
Evidence for this hypothesis from run L2.t12 is presented in Figure \[fig:radampstrunc\]. The right panel shows the frequency spectrum of $m=2$ perturbations. The bar’s pattern speed is $\om \simeq 0.29$, while that of the spirals is $\om \simeq 0.18$. For the rotation curve of this system, this puts the corotation radius of the bar at about the inner 4:1 resonance radius of the spirals. The outer Lindblad resonance (OLR) of these spirals is at $R \simeq 7.5\rd$. The break sets in at $R\simeq 6\rd$, suggesting that it develops interior to the spiral OLR, at which spiral waves are absorbed. The left panel shows that the peak amplitude of $m=2$ perturbations decreases dramatically between $7\rd \leq R \leq 9\rd$, just outside where the break develops (see Figure \[fig:breakrad\]). We stress that other spirals with different pattern speeds can and do propagate to even larger radii. Resonantly coupled spirals are favored in that they are stronger, so they transmit more efficiently the angular momentum shed by the bar during its formation.
For an independent test of this hypothesis we evolved a model with very different resonance radii: in simulation T1.t12 we forced a smaller bar with a larger pattern speed by making the disk more massive in the center (while still exponential in the outer parts). Assuming a corotation-4:1 bar-spiral coupling, this would bring in the spiral OLR to roughly $4.8\rd$; indeed we found a break at $R_{br} \simeq
4.5\rd$.
Thus, we propose that profile breaks develop interior to where the angular momentum shed by the bar and carried away by resonantly-coupled spirals is deposited.
Bulge+disk decompositions {#sec:b+d}
=========================
We compare the simulations with observations of bulges using the parameters of one-dimensional bulge+disk (bulge+disk) decompositions and $V_p/\bar{\sigma}$ at a given flattening. Here $V_p$ is the peak line-of-sight velocity within some same radial range on the disk major-axis and $\bar{\sigma}$ is the line-of-sight velocity dispersion (averaged within the same radial range).
We decomposed the face-on, azimuthally-averaged radial mass profiles of our simulations into a central Sérsic and an outer exponential component, which we will refer to as “bulge” and “disk,” respectively. These decompositions are characterized by five parameters: $\Sigma_{0,d}$, $\Sigma_{0,b}$ (the exponential and Sérsic central surface density, respectively), $R_{d,f}$ (final exponential scale-length), $R_{b,eff}$ (the Sérsic effective radius), and $n_b$ (the index of the Sérsic profile). In our bulge+disk fitting, we computed the fits at fixed $n_b$ and obtained the best fit, including $n_b$, by repeating the fits for $n_b$ in the range 0.1-4 in steps of 0.1, then selecting the fit with the smallest $\chi^2$. We did not distinguish between Freeman types in our bulge-disk decompositions. In several cases, therefore, the profile fits represent a best-fitting [*average*]{} between small and large radii.
We compare our simulations with observed galaxies in the dimensionless space spanned by the parameters $R_{b,eff}/R_{d,f}$, $n_b$, and $B/D$, the bulge-to-disk mass ratios for profiles extrapolated to infinity. The photometric data came from two separate studies. Our first sample comes from MacArthur (2003), who presented bulge+disk decompositions for a sample of 121 predominantly late-type galaxies of various inclinations, observed in $B$, $V$, $R$, and $H$ bands. This study considered only systems with Freeman type I ( exponential) disk profiles in all bands. As our systems often exhibited transient type II phases, we also used the decompositions of Graham (2001, 2003). These were obtained from the diameter-limited sample of 86 low-inclination disk galaxies of all Hubble types observed in the $B$, $R$, $I$, and $K$ bands by de Jong & van der Kruit (1994). It is well known that bulge+disk structural parameters depend on the filter used ( Möllenhoff 2004). We compared directly with the data in all passbands; thus, any discrepancies we find between simulations and observations are not likely to be due to any differences in mass-to-light ratios of disks and bulges.
Paper I presented the decompositions of the rigid-halo models where we showed that the models partly overlap with the observations but are mismatched elsewhere. This mismatch can be diminished if only those bulges rounder than the disk are considered, but at the cost of requiring inclinations $i\gtsim 60\degrees$. In Figure \[fig:livedecs\] we present the face-on bulge+disk decompositions for live-halo models after bar formation. Several trends are worth noting. Compared with Paper I, the main changes are the generally smaller $n_b$ and $B/D$ values, as well as the smaller discrepancy with observations, although a small discrepancy is still evident in the $(B/D$,$n_b)$-plane. All simulations fall in the same space as observed galaxies in the $(B/D,R_{b,eff}/R_{d,f})$-plane plane. In the absence of gas, the values of $B/D$ tend to their largest values. Moreover, this quantity increases with time, a result of the continuing loss of disk angular momentum to the halo, leading to denser disk centers. Indeed, we find that the density of the inner region increases through most of the simulation. Introducing $10\%$ gas leads to a lower $B/D$ when the gas can cool (NG1) because the resulting central gas concentration leads to a weaker bar. When the gas is adiabatic (NG2), the central mass that grows is significantly smaller, the bar amplitude is not much different from the collisionless case, and the $B/D$ ratio is about as large as in the collisionless systems. When star formation is included (NG3), the $B/D$ ratio is intermediate between NG1 and NG2 and is continually increasing, changing by almost an order of magnitude as gas is turned into stars. When gas accounts for $50\%$ of the disk mass, the $B/D$ ratio is again high when gas is adiabatic (NG4) but remains smaller when it can cool (NG5), in which case a bulge is built via the clump instability (Noguchi 1999; Immeli 2004).
Some overall trends can be noticed. Generally the systems evolve parallel to the mean observed correlation between $B/D$ and $R_{b,eff}/R_{d,f}$ largely because $n_b$ does not evolve much. In the $(n_b,R_{b,eff}/R_{d,f})$- and $(B/D,n_b)$-planes, the simulations show a slight tendency to fall outside the observed range, with $B/D$ evolving toward values larger than observed. The exception is model NG3, which evolves parallel to the mean relation between these two parameters. This model is also comfortably within the range of observed bulges in the $(B/D,R_{b,eff}/R_{d,f})$-plane. Thus, dissipation with star formation is an important ingredient in the secular assembly of the bulges seen today.
We compared the kinematics of the bulges with observations in the $(V/\sigma$,$\epsilon)$-plane at various orientations. Because of the smaller number of particles compared with the rigid-halo simulations, we were not able to fit ellipses as in Paper I. We therefore obtained the effective radius of the inclined system through a Sérsic bulge$+$exponential disk fit to the mass distribution along the major ( inclination) axis. As in Paper I, because our bulges are poorly fitted by a de Vaucouleurs profile, we measure kinematic quantities and mean ellipticities at both one-half and one effective radius and use the differences as an error estimate. Ellipticities were measured from the two-dimensional mass moments of the projected mass distribution at these two radii, with the difference between the two giving an error estimate. Each of the final states of the simulations is viewed at inclination of $i= 30\degrees$ or $60\degrees$ and for position angles $\phi_b =0, 45\degrees$, or $90\degrees$.
In Figure \[fig:vsigmanew\] we report the results. The larger scatter in the live-halo simulations compared with the rigid-halo simulations (shown in Paper I) is mainly due to the lower number of particles and related discreteness noise of the mass distribution at radii only a few times larger than the softening length. Overall live and rigid-halo simulations occupy very similar locations, with many systems below the locus of oblate isotropic systems flattened by rotation. Indeed, the simulations include systems significantly flatter than the observations. These are usually systems in which the peanut is less pronounced due to weak buckling and the bar still very strong (a clear example is run NG3), and they are viewed at small position angles and high inclinations; these objects would appear markedly bar-like and thus probably would not be included in surveys of bulges. Anisotropy clearly plays a role in determining the flattening, which is unsurprising given that almost all bars survive. Systems with gas tend to have higher $V/\sigma$ for a given value of the ellipticity, and the most gas-rich systems are those that lie closest to the locus of oblate isotropic rotators. This is expected if the gas falling toward the center sheds a fraction of its angular momentum to the stars and spins them up. Run NG3 produces a system with the highest flattening due to a combination of a strong bar and the suppression of the buckling of the bar.
The pseudo-bulge formed by the clump instability
------------------------------------------------
The gas sinking to the center of run NG5 was sufficient to destroy the bar. Because we have no star formation in this simulation, we are left with a rotationally supported massive central gas disk. Star formation would have presumably led to a thin, rotationally supported pseudo-bulge. The mass of gas within one $R_{b,eff} = 0.83$ kpc (which is measured face-on from only the stellar component) is $1.2
\times 10^{10} M_\odot$. The associated aperture dispersion within $R_{b,eff}$ is $\sigma = 141$ . In order for the resulting bulge to sit on the $M_\bullet - \sigma$ relation (Gebhardt 2000; Ferrarese & Merritt 2000; Tremaine 2002), less than $0.3\%$ of this gas needs to collapse into a black hole. This shows that disk instabilities are another way, in addition to gas-rich mergers (Kazantzidis et al. 2005), by which a significant reservoir of gas can be built to feed an already existing central supermassive black hole or perhaps produce a new one. Since violent gravitational instabilities in the gas disk need a high gas mass fraction, such a mechanism might have played a role in the formation of seed black holes at high redshift.
Discussion and conclusions
==========================
In this paper we explored the secular evolution of disk structural parameters using simulations. The initial galaxy model is similar to a “Milky Way”-type galaxy that might form within a $\Lambda$CDM universe. We have found that bar formation leads to a significant mass redistribution both in and away from the disk plane. The evolution of stellar surface density profiles and the formation of peanut-shaped, dynamically hot stellar structures in the central regions of the disk are both consequences of bar formation. Angular momentum redistribution can lead to large changes in density profiles, resulting in profiles that can be reasonably fitted by a central Sérsic and an outer exponential component. These can be identified as bulge and disk components, respectively, although fundamentally they remain bars. On purely photometric grounds it is difficult to distinguish these profiles from bulge+disk decompositions for real galaxies; however, kinematically these secular bulges clearly fall below the oblate isotropic rotators. Real bulges are observed to be at or above this line.
We have shown that when bars occur in a typical bright spiral galaxy, they are difficult to destroy. The buckling instability, which is one way a peanut-shaped bulge can form, is suppressed when gas is highly concentrated in the center of the disk. A peanut can form other than through a major buckling event in such systems. It is interesting to note that when buckling occurs, it peaks typically $\sim 2$ Gyr after the bar forms. It is only at this point that the peanut-shaped bulge appears.
Our main results can be summarized as follows:
1. While strong buckling always leads to significant vertical heating, we found that heating does not always result in a peanut structure. When buckling occurs, the central density may increase; in one simulation this increase more than doubled the central density. As a consequence, the Sérsic index of the resulting bulge+disk decomposition increases somewhat.
2. The effect of gas on the buckling instability depends on the gas physics. When the gas evolution included radiative cooling, buckling was not possible and no peanuts formed. When the gas is adiabatic, buckling can occur and peanuts form. This difference might arise because the higher central gas concentration suppresses bending modes, as suggested by the correlation between strength of buckling and central gas mass concentration. However, a clear test is needed to discriminate between this scenario and one in which buckling does not occur because the energy in the modes is dissipated by radiative cooling. Peanuts can still be recognized by the negative minimum in the $s_4$ criterion (Debattista 2005) even when gas is present because it sinks to small radii, with the peanut at larger radii.
3. We found no case in which buckling destroyed a bar. In Paper I we demonstrated this with the rigid-halo simulations. In this paper we showed that this result continues to hold when a live halo is included. The most damaging buckling events we saw were induced in the rigid-halo simulations by slowing bars, but even in those cases a bar survived. Sometimes, however, the surviving bar is weak and may be better described as SAB rather than SB.
4. Density profiles may evolve substantially under the action of a bar. Reasonable bulge+disk decompositions can be fitted to the resulting profiles. When comparing the fits with observed galaxies, purely collisionless secular evolution gives rise to systems marginally consistent with bulges in nature. The presence of a modest ($10\%$) amount of gas produces systems that are better able to match observations. Star formation helps further and leads to an evolution of structural parameters parallel to their locus for observed galaxies. Secular evolution generally gives rise to nearly exponential inner profiles. Kinematically, however, the central bulge-like components of our simulations clearly fall below the locus of oblate isotropic rotators (at or above which real bulges occur in nature), reflecting the fact that they are still, fundamentally, bars.
5. The amount of evolution of a density profile following bar formation depends sensitively on the Toomre-$Q$ of the initial disk. When this is small, the inner disk needs to shed a large amount of angular momentum to form a bar and the central density steepens considerably. When the disk is hotter, the density change is smaller. As a result, the exponential scale-length of the disk outside the bar region depends on the initial disk kinematics. Thus, a distribution of dark matter halo specific angular momenta cannot trivially be related to a distribution of disk scale-lengths, as is often assumed.
6. Angular momentum redistribution also leads to realistic breaks in the surface density of disks. The radius at which breaks occur is interior to the outer Lindblad resonance of spirals resonantly coupled to the bar. When the initial disk is hot, little angular momentum redistribution occurs and no density breaks occur; thus, secular evolution can also account for galaxies that do not exhibit any breaks. The breaks that result in these simulations are in very good agreement with observations, including not only the break radii in units of inner disk scale length but also outer scale-lengths and the difference between central surface brightnesses of the two exponentials. On the other hand, we cannot exclude that angular momentum redistribution driven by other than bars does not account for some or most of the observed breaks. In particular, spirals excited by interactions may constitute another channel by which such breaks may form; the presence of breaks in unbarred galaxies may require such a mechanism. Since our disk break formation simulations were all collisionless, we cannot address whether star formation thresholds play any role in break formation, but two results here suggest that these may not play a prominent role. First, we are able to produce disks without truncations, which models invoking star formation thresholds may have difficulty in producing. Secondly, it is clear that the breaks that do form in our simulations are quite insensitive to extent of the initial disks. The ease with which angular momentum redistribution gives rise to realistic profile breaks, together with the ability to produce also profiles without breaks, provides strong incentives for exploring such models further. Moreover, the type of angular momentum exchange we are advocating here as leading to breaks need not be necessarily driven by bars and external perturbations can also play a role. In that case, depending on the frequency of the perturbation, anti-truncations (Erwin 2005) may also be possible in a unified picture.
The fraction of disks with bars is $\sim 30\%$ at both low (Sellwood & Wilkinson 1993) and high (Jogee 2004) redshift, and this increases to $\sim 70 \%$ at low redshift when measured via dust-penetrating infra-red observations (Knapen 1999; Eskridge 2000). Since, as we show, bars are long-lived, we are forced to the conclusion that disk galaxies know at an early epoch whether or not they will form a bar and there is simply little room for continued bar formation as a function of cosmic time. Thus, secular evolution has had a long time to act on galaxies.
The results of this paper demonstrate a strong coupling between the properties and evolution of disk galaxies and their associated inner and outer morphological and structural parameters. In contrast, semi-analytic models of disk structural parameters that invoke specific angular momentum conservation miss the important effects of bar formation on disk structure. As noted in Paper I, a more nuanced analysis that takes into account secular evolution may help to alleviate discrepancies between predictions for disk galaxy structure from cosmological models ( Mo 1998) and observations (De Jong & Lacey 2000).
Nonetheless, secular evolution cannot account for all discrepancies between theory and observation. An example is the difficulty cosmological models have in forming bulgeless disks as extended as those observed (D’Onghia & Burkert 2004, hereafter DB04). The $N$-body simulations of DB04 showed that halos with a quiet merging history since $z = 3$ (which are expected to lead to bulgeless disks, but see also Springel & Hernquist \[2005\]) have a median $\lambda \simeq 0.023$ with a scatter $\sigma_{\ln \lambda} \simeq
0.3$. On the other hand, van den Bosch (2001) measured their observed mean at $\lambda \simeq 0.067$. Thus, the mean of $\lambda$ in the cosmological simulations is more than $3\sigma$ smaller than observed. Our simulations indicate that the fraction of galaxies that form pure exponential disks must be higher (and extends to higher mass galaxies such as our own Milky Way) before secular evolution turns them into more disk$+$bulge-like systems. However, even if all of these systems would have lower $\lambda$-valuess, the distribution of $\lambda$-values of quiet merger halos makes it unlikely that secular evolution can account for the discrepancy between predictions and observations. An interesting test of $\Lambda$CDM will be whether enough halos with quiet merger histories form to account for a higher fraction of bulgeless galaxies.
Discussions with Stéphane Courteau, Aaron Dutton, Peter Erwin, Lauren MacArthur, Michael Pohlen, Juntai Shen, and especially Frank van den Bosch were useful. Thanks also to Michael Pohlen for sharing his data with us in advance of publication. T.Q. and V.P.D. acknowledge partial support from NSF ITR grant PHY-0205413. We thank the anonymous referee for a very careful reading and a detailed report that helped improve this paper.
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[^1]: In order that models remain centered at the origin, we excluded the $m=1$ term in the expansion. Including this term would result in large but artificial offsets between the bar and the center, similar to the ones found by McMillan & Dehnen (2005).
[^2]: We use Gauss-Hermite moments in which the velocity scale is the rms. In order to distinguish this from the more commonly used best-fit Gaussian scale, we refer to this moment as $s_4$ rather than $h_4$.
|
---
abstract: 'We introduce a three-state model for a single DNA chain under tension that distinguishes between [B]{}-DNA, [S]{}-DNA, and [M]{} (molten or denatured) segments and at the same time correctly accounts for the entropy of molten loops, characterized by the exponent ${c}{}$ in the asymptotic expression $S\sim -{c}{} \ln n$ for the entropy of a loop of length $n$. Force extension curves are derived exactly employing a generalized Poland-Scheraga approach and compared to experimental data. Simultaneous fitting to force-extension data at room temperature and to the denaturation phase transition at zero force is possible and allows to establish a global phase diagram in the force-temperature plane. Under a stretching force, the effects of the stacking energy, entering as a domain-wall energy between paired and unpaired bases, and the loop entropy are separated. Therefore we can estimate the loop exponent ${c}{}$ independently from the precise value of the stacking energy. The fitted value for ${c}{}$ is small, suggesting that nicks dominate the experimental force extension traces of natural DNA.'
author:
- 'Thomas R. Einert'
- 'Douglas B. Staple'
- 'Hans-Jürgen Kreuzer'
- 'Roland R. Netz'
bibliography:
- 'dna\_BSL-bib.bib'
title: '[A three-state model with loop entropy for the over-stretching transition of DNA]{}'
---
Introduction {#sec:introduction}
============
DNA continuously stays in focus of polymer scientists due to its unique mechanical and structural properties. In particular the possibility to trigger phase transformations in this one-dimensional system has intrigued theorists from different areas [@Poland1966]. [In fact, the thermal denaturation or melting transition of DNA was shown to correspond to a true phase transition, brought about by a logarithmic contribution to the configurational entropy of molten loops or bubbles, $S\sim -{c}{} \ln n$, as a function of the loop size $n$ [@Poland1966a].]{} The value of the exponent ${c}{}$ is crucial since it determines the resulting transition characteristics. For ${c}{}=3/2$, the value for a phantom chain without self-avoidance, the transition is continuous, while self-avoidance increases ${c}{}$ slightly beyond the threshold ${c}{}=2$ above which the transition becomes discontinuous [@Kafri2002; @Poland1966a]. A distinct mechanism for transforming DNA involves the application of an extensional force. For forces around ${F}\approx\unit{65}{\pico\newton}$ DNA displays a highly cooperative transition and its contour length increases by a factor of roughly 1.7 to 2.1 over a narrow force range [@Smith1996; @Bensimon1995; @Cluzel1996]. [These experiments sparked a still ongoing debate on whether this over-stretching transition produces a distinct DNA state, named [S]{}-DNA, or merely the denatured state under external tension. According to the first view [S]{}-DNA is a highly stretched state with paired bases but disrupted base stacking [@Cocco2004; @Leger1999; @Whitelam2008; @Konrad1996; @Kosikov1999; @Li2009; @Lebrun1996]. In the other view the over-stretched state consists of two non-interacting strands [@Williams2001a; @Rouzina2001; @Shokri2008; @Mameren2009]. Evidence for the existence of a distinct [S]{}-state comes from theoretical models [@Cocco2004; @Whitelam2008], molecular dynamics simulations [@Konrad1996; @Kosikov1999; @Lebrun1996; @Li2009]]{} and from AFM experiments of @Rief1999 [@Rief1999; @ClausenSchaumann2000; @Krautbauer2000] where in addition to the over-stretching transition a second weak transition at forces between and is discerned, which has been interpreted as a force induced melting of the [S]{}-state. The critical force of both transitions depends on the actual sequence [@ClausenSchaumann2000] and the salt concentration [@Baumann1997], but the interpretation of the second transition is complicated by the occurrence of pronounced hysteresis effects that depend on various parameters such as pulling velocity, salt concentration, or presence of co-solutes such as cisplatin [@ClausenSchaumann2000; @Krautbauer2000]. [On the other hand, support for the view according to which [S]{}-DNA is not a distinct state comes from theoretical models [@Rouzina2001; @Williams2001a], simulations [@Piana2005] and recent experiments by @Shokri2008 and @Mameren2009.]{}
[Apart from simulations [@Konrad1996; @Kosikov1999; @Lebrun1996; @Li2009], existing theoretical works that grapple with experimental force traces or DNA melting fall into three categories with increasing computational complexity, for reviews see refs. [@Cocco2002a; @Peyrard2008; @Wartell1985]. In the first group are Ising-like models for DNA under tension which give excellent fitting of the over-stretching transition but by construction cannot yield the denaturing transition [@Cizeau1997; @Ahsan1998; @Storm2003a; @Rouzina2001]. The work of @Marko1998 is similar but employs a continuous axial strain variable. In the second group are models that include a logarithmic entropy contribution of molten loops in the spirit of the classical model by @Poland1966a [@Hanke2008; @Rudnick2008; @Whitelam2008; @Garel2004; @Kafri2002; @Blake1999; @Carlon2002]. This gives rise to effectively long-ranged interactions between base pairs and thus to a true phase transition. The third group consists of models which explicitly consider two strands [@Rahi2008a; @Palmeri2007; @Jeon2006; @Peyrard2004; @Cule1997]. Those models thereby account for the configurational entropy of loops – at the cost of considerable calculational efforts – and correspond to loop exponents ${c}{}=d/2$ in the absence of self-avoidance effects, where the dimensionality of the model is $d = 3$ for ref. [@Rahi2008a; @Palmeri2007; @Jeon2006] and $1$ for ref. [@Peyrard2004; @Cule1997]. All these above mentioned works consider only two different base states (paired versus unpaired) and thus do not allow to distinguish between [B]{}-DNA, [S]{}-DNA and denatured bases. Recently, three-state models were introduced that yield very good fits of experimental force traces at ambient temperatures. However, in previous analytic treatments of such three-state models [@Cocco2004; @Ho2009], the loop entropy was neglected and therefore the temperature-induced denaturation in the absence of force cannot be properly obtained, while the loop entropy was included in a simulation study where most attention was given to dynamic effects [@Whitelam2008].]{}
[In this paper we combine the Poland-Scheraga formalism with a three-state transfer matrix approach which enables us to include three distinct local base pairing states and at the same time to correctly account for long-ranged interactions due to the configurational entropy of molten DNA bubbles. Our approach thus allows for a consistent description of thermal denaturation and the force induced [B]{}[S]{}-transition within one framework and yields the global phase diagram in the force-temperature plane. We derive a closed form expression for the partition function of three-state DNA under tension. This allows to systematically investigate the full parameter range characterizing the three states and the DNA response to temperature and external force. In our model we allow for the existence of [S]{}-DNA but stress that the actual occurrence of [S]{}-DNA is governed by the model parameters. By assuming such a general point of view, our work is able to shed new light on the question of the existence of [S]{}-DNA. The extensible worm-like chain model is employed for the stretching response of each state. The loop exponent is found to have quite drastic effects on the force extension curve.]{} For realistic parameters for the stacking energy, the experimental force extension curves are fitted best for small loop exponents $0\leq{{c}}{}\leq1$, hinting that the DNA in the experiments contained nicks. Loop exponents ${c}{}>1$, which give rise to a genuine phase transition, are not compatible with experimental force-distance curves. Under external force, the effects of stacking energy and loop exponent are largely decoupled, since the stacking energy only determines the cooperativity of the [B]{}[S]{}-transition while the loop exponent influences the second [S]{}[M]{}-transition found at higher forces. This allows to disentangle these two parameters, in contrast to the denaturation transition at zero force where the effects of these two parameters are essentially convoluted. [The precise value of ${c}{}$ is important also from a practical point of view, as it impacts the kinetics of DNA melting [@Whitelam2008; @Bar2007], which is omnipresent in biological and bio-technological processes.]{}
Three-state model {#sec:three-state-model}
=================
Double-stranded DNA is modeled as a one-dimensional chain with bases or segments that can be in three different states, namely paired and in the native [B]{}-state, in the paired stretched [S]{}-state, or in the molten [M]{}-state. The free energy of a region of $n$ segments in the same state reads $$\label{eq:1}
{E_{i}}(n,{F}) = n{\cdot}g_i ({F}) -{\delta_{i,\mathrm{{M}}}}{\cdot}{\mathrm{k_B}}T \ln
n^{-{c}} {\;},$$ with $i=\mathrm{{B},{S},{M}}$. The force ${F}$ dependent contribution $$\label{eq:2}
{{g}_{i}}({F})={{g}_{i}^{0}}+ {{g}_{i}^{\mathrm{stretch}}}({F}) +{{g}_{i}^{\mathrm{WLC}}}({F})$$ is split into three parts. ${{g}_{i}^{0}}$ is a constant that accounts for the base pairing as well as the difference of reference states of the worm-like stretching energy, [*cf. *]{}supporting material [eq.]{} . [The stacking energy of neighboring bases in the same state is absorbed into ${{g}_{i}^{0}}$, too, so that the stacking energy will appear explicitly only as an interfacial energy ${V_{ij}}$ between two regions which are in a different state.]{} The second term ${{g}_{i}^{\mathrm{stretch}}}= - {F}^2
{l}_i/(2{\cdot}{\kappa}_i)$ takes into account stretching along the contour with ${l}_i$ and ${\kappa}_i$ the segmental contour length and the elastic stretch modulus. Finally, ${{g}_{i}^{\mathrm{WLC}}}({F})$ is the free energy of a worm-like chain (WLC) in the Gibbs ensemble (constant force ${F}$), based on the heuristic relation between force ${F}$ and projected extension $x$ [@Marko1995] $ {{F}^{\mathrm{WLC}}}_i(x) {\cdot}{\xi}_i / {\mathrm{k_B}}T = \bigl(1-
x/(n{l}_i)\bigr)^{-2}/4 + x/(n{l}_i) - 1/4$ where ${\xi}_i$ is the persistence length and $n$ the number of segments. The Gibbs free energy $n{\cdot}{{g}_{i}^{\mathrm{WLC}}}({F})$ of a stretch of $n$ segments is extensive in $n$ and follows [*via* ]{}integration, see supporting material section \[appendix-sec:gibbs-free-energy\]. We note that this is only valid if the persistence length is smaller than the contour length of a region, ${\xi}_i<n{l}_i$, which is a plausible assumption because of the high domain wall energies. Likewise, the decoupling of the free energy into contour stretching elasticity and worm-like chain elasticity is only approximate [@Netz2001a; @Livadaru2003] but quite accurate for our parameter values [@Odijk1995]: For small force WLC bending fluctuations dominate and the contour extensibility is negligible, while contour stretching sets in only when the WLC is almost completely straightened out. [The last term in [eq.]{} is the logarithmic configurational entropy of a molten loop ($i=\mathrm{{M}}$), characterized by an exponent ${c}{}$ [@Kafri2002; @Hanke2008; @Einert2008], see supporting material section \[appendix-sec:orig-logar-loop\].]{} The exponent is ${c}{}=3/2$ for an ideal polymer [@Gennes1979] and $2.1$ for a self avoiding loop with two attached helices [@Duplantier1986; @Kafri2002]. If the DNA loop contains a nick the exponent is reduced to ${{c}}{}=0$ for an ideal polymer and $0.092$ for a self avoiding polymer [@Kafri2002]. We consider the simple case ${c}{}=0$, where transfer matrix methods can be used to yield results in the canonical ensemble with a fixed number of segments $N$ [@Cocco2004], as well as the case of finite ${c}{}$ where we introduce a modified Poland-Scheraga method to obtain results in the grand canonical ensemble.
Partition function {#sec:poland-scher-appr}
==================
Modified Poland-Scheraga approach for ${c}{}\neq 0$ {#sec:gener-form-canon}
----------------------------------------------------
\[sec:grand-canon-part\]
The molecule is viewed as an alternating sequence of different regions each characterized by grand canonical partition functions. Various techniques for going back to the canonical ensemble are discussed below. The canonical partition function of a stretch of $n$ segments all in state $i=\text{{B}, {S}, or {M}}$ is $${{\mathcal{Q}}}_i(n)=\exp({-\beta{E_{i}}(n)})\label{eq:3}{\;},$$ where $\beta = ({\mathrm{k_B}}T)^{-1}$ is the inverse thermal energy. The grand canonical partition functions are defined as ${{\mathcal{Z}}}_i = \sum_{n=1}^\infty\lambda^n {{\mathcal{Q}}}_i(n)$ with $\lambda =
\exp(\beta\mu)$ the fugacity and $\mu$ the chemical potential. The grand canonical partition function of the whole DNA chain which contains an arbitrary number of consecutive [B]{}, [S]{}, and [M]{} stretches reads $$\label{eq:4}
{{\mathcal{Z}}}= \sum_{k=0}^\infty{{\bm{\mathrm{v}}}^{\mathrm{T}}}{\cdot}({\bm{\mathrm{M}}}_{\mathrm{PS}} {\bm{\mathrm{V}}}_{\mathrm{PS}})^k{\bm{\mathrm{M}}}_{\mathrm{PS}}{\cdot}{\bm{\mathrm{v}}}
={{\bm{\mathrm{v}}}^{\mathrm{T}}}{\cdot}({\bm{\mathrm{1}}} - {\bm{\mathrm{M}}}_{\mathrm{PS}}{\bm{\mathrm{V}}}_{\mathrm{PS}})^{-1}{\bm{\mathrm{M}}}_{\mathrm{PS}}{\cdot}{\bm{\mathrm{v}}},$$ [with the matrices in this Poland-Scheraga approach given by]{} $$\label{eq:5}
{\bm{\mathrm{M}}}_{\mathrm{PS}} =\begin{pmatrix}{{{\mathcal{Z}}}_\mathrm{{B}}}&0&0\\0&{{{\mathcal{Z}}}_\mathrm{{S}}}&0\\0&0&{{{\mathcal{Z}}}_\mathrm{{M}}}\end{pmatrix}{\;},
\quad
{\bm{\mathrm{V}}}_{\mathrm{PS}}=\begin{pmatrix}
0& {\mathrm{e}}^{-\beta{{V_{\mathrm{{B}}\mathrm{{S}}}}}}& {\mathrm{e}}^{-\beta{{V_{\mathrm{{B}}\mathrm{{M}}}}}}\\
{\mathrm{e}}^{-\beta{{V_{\mathrm{{S}}\mathrm{{B}}}}}}& 0 & {\mathrm{e}}^{-\beta{{V_{\mathrm{{S}}\mathrm{{M}}}}}}\\
{\mathrm{e}}^{-\beta{{V_{\mathrm{{M}}\mathrm{{B}}}}}}& {\mathrm{e}}^{-\beta{{V_{\mathrm{{M}}\mathrm{{S}}}}}}& 0
\end{pmatrix}{\;},
\quad
{\bm{\mathrm{v}}} =\begin{pmatrix}1\\1\\1\end{pmatrix}$$ and where ${\bm{\mathrm{1}}}$ is the unity matrix. The energies $V_{ij}$ are the interfacial energies to have neighboring segments in different states and are dominated by unfavorable base pair un-stacking. The diagonal elements of ${\bm{\mathrm{V}}}_{\mathrm{PS}}$ are zero which ensures that two neighboring regions are not of the same type and thus prevents double counting. The explicit form of ${{\mathcal{Z}}}$ is given in the supporting material, see [eq.]{} . [The partition function in [eq.]{} is general and useful for testing arbitrary models for the three DNA states as given by the different ${{\mathcal{Z}}}_i$. This approach is also easily generalized to higher numbers of different states. Using the parameterization [eq.]{} for vanishing loop exponent ${c}{}=0$ the partition functions of the different regions are given by]{} $$\label{eq:6}
{{\mathcal{Z}}}_i=\sum_{n=1}^\infty\lambda^n{{\mathcal{Q}}}_i(n)=\frac{\lambda {\mathrm{e}}^{-\beta {{g}_{i}}}}{1-\lambda{\mathrm{e}}^{-\beta {{g}_{i}}}}
{\;}, {\quad\quad\text{for $\lambda {\mathrm{e}}^{-\beta {{g}_{i}}}<1$,}}$$ $i=\mathrm{{B},{S},{M}}$. Insertion into [eq.]{} yields $$\label{eq:7}
{{{\mathcal{Z}}}}_{{c}=0} = \frac{a_1 \lambda + a_2 \lambda^2 + a_3\lambda^3}{a_4 + a_5 \lambda + a_6 \lambda^2 + a_7\lambda^3}{\;}.$$ which is a rational function of the fugacity $\lambda$, whose coefficients $a_i$ – determined by [eqs.]{} and – are smooth functions of the force ${F}$ and the temperature $T$. For ${c}{}\neq0$ the partition function of a molten stretch is modified to $$\label{eq:8}
\begin{split}
{{{\mathcal{Z}}}_\mathrm{{M}}}& = \sum_{n=1}^\infty\lambda^n{{{\mathcal{Q}}}_\mathrm{{M}}}(n) =
\sum_{n=1}^\infty \lambda^n\left({\mathrm{e}}^{-\beta {{g}_{\mathrm{{M}}}}}\right)^n \frac1{n^{c}}\\
& = {\mathrm{Li}_{{c}}\left(\lambda{\mathrm{e}}^{-\beta {{g}_{\mathrm{{M}}}}}\right)} {\;}, {\quad\quad\text{for $\lambda
{\mathrm{e}}^{-\beta {{g}_{\mathrm{{M}}}}}<1$}}
\end{split}$$ where ${\mathrm{Li}_{{c}}\left(z\right)} = \sum_{n=1}^\infty z^nn^{-{c}}$ for $z<1$ is the polylogarithm [@Erdelyi1953] and exhibits a branch point at $z=1$. The functional form of the grand canonical partition function for ${c}{}\neq0$ reads $$\label{eq:9}
{{{\mathcal{Z}}}}_{{c}\neq0} = \frac{
b_0 \lambda + b_1\lambda^2 + b_2 {\mathrm{Li}_{{c}}\left(\lambda /{\lambda_{\mathrm{b}}}\right)} +
b_3 \lambda {\mathrm{Li}_{{c}}\left(\lambda /{\lambda_{\mathrm{b}}}\right)} +
b_4\lambda^2 {\mathrm{Li}_{{c}}\left(\lambda /{\lambda_{\mathrm{b}}}\right)}
}{
b_5 + b_6 \lambda + b_7\lambda^2 + b_8 \lambda {\mathrm{Li}_{{c}}\left(\lambda /{\lambda_{\mathrm{b}}}\right)} +
b_9\lambda^2 {\mathrm{Li}_{{c}}\left(\lambda /{\lambda_{\mathrm{b}}}\right)}
}
{\;},$$ where ${\lambda_{\mathrm{b}}}={\mathrm{e}}^{\beta {{g}_{\mathrm{{M}}}}}$ denotes the position of the branch point and the coefficients $b_i$, determined by [eqs.]{} , and , are smooth functions of ${F}$ and $T$.
\[sec:canon-part-funct\]
[The grand canonical ensemble where $N$, the total number of segments fluctuates, does not properly describe a DNA chain of fixed length. We therefore have to investigate the back-transformation into the canonical ensemble where the number of segments $N$ is fixed. For the back-transformation there are three options:]{}
### Calculus of residues route: {#sec:calc-resid-route}
The grand-canonical partition function ${{\mathcal{Z}}}(\lambda) = \sum_{N=1}^\infty \lambda^N {{\mathcal{Q}}}(N)$ can be viewed as a Laurent series, the coefficients of which are the canonical partition functions ${{\mathcal{Q}}}(N)$ determined exactly by $$\label{eq:10}
{{\mathcal{Q}}}(N) = \frac1{2\pi{\mathrm{i}}}\oint_{\mathcal C}\frac{{{\mathcal{Z}}}(\lambda)}{\lambda^{N+1}}{\mathrm d}\lambda{\;}.$$ The contour $\mathcal C = \lambda_0 {\mathrm{e}}^{2 \pi{\mathrm{i}}t}$, $0\leq t\leq1$, is a circle in the complex plane around the origin with all singularities of ${{\mathcal{Z}}}(\lambda)$ lying outside. This complex contour integral can be evaluated using calculus of residues [@Arfken2001] which becomes technically involved for large $N$ and thus limits the practical relevance of this route.
### Legendre transformation route: {#sec:legendre-transf-rout}
The canonical Gibbs free energy $$\label{eq:11}
{\mathcal G}(N)= - {\mathrm{k_B}}T \ln {{\mathcal{Q}}}(N)$$ and the grand potential $$\label{eq:12}
{\Phi}(\mu) = - {\mathrm{k_B}}T \ln {{\mathcal{Z}}}(\lambda){\;},$$ are related [*via* ]{}a Legendre transformation $$\label{eq:13}
{\mathcal G}(N) = {\Phi}(\mu(N)) + N{\cdot}\mu (N){\;}.$$ The chemical potential $\mu$ as a function of the segment number $N$ is obtained by inverting the relation $$\label{eq:14}
N(\mu) =
-\frac{\partial{\Phi}(\mu)}{\partial \mu}{\;}.$$ Let us briefly review the origin of [eqs.]{} and in the present context. Changing the integration variable in [eq.]{} from $\lambda$ to $\mu =
\ln(\lambda)/\beta$, the complex path integral can be transformed into $$\label{eq:15}
{{\mathcal{Q}}}(N) = \int_{\mathcal{C'}}{\mathrm{e}}^{-\beta {\Phi}(\mu) - \beta N \mu}{\mathrm d}\mu
\approx{\mathrm{e}}^{-\beta{\Phi}(\mu_{\mathrm{sp}}(N)) - \beta N\mu_{\mathrm{sp}}(N)}{\;},$$ with the contour $\mathcal{C'} = \mu_0 + 2\pi{\mathrm{i}}t/\beta$, $0\leq t\leq 1$ and $\mu_0={\mathrm{k_B}}T\ln\lambda_0$. The integral in [eq.]{} has been approximated by the method of steepest descent, where the contour $\mathcal{C'}$ is deformed such that it passes through the saddle point $\mu_{\mathrm{sp}}$ [@Arfken2001] determined by equation . If ${\Phi}$ features singularities deformation of the contour $\mathcal{C'}$ requires extra care. In the present case the presence of a pole ${\lambda_{\mathrm{p}}}=\exp(\beta\mu_{\mathrm{p}})$ of ${{\mathcal{Z}}}(\lambda)$ produces no problem as $\mu_{\mathrm{sp}} < \mu_{\mathrm{p}}$ holds, meaning that the deformed contour does not enclose the pole singularity. This is different for the branch point singularity $\mu_{\mathrm{b}}$ where we will encounter the case $\mu_{\mathrm{b}}<\mu_{\mathrm{sp}}$ for large ${c}{} >2$.
### Dominating singularity route: {#sec:domin-sing-route}
For large systems, [*i.e. *]{}$N\gg1$, one approximately has $-\ln{{\mathcal{Q}}}(N)\sim N\ln{\lambda_{\mathrm{d}}}$, where the dominant singularity ${\lambda_{\mathrm{d}}}=\exp(\beta\mu_{\mathrm{d}})$ is the singularity (in the general case a pole or a branch point) of ${{\mathcal{Z}}}(\lambda)$ which has the smallest modulus. One thus finds $$\label{eq:16}
{\mathcal G}(N) = {\mathrm{k_B}}T N \ln {\lambda_{\mathrm{d}}}{\;}.$$ This easily follows from [eq.]{} : In the limit of $N\gg1$ the integral can be approximated by expanding ${{\mathcal{Z}}}(\lambda)$ around ${\lambda_{\mathrm{d}}}$ and deforming $\mathcal{C}$ to a Hankel contour which encircles ${\lambda_{\mathrm{d}}}$ [@Flajolet1990]. For the case where $N(\mu)
\propto (\mu_{\mathrm{d}} - \mu)^{-\alpha}$, $\alpha>0$, this can be understood also in the context of a Legendre transform. As $N=-\partial{\Phi}/\partial\mu$ one has ${\Phi}\propto(\mu_{\mathrm{d}} - \mu)^{-\alpha+1}$ and therefore the first term of [eq.]{} scales like ${\Phi}(\mu(N))\propto N^{1-1/\alpha}$. Thus, the second term $N\mu(N)\propto N \mu_{\mathrm{d}} - N^{1-1/\alpha} \propto N \mu_{\mathrm{d}}$ is dominant. Since the saddle point behaves as $\mu_{\mathrm{sp}} = \mu(N) \rightarrow
\mu_{\mathrm{d}}$ for $N\rightarrow\infty$, it follows that the dominating singularity expression [eq.]{} equals the Legendre transform [eq.]{} in the thermodynamic limit $N\rightarrow\infty$.
Transfer matrix approach for $\mathbf{{c}=0}$ {#sec:transf-matr-appr}
---------------------------------------------
For ${c}{}=0$ only interactions between nearest neighbors are present and straight transfer matrix techniques are applicable. We introduce a spin variable $i_n$ for each segment which can have the values $i_n=\mathrm{{B},{S},{M}}$. The energetics are given by the Hamiltonian $$\label{eq:17}
H(i_{1},i_{2},...,i_{N}) =\sum_{n=1}^{N}{g}_{i_n}+\sum_{n=1}^{N-1}{V_{i_{n}i_{n+1}}}$$ where ${g}_{i_n}$ and ${V_{i_{n}i_{n+1}}}$ are the previously introduced parameters for the segment and interfacial free energies. The canonical partition function of the molecule can be written as $$\label{eq:18}
{{\mathcal{Q}}}(N) = \sum_{i_1, \ldots,i_N} {\mathrm{e}}^{-\beta H(i_{1},i_{2},...,i_{N}) }=
{{\bm{\mathrm{v}}}^{\mathrm{T}}}
{\cdot}{\bm{\mathrm{T}}}^{N-1}{\bm{\mathrm{M}}}_{\mathrm{TM}}
{\cdot}{\bm{\mathrm{v}}}
{\;},$$ where we introduced the transfer matrix $ {\bm{\mathrm{T}}} = {\bm{\mathrm{M}}}_{\mathrm{TM}}
{\bm{\mathrm{V}}}_{\mathrm{TM}}$ and $$\label{eq:19}
{\bm{\mathrm{M}}}_{\mathrm{TM}}=
\begin{pmatrix}{\mathrm{e}}^{-\beta{{g}_{\mathrm{{B}}}}}&0&0\\0&{\mathrm{e}}^{-\beta{{g}_{\mathrm{{S}}}}}&0\\0&0&{\mathrm{e}}^{-\beta{{g}_{\mathrm{{M}}}}}\end{pmatrix}
{\;},\quad
{\bm{\mathrm{V}}}_{\mathrm{TM}}=
\begin{pmatrix}
1& {\mathrm{e}}^{-\beta{{V_{\mathrm{{B}}\mathrm{{S}}}}}}& {\mathrm{e}}^{-\beta{{V_{\mathrm{{B}}\mathrm{{M}}}}}}\\
{\mathrm{e}}^{-\beta{{V_{\mathrm{{S}}\mathrm{{B}}}}}}& 1 & {\mathrm{e}}^{-\beta{{V_{\mathrm{{S}}\mathrm{{M}}}}}}\\
{\mathrm{e}}^{-\beta{{V_{\mathrm{{M}}\mathrm{{B}}}}}}& {\mathrm{e}}^{-\beta{{V_{\mathrm{{M}}\mathrm{{S}}}}}}& 1
\end{pmatrix}{\;},
\quad
{\bm{\mathrm{v}}} =\begin{pmatrix}1\\1\\1\end{pmatrix}{\;}.$$ ${{\mathcal{Q}}}(N)$ is calculated readily by diagonalizing $ {\bm{\mathrm{T}}} $ $$\label{eq:20}
{{\mathcal{Q}}}(N) = {{\bm{\mathrm{v}}}^{\mathrm{T}}} {\cdot}{\bm{\mathrm{U}}} {\bm{\mathrm{D}}}^{N-1} {\bm{\mathrm{U}}}^{-1} {\bm{\mathrm{M}}}_{\mathrm{TM}} {\cdot}{\bm{\mathrm{v}}} =
{{{\bm{\mathrm{v}}}}^{\mathrm{T}}}_{\mathrm l}{\cdot}{\bm{\mathrm{D}}}^{N-1}{\cdot}{{\bm{\mathrm{v}}}}_{\mathrm r}
= \sum_{i=1}^3 v_{{\mathrm l},i} v_{{\mathrm r},i} x_i^{N-1}
{\;}.$$ where ${\bm{\mathrm{D}}} = {\bm{\mathrm{U}}}^{-1} {\bm{\mathrm{T}}} {\bm{\mathrm{U}}}$ is a diagonal matrix with eigenvalues $x_i$, the columns of ${\bm{\mathrm{U}}}$ are the right eigenvectors of $ {\bm{\mathrm{T}}} $, ${{{\bm{\mathrm{v}}}}_{\mathrm l}^{\mathrm{T}}} = {{\bm{\mathrm{v}}}^{\mathrm{T}}}{\cdot}{\bm{\mathrm{U}}}$ and ${{{\bm{\mathrm{v}}}}_{\mathrm
r}}={\bm{\mathrm{U}}}^{-1}{\bm{\mathrm{M}}}_{\mathrm{TM}} {\cdot}{\bm{\mathrm{v}}}$. By virtue of the Perron-Frobenius theorem one eigenvalue $x_{\mathrm{max}}$ is larger than the other eigenvalues and thus the free energy is in the thermodynamic limit dominated by $x_{\mathrm{max}}$ and reads $$\label{eq:21b}
{\mathcal G}= -{\mathrm{k_B}}T \ln {{\mathcal{Q}}}(N) \approx-{\mathrm{k_B}}T N \ln x_{\mathrm{max}}{\;}.$$ The transfer-matrix eigenvalues $x_i$ and the poles ${\lambda_{\mathrm{p}}}$ of ${{{\mathcal{Z}}}}_{{c}=0}$ [eq.]{} are related [*via* ]{}$x_i = 1/{\lambda_{\mathrm{p}}}$. As expected, the free energies from the transfer matrix approach [eq.]{} and from the Poland-Scheraga approach for ${c}{}=0$ and using the dominating singularity approximation [eqs.]{} are identical in the limit $N\rightarrow \infty$. Clearly, for ${c}{}\neq 0$ the modified Poland-Scheraga approach yields new physics that deviates from the transfer-matrix results.
Although not pursued in this paper, the transfer matrix approach allows to calculate correlators. For example the probability $p_{{M}{}}(k,m)$ of a denatured region with $k$ consecutive molten base pairs starting at base $m$ is given by $$\label{eq:21}
\begin{split}
p_{{M}{}}(k,m)
={{\mathcal{Q}}}(N)^{-1}{\cdot}{{\bm{\mathrm{v}}}^{\mathrm{T}}}{\cdot}{\bm{\mathrm{T}}}^{m-2}({\bm{\mathrm{P}}}_{\mathrm{{B}}}{\bm{\mathrm{T}}} + {\bm{\mathrm{P}}}_{\mathrm{{S}}}{\bm{\mathrm{T}}})
({\bm{\mathrm{P}}}_{\mathrm{{M}}} {\bm{\mathrm{T}}} )^k ({\bm{\mathrm{P}}}_{\mathrm{{B}}}{\bm{\mathrm{T}}} +
{\bm{\mathrm{P}}}_{\mathrm{{S}}}{\bm{\mathrm{T}}}) {\bm{\mathrm{T}}}^{N-m-k-1}{\bm{\mathrm{M}}}_{\mathrm{TM}}{\cdot}{\bm{\mathrm{v}}} {\;}.
\end{split}$$ The ${\bm{\mathrm{P}}}_i$-matrices, which project a segment onto a certain state, are defined as $$\label{eq:22}
{\bm{\mathrm{P}}}_{\mathrm{{B}}}=\begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\end{pmatrix}{\;},\quad
{\bm{\mathrm{P}}}_{\mathrm{{S}}}=\begin{pmatrix} 0&0&0\\0&1&0\\0&0&0\end{pmatrix}{\;},\text{ and} \quad
{\bm{\mathrm{P}}}_{\mathrm{{M}}}=\begin{pmatrix} 0&0&0\\0&0&0\\0&0&1\end{pmatrix}{\;}.$$
Force extension curves {#sec:force-extens-curve-1}
======================
[The central quantity is ${\mathcal G}({F},T,N)$, the Gibbs free energy of a DNA chain with $N$ base pairs, subject to a force ${F}$ and temperature $T$.]{} From ${\mathcal G}({F},T, N)$, obtained via the Legendre transform, [eq.]{} , the dominating singularity, [eq.]{} , or the exact transfer matrix partition function, [eq.]{} , we can calculate observables by performing appropriate derivatives. The number of segments in state $i=\mathrm{{B},{S},{M}}$ is obtained by $$\label{eq:23}
N_i = \left.\frac{\partial {\mathcal G}}{\partial {{g}_{i}}}\right|_{T,N,{F}}{\;}.$$ The force extension curve is readily calculated [*via* ]{}$$\label{eq:24}
x({F}) = \left.-\frac{\partial {\mathcal G}}{\partial {F}}\right|_{T,N} =
- \sum_{i=\mathrm{{B},{S},{M}}} \frac{\partial {\mathcal G}}{\partial
{{g}_{i}}}\frac{\partial {{g}_{i}}}{\partial {F}}
= \sum_{i=\mathrm{{B},{S},{M}}} N_i\left(x^{\mathrm{WLC}}_i({F}) +
{F}{\cdot}{l}_i/{\kappa}_i\right)$$ where $x_i^{\mathrm{WLC}}({F})$ is the stretching response of a worm-like chain and given explicitly in the supplementary information [eq.]{} .
Vanishing loop entropy, ${c}{}= 0$ {#sec:affine-free-energies-1}
----------------------------------
In this section we compare the prediction for vanishing loop exponent ${c}{}=0$ to experimental data and obtain estimates of the various parameters. We also demonstrate the equivalence of the grand canonical and canonical ensembles even for small chain length $N$. \[sec:parameters\]
In order to reduce the number of free fitting parameters we extract as many reasonable values from literature as possible. For the helical rise, the stretch modulus, and persistence length of [B]{}-DNA we use ${{l}_{\mathrm{{B}}}}=\unit{3.4}{\angstrom}$, ${{\kappa}_{\mathrm{{B}}}}=\unit{1}{\nano\newton}$, and ${{\xi}_{\mathrm{{B}}}}=\unit{48}{\nano\meter}$ [@Wenner2002]. [For the [M]{}-state, which is essentially single stranded DNA (ssDNA), *ab initio* calculations yield ${{l}_{\mathrm{{M}}}}=\unit{7.1}{\angstrom}$ and ${{\kappa}_{\mathrm{{M}}}}=\unit{2{\cdot}9.4}{\nano\newton}$ [@Hugel2005], where ${{\kappa}_{\mathrm{{M}}}}$ is valid for small forces ${F}<\unit{400}{\pico\newton}$ and the factor $2$ accounts for the presence of two ssDNA strands. Our value for the stretch modulus is considerably larger than previous experimental fit estimates [@Smith1996; @Dessinges2002] which might be related to the fact that experimental estimates depend crucially on the model used to account for conformational fluctuation effects; however, the actual value of ${{\kappa}_{\mathrm{{M}}}}$ is of minor importance for the stretching response, see supporting material. The persistence length of ssDNA is given by ${{\xi}_{\mathrm{{M}}}}\approx\unit{3}{\nano\meter}$ [@Murphy2004]. It turns out that the quality of the fit as well as the values of the other fit parameters are not very sensitive to the exact value of the persistence length ${{\xi}_{\mathrm{{S}}}}$ and the stretch modulus ${{\kappa}_{\mathrm{{S}}}}$ of the [S]{}-state as long as $\unit{10}{\nano\meter}\lesssim{{\xi}_{\mathrm{{S}}}}\lesssim\unit{50}{\nano\meter}$ and ${{\kappa}_{\mathrm{{S}}}}$ is of the order of ${{\kappa}_{\mathrm{{M}}}}$, see supporting material section \[appendix-sec:effect-param-feat\]]{}. Therefore we set ${{\xi}_{\mathrm{{S}}}}=\unit{25}{\nano\meter}$, which is an intermediate value between the persistence lengths of ssDNA and [B]{}-DNA, and ${{\kappa}_{\mathrm{{S}}}}= {{\kappa}_{\mathrm{{M}}}}=
\unit{2{\cdot}9.4}{\nano\newton}$ [@Cocco2004]. The segment length of the [S]{}-state ${{l}_{\mathrm{{S}}}}$ will be a fit parameter.
The chemical potentials ${{g}_{i}^{0}}$, $i=\mathrm{{B},{S},{M}}$, account for the free energy of base pairing and, since we set the interaction energies between neighboring segments of the same type to zero, ${V_{ii}}= 0$, also for the free energy gain due to base pair stacking [@SantaLucia1998]. They also correct for the fact that the reference state of the three different WLCs, which is $x = 0$ in the Helmholtz ensemble (constant extension $x$), [*cf. *]{}[eq.]{} , is not the same as contour and persistence lengths differ for [B]{}-, [S]{}-, and molten [M]{}-DNA. We choose ${{g}_{\mathrm{{B}}}^{0}}=0$ and treat ${{g}_{\mathrm{{S}}}^{0}}$ and ${{g}_{\mathrm{{M}}}^{0}}$ as fitting parameters.
[Each of these parameters controls a distinct feature of the force-extension curve: The chemical potentials ${{g}_{i}^{0}}$ determine the critical forces, the segment lengths ${l}_i$ affect the maximal extensions of each state and the off-diagonal ${V_{ij}}$ control the cooperativity of the transitions, see supporting material section \[appendix-sec:effect-param-feat\] for an illustration and section \[appendix-sec:tables-with-model\] for a summary of all parameter values.]{}
### Force extension curve {#sec:force-extens-curve-2}
![Comparison of force extension curves obtained by different methods for ${c}{}=0$. The curve obtained [*via* ]{}the exact transfer matrix calculation [eq.]{} is already for $N=2$ accurately reproduced by the approximate Legendre transformation [eq.]{} . The dominating singularity method [eqs.]{} or – equivalently – is strictly valid in the thermodynamic limit but agrees with the Legendre transform already for a modest value of $N=10$. [The units of the abscissa is extension per base pair (bp).]{} Parameters for $\lambda$-DNA in the absence of DDP are used, see supporting material section \[appendix-sec:tables-with-model\].[]{data-label="fig:1"}](018_bsl_comp_Legendre_TM_singularity_gs_image)
In [fig.]{} \[fig:1\] force extension curves based on three different levels of approximation are compared, using the same parameters that we extracted from DNA stretching data as will be detailed below. It turns out that the force extension curve obtained [*via* ]{}the Legendre transformation route [eq.]{} (dashed line) is a very good approximation of the results obtained from the exact transfer matrix results [eq.]{} (dotted line) already for $N=2$. For $N=10$ and larger virtually no differences between these two approaches are detectable. The deviations from the dominating singularity route [eq.]{} (solid line), which gives a result independent of $N$, are somewhat larger. But one sees that already the Legendre transform for $N=10$ (dash-dotted line) matches the dominating singularity result very closely. Therefore the use of the dominating singularity, [eq.]{} , or the largest transfer-matrix eigenvalue, [eq.]{} , is a very good approximation already for oligo-nucleotides and will be used in the rest of this work.
In [fig.]{} \[fig:2\] experimental stretching curves of $\lambda$-DNA with and without DDP (cisplatin) are presented [@Krautbauer2000]. When [B]{}-DNA is converted into [S]{}-DNA or [M]{}-DNA the base stacking is interrupted which gives rise to an interfacial energy between [B]{} and [S]{} as well as between [B]{} and [M]{} of the order of the stacking energy [@Konrad1996; @Kosikov1999]. [For untreated DNA, we use the value ${{V_{\mathrm{{B}}\mathrm{{S}}}}}={{V_{\mathrm{{S}}\mathrm{{B}}}}}={{V_{\mathrm{{B}}\mathrm{{M}}}}}={{V_{\mathrm{{M}}\mathrm{{B}}}}}=
\unit{1.2{\cdot}10^{-20}}{\joule}$ and show in the supplement that variations down to $\unit{0.8{\cdot}10^{-20}}{\joule}$ do not change the resulting curves much. ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$ is presumably small as the stabilizing stacking interactions are already disrupted [@Cocco2004], we thus set ${{V_{\mathrm{{S}}\mathrm{{M}}}}}={{V_{\mathrm{{M}}\mathrm{{S}}}}}=0$ for the fits in [fig.]{} \[fig:2\] – but we will come back to the issue of a non-zero ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$ later on.]{} Cisplatin is thought to disrupt the stacking interaction between successive base pairs and thereby to reduce the cooperativity of the [B]{}[S]{}-transition. This fact we incorporate by setting all interfacial energies to zero, ${V_{ij}}= 0$, for DDP treated DNA. The three remaining undetermined parameters (${{l}_{\mathrm{{S}}}}$, ${{g}_{\mathrm{{S}}}^{0}}$, ${{g}_{\mathrm{{M}}}^{0}}$) have distinct consequences on the force-extension curve. The segment length ${{l}_{\mathrm{{S}}}}$ and the chemical potential ${{g}_{\mathrm{{S}}}^{0}}$ determine the position of the [B]{}[S]{}-plateau with respect to the polymer extension and applied force, respectively, while the chemical potential ${{g}_{\mathrm{{M}}}^{0}}$ controls the force at which the second transition appears. Fitting to experimental data is thus straightforward and yields for untreated $\lambda$-DNA ${{l}_{\mathrm{{S}}}}= \unit{6.1}{\angstrom}$, ${{g}_{\mathrm{{S}}}^{0}}= \unit{1.6{\cdot}10^{-20}}{\joule}$, ${{g}_{\mathrm{{M}}}^{0}}= \unit{2.4{\cdot}10^{-20}}{\joule}$ and for $\lambda$-DNA in the presence of DDP (cisplatin) ${{l}_{\mathrm{{S}}}}= \unit{6.0}{\angstrom}$, ${{g}_{\mathrm{{S}}}^{0}}=
\unit{1.2{\cdot}10^{-20}}{\joule}$, ${{g}_{\mathrm{{M}}}^{0}}= \unit{2.8{\cdot}10^{-20}}{\joule}$, see [fig.]{} \[fig:2\]. We also fit the number of monomers $N$ and allow for an overall shift along the $x$-axis. [The main difference between the two stretching curves is the cooperativity of the [B]{}[S]{}-transition, which is controlled by the interfacial energies ${{V_{\mathrm{{B}}\mathrm{{S}}}}}$ and ${{V_{\mathrm{{B}}\mathrm{{M}}}}}$. Note that, although the over-stretching transition is quite sharp for DNA without DDP, it is not a phase transition in the strict statistical mechanics sense. A true phase transition arises only for ${c}{}>1$, as will be shown in the next section.]{} The fitted value of ${{g}_{\mathrm{{M}}}^{0}}$ is about two times larger than typical binding energies [@SantaLucia1998] for pure DNA. [As a possible explanation, we note that the force extension curve of DNA without DDP exhibits pronounced hysteresis (especially at higher force) which will increase the apparent binding energy due to dissipation effects [@Ho2009]. Any statements as to the stability of [S]{}-DNA based on our fitting procedures are thus tentative. However, such complications are apparently absent in the presence of DDP [@Krautbauer2000] which rules out kinetic effects as the reason for our relatively high fit values of ${{g}_{\mathrm{{M}}}^{0}}$ and the stability of [S]{}-DNA. Cisplatin most likely stabilizes base pairs due to cross-linking and thus shifts the subtle balance between [B]{}-, [S]{}-, and [M]{}-DNA. Therefore, the relative stability of [B]{}-, [S]{}-, and [M]{}-DNA is sensitively influenced by co-solute effects. We note that even with ${c}{}=0$ a good fit of the data is possible. In the top panel of [fig.]{} \[fig:2\] we show the fraction of segments in [B]{}-, [S]{}-, and [M]{}-states for untreated $\lambda$-DNA. There is a balanced distribution of bases in all three states across the full force range, in agreement with previous results [@Cocco2004].]{}
![Bottom panel: Force extension curve of double-stranded $\lambda$-DNA with and without DDP. Symbols denote experimental data [@Krautbauer2000], lines are fits with the three-state model for ${c}{}=0$. The main difference between the two curves is the lack of cooperativity in the [B]{}[S]{}-transition in the presence of DDP which we take into account by choosing vanishing interaction energies ${V_{ij}}=0$, $i,j=\mathrm{{B},{S},{M}}$. Top panel: Fraction $N_i/N$ of segments in the different states as follows from [eq.]{} in the absence of DDP. []{data-label="fig:2"}](008_bsl_experiment_dna_force_extension_gs_image)
Non-vanishing loop entropy, ${c}{}\neq0$ {#sec:logar-loop-free}
----------------------------------------
We now turn to non-zero loop exponents ${c}{} \neq 0$ and in specific try to estimate ${c}{}$ from the experimental stretching data. The partition function ${{\mathcal{Z}}}_{{c}\neq 0}$ in [eq.]{} exhibits two types of singularities. First, simple poles at $\lambda={\lambda_{\mathrm{p}}}$, which are the zeros of the denominator of [eq.]{} and which are determined as the roots of the equation $$\label{eq:25}
-\frac{b_5 + b_6 \lambda + b_7\lambda^2}{b_8 \lambda + b_9\lambda^2} = {\mathrm{Li}_{{c}}\left(\lambda /{\lambda_{\mathrm{b}}}\right)}
{\;}.$$ Second, a branch point that occurs at $$\label{eq:26}
\lambda = {\lambda_{\mathrm{b}}}= {\mathrm{e}}^{\beta {{g}_{\mathrm{{M}}}}}
{\;}.$$ The singularity with the smallest modulus is the dominant one [@Poland1966; @Flajolet1990], and we define the critical force ${{F}_{\mathrm{c}}}$ as the force where both equations, [eq.]{} and , hold simultaneously. For ${c}\leq1$ the dominant singularity is always given by the pole ${\lambda_{\mathrm{p}}}$ and thus no phase transition is possible. For $1<{c}\leq2$ a continuous phase transition occurs. By expanding [eq.]{} around ${{F}_{\mathrm{c}}}$ one can show that all derivatives of the free energy up to order $n$ are continuous, where $n\in{\mathbb{N}}$ is defined as the largest integer with $n<({c}-1)^{-1}$ [@Kafri2002]. For instance, ${c}=3/2$ leads to a kink in the force extension curve. For ${c}=1.2$ this leads to a kink in $x'''({F})$. If ${c}>2$ the transition becomes first order and the force extension curve exhibits a discontinuity at ${F}={{F}_{\mathrm{c}}}$. In [fig.]{} \[fig:3\]a we plot force extension curves for different values of the loop exponent ${c}$ with all other parameter fixed at the values fitted for untreated DNA. It is seen that finite ${c}{}$ leads to changes of the force extension curves only at rather elevated forces. In order to see whether a finite ${c}{}$ improves the comparison with the experiment and whether it is possible to extract the value of ${c}{}$ from the data, we in [fig.]{} \[fig:3\]b compare the untreated DNA data with a few different model calculations for which we keep the parameters ${{l}_{\mathrm{{S}}}}$, ${{g}_{\mathrm{{S}}}^{0}}$, ${{g}_{\mathrm{{M}}}^{0}}$, ${{V_{\mathrm{{B}}\mathrm{{S}}}}}$, ${{V_{\mathrm{{B}}\mathrm{{M}}}}}$ fixed at the values used for the fit with ${c}{}=0$ in [fig.]{} \[fig:2\]. Allowing for finite ${c}{}$ but fixing a zero domain wall energy between [S]{}- and [M]{}-regions, ${{V_{\mathrm{{S}}\mathrm{{M}}}}}=0$, leads to an optimal exponent ${c}{}=0.6$ and slightly improves the fit to the data which show the onset of a plateau at a force of about $\unit{100}{\pico\newton}$. The same effect, however, can be produced by fixing ${c}{}=0$ and allowing for a finite ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$, which yields the optimal value of ${{V_{\mathrm{{S}}\mathrm{{M}}}}}= \unit{1.1{\cdot}10^{-21}}{\joule}$. Finally, fixing ${{V_{\mathrm{{S}}\mathrm{{M}}}}}= \unit{1.1{\cdot}10^{-21}}{\joule}$ and optimizing ${c}{}$ yields in this case ${c}{}=0.3$ and perfect agreement with the experimental data. However, the significance of this improvement is not high, as the experimental data are quite noisy and possibly plagued by kinetic effects. What the various curves illustrate quite clearly, however, is that a non-zero exponent ${c}{}$ leads to modifications of the stretching curves that are similar to the effects of a non-vanishing domain wall energy ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$. Although ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$ should be considerably smaller than ${{V_{\mathrm{{B}}\mathrm{{M}}}}}$ or ${{V_{\mathrm{{B}}\mathrm{{S}}}}}$, a finite value of ${{V_{\mathrm{{S}}\mathrm{{M}}}}}=
\unit{1.1{\cdot}10^{-21}}{\joule}$ as found in the fit is reasonable and cannot be ruled out on general grounds. The maximal value of ${c}{}$ is obtained for vanishing ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$ and amounts to about ${c}{}=0.6$. A value of ${c}{}=2.1$, which would be expected based on the entropy of internal DNA loops [@Kafri2002], on the other hand does not seem compatible with the experimental data, as follows from [fig.]{} \[fig:3\]a. This might have to do with the presence of nicks. [Nicks in the DNA drastically change the topology of loops and result in a reduced loop exponent which is ${{c}}{}=0$ for an ideal polymer and $0.092$ for a self avoiding polymer [@Kafri2002]. Therefore, the low value of ${c}{}$ we extract from experimental data might be a signature of nicked DNA. Additional effects such as salt or co-solute binding to loops are also important. Therefore ${c}$ can be viewed as a heuristic parameter accounting for such non-universal effects as well. We note in passing that ${c}$ only slightly affects the [B]{}[S]{}-transition, as seen in [fig.]{} \[fig:3\]a.]{}
![ Various force-extension curves of the three-state model with fit parameters for $\lambda$-DNA without DDP. **a)** Lower panel: Force extension curves for different values of the loop exponent ${c}$, showing no phase transition (${c}\leq1$), a continuous ($1<{c}\leq2$), or a first order phase transition (${c}>2$). The critical forces are denoted by open circles. The inset is a magnification of the region around the transition. Upper panel: Fraction of bases in the three states for ${c}{}=3/2$. The critical transition, above which all bases are in the molten [M]{}-state, is discerned as a kink in the curves. **b)** Comparison of experimental data (circles) and theory for ${c}{}\neq 0$. The curve for ${c}{}=0$ and ${{V_{\mathrm{{S}}\mathrm{{M}}}}}=0$, already shown in [fig.]{} \[fig:2\], is obtained by fitting ${{l}_{\mathrm{{S}}}}$, ${{g}_{\mathrm{{S}}}^{0}}$, ${{g}_{\mathrm{{M}}}^{0}}$ to the experimental data, the values of which are kept fixed for all curves shown. The curve for ${{V_{\mathrm{{S}}\mathrm{{M}}}}}=0$ and ${c}{}=0.6$ results by fitting ${c}{}$ and slightly improves the fit quality. The curve ${c}{}=0$ and ${{V_{\mathrm{{S}}\mathrm{{M}}}}}= \unit{1.1{\cdot}10^{-21}}{\joule}$ is obtained by fitting ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$. The curve for ${{V_{\mathrm{{S}}\mathrm{{M}}}}}= \unit{1.1{\cdot}10^{-21}}{\joule}$ and ${c}{}=0.3$ is obtained by fitting ${c}{}$ and keeping ${{V_{\mathrm{{S}}\mathrm{{M}}}}}$ fixed. The inset shows the first derivative of $x({F})$ and illustrates that increasing ${c}{}$ leads to a growing asymmetry around the transition region. **c)** Temperature dependence of the force extension curves. Increasing temperature leads to a decrease of the [B]{}[S]{}-plateau force. In the presence of a true denaturing transition, i.e. for ${c}{}>1$, the critical force ${{F}_{\mathrm{c}}}$ decreases with increasing temperature and for ${F}>{{F}_{\mathrm{c}}}$ the force extension curve follows a pure WLC behavior. []{data-label="fig:3"}](022_bsl_combine_019-021-020_gs_image)
Finite temperature effects {#sec:phase-diagram}
==========================
The temperature dependence of all parameters is chosen such that the force extension curves at $T=\unit{20}{\degreecelsius}$ that were discussed up to now remain unchanged. The persistence lengths are modeled as ${\xi}_i(T)={\xi}_i{\cdot}(T/\unit{293}{\kelvin})^{-1}$. The [S]{}-state free energy is split into enthalpic and entropic parts as ${{g}_{\mathrm{{S}}}^{0}}(T)
={\tau_{\mathrm{{S}}}}(h_{\mathrm{{S}}} - Ts_{\mathrm{{S}}})$ where we use $h_{\mathrm{{S}}}=\unit{7.14{\cdot}10^{-20}}{\joule}$ and $s_{\mathrm{{S}}}=\unit{1.88{\cdot}10^{-22}}{\joule\per\kelvin}$ from @ClausenSchaumann2000. The correction factor ${\tau_{\mathrm{{S}}}}= 0.98$ accounts for slight differences in the experimental setups and is determined such that ${{g}_{\mathrm{{S}}}^{0}}(T=\unit{20}{\degreecelsius})$ equals the previously fitted value. The molten state energy ${{g}_{\mathrm{{M}}}^{0}}(T)=(h_{\mathrm{{M}}} - Ts_{\mathrm{{M}}})$ is also chosen such that ${{g}_{\mathrm{{M}}}^{0}}(T=\unit{20}{\degreecelsius})$ agrees with the previous fit value and that the resulting denaturing temperature in the absence of force agrees with experimental data. Assuming a melting temperature of ${T_{\mathrm{c}}}= \unit{348}{\kelvin}$ for $\lambda$-DNA [@Gotoh1976], we obtain ${{g}_{\mathrm{{M}}}^{0}}(T) =
\unit{1.5{\cdot}10^{-19}}{\joule}-T{\cdot}\unit{4.2{\cdot}10^{-22}}{\joule\per\kelvin}$ for ${c}{} = 0$ and ${{g}_{\mathrm{{M}}}^{0}}(T) =
\unit{1.6{\cdot}10^{-19}}{\joule}-T{\cdot}\unit{4.6{\cdot}10^{-22}}{\joule\per\kelvin}$ for ${c}{} = 3/2$. In [fig.]{} \[fig:3\]c we plot a few representative stretching curves for different temperatures. It is seen that increasing temperature lowers the [B]{}[S]{}-plateau and makes this transition less cooperative. Differences between ${c}{}=0$ and ${c}{}=3/2$ are only observed at elevated forces, where for ${c}{}=3/2$ one encounters a singularity characterized by a kink in the extension curves.
For ${c}{}>1$ the critical force ${{F}_{\mathrm{c}}}$ is defined as the force where the pole and the branch point coincide and [eqs.]{} and are simultaneously satisfied. The phase boundary in the force-temperature plane is thus defined by $$\label{eq:27}
-\frac{b_5 + b_6 {\lambda_{\mathrm{b}}}+ b_7{\lambda_{\mathrm{b}}}^2}{b_8 {\lambda_{\mathrm{b}}}+ b_9{\lambda_{\mathrm{b}}}^2}=\zeta({c})
{\;},$$ where $b_i$ and ${\lambda_{\mathrm{b}}}$ depend on $T$ and ${F}$ and $\zeta({c}{}) =
{\mathrm{Li}_{{c}}\left(1\right)}$ is the Riemann zeta function. The phase boundary ${{F}_{\mathrm{c}}}(T)$ for ${c}{}=3/2$ is shown in [fig.]{} \[fig:4\] and agrees qualitatively with the experimental data of ref. [@Williams2001a]. For exponents ${c}{}<1$ no true phase transition exists. We therefore define crossover forces as the force at which half of the segments are in the molten state or in the [B]{}-state, i.e. $N_{\mathrm{{M}}}/N=1/2$ or $N_{\mathrm{{B}}}/N=1/2$. In [fig.]{} \[fig:4\] we show these lines for both ${c}{}=0$ and ${c}{}=3/2$. Note that the parameters for the ${c}{}=0$ case have been adjusted so that $N_{\mathrm{{M}}}/N=1/2$ at $T=\unit{348}{\kelvin}$ and ${F}=0$. The broken lines on which $N_{\mathrm{{B}}}/N=1/2$ for ${c}{}=0$ and ${c}{}=3/2$ are virtually the same, showing again that loop entropy is irrelevant for the [B]{}[S]{}-transition. The [S]{}-state is populated in the area between the $N_{\mathrm{{B}}}/N=1/2$ and $N_{\mathrm{{M}}}/N=1/2$ lines, which for $T>\unit{330}{\kelvin}$ almost coalesce, meaning that at elevated temperatures the [S]{}-state is largely irrelevant. Force-induced re-entrance at constant temperature is found in agreement with previous two-state models [@Hanke2008; @Rahi2008a]. Re-entrance at constant force as found for a Gaussian model [@Hanke2008] is not reproduced, in agreement with results for a non-extensible chain [@Rahi2008a].
![ [The solid line shows the critical force ${{F}_{\mathrm{c}}}$ for ${c}{}=3/2$, at which a singularity occurs according to [eq.]{} . Phase boundaries for ${c}{}=0$ (thick lines) and ${c}{}=3/2$ (thin lines) are defined by $N_{\mathrm{{M}}}/N=0.5$ (dotted) and $N_{\mathrm{{B}}}/N=0.5$ (dashed). The melting temperature ${T_{\mathrm{c}}}$ is denoted by the dot. The insets show the behavior of the phase boundaries near the melting temperature, ${F}\propto \sqrt{{T_{\mathrm{c}}}-T}$.]{} Parameters for $\lambda$-DNA without DDP are used. []{data-label="fig:4"}](012_bsl_dna_phase_diagram_gs_image)
![Relative fraction of segments in the [B]{}-state, $N_{\mathrm{B}}/N$, as a function of temperature for different loop exponents ${c}=0,\ 1.5,\
2.1$ and for finite [B]{}[S]{} interfacial energy ${{V_{\mathrm{{B}}\mathrm{{S}}}}}={{V_{\mathrm{{B}}\mathrm{{M}}}}}=\unit{1.2{\cdot}10^{-20}}{\joule}$ (bold lines) and for ${{V_{\mathrm{{B}}\mathrm{{S}}}}}={{V_{\mathrm{{B}}\mathrm{{M}}}}}=0$ (thin lines). Circles indicate the positions of the phase transition. For all curves parameters for $\lambda$-DNA without DDP have been used, ${{g}_{\mathrm{{M}}}^{0}}(T) =
\unit{1.5{\cdot}10^{-19}}{\joule}-T{\cdot}\unit{4.2{\cdot}10^{-22}}{\joule\per\kelvin}$ for ${c}{} = 0$ and ${{g}_{\mathrm{{M}}}^{0}}(T) =
\unit{1.6{\cdot}10^{-19}}{\joule}-T{\cdot}\unit{4.6{\cdot}10^{-22}}{\joule\per\kelvin}$ for ${c}{} > 0$. []{data-label="fig:5"}](023_bsl_T-NB-c_fraction_of_states_temperature_dependence_gs_image)
As we have shown so far, a non-zero loop exponent ${c}{}$ only slightly improves the fit of the experimental stretching data and the optimal value found is less than unity. This at first sight seems at conflict with recent theoretical work that argued that a loop exponent larger than ${c}{}=2$ is needed in order to produce denaturation curves (at zero force) that resemble experimental curves in terms of the steepness or cooperativity of the transition [@Kafri2002]. To look into this issue, we plot in [fig.]{} \[fig:5\] the fraction of native base pairs, $N_{\mathrm{{B}}}/N$ as a function of temperature for zero force and different parameters. As soon as the domain-wall energies ${{V_{\mathrm{{B}}\mathrm{{S}}}}}$ and ${{V_{\mathrm{{B}}\mathrm{{M}}}}}$ are finite, the transition is quite abrupt, even for vanishing exponent ${c}{}$. Therefore even loop exponents ${c}<1$ yield melting curves which are consistent with experimental data, where melting occurs over a range of the order of $\unit{10}{\kelvin}$ [@Blossey2003; @Gotoh1976].
Conclusions {#sec:conclusions}
===========
The fact that the domain-wall energy due to the disruption of base pair stacking, ${{V_{\mathrm{{B}}\mathrm{{S}}}}}$ and ${{V_{\mathrm{{B}}\mathrm{{M}}}}}$, and the loop entropy embodied in the exponent ${c}{}$, give rise to similar trends and sharpen the melting transition has been realized and discussed before [@Blossey2003; @Whitelam2008]. The present three-state model and the simultaneous description of experimental data where the denaturation is induced by application of force and by temperature allows to disentangle the influence of these two important effects. By the application of a force, the de-stacking and the loop formation occur subsequently, allowing to fit both parameters separately. As our main finding, we see that for a finite domain-wall energy ${{V_{\mathrm{{B}}\mathrm{{S}}}}}={{V_{\mathrm{{B}}\mathrm{{M}}}}}$, the additional influence of the loop exponent on the force stretching curves and the denaturation curves is small. In fact, the optimal value for ${c}{}$ turns out to be of the order of ${c}{}
\approx 0.3 - 0.6$, even if we choose a vanishing value ${{V_{\mathrm{{S}}\mathrm{{M}}}}}=0$. This estimate for ${c}{}$ is smaller than previous estimates. One reason for this might be nicks in the DNA. So it would be highly desirable to redo stretching experiments with un-nicked DNA [from which the value of ${c}{}$ under tension could be determined. A second transition at high forces of about ${F}\simeq \unit{200}{\pico\newton}$ which is seen in the experimental data used in the paper, inevitably leads via the fitting within our three-state model to an intermediate [S]{}-DNA state. But we stress that the occurrence of such an intermediate [S]{}-state depends on the fine-tuning of all model parameters involved, which suggests that in experiments the [S]{}-state stability should also sensitively depend on the precise conditions.]{} One drawback of the current model is that sequence effects are not taken into account. This means that our fitted parameters have to be interpreted as coarse-grained quantities which average over sequence disorder. Calculations with explicit sequences have been done for short DNA strands but should in the future be doable for longer DNA as well.
Acknowledgments {#sec:acknowledgements}
===============
The authors express their gratitude to Hauke Clausen-Schaumann and Rupert Krautbauer for providing the experimental data and helpful comments and thank Ralf Metzler, Matthias Erdmann, and Dominik Ho for fruitful discussions. Financial support comes from the DFG [*via* ]{}grants NE 810/7 and SFB 863. T.R.E. acknowledges support from the *Elitenetzwerk Bayern* within the framework of *CompInt*.
|
---
abstract: |
This article is devoted to the study of an incompressible viscous flow of a fluid partly enclosed in a cylindrical container with an open top surface and driven by the constant rotation of the bottom wall. Such type of flows belongs to a group of recirculating lid-driven cavity flows with geometrical axisymmetry and of the prescribed boundary conditions of Dirichlet type—no-slip on the cavity walls. The top surface of the cylindrical cavity is left open with an imposed stress-free boundary condition, while a no-slip condition with a prescribed rotational velocity is imposed on the bottom wall. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. The approach taken here revealed new flow states that were investigated based on a fully three-dimensional solution of the Navier–Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. Theses solutions are obtained through direct numerical simulations based on a Legendre spectral element method.
Swirling flow,free surface,vortex breakdown,lid-driven cavity
address:
- |
Laboratory of Computational Engineering,\
École Polytechnique Fédérale de Lausanne,\
STI–IGM–LIN, Station 9,\
CH–1015 Lausanne, Switzerland
- |
Present address: Massachusetts Institute of Technology,\
Department of Mechanical Engineering,\
77 Massachusetts Avenue, Bldg 5–326,\
Cambridge, MA 02139
- |
Department of Mechanical and Aerospace Engineering,\
Monash University,\
Victoria, 3800 Melbourne, Australia
author:
- Roland Bouffanais
- David Lo Jacono
title: Transitional cylindrical swirling flow in presence of a flat free surface
---
,
Introduction {#sec:introduction}
============
Besides the differences in terms of geometry, the lid-driven cubical cavity flow [@shankar00:_fluid_mechan_driven_cavit] and the cylindrical swirling flow investigated in this article, present similar features typical of shear-driven recirculating flows such as intense wall-jets, shear layers in the vicinity of the driven wall, and secondary recirculating flows, all of which are very dependent on the flow parameters. Nevertheless, the geometry—cubical on one hand and cylindrical on the other hand—dramatically influences the nature and structure of these secondary flows: corner eddies for the cubical cavity and recirculation bubbles or vortex breakdown in the cylindrical case.
General considerations {#sec:general-considerations}
----------------------
Following the pioneering work of Bogatyrev & Gorin [@bogatyrev78:_end] and Koseff & Street [@koseff84; @koseff84:_visual], it was shown that contrary to intuition, the lid-driven *cubical* cavity flow is essentially three-dimensional, even when considering large aspect ratio. It is only recently that the three-dimensionality of the lid-driven *cylindrical* cavity flow was confirmed numerically by Blackburn & Lopez [@blackburn00:_symmet; @blackburn02:_modul] after it was suggested but not fully proved experimentally by S[ø]{}rensen [@soerensen92:_visual], Spohn [@spohn98:_exper], Sotiropoulos & Ventikos [@sotiropoulos98:_trans], and Pereira & Sousa [@pereira99:_confin]. In 2001, Sotiropoulos & Ventikos [@sotiropoulos01] gave full experimental evidence of the three-dimensional character of the flow with the onset of non-axisymmetric modes. The three-dimensional nature of these driven cavity flows therefore appears as a general characteristic of internal recirculating shear-driven flows.
In the sequel, we will only consider the cylindrical lid-driven cavity flow also referred to as “swirling” flow without any additional precision. The first experiments by Vogel [@vogel68:_exper_ergeb_str_geh] and later Ronnenberg [@ronnenberg77:_kompon_laser_vergeic_unter_drehs_r] showed that Ekman suction and pumping, induced by the Ekman layers on the rotating and stationary disks, lead to the formation of a concentrated vortex core along the axis in the closed cavity case. The two dimensionless numbers characterizing this flow are the height-to-radius aspect ratio $\Lambda=H/R$ and the Reynolds number ${\textrm{Re}}=R^2\Omega/\nu$, where $H$ and $R$ are the height and radius of the cylinder respectively, $\Omega$ the constant angular velocity of the bottom end-wall, and $\nu$ the kinematic viscosity of the Newtonian fluid. For specific values of the aspect ratio $\Lambda$, and above a critical Reynolds number, the vortex core breaks down in the form of one or more recirculation bubbles which are on-axis in the closed cavity case and on- or off-axis in the open cavity case. Owing to the enormous extent of work in the area of vortex breakdown (VB), (see reviews by Hall [@hall72:_vortex], Leibovich [@leibovich78:_struc], Shtern & Hussain [@shtern99:_collap], Kerswell [@kerswell02:_ellip], and Arndt [@arndt02:_cavit]), we will only briefly recall the central features of VB. As defined by Leibovich in its review on the structure of vortex breakdown [@leibovich78:_struc], the term “vortex breakdown” refers to a disturbance characterized by the formation of an internal stagnation point on the vortex axis, followed by reversed flow in a region of limited axial extent. Two forms of vortex breakdown predominate, one called “near-axisymmetric” (sometimes “axisymmetric” or “bubble-like”), and the other called “spiral”.
The practical importance of vortex breakdown lies mainly in the field of aeronautics, where they can be observed over wings—mainly delta wings—with highly swept leading edges when the angle of incidence exceeds a critical value. Vortex breakdown can be a limiting factor on the operating altitude of slender-winged flying vehicles. Moreover, the occurrence of VB in the wake of a large aircraft is relevant to the safety of flight in dense air-traffic, which is becoming more and more frequent with the constant increase in air-traffic over the years. VB is also important in other fields for example it has been observed in the swirling flows through nozzles and diffusers [@faler77:_disrup], and in the field of reactive flows, in combustion chambers. Besides the tremendous importance of VB in engineering applications, it is also a prototypical phenomenon allowing to elucidate the fundamental aspects of the bubble mode.
The lid-driven cylindrical cavity flow {#sec:closed-cavity}
--------------------------------------
The first comprehensive experimental study of the closed cylindrical container case was undertaken by Escudier [@escudier84:_obser], and Escudier & Keller [@escudier85:_recir], who extended the earlier results of Vogel [@vogel68:_exper_ergeb_str_geh] and Ronnenberg [@ronnenberg77:_kompon_laser_vergeic_unter_drehs_r] to obtain the first map of VB transitions with respect to the aspect ratio $\Lambda$ and the Reynolds number. Escudier [@escudier84:_obser] revealed flow states with one, two or even three successive breakdowns, as well as a transition from steadiness to unsteadiness. S[ø]{}rensen [@soerensen92:_visual] extended to a broader range of Reynolds number in the same experiment as Escudier [@escudier84:_obser] for the closed container, and inferred that above a critical Reynolds number in the unsteady flow regime, the meridional flow becomes highly asymmetric. The first experimental study of the open cylindrical container case with a free surface on the top, was undertaken by Spohn [@spohn93:_obser], who highlighted the significant change in the structure, the occurrence and the location of the breakdown bubbles. In the steady closed cylinder case, Hourigan [@hourigan95:_spiral] investigated the asymmetric spiraling effects along the cylinder axis prior to the first vortex breakdown. They argued that the observed asymmetry was purely an experimental artifact and not an evidence of the three-dimensional nature of the flow. Spohn [@spohn98:_exper] were the first to investigate experimentally the origin of possible asymmetric features of the instabilities at their onset. The steady breakdown bubbles reported by Spohn [@spohn98:_exper] showcase asymmetric features comparable to earlier measurements, and also to unsteady bubbles observed in circular diffusers by Faler & Leibovich [@faler77:_disrup]. As a matter of fact, the work of Spohn [@spohn98:_exper] is really a pioneering work in the acceptance of the axisymmetry breaking, amongst fluid experimentalists, see Br[ø]{}ns [@broens08:_dye_visual_near_three_dimen_stagn_point]. It is noteworthy at this point, that the complex physics associated with these intricate phenomena occurring in closed/open rotating cylindrical container is still not clearly understood.
Like for the lid-driven cubical cavity flow, and in relation with the simple geometry of the flow, the rotating cylindrical cavity flow has been extensively studied using direct numerical simulations. It is important to note that since the early seventies, the method of choice has consisted in solving the streamfunction-vorticity formulation of the axisymmetric incompressible Navier–Stokes equations. Without being exhaustive, the following list of references gives an overview of the numerical simulation of the closed lid-driven cylindrical flow over three decades: Pao [@pao70], Lugt & Haussling [@lugt73:_devel; @lugt82:_axisy], Dijkstra & van Heijst [@dijkstra83], Lugt & Abboud [@lugt87:_axisy], Neitzel [@neitzel88:_streak], Daube & S[ø]{}rensen [@daube89:_numer], Lopez [@lopez90:_axisy], Brown & Lopez [@brown90:_axisy], Lopez & Perry [@lopez92:_axisy], S[ø]{}rensen & Christensen [@soerensen95:_direc], Watson & Neitzel [@watson96:_numer], Gelfgat [@gelfgat96:_stead; @gelfgat96:_stabil], Tsitverblit & Kit [@tsitverblit98], and Br[ø]{}ns [@brons99:_stream]. All these works were able to reproduce with a reasonable accuracy, the basic features observed experimentally and reported earlier including the size, shape and number of recirculation bubbles. The onset of vortex breakdown was to some extent captured by several of these numerical simulations. Lopez [@lopez90:_axisy], and Brown & Lopez [@brown90:_axisy] suggested a physical mechanism for the intricate phenomena observed. They prove the existence of a standing centrifugal wave, whose amplitude increases with the Reynolds number and which can create a stagnation point on the cylinder axis, initiating the breakdown process. It is worth recalling that the streamfunction-vorticity formulation is adequate and appropriate only for the study of flow dynamics preserving the property of axisymmetry. At the inception of any instability breaking the axisymmetry of the flow, a three-dimensional solution of the Navier–Stokes equations is required, thereby increasing considerably the complexity of the task. The last remark justifies the observed changes in terms of numerical modeling of Lopez’ group and S[ø]{}rensen’s group, to allow them to investigate axisymmetry breaking in the closed cylinder case [@blackburn00:_symmet; @blackburn02:_modul; @blackburn03; @bisgaard06:_vortex; @shen06:_numer]. Therefore, three-dimensional flow structures have started being simulated more recently, see Gelfgat [@gelfgat01:_three], Sotiropoulos & Ventikos [@sotiropoulos01], Sotiropoulos [@sotiropoulos01:_chaot], Marques & Lopez [@marques01:_preces], Blackburn & Lopez [@blackburn00:_symmet; @blackburn02:_modul], Serre & Bontoux [@serre02:_vortex], Blackburn [@blackburn02:_three], and Lopez [@lopez06:_rotat].
Apart from the canonical case with a single driving lid in rotation at a constant angular velocity, different variations of the problem have been extensively studied in the past years: cylinder with co- and counter-rotating end-covers by Br[ø]{}ns [@brons99:_stream], steady axisymmetric flow in an open cylindrical container with a partially rotating bottom wall by Piva & Meiburg [@piva05:_stead], vortex scenario and bubble generation in a cylindrical cavity with rotating top and bottom by Okulov [@okulov05:_vortex]. Mullin [@mullin00] also included a rod at the axis to control the breakdown, and Pereira & Sousa [@pereira99:_confin] significantly changed the configuration by replacing the flat rotating bottom cover by a cone. As noted by Br[ø]{}ns [@brons01:_topol], all these studies show a large set of flow structures which are quite sensitive to variations of external parameters. Mununga [@mununga04:_confin_flow_vortex_break_contr] and Lo Jacono [@lo08:_contr_of_vortex_break_in] investigated different strategies for the control of vortex breakdown.
Open swirling flow {#sec:open-swirling-flow}
------------------
The focus in the present article is on the canonical problem of a cylinder with a rotating bottom end-wall but replacing the stationary solid top end-wall by a free surface. The flow associated with this problem was first studied experimentally by Spohn [@spohn93:_obser; @spohn98:_exper]. They observed the influence of the top free surface—assuredly clean of surfactants—on the onset, structure, nature and number of recirculating bubbles. Their central observations are that breakdown bubbles still appear, but are off-axis and may be attached to the free surface, depending on the aspect ratio $\Lambda$ and the Reynolds number. Of course, such structures could not be observed in the closed case because of the no-slip condition imposed on the top wall. All the past simulations of free-surface swirling flows rely on the central assumptions that the free surface is flat and clean, which means that the Froude number is very small and that surface tension effects are negligible. With these assumptions, the flow is identical to the flow in the lower half part of a cylinder with two solid covers in co-rotation, rotating at the same angular velocity. Br[ø]{}ns reported a wide range of topologies of vortex breakdown bubbles in a bottom-driven cylinder with a free surface. Valentine & Jahnke [@valentine94:_flows], observed in their simulations the existence of one or two toroidal-like types of recirculation bubble having their stagnation lines attached to the free surface, depending on the value of the Reynolds number. Their study was complemented by the work of Lopez [@lopez95:_unstead] for oscillating unsteady flows. Information relevant to the present problem with a free surface all indicate consistent flow behavior at small aspect ratio , $0.5\leq \Lambda \leq 1.0$ in that stagnation occurs off-axis and associated secondary flow creates a toroidal recirculation bubble. Steady free-surface flows have been computed by Iwatsu [@iwatsu05:_numer; @iwatsu04:_analy] providing flow state classifications with new flow patterns not revealed in the previous studies.
Motivations and objectives {#sec:Motivations_objectives}
--------------------------
The present study is motivated by several factors. Firstly, compared to the closed cylinder case, only some limited aspects of the open swirling flow have been investigated so far. The study of this intricate problem is relatively new and consequently the body of knowledge in some $(\Lambda , {\textrm{Re}})$-parameter regions appears fairly limited. Secondly, most of the past studies involving numerical simulations of this free-surface swirling flow, used axisymmetric streamfunction-vorticity formulations: Br[ø]{}ns [@brons01:_topol], Iwatsu [@iwatsu05:_numer; @iwatsu04:_analy], and Piva & Meiburg [@piva05:_stead]. To our knowledge, the only fully three-dimensional numerical simulations is due to Lopez [@lopez04:_symmet], who investigated mainly symmetry breaking issues.
In the present article, new flow states are investigated based on a fully three-dimensional solution of the Navier–Stokes equations without the need to resort to symmetry properties by doubling the computational domain and enforcing co-rotation of both end-walls. To our knowledge, the present study provides the most general available results for this flat-free-surface problem. Both, steady and unsteady flows are considered for different sets of governing parameters $(\Lambda, {\textrm{Re}})$. A Legendre spectral element method is used to provide an accurate solution of the governing equations, while the stress-free boundary condition is naturally enforced into the weak formulation of the problem.
The mathematical model and the problem formulation are detailed in Sec. \[sec:formulation\], while the original computational approach of this study is presented in Sec. \[sec:computational\]. Subsequently, Sec. \[sec:numerical\] contains all the numerical results corresponding to various physical situations and flow states. Finally, the article ends with Sec. \[sec:conclusions\] providing summary and conclusions on the present work.
Mathematical model and problem formulation {#sec:formulation}
==========================================
Mathematical description of the problem
---------------------------------------
![Schematic of the geometry studied with the set of coordinates employed.[]{data-label="fig:sketch"}](cylindrical-cavity.eps){width="55.00000%"}
The fluid enclosed in the cylindrical cavity is assumed to be incompressible, Newtonian with uniform density and temperature. The flow is governed by the Navier–Stokes equations $$\begin{aligned}
\frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial
x_j} & =
\frac{\partial \sigma_{ij}^*}{\partial x_j} + g_i, \label{eq:NS-1}\\
\frac{\partial u_j}{\partial x_j} & = 0, \label{eq:NS-2}\end{aligned}$$ where $\sigma_{ij}^* = -p \delta_{ij}+2 \nu D_{ij}$ is the reduced Cauchy stress tensor of the fluid, $p$ the static or reduced pressure, $D_{ij}$ the rate-of-deformation tensor, $\nu$ the assuredly constant and uniform kinematic viscosity, and $g_i$ the components of the acceleration of gravity ($g_1=g_2=0$ and $g_3=-g$). Inside the fluid domain denoted by ${{\mathcal V}}$, no-slip boundary conditions are imposed on all cavity walls: the tubular side-wall, the bottom end-wall in steady rotation, see Fig. \[fig:sketch\]. The mathematical expression of the no-slip condition on the tubular side-wall simply reads $$\label{eq:no-slip-tubular}
u(r=R,\theta,z,t) = v(r=R,\theta,z,t) = w(r=R,\theta,z,t) = 0,
\quad 0 \leq \theta \leq 2\pi, \ 0\leq z \leq H.$$ The flow is driven by imposing a prescribed angular velocity distribution of the bottom end-wall, which transfers its kinetic energy to the fluid above. The details regarding the imposition of this Dirichlet boundary condition for the velocity field at the lid are discussed in the next section \[sec:lid\]. The top surface is left open and is modeled as a flat, fixed and clean free surface. The details regarding the imposition of this stress-free condition on the free surface are discussed in Sec. \[sec:free-surface\]. As mentioned in the Introduction, Sec. \[sec:general-considerations\], two parameters that determine completely the flow state are the height-to-radius aspect ratio $\Lambda=H/R$ and the Reynolds number ${\textrm{Re}}=R^2\Omega_0/\nu$, based on the maximal prescribed angular velocity $\Omega_0$ of the bottom end-wall.
In the sequel, the length, time, velocity, vorticity, helicity, streamfunction, (reduced) pressure and kinetic energy, and enstrophy are non-dimensionalized with respect to the reference scales $R$, $\Omega_0^{-1}$, $R\Omega_0$, $\Omega_0$, $R\Omega_0^2$, $R^3\Omega_0$, $R^2\Omega_0^2$, $\Omega_0^2$ respectively.
Angular velocity distribution {#sec:lid}
-----------------------------
As already mentioned in Bouffanais [@bouffanais06:_large] for the study of the lid-driven cubical cavity flow, imposing a given angular velocity distribution on the bottom end-wall of a cavity is not an easy task numerically. Indeed, imposing a constant angular velocity profile leads to a singularity (discontinuous behavior in the velocity boundary conditions) at the circular edge between the bottom end-wall and the tubular side-wall, see Fig. \[fig:sketch\]. Without adequate treatment, this discontinuous behavior will undermine the convergence and the accuracy of any numerical method in the vicinity of the lid. The same remedy as in the lid-driven cubical cavity problem in [@bouffanais06:_large] is used here for the same reasons and with analogous justifications. A regularized angular velocity profile is employed by prescribing the following high-order polynomial expansion which vanishes along its first derivatives on the circular edge $$\label{eq:angular-velocity-distribution}
{\bm \Omega}(r,\theta,z=0,t) = \Omega_0 \left[ 1-\left(\frac{r}{R}
\right)^{16} \right]^2 \, \mathbf{e}_z,$$ which leads to the following expressions in Cartesian coordinates of the components of the prescribed velocity field on the bottom end-wall $$\begin{aligned}
u(x,y,z=0,t) = u_x(x,y,z=0,t) & = -y \Omega_0 \left[ 1-\left(
\sqrt{x^2+y^2}/R\right)^{16} \right]^2,\label{eq:velocity-distribution-1}\\
v(x,y,z=0,t) = u_y(x,y,z=0,t) & = +x \Omega_0 \left[ 1-\left(
\sqrt{x^2+y^2}/R\right)^{16} \right]^2,\label{eq:velocity-distribution-2}\\
w(x,y,z=0,t) = u_z(x,y,z=0,t) & =
0,\label{eq:velocity-distribution-3}\end{aligned}$$ where $x=(r,\theta=0)$ and $y=(r,\theta=\pi/2)$. This profile flattens very quickly near the circular edge $(r/R=1,\theta,z/H=0)$ while away from it, it grows rapidly to a constant value $\Omega_0$ of the angular velocity over a short distance. The highest polynomial degree of this distribution is 32. Such high-order polynomial expansions lead to steep velocity gradients in the vicinity of the circular edge of the bottom end-wall. The grid refinement, in terms of spectral element distribution near the disk will be presented in greater details in Sec. \[sec:computational\]. One of the constraint in the grid design is to ensure the proper resolution of the lid velocity distribution by the spectral element decomposition.
Free-surface modeling {#sec:free-surface}
---------------------
The analysis of a two-phase flow is based on the coupled hydrodynamics interactions between adjacent layers over a broad range of space and time scales. This analysis can be significantly simplified if the dynamics of the interface is almost entirely dependent—from the hydrodynamics and physico-chemistry viewpoints—on one phase, a liquid phase, and almost independent of the dynamics of the second phase, a gas phase. Based on this hypothesis, the surface is said to be *free*. Consequently, the two fluid phases can only exert constant normal stresses. Sarpkaya in his review entitled “Vorticity, free surface and surfactants” [@sarpkaya96:_vortic] gives a clear characterization and definition of a free surface: “Although, the exterior of a free surface is free from externally imposed shear, the interior is not necessarily free from the shear generated internally. In fact, surface deformations and contaminants give rise to surface-gradients and tangential stresses in the internal side of the bounding interface. From a mathematical viewpoint, a free surface means that the density and the viscosity of the upper fluid are zero and that the existence of a continuum above the interface is inconsequential. From a practical point of view, the free surface means that the dynamics of the continuum above the interface has negligible influence on the lower phase, a free surface is a simplifying approximation for an *almost free surface*.”
In the present study, the modeling of the interface between the fluids in the cylindrical cavity as a free surface is supplemented by an additional simplifying approximation: the free surface is supposed to remain flat and fixed all along the dynamic range of investigation. In general, the dynamics of the free surface depends on the non-dimensional Froude number defined here as $$\label{eq:froude-number}
\textrm{Fr}=\frac{R^2\Omega_0^2}{gH},$$ which measures the relative importance of the inertial effects compared to the stabilizing gravitational effects. Therefore, assuming a flat free surface corresponds mathematically to a Froude number identically zero. As a consequence and in consistency with the latter assumption, the axial component of the velocity $w=u_z$ needs to vanish at the free surface $z/H=1$ $$\label{eq:w}
w(x,y,z=H,t) = 0,\qquad x^2+y^2 \leq R^2,$$ thereby expressing the kinematic boundary condition at the free surface. This latter condition on the axial velocity $w$ is to be supplemented with the stress-free condition at the free surface $$\label{eq:stress-free}
\sigma^*_{ij}\hat{n}_j = -p \delta_{ij}\hat{n}_j + 2\nu D_{ij}\hat{n}_j = 0,$$ where $\hat{\mathbf{n}}$ is the local outward unit vector normal at the free surface, which in the present particular situation is the unit normal vector $\mathbf{e}_z$ in the $z$-direction. Consequently, the stress-free condition reduces to $\sigma^*_{i3}=\sigma^*_{iz}=0$, $i=1,2,3$ at $z=H$, and is explicitly stated as $$\begin{aligned}
\sigma^*_{13} = 2 \nu D_{13} &= 0 ,\\
\sigma^*_{23} = 2 \nu D_{23} &= 0 ,\\
\sigma^*_{33} = -p+2\nu D_{33} &= 0 ,\end{aligned}$$ which under the zero-deformation condition , simplifies to $$\begin{aligned}
\left(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x}
\right) = \frac{\partial u}{\partial z}&=0,\label{eq:stress-free-1}\\
\left(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y}
\right) = \frac{\partial v}{\partial z}&=0,\label{eq:stress-free-2}\\
-p+2\nu \frac{\partial w}{\partial z} &=0.\label{eq:stress-free-3}\end{aligned}$$
Computational approach {#sec:computational}
======================
Space discretization {#sec:space-discretization}
--------------------
The Navier–Stokes equations –, supplemented with the boundary conditions , –, , –, constitute the set of governing equations for this free-surface swirling flow to be discretized and ultimately solved. The numerical method treats Eqs. – within the weak Galerkin formulation framework. The spatial discretization uses Lagrange–Legendre polynomial interpolants. The reader is referred to the monograph by Deville [@deville02:_high] for full details. The velocity and pressure are expressed in the $\mathbb{P}_N-\mathbb{P}_{N-2}$ functional spaces where $\mathbb{P}_N$ is the set of polynomials of degree lower than $N$ in each space direction. This spectral element method avoids the presence of spurious pressure modes as it was proved by Maday & Patera [@maday89:_spect_navier_stokes] and Maday [@maday93:_analy_stokes_applic]. The quadrature rules are based on a Gauss–Lobatto–Legendre (GLL) grid for the velocity nodes and a Gauss–Legendre grid (GL) for the pressure nodes. The spectral element grid used for all simulations is presented on Fig. \[fig:grid-cuts\], in the particular case $\Lambda=1$. This mesh comprises 440 spectral elements distributed into 10 cylindrical layers of different heights, but all made of the same distribution of 44 spectral elements, see Fig. \[fig:grid-cuts\] (right). In order to resolve the boundary layer along the tubular side-wall, the Ekman shear layer above the rotating bottom end-wall and the surface shear layer below the free surface, the spectral elements are unevenly distributed as can be seen in Figs. \[fig:grid-cuts\]. The choice of polynomial degree in the three space directions, defining the inner GLL and GL grid into each spectral element is deferred to Sec. \[sec:convergence\].
![Typical meshes view used throughout this study. Left: half-meridional grid. Center: spectral-element grid in any plane normal to the $z$-direction. Right: Three-dimensional grid comprising 10 cylindrical layers of nonuniform heights made of 44 spectral elements each. Case $\Lambda=1$.[]{data-label="fig:grid-cuts"}](cylindrical-meridional-grid.eps "fig:"){width="32.00000%"} ![Typical meshes view used throughout this study. Left: half-meridional grid. Center: spectral-element grid in any plane normal to the $z$-direction. Right: Three-dimensional grid comprising 10 cylindrical layers of nonuniform heights made of 44 spectral elements each. Case $\Lambda=1$.[]{data-label="fig:grid-cuts"}](cylindrical-circular-grid.eps "fig:"){width="32.00000%"} ![Typical meshes view used throughout this study. Left: half-meridional grid. Center: spectral-element grid in any plane normal to the $z$-direction. Right: Three-dimensional grid comprising 10 cylindrical layers of nonuniform heights made of 44 spectral elements each. Case $\Lambda=1$.[]{data-label="fig:grid-cuts"}](cylindrical-3D-grid.eps "fig:"){width="32.00000%"}
The essential Dirichlet boundary conditions—homogeneous for ${{\mathbf u}}$ on the tubular side-wall as expressed by Eq. , homogeneous for $w=u_z$ on the free surface as expressed by Eq. , and non-homogeneous for ${{\mathbf u}}$ on the rotating bottom end-wall as expressed by Eqs. –—are embodied into the choice of test and trial functions chosen for the velocity field.
The stress-free condition at the free surface, further expressed by Eqs. –, appears as a homogeneous natural boundary condition in the weak Galerkin framework. This central point is discussed in detail and in a more general framework, in Bodard [@bodard06:_solut], where a non-homogenous natural boundary condition is accounted for in the weak formulation of the problem. Based on this previous analysis, the treatment of the present stress-free condition at the free surface appears straightforward and is automatically incorporated into the weak formulation of the problem.
Borrowing the notation from Deville [@deville02:_high], the semi-discrete Navier–Stokes equations resulting from space discretization are $$\begin{aligned}
{{\mathbf M}}\frac{{\text{d}}\underline{{{\mathbf u}}}}{{\text{d}}t}+ {{\mathbf C}}\underline{{{\mathbf u}}}
+\nu \AA \underline{{{\mathbf u}}} -{{\mathbf D}}^T \underline{p}&=0,\label{eq:odes}\\
-{{\mathbf D}}\underline{{{\mathbf u}}} &=0\label{eq:constr}.\end{aligned}$$ The diagonal mass matrix ${{\mathbf M}}$ is composed of three blocks, namely the mass matrices $M$. The global vector $\underline{{{\mathbf u}}}$ contains all the nodal velocity components while $\underline{p}$ is made of all nodal pressures. The matrices $\AA$, ${{\mathbf D}}^T$, ${{\mathbf D}}$ are the discrete Laplacian, gradient and divergence operators, respectively. The matrix operator ${{\mathbf C}}$ represents the action of the non-linear term written in convective form $\underline{{{\mathbf u}}}\cdot {{\bm \nabla}}$, on the velocity field and depends on $\underline{{{\mathbf u}}}$ itself. The semi-discrete equations constitute a set of non-linear ordinary differential equations subject to the incompressibility condition .
Time integration
----------------
The time discretization of the semi-discrete set of governing equations – is the same as the one previously used in Bouffanais [@bouffanais06:_large] and Habisreutinger [@habisreutinger06]. We only briefly recall here the fundamentals of the method. The set of semi-discrete equations – is discretized in time using finite-difference schemes in a decoupled approach. The computation of the linear Helmholtz problem—corresponding to the stiffness matrix $\AA$—is integrated based on an implicit backward differentiation formula of order 2, the nonlinear convective term—corresponding to the operator ${{\mathbf C}}$—is integrated based on a relatively simple extrapolation method of order 2, introduced by Karniadakis [@karniadakis91:_high_navier], see Bouffanais [@bouffanais07:_simul] for full details.
Convergence tests {#sec:convergence}
-----------------
In order to demonstrate the spatial and temporal convergence of the simulation method, time series data have been analyzed, while varying separately the time-step $\Delta t$ and the polynomial degree $N$ of the GLL basis in each space direction, at the upper bound in Reynolds number ${\textrm{Re}}=6\,000$, and with $\Lambda=H/R=1$. As no experimental, nor numerical reference results are available for the present problem, three integral and one local quantities have been computed and compared. These three integral quantities are the total kinetic energy $Q$, enstrophy $E$ and helicity ${{\mathcal H}}$ of the flow, which definitions are recalled $$\begin{aligned}
Q & = \frac{1}{2} \int_{{\mathcal V}}{{\mathbf u}}\cdot {{\mathbf u}}\, {\text{d}}{{\mathcal V}},\\
E & = \frac{1}{2} \int_{{\mathcal V}}{\bm \omega}\cdot {\bm \omega}\, {\text{d}}{{\mathcal V}},\\
{{\mathcal H}}& = \int_{{\mathcal V}}{{\mathbf u}}\cdot {\bm \omega}\, {\text{d}}{{\mathcal V}}.\end{aligned}$$ The local quantity monitored is the axial velocity component $w=u_z$ at the point $\Pi_0$ of coordinates $(r/R=0,z/H=0.75)$, located along the cylinder axis. The location choice of this monitoring point is motivated by the study of Piva & Meiburg [@piva05:_stead] for a very similar configuration but at smaller Reynolds numbers. They show that in the vicinity of this point, the axial velocity component reaches a local maximum. Given the relatively high Reynolds number of our benchmark simulation, a quite long transient—approximately 500 time units in $\Omega_0^{-1}$ units—is observed. Performing convergence tests involving a simulation time of the order of this transient time would simply be prohibitive. Consequently, it was chosen to assess the convergence after only 50 time units of simulations, which corresponds to the appearance of the first vortex-breakdown recirculation bubble in the fluid domain. First, the spatial convergence is assessed by varying the polynomial degree in the range $6\leq N \leq 10$, while keeping the time-step values constant $\Delta t= 0.0025$. Results are reported in Table \[tab:spatial-convergence\], and suggest that the spatial convergence is achieved using a polynomial degree $N=8$ in all three space directions. This value is used for all the direct numerical simulations presented in the sequel, except for one single case corresponding to the steady laminar case $(\Lambda = 1,
{\textrm{Re}}=900)$, for which $N=7$ is chosen.
$N$ $Q$ $E$ ${{\mathcal H}}$ $w(\Pi_0)$
----- --------------- --------------- ------------------ ---------------
6 `2.02745e-02` `2.11900e+01` `1.29876e-01` `3.28923e-03`
7 `2.08244e-02` `2.18923e+01` `1.33612e-01` `3.40034e-03`
8 `2.19036e-02` `2.37953e+01` `1.66448e-01` `3.81373e-03`
9 `2.19034e-02` `2.37957e+01` `1.66450e-01` `3.81376e-03`
10 `2.19035e-02` `2.37955e+01` `1.66447e-01` `3.81375e-03`
: Spatial convergence analysis for the case $({\textrm{Re}}=6\,000,\Lambda=1)$ with $\Delta t=
0.0025\,\Omega_0^{-1}$. $Q$ in $R^3(R\Omega_0)^2$ units, $E$ in $R^3\Omega_0^2$ units, ${{\mathcal H}}$ in $R^4\Omega_0^2$ units, and $w$ in $R\Omega_0$ units. Instant $t=50\Omega_0^{-1}$.[]{data-label="tab:spatial-convergence"}
Finally, the temporal convergence is assessed by varying the time-step, while keeping the polynomial degree constant at the value $N=8$, in agreement with the earlier spatial convergence analysis. Results are reported in Table \[tab:temporal-convergence\] below, and suggest that the temporal convergence is achieved using a time-step $\Delta t =0.0025$. At a reduced Reynolds number compared to the one employed for this convergence analysis ${\textrm{Re}}=6\,000$, greater values of the time-step have been chosen in relation with the more laminar nature of the flow without affecting the convergence of the simulations.
$\Delta t$ $Q$ $E$ ${{\mathcal H}}$ $w(\Pi_0)$
------------ --------------- --------------- ------------------ ---------------
0.0050 `2.08574e-02` `2.14302e+01` `1.36342e-01` `3.49221e-03`
0.0035 `2.11896e-02` `2.25311e+01` `1.48303e-01` `3.61923e-03`
0.0025 `2.19036e-02` `2.37953e+01` `1.66448e-01` `3.81373e-03`
0.0010 `2.19034e-02` `2.37960e+01` `1.66446e-01` `3.81379e-03`
: Temporal convergence analysis for the case $({\textrm{Re}}=6\,000,\Lambda=1)$ with $N=8$. $\Delta t$ in $\Omega_0^{-1}$ units, $Q$ in $R^3(R\Omega_0)^2$ units, $E$ in $R^3\Omega_0^2$ units, ${{\mathcal H}}$ in $R^4\Omega_0^2$ units, and $w$ in $R\Omega_0$ units. Instant $t=50\Omega_0^{-1}$.[]{data-label="tab:temporal-convergence"}
Numerical simulations and results {#sec:numerical}
=================================
General physical characteristics of the flow {#sec:general}
--------------------------------------------
The central characteristics of the flow in a closed cylindrical container with a bottom rotating end-wall is a large recirculation of the fluid. The features of the intense shear layer induced by the rotation of the bottom wall can be obtained from the analogy with the analysis by von Kármán for the flow generated by a spinning plate in an unbounded fluid domain, see the review by Zandbergen & Dijkstra [@zandbergen87:_vonkar] for full details. The rotation of the bottom wall has a suction effect on the fluid in the near-axis region and a pumping effect, while accelerating the fluid radially outwards in an Ekman shear layer of thickness $O({\textrm{Re}}^{-1/2})$. In the framework of our problem, this Ekman layer is bounded by the tubular cylinder side-wall, which forces the recirculation of the fluid in the upward direction along the side-wall, and towards the top wall. As the fluid approaches the curved corner, the radial velocity contribution to the kinetic energy is progressively transformed into an axial velocity contribution to the kinetic energy. It has been observed that the fluid turns and subsequently spirals upward along the tubular side-wall.
As mentioned in Sec. \[sec:open-swirling-flow\], replacing the fixed top solid wall with a free surface changes significantly the physics of the flow and the recirculation mechanisms. In absence of tangential stresses at the free surface, the boundary layer is replaced by a surface layer in the sense of Shen [@shen99; @shen00:_turbul]. In addition, the inward spiraling fluid elements conserve their angular momentum at the free surface. When the related centrifugal force is large enough to balance the radial pressure gradient, the flow separates from the free surface and leads to the generation of a vortex breakdown bubble. The most striking difference between the flow patterns observed in the present open cylinder case and compared to the close cylinder one are the possible appearances of recirculation bubbles, which are generally attached to the free surface. Such flow patterns are simply impossible in presence of the no-slip condition imposed on the top wall in the closed cylinder case. More precisely, Iwatsu [@iwatsu05:_numer] determined 24 different flow states in the steady regime according to the meridional streamline patterns observed. Spohn [@spohn98:_exper] summarized those flow states in a simplified bifurcation diagram in the $(\Lambda, {\textrm{Re}})$ space. Based on his extensive and comprehensive study, Iwatsu [@iwatsu05:_numer] came out with a more detailed and complex bifurcation diagram.
In subsequent studies, the stability of those steady axisymmetric flows were investigated. Young [@young95:_period], Miraghaie [@miraghaie03:_flow], Lopez [@lopez04:_symmet], and Lopez & Marques [@lopez04:_mode] observed unstable azimuthal modes which are triggered at different values of the Reynolds number depending on the nature “shallow” ($\Lambda < 1$) or “deep” ($\Lambda >1$) of the system. Valentine & Jahnke [@valentine94:_flows], Lopez [@lopez95:_unstead], and Br[ø]{}ns [@brons01:_topol] associated the axisymmetry breakage to instability modes. These modes appear following a Hopf bifurcation which generally occurs at relatively high Reynolds number.
Cases studied {#sec:cases}
-------------
As mentioned in the previous sections, the physics of these free-surface swirling flows depends critically on the Reynolds number. Nevertheless, the height-to-radius aspect ratio $\Lambda$ also has considerable impact on the observed nature of the flow. Very often, situations corresponding to extreme values of $\Lambda$ have been studied, as they generally lead to simplified flow mechanisms. For instance, shallow systems ($\Lambda<1$) are often referred to as “rotor-stator” configurations, in which the fluid is almost in a state of solid-body rotation. On the other hand, deep systems associated with large values of $\Lambda$, generate recirculation bubbles away from the free surface and generally on the cylinder axis. Consequently, systems corresponding to values of $\Lambda$ close to the unity are intermediate in the sense that the physics of the flow observed is a complex combination of the general features reported for the shallow and deep systems.
Case $\Lambda=H/R$ Time-step $\Delta t$ Time evolution Vortex breakdown
------- ---------- --------------- ---------------------- ---------------- ------------------------------
$(a)$ $900$ $1.0$ 0.0050 steady one attached bubble
$(b)$ $1\,500$ $1.0$ 0.0050 steady one attached toroidal bubble
$(c)$ $6\,000$ $1.0$ 0.0025 unsteady complex dynamics
$(d)$ $2\,000$ $1/3$ 0.0040 steady two long attached bubbles
$(e)$ $2\,000$ $3.0$ 0.0040 steady one short detached bubble
: Parameters and characteristics of the cases considered. The time-step $\Delta t$ is expressed in $\Omega_0^{-1}$ units.[]{data-label="tab:parameters"}
![Time history of the volume integral of the kinetic energy $Q$ of the flow, in $R^5\Omega_0^2$ units for cases $(a)$–$(e)$.[]{data-label="fig:T-K-O-H"}](T-K-O-H.eps){width="60.00000%"}
The details related to the five cases considered in this article are summarized in Table \[tab:parameters\]. The primary focus is on the flow defined by ${\textrm{Re}}=6\,000$ and $\Lambda=1$, and corresponding to case $(c)$. The value of the Reynolds number is intentionally set to a high value compared to previous studies—the highest to our knowledge, in order to obtain fields of a relative intensity at the free surface. The choice of the value of $\Lambda$ follows the earlier comment on systems being intermediate between shallow and deep. This central case $(\Lambda=1,{\textrm{Re}}=6\,000)$ is supplemented with four secondary cases described in Table \[tab:parameters\]. The study of those secondary flows is of prime importance for the understanding of the complex dynamics of the primary case $(\Lambda=1,{\textrm{Re}}=6\,000)$.
In terms of initial conditions, the steady rotation is impulsively started from a quiescent fluid state for all cases presented in the sequel. At this point, it is worth noting the timescales of the evolution of these flows. Figure \[fig:T-K-O-H\] displays the time history of the volume integral of the kinetic energy of the flow $Q$. For all cases except case $(c)$, the flow reaches a steady state after a given time scale, which is, as expected, shorter for shallow systems. Case $(c)$ leads to an unsteady flow which does not display any oscillatory evolution. The value of the Reynolds number for this case is large enough to produce a non-trivial evolution of the recirculation zones as will be seen in the sequel.
The time histories of the volume integral of the kinetic energy $Q$ for the five cases $(a)-(e)$ can be compared to the ones, reported in Bouffanais [@bouffanais07:_simul] for the closed swirling flow problem with $\Lambda=2.5$, see Fig B.4. It should be noted that for a fixed value of $\Lambda$, the total kinetic energy $Q$ of the flow decreases with the Reynolds number for the closed cylinder case, while it increases in the open cylinder case. This decreasing trend for $Q$ in the closed cylinder case can easily be resolved by transposing the analysis given by Leriche & Gavrilakis [@leriche00:_direc] in their study of the closed lid-driven cubical cavity flow. Leriche & Gavrilakis argue that the most significant part of the kinetic energy of the flow is contained in the viscous layer developing on the driving wall. Consequently, the total energy varies like the energy contained in this viscous layer which can approximately be expressed as $U_0^2{{\mathcal V}}{\textrm{Re}}^{-1/2}$, where $U_0$ is the characteristic velocity of the driving wall and ${{\mathcal V}}$ the volume of the cavity. Such argument and estimate can easily be transposed for the closed swirling flows, and explains the decreasing trend for $Q$ with respect to . Furthermore this argument is confirmed by the measurements of the kinetic energy $Q(L_1)$ of the cylindrical layer of fluid $L_1$ located right above the spinning disk and of height $0.015H$, reported in Table \[tab:Q-E\]. This thin layer of fluid which only represents $1.5\%$ of the total volume of fluid, contributes for approximately $10\%$ to the total kinetic energy of the flow. Concurrently, its contribution to the total kinetic energy of the flow decreases with the Reynolds number.
[lccc]{} & $Q(L_1)$ & $E(L_1)$ & $E_z(L_1)$\
$900$ & `7.42243e-03` & `3.04128e+00` & `3.99817e-01`\
$1\,500$ & `7.05011e-03` & `4.46885e+00` & `4.03147e-01`\
$6\,000$ & `5.71943e-03` & `1.18305e+01` & `4.25622e-01`\
\
& $Q(L_{10})$ & $E(L_{10})$ & $E_z(L_{10})$\
$900$ & `3.88037e-04` & `6.14708e-03` & `5.99848e-03`\
$1\,500$ & `5.47285e-04` & `9.04519e-03` & `8.25633e-03`\
$6\,000$ & `1.21966e-03` & `6.43344e-02` & `2.00737e-02`\
On the contrary, a reverse trend is observed for the variations of $Q$ with respect to in the open swirling flow. It therefore requires another physical justification. Nevertheless, the previous energetic argument associated with the viscous layer still holds for the viscous layer above the spinning disk and near the tubular side-wall in the open cylinder swirling flow. Below the surface at $z=H$, the viscous layer in the closed cylinder case is replaced by an intense shear layer. As mentioned in Sec. \[sec:free-surface\], in the present flat-free-surface problem, the axial vorticity $\omega_z$ is the only component of the vorticity field which is not vanishing at the free surface and it provides a measure of the internal shear at the free surface. The part $E_z$ of the enstrophy associated with the axial vorticity component is measured in the cylindrical layer of fluid $L_{10}$, of height $0.02H$ and located below the free surface. Results are reported in Table \[tab:Q-E\] and clearly show a significant increase of $E_z$ with respect to in the layer $L_{10}$, while it is almost constant in $L_1$. These results allow us to infer that the shear layer below the free surface becomes more and more intense and energetic—see $Q(L_{10})$— when increasing the Reynolds number. But this observed energetic trend of the free-surface shear layer is not the only factor responsible for counterbalancing the decreasing trend of the viscous layers. The internal structure of the free-surface layer is itself physically different as reported by Shen [@shen99; @shen00:_turbul]. The so-called surface layer corresponds to a thin region adjacent to the free surface characterized by fast variations of the tangential vorticity components. This surface layer is caused by the dynamic zero-stress boundary conditions at the free surface and lies inside a thicker blockage (or “source”) layer, which is due to the kinematic boundary condition at the free surface. The importance of the outer blockage layer is manifested mainly in the redistribution of the kinetic energy, in the increase of the kinetic energy of the tangential velocity components at the expense of the kinetic energy of the axial velocity component. This point is to be further discussed in Sec. \[sec:unsteady\], where a comprehensive comparison of the flow below the surface $z=H$ for the case $(c)$ with a free surface and the equivalent closed case is given.
Physical description of flow states {#sec:physical}
-----------------------------------
### Steady flows {#sec:steady}
As a first step, we present the two steady flows for $\Lambda=1$ at ${\textrm{Re}}=900$ and $\ 1\,500$ corresponding to cases $(a)$ and $(b)$ respectively. Figure \[fig:streamlines-900-1500\] displays the streamlines of these flows into any meridional plane. Both of these flows present a large axisymmetric vortex breakdown bubble attached to the free surface in agreement with the experimental results from Spohn [@spohn98:_exper] summarized in their bifurcation diagram. These recirculation zones are characteristic of these swirling flows due to the conjugate effects of the centrifugal force and the overturning flow induced by the presence of the tubular side-wall. The central difference between the low-Reynolds-number cases $(a)$ and $(b)$ is the shape of the recirculation, which becomes toroidal after leaving the axis when the Reynolds number is increased from $900$ up to $1\,500$. These results can be further validated by comparing them to the experimental results (dye visualizations) obtained by Piva for ${\textrm{Re}}= 1\,120$. Finally, case $(a)=(\Lambda=1,{\textrm{Re}}=900)$ has also been computed by Piva & Meiburg [@piva05:_stead].
![Contours of streamlines in a meridional plane, case $\Lambda=1$. Left: case ${\textrm{Re}}= 900$; Right: case ${\textrm{Re}}=1\,500$. The 30 contours are non-uniformly spaced for visualization purposes, 20 equally-spaced negative contours and 10 equally-spaced positive contours for ${\textrm{Re}}=900$ and ${\textrm{Re}}=1\,500$.[]{data-label="fig:streamlines-900-1500"}](SF-H1-Re900-single-meridian-pT-BW.eps "fig:"){width="39.00000%"} ![Contours of streamlines in a meridional plane, case $\Lambda=1$. Left: case ${\textrm{Re}}= 900$; Right: case ${\textrm{Re}}=1\,500$. The 30 contours are non-uniformly spaced for visualization purposes, 20 equally-spaced negative contours and 10 equally-spaced positive contours for ${\textrm{Re}}=900$ and ${\textrm{Re}}=1\,500$.[]{data-label="fig:streamlines-900-1500"}](SF-H1-Re1500-single-meridian-pT-BW.eps "fig:"){width="39.00000%"}\
As a second step, the contours of the radial, azimuthal and axial velocity components in any meridional plane are given in Fig. \[fig:vr-vz-vt-gamma-900-1500\]. These data are supplemented with the contours of the axial component of the angular momentum $\Gamma=r u_\theta$ still in Fig. \[fig:vr-vz-vt-gamma-900-1500\], extreme right column. The interest for $\Gamma$ lies in the fact that it plays the role of a streamfunction for the part of the velocity field comprised in any meridional plane, see Bragg & Hawthorne [@bragg50:_some] and Keller [@keller96] for full details. Therefore, the contours of $\Gamma$ deliver the intersection of vortex surfaces with the corresponding meridional plane where they are drawn, and as such provide us with the local direction of the meridional vorticity field. One can notice from the velocity components and axial angular momentum component that the meridional structure of these flows is far from being trivial. It consists of an intense boundary layer above the spinning bottom end-wall that is turned into the interior by the presence of the tubular side-wall, forming a shear layer having a jet-like velocity profile in the azimuthal direction. The contour lines of the axial component of the angular momentum shown in Fig. \[fig:vr-vz-vt-gamma-900-1500\] (extreme right column) simply represent the vortex lines, which all emanate from the rotating disk; the structure of the shear layer is apparent. It is worth noting here that the vortex lines distribution at their origin varies like $r^2$. As a consequence of the regularized profile of angular velocity of the rotating disk—see Sec. \[sec:lid\], this distribution in $r^2$ is slightly affected in the vicinity of $r=R$. This regularization of this profile has the advantage of preventing the appearance of vortex lines terminating at the circular corner $(r=R,z=0)$. The overturning nature of these flows is also apparent in the vicinity of the tubular side-wall, which is the vortex surface corresponding to $\Gamma=0$, together with the cylinder axis. As non-zero azimuthal velocities are possible at the free surface, vortex lines emanating from the rotating end-wall have the option of terminating orthogonally to the free surface. This observation is one of the major difference with the closed cylinder swirling flow where all vortex lines have to terminate in the corner. Furthermore, the termination of vortex lines at the free surface is responsible for the possibility of having vortex breakdown bubbles being attached to the free surface as observed in Fig. \[fig:streamlines-900-1500\]. A careful analysis of Fig. \[fig:vr-vz-vt-gamma-900-1500\] (extreme right column) reveals that one vortex line marks the limit between an inner region comprising only vortex lines terminating at the free surface, and an outer region, where they terminate near the circular corner, like in the closed cylinder case. In summary, it appears that the main effect of this overturning flow is to bring high-angular-momentum fluid towards the cylinder axis.
The results for case $(a)=(\Lambda=1, {\textrm{Re}}=900)$ presented in Fig. \[fig:streamlines-900-1500\] (left) and Fig. \[fig:vr-vz-vt-gamma-900-1500\] (top row) show a qualitative good agreement with the numerical results of Piva & Meiburg [@piva05:_stead]. The differences related to the features of the recirculation bubbles for cases $(a)$ and $(b)$ have been discussed earlier. One can notice in Fig. \[fig:vr-vz-vt-gamma-900-1500\] (two first left columns), that the thicknesses of the intense radial velocity layer as well as the axial wall jet are reduced when the Reynolds number is increased from $900$ to $1\,500$, as expected. The contours of the axial velocity component reveal that the downward-directed flow induced by the suction effect of the Ekman layer, is more intense at higher Reynolds number. In addition, the region of the flow where $w=u_z$ has a negative extremum tends to move closer to the free surface when increasing . Regarding the vortex lines shown in the extreme right column, their bending towards the cylinder axis is more pronounced at the higher Reynolds number of $1\,500$. In relation with the previous analysis, this latter observation highlights the fact that more high-angular-momentum fluid is brought towards the axis when increasing .
As mentioned in Sec. \[sec:cases\], our primary interest lies in case $(c)=(\Lambda=1,{\textrm{Re}}= 6\,000 )$, thereby justifying the study of cases $(a)$ and $(b)$, having the same aspect ratio $\Lambda$ but corresponding to laminar cases. Nevertheless, the study of cases $(d)$ and $(e)$, which both correspond to “extreme” cases in terms of height-to-radius aspect ratio, illustrate some essential features of the open swirling flow. In case $(c)$ some of these features may prevail only in specific regions of the flow as it corresponds to an intermediate case between a shallow system characterized by case $(d)$ and a deep system characterized by case $(e)$. These features are as follows:
- solid-body rotation of the inner core region, predominantly for small $\Lambda$;
- radial jet of angular momentum at the free surface;
- lateral jet-like shear layer along the tubular side-wall;
![Contours of streamlines in a meridional plane. Left: case $(d):\ (\Lambda=1/3,{\textrm{Re}}= 2\,000)$; Right: case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$. The 30 contours are non-uniformly spaced for visualization purposes, 20 equally-spaced negative contours and 10 equally-spaced positive contours for $(d)$ and $(e)$.[]{data-label="fig:streamlines-H13-H3"}](SF-H13-Re2000-meridian-pT-BW.eps "fig:"){width="39.00000%"} ![Contours of streamlines in a meridional plane. Left: case $(d):\ (\Lambda=1/3,{\textrm{Re}}= 2\,000)$; Right: case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$. The 30 contours are non-uniformly spaced for visualization purposes, 20 equally-spaced negative contours and 10 equally-spaced positive contours for $(d)$ and $(e)$.[]{data-label="fig:streamlines-H13-H3"}](SF-H3-Re2000-meridian-pT-BW.eps "fig:"){width="39.00000%"}
The very different flow patterns developed by both cases $(d)$ and $(e)$ are highlighted by the very distinctive streamlines shown in Fig. \[fig:streamlines-H13-H3\]. The shallow case $(d)$ yields two vortex breakdown bubbles, which are off the cylinder axis while remaining attached to the free surface. The recirculation is more intense in the largest bubble, which is elongated enough to produce a recirculation of the fluid from the free surface all the way down to the rotating disk, and so forth. Consequently, this elongated bubble completely separate the outer region of the flow ($r/R\geq 0.5$) from the inner core, where the second less intense, vortex breakdown occurs. Comparatively, the recirculation in case $(e)$ is fairly limited. A vortex breakdown still occurs in this case, leading to the formation of a small on-axis bubble, which is detached from the free surface. One can notice from the bending of the streamlines near the axis at the height $z/H=0.8$, that another vortex breakdowns is in preparation—compare this effect to the similar effect on the streamlines prior the vortex breakdown in the closed swirling flow at ${\textrm{Re}}=1\,900$, see Bouffanais [@bouffanais07:_simul].
The shallow system $(d)=(\Lambda=1/3,{\textrm{Re}}=2\,000)$ possesses some very distinctive features as can be seen in Fig. \[fig:all-v-H13-H3\] (top row). The vortex lines for $r/R<0.4$ being aligned with the rotation axis, one can easily conclude that the flow is essentially in solid-body rotation in this inner core region of the cavity. The meridional flow in this inner part of the cavity has a very weak intensity as attested by the values close to zero of the contours of the axial and radial velocity components—solid contour lines are positive and negative contour lines being dashed. In contrast, for $r>0.4$ the primary recirculation of the flow is intense and predominates. The vortex lines bending is limited to this region and again is at the origin of the vortex breakdown appearing near $r/R=0.4$. The boundary layer on the rotating disk is limited to the region $r/R>0.5$ and the internal jet-like shear layer close the tubular side-wall has a structure quite different from the cases with $\Lambda=1$. Indeed in this shear layer, the axial velocity is relatively intense all along the tubular side-wall, unlike cases $(a)$, $(b)$ and $(e)$, where the axial velocity $u_z$ decreases rapidly with $z/H$. This observation is easily explained by the shorter height in the case studied, but has several considerable consequences on the flow itself. A more intense wall-jet implies a more intense angular momentum jet at the free surface, which facilitates the vortex breakdown phenomena. The intense angular momentum free-surface jet produces an elongated recirculation bubble located as seen earlier, around $r/R=0.4$. In this elongated bubble, the axial velocity field is globally positive, thereby producing an effect similar to the jet-like shear layer near the tubular side-wall. In turn it generates a secondary angular momentum free-surface wall jet responsible for the second vortex breakdown.
Increasing the height-to-radius aspect ratio to $\Lambda=3$, modifies considerably the flow dynamics as can be seen in Fig.\[fig:all-v-H13-H3\] (bottom row). It seems clear from the previous analysis for the shallow case $(d)$, that the influence of the free surface on the flow is more important when $\Lambda$ is small. The proximity between the driving disk, which generates the primary flow and the free surface with its specific boundary conditions, leads to the complex flow dynamics earlier explained. Conversely, for large values of $\Lambda$ the important distance between the spinning disk and the free surface is so important that it significantly reduces the effect of the presence of the free surface. The flow pattern presents *in fine* a structure very similar to the flow pattern observed in the closed cylinder swirling flow, except very close to the free surface. As mentioned earlier the recirculation bubble itself is fairly small and located on the cylinder axis likewise in the closed cylinder case. Finally, it is worth adding that the region of solid-body rotation is almost completely eliminated. Even the closest-to-the-axis vortex lines present some bending.
As a brief conclusion of the previous study of the steady and laminar free surface swirling flows, it appears that the choice $\Lambda=1$ for the height-to-radius aspect ratio of the cavity in presence of a free surface, ensures us to deal with a complex flow dynamics. Different mechanisms are in competition in different regions of the cavity, and in the end make the cases with $\Lambda=1$ physically more challenging and more interesting. This conclusion—valid in presence of a free surface—stops being valid for the closed cylinder swirling flow, and thereby explains the focus in the literature on cases with $\Lambda\geq 2$.
### Unsteady flow {#sec:unsteady}
For sufficiently small Reynolds number and irrespective of $\Lambda$, the basic flow state is stable. As noted by Lopez [@lopez04:_symmet], when is increased, the basic flow state loses stability via a variety of Hopf bifurcations. It is worth noting that when tends to infinity, the stream surfaces and vortex surfaces—giving the streamlines and vortex lines by intersection with a meridional plane—must coincide. At this point, the presence of a flat free surface poses problem because of the constraint of having orthogonal streamlines and vortex lines on it. This apparent paradox is unraveled by simply letting the free surface move. Nevertheless, we know from the experiments carried out by Spohn [@spohn98:_exper; @spohn93:_obser], that even at a ${\textrm{Re}}=6\,000$ the tangential flow is extremely intense compared to the normal one, leading to small free-surface deformations. It is very likely that these small amplitude deformations are not sufficient to solve our apparent paradox. At low Reynolds number, like those of cases $(a)$, $(b)$, $(d)$, and $(e)$, the viscosity acts on the velocity field to allow the latter condition of orthogonality to be fulfilled. But when the Reynolds number is increased, the action of viscosity and the limited deformation of the free surface are not sufficient to bring back the orthogonality of the two sets of lines. Therefore, the flow must either lose its axisymmetry or become unsteady in order to allow to drop the orthogonality condition. The experiments by Spohn [@spohn98:_exper] suggest that the open swirling flow first go through the unsteady path.
![Contours of streamlines in two orthogonal meridional planes, case $\Lambda=1$ and ${\textrm{Re}}=6\,000$. Top row: instantaneous flow; Bottom row: mean flow. Left column: meridional plane $y/R=0$; Right column: meridional plane $x/R=0$. The 30 contours are non-uniformly spaced for visualization purposes, 20 equally-spaced negative contours and 10 equally-spaced positive contours.[]{data-label="fig:orthogonal-meridian"}](SF-H1-Re6000-two-orthogonal-meridian-asymmetry-pT-BW.eps "fig:"){width="75.00000%"} ![Contours of streamlines in two orthogonal meridional planes, case $\Lambda=1$ and ${\textrm{Re}}=6\,000$. Top row: instantaneous flow; Bottom row: mean flow. Left column: meridional plane $y/R=0$; Right column: meridional plane $x/R=0$. The 30 contours are non-uniformly spaced for visualization purposes, 20 equally-spaced negative contours and 10 equally-spaced positive contours.[]{data-label="fig:orthogonal-meridian"}](SF-H1-Re6000-two-orthogonal-meridian-asymmetry-pT-Mean-Flow-BW.eps "fig:"){width="75.00000%"}
In this section, the study is focused on the unsteady swirling flow corresponding to case $(c)=(\Lambda=1,{\textrm{Re}}= 6\,000)$. To our knowledge, such transitional regime at this relatively high has never been investigated nor reported in the literature. At this Reynolds number the loss of axisymmetry in this flat-free-surface case is evident from the observation of the contours of streamlines in Fig. \[fig:orthogonal-meridian\] (top row). As the flow is unsteady, these recirculation bubbles are instantaneous and correspond to a flow sample taken in the statistically-steady regime for $t>600$ in $\Omega_0^{-1}$ units, see Fig. \[fig:T-K-O-H\]. The streamlines of this flow sample are represented in two orthogonal meridian planes corresponding to $y/R=0$ and $x/R=0$, in Fig. \[fig:orthogonal-meridian\] (top row). Once again, the loss of axisymmetry appears clearly from the complex and nonaxisymmetric structure of the recirculation bubbles. Compared to the laminar and steady cases $(a)$ and $(b)$, the recirculation bubbles have their own dynamics and evolution. In a common approach to such unsteady problems, this complex dynamics is analyzed by the means of an averaging process, which is supplemented with an analysis of instantaneous flow samples equally-spaced in time. The mean flow is obtained by averaging 500 flow samples corresponding to successive flow states extracted every 0.25 times units (or equivalently every 100 iterations). Subsequently, the root-mean-square (rms) fluctuations of flow fields are calculated using the same extracted flow samples and the mean flow field obtained earlier. The streamlines associated with the mean flow are shown in Fig. \[fig:orthogonal-meridian\] (bottom row). The streamlines of the mean flow reveal the existence of a toroidal recirculation bubble, located off the cylinder axis and more surprisingly detached from the free surface. The toroidal shape and off-axis location of the mean recirculation bubble is in agreement with the increased- trend observed with cases $(a)$ and $(b)$ in Sec. \[sec:steady\]. Regarding the detachment from the free surface of the mean bubble, it is more relevant here to notice that the instantaneous bubbles are still attached to the free surface. More precisely, one may notice two points:
- in the meridional plane $y/R=0$, a small recirculation zone appears attached to the rotating disk for $r/R\simeq 0.2$;
- in the meridional plane $x/R=0$, the recirculation bubble is stretched from the free surface $z/H=1$ down to $z/H=0.15$, in a radial position $r/R\simeq 0.4$.
These two observations remind the streamline patterns described in the case $(d)=(\Lambda =1/3 , {\textrm{Re}}= 2\,000)$, with a long bubble stretching from the free surface down to the driving disk. The previous analysis is further confirmed by the contours of the three velocity components and of the axial angular momentum for both an instantaneous flow sample and the mean flow, presented in Fig. \[fig:ur-ut-uz-6000\], in the meridional plane $x/R=0$. A careful analysis of the vortex lines for the instantaneous flow sample shows a bending in the whole meridian plane. This bending is very significant in the region $0.3 \leq r/R \leq 0.8$ and $0.4 \leq z/H
\leq 1$, which corresponds to the limit between the primary recirculation of the flow and the secondary recirculation bubble. On the contrary, the vortex lines structure of the mean flow is as expected much more regular. The inner core region of the flow $r/R
\leq 0.4$ displays a state of solid-body rotation. For both the instantaneous and mean flow, the jet-like shear layer along the tubular side-wall is turned into the interior of the flow by the free surface. Compared to the previous cases $(a)$, $(b)$, $(d)$, $(e)$, and also the closed swirling flow $(\Lambda=1, {\textrm{Re}}=6\,000)$ in Bouffanais [@bouffanais07:_simul], the structure of this shear layer at ${\textrm{Re}}=6\,000$ reveals the presence of an intense radial jet of angular momentum at the free surface.
The fluctuations of the flow with respect to its mean state have been calculated with the same flow samples as before. It should be noted that the fluctuation level corresponds to less than $5\%$ of the maximal intensity of the respective mean flow fields. Despite the relatively low level of fluctuation encountered, these fluctuations are very localized in space as can be seen in Fig. \[fig:rms-ur-uz-ut-gamma-6000\]. Similarly to the mean flow fields, the rms fluctuations of the velocity field and of $\Gamma$ appear to be slightly nonaxisymmetric. All the three velocity components present a noticeable level of fluctuation near the free surface for radii close to $0.4$. In this region, the free-surface radial jet of angular momentum reaches the inner flow, which is solid-body rotation. These fluctuations are therefore located in the vicinity of the stagnation point where the vortex breakdown is initiated. The rms-fluctuations of the vortex lines, $\Gamma$, are the highest in the corner region between the free surface and the tubular side-wall. It is in this corner, where the shear layer is turned into the interior by the presence of the free surface.
One-dimensional momentum budgets {#sec:one}
--------------------------------
This section is devoted to the careful analysis of the momentum balance for the radial, azimuthal and axial components. This study is performed along different radial and axial lines within the cavity. It is of interest to determine the predominant physical terms, which are responsible for the complex flow dynamics depicted in the previous sections. As noticed in these previous sections, the structure of the flow in the inner core region is far different from the one close to the tubular side-wall. Similarly, the flow above the rotating driving disk has properties, which are not comparable to the ones below the flat free surface. For the sake of conciseness, this momentum balance analysis is limited to cases $(a)-(c)$ for which $\Lambda=1$.
### General considerations {#general-considerations}
The numerical integration of the Navier–Stokes equations using the spectral element method as described in Sec. \[sec:computational\], is performed in Cartesian coordinates $(x,y,z)$ for the velocity components $(u,v,w)$. Nevertheless, the axisymmetric nature of the container and of the boundary conditions imposed to the flow suggests the use of cylindrical coordinates. Indeed, the different physical terms involved in the momentum equation represented here by the Navier–Stokes equations—nonlinear advective term, viscous strain, pressure gradient, etc.—are better apprehended when expressed in cylindrical coordinates. Accordingly, all vectors and physical terms are recast as functions of $(r,\theta,z )$, and for instance the velocity components are $(u_r,u_\theta,u_z)$.
The complete expression of the momentum equations in cylindrical coordinates reads $$\begin{aligned}
{\frac{\partial u_r}{\partial t}} + u_r {\frac{\partial u_r}{\partial r}} + \frac{u_\theta}{r} {\frac{\partial u_r}{\partial \theta}} -\frac{u_\theta^2}{r}+ u_z {\frac{\partial u_r}{\partial z}} & = -{\frac{\partial p}{\partial r}} +\frac{1}{{\textrm{Re}}} \left[\frac{\partial}{\partial r}\left(\frac{1}{r}{\frac{\partial (ru_r)}{\partial r}} \right)+\frac{1}{r^2} \frac{\partial^2 u_r}{\partial \theta^2} -\frac{2}{r^2}{\frac{\partial u_\theta}{\partial \theta}} + \frac{\partial^2 u_r}{\partial z^2} \right],\label{eq:momentum-1}\\
{\frac{\partial u_\theta}{\partial t}} + u_r {\frac{\partial u_\theta}{\partial r}} + \frac{u_\theta}{r} {\frac{\partial u_\theta}{\partial \theta}} +\frac{u_r u_\theta}{r}+ u_z {\frac{\partial u_\theta}{\partial z}} & = -\frac{1}{r}{\frac{\partial p}{\partial \theta}}+\frac{1}{{\textrm{Re}}} \left[\frac{\partial}{\partial r}\left(\frac{1}{r}{\frac{\partial (ru_\theta)}{\partial r}} \right) +\frac{1}{r^2} \frac{\partial^2 u_\theta}{\partial \theta^2}+\frac{2}{r^2}{\frac{\partial u_r}{\partial \theta}} + \frac{\partial^2 u_\theta}{\partial z^2} \right],\label{eq:momentum-2}\\
{\frac{\partial u_z}{\partial t}} + u_r {\frac{\partial u_z}{\partial r}} + \frac{u_\theta}{r}
{\frac{\partial u_z}{\partial \theta}} + u_z {\frac{\partial u_z}{\partial z}} & = -{\frac{\partial p}{\partial z}}+\frac{1}{{\textrm{Re}}}
\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r{\frac{\partial u_z}{\partial r}}
\right) +\frac{1}{r^2} \frac{\partial^2 u_z}{\partial \theta^2} +
\frac{\partial^2 u_z}{\partial z^2} \right],\label{eq:momentum-3}\end{aligned}$$ where successively appears, the velocity time derivative, the nonlinear advective term, the pressure gradient and the viscous strain. The central objective of this study is to compare the relative importance of some of these terms along different lines. Equation (resp. ) represents the momentum balance in the radial (resp. azimuthal) direction, and is analyzed along four radial lines at four different heights $z/H=0.03,\
0.64,\ 0.95,\ 1$, ranging from right above the rotating disk up to the free surface. Equation represents the momentum balance in the axial direction, and is analyzed along three different axial vertical lines at three radial positions $r/R=0.08,\ 0.48,\
0.98$, ranging from near the cylinder axis to near the tubular side-wall.
For the sake of simplicity, some of the terms appearing in – are identified and denoted specifically in Table \[tab:name-terms\]. In the sequel, the various graphs reporting the variations of these terms will use this nomenclature.
Name Expression Name Expression Name Expression
----------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------- --------------------------------------------------------------------------------------------------------------------------------------------------- -------- --------------------------------------------------------------------------------------------------------------------------------------------
\[-1ex\] NL$r1$ $\displaystyle{- u_r {\frac{\partial u_r}{\partial r}}}$ NL$t1$ $\displaystyle{-u_r {\frac{\partial u_\theta}{\partial r}}}$ NL$a1$ $\displaystyle{-u_r {\frac{\partial u_z}{\partial r}}}$
\[2ex\] NL$r2$ $\displaystyle{+\frac{u_\theta^2}{r}}$ NL$t2$ $\displaystyle{-\frac{u_r u_\theta}{r}}$ NL$a2$ $\displaystyle{-u_z {\frac{\partial u_z}{\partial z}}}$
\[2ex\] NL$r3$ $\displaystyle{-u_z {\frac{\partial u_r}{\partial z}}}$ NL$t3$ $\displaystyle{- u_z {\frac{\partial u_\theta}{\partial z}}}$ PG$a$ $\displaystyle{-{\frac{\partial p}{\partial z}}}$
\[2ex\] PG$r$ $\displaystyle{-{\frac{\partial p}{\partial r}}}$ VS$t1$ $\displaystyle{\frac{1}{{\textrm{Re}}}\left[\frac{\partial}{\partial r}\left(\frac{1}{r}{\frac{\partial (ru_\theta)}{\partial r}}\right)\right]}$ VS$a1$ $\displaystyle{\frac{1}{{\textrm{Re}}}\left[\frac{1}{r}\frac{\partial}{\partial r}\left(r{\frac{\partial u_z}{\partial r}}\right)\right]}$
\[2ex\] VS$r$ $\displaystyle{\frac{1}{{\textrm{Re}}} \left[\frac{\partial}{\partial r}\left(\frac{1}{r}{\frac{\partial (ru_r)}{\partial r}}\right)+\frac{\partial^2 u_r}{\partial z^2}\right]}$ VS$t2$ $\displaystyle{\frac{1}{{\textrm{Re}}}\left(\frac{\partial^2 u_\theta}{\partial z^2}\right)}$ VS$a2$ $\displaystyle{\frac{1}{{\textrm{Re}}}\left(\frac{\partial^2 u_z}{\partial z^2}\right)}$
\[2ex\]
: Name definitions of different terms appearing in the momentum budget equations –.[]{data-label="tab:name-terms"}
### Steady flows {#sec:momentum-steady}
As discussed in Sec. \[sec:steady\], the steady swirling flows are also fully axisymmetric. The first-order time derivative of the velocity fields is identically zero in Eqs. –. The axisymmetric property makes the velocity field independent of the azimuthal angle $\theta$, and consequently all partial derivatives with respect to this variable vanish. Therefore, the balance in the momentum equation solely involves the terms described in Table \[tab:name-terms\].
As a first step, the momentum balance in the radial direction is presented in Fig. \[fig:radial-900-1500\] for case $(a)$ (left column) and case $(b)$ (right column). At the free surface $z/H=1$ (top row), the viscous terms are insignificant, and the flow is driven by the radial pressure deceleration, which is mainly counterbalanced by the centrifugal acceleration NL$r2=u_\theta^2/r$ and to a certain extent by NL$r$1. This analysis at the free surface still holds below the free surface at $z/H=0.94$ and at $z/H=0.64$. At this latter height, both the radial pressure deceleration and the centrifugal acceleration NL$r2$ have a lower magnitude than at the free surface but their magnitude is less localized than at the free surface. As expected, above the disk, at $z/H=0.03$, all the terms have a higher magnitude and the momentum balance is more complex as only one single term NL$r3$ does not really contribute to the balance. The centrifugal acceleration NL$r2$ keeps its predominant position, but its maximum is now shifted towards the outer radial region, which corresponds to the region of highest angular momentum $0.8\leq r/R \leq 1$. The other acceleration term NL$r1$ becomes relatively important. Both of these acceleration terms are counterbalanced by the radial pressure gradient and now also by the viscous strain. The importance of the viscous strain at this height $z/H=0.03$ can easily be understood, as we are located in the viscous layer generated by the motion of the disk. One may add that in the inner core region of the flow, say $r/R<0.15$, the linear trend observed for both the centrifugal acceleration and the pressure deceleration are well-know features of a flow in solid-body rotation, as observed with the vertical vortex lines in Fig. \[fig:vr-vz-vt-gamma-900-1500\] (extreme right column).
The evolution of those momentum balances along radial lines at different heights does not really change when increasing from 900 up to $1\,500$. But some noticeable trends are observable. For instance, even if the viscous strain does not play a central role at the free surface, it is worth noting that its effect is increased with while conversely it is decreased when getting closer to the rotating bottom end-wall.
As a second step, we aim at analyzing the momentum balance in the azimuthal direction along radial lines at the same different heights as before. The results are presented in Fig. \[fig:azimuthal-900-1500\] for case $(a)$ (left column) and case $(b)$ (right column). It is important to note at this point that the two terms NL$t1$ and VS$t1$ involve partial derivatives with respect to the radial variable $r$. Given the fact that our solution is continuous and first-order differentiable within a spectral element and only continuous at the element edges, one expects some slight unphysical deformations of the plots associated with these two terms. Along the radial lines of interest, the spectral element edges are located at $r/R=0.2,\ 0.4,\ 0.6,\ 0.8,\ 0.97$. As a consequence, some rapid variations of the terms VS$t1$ and NL$t1$ are going to be simply disregarded in the coming discussions.
A rapid glance at all the plots in Fig. \[fig:azimuthal-900-1500\] allows to conclude that the importance of the azimuthal momentum transfers resides in the near bottom end-wall region. The magnitude of all terms is over ten times smaller at $z/H=0.64,\ 0.95,\ 1$, compared to $z/H=0.03$. Given the solid-body rotation in the inner core region of the flow $r/R<0.15$, most of the terms are vanishing small—excluding the unphysical values of VS$t1$.
At $z/H=0.03$, one can notice the vigorous action of the viscous strain term VS$t2$ which literally drives the fluid in the viscous Ekman layer. This driving viscous term is being compensated by the convective terms NL$t1$ and NL$t3$, and by the Coriolis term NL$t2$. As one gets closer to the corner between the rotating disk and the tubular side-wall, say $r/R\geq 0.8$, the interplay between the various terms is being reversed. The term VS$t2$, which is driving the fluid in the inner region of the cavity is now a dissipative term in the jet-like shear layer. Conversely, the convective term NL$t1$ becomes large and is driving the fluid in the shear layer. Very close to the tubular side-wall, this term starts being counterbalanced by the second viscous term VS$t1$.
The viscous driving effect of the term VS$t2$ becomes insignificant at $z/H=0.64$, but when $z/H$ is increased, VS$t2$ starts growing again to reach a local maximum value at the free surface, but with a magnitude slightly smaller than the two other nonlinear convective terms NL$t1$ and NL$t2$. Close to the free surface and at the free surface, the flow is primarily driven by the Coriolis term NL$t2$ together with the viscous term VS$t2$. Their global action is counterbalanced by the nonlinear convective term NL$t1$. This momentum balance at the free surface in the outer region $r/R\geq 0.4$ reflects the central effect of the free-surface jet of angular momentum. Again the viscous effects are more intense at the free surface when is increased from $900$ to $1\,500$, despite the presence of the kinematic viscosity term $1/{\textrm{Re}}$ in their definitions.
![Momentum balance in radial direction plotted along horizontal radial lines at four different vertical positions. First row: $z/H=1$; second row: $z/H=0.95$; third row: $z/H=0.64$; fourth row: $z/H=0.03$. Left column: case ${\textrm{Re}}=900$; Right column: case ${\textrm{Re}}=1\,500$. Case $\Lambda=1$. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:radial-900-1500"}](full-radial-budget-laminar.eps){width="60.00000%"}
![Momentum balance in azimuthal direction plotted along horizontal radial lines at four different vertical positions. First row: $z/H=1$; second row: $z/H=0.95$; third row: $z/H=0.64$; fourth row: $z/H=0.03$. Left column: case ${\textrm{Re}}=900$; Right column: case ${\textrm{Re}}=1\,500$. Case $\Lambda=1$. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:azimuthal-900-1500"}](full-azimuthal-budget-laminar.eps){width="60.00000%"}
![Momentum balance in axial direction plotted along vertical lines at three different radial positions. First row: $r/R=0.98$; second row: $r/R=0.48$; third row: $r/R=0.08$. Left column: case ${\textrm{Re}}=900$; Right column: case ${\textrm{Re}}=1\,500$. Case $\Lambda=1$. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:axial-900-1500"}](full-axial-budget-laminar.eps){width="60.00000%"}
As a last step for these two steady flows $(a)$ and $(b)$, we investigate the momentum transfer in the axial direction, but now along three different vertical lines corresponding to three different radii $r/R=0.08,\ 0.48,\ 0.98$. The graphs for the five different terms involved are reported in Fig. \[fig:axial-900-1500\]. We start from the top row, which is associated with the results for $r/R=0.98$ in the jet-like shear layer and which present the highest magnitudes of all radii considered. The lack of axial momentum transfers is clearly visible in the region $z/H\geq 0.4$ and even inexistent at the free surface. Conversely, in the corner between the rotating bottom end-wall and the tubular side-wall the flow is driven by the axial pressure gradient, and to some extent by the two nonlinear terms NL$a1$ and NL$a2$ independently. The viscous term VS$a1$ is primarily responsible for counterbalancing the driving pressure effects. The second viscous term VS$a2$ has a less important effect in terms of magnitude but is driving very close to the disk (viscous Ekman layer), when $z/H$ is increased it becomes dissipative as we are out of the Ekman layer but within the wall-jet shear layer.
Near the medium radial position $r/R=0.48$, the axial momentum transfers near the free surface show the relative importance of the two nonlinear terms NL$a1$ and NL$a2$, which are driving the fluid against the axial pressure gradient, which is negative as a consequence of the kinematic boundary condition imposing a vanishing axial velocity component at the free surface.
Near the cylinder axis, for $r/R=0.08$, the axial momentum transfers are limited and relatively simples. Nonlinear convective effects seem almost insignificant and the axial component of the flow is obtained from the balance between the axial pressure gradient and the two viscous terms: VS$a2$ in the bottom of the cavity and VS$a1$ in the top of the cavity.
### Unsteady flow {#unsteady-flow}
The momentum balance analysis developed in the previous section for the two steady flows $(a)$ and $(b)$ cannot be directly transposed to the unsteady case $(c)$. The two reasons for that are first the unsteady character requiring to account for the first-order time derivative $\partial {{\mathbf u}}/\partial t$. The second reason is the loss of axisymmetry of case $(c)$ imposing to account for all the terms involving a partial derivative with respect to $\theta$ in Eqs. –. One can overcome the issue associated with $\partial {{\mathbf u}}/\partial t$ by performing the analysis on the mean flow, which is obtained from the statistically steady regime. On the other hand, the momentum equations for the mean flow involve the Reynolds stress terms, expressing the influence of the fluctuating velocity field onto the dynamics of the mean field. Despite all these considerations and issues, we have deliberately omitted the terms involving derivatives with respect to the time and to the azimuthal coordinate, and we have calculated the values of the terms in Table \[tab:name-terms\] on the same lines as in Sec. \[sec:momentum-steady\]. By doing so, the objective is not to reproduce a similar analysis as with the steady cases, but more to investigate the evolution of the different terms for the instantaneous flow and the mean flow as compared to the laminar cases $(a)$ and $(b)$.
We compare the radial terms for the instantaneous and mean flows presented in Fig. \[fig:radial-6000\], to their laminar and steady counterparts in Fig. \[fig:radial-900-1500\]. In terms of magnitude, the leading terms have a slightly higher magnitude at ${\textrm{Re}}=6\,000$. The general observations given in Sec. \[sec:momentum-steady\] remain valid here for $z/H=0.64,\
0.95,\ 1$. However, the influence of the intense recirculation bubble modifies locally and significantly the terms in the region $0.2 \leq
r/R\leq 0.4$, for the instantaneous flow. Indeed, in this interval around the radial position $r/R=0.3$ and close to the free surface, the centrifugal acceleration NL$r2$ presents a local minimum, while the decelerating radial pressure gradient presents a local maximum. These localized effects are directly related to the presence of the recirculation bubble as can be seen in Fig. \[fig:orthogonal-meridian\]. A similar observation can be done for the mean flow but the effect is much less visible.
Close to the rotating disk, at $z/H=0.03$, the variations of the five axial terms are notably different from their laminar counterparts, but extremely similar for the instantaneous and mean flows. More precisely, the viscous term VS$r$ has mainly a dissipative action in the laminar regime, while it is slightly driving the flow at ${\textrm{Re}}=6\,000$, except very close to the tubular side-wall where it gets back its dissipative action in the jet-like shear layer. Moreover, the two leading terms, namely the centrifugal acceleration NL$r2$ and the radial pressure gradient, both presents a local maximum in the region $0.2\leq r/R \leq 0.3$, for the case $(c)$, while it keeps growing in the laminar regime. This particular observation is again related to the presence of the recirculation bubbles in this region, which locally strongly modifies the momentum transfers. Finally, the third radial convective term NL$r3$, which is very small in the laminar regime, acquires a magnitude as important as the two other convective terms for $r/R\geq 0.9$.
Let us consider now the radial variations of the five azimuthal terms as shown in Fig. \[fig:azimuthal-6000\]. A rapid overlook of all variations for the instantaneous flow sample (left column) allows to conclude to a general agreement with the results obtained in the laminar cases $(a)$ and $(b)$. The variations of the different terms are similar for the instantaneous and mean flows in the outer radial region, which implies again a relative steadiness of those terms for $r/R\geq 0.7$. On the other hand, the mean flow yields vanishingly small terms in the inner core region $r/R\leq 0.4$, where the instantaneous flow have the three nonlinear convective terms NL$t1$–NL$t3$ with a relatively high magnitude. The unsteady activity of those three nonlinear terms and the intense fluctuating activity generated by them is further discussed in Sec. \[sec:modes\].
As a last step, we compare the axial terms for the instantaneous and mean flows presented in Fig. \[fig:axial-6000\] to their laminar and steady counterparts in Fig. \[fig:axial-900-1500\]. We start from the outer radial line $r/R=0.98$, where the results for the instantaneous flow and the mean flow are extremely close, revealing an almost steady behavior of the jet-like wall shear layer surrounding the tubular side-wall. The comparison of these results with those of cases $(a)$ and $(b)$ leads to several comments. First, the variations of all the terms are limited to a smaller zone above the disk at ${\textrm{Re}}=6\,000$. The magnitude of the axial pressure gradient is increased with . More surprisingly the axial pressure gradient is no longer counterbalanced by the convective term NL$a1$, but is now counterbalanced by NL$a2$. By extension, one can infer that the jet-like shear layer is dominated by axial effects at high Reynolds number.
![Momentum balance in radial direction plotted along horizontal radial lines at four different vertical positions. First row: $z/H=1$; second row: $z/H=0.95$; third row: $z/H=0.64$; fourth row: $z/H=0.03$. Left column: instantaneous flow; Right column: mean flow. Case $\Lambda=1$ and ${\textrm{Re}}=6\,000$. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:radial-6000"}](full-radial-budget-transitional.eps){width="60.00000%"}
![Momentum balance in azimuthal direction plotted along horizontal radial lines at four different vertical positions. First row: $z/H=1$; second row: $z/H=0.95$; third row: $z/H=0.64$; fourth row: $z/H=0.03$. Left column: instantaneous flow; Right column: mean flow. Case $\Lambda=1$ and ${\textrm{Re}}=6\,000$. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:azimuthal-6000"}](full-azimuthal-budget-transitional.eps){width="60.00000%"}
For the two other radii $r/R=0.08,\ 0.48$, the instantaneous flow terms are far different from their mean counterparts. As consequence, the flow in the inner core region appears much more unsteady than the flow in the outer region of the cavity. In addition, given the high level of fluctuation in the inner core region of the flow—see Fig. \[fig:rms-ur-uz-ut-gamma-6000\]—it appears irrelevant to further analyze the results for the mean flow. On the other hand, variations of the different axial terms for the instantaneous flow reveals that the axial momentum transfers are more important at high and are predominant in the top half of the cavity, including below the free surface. Also, not shown here, the viscous terms are still insignificant and the two axial convective terms NL$a1$ and NL$a2$, and the axial pressure gradient dominate the transfers with other unsteady and nonaxisymmetric terms.
![Momentum balance in axial direction plotted along vertical lines at three different radial positions. First row: $r/R=0.98$; second row: $r/R=0.48$; third row: $r/R=0.08$. Left column: instantaneous flow; Right column: mean flow. Case $\Lambda=1$ and ${\textrm{Re}}=6\,000$. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:axial-6000"}](full-axial-budget-transitional.eps){width="60.00000%"}
Nonaxisymmetric modes in the unsteady transitional flow {#sec:modes}
-------------------------------------------------------
All the previous results dealing with the transitional case $(c)$ reveals a complex flow dynamics due to instabilities developing from a steady stable flow similar to the steady laminar cases $(a)$ and $(b)$. The objective of this section is to discuss the physical origin of this instability based on the results presented earlier and also to characterize, at least qualitatively, its effect on the flow field.
As mentioned on several occasions in Sec. \[sec:physical\] and Sec. \[sec:one\], the inner core region of the flow $r/R\leq 0.2$ is globally governed by a quasi-solid-body rotation and in the outer radial region, a wall-jet along the tubular side-wall drives the flow. As already discussed, this latter axial wall-jet is turned into a radial surface jet by the presence of the free surface. It seems therefore legitimate to consider the interfacial zone—denoted $\Upsilon$ in the sequel—between the inner core region and the radially-inward coming jet as prone to developing instabilities. Given the stress-free condition imposed on the free surface, the effect of the instabilities should persist all the way to the free surface itself. Consequently, we consider the variations at the free surface $z/H=1$ of the radial and azimuthal velocity components—the axial component vanishes at the free surface because of the kinematic boundary condition on it—and of the axial angular momentum $\Gamma=ru_\theta$. These variations for the instantaneous flow are shown in Fig. \[fig:free-surface-6000\] (top row), while the corresponding rms fluctuations are presented in the row below.
The most significative feature highlighted in these graphs is the presence of an annular region with $0.3\leq r/R \leq 0.4$, where the fluctuations of $u_r$, $u_\theta$ and $\Gamma$ are intense. Such intense fluctuating activity brings the interfacial zone $\Upsilon$ to light. Focusing now on the rms fluctuations of the axial angular momentum, one may notice the presence of a second outer annular region comprised in the interval $0.6 \leq r/R \leq 0.9$, which is nonaxisymmetric and relatively intense. This outer region of intense rms fluctuations for $\Gamma$ correspond to the zone where the jet-like shear layer is turned into the interior by the free surface. This outer wall jet injects high-angular-momentum fluid towards the cylinder axis. This radial jet impinges on the cylindrical core of the fluid that is in solid-body rotation. When increasing the Reynolds number, the radius of the cylindrical core in solid-body rotation is reduced, while the intensity of the radial jet is increased. Above a given value of the Reynolds number, the action of the impingement of the radial jet on the inner core region starts developing unstable modes.
The origin of these unstable modes is to be found in the analysis of the momentum transfers performed in the previous section. Returning on the radial variations of the five azimuthal terms below the free surface at $z/H=0.95$, Fig. \[fig:azimuthal-instable\] reproduces these variations for ${\textrm{Re}}=900,\ 1\,500,\ 6\,000$ (instantaneous flow), $6\,000$ (mean flow) from top to bottom. All these results have been shown separately before and are now shown together to facilitate the discussion. The flow in the outer region $r/R>0.5$ has a strong steady character given the fact that the variations for the mean flow are fairly close to those of the instantaneous flow. The Coriolis term NL$t2=-u_ru_\theta /r$ keeps the same radially-outward decreasing trend. On the other hand, the two other nonlinear terms NL$t1$ and NL$t3$ develop opposed and equally-intense peaks around $r/R=0.9$. These opposite peaks have an increasing intensity with the Reynolds number. In the interfacial zone $\Upsilon$, all the terms involved present brutal variations and changes of behavior, which give another characterization of this interfacial zone $\Upsilon$. In the inner core region $r/R\leq 0.4$, the flow possesses a strong unsteady character brought to light by the vanishingly small values of the various terms for the mean flow. Consequently, the unstable azimuthal modes are to be found into this inner cylindrical region. Indeed, one may notice that the convective term NL$t1=-u_r\partial u_\theta
/\partial r$ and the Coriolis term NL$t2=-u_r u_\theta /r$ are negative and have a low magnitude in the laminar cases $(a)$ and $(b)$, but acquires large positive values at ${\textrm{Re}}=6\,000$. Moreover, all the three other terms NL$t3$, VS$t1$ and VS$t2$ have very low magnitudes, and thus cannot counterbalance the azimuthal momentum injected by NL$t1$ and the Coriolis term NL$t2$. Only an unsteady and nonaxisymmetric flow can support such azimuthal momentum effects. The effect of these two destabilizing terms on the flow apparently leads to the formation of azimuthal rotating waves superimposed to the stable base flow. The variations at the free surface $z/H=1$, of the radial and azimuthal velocity components, and of the axial angular momentum shown in Fig. \[fig:free-surface-6000\] (top row), suggest the conjugate effect of several rotating waves. These rotating waves correspond to even azimuthal Fourier modes, mainly $n=2$ and $n=4$.
![Momentum terms in azimuthal direction plotted along the horizontal radial line at $z/H=0.95$. From top to bottom: case $(a)$; case $(b)$; case $(c)$ instantaneous; case $(c)$ mean flow. The terminology refers to Tab. \[tab:name-terms\].[]{data-label="fig:azimuthal-instable"}](instability-budget.eps){width="40.00000%"}
Conclusions {#sec:conclusions}
===========
The incompressible flow of a viscous fluid enclosed in a cylindrical container with an open top flat surface and driven by the constant rotation of the bottom wall has been thoroughly investigated. The top surface of the cylindrical cavity is left open with a stress-free boundary condition imposed on it. No-slip condition is imposed on the side-wall and also on the rotating bottom end-wall by means of a regularized angular velocity profile. More specifically, the stress-free top surface is maintained fixed and flat.
New flow states have been investigated based on a fully three-dimensional solution of the Navier–Stokes equations for the free-surface cylindrical swirling flow, without resorting to any symmetry property unlike all other results available in the literature. To our knowledge, the present study delivers the most general available results for this flat-free-surface problem due to its original mathematical treatment.
Five different cases corresponding to different pairs of governing parameters $(\Lambda,{\textrm{Re}})$ have been considered. The Reynolds regime corresponds to transitional flows with some incursions in the fully laminar regime. Both steady and unsteady non-oscillatory swirling flows are considered with a particular emphasis on the case $(\Lambda
=1 , {\textrm{Re}}=6\, 000)$. Of great concern to this study is the question of space resolution. This is particularly important for the bifurcated case at ${\textrm{Re}}= 6\,000$. Convergence tests in space and time have been carried out on this upmost problematic case, and optimal values of the polynomial degree and time-step have been deduced.
The evolution of the total kinetic energy of this open flow has been carefully studied for increasing Reynolds numbers and has been compared to the results for the closed swirling flow. The presence of the free surface on the top of the cylinder is found to strongly modify the observed trend: the total kinetic energy is increased with in the open cylinder case, while the converse is observed in the closed cylinder case. A physical analysis of the energetic action of the surface layer below the free surface allows to justify the above results. A comprehensive physical description of all flow states has been given with particular emphasis on the vortex breakdown bubbles and on the structure of the vortex lines. The unsteady case at ${\textrm{Re}}=6\,000$ has retained more attention, given its unsteady transitional character. The mean flow and the corresponding rms fluctuations have been calculated and the results analyzed accordingly. The momentum transfers in the radial, azimuthal and axial directions have been studied along various one-dimensional lines. For the transitional case at ${\textrm{Re}}=6\,000$, the flow in an inner cylindrical core is in solid-body rotation, while the outer radial layer is dominated by the jet-like shear layer along the tubular side-wall. This axial wall-jet is turned into a radial jet of angular momentum, which prevails all the way up to the free surface. The impingement of this radial jet onto the inner cylindrical core in solid-body rotation leads to the development of unstable azimuthal modes. The nonlinear terms, which includes a Coriolis effect, responsible for the development of these unstable modes have been found using the azimuthal momentum imbalance below the free surface. These unstable modes take the form of even-order azimuthal rotating waves.
This research is being partially funded by a Swiss National Science Foundation Grant (No. 200020–101707) whose supports are gratefully acknowledged.
The results were obtained on supercomputing facilities on the Pleiades clusters at EPFL–IGM.
[10]{} url \#1[`#1`]{}urlprefix
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![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re900-single-meridian-vR.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re900-single-meridian-vZ.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re900-single-meridian-vT.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re900-single-meridian-angular-momentum.eps "fig:"){width="34.00000%"}\
![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re1500-single-meridian-vR.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re1500-single-meridian-vZ.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re1500-single-meridian-vT.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$. Top row: case ${\textrm{Re}}= 900$; Bottom row: case ${\textrm{Re}}= 1\,500$. From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. The 35 contours are uniformly spaced, between $-0.1$ and $0.145$ for $u_r$; and between $-0.08$ and $0.115$ for $u_z$; The 50 contours are uniformly spaced, between $0$ and $1$ for $u_\theta$ and $\Gamma$.[]{data-label="fig:vr-vz-vt-gamma-900-1500"}](SF-H1-Re1500-single-meridian-angular-momentum.eps "fig:"){width="34.00000%"}
![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](SF-H13-Re2000-meridian-vR.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](SF-H13-Re2000-meridian-vZ.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](SF-H13-Re2000-meridian-vT.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](SF-H13-Re2000-meridian-Gamma.eps "fig:"){width="34.00000%"}\
![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](reinter-SF-H3-Re2000-meridian-vR.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](reinter-SF-H3-Re2000-meridian-vZ.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](reinter-SF-H3-Re2000-meridian-vT.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $(d):\
(\Lambda=1/3,{\textrm{Re}}= 2\,000)$ (top row) and case $(e):\
(\Lambda=3,{\textrm{Re}}= 2\,000)$ (bottom row). From left column to right column: radial velocity component $u_r$; axial velocity component $w=u_z$; azimuthal velocity component $u_\theta$; axial angular momentum component $\Gamma = r u_\theta$. All contours are uniformly spaced; 35 contours between $-0.17$ and $0.15$ for $u_r$ case $(d)$; 90 contours between $-0.03$ and $0.15$ for $u_r$ case $(e)$; 35 contours between $-0.1$ and $0.12$ for $u_z$ case $(d)$; 35 contours between $-0.05$ and $0.12$ for $u_z$ case $(e)$; 50 contours between $0$ and $1$ for $u_\theta$ and $\Gamma$, cases $(d)$ and $(e)$.[]{data-label="fig:all-v-H13-H3"}](reinter-SF-H3-Re2000-meridian-Gamma.eps "fig:"){width="34.00000%"}
![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-vR-ref.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-vZ-ref.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-vT-ref.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-Gamma-ref.eps "fig:"){width="34.00000%"}\
![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-vR-Mean-Flow.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-vZ-Mean-Flow.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-vT-Mean-Flow.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: mean flow. From left column to right column: radial velocity component $u_r$; axial velocity component $u_z$; azimuthal velocity component $u_\theta$; axial angular momentum $\Gamma=ru_\theta$. The 50 contours are uniformly spaced, between $-0.13$ and $0.16$ for $u_r$; between $-0.09$ and $0.14$ for $u_z$; and between $0$ and $1$ for $u_\theta$. The 100 contours are uniformly spaced between $0$ and $1$ for $\Gamma$.[]{data-label="fig:ur-ut-uz-6000"}](SF-H1-Re6000-meridian-Gamma-Mean-Flow.eps "fig:"){width="34.00000%"}
![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-vR-rms.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-vZ-rms.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-vT-rms.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-Gamma-rms.eps "fig:"){width="34.00000%"}\
![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-vR-rms-orthogonal.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-vZ-rms-orthogonal.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-vT-rms-orthogonal.eps "fig:"){width="34.00000%"} ![Contours in a meridional plane for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: in the meridian plane $y/R=0$; Bottom row: in the meridian plane $x/R=0$. From left column to right column: rms fluctuations of radial velocity component $u_r$; rms fluctuations of the axial velocity component $u_z$; rms fluctuations of the azimuthal velocity component $u_\theta$; and rms fluctuations of the axial angular momentum $\Gamma=ru_\theta$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.0024$ for rms-$u_z$; between $0$ and $0.004$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:rms-ur-uz-ut-gamma-6000"}](SF-H1-Re6000-meridian-Gamma-rms-orthogonal.eps "fig:"){width="34.00000%"}\
![Contours on the free surface $z/H=1$ for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: rms fluctuations. Left column: radial velocity component $u_r$; Central column: azimuthal velocity component $u_\theta$; Right column: axial component of the angular momentum $\Gamma= r u_\theta$. The 15 contours are uniformly spaced, between $-0.13$ and $0.03$ for $u_r$; between $0$ and $0.35$ for $u_\theta$; and between $0$ and $0.2$ for $\Gamma$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.024$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:free-surface-6000"}](SF-H1-Re6000-circular-free-surface-vR.eps "fig:"){width="40.00000%"} ![Contours on the free surface $z/H=1$ for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: rms fluctuations. Left column: radial velocity component $u_r$; Central column: azimuthal velocity component $u_\theta$; Right column: axial component of the angular momentum $\Gamma= r u_\theta$. The 15 contours are uniformly spaced, between $-0.13$ and $0.03$ for $u_r$; between $0$ and $0.35$ for $u_\theta$; and between $0$ and $0.2$ for $\Gamma$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.024$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:free-surface-6000"}](SF-H1-Re6000-circular-free-surface-vT.eps "fig:"){width="40.00000%"} ![Contours on the free surface $z/H=1$ for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: rms fluctuations. Left column: radial velocity component $u_r$; Central column: azimuthal velocity component $u_\theta$; Right column: axial component of the angular momentum $\Gamma= r u_\theta$. The 15 contours are uniformly spaced, between $-0.13$ and $0.03$ for $u_r$; between $0$ and $0.35$ for $u_\theta$; and between $0$ and $0.2$ for $\Gamma$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.024$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:free-surface-6000"}](SF-H1-Re6000-circular-free-surface-angular-momentum.eps "fig:"){width="40.00000%"}\
![Contours on the free surface $z/H=1$ for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: rms fluctuations. Left column: radial velocity component $u_r$; Central column: azimuthal velocity component $u_\theta$; Right column: axial component of the angular momentum $\Gamma= r u_\theta$. The 15 contours are uniformly spaced, between $-0.13$ and $0.03$ for $u_r$; between $0$ and $0.35$ for $u_\theta$; and between $0$ and $0.2$ for $\Gamma$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.024$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:free-surface-6000"}](SF-H1-Re6000-circular-free-surface-vR-rms.eps "fig:"){width="40.00000%"} ![Contours on the free surface $z/H=1$ for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: rms fluctuations. Left column: radial velocity component $u_r$; Central column: azimuthal velocity component $u_\theta$; Right column: axial component of the angular momentum $\Gamma= r u_\theta$. The 15 contours are uniformly spaced, between $-0.13$ and $0.03$ for $u_r$; between $0$ and $0.35$ for $u_\theta$; and between $0$ and $0.2$ for $\Gamma$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.024$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:free-surface-6000"}](SF-H1-Re6000-circular-free-surface-vT-rms.eps "fig:"){width="40.00000%"} ![Contours on the free surface $z/H=1$ for the case $\Lambda=1$ and ${\textrm{Re}}= 6\,000$. Top row: instantaneous flow; Bottom row: rms fluctuations. Left column: radial velocity component $u_r$; Central column: azimuthal velocity component $u_\theta$; Right column: axial component of the angular momentum $\Gamma= r u_\theta$. The 15 contours are uniformly spaced, between $-0.13$ and $0.03$ for $u_r$; between $0$ and $0.35$ for $u_\theta$; and between $0$ and $0.2$ for $\Gamma$. The 20 contours are uniformly spaced, between $0$ and $0.002$ for rms-$u_r$; between $0$ and $0.024$ for rms-$u_\theta$; and between $0$ and $0.001$ for rms-$\Gamma$.[]{data-label="fig:free-surface-6000"}](SF-H1-Re6000-circular-free-surface-angular-momentum-rms.eps "fig:"){width="40.00000%"}
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---
abstract: 'It is known since the work of [@AA14] that for any permutation symmetric function $f$, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity, more precisely that $R(f) = \widetilde{O}\left(Q^7(f)\right)$. In this paper, we improve this result and show that $R(f) = {O}\left(Q^3(f)\right)$ for a more general class of symmetric functions. Our proof is constructive and relies largely on the quantum hardness of distinguishing a random permutation from a random function with small range from Zhandry [@Zha15].'
author:
- 'Andr[é]{} Chailloux[^1]'
bibliography:
- 'paper.bib'
title: A note on the quantum query complexity of permutation symmetric functions
---
Introduction
============
$\indent$The black box model has been a very fruitful model for understanding the possibilities and limitations of quantum algorithms. In this model, we can prove some exponential speedups for quantum algorithms, which is notoriously hard to do in standard complexity theory. Famous examples are the Deutsch-Josza problem [@DJ92] and Simon’s problem [@Sim94]. There has been a great line of work to understand quantum query complexity, which developed some of the most advanced algorithms techniques. Even Shor’s algorithm [@Sho94] for factoring fundamentally relies on a black box algorithm for period finding.
We describe here the query complexity model in a nutshell. The idea is that we have to compute $f(x_1,\dots,x_n)$ where each $x_i \in [M]$ can be accessed via a query. We consider decision problems meaning that $f : S \rightarrow {\{0,1\}}$ with $S \subseteq [M]^n$. In this paper, we will consider inputs $x \in [M]^n$ equivalently as functions from $[n] \rightarrow [M]$. We are not interested in the running time of our algorithm but only want to minimize the number of queries to $x$, which in the quantum setting consists of applying the unitary ${\mathscr{O}}_x : {|i\rangle}{|j\rangle} \rightarrow {|i\rangle}{|j + x_i\rangle}$. $D(f),R(f)$ and $Q(f)$ represent the minimal amount of queries to compute $f$ with probability greater than $2/3$ (or = 1 for the case of $D(f)$) using respectively a deterministic algorithm with classical queries, a randomized algorithm with classical queries and a quantum algorithm with quantum queries.
As we said before, the query complexity is great for designing new quantum algortihms. It is also very useful for providing black box limitations for quantum algorithms. There are some cases in particular where we can prove that the quantum query complexity of $f$ is at most polynomially smaller than classical (deterministic or randomized) query complexity. For example:
- for specific functions such as search [@BBBV97] or element distinctness [@AS04; @Kut05; @Amb05], we have respectively $Q(Search) = \Theta(n^{1/2}), D({Search}) = \Theta(n)$ and $Q(ED) = \Theta(n^{2/3}), D(\textrm{{ED}}) = \Theta(n)$.
- For any total function $f$ [*i.e.*]{} when its domain $S = [M]^n$, Beals [*et al. *]{} [@BBC+01] proved using the polynomial method that $D(f) \le O(Q^6(f))$.
Another case of interest where we can lower bound the quantum query complexity is the case of permutation symmetric functions. There are several ways of defining such functions and we will be interested in the following definitions for a function $f : S \rightarrow {\{0,1\}}$ with $S \subseteq [M]^n$.
$ \ $
- $f$ permutation symmetric of the first type iff. $\forall \pi \in S_n, \ f(x) = f(x \circ \pi)$.
- $f$ is permutation symmetric of the second type iff. $\forall \pi \in S_n, \ \forall \sigma \in S_M, \\ f(x) = f(\sigma \circ x \circ \pi)$.
where $S_n$ (resp. $S_M$) represents the set of permutations on $[n]$ (resp. $[M]$).
Here, recall that we consider strings $x \in [M]^n$ as functions from $[n] \rightarrow [M]$. Notice also that this definition implies that $S$ is stable by permutation, meaning that $x \in S \Leftrightarrow \forall \pi \in S_n, \ x \circ \pi \in S$. We already know from the work of Aaronson and Ambainis the following result:
\[Theorem:Old\] For any permutation invariant function $f$ of the second type, $R(f) \le \widetilde{O}(Q^7(f))$.
In a recent survey on quantum query complexity and quantum algorithms [@Amb17], Ambainis writes:
> “It has been conjectured since about $2000$ that a similar result also holds for $f$ with a symmetry of the first type.”
#### Contribution.
The contribution of this paper is to prove the above conjecture. We show the following:
\[Theorem:Main\] For any permutation invariant function $f$ of the first type, $R(f) \le O(Q^3(f))$.
This result not only generalizes the result for a more general class of permutation symmetric function, but also improves the exponent from $7$ to $3$. In the case where $M = 2$, this result was already known [@AA14] with an exponent of $2$, which is tight from Grover’s algorithm.
The proof technique is arguably simple, constructive and relies primarily on the quantum hardness of distinguishing a random permutation from a random function with small range from Zhandry [@Zha15]. We start from a permutation symmetric function $f$. At high level, the proof goes as follows:
- We start from an algorithm ${\mathscr{A}}$ that outputs $f(x)$ for all $x$ with high (constant) probability. Let $q$ the number of quantum queries to ${\mathscr{O}}_x$ performed by ${\mathscr{A}}$.
- Instead of running ${\mathscr{A}}$ on input $x$, we choose a random function $C : [n] \rightarrow [n]$ with small range $r$ (from a distribution specified later in the paper) and apply the algorithm ${\mathscr{A}}$ where we replace calls to ${\mathscr{O}}_x$ with calls to ${\mathscr{O}}_{x \circ C}$. We note that there is a simple procedure to compute ${\mathscr{O}}_{x \circ C}$ from ${\mathscr{O}}_x$ and ${\mathscr{O}}_C$.
- If we take $r = \Theta(q^3)$, we can use Zhandry’s lower bound, we show that for each $x$, the output will be close to the output of the algorithm ${\mathscr{A}}$ where we replace calls to ${\mathscr{O}}_{x \circ C}$ with calls to ${\mathscr{O}}_{x \circ \pi}$ for a random permutation $\pi$. Using the fact that $f$ is permutation symmetric, the latter algorithm will output with high probability $f(x \circ \pi) = f(x)$. In other words, if the algorithm ${\mathscr{A}}$ that calls ${\mathscr{O}}_{x \circ C}$ wouldn’t output $f(x)$ for a random $C$ and a fixed $x$ then we would find a distinguisher between a random $C$ and a random permutation $\pi$, which is hard from Zhandry’s lower bound.
- The above tells us that applying ${\mathscr{A}}$ where we replace calls to ${\mathscr{O}}_x$ with calls to ${\mathscr{O}}_{x \circ C}$ gives us output $f(x)$ with high probability. Knowing $C$, we can construct the whole string $x \circ C$ by querying $x$ on inputs $i \in Im(C)$ which can be done with $Im(C) \le r$ classical queries which allows us to construct the unitary ${\mathscr{O}}_{x \circ C}$. This means we can emulate ${\mathscr{A}}$ on input $x \circ C$ with $r$ classical queries to $x$ and this gives us $f(x)$ with high probability.
After presenting a few notations, we dive directly into the proof of our theorem.
Preliminaries
=============
Notations
---------
For any function $f$ we denote by $Dom(f)$ its domain and by $Im(f)$ its range (or image).
#### Query algorithms.
A query algorithm ${\mathscr{A}}^{{\mathscr{O}}}$ is described by an algorithm that calls another function ${\mathscr{O}}$ in a black box fashion. We will never be interested in the running time or the size of ${\mathscr{A}}$ but only in the number of calls, or queries, to ${\mathscr{O}}$. We will consider both the cases where the algorithm ${\mathscr{A}}^{{\mathscr{O}}}$ is classical and quantum. In the latter ${\mathscr{O}}$ will be a quantum unitary. In both cases, we only consider algorithms that output a single bit.
#### Oracles.
We use oracles to perform black box queries to a function. For any function $g$, ${\mathscr{O}}_g^{\textrm{Classical}}$ is a black box that on input $i$ outputs $g(i)$ while ${\mathscr{O}}_g$ (without any superscript) is the quantum unitary satisfying $${\mathscr{O}}_g : {|i\rangle}{|j\rangle} \rightarrow {|i\rangle}{|j + g(i)\rangle}.$$
#### Query complexity.
Fix a function $f : S \rightarrow {\{0,1\}}$ where $S \subseteq [M]^n$.
The randomized query complexity $R(f)$ of $f$ is the smallest integer $q$ such that there exists a classical randomized algorithm ${\mathscr{A}}^{{\mathscr{O}}}$ performing $q$ queries to ${\mathscr{O}}$ satisfying: $$\forall x \in S, \ \Pr[{\mathscr{A}}^{{\mathscr{O}}_x^{\textrm{Classical}}} \textrm{ outputs } f(x)] \ge 2/3.$$
The quantum query complexity $Q(f)$ of $f$ is the smallest integer $q$ such that there exists a quantum algorithm ${\mathscr{A}}^{{\mathscr{O}}}$ performing $q$ queries to ${\mathscr{O}}$ satisfying: $$\forall x \in S, \ \Pr[{\mathscr{A}}^{{\mathscr{O}}_x} \textrm{ outputs } f(x)] \ge 2/3.$$
Hardness of distinguishing a random permutation from a random function with small range
---------------------------------------------------------------------------------------
Our proof will use a quantum lower bound on distinguishing a random permutation from a random function with small range proven in [@Zha15]. Following this paper, we define, for any $r \in [n]$, the following distribution $D_r$ on functions from $[n]$ to $[n]$ from which can be sampled as follows.
- Draw a random function $g$ from $[n] \rightarrow [r]$.
- Draw a random injective function $h$ from $[r] \rightarrow [n]$.
- Output the composition $h \circ g$.
Notice that any function $f$ drawn from $D_r$ is of small range and satisfies $|Im(f)| \le r$. Let also $D_{\textrm{perm}}$ be the uniform distribution on permutations on $[n]$. Zhandry’s lower bound can be stated as follows:
\[Proposition:LBCollision\] There exists an absolute constant $\Lambda$ such that for any $r \in [n]$ and any quantum query algorithm $\mathscr{B}^{\mathscr{O}}$ performing at most $\lceil \Lambda {r}^{1/3}\rceil$ queries to ${\mathscr{O}}$: $$\forall b \in {\{0,1\}}, \ \left|{\mathbb{E}}_{\pi \leftarrow D_{\textrm{perm}}}\Pr[\mathscr{B^{{\mathscr{O}}_\pi}} \textrm{ outputs } b] - {\mathbb{E}}_{C \leftarrow D_r}\Pr[\mathscr{B}^{{\mathscr{O}}_C} \textrm{ outputs } b]\right| \le \frac{2}{27}.$$
This is obtained immediately by combining Theorem $8$ and Lemma $1$ of $\cite{Zha15}$[^2].
Proving our main theorem
========================
The goal of this section is to prove Theorem \[Theorem:Main\]. Fix a function $f : S \rightarrow {\{0,1\}}$ where $S \subseteq [M]^n$ with $Q(f) =q$. This means there exists a quantum query algorithm ${\mathscr{A}}^{\mathscr{O}}$ performing $q$ queries to ${\mathscr{O}}$ such that $$\forall x \in S, \ \Pr[{\mathscr{A}}^{{\mathscr{O}}_x} \textrm{ outputs } f(x)] \ge 2/3.$$ We first amplify the success probability to $20/27$.
\[Lemma\] There exists a quantum query algorithm ${\mathscr{A}}^{\mathscr{O}}_3$ that performs $3q$ queries to ${\mathscr{O}}$ such that $$\forall x \in S, \ \Pr[{\mathscr{A}}^{{\mathscr{O}}_x}_3 \textrm{ outputs } f(x)] \ge \frac{20}{27}.$$
${\mathscr{A}}^{\mathscr{O}}_3$ will consist of the following: run ${\mathscr{A}}^{\mathscr{O}}$ independently $3$ times and take the output that occurs the most. For each $x$, each run of ${\mathscr{A}}^{{\mathscr{O}}_x}$ outputs $f(x)$ with probability at least $2/3$. The probability that the correct $f(x)$ appears at least twice out of the $3$ results is therefore greater than $\frac{8}{27} + 3 \cdot \frac{4}{27} = \frac{20}{27}$.
Using the fact that $f$ is permutation symmetric, we get the following corrolary:
\[Corollary\] $$\forall x \in S, \ \forall \pi \in S_n, \
\Pr[{\mathscr{A}}^{{\mathscr{O}}_{x \circ \pi}}_3 \textrm{ outputs } f(x)]
= \Pr[{\mathscr{A}}^{{\mathscr{O}}_{x \circ \pi}}_3 \textrm{ outputs } f(x \circ \pi)] \ge \frac{20}{27}.$$
Looking at a small number of indices of x
-----------------------------------------
The main idea of the proof is to show that ${\mathscr{A}}_3$ will output $f(x)$ with high probability when replacing queries to ${\mathscr{O}}_x$ with queries to ${\mathscr{O}}_{x \circ C}$ for $C$ chosen uniformly from $D_r$ for some $r = \Theta(Q^3(f))$. First notice that for any $x : [n] \rightarrow [M]$ and any $g : [n] \rightarrow [n]$, it is possible to apply $\mathscr{O}_{x \circ g}$ with $2$ calls to ${\mathscr{O}}_g$ and $1$ call to ${\mathscr{O}}_x$ with the following procedure: $$\begin{aligned}
{|i\rangle}{|j\rangle}{|0\rangle}
\rightarrow {|i\rangle}{|j\rangle}{|g(i)\rangle}
\rightarrow {|i\rangle}{|j + (x \circ g) (i)\rangle}{|g(i)\rangle}
\rightarrow {|i\rangle}{|j + (x \circ g)(i)\rangle}{|0\rangle}
\end{aligned}$$ where we respectively apply ${\mathscr{O}}_g$ on registers $(1,3)$ ; ${\mathscr{O}}_x$ on registers $(3,2)$ and ${\mathscr{O}}_g^\dagger$ on registers $(1,3)$.
Therefore, for any fixed (and known) $x$, for any function $g : [n] \rightarrow [n]$, we can look at ${\mathscr{A}}_3^{{\mathscr{O}}_{x \circ g}}$ as a quantum query algorithm that queries ${\mathscr{O}}_g$. In other words, for each $x \in S$, there is a quantum query algorithm ${\mathscr{B}}_x^{\mathscr{O}}$ such that ${\mathscr{B}}_x^{{\mathscr{O}}_g} = {\mathscr{A}}^{{\mathscr{O}}_{x \circ g}}$ for any function $g : [n] \rightarrow [n]$. Notice also that since a query to ${\mathscr{O}}_{x \circ g}$ is done by doing $2$ queries to ${\mathscr{O}}_g$, we have that ${\mathscr{B}}^{\mathscr{O}}$ uses twice as many queries than ${\mathscr{A}}_3^{\mathscr{O}}$.
We can now prove our main proposition that shows that we can compute $f(x)$ by looking only at $x \circ C$ meaning that we only need to look at $Im(C) \le r$ random.
\[Proposition:Main\] Let $f : [M]^n \rightarrow {\{0,1\}}$ with $Q(f) = q$ and $r = \lceil216q^3\Lambda^{-3}\rceil$ where $\Lambda$ is the absolute constant from Proposition \[Proposition:LBCollision\]. $$\forall x \in S, \ {\mathbb{E}}_{C \leftarrow D_r}\Pr[\mathscr{A}^{{\mathscr{O}}_{x \circ C}}_3 \textrm{ outputs } f(x)] \ge 2/3.$$
For each $x \in S$, we consider the algorithm ${\mathscr{B}}_x^{\mathscr{O}}$ described above. Recall that for all $g : [n] \rightarrow [n]$, ${\mathscr{B}}_x^{{\mathscr{O}}_g}= {\mathscr{A}}_3^{{\mathscr{O}}_{x \circ g}}$. Since ${\mathscr{A}}_3^{\mathscr{O}}$ uses $3q$ queries, ${\mathscr{B}}_x^{\mathscr{O}}$ uses $6q$ queries. We first consider the case where $g$ is a random permutation. Using Corollary \[Corollary\]:
$$\begin{aligned}
\forall x \in S, \ {\mathbb{E}}_{\pi \leftarrow D_{\textrm{perm}}}\Pr[\mathscr{B}^{{\mathscr{O}}_\pi}_x \textrm{ outputs } 0] & =
{\mathbb{E}}_{\pi \leftarrow D_{\textrm{perm}}}\Pr[\mathscr{A}^{{\mathscr{O}}_{x \circ \pi}}_3 \textrm{ outputs } f(x)] \ge \frac{20}{27}\end{aligned}$$
Using the lower bound of Proposition \[Proposition:LBCollision\] noticing that $6q \le \Lambda r^{1/3}$, we have
$$\forall x \in S, \ \left|{\mathbb{E}}_{\pi \leftarrow D_{\textrm{perm}}}\Pr[{\mathscr{B}}^{{\mathscr{O}}_\pi}_x \textrm{ outputs } f(x)] - {\mathbb{E}}_{C \leftarrow D_r}\Pr[\mathscr{B}^{{\mathscr{O}}_C}_x \textrm{ outputs } f(x)]\right| \le \frac{2}{27}.$$ which gives us $$\forall x \in S, \ {\mathbb{E}}_{C \leftarrow D_{r}}\Pr[\mathscr{B}^{{\mathscr{O}}_C}_x \textrm{ outputs } f(x)] \ge \frac{20}{27} - \frac{2}{27} = 2/3.$$ Since for each $x \in S$, $\mathscr{B}^{{\mathscr{O}}_C}_x = {\mathscr{A}}_3^{x \circ C}$, we can therefore conclude $$\forall x \in S, \ {\mathbb{E}}_{C \leftarrow D_r}\Pr[\mathscr{A}^{{\mathscr{O}}_{x \circ C}}_3 \textrm{ outputs } f(x)] \ge 2/3.$$
Constructing a classical query algorithm for f
----------------------------------------------
We can now use the above proposition to prove our main theorem.
For any permutation invariant function $f$ of the first type, $R(f) \le O(Q^3(f))$.
Fix a function $f : S \rightarrow {\{0,1\}}$ where $S \subseteq [M]^n$ with $Q(f) =q$. This means there exists a quantum query algorithm ${\mathscr{A}}^{\mathscr{O}}$ performing $q$ queries to ${\mathscr{O}}$ such that $$\forall x \in S, \ \Pr[{\mathscr{A}}^{{\mathscr{O}}_x} \textrm{ outputs } f(x)] \ge 2/3.$$ We construct a randomized algorithm that performs $r = \lceil216q^3\Lambda^{-3}\rceil$ classical queries to ${\mathscr{O}}_{x}^{\textrm{Classical}}$ as follows:
1. Choose a random $C$ according to distribution $D_r$.
2. Query ${\mathscr{O}}_x^{\textrm{Classical}}$ to get all values $x_i$ for $i \in Im(C)$. This requires $|Im(C)| \le r$ queries to ${\mathscr{O}}_x^{\textrm{Classical}}$. These queries fully characterize the function $x \circ C$, hence the quantum unitary ${\mathscr{O}}_{x \circ C}$.
3. From ${\mathscr{A}}^{\mathscr{O}}$, construct the quantum algorithm ${\mathscr{A}}_3^{\mathscr{O}}$ as in Lemma \[Lemma\]. Recall that ${\mathscr{A}}_3^{\mathscr{O}}$ just consists of applying ${\mathscr{A}}^{\mathscr{O}}$ independently $3$ times and output the majority outcome.
4. We consider ${\mathscr{A}}_3^{{\mathscr{O}}_{x \circ C}}$ as a quantum unitary circuit acting on $t$ qubits. At each step of the algorithm, we store the $2^{t}$ amplitudes. When ${\mathscr{O}}_{x \circ C}$ is called, we use its representation from step $2$ to calculate its action on the $2^t$ amplitudes. Other parts of ${\mathscr{A}}_3^{{\mathscr{O}}_{x \circ C}}$ are treated the same way. While this uses a lot of computing power, it does not require any queries to ${\mathscr{O}}_x^{\textrm{Classical}}$ or ${\mathscr{O}}_x$ other than those used at step $2$.
Step $4$ outputs the same output distribution than the quantum algorithm ${\mathscr{A}}_3^{{\mathscr{O}}_{x \circ C}}$. Using Proposition \[Proposition:Main\], for all $x \in S$, this algorithm outputs, $f(x)$ with probability greater than $2/3$, which implies $$R(f) \le r = \lceil216Q^3(f)\Lambda^{-3}\rceil.$$
Notice that after step $2$, it is not possible to just compute $f(x \circ C)$, and try to show that it is equal to $f(x)$ since we don’t even always have $x \circ C \in S$. This is yet another example in query complexity where we use the behavior of a query algorithm on inputs not necessarily in the domain of $f$.
[^1]: Inria de Paris, EPI SECRET, `[email protected]`
[^2]: Equivalently, this is obtained immediately by combining Lemma 3.2 and Lemma 3.4 from the arXiv version `quant-ph:1312.1027.`
|
---
abstract: 'Motivated by the need of processing functional-valued data, or more general, operator-valued data, we introduce the notion of the operator reproducing kernel Hilbert space (ORKHS). This space admits a unique operator reproducing kernel which reproduces a family of continuous linear operators on the space. The theory of ORKHSs and the associated operator reproducing kernels are established. A special class of ORKHSs, known as the perfect ORKHSs, are studied, which reproduce the family of the standard point-evaluation operators and at the same time another different family of continuous linear operators. The perfect ORKHSs are characterized in terms of features, especially for those with respect to integral operators. In particular, several specific examples of the perfect ORKHSs are presented. We apply the theory of ORKHSs to sampling and regularized learning, where operator-valued data are considered. Specifically, a general complete reconstruction formula from linear operators values is established in the framework of ORKHSs. The average sampling and the reconstruction of vector-valued functions are considered in specific ORKHSs. We also investigate in the ORKHSs setting the regularized learning schemes, which learn a target element from operator-valued data. The desired representer theorems of the learning problems are established to demonstrate the key roles played by the ORKHSs and the operator reproducing kernels in machine learning from operator-valued data. We finally point out that the continuity of linear operators, used to obtain the operator-valued data, on an ORKHS is necessary for the stability of the numerical reconstruction algorithm using the resulting data.'
author:
- 'Rui Wang[^1] and Yuesheng Xu[^2]'
title: '**Operator Reproducing Kernel Hilbert Spaces**'
---
**Key words**: Operator reproducing kernel Hilbert space, operator reproducing kernel, average sampling, regularized learning, non-point-evaluation data
**2010 Mathematics Subject Classification:** 46E22, 68T05, 94A20
Introduction
============
The main purpose of this paper is to introduce the notion of the operator reproducing kernel Hilbert space and discuss the essential ideas behind such spaces along with their applications to sampling and machine learning. The introduction of such a notion is motivated by the need of processing various data (not necessarily the point-evaluations of a function) emerging in practical applications.
Many scientific problems come down to understanding a function from a finite set of its sampled data. Commonly used sampled data of a function to be constructed are its function values. Through out this paper, we reserve the term “function value" for the evaluation of a function at a point in its domain. We call such data the point-evaluation data of the function. The stability of the sampling process requires that the point-evaluation functionals must be continuous. This leads to the notion of reproducing kernel Hilbert spaces (RKHSs) [@Ar] which has become commonly used spaces for processing point-evaluation data. There is a bijective correspondence between a RKHS and its reproducing kernel [@Mercer]. Moreover, reproducing kernels are used to measure the similarity between elements in an input space. In machine learning, many effective schemes based upon reproducing kernels were developed and proved to be useful [@CS; @PS; @SS; @SC; @V]. The theory of vector-valued RKHSs [@P] received considerable attention in multi-task learning which concerns estimating a vector-valued function from its function values. In the framework of vector-valued RKHSs, kernel methods were proposed to learn multiple related tasks simultaneously, [@EMP; @MP05]. Recently, to learn a function in a Banach space which has more geometric structures than a Hilbert space, the notion of reproducing kernel Banach spaces (RKBSs) was introduced in [@ZXZ] and further developed in [@SZH; @XQ; @ZZ12; @ZZ13]. The celebrated Shannon sampling formula can be regarded as an orthonormal expansion of a function in the Paley-Wiener space, which is a typical RKHS. Motivated by this fact, many general sampling theorems were established in RKHSs through frames or Riesz bases composed by reproducing kernels, [@H07; @H09; @HKK; @MNS; @NW].
There are many practical occasions where point-evaluation data are not readily available. Sampling is to convert an analog signal into a sequence of numbers, which can then be processed digitally on a computer. These numbers are expected to be the values of the original signal at points in the domain of the signal. However, in most physical circumstances, one cannot measure the value $f(x)$ of a signal $f$ at the location $x$ exactly, due to the non-ideal acquisition device at the sampling location. A more suitable form of the sampled data should be the weighted-average value of $f$ in the neighborhood of the location $x$. The average functions may reflect the characteristic of the sampling devices. Such a sampling process is called an average sampling process [@A; @S]. In signal analysis, sometimes what is known is the frequency information of a signal. These data, represented mathematically by the Fourier transform or wavelet transform, are clearly not the function values. Another example of non-point-evaluation sampled data concerns EXAFS (extended x-ray absorption fine structure) spectroscopy, which is a useful atomic-scale probe for studying the environment of an atomic species in a chemical system. One key problem in EXAFS is to obtain the atomic radial distribution function, which describes how density varies as a function of the distance from a reference particle to the points in the space. The measured quantity used to reconstruct the atomic radial distribution function is the x-ray absorption coefficient as a function of the wave vector modulus for the photoelectron. Theoretically, these observed information is determined by an integral of the product of the atomic radial distribution function and functions reflecting the characteristics of the sample and the physics of the electron scattering process [@CNSU; @MP04]. In the areas of machine learning and functional data analysis, we are required to process measured information rather than function values such as curves, surfaces and other geometric shapes [@RS]. All above mentioned non-point-evaluation sampled data can be formulated mathematically as linear functional values or linear operator values of a function. The term “functional value" is reserved in this paper for the value of a functional applied to a function in its domain. Function values are a special case of functional values when the linear functional is the point-evaluation functional. Likewise, we also reserve the term “operator value" for the value of an operator at a function in its domain. Large amount of such sampled data motivate us to consider constructing a function from its linear functional values or its linear operator values other than its function values.
There is considerable amount of work on processing non-point-evaluation sampled data in the literature. The reconstruction of a function in the band-limited space, the finitely generated shift-invariant spaces and more generally the spaces of signals with finite rate of innovation from its local average values were investigated, respectively, in [@G; @SZ], [@A; @AST] and [@S]. Sampling and reconstruction of vector-valued functions were considered in [@AI08], where the sampled data were defined as the inner product of the function values and some given vectors. It is clear that these sampled data can also be treated as the linear functional values of the vector-valued function. A number of alternative estimators were introduced for the functional linear regression based upon the functional principal component analysis [@CH; @RS; @YMW] or a regularization method in RKHSs [@YC]. Optimal approximation of a function $f$ from a general RKHS by algorithms using only its finitely many linear functional values was widely studied in the area of information-based complexity [@NoWo; @TWW]. In machine learning, regularized learning schemes were proposed for functional-valued data. For example, learning a function from local average integration functional values was investigated in [@V] in the framework of the supported vector machine. Regularized learning from general linear functional values in Banach spaces was considered in [@MP04]. Regularized learning in the context of RKBSs was studied in [@ZXZ; @ZZ12] from both function-valued data and functional-valued data.
Reconstructing a function in a usual RKHS (where only the point-evaluation functionals are continuous) from its non-point-evaluation data is not appropriate in general since the non-point-evaluation functionals used in the processing may not be continuous in the underlying RKHS. When constructing a function from its linear functional values, the sampled data are generally obtained with noise. To reduce the effects of the noise, we require that the sampling process must be stable. Clearly, a usual RKHS that enjoys only the continuity of point-evaluation functionals is not a suitable candidate for processing a class of non-point-evaluation functionals. This demands the availability of a Hilbert space where the specific non-point-evaluation functionals used in the processing are continuous. Such spaces are presently not available in the literature. Therefore, it is desirable to construct a Hilbert space where a specific class of linear functionals or linear operators on it are continuous.
The goal of this paper is to study the operator reproducing kernel Hilbert space. We shall introduce the notion of the operator reproducing kernel Hilbert space by reviewing examples of data forms available in applications. Such a space ensures the [*continuity*]{} of a family of prescribed linear operators on it. An operator reproducing kernel Hilbert space is expected to admit a reproducing kernel, which reproduces the linear operators defining the space. We shall show that every operator reproducing kernel Hilbert space is isometrically isomorphic to a usual vector-valued RKHS. Although this does not mean that the study of operator reproducing kernel Hilbert spaces can be trivially reduced to that of vector-valued RKHSs, several important results about vector-valued RKHSs can be transferred to operator reproducing kernel Hilbert spaces by the isometric isomorphism procedure. In the literature, the average sampling problem was usually investigated in spaces which are typical RKHSs. We shall point out that these spaces are in fact special operator reproducing kernel Hilbert spaces. We shall call such a space a [*perfect*]{} operator reproducing kernel Hilbert space, since it admits two different families of continuous linear operators, among which one is the set of the standard point-evaluation operators. The perfect operator reproducing kernel Hilbert spaces, especially the ones with respect to integral operators, bring us special interest. The existing results about average sampling are special cases of the sampling theorem which we establish in the operator reproducing kernel Hilbert space setting, with additional information on the continuity of the used integral functionals on the space. In machine learning, learning a function from its functional values was also considered in RKHSs or RKBSs. In the framework of the new spaces, we study the regularized learning schemes for learning an element from non-point-evaluation operator values. By establishing the representer theorem, we shall show that the operator reproducing kernel Hilbert spaces are more suitable for learning from non-point-evaluation operator values than the usual RKHSs which are used in the existing literature. The theoretic results and specific examples in this paper all demonstrate that operator reproducing kernel Hilbert spaces provide a right framework for constructing an element from its non-point-evaluation continuous linear functional or operator values.
This paper is organized in nine sections. We review in section 2 several non-point-evaluation data often used in practical applications. Motivated by processing such forms of data, we introduce the notion of the operator reproducing kernel Hilbert space in section 3. We also identify an operator reproducing kernel with each operator reproducing kernel Hilbert space and reveal the relation between the new type of spaces and the usual vector-valued RKHSs. In section 4, we characterize the perfect operator reproducing kernel Hilbert spaces and the corresponding operator reproducing kernels and study the universality of these kernels. Motivated by the local averages of functions, we investigate in section 5 the perfect operator reproducing kernel Hilbert space with respect to a family of integral operators and present two specific examples. Based upon the framework of operator reproducing kernel Hilbert spaces, we establish a general reconstruction formula of an element from its operator values in section 6 and develop the regularized learning algorithm for learning from non-point-evaluation operator values in section 7. In section 8, we comment on stability of a numerical reconstruction algorithm using operator values in an operator reproducing kernel Hilbert space. Finally, we draw a conclusion in section 9.
Non-Point-Evaluation Data
=========================
Point-evaluation function values are often used to construct a function in the classical RKHS setting. We encounter in both theoretical analysis and practical applications huge amount of data which are not point-evaluation function values such as integral values of lines and surfaces, geometric objects, and operator values. The purpose of this section is to review several such forms of data, which serve as motivation of introducing the notion of operator reproducing kernel Hilbert space.
Classical methods to construct a function are based on its finite samples. The finite empirical data of a function $f$ from points in its domain $\mathcal{X}$ to $\mathbb{R}$ or $\mathbb{C}$ are usually taken as the values of $f$ on the finite set $\{x_j:j\in\mathbb{N}_m\}\subseteq\mathcal{X}$, where $\mathbb{N}_m:=\{1,2,\dots, m\}$. We call them the point-evaluation data. Various approaches were developed for processing the point-evaluation data in the literature. The classical RKHS is the function space suitable for the point-evaluation data. However, because of a variety of reasons, the available data in practical applications are not always in the form of point evaluations. We shall call data that are not point-evaluation function values the non-point-evaluation data.
The first type of non-point-evaluation data concerns the functional-valued data. In most physical circumstances, although the values of a function $f$ are expected to be sampled and processed on computers, one can not measure the value $f(x)$ at the location $x$ exactly because of the non-ideal acquisition device [@A; @S]. Instead of the function values at the sample points, one obtains the finite local average values of $f$ in the neighborhood of the sample points. Specifically, we suppose that $\{x_j:j\in\mathbb{N}_m\}$ is a finite set of $\mathbb{R}$ and $\{u_j: j\in\mathbb{N}_m\}$ is a set of nonnegative functions on $\mathbb{R}$, where for each $j\in\mathbb{N}_m$, the function $u_j$ supports in a neighborhood of $x_j$. The sampled data have the form $\{L_j(f):j\in\mathbb{N}_m\}$, where for each $j\in\mathbb{N}_m$, $L_j$ is defined by $$L_{j}(f):=\int_{\mathbb{R}}f(x)u_{j}(x)dx,$$ which can be viewed as a functional of $f$. We note that when the support of $u_j$ is small, the functional value $L_j(f)$ can be treated as an approximation of the point-evaluation function value of $f$ at the location $x_j$.
There are other practical occasions, where the available information is not point-evaluation data or the local-average-valued data. The frequency information of a signal are such non-point-evaluation data. Specifically, suppose that $f$ is a $2\pi$-periodic signal. The Fourier coefficients of $f$ are defined by $$c_j(f):=\frac{1}{\sqrt{2\pi}}\int_{0}^{2\pi}f(x)e^{-ijx}dx,
\ j\in\mathbb{Z}.$$ These coefficients, taken as the functional-valued data, contain the information of the signal in the frequency domain. Such sampled data are usually used to recover the signal in the time domain.
Another example concerns the x-ray absorption coefficients of an atomic radial distribution function in EXAFS spectroscopy [@CNSU; @MP04]. Within the EXAFS theory accepted generally, the x-ray absorption coefficient of the atomic radial distribution function $f$ is described by $$\label{EXAFS}
\chi(\kappa):=\int_{a}^{b}G(\kappa,r)f(r)dr,$$ where $G$ is the kernel defined by $$G(\kappa,r):=\displaystyle{4\pi\rho\phi(\kappa) e^{-\frac{2r}{\lambda(\kappa)}}\sin(2\kappa r
+\psi(\kappa))}.$$ Here, the constant $\rho$ and the functions $\lambda, \phi, \psi$ are used to describe physical parameters of the Radon transform. Specifically, $\rho$ is the sample density, $\lambda$ is the mean-free-path of the photo-electron, $\phi$ is the scattering amplitude of the electron scattering process and $\psi$ is the phase-shifts of the electron scattering process. To determine the atomic radial distribution function $f$, measurements are made giving values $\chi(\kappa_j)$, $j\in\mathbb{N}_m$. There were many methods developed in the literature to learn the atomic radial distribution function from such sampled data. It is clear that for each $\kappa$, the quantity $\chi(\kappa)$ can be viewed as a functional value of the atomic radial distribution function $f$.
The sampled data emerge in the cases discussed above are not the point-evaluation data but the functional-valued ones. This will motivate us to consider estimating the target function $f$ from the finite information $\{L_j(f): j\in\mathbb{N}_m\}$, where $L_j$, $j\in\mathbb{N}_m,$ are linear functionals of $f$.
Operator-valued data often appear in applications. In machine learning, multiple related learning tasks need to be learned simultaneously. For example, multi-modal human computer interface requires modeling of both speech and vision. This can improve performance relative to learning each task independently, which can not capture the relations between these tasks. Empirical studies have shown the benefits of such multi-task learning [@BH; @Ca; @EMP]. This leads us to learning functions taking their range in a finite-dimensional Euclidean space $\mathbb{C}^n$, or more generally, a Hilbert space $\mathcal{Y}$. In this case, the measured information to be processed is the finite function values $f(x_j)$, $j\in\mathbb{N}_m,$ on a discrete set in the domain. However, they are vectors in a Hilbert space $\mathcal{Y}$ rather than scalars in $\mathbb{C}$. The functional data, considered in the area of functional data analysis, consist of functions. Examples of this type range from the shapes of bones excavated by archaeologists, to economic data collected over many years, to the path traced out by a juggler¡¯s finger [@RS]. In these occasions, the known information can be seen as the operator-valued data. This will lead us to a study of estimating a function $f$ from finite sampled data $\{L_j(f): j\in\mathbb{N}_m\}$, where $L_j$, $j\in\mathbb{N}_m,$ are linear operators of $f$, taking values in a Hilbert space $\mathcal{Y}$.
Finally, we describe types of data used in sampling. Motivated by the average sampling, one may consider reconstruction of a function $f$ from functional-valued data $\{L_j(f): j\in\mathbb{J}\}$, where $\mathbb{J}$ is an index set and $L_j$, $j\in\mathbb{J}$, is a sequence of linear functionals of $f$. For vector-valued functions, there are two types of data used in the reconstruction formula. One concerns the point-evaluation data $\{f(x_j): j\in\mathbb{J}\}$. Another one consists linear combinations of the components of each point-evaluation data $f(x_j)$. Following [@AI08], we shall consider reconstructing a function $f$ from the finite sampled data, defined as the inner product $$\langle L_j(f),\xi_j\rangle_{\mathcal{Y}},
\ \ j\in\mathbb{J},$$ where $\mathbb{J}$ is an index set, $L_j$, $j\in\mathbb{J},$ are linear operators of $f$ taking values in a Hilbert space $\mathcal{Y}$ and $\{\xi_j: j\in\mathbb{J}\}$ is a subset in $\mathcal{Y}$. In fact, we shall only consider the sampling and the reconstruction problem with respect to the functional values. This is because if we introduce a family of functionals $\widetilde{L}_{j}, j\in\mathbb{J},$ as $$\widetilde{L}_{j}(f)=\langle L_{j}(f), \xi_j
\rangle_{\mathcal{Y}},\ j\in\mathbb{J},$$ the sampling problem with respect to the operator values can also be taken as reconstructing an element $f$ from its functional values $\widetilde{L}_{j}(f),
\ j\in\mathbb{J}$.
It is known that RKHSs are ideal spaces for processing the point-evaluation data. However, these spaces are no longer suitable for processing non-point-evaluation data. Hence, it is desirable to introduce spaces suitable for processing non-point-evaluation data. This is the focus of this paper.
The Notion of the Operator Reproducing Kernel Hilbert Space
===========================================================
We introduce in this section the notion of the operator reproducing kernel Hilbert space. We identify the classical scalar-valued and vector-valued RKHSs as its special examples. We also present nontrivial examples of operator reproducing kernel Hilbert spaces with respect to integral operators. In a way similar to RKHSs, each operator reproducing kernel Hilbert space has exactly one operator reproducing kernel to reproduce the operator values of elements in the space. Based upon the isometric isomorphism between an operator reproducing kernel Hilbert space and a vector-valued RKHS, we characterize an operator reproducing kernel by its features.
Let $\mathcal{H}$ denote a Hilbert space with inner product $\left<\cdot, \cdot\right>_\mathcal{H}$ and norm $\|\cdot\|_{\mathcal{H}}:=\left<\cdot, \cdot
\right>_\mathcal{H}^{1/2}$. For a Hilbert space $\mathcal{Y}$ and a family $\mathcal{L}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$, we say that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$ if for each $f\in \mathcal{H}$, $L(f)=0$ for all $L\in\mathcal{L}$ if and only if $\|f\|_{\mathcal{H}}=0$.
\[def\_ORKHS\] Let $\mathcal{H}$ be a Hilbert space and $\mathcal{L}$ a family of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$. Space $\mathcal{H}$ is called an [*operator reproducing kernel Hilbert space*]{} with respect to $\mathcal{L}$ if the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$ and each linear operator in $\mathcal{L}$ is continuous on $\mathcal{H}$.
In this paper, we shall use the abbreviation ORKHS to represent an operator reproducing kernel Hilbert space. We remark that unlike the typical RKHS, ORKHS is not always composed of functions, but general vectors. We shall see that an ORKHS is associated with a kernel which reproduces the linear operators in $\mathcal{L}$. It is convenient to associate the family $\mathcal{L}$ of linear operators with an index set $\Lambda$, mainly, $\mathcal{L}:=\{L_{\alpha}:\alpha\in \Lambda\}$.
In the scalar case when $\mathcal{Y}=\mathbb{C}$, where $\mathbb{C}$ denotes the set of complex numbers, if a Hilbert space $\mathcal{H}$ and a family $\mathcal{F}$ of linear functionals on $\mathcal{H}$ satisfy Definition \[def\_ORKHS\], we shall call $\mathcal{H}$ a [*functional reproducing kernel Hilbert space*]{} with respect to $\mathcal{F}$. We shall use the abbreviation FRKHS to represent a functional reproducing kernel Hilbert space. We remark that for any Hilbert space $\mathcal{H}$ there always exists a family $\mathcal{F}$ of linear functionals such that $\mathcal{H}$ becomes an FRKHS with respect to $\mathcal{F}$. In fact, we can choose the family $\mathcal{F}$ of linear functionals as the set of all the continuous linear functionals on $\mathcal{H}$. That is, $\mathcal{F}=\mathcal{H}^{*}$, the dual space of $\mathcal{H}$. Since $\mathcal{H}^{*}$ is isometrically anti-isomorphic to $\mathcal{H}$, we can easily obtain that the norm of $\mathcal{H}$ is compatible with $\mathcal{F}$ and then $\mathcal{H}$ is an FRKHS with respect to $\mathcal{F}$. Although any Hilbert space is an FRKHS with respect to some family of linear functionals, we are interested in finding an FRKHS with respect to a specifically given family of linear functionals. This is nontrivial and useful in practical applications.
We begin with presenting several specific examples of ORKHSs and demonstrating that the notion of ORKHSs is not a trivial extension, but a useful development of the concept of RKHSs. Two typical RKHSs are special examples of ORKHSs. Suppose that $\mathcal{X}$ is a set and $\mathcal{Y}$ is a Hilbert space. The space $\mathcal{H}$ is chosen as a Hilbert space of functions from $\mathcal{X}$ to $\mathcal{Y}$ and the family $\mathcal{L}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ is chosen as the set of the point-evaluation operators defined by $L_x(f):=f(x),\ x\in \mathcal{X}$. Then it is clear that the norm of $f\in\mathcal{H}$ vanishes if and only if $f$, as a function, vanishes everywhere on $\mathcal{X}$, which indicates that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. If each point-evaluation operator in $\mathcal{L}$ is continuous on $\mathcal{H}$, the corresponding ORKHS $\mathcal{H}$ reduces to the vector-valued RKHS. In the special case when $\mathcal{Y}=\mathbb{C}$, we choose $\mathcal{H}$ as a Hilbert space of functions on a set $\mathcal{X}$ and family $\mathcal{F}$ of linear functionals as the collection of the point-evaluation functionals, the corresponding FRKHS $\mathcal{H}$ also reduces to the classical scalar-valued RKHS.
We are particularly interested in ORKHSs with respect to a family of linear operators not restricted to point-evaluation operators. The next three examples concerning integral operators show that the notion of ORKHSs is a useful development of the concept of classical RKHSs. To this end, we recall the Bochner integral of a vector-valued function [@DU]. Let $(\mathcal{X},\mathcal{M},\mu)$ be a measure space and $\mathcal{Y}$ a Hilbert space. To define the integral of a function from $\mathcal{X}$ to $\mathcal{Y}$ with respect to $\mu$, we first introduce the integral of a simple function with the form $f:=\sum_{j\in \mathbb{N}_n}\chi_{E_j}\xi_j,$ where $\xi_j,j\in \mathbb{N}_n,$ are distinct elements of $\mathcal{Y}$, $E_j$, $j\in \mathbb{N}_n$, are pairwise disjoint measurable sets and $\chi_{E_j}$ denotes the characteristic function of $E_j$. If $\mu(E_j)$ is finite whenever $\xi_j\neq0$, the integral of $f$ on a measurable set $E$ is defined by $$\int_{E}f(x)d\mu(x):=\sum_{j\in \mathbb{N}_n}
\mu(E_j\cap E)\xi_j.$$ A function $f:\mathcal{X}\rightarrow \mathcal{Y}$ is called measurable if there exists a sequence of simple functions $f_n:\mathcal{X}\rightarrow \mathcal{Y}$ such that $$\lim_{n\rightarrow \infty}\|f_n(x)-f(x)\|_\mathcal{Y}=0,
\ \ \mbox{almost everywhere with respect to} \ \mu\ \mbox{on}\ \mathcal{X}.$$ A measurable function $f:\mathcal{X}\rightarrow \mathcal{Y}$ is called Bochner integrable if there exists a sequence of simple functions $f_n:\mathcal{X}\rightarrow \mathcal{Y}$ such that $$\lim_{n\rightarrow \infty}\int_{\mathcal{X}}
\|f_n(x)-f(x)\|_\mathcal{Y}d\mu(x)=0.$$ One can get by the above equation for any measurable set $E$ that the sequence $\int_{E}f_n(x)d\mu(x)$ is a Cauchy sequence in $\mathcal{Y}$. This together with the completeness of $\mathcal{Y}$ allows us to define the integral of $f$ on $E$ by $$\int_{E}f(x)d\mu(x):=\lim_{n\rightarrow\infty}
\int_{E}f_n(x)d\mu(x).$$ It is known that a measurable function $f:\mathcal{X}\rightarrow \mathcal{Y}$ is Bochner integrable if and only if there holds $$\int_{\mathcal{X}}\|f(x)\|_{\mathcal{Y}}d\mu(x)<+\infty.$$ It follows that $$\left|\int_{\mathcal{X}}\langle \xi,f(x)
\rangle_{\mathcal{Y}}d\mu(x)\right|
\leq\|\xi\|_{\mathcal{Y}}\int_{\mathcal{X}}
\|f(x)\|_{\mathcal{Y}}d\mu(x),
\ \mbox{for any}\ \xi\in \mathcal{Y},$$ yielding that the functional $\int_{\mathcal{X}}\langle\cdot,f(x)\rangle_{\mathcal{Y}} d\mu(x)$ is continuous on $\mathcal{Y}$. Consequently, we can comprehend the integral $\int_{\mathcal{X}}f(x)d\mu(x)$ as an element of $\mathcal{Y}$ satisfying $$\left\langle \xi, \int_{\mathcal{X}}f(x)d\mu(x)
\right\rangle_\mathcal{Y}=\int_{\mathcal{X}}
\langle \xi,f(x)\rangle_\mathcal{Y} d\mu(x),
\ \mbox{for any}\ \xi\in \mathcal{Y}.$$ For $1\leq p<+\infty$, we denote by $L^p(\mathcal{X},\mathcal{Y},\mu)$ the space of all the measurable functions $f:\mathcal{X}\rightarrow \mathcal{Y}$ such that $$\|f\|_{L^p(\mathcal{X},\mathcal{Y},\mu)}:=
\left(\int_{\mathcal{X}}\|f(x)\|_{\mathcal{Y}}^pd\mu(x)
\right)^{1/p}<+\infty.$$ In the case that $\mu$ is the Lebesgue measure on $\mathbb{R}^d$ and $\mathcal{X}$ is a measurable set of $\mathbb{R}^d$, $L^p(\mathcal{X},\mathcal{Y},\mu)$ will be abbreviated as $L^p(\mathcal{X},\mathcal{Y})$. Moreover, if $\mathcal{Y}=\mathbb{C}$, we also abbreviate $L^p(\mathcal{X},\mathcal{Y})$ as $L^p(\mathcal{X})$. Our interest in integral operators is motivated by a practical scenario that the point-evaluation function values can not be measured exactly in practice due to inevitable measurement errors. Alternatively, it may be of practical advantage to use local averages of $f$ as observed information [@A; @MP04; @S; @SZ; @V].
The first two concrete examples of ORKHSs with respect to integral operators is related to $L^2(\mathcal{X},\mathbb{C}^{n})$, where $\mathcal{X}$ is a measurable set of $\mathbb{R}^d$ and $n$ is a positive integer. Let $\Lambda$ be an index set. We suppose that a family of functions $\{u_{\alpha}:=[u_{\alpha,j}:j\in\mathbb{N}_{n}]: \alpha
\in\Lambda\}\subset L^2(\mathcal{X},\mathbb{C}^{n})$ has the property that for each $j\in\mathbb{N}_{n}$ the linear span of $\{u_{\alpha,j}: \alpha\in\Lambda\}$ is dense in $L^2(\mathcal{X})$. For each $\alpha\in \Lambda$ we define a linear operator $L_{\alpha}$ for $f\in L^2(\mathcal{X},\mathbb{C}^{n})$ by $$\label{example_integration_functional}
L_{\alpha}(f):=\int_{\mathcal{X}}f(x)\circ
\overline{u_{\alpha}(x)}dx,$$ and set $\mathcal{L}:=\{L_{\alpha}: \alpha\in\Lambda\}$. Here, we denote by $u\circ v$ the Hadamard product of two functions $u$ and $v$ from $\mathcal{X}$ to $\mathbb{C}^{n}$. We first show that for each $\alpha\in \Lambda$, $L_{\alpha}$ is well-defined. Note that $f=[f_j:j\in\mathbb{N}_{n}]\in L^2(\mathcal{X},\mathbb{C}^{n})$ if and only if $f_j\in L^2(\mathcal{X})$ for all $j\in\mathbb{N}_{n}$. For each $\alpha\in\Lambda$ and each $f\in L^2(\mathcal{X},\mathbb{C}^{n})$ there holds $$\int_{\mathcal{X}}\|f(x)\circ\overline{u_{\alpha}(x)}
\|_{\mathbb{C}^{n}}dx=\int_{\mathcal{X}}
\left(\sum_{j\in\mathbb{N}_{n}}|f_j(x)u_{\alpha,j}(x)|^2
\right)^{1/2}dx\leq\sum_{j\in\mathbb{N}_{n}}
\int_{\mathcal{X}}|f_j(x)u_{\alpha,j}(x)|dx.$$ Since $f_j, u_{\alpha,j}\in L^2(\mathcal{X})$ for each $j\in\mathbb{N}_{n}$, by the Hölder’s inequality, we have that $$\int_{\mathcal{X}}\|f(x)\circ\overline{u_{\alpha}(x)}
\|_{\mathbb{C}^{n}}dx\leq\sum_{j\in\mathbb{N}_{n}}
\|f_j\|_{L^2(\mathcal{X})}\|u_{\alpha,j}
\|_{L^2(\mathcal{X})}<+\infty.$$ This shows that $f\circ\overline{u}_{\alpha}$ is Bochner integrable, which guarantees the well-definedness of the operator $L_{\alpha}$. We next verify that $L^2(\mathcal{X},\mathbb{C}^{n})$ is an ORKHS with respect to $\mathcal{L}$. By the definition of the Bochner integral, we have for each $\alpha\in\Lambda$ and $f=[f_j:j\in\mathbb{N}_{n}]\in L^2(\mathcal{X},\mathbb{C}^{n})$ that $$L_{\alpha}(f)=\left[\int_{\mathcal{X}}f_j(x)
\overline{u_{\alpha,j}(x)}dx:j\in\mathbb{N}_{n}\right].$$ This together with the density of the linear span of $u_{\alpha,j}\in L^2(\mathcal{X}),\ \alpha\in\Lambda$, leads to the statement that $L_{\alpha}(f)=0$ for all $\alpha\in\Lambda$ if and only if $\|f\|_{L^2(\mathcal{X},\mathbb{C}^{n})}=0$. Hence, the norm of $L^2(\mathcal{X},\mathbb{C}^{n})$ is compatible with $\mathcal{L}$. Moreover, with the help of the Hölder’s inequality, we have for each $\alpha\in\Lambda$ and $f\in L^2(\mathcal{X},\mathbb{C}^{n})$ that $$\|L_{\alpha}(f)\|_{\mathbb{C}^{n}}
\leq\left(\sum_{j\in\mathbb{N}_{n}}
\|f_j\|^2_{L^2(\mathcal{X})}
\|u_{\alpha,j}\|^2_{L^2(\mathcal{X})}
\right)^{1/2}\leq c_{\alpha}
\|f\|_{L^2(\mathcal{X},\mathbb{C}^{n})},$$ where $c_{\alpha}:=\max\{\|u_{\alpha,j}
\|_{L^2(\mathcal{X})}:j\in\mathbb{N}_{n}\}$. This ensures that the operator $L_{\alpha}$ defined by (\[example\_integration\_functional\]) is continuous on $L^2(\mathcal{X},\mathbb{C}^{n})$. Consequently, by Definition \[def\_ORKHS\] we conclude the following result.
\[ORKHS1\] If the family of functions $\{u_{\alpha}:=[u_{\alpha,j}:j\in\mathbb{N}_{n}]: \alpha
\in\Lambda\}\subset L^2(\mathcal{X},\mathbb{C}^{n})$ has the property that for each $j\in\mathbb{N}_{n}$ the linear span of $\{u_{\alpha,j}: \alpha\in\Lambda\}$ is dense in $L^2(\mathcal{X})$, then the space $L^2(\mathcal{X},\mathbb{C}^{n})$ is an ORKHS with respect to $\mathcal{L}$ defined as in in terms of $u_{\alpha}$.
Two specific choices of the family of functions $\{u_{\alpha}:\alpha\in\Lambda\}$ lead to two ORKHSs which may be potentially important in practical applications. The first one is motivated by reconstructing a signal from its frequency components, which is one of the main themes in signal analysis. Fourier analysis has been proved to be an efficient approach for providing global energy-frequency distributions of signals. Because of its prowess and simplicity, Fourier analysis is considered as a popular tool in signal analysis and other areas of applications. In this example, we restrict ourselves to the case of periodic functions on $\mathbb{R}$, where the frequency information of such a signal is represented by its Fourier coefficients. Specifically, we consider the Hilbert space $L^2([0,2\pi])$. Set $\Lambda:=\mathbb{Z}$ and choose the functions $u_j,\ j\in\Lambda,$ as the Fourier basis functions. That is, $$u_j:=\frac{1}{\sqrt{2\pi}}e^{ij(\cdot)},\ \ j\in\Lambda.$$ Associated with these functions, we introduce a family of linear functionals on $L^2([0,2\pi])$ as $$\label{integration_functional_Fourier}
L_j(f):=\int_{0}^{2\pi}f(x)\overline{u_j(x)}dx,\ f\in L^2([0,2\pi]),\ j\in\Lambda,$$ and set $\mathcal{L}:=\{L_j:j\in\Lambda\}$. It follows from Proposition \[ORKHS1\] and the density of the Fourier basis functions that the space $L^2([0,2\pi])$ is an FRKHS with respect to $\mathcal{L}$.
Wavelet analysis is an important mathematical tool in various applications, such as signal processing, communication and numerical analysis. Various wavelet functions have been constructed for practical purpose in the literature [@CMX; @D; @Mallat; @M; @MX]. Particularly, wavelet analysis provides an alternative representation of signals. The wavelet transform contains the complete information of a signal in both time and frequency domains at the same time. Our next example comes from the problem of recovering a signal from finite wavelet coefficients, which can be seen as the linear functional values of the signal. Suppose that $\psi\in L^2(\mathbb{R}^d)$ is a wavelet function such that the sequence $\psi_{j,k},\ j\in\mathbb{Z},\ k\in\mathbb{Z}^d$, defined by $$\psi_{j,k}(x):=2^{\frac{jd}{2}}\psi(2^jx-k), \ \ x\in\mathbb{R}^d, \ \ j\in\mathbb{Z},\ k\in\mathbb{Z}^d,$$ constitutes an orthonormal basis for $L^2(\mathbb{R}^d)$. We consider the space $L^2(\mathbb{R}^d)$. To introduce a family of linear functionals, we let $\Lambda:=\mathbb{Z}\times\mathbb{Z}^{d}$ and for each $(j,k)\in\Lambda$ set $u_{(j,k)}:=\psi_{j,k}$. Accordingly, the linear functionals are defined by $$\label{integration_functional_wavelet}
L_{(j,k)}(f):=\int_{\mathbb{R}^d}f(x)\overline{u_{(j,k)}(x)}
dx,\ f\in L^2(\mathbb{R}^d),\ (j,k)\in\Lambda.$$ By Proposition \[ORKHS1\], we obtain directly that the space $L^2(\mathbb{R}^d)$ is an FRKHS with respect to the family $\mathcal{L}:=\{L_{(j,k)}:\ (j,k)\in\Lambda\}$ of continuous linear functionals.
As a third concrete example of ORHKSs with respect to integral operators, we consider the Paley-Wiener space of functions from $\mathbb{R}^d$ to $\mathbb{C}^{n}$. For a set of positive numbers $\Delta:=\{\delta_j:j \in\mathbb{N}_d\}$, we denote by $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ the closed subspaces of functions in $L^2(\mathbb{R}^d,\mathbb{C}^{n})$ which are bandlimited to the region $$I_{\Delta}:=\prod_{j\in\mathbb{N}_d}[-\delta_j,\delta_j].$$ That is, $$\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n}):=
\left\{f\in L^2(\mathbb{R}^d,\mathbb{C}^{n}): \ {\rm supp}\hat f\subseteq I_{\Delta}\right\},$$ with the inner product $$\langle f,g\rangle_{\mathcal{B}_{\Delta}
(\mathbb{R}^d,\mathbb{C}^{n})}:=
\sum_{j\in\mathbb{N}_{n}}\langle f_j,g_j\rangle_{L^{2}(\mathbb{R}^d)}.$$ Here, we define the Fourier transform $\hat{h}$ and the inverse Fourier transform $\check{h}$ of $h\in L^1(\mathbb{R}^d,\mathbb{C}^n)$, respectively, by $$\hat h(\omega):=\int_{\mathbb{R}^d}
h(t)e^{-i(t,\omega)}dt,\ \omega\in\mathbb{R}^d,$$ and $$\check{h}(\omega):=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}
h(t)e^{i(t,\omega)}dt,\ \omega\in\mathbb{R}^d,$$ with $(\cdot,\cdot)$ being the standard inner product on $\mathbb{R}^d$. By standard approximation process, the Fourier transform and its inverse can be extended to the functions in $L^2(\mathbb{R}^d,\mathbb{C}^n)$. It is well-known that the space $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ is a vector-valued RKHS. In other words, it is an ORKHS with respect to the family of the point-evaluation operators. We shall show that it is also an ORKHS with respect to a family of the integral operators to be defined next. Specifically, we suppose that $u\in L^2(\mathbb{R}^d)$ satisfies the property that $\hat{u}(\omega)\neq0, \ {\rm a.e.}\ \omega\in I_{\Delta}$. For each $x\in\mathbb{R}^d$, we set $u_x:=u(\cdot-x)$ and introduce the operator $$\label{example_integration_functional1}
L_x(f):=\int_{\mathbb{R}^d}f(t)\overline{u_x(t)}dt$$ and define $\mathcal{L}:=\{L_x: x\in \Lambda\}$, with $\Lambda:=\mathbb{R}^d$. We first show that the norm of $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ is compatible with $\mathcal{L}$. It suffices to verify for each $f\in\mathcal{B}_{\Delta}
(\mathbb{R}^d,\mathbb{C}^{n})$ that if $L_x(f)=0$ for all $x\in\mathbb{R}^d$, then $\|f\|_{L^2(\mathbb{R}^d, \mathbb{C}^{n})}=0$. By the Plancherel identity we have for any $x\in \mathbb{R}^d$ and any $\xi\in \mathbb{C}^n$ that $$\langle L_x(f),\xi\rangle_{\mathbb{C}^{n}}=
\int_{\mathbb{R}^d}\langle f(t),\xi
\rangle_{\mathbb{C}^{n}}\overline{u_x(t)}dt
=\int_{I_{\Delta}}\sum_{j\in\mathbb{N}_{n}}
\overline{\xi_j}\hat{f}_j(\omega)
\overline{\hat{u}(\omega)}e^{i(x,\omega)}d\omega.$$ The density $\overline{\span}\{e^{i(x,\cdot)}:
x\in\mathbb{R}^d\}=L^2(I_{\Delta})$ ensures that the condition $L_x(f)=0$ for all $x\in\mathbb{R}^d$ is equivalent to that $$\label{new_sum}
\sum_{j\in\mathbb{N}_{n}}\overline{\xi_j}
\hat{f}_j(\omega)\overline{\hat{u}(\omega)}=0,\ \mbox{a.e.} \ \omega\in I_{\Delta},
\ \mbox{for all}\ \ \xi\in\mathbb{C}^{n}.$$ By the fact that $\hat{u}(\omega)\neq0, \ {\rm a.e.}\ \omega\in I_{\Delta}$, holds if and only if $\|f_j\|_{L^2(\mathbb{R}^d)}=0,j\in\mathbb{N}_{n}$, which is equivalent to $\|f\|_{L^2(\mathbb{R}^d, \mathbb{C}^{n})}=0$. Thus, we have proved that the norm of $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ is compatible with $\mathcal{L}$. Moreover, applying the Hölder’s inequality to the definition of operator $L_x$ yields that for each $x\in\mathbb{R}^d$ the operator $L_x$ is continuous on $\mathcal{B}_{\Delta}
(\mathbb{R}^d,\mathbb{C}^{n})$. According to Definition \[def\_ORKHS\], we obtain the following result regarding space $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$.
\[Paley\] If $u\in L^2(\mathbb{R}^d)$ satisfies $\hat{u}(\omega)\neq0, \ {\rm a.e.}\ \omega\in I_{\Delta}$, then $\mathcal{B}_{\Delta}
(\mathbb{R}^d, \mathbb{C}^{n})$ is an ORKHS with respect to $\mathcal{L}$ defined as in .
Finally, we describe a finite-dimensional ORKHS. Let $\mathcal{H}$ be a Hilbert space of dimension $n$ and $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ a family of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$. We suppose that there exists a finite set $\{\alpha_j:j\in\mathbb{N}_n\}$ of $\Lambda$ such that $L_{\alpha_j},\ j\in\mathbb{N}_n,$ are linearly independent. That is, if for $\{\xi_j:j\in\mathbb{N}_n\}\subseteq\mathcal{Y}$ there holds $$\sum_{j\in\mathbb{N}_n}\langle L_{\alpha_j}(f),\xi_j\rangle_{\mathcal{Y}}=0,\ \mbox{for all} \ f\in\mathcal{H},$$ then $\xi_j=0, j\in\mathbb{N}_n.$ It is known that any linear operator from a finite-dimensional normed space to a normed space is continuous. Hence, the continuity of the linear operators in $\mathcal{L}$ is clear. It suffices to verify that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. Let $\xi$ be a nonzero element in $\mathcal{Y}$. As $L_{\alpha_j}, j\in\mathbb{N}_n,$ are continuous on $\mathcal{H}$, there exists a finite set $\{\phi_j: j\in\mathbb{N}_n\}\subseteq\mathcal{H}$ such that $$\label{finitebasis}
\langle L_{\alpha_j}(f),\xi\rangle_{\mathcal{Y}}=
\langle f,\phi_j\rangle_{\mathcal{H}}, \ \mbox{for all}\ f\in\mathcal{H}.$$ We shall show that $\phi_j, j\in\mathbb{N}_n$, are linear independent. Assume that there holds for some scalars $c_j,j\in\mathbb{N}_n$, that $\sum_{ j\in\mathbb{N}_n}
c_j\phi_j=0$. This together with (\[finitebasis\]) leads to $$\sum_{j\in\mathbb{N}_n}\langle L_{\alpha_j}(f), \overline{c_j}\xi\rangle_{\mathcal{Y}}=\left< f,\sum_{ j\in\mathbb{N}_n}c_j\phi_j\right>_{\mathcal{H}}=0, \ \mbox{for all}\ f\in\mathcal{H}.$$ By the linear independency of $L_{\alpha_j}, j\in\mathbb{N}_n$, we have that $c_j=0$ for all $j\in\mathbb{N}_n.$ That is, $\phi_j, j\in\mathbb{N}_n$, are linear independent, which implies $\span\{\phi_j: j\in\mathbb{N}_n\}=\mathcal{H}$. Suppose that $f\in\mathcal{H}$ satisfies $L_{\alpha_j}(f)=0$ for all $j\in\mathbb{N}_n$. We then get by (\[finitebasis\]) that $\langle f,\phi_j\rangle_{\mathcal{H}}=0, j\in\mathbb{N}_n$, which yields $\|f\|_{\mathcal{H}}=0$. Consequently, we have proved that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. By Definition \[def\_ORKHS\], we conclude that $\mathcal{H}$ is an ORKHS with respect to $\mathcal{L}$, which is stated in the following proposition.
\[Finitedimension\] Let $\mathcal{H}$ be a Hilbert space of dimension $n$ and $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ a family of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$. If there exist $n$ linearly independent linear operators in $\mathcal{L}$, then $\mathcal{H}$ is an ORKHS with respect to $\mathcal{L}$.
Observing from the examples presented above, we point out that the spaces $L^2([0,2\pi])$ and $L^2(\mathbb{R}^d)$ are FRKHSs but not classical RKHSs as they consist of equivalent classes of functions with respect to the Lebesgue measure. However, the space $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^n)$ admits two different families of linear operators from itself to $\mathbb{C}^n$, one being the standard point-evaluation operators and the other being the integral operators having the form of (\[example\_integration\_functional1\]). If the finite-dimensional ORKHS $\mathcal{H}$ consists of functions from an input set $\mathcal{X}$ to $\mathcal{Y}$, then all the point-evaluation operators are continuous on $\mathcal{H}$. Hence, the finite dimensional space $\mathcal{H}$ also admits the family of point-evaluation operators, besides the family $\mathcal{L}$. The ORKHSs having this property are especially desirable, which we shall study in the next two sections.
We now turn to investigating the properties of general ORKHSs. A core in the theory of RKHSs lies in the celebrated result that there is a bijective correspondence between the reproducing kernel and the corresponding RKHS. We next establish a similar bijection in the framework of ORKHSs.
We first identify the operator reproducing kernel to each ORKHS. Recall that a vector-valued RKHS admits a reproducing kernel and the functional values $\langle f(x), \xi\rangle_{\mathcal{Y}}$, $x\in \mathcal{X}, \xi\in\mathcal{Y}$, of a function $f$ in the vector-valued RKHS can be reproduced via its inner product with the kernel. In a manner similar to the vector-valued RKHS, we show that for a general ORKHS there exists a kernel which can be used to reproduce the functional values $\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}$, $\alpha\in \Lambda,\xi\in\mathcal{Y}$. For two Hilbert spaces $\mathcal{H}_1,\mathcal{H}_2$, we denote by $\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ the set of bounded linear operators from $\mathcal{H}_1$ to $\mathcal{H}_2$. For each $L\in \mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$, we let $\|L\|_{\mathcal{B}(\mathcal{H}_1, \mathcal{H}_2)}$ denote the operator norm of $L$ in $\mathcal{B}(\mathcal{H}_1,\mathcal{H}_2)$ and $L^{*}$ denote the adjoint of $L$.
\[property\] Suppose that $\mathcal{H}$ is a Hilbert space and $\mathcal{L}:=\{L_{\alpha}: \alpha\in \Lambda\}$ is a family of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$. If $\mathcal{H}$ is an ORKHS with respect to $\mathcal{L}$, then there exists a unique operator $K:\Lambda\rightarrow
\mathcal{B}(\mathcal{Y},\mathcal{H})$ such that the following statements hold.
\(1) For each $\alpha\in \Lambda$ and each $\xi\in\mathcal{Y}$, $K(\alpha)\xi\in \mathcal{H}$ and $$\label{reproducing_property}
\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\langle f,K(\alpha)\xi\rangle_\mathcal{H},
\ \ \mbox{for all}\ \ f\in \mathcal{H}.$$
\(2) The linear span $\mathcal{S}_{K}:=\span\{K(\alpha)\xi:
\alpha\in\Lambda,\xi\in\mathcal{Y}\}$ is dense in $\mathcal{H}$.
\(3) For each $n\in\mathbb{N}$, each pair of finite sets $\Lambda_n:=\{\alpha_j:j\in\mathbb{N}_n\}
\subseteq\Lambda$ and $\mathcal{Y}_n:=\{\xi_j:
j\in\mathbb{N}_n\}\subseteq\mathcal{Y}$ there holds $$\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_n}
\langle L_{\alpha_k}(K(\alpha_j)\xi_j),\xi_k
\rangle_{\mathcal{Y}}\geq0.$$
We construct the operator $K:\Lambda\rightarrow\mathcal{B}(\mathcal{Y},\mathcal{H})$ which satisfies (1)-(3). For each $\alpha\in \Lambda$, since $L_{\alpha}\in\mathcal{B}(\mathcal{H},\mathcal{Y})$, the adjoint $L_{\alpha}^{*}$ is the unique operator in $\mathcal{B}(\mathcal{Y},\mathcal{H})$ satisfying $$\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\langle f,L_{\alpha}^{*}\xi\rangle_\mathcal{H},\ \mbox{for all} \ f\in \mathcal{H}\ \mbox{and}\ \xi\in\mathcal{Y}.$$ We define the operator $K$ from $\Lambda$ to $\mathcal{B}(\mathcal{Y},\mathcal{H})$ by $K(\alpha):=L_{\alpha}^{*}$, $\alpha\in\Lambda$. Then, clearly, $K$ satisfies statement (1).
To prove statement (2), we suppose that $f\in\mathcal{H}$ satisfies $\langle f,K(\alpha)\xi\rangle_{\mathcal{H}}=0$, for all $\alpha\in \Lambda, \xi\in\mathcal{Y}$. This together with (\[reproducing\_property\]) ensures that $L_{\alpha}(f)=0$ for all $\alpha\in \Lambda$. By the hypothesis of this theorem, $\mathcal{H}$ is an ORKHS with respect to $\mathcal{L}$. It follows that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. Thus, we have that $f=0$. This implies that $\overline{\mathcal{S}_{K}}=\mathcal{H}.$
We next show statement (3). For each $n\in\mathbb{N}$ and for $\{\alpha_j:j\in\mathbb{N}_n\}\subseteq\Lambda$ and $\{\xi_j:j\in\mathbb{N}_n\}\subseteq\mathcal{Y}$, we let $f_n:=\sum_{j\in\mathbb{N}_n} K(\alpha_j)\xi_j$. We then observe from equation (\[reproducing\_property\]) that $$\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_n}
\langle L_{\alpha_k}(K(\alpha_j)\xi_j),
\xi_k\rangle_{\mathcal{Y}}
=\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_n}
\langle K(\alpha_j)\xi_j, K(\alpha_k)\xi_k
\rangle_{\mathcal{H}}
=\left<f_n,f_n\right>_{\mathcal{H}}\geq0,$$ proving the desired result.
It is known from equation (\[reproducing\_property\]) that one can use the operator $K$ to reproduce the functional values $\langle L_{\alpha}(f),
\xi\rangle_{\mathcal{Y}},\ \alpha\in\Lambda,
\xi\in\mathcal{Y}$. Hence, we shall call equation (\[reproducing\_property\]) the reproducing property and $K$ the operator reproducing kernel for ORKHS $\mathcal{H}$. We remark on the term “operator reproducing kernel". The reproducing kernel $\mathcal{K}$ for a vector-valued RKHS, consisting of functions from $\mathcal{X}$ to a Hilbert space $\mathcal{Y}$, is called in the literature an operator-valued reproducing kernel. It is because for each $(x,y)\in\mathcal{X}\times \mathcal{X}$, $\mathcal{K}(x,y)$ is an operator from $\mathcal{Y}$ to itself. However, the term “operator" in the definition of an operator reproducing kernel for an ORKHS $\mathcal{H}$ is used to demonstrate the family of operators, with respect to which $\mathcal{H}$ is an ORKHS.
As a special case, the conclusion in Theorem \[property\] in the scalar case when $\mathcal{Y}=\mathbb{C}$ can be stated as the following corollary.
\[FRKHS\_property\] Suppose that $\mathcal{H}$ is a Hilbert space and $\mathcal{F}:=\{L_{\alpha}: \alpha\in \Lambda\}$ is a family of linear functionals on $\mathcal{H}$. If $\mathcal{H}$ is an FRKHS with respect to $\mathcal{F}$, then there exists a unique operator $K:\Lambda\rightarrow \mathcal{H}$ such that the linear span $\mathcal{S}_{K}:=\span\{K(\alpha):\alpha\in \Lambda\}$ is dense in $\mathcal{H}$ and for each $\alpha\in \Lambda$, there holds the reproducing property $$\label{FRKHS_reproducing_property}
L_{\alpha}(f)=\langle f,K(\alpha)\rangle_\mathcal{H},
\ \ \mbox{for all}\ \ f\in \mathcal{H}.$$ Moreover, for each $n\in\mathbb{N}$ and each finite set $\Lambda_n:=\{\alpha_j:j\in \mathbb{N}_n\}
\subseteq\Lambda$ the matrix ${\bf F}_{\Lambda_n}:=[L_{\alpha_k}(K(\alpha_j)):
j,k\in\mathbb{N}_n]$ is hermitian and positive semi-definite.
We present the operator reproducing kernels for the specific ORKHSs discussed earlier. We first consider the ORKHS identical to the vector-valued RKHS. It is known [@MP05] that a vector-valued RKHS $\mathcal{H}$ consisting of functions from $\mathcal{X}$ to $\mathcal{Y}$ has a reproducing kernel $\mathcal{K}$ from $\mathcal{X}\times \mathcal{X}$ to $\mathcal{B}(\mathcal{Y},\mathcal{Y})$ such that $$\langle f(x),\xi\rangle_{\mathcal{Y}}
=\langle f,\mathcal{K}(x,\cdot)\xi\rangle_{\mathcal{H}},
\ f\in\mathcal{H},\ x\in \mathcal{X},\ \xi\in\mathcal{Y}.$$ Then the operator reproducing kernel $K$ may be represented via $\mathcal{K}$ by $$K(x)\xi:=\mathcal{K}(x,\cdot)\xi,\ x\in \mathcal{X},
\ \xi\in\mathcal{Y}.$$ It follows from $$\|K(x)\xi\|_{\mathcal{H}}
=\langle \mathcal{K}(x,\cdot)\xi,\mathcal{K}(x,\cdot)
\xi\rangle_{\mathcal{H}}^{1/2}
=\langle \mathcal{K}(x,x)\xi,\xi
\rangle_{\mathcal{Y}}^{1/2}
\leq\|\mathcal{K}(x,x)\|^{1/2}_{\mathcal{B}
(\mathcal{Y},\mathcal{Y})}\|\xi\|_{\mathcal{Y}}$$ that $K(x)\in \mathcal{B}(\mathcal{Y},\mathcal{H})$. Moreover, when $\mathcal{Y}=\mathbb{C}$, the functional reproducing kernel can be represented via the classical reproducing kernel $\mathcal{K}:\mathcal{X}
\times \mathcal{X}\rightarrow \mathbb{C}$ by $K(x):=\mathcal{K}(x,\cdot),\ x\in \mathcal{X}$. In the remaining part of this paper, we shall use the conventional notation of the reproducing kernels for the classical scalar-valued or vector-valued RKHSs.
The kernel for the ORKHS $L^2(\mathcal{X},\mathbb{C}^{n})$ with respect to the family $\mathcal{L}$ of the integral operators can be represented by the functions $u_{\alpha},\ \alpha\in\Lambda,$ appearing in (\[example\_integration\_functional\]). To see this, we let $\mathcal{L}$ be the family of the integral operators defined as in (\[example\_integration\_functional\]) and recall that if for each $j\in\mathbb{N}_{n}$ the linear span of $u_{\alpha,j}, \alpha\in\Lambda$, is dense in $L^2(\mathcal{X})$, then $L^2(\mathcal{X},\mathbb{C}^{n})$ is an ORKHS with respect to $\mathcal{L}$. Observing from $$\langle L_{\alpha}(f),\xi\rangle_{\mathbb{C}^{n}}
=\int_{\mathcal{X}}\langle f(x),u_{\alpha}(x)
\circ\xi\rangle_{\mathbb{C}^{n}}dx
=\langle f,u_{\alpha}\circ\xi
\rangle_{L^2(\mathcal{X},\mathbb{C}^{n})},$$ we get that the operator reproducing kernel for the ORKHS $L^2(\mathcal{X},\mathbb{C}^{n})$ has the form $$K(\alpha)\xi:=u_{\alpha}\circ\xi,
\ \alpha\in\Lambda,\ \xi\in\mathbb{C}^{n}.$$ Particularly, the functional reproducing kernel for $L^2([0,2\pi])$ with respect to the family $\mathcal{L}$ of the linear functionals defined by (\[integration\_functional\_Fourier\]) is determined by $$K(j):=\frac{1}{\sqrt{2\pi}}e^{ij(\cdot)},
\ j\in\mathbb{Z}.$$ Similarly, the functional reproducing kernel for $L^2(\mathbb{R}^d)$ with respect to the family $\mathcal{L}$ of the linear functionals defined by (\[integration\_functional\_wavelet\]) coincides with the wavelet basis functions. That is, $$K(j,k):=2^{\frac{jd}{2}}\psi(2^j(\cdot)-k),
\ j\in\mathbb{Z},k\in\mathbb{Z}^d.$$
Another example of ORKHSs with respect to the family of the integral operators is the Paley-Wiener space of functions from $\mathbb{R}^d$ to $\mathbb{C}^{n}$. Recall that $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ is a vector-valued RKHS with the translation-invariant reproducing kernel $$\label{sinc}
\mathcal{K}(x,y):={\rm diag}(\widetilde{\mathcal{K}}(x,y):
l\in\mathbb{N}_{n}),$$ where $$\widetilde{\mathcal{K}}(x,y):=\prod_{j\in\mathbb{N}_d}
\frac{\sin \delta_j(x_j-y_j)}{\pi(x_j-y_j)},\ x,y\in \mathbb{R}^d.$$ Let $\mathcal{L}$ be the family of the integral operators defined as in (\[example\_integration\_functional1\]). We also verified earlier in this section that $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ is an ORKHS with respect to such a set $\mathcal{L}$. Let $\mathscr{P}_{\mathcal{B}_{\Delta}}$ denote the orthogonal projection from $L^2(\mathbb{R}^d,\mathbb{C}^{n})$ onto $\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$. Then, there holds for each $x\in \mathbb{R}^d$ and each $f\in\mathcal{B}_{\Delta}(\mathbb{R}^d,\mathbb{C}^{n})$ $$\langle L_x(f),\xi\rangle_{\mathbb{C}^n}=\langle f,u(\cdot-x)\xi\rangle_{L^2(\mathbb{R}^d,\mathbb{C}^{n})}
=\langle f,\mathscr{P}_{\mathcal{B}_{\Delta}}
(u(\cdot-x)\xi)\rangle_{L^2(\mathbb{R}^d,\mathbb{C}^{n})}.$$ This leads to $K(x)\xi=\mathscr{P}_{\mathcal{B}_{\Delta}}
(u(\cdot-x)\xi)$, $x\in\mathbb{R}^d,\xi\in\mathbb{C}^{n}$.
Our last example concerns the finite-dimensional ORKHS. If there exist $n$ linearly independent elements in the family $\mathcal{L}$ of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$, then the finite-dimensional Hilbert space $\mathcal{H}$ was proved to be an ORKHS with respect to such a family $\mathcal{L}$. To present the operator reproducing kernel, we suppose that $\{\phi_j: j\in\mathbb{N}_n\}$ is a linearly independent sequence in $\mathcal{H}$. For each $\alpha\in\Lambda$ and each $\xi\in\mathcal{Y}$, we can represent $K(\alpha)\xi\in\mathcal{H}$ as $$K(\alpha)\xi=\sum_{j\in\mathbb{N}_n}c_j\phi_j,\
\ \mbox{for some sequence}\ \ \{c_j:j\in\mathbb{N}_n\}\subseteq\mathbb{C}.$$ It follows from the reproducing property (\[reproducing\_property\]) that the sequence $\{c_j:j\in\mathbb{N}_n\}$ satisfies the following system $$\langle \xi, L_{\alpha}(\phi_k)\rangle_{\mathcal{Y}}
=\sum_{j\in\mathbb{N}_n}c_j
\langle\phi_j,\phi_k\rangle_\mathcal{H},\ k\in\mathbb{N}_n.$$ We denote by $\mathbf{A}$ the coefficients matrix, that is, $\mathbf{A}:=[\langle\phi_k,\phi_j\rangle_\mathcal{H}:
j,k\in\mathbb{N}_n]$. Due to the linear independence of $\phi_j,\ j\in\mathbb{N}_n,$ matrix $\mathbf{A}$ is positive definite. Denoting by $\mathbf{B}:=[b_{j,k}:
j,k\in\mathbb{N}_n]$ the inverse matrix of $\mathbf{A}$, we obtain the operator reproducing kernel as follows: $$K(\alpha)\xi=\sum_{j\in\mathbb{N}_n}
\sum_{k\in\mathbb{N}_n}b_{j,k}\langle \xi, L_{\alpha}(\phi_k)\rangle_{\mathcal{Y}}\phi_j,
\ \alpha\in\Lambda, \xi\in\mathcal{Y}.$$
Theorem \[property\] indicates that an ORKHS has a unique operator reproducing kernel. We now turn to the reverse problem. That is, associated with a given operator reproducing kernel, there exists a unique ORKHS. To show this, we introduce an alternative definition for the operator reproducing kernel according to Property (3) in Theorem \[property\] and prove its equivalence to the original definition. For two vector space $V_1,V_2$, we denote by $\mathcal{L}(V_1,V_2)$ the set of linear operators from $V_1$ to $V_2$.
\[Alter-Defn\] Suppose that $\Lambda$ is a set, $\mathcal{Z}$ is a vector space and $\mathcal{Y}$ is a Hilbert space. Let $K$ be an operator from $\Lambda$ to $\mathcal{L}(\mathcal{Y},\mathcal{Z})$ and $\mathcal{L}:=\{L_{\alpha}:\alpha\in \Lambda\}$ be a family of linear operators from the linear span $\mathcal{S}_{K}:=\span\{K(\alpha)\xi:\alpha\in \Lambda,\xi\in\mathcal{Y}\}\subseteq\mathcal{Z}$ to $\mathcal{Y}$. We call $K$ an operator reproducing kernel with respect to $\mathcal{L}$ if for each $n\in\mathbb{N}$ and each pair of finite sets $\{\alpha_j:j\in \mathbb{N}_n\}\subseteq\Lambda$, $\{\xi_j:j\in \mathbb{N}_n\}\subseteq\mathcal{Y}$ there holds $$\label{Operator-Kernel}
\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_n}
\langle L_{\alpha_k}(K(\alpha_j)\xi_j),
\xi_k\rangle_{\mathcal{Y}}\geq0.$$
In the scalar case that $\mathcal{Y}=\mathbb{C}$, the operator reproducing kernel $K:\Lambda\rightarrow \mathcal{Z}$ becomes the functional reproducing kernel with respect to a family $\mathcal{F}:=\{L_{\alpha}:\alpha\in \Lambda\}$ of linear functionals on $\mathcal{S}_{K}:=\span\{K(\alpha):\alpha\in\Lambda\}$. Moreover, in this case, condition (\[Operator-Kernel\]) reduces to that for each $n\in\mathbb{N}$ and each finite set $\Lambda_n:=\{\alpha_j:j\in \mathbb{N}_n\}
\subseteq\Lambda$, the matrix $\mathbf{F}_{\Lambda_n}:=
[L_{\alpha_k}(K(\alpha_j)):j,k\in\mathbb{N}_n]$ is hermitian and positive semi-definite.
The next theorem shows that corresponding to each operator reproducing kernel, there associates an ORKHS.
Suppose that $K$ is an operator from $\Lambda$ to $\mathcal{L}(\mathcal{Y},\mathcal{Z})$. If $K$ is an operator reproducing kernel with respect to a family $\mathcal{L}:=\{L_{\alpha}:\ \alpha\in \Lambda\}$ of linear operators from the linear span $\mathcal{S}_{K}$ to $\mathcal{Y}$, then there exists a unique ORKHS $\mathcal{H}$ with respect to $\mathcal{L}$ such that $\overline{\mathcal{S}_{K}}=\mathcal{H}$ and for each $f\in\mathcal{H}$ and each $\alpha\in \Lambda,\xi\in\mathcal{Y}$, $\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\langle f,K(\alpha)\xi\rangle_\mathcal{H}$.
We specifically construct the ORKHS and verify that it has the desired properties. We first construct a vector space based upon the given operator reproducing kernel. To this end, we define equivalent classes in the linear span $\mathcal{S}_{K}$. Specifically, two functions $f_1,f_2\in\mathcal{S}_{K}$ are said to be equivalent provided that $L_{\alpha}(f_1)=L_{\alpha}(f_2)$ holds for all $\alpha\in\Lambda$. Accordingly, they induce a partition of $\mathcal{S}_{K}$ by which $\mathcal{S}_{K}$ is partitioned into a collection of disjoint equivalent classes. For a function $f\in\mathcal{S}_{K}$, we denote by $\tilde{f}$ the class equivalent to $f$ and introduce the vector space $\mathcal{H}_0:=\{\tilde{f}:
f\in\mathcal{S}_{K}\}$.
We then equip the vector space $\mathcal{H}_0$ with an inner product so that it becomes an inner product space. For this purpose, we define a bilinear mapping $\langle \cdot,\cdot\rangle_{\mathcal{H}_0}:\mathcal{H}_0\times \mathcal{H}_0\rightarrow \mathbb{C}$ for $f:=\sum_{j\in\mathbb{N}_n}K(\alpha_j)\xi_j$ and $g:=\sum_{k\in\mathbb{N}_m}K(\beta_k)\eta_k$ by $$\label{inner-product}
\langle \tilde{f},\tilde{g}\rangle_{\mathcal{H}_0}
:=\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_m}\langle
L_{\beta_k}(K(\alpha_j)\xi_j),\eta_k\rangle_{\mathcal{Y}}.$$ From the representation $$\langle \tilde{f},\tilde{g}\rangle_{\mathcal{H}_0}
=\sum_{k\in\mathbb{N}_m}\left\langle L_{\beta_k}
\left(\sum_{j\in\mathbb{N}_n}K(\alpha_j)\xi_j\right),
\eta_k\right\rangle_{\mathcal{Y}}
=\sum_{k\in\mathbb{N}_m}\langle L_{\beta_k}(f),
\eta_k\rangle_{\mathcal{Y}},$$ we observe that the bilinear mapping $\langle \cdot,\cdot\rangle_{\mathcal{H}_0}$ is independent of the choice of the equivalent class representation of elements in $\mathcal{H}_0$ for the first variable. Likewise, we may show that the bilinear mapping $\langle \cdot,\cdot\rangle_{\mathcal{H}_0}$ is also independent of the choice of the equivalent class representation of elements in $\mathcal{H}_0$ for the second variable. Thus, we conclude that the bilinear mapping (\[inner-product\]) is well-defined.
We next verify that the bilinear mapping $\langle \cdot,\cdot\rangle_{\mathcal{H}_0}$ is indeed an inner product on $\mathcal{H}_0$. It follows that for any $f:=\sum_{j\in\mathbb{N}_n}K(\alpha_j)\xi_j$ and $g:=\sum_{k\in\mathbb{N}_m}K(\beta_k)\eta_k$, there holds $$\langle \tilde{f},\tilde{g}\rangle_{\mathcal{H}_0}
=\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_m}
\langle L_{\beta_k}(K(\alpha_j)\xi_j),
\eta_k\rangle_{\mathcal{Y}}
=\overline{\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_m}
\langle L_{\alpha_j}(K(\beta_k)\eta_k),\xi_j
\rangle_{\mathcal{Y}}}=\overline{\langle\tilde{g},
\tilde{f}\rangle_{\mathcal{H}_0}}.$$ Moreover, inequality (\[Operator-Kernel\]) leads directly to $$\langle \tilde{f},\tilde{f}\rangle_{\mathcal{H}_0}
=\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_m}\langle
L_{\alpha_k}(K(\alpha_j)\xi_j),\xi_k\rangle_{\mathcal{Y}}
\geq 0.$$ Hence, the bilinear mapping $\langle\cdot,\cdot\rangle_{\mathcal{H}_0}$ is a semi-inner product on $\mathcal{H}_0$. It remains to verify that if $\langle \tilde{f},\tilde{f}
\rangle_{\mathcal{H}_0}=0$ then $\tilde{f}=0$. Suppose that $f:=\sum_{j\in\mathbb{N}_n}K(\alpha_j)\xi_j$ satisfies $\langle\tilde{f},\tilde{f}
\rangle_{\mathcal{H}_0}=0$. By the Cauchy-Schwarz inequality we have for any $\alpha\in\Lambda,\xi\in\mathcal{Y}$ that $$|\langle \tilde{f},\widetilde{K(\alpha)\xi}
\rangle_{\mathcal{H}_0}|\leq
\langle \tilde{f},\tilde{f}\rangle_{\mathcal{H}_0}
\langle\widetilde{K(\alpha)\xi},
\widetilde{K(\alpha)\xi}\rangle_{\mathcal{H}_0}=0.$$ Combining this with (\[inner-product\]), we have for any $\alpha\in\Lambda,\xi\in\mathcal{Y}$ that $$\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\sum_{j\in\mathbb{N}_n}\langle L_{\alpha}(K(\alpha_j)\xi_j),\xi\rangle
=\langle \tilde{f},\widetilde{K(\alpha)\xi}
\rangle_{\mathcal{H}_0}=0,$$ which implies that $L_{\alpha}(f)=0$ holds for all $\alpha\in\Lambda$. The definition of the equivalent classes in $\mathcal{S}_{K}$ implies that $\tilde{f}=0$. Consequently, we have established that $\mathcal{H}_0$ is an inner product space endowed with the inner product $\langle \cdot,\cdot\rangle_{\mathcal{H}_0}$.
Let $\mathcal{H}$ be the completion of $\mathcal{H}_0$ upon the inner product $\langle \cdot,\cdot\rangle_{\mathcal{H}_0}.$ We finally show that $\mathcal{H}$ is an ORKHS. For each $\tilde{f}\in\mathcal{H}_0$ and each $\alpha\in \Lambda$, we set $L_{\alpha}(\tilde{f}):=L_{\alpha}(f)$. Then, for any $\alpha\in\Lambda, \xi\in\mathcal{Y}$ and any $\tilde{f}\in\mathcal{H}_0$, $\langle L_{\alpha}(\tilde{f}),\xi\rangle_{\mathcal{Y}}
=\langle \tilde{f},\widetilde{K(\alpha)\xi}
\rangle_{\mathcal{H}_0}$. This implies that the linear functional $\langle L_{\alpha}(\cdot), \xi\rangle_{\mathcal{Y}}$ is continuous on $\mathcal{H}_0$. For any $F\in \mathcal{H}$, there exists a sequence $\{\tilde{f}_n:\ n\in\mathbb{N}\}\subset \mathcal{H}_0$ converging to $F$ as $n\rightarrow \infty$. By the continuity of the linear functional $\langle L_{\alpha}(\cdot), \xi\rangle_{\mathcal{Y}}$, we obtain that $\langle L_{\alpha}(\tilde{f}_n), \xi\rangle_{\mathcal{Y}}, n\in\mathbb{N},$ is a Cauchy sequence. We then define $L_{\alpha}(F)$ as an element in $\mathcal{Y}$ such that $\langle L_{\alpha}(F), \xi\rangle_{\mathcal{Y}}=\lim_{n\rightarrow \infty}\langle L_{\alpha}(\tilde{f}_n),\xi\rangle_{\mathcal{Y}}$ for any $\xi\in\mathcal{Y}$. That is, the linear operators $L_{\alpha}$, $\alpha\in \Lambda,$ can be extended to the Hilbert space $\mathcal{H}$. Moreover, it follows that $$\langle L_{\alpha}(F),\xi\rangle_{\mathcal{Y}}
=\lim_{n\rightarrow \infty}\langle L_{\alpha}(\tilde{f}_n),\xi\rangle_{\mathcal{Y}}
=\lim_{n\rightarrow \infty}\langle \tilde{f}_n,\widetilde{K(\alpha)\xi}
\rangle_{\mathcal{H}_0}
=\langle F, \widetilde{K(\alpha)\xi}\rangle_{\mathcal{H}}.$$ This together with the density $\overline{\mathcal{H}_0} =\mathcal{H}$ yields that if $L_{\alpha}(F)=0$ holds for all $\alpha\in \Lambda$ then $\|F\|_{\mathcal{H}}=0$. This shows that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. To show for each $\alpha\in\Lambda,$ $L_{\alpha}$ is continuous from $\mathcal{H}$ to $\mathcal{Y}$, we note that $$\|L_{\alpha}(F)\|_{\mathcal{Y}}
=\sup_{\|\xi\|_{\mathcal{Y}}\leq1}|
\langle L_{\alpha}(F),\xi\rangle_{\mathcal{Y}}|
\leq\|F\|_{\mathcal{H}}\sup_{\|\xi\|_{\mathcal{Y}}\leq1}
\|\widetilde{K(\alpha)\xi}\|_{\mathcal{H}}.$$ For each $G\in\mathcal{H},$ there holds $$\sup_{\|\xi\|_{\mathcal{Y}}\leq1}|\langle G, \widetilde{K(\alpha)\xi}\rangle_{\mathcal{H}}|
=\sup_{\|\xi\|_{\mathcal{Y}}\leq1}|\langle L_{\alpha}(G),
\xi\rangle_{\mathcal{Y}}|=\|L_{\alpha}(G)\|_{\mathcal{Y}}.$$ By the principle of uniform boundedness, we have that the subset $\{\widetilde{K(\alpha)\xi}:\xi\in\mathcal{Y},
\|\xi\|_{\mathcal{Y}}\leq1\}$ of $\mathcal{H}$ is bounded, which leads to the existence of a positive constant $c$ such that $\|L_{\alpha}(F)\|_{\mathcal{Y}}\leq c\|F\|_{\mathcal{H}}$. Consequently, we conclude that $\mathcal{H}$ is an ORKHS with respect to $\mathcal{L}$. We abuse the notation by denoting the equivalent class of $K(\alpha)\xi$ by $K(\alpha)\xi$. Then we have that $\overline{\mathcal{S}_{K}}=\mathcal{H}$ and for each $F\in\mathcal{H}$, each $\alpha\in \Lambda$ and each $\xi\in\mathcal{Y}$, $\langle L_{\alpha}(F),\xi\rangle_{\mathcal{Y}}=\langle F,K(\alpha)\xi\rangle_\mathcal{H}$. In addition, by the density, we also have the uniqueness of $\mathcal{H}$.
As a direct consequence, we identifies a functional reproducing kernel with an FRKHS in the following result.
Suppose that $K$ is an operator from $\Lambda$ to $\mathcal{Z}$. If $K$ is a functional reproducing kernel with respect to a family $\mathcal{F}:=\{L_{\alpha}:
\ \alpha\in \Lambda\}$ of linear functionals on $\mathcal{S}_{K}$, then there exists a unique FRKHS $\mathcal{H}$ with respect to $\mathcal{F}$ such that $\overline{\mathcal{S}_{K}}=\mathcal{H}$ and for each $f\in\mathcal{H}$ and each $\alpha\in \Lambda$, $L_{\alpha}(f)=\langle f,K(\alpha)\rangle_\mathcal{H}$.
Vector-valued RKHSs are special cases of ORKHSs. We shall show that there exists an isometric isomorphism between an ORKHS and a vector-valued RKHS. Suppose that $\mathcal{H}$ is an ORKHS with respect to a family $\mathcal{L}:=\{L_\alpha: \alpha\in \Lambda\}$ of linear operators form $\mathcal{H}$ to $\mathcal{Y}$ and $K$ is the corresponding operator reproducing kernel. Associated with $K$, we introduce a function $\mathcal{K}:\Lambda\times \Lambda\rightarrow \mathcal{B}(\mathcal{Y},\mathcal{Y})$ by $$\label{kernel_representation}
\mathcal{K}(\alpha,\beta)\xi:=L_{\beta}(K(\alpha)\xi),\ \alpha,\beta\in \Lambda,\ \xi\in\mathcal{Y}.$$ We next show that the function so defined is a reproducing kernel on $\Lambda$. For this purpose, we review the notion of feature maps in the theory of RKHSs. It is known that every reproducing kernel has a feature map representation, which is the reason that the reproducing kernels can be used to measure the similarity of any two inputs in machine learning. Specifically, $\mathcal{K}:\mathcal{X}\times\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},\mathcal{Y})$ is a reproducing kernel on an input set $\mathcal{X}$ if and only if there exist a Hilbert space $W$ and a mapping $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ such that $$\mathcal{K}(x,y)=\Phi(y)^{*}\Phi(x),\ x,y\in \mathcal{X}.$$ If $\overline{\span}\{\Phi(x)\xi:\ x\in \mathcal{X},\xi\in\mathcal{Y}\}=W$, the vector-valued RKHS of $\mathcal{K}$ can be determined by $$\begin{aligned}
\label{vector-feature_representation_HK}
\mathcal{H}:=\{\Phi(\cdot)^{*}w:\ w\in W\}\end{aligned}$$ with the inner product $
\big\langle\Phi(\cdot)^{*}w,\Phi(\cdot)^{*}v
\big\rangle_{\mathcal{H}}:=\langle w,v\rangle_W.
$
\[isometric-isomorphism\] Let $\mathcal{H}$ be an ORKHS with respect to a family $\mathcal{L}=\{L_{\alpha}:\ \alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ and $K$ the operator reproducing kernel for $\mathcal{H}$. If $\mathcal{K}$ is the function on $\Lambda\times \Lambda$ defined as in , then $\mathcal{K}$ is a reproducing kernel on $\Lambda$ and the vector-valued RKHS of $\mathcal{K}$ is determined by $$\label{classicalRKHS1}
\mathcal{H}_{\mathcal{K}}:=\{L_{\alpha}(f):
\ \alpha\in\Lambda,\ f\in \mathcal{H}\}$$ with the inner product defined for $\widetilde{f}(\alpha):=L_{\alpha}(f)$ and $\widetilde{g}(\alpha):=L_{\alpha}(g),
\ \alpha\in \Lambda$, by $$\label{classical_inner_product1}
\langle\tilde{f},\tilde{g}\rangle_{\mathcal{H}_{
\mathcal{K}}}:=\langle f,g\rangle_{\mathcal{H}}.$$
We prove the desired result by defining the feature map representation of $\mathcal{K}$. We define $\Phi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},
\mathcal{H})$ for $\alpha\in\Lambda$ by $\Phi(\alpha):=K(\alpha)$. By the definition of $\mathcal{K}$, we have for each $\alpha,\beta\in\Lambda$ that $$\langle \mathcal{K}(\alpha,\beta)\xi,
\eta\rangle_{\mathcal{Y}}
=\langle L_{\beta}(K(\alpha)\xi),
\eta\rangle_{\mathcal{Y}},
\ \xi,\eta\in\mathcal{Y}.$$ This together with the reproducing property (\[reproducing\_property\]) leads to $$\langle \mathcal{K}(\alpha,\beta)\xi,
\eta\rangle_{\mathcal{Y}}
=\langle K(\alpha)\xi,K(\beta)\eta\rangle_{\mathcal{H}},
\ \xi,\eta\in\mathcal{Y}.$$ According to the definition of $\Phi$, we then get for any $\xi,\eta\in\mathcal{Y}$ that $$\langle \mathcal{K}(\alpha,\beta)\xi,
\eta\rangle_{\mathcal{Y}}
=\langle \Phi(\alpha)\xi,\Phi(\beta)\eta \rangle_{\mathcal{H}}
=\langle \Phi(\beta)^{*}\Phi(\alpha)\xi,
\eta\rangle_{\mathcal{Y}}.$$ This leads to $\mathcal{K}(\alpha,\beta)
=\Phi(\beta)^{*}\Phi(\alpha).$ Hence, we have that $\mathcal{K}$ is a reproducing kernel on $\Lambda$. Moreover, $\Phi$ and $\mathcal{H}$ are, respectively, the feature map and the feature space of $\mathcal{K}$, and there holds the density $$\overline{\span}\{\Phi(\alpha)\xi:\ \alpha\in \Lambda,\xi\in\mathcal{Y}\}=\mathcal{H}.$$ By the feature map representation (\[vector-feature\_representation\_HK\]) of the vector-valued RKHS, we have for each $\tilde{f}$ in the vector-valued RKHS of $\mathcal{K}$ that there exists $f\in\mathcal{H}$ such that $$\langle\tilde{f}(\alpha),\xi\rangle_{\mathcal{Y}}
=\langle\Phi(\alpha)^{*}f,\xi\rangle_{\mathcal{Y}}
=\langle f,K(\alpha)\xi\rangle_{\mathcal{H}},
\ \alpha\in \Lambda,$$ and there holds $\langle\tilde{f},\tilde{g}\rangle
=\langle f,g\rangle_{\mathcal{H}},$ for $\tilde{g}(\alpha)=\Phi(\alpha)^{*}g, \alpha\in \Lambda$. This together with the reproducing property (\[reproducing\_property\]) leads to the representation of $\tilde{f}$ as $\tilde{f}(\alpha)=L_{\alpha}(f),\ \alpha\in\Lambda$. This proves the desired result.
Proposition \[isometric-isomorphism\] reveals an isometric isomorphism between the ORKHS $\mathcal{H}$ and $\mathcal{H}_{\mathcal{K}}$, which is established in the next theorem.
\[isometric-isomorphism1\] Suppose that $\mathcal{H}$ is an ORKHS with respect to a family $\mathcal{L}=\{L_{\alpha}:\ \alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ and $K$ is the operator reproducing kernel for $\mathcal{H}$. If $\mathcal{K}$ is the reproducing kernel on $\Lambda$ defined as in , then there is an isometric isomorphism between $\mathcal{H}$ and the RKHS $\mathcal{H}_{\mathcal{K}}$ of $\mathcal{K}$.
We introduce the operator $\mathcal{T}:\mathcal{H}
\rightarrow \mathcal{H}_{\mathcal{K}}$ for each $f\in \mathcal{H}$ by $(\mathcal{T}f)(\alpha):=L_{\alpha}(f),
\ \alpha\in \Lambda$. The representation (\[classicalRKHS1\]) leads to that $\mathcal{T}$ is a surjection. Moreover, it follows from (\[classical\_inner\_product1\]) that there holds $\|\mathcal{T}f\|_{\mathcal{H}_{\mathcal{K}}}
=\|f\|_{\mathcal{H}}$, which yields that $\mathcal{T}$ is an isometric isomorphism between $\mathcal{H}$ and $\mathcal{H}_{\mathcal{K}}$.
When Theorem \[isometric-isomorphism1\] is specialized to an FRKHS and a scalar-valued RKHS, it gives an isometric isomorphism between an FRKHS and a scalar-valued RKHS. We state this result in the next corollary.
\[FRKHS\_isometric-isomorphism\] Suppose that $\mathcal{H}$ is an FRKHS with respect to a family $\mathcal{F}=\{L_{\alpha}:\ \alpha\in \Lambda\}$ of linear functionals and $K$ is the functional reproducing kernel for $\mathcal{H}$. If $\mathcal{K}:\Lambda\times \Lambda\rightarrow\mathbb{C}$ is defined by $\mathcal{K}(\alpha,\beta):=L_{\beta}(K(\alpha))$, $\alpha,\beta\in \Lambda$, then $\mathcal{K}$ is a reproducing kernel on $\Lambda$ and the RKHS of $\mathcal{K}$ is determined by $$\label{classicalRKHS}
\mathcal{H}_{\mathcal{K}}:=\{L_{\alpha}(f):
\ \alpha\in\Lambda,\ f\in \mathcal{H}\}$$ with the inner product defined for $\widetilde{f}(\alpha):=L_{\alpha}(f)$ and $\widetilde{g}(\alpha):=L_{\alpha}(g),
\ \alpha\in \Lambda$, by $$\label{classical_inner_product}
\langle\tilde{f},\tilde{g}\rangle_{\mathcal{H}_{
\mathcal{K}}}:=\langle f,g\rangle_{\mathcal{H}}.$$ Moreover, there is an isometric isomorphism between $\mathcal{H}$ and $\mathcal{H}_{\mathcal{K}}$.
Even though an ORKHS is isometrically isomorphic to a usual vector-valued RKHS, one cannot reduce trivially the study of ORKHSs to that of vector-valued RKHSs. Through the isometric isomorphism procedure, the new input space $\Lambda$ may not have any useful structure. Moreover, we observe from the above discussion that the functions in the resulting RKHS and the corresponding reproducing kernel are obtained by taking the linear operators on the original ones. The use of general operators such as the integral operators will bring difficulties to analyzing the new RKHS and its reproducing kernel.
To close this section, we present the characterization of an operator reproducing kernel in terms of its feature map and feature space.
\[feature\_representation\_KL\] An operator $K:\Lambda\to\mathcal{L}(\mathcal{Y},
\mathcal{Z})$ is an operator reproducing kernel with respect to a family $\mathcal{L}=\{L_{\alpha}:
\ \alpha\in\Lambda\}$ of linear operators from $\mathcal{S}_{K}\subseteq\mathcal{Z}$ to $\mathcal{Y}$ if and only if there exists a Hilbert space $W$ and a mapping $\Phi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},W)$ such that $$\label{feature_representation_KL1}
L_{\beta}(K(\alpha)\xi)=\Phi(\beta)^{*}\Phi(\alpha)\xi,
\ \alpha,\beta\in \Lambda,\xi\in\mathcal{Y}.$$
If $K$ is an operator reproducing kernel, we obtain (\[feature\_representation\_KL1\]) by choosing $W$ as the ORKHS $\mathcal{H}$ of $K$ and $\Phi$ as the operator $K$.
Conversely, we assume that (\[feature\_representation\_KL1\]) holds for any $\alpha,\beta\in \Lambda$ and any $\xi\in\mathcal{Y}$. Hence, for all $n\in\mathbb{N}$ and $\alpha_j\in \Lambda,\xi_j\in\mathcal{Y},\ j\in \mathbb{N}_n$, we get that $$\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_n}
\langle L_{\alpha_k}(K(\alpha_j)\xi_j),
\xi_k\rangle_{\mathcal{Y}}
=\left\langle \sum_{j\in\mathbb{N}_n}\Phi(\alpha_j)\xi_j,
\sum_{k\in\mathbb{N}_n}\Phi(\alpha_k)\xi_k
\right\rangle_W\geq 0.$$ This together with Definition \[Alter-Defn\] yields that $K$ is an operator reproducing kernel with respect to $\mathcal{L}$.
As in the theory of RKHSs, a Hilbert space $W$ and a map $\Phi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},W)$ satisfying (\[feature\_representation\_KL1\]) are called, respectively, the feature space and the feature map of the operator reproducing kernel $K$.
In the scalar case, the feature map representation of a reproducing kernel $\mathcal{K}$ on $\mathcal{X}$ has the form $$\mathcal{K}(x,y)=\langle \Phi(x),\Phi(y)\rangle_W,
\ \ x,y\in \mathcal{X},$$ with $W$ being a Hilbert space and $\Phi$ a mapping from $\mathcal{X}$ to $W$. If there holds $\overline{\span}\{\Phi(x):\ x\in \mathcal{X}\}=W$, the RKHS of $\mathcal{K}$ can be determined by $$\begin{aligned}
\mathcal{H}_{\mathcal{K}}
:=\{\langle w,\Phi(\cdot)\rangle_W:\ w\in W\}\end{aligned}$$ with the inner product $$\big\langle\langle w,\Phi(\cdot)\rangle_W,
\langle v,\Phi(\cdot)\rangle_W
\big\rangle_{\mathcal{H}_{\mathcal{K}}}
:=\langle w,v\rangle_W.$$ Due to above feature map representation and Corollary \[FRKHS\_isometric-isomorphism\], we have the following special result regarding the feature map representation of a functional reproducing kernel.
\[FRKHS\_feature\_representation\_KL\] The operator $K:\Lambda\to\mathcal{Z}$ is a functional reproducing kernel with respect to a family $\mathcal{F}=\{L_{\alpha}:\ \alpha\in\Lambda\}$ of linear functionals on $\mathcal{S}_{K}\subseteq \mathcal{Z}$ if and only if there exists a Hilbert space $W$ and a mapping $\Phi:\Lambda\rightarrow W$ such that $$\label{FRKHS_feature_representation_KL1}
L_{\beta}(K(\alpha))=\langle \Phi(\alpha),
\Phi(\beta)\rangle_W,\ \alpha,\beta\in \Lambda.$$
Perfect ORKHSs
==============
In this section, we study a special interesting class of ORKHSs (of functions from $\mathcal{X}$ to a Hilbert space $\mathcal{Y}$) with respect to two different families of linear operators, one of which is the family of the standard point-evaluation operators. We shall characterize the operator reproducing kernels of this type and study their universality.
For two vector spaces $V_1$ and $V_2$ we say two families of linear operators from $V_1$ to $V_2$ are not equivalent if each linear operator in one family can not be represented by any finite linear combination of linear operators in the other family. Throughout this paper we denote by $\mathcal{L}_0$ the family of point-evaluation operators.
\[perfect\] Let a Hilbert space $\mathcal{H}$ of functions from $\mathcal{X}$ to a Hilbert space $\mathcal{Y}$ be an ORKHS with respect to a family $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$, which is not equivalent to $\mathcal{L}_0$. We call $\mathcal{H}$ a [*perfect*]{} ORKHS with respect to $\mathcal{L}$ if it is also an ORKHS with respect to $\mathcal{L}_0$.
In the scalar case when $\mathcal{Y}=\mathbb{C}$, if $\mathcal{H}$ is an FRKHS with respect to not only the family of point-evaluation functionals but also a family of linear functionals, not equivalent to the family of point-evaluation functionals, we shall call it a perfect FRKHS.
It follows from Theorem \[property\] that a perfect ORKHS admits two operator reproducing kernels, which reproduce two different families of operators, one being $\mathcal{L}_0$. The following proposition gives the relation between the two operator reproducing kernels.
If $\mathcal{H}$ is a perfect ORKHS with respect to a family $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ and if $K$ and $\mathcal{K}$ are the corresponding operator reproducing kernels with respect to $\mathcal{L}$ and $\mathcal{L}_0$, respectively, then $$\label{relation-kernels}
\langle(K(\alpha)\xi)(x),\eta\rangle_{\mathcal{Y}}
=\langle \xi, L_{\alpha}(\mathcal{K}(x,\cdot)\eta)
\rangle_{\mathcal{Y}},\ x\in \mathcal{X},
\ \alpha\in\Lambda,\ \xi,\eta\in\mathcal{Y}.$$
It follows from the reproducing properties of the two operator reproducing kernels that for each $f\in\mathcal{H}$ there hold $$\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\langle f,K(\alpha)\xi\rangle_{\mathcal{H}},
\ \ \alpha\in\Lambda,\ \xi\in\mathcal{Y}$$ and $$\langle f(x),\eta\rangle_{\mathcal{Y}}
=\langle f,\mathcal{K}(x,\cdot)\eta\rangle_{\mathcal{H}},
\ \ x\in \mathcal{X},\ \eta\in\mathcal{Y}.$$ In particular, in the first equation we choose $$f:=\mathcal{K}(x,\cdot)\eta,\ \ \mbox{for}\ \ x\in \mathcal{X},\ \eta\in\mathcal{Y}$$ and we obtain that $$\langle L_{\alpha}(\mathcal{K}(x,\cdot)\eta),
\xi\rangle_{\mathcal{Y}}
=\langle \mathcal{K}(x,\cdot)\eta,
K(\alpha)\xi\rangle_{\mathcal{H}},
\ \ x\in \mathcal{X},\ \alpha\in\Lambda,
\ \xi,\eta\in\mathcal{Y}.$$ Likewise, in the second equation we choose $$f:=K(\alpha)\xi,\ \ \mbox{for}\ \ \alpha\in\Lambda,\ \xi\in\mathcal{Y}$$ and we obtain that $$\langle (K(\alpha)\xi)(x),\eta\rangle_{\mathcal{Y}}
=\langle K(\alpha)\xi,\mathcal{K}(x,\cdot)
\eta\rangle_{\mathcal{H}},\ \ x\in \mathcal{X},
\ \alpha\in\Lambda,\ \xi,\eta\in\mathcal{Y}.$$ The desired result of this proposition follows directly from the above two equations.
In the special case when $\mathcal{H}$ is a perfect FRKHS with respect to the family $\mathcal{F}$ of linear functionals, equation (\[relation-kernels\]) reduces to $$\label{relation-kernels1}
K(\alpha)(x)=\overline{L_{\alpha}(\mathcal{K}(x,\cdot))},
\ x\in \mathcal{X},\ \alpha\in\Lambda,$$ where $K$ is the functional reproducing kernel with respect to $\mathcal{F}$ and $\mathcal{K}$ is the classical reproducing kernel on $\mathcal{X}$.
We next represent the operator reproducing kernel $K$ for a perfect ORKHS $\mathcal{H}$ by the feature map. For this purpose, we let $W$ and $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ be the feature space and the feature map of $\mathcal{K}$, respectively. We assume that there holds the density condition $$\label{density_phi}
\overline{\span}\{\Phi(x)\xi:\ x\in \mathcal{X},
\xi\in\mathcal{Y}\}=W.$$ Associated with the features $\Phi$ and $W$ of $\mathcal{K}$, we introduce a map $\Psi$ from $\Lambda$ to $\mathcal{B}(\mathcal{Y},W)$ as follows. Noting that for each $\alpha\in\Lambda$, the linear operator $\widetilde{L}_{\alpha}$ defined on $W$ for each $u\in W$ by $$\widetilde{L}_{\alpha}(u):=L_{\alpha}(\Phi(\cdot)^{*}u)$$ is continuous since there holds for each $u\in W$, $$\|\widetilde{L}_{\alpha}(u)\|_{\mathcal{Y}}
\leq \|L_{\alpha}\|_{\mathcal{B}(\mathcal{H},\mathcal{Y})}
\|\Phi(\cdot)^{*}u\|_{\mathcal{H}}
=\|L_{\alpha}\|_{\mathcal{B}
(\mathcal{H},\mathcal{Y})}\|u\|_W.$$ This allows us to define $\Psi$ by $$\label{Psi}
\Psi(\alpha):=\widetilde{L}^{*}_{\alpha},
\ \ \alpha\in\Lambda.$$ In terms of the maps $\Phi$ and $\Psi$, we can represent $K$ as in the following proposition.
\[feature\_representation\_perfect\] Suppose that $\mathcal{H}$ is a perfect ORKHS with respect to a family $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$. Let $K$ and $\mathcal{K}$ denote the operator reproducing kernels with respect to $\mathcal{L}$ and $\mathcal{L}_0$, respectively. If $W$ is the feature space of $\mathcal{K}$, $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ is its feature map with the density condition and $\Psi:\Lambda\rightarrow\mathcal{B}(\mathcal{Y},W)$ is defined by , then $$K(\alpha)=\Phi(\cdot)^{*}\Psi(\alpha),
\ \ \alpha\in\Lambda.$$ Moreover, $W$ and $\Psi$ are, respectively, the feature space and the feature map of $K$, satisfying the density condition $$\label{density_psi}
\overline{\span}\{\Psi(\alpha)\xi:\ \alpha\in \Lambda,\xi\in\mathcal{Y}\}=W.$$
It follows from the feature map representation $\mathcal{K}(x,y)=\Phi(y)^{*}\Phi(x),\ x,y\in\mathcal{X},$ that $$L_{\alpha}(\mathcal{K}(x,\cdot)\eta)
=L_{\alpha}(\Phi(\cdot)^{*}\Phi(x)\eta)
=\widetilde{L}_{\alpha}(\Phi(x)\eta),
\ x\in \mathcal{X},\ \alpha\in\Lambda,
\ \eta\in\mathcal{Y}.$$ Substituting the above equation into (\[relation-kernels\]), we obtain that $$\langle(K(\alpha)\xi)(x), \eta\rangle_{\mathcal{Y}}
=\langle \xi, \widetilde{L}_{\alpha}
(\Phi(x)\eta)\rangle_{\mathcal{Y}}.$$ Then by the definition of $\Psi$ we get that $$\langle(K(\alpha)\xi)(x),\eta\rangle_{\mathcal{Y}}
=\langle \Psi(\alpha)\xi,\Phi(x)\eta\rangle_W
=\langle \Phi(x)^{*}\Psi(\alpha)
\xi,\eta\rangle_{\mathcal{Y}},$$ which implies $K(\alpha)=\Phi(\cdot)^{*}\Psi(\alpha), \alpha\in\Lambda.$
Furthermore, for any $\alpha,\beta\in\Lambda$, and any $\xi,\eta\in\mathcal{Y}$, we have that $$\langle L_{\beta}(K(\alpha)\xi),\eta\rangle_{\mathcal{Y}}
=\langle\Psi(\alpha)\xi,\Psi(\beta)\eta\rangle_W
=\langle\Psi(\beta)^{*}\Psi(\alpha)\xi,
\eta\rangle_{\mathcal{Y}},$$ which yields $L_{\beta}(K(\alpha)\xi)
=\Psi(\beta)^{*}\Psi(\alpha)\xi$. We conclude from this that $W$ and $\Psi$ are, respectively, the feature space and the feature map of $K$.
To verify the density condition (\[density\_psi\]), it suffices to show that if $u\in W$ satisfies $\langle u,\Psi(\alpha)\xi\rangle_W=0$ for all $\alpha\in \Lambda$ and all $\xi\in\mathcal{Y}$, then $u=0$. Since there holds for all $\alpha\in \Lambda$ and all $\xi\in\mathcal{Y}$, $$\langle\widetilde{L}_{\alpha}(u),\xi\rangle_{\mathcal{Y}}
=\langle u,\Psi(\alpha)\xi\rangle_W,$$ we have that $\widetilde{L}_{\alpha}(u)=0$ for all $\alpha\in\Lambda$. By the definition $\widetilde{L}_{\alpha}$, we get that $L_{\alpha}(\Phi(\cdot)^{*}u)=0$ for all $\alpha\in\Lambda$. Since the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$, we observe that $\|\Phi(\cdot)^{*}u\|_{\mathcal{H}}=0$, which is equivalent to $\|u\|_W=0$.
Motivated by the representation of an operator reproducing kernel for a perfect ORKHS in Proposition \[feature\_representation\_perfect\], we now turn to characterizing the operator reproducing kernels of this type, which is useful in constructing operator reproducing kernels.
\[L\_RKHS\_Thm\] An operator $K:\Lambda\rightarrow\mathcal{B}
(\mathcal{Y},\mathcal{H})$ is an operator reproducing kernel for a perfect ORKHS $\mathcal{H}$ with respect to a family $\mathcal{L}=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ if and only if there exist a Hilbert space $W$ and two mappings $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ with the density condition and $\Psi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},W)$ with the density condition such that $$\label{kernel-perfectRKHS}
K(\alpha)=\Phi(\cdot)^{*}\Psi(\alpha),
\ \ \alpha\in\Lambda.$$ Moreover, the perfect ORKHS of $K$ has the form $\mathcal{H}:=\{\Phi(\cdot)^{*}u:u\in W\}$ with the inner product $$\langle \Phi(\cdot)^{*}u,\Phi(\cdot)^{*}v
\rangle_{\mathcal{H}}:=\langle u,v\rangle_W.$$
Suppose that $K:\Lambda\rightarrow\mathcal{B}(\mathcal{Y},\mathcal{H})$ is an operator reproducing kernel for a perfect ORKHS $\mathcal{H}$ with respect to a family $\mathcal{L}=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$. According to the definition of the perfect ORKHS, $\mathcal{H}$ is a reproducing kernel Hilbert space with its reproducing kernel $\mathcal{K}$. By theory of the classical reproducing kernel Hilbert space, we can choose a Hilbert space $W$ as the feature space of the reproducing kernel $\mathcal{K}$ and a mapping $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ which satisfying the density condition as its feature map. We then define the map $\Psi$ according to (\[Psi\]). Hence, by Proposition \[feature\_representation\_perfect\], we get the representation of $K$ as in (\[kernel-perfectRKHS\]).
Conversely, we suppose that $W$ is a Hilbert space, $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ satisfies the density condition and $\Psi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},W)$ satisfies the density condition , and $K$ has the form . Associated with $W$ and $\Phi$ we introduce a space $\mathcal{H}:=\{\Phi(\cdot)^{*}u:u\in W\}$ of functions from $\mathcal{X}$ to $\mathcal{Y}$ and define a bilinear mapping on $\mathcal{H}$ by $$\langle \Phi(\cdot)^{*}u,\Phi(\cdot)^{*}
v\rangle_{\mathcal{H}}:=\langle u,v\rangle_W.$$ By the theory of classical RKHSs, we conclude that $\mathcal{H}$ is an RKHS. We shall show that $\mathcal{H}$ is a perfect ORKHS with respect to some family of linear operators and the operator $K$ is exactly the operator reproducing kernel for $\mathcal{H}$.
We first introduce a family of linear operators from $\mathcal{H}$ to $\mathcal{Y}$. For each $\alpha\in\Lambda$ we define operator $L_{\alpha}$ on $\mathcal{H}$ by $$L_{\alpha}(\Phi(\cdot)^{*}u):=\Psi(\alpha)^{*}u,\ u\in W.$$ It is clear that $L_{\alpha}$ is linear. It follows that $$\|L_{\alpha}(\Phi(\cdot)^{*}u)\|_{\mathcal{Y}}
\leq\|\Psi(\alpha)\|_{\mathcal{B}(\mathcal{Y},W)}\|u\|_W
=\|\Psi(\alpha)\|_{\mathcal{B}(\mathcal{Y},W)}
\|\Phi(\cdot)^{*}u\|_{\mathcal{H}},$$ which ensures that $L_{\alpha}$ is continuous on $\mathcal{H}$. Set $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$. Then $\mathcal{L}$ is a family of continuous linear operators. We next verify that the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. Suppose that $f:=\Phi(\cdot)^{*}u\in\mathcal{H}$ satisfies $L_{\alpha}(f)=0$ for all $\alpha\in\Lambda$. By the definition of operators $L_{\alpha}$, we have for all $\alpha\in\Lambda$ and all $\xi\in\mathcal{Y}$ that $$\langle u, \Psi(\alpha)\xi\rangle_W
=\langle \Psi(\alpha)^{*}u, \xi\rangle_{\mathcal{Y}}
=\langle L_{\alpha}(\Phi(\cdot)^{*}u),
\xi\rangle_{\mathcal{Y}}=0.$$ This together with the density condition (\[density\_psi\]) leads to $\|u\|_{W}=0$ and then $\|f\|_{\mathcal{H}}=0$. That is, the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$. We then conclude that $\mathcal{H}$ is a perfect ORKHS with respect to $\mathcal{L}$.
It remains to prove that the operator $K$ with the form (\[kernel-perfectRKHS\]) is the operator reproducing kernel for $\mathcal{H}$. It follows from $$\|K(\alpha)\xi\|_{\mathcal{H}}
=\|\Psi(\alpha)\xi\|_W\leq\|\Psi(\alpha)
\|_{\mathcal{B}(\mathcal{Y},W)}\|\xi\|_{\mathcal{Y}},
\ \xi\in\mathcal{Y},$$ that $K(\alpha)\in\mathcal{B}(\mathcal{Y},\mathcal{H})$. For each $\alpha\in\Lambda,\xi\in\mathcal{Y}$ and each $f:=\Phi(\cdot)^{*}u\in\mathcal{H}$, there holds $$\langle f,K(\alpha)\xi\rangle_{\mathcal{H}}
=\langle\Phi(\cdot)^{*}u,\Phi(\cdot)^{*}
\Psi(\alpha)\xi\rangle_{\mathcal{H}}
=\langle u,\Psi(\alpha)\xi\rangle_W
=\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}.$$ This ensures that $K$ is the operator reproducing kernel for $\mathcal{H}$.
We remark on the representation (\[kernel-perfectRKHS\]) of an operator reproducing kernel for a perfect ORKHS. For a general operator reproducing kernel, the feature map can only give the representation (\[feature\_representation\_KL1\]) for the values of the operators $L_{\beta}$, $\beta\in \Lambda$, at the kernel sections $K(\alpha)\xi, \alpha\in\Lambda, \xi\in\mathcal{Y}$. Due to the special structure of the perfect ORKHS, we can give a more precise representation (\[kernel-perfectRKHS\]) for its operator reproducing kernel.
We next present a specific example to illustrate the representation (\[kernel-perfectRKHS\]) of the operator reproducing kernel for a perfect ORKHS. To this end, we recall the RKHS $\mathcal{B}_{\Delta}
(\mathbb{R}^d,\mathbb{C}^{n})$. Proposition \[Paley\] shows that the space $\mathcal{B}_{\Delta}
(\mathbb{R}^d,\mathbb{C}^{n})$ is also an ORKHS with respect to the family $\mathcal{L}$ of the integral operators defined as in (\[example\_integration\_functional1\]). Hence, it is a perfect ORKHS. It was pointed out in section 3 that the operator reproducing kernel for $\mathcal{B}_{\Delta}
(\mathbb{R}^d,\mathbb{C}^{n})$ has the form $$\label{example-feature-representation}
K(x)\xi=\mathscr{P}_{\mathcal{B}_{\Delta}}(u(\cdot-x)\xi),
\ \ x\in \mathbb{R}^d, \ \ \xi\in\mathbb{C}^{n}.$$ To give the feature representation (\[kernel-perfectRKHS\]) of $K$, we first describe the choice of the space $W$ as the feature space of the kernel $\mathcal{K}$ defined as in (\[sinc\]) and the mapping $\Phi$ as its feature map. Let $W:=L^2(I_{\Delta},\mathbb{C}^n)$. We define $\Phi:\mathbb{R}^d\to\mathcal{B}(\mathbb{C}^n,W)$ for each $x\in\mathbb{R}^d$ by $$\Phi(x):=\frac{1}{(2\pi)^{d/2}}
{\rm diag}(e^{i(x,\cdot)}:j\in\mathbb{N}_{n}).$$ Then the adjoint of $\Phi(x)$ can be represented for each $w:=[w_j:j\in\mathbb{N}_n]\in W$ as $$\Phi(x)^{*}w=\frac{1}{(2\pi)^{d/2}}
[\langle w_j,e^{i(x,\cdot)}\rangle_{L^2(I_{\Delta})}:
j\in\mathbb{N}_{n}].$$ Associated with $W$ and $\Phi$, the kernel $\mathcal{K}$ having the form (\[sinc\]) may be rewritten as $$\mathcal{K}(x,y)=\Phi(y)^{*}\Phi(x),
\ \ x,y\in\mathbb{R}^d.$$ This shows that $W$ and $\Phi$ are, respectively, the feature space and the feature map of $\mathcal{K}$. We next present the other map $\Psi$ needed for the construction of $K$ in terms of the function $u$ appearing in (\[example\_integration\_functional1\]). Specifically, for each $x\in\mathbb{R}^d$, we set $$\Psi(x):=(2\pi)^{d/2}{\rm diag}
(\check{u}e^{i(x,\cdot)}:j\in\mathbb{N}_{n}).$$ By equation (\[example-feature-representation\]), we have for each $x,y\in\mathbb{R}^d$ and each $\xi\in\mathbb{C}^n$ that $$\begin{aligned}
(K(x)\xi)(y)=(\hat{u}e^{-i(x,\cdot)}\xi
\chi_{I_{\Delta}})^{\vee}(y)
=\langle \check{u}e^{i(x,\cdot)},e^{i(y,\cdot)}
\rangle_{L^2(I_{\Delta})}\xi.\end{aligned}$$ This together with the definition of $\Phi(y)^{*}$ and $\Psi(x)$ leads to $(K(x)\xi)(y)=\Phi(y)^{*}\Psi(x)\xi.$ That is, $K(x)=\Phi(\cdot)^{*}\Psi(x)$.
The following corollary gives a characterization of a functional reproducing kernel for a perfect FRKHS.
An operator $K:\Lambda\to\mathcal{H}$ is a functional reproducing kernel for a perfect FRKHS $\mathcal{H}$ with respect to a family $\mathcal{F}=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear functionals on $\mathcal{H}$ if and only if there exist a Hilbert space $W$ and two mappings $\Phi:\mathcal{X}\rightarrow W$ with $\overline{\span}\{\Phi(x):x\in \mathcal{X}\}=W$ and $\Psi:\Lambda\rightarrow W$ with $\overline{\span}\{\Psi(\alpha):\alpha\in \Lambda\}=W$ such that $$\label{FRKHS_kernel-perfectRKHS}
K(\alpha)=\langle\Psi(\alpha),\Phi(\cdot)\rangle_W,
\ \ \alpha\in\Lambda.$$ Moreover, the perfect FRKHS of $K$ has the form $\mathcal{H}:=\{\langle u,\Phi(\cdot)\rangle_W:u\in W\}$ with the inner product $$\langle \langle u,\Phi(\cdot)\rangle_W,\langle v,\Phi(\cdot)\rangle_W\rangle_{\mathcal{H}}
:=\langle u,v\rangle_W.$$
Associated with an operator reproducing kernel in the form (\[kernel-perfectRKHS\]) for a perfect ORKHS, there are two reproducing kernels $$\mathcal{K}(x,y)=\Phi(y)^{*}\Phi(x),\ x,y\in\mathcal{X}
\ \ \mbox{and}\ \ \widetilde{\mathcal{K}}(\alpha,\beta)
=\Psi(\beta)^{*}\Psi(\alpha),\ \alpha,\beta\in\Lambda.$$ As a direct consequence of Theorems \[isometric-isomorphism1\] and \[L\_RKHS\_Thm\], there exists an isometric isomorphism between the RKHSs of $\mathcal{K}$ and $\widetilde{\mathcal{K}}$. We present this result below.
\[K\_L\_K\] Suppose that $W$ is a Hilbert space and two mappings $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ and $\Psi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},W)$ satisfy the density conditions and , respectively. If the mapping $K$ has the form and two mappings $\mathcal{K}$ and $\widetilde{\mathcal{K}}$ are defined as above, then $K$ and $\mathcal{K}$ are the operator reproducing kernel and the reproducing kernel, respectively, for a perfect ORKHS $\mathcal{H}$ and $\widetilde{\mathcal{K}}$ is a reproducing kernel for an RKHS $\mathcal{\widetilde{H}}$. Moreover, there is an isometric isomorphism between $\mathcal{H}$ and $\mathcal{\widetilde{H}}$.
It is known from Theorem \[L\_RKHS\_Thm\] that $K$ is an operator reproducing kernel for a perfect ORKHS $\mathcal{H}$ with respect $\mathcal{L}=\{L_{\alpha}:\ \alpha\in\Lambda\}$ defined by $$L_{\alpha}(\Phi(\cdot)^{*}u):=\Psi(\alpha)^{*}u,\ u\in W.$$ Moreover, $\mathcal{K}$ is the corresponding reproducing kernel for $\mathcal{H}$ with respect to the family of the point-evaluation operators. The feature map representation of $\widetilde{\mathcal{K}}$ leads directly to that $\widetilde{\mathcal{K}}$ is a reproducing kernel on $\Lambda$ for the RKHS $\mathcal{\widetilde{H}}:=\{\Psi(\cdot)^{*}u:u\in W\}$. By definitions of $K, \widetilde{\mathcal{K}}$ and $L_{\beta}$, we get for all $\alpha,\beta\in\Lambda$ and all $\xi\in\mathcal{Y}$ that $$\widetilde{\mathcal{K}}(\alpha,\beta)\xi
=\Psi(\beta)^{*}\Psi(\alpha)\xi
=L_{\beta}(\Phi(\cdot)^{*}\Psi(\alpha)\xi)
=L_{\beta}(K(\alpha)\xi).$$ We then conclude from Theorem \[isometric-isomorphism1\] that there is an isometric isomorphism between $\mathcal{H}$ and $\mathcal{\widetilde{H}}$.
To close this section, we consider universality of an operator reproducing kernel for a perfect ORKHS. Universality of classical reproducing kernels has attracted much attention in theory of machine learning [@CMPY; @MXZ; @PMRRV; @St]. This important property ensures that a continuous target function can be uniformly approximated on a compact subset of the input space by the linear span of kernel sections. It is crucial for the consistence of the kernel-based learning algorithms, which generally use the representer theorem to learn a target function in the linear span of kernel sections. For learning a function from its operator values, we also have similar representer theorems, which will be presented in section 7. Below, we discuss universality of an operator reproducing kernel for a perfect ORKHS.
Let Hilbert space $\mathcal{H}$ of $\mathcal{Y}$-valued functions on a metric space $\mathcal{X}$ is a perfect ORKHS with respect to a family $\mathcal{L}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$. Suppose that the operator reproducing kernel $K$ for $\mathcal{H}$ satisfies that for each $\alpha\in\Lambda$ and each $\xi\in\mathcal{Y}$, $K(\alpha)\xi$ is continuous on $\mathcal{X}$. We denote by $C(\mathcal{Z},\mathcal{Y})$ the Banach space of continuous $\mathcal{Y}$-valued functions on compact subset $\mathcal{Z}$ of $\mathcal{X}$ with the norm defined by $\|f\|_{C(\mathcal{Z},\mathcal{Y})}
:=\sup_{x\in\mathcal{Z}}\|f(x)\|_{\mathcal{Y}}$. We say that $K$ is universal if for each compact subset $\mathcal{Z}$ of $\mathcal{X}$, $$\label{universal}
\overline{\span}\{K(\alpha)\xi:
\alpha\in\Lambda,\ \xi\in\mathcal{Y}\}
=C(\mathcal{Z},\mathcal{Y}),$$ where for each $\alpha\in\Lambda$ and each $\xi\in\mathcal{Y}$, $K(\alpha)\xi$ is taken as a $\mathcal{Y}$-valued function on $\mathcal{Z}$ and the closure is taken in the maximum norm. To characterize universality of the operator reproducing kernel having the form (\[kernel-perfectRKHS\]), we first recall a convergence property of functions in an RKHS. That is, if a sequence $f_n$ converges to $f$ in an RKHS, then $f_n$, as a sequence of functions on $\mathcal{X}$, converges pointwisely to $f$. As a direct consequence, we have the following result concerning the density in $\mathcal{H}$.
\[density-H-C\] Let $\mathcal{H}$ be an RKHS of continuous $\mathcal{Y}$-valued functions on a metric space $\mathcal{X}$ and $\mathcal{M}$ a subset of $\mathcal{H}$. If $\mathcal{M}$ is dense in $\mathcal{H}$, then for any compact subset $\mathcal{Z}$ of $\mathcal{X}$ there holds $\overline{\mathcal{M}}=\overline{\mathcal{H}}$, where the closure is taken in the maximum norm of $C(\mathcal{Z},\mathcal{Y})$.
It suffices to prove that for any compact subset $\mathcal{Z}$ there holds $\mathcal{H}\subseteq \overline{\mathcal{M}},$ where the closure is taken in the maximum norm of $C(\mathcal{Z},\mathcal{Y})$. Let $f$ be any element in $\mathcal{H}$. By the density of $\mathcal{M}$ in Hilbert space $\mathcal{H}$, there exists a sequence $f_n$ converging to $f$. Due to the convergence property in RKHSs, we have for any compact subset $\mathcal{Z}$ of $\mathcal{X}$ that $f_n$ converges to $f$ in $C(\mathcal{Z},\mathcal{Y})$. According to the arbitrary of $f$ in $\mathcal{H}$, we get $\mathcal{H}\subseteq \overline{\mathcal{M}}.$
By the above lemma, we give a characterization of universality of the operator reproducing kernel with the form (\[kernel-perfectRKHS\]) as follows.
\[universal-characterize\] Let $\mathcal{X}$ be a metric space, $\Lambda$ a set and $W,\mathcal{Y}$ both Hilbert spaces. Suppose that $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ is continuous and satisfies the density condition and $\Psi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},W)$ satisfies the density condition . Then the operator reproducing kernel $K$ defined by is universal if and only if there holds for any compact subset $\mathcal{Z}$ of $\mathcal{X}$, $$\label{universal1}
\overline{\span}\{\Phi(\cdot)^{*}u:
u\in W\}=C(\mathcal{Z},\mathcal{Y}).$$
It follows from Theorem \[L\_RKHS\_Thm\] that the perfect ORKHS of the operator reproducing kernel $K$ defined by has the form $\mathcal{H}:=\{\Phi(\cdot)^{*}u:u\in W\}$ with the inner product $\langle \Phi(\cdot)^{*}u,\Phi(\cdot)^{*}v
\rangle_{\mathcal{H}}:=\langle u,v\rangle_W.$ Since $\Phi$ is continuous, the functions in $\mathcal{H}$ are all continuous. Let $\mathcal{M}$ denote the linear span $\span\{\Phi(\cdot)^{*}\Psi(\alpha)\xi:\alpha\in \Lambda,
\xi\in\mathcal{Y}\}$. By the density condition , we have that $\mathcal{M}$ is dense in $\mathcal{H}$. Then according to Lemma \[density-H-C\], we get for any compact subset $\mathcal{Z}$ of $\mathcal{X}$ that $$\overline{\span}\{\Phi(\cdot)^{*}\Psi(\alpha)\xi:
\xi\in\mathcal{Y}\}=\overline{\span}\{\Phi(\cdot)^{*}u: u\in W\},$$ where the closure is taken in the maximum norm of $C(\mathcal{Z},\mathcal{Y})$. The above equation together with leads to $$\overline{\span}\{K(\alpha)\xi: \xi\in\mathcal{Y}\}
=\overline{\span}\{\Phi(\cdot)^{*}u: u\in W\},$$ which completes the proof.
Besides the operator reproducing kernel $K$, the perfect ORKHS also enjoys a classical reproducing kernel on $\mathcal{X}$ with the form $\mathcal{K}(x,y)=\Phi(y)^{*}\Phi(x)$, $x,y\in \mathcal{X}$. The characterization of universality of classical reproducing kernels in [@CMPY; @MXZ] shows that $\mathcal{K}$ is universal if and only if (\[universal1\]) holds for any compact subset $\mathcal{Z}$ of $\mathcal{X}$. Hence, Theorem \[universal-characterize\] tells us that for a perfect ORKHS the universality of its two kernels is identical.
Perfect ORKHSs with respect to Integral Operators
=================================================
Local average values of a function are often used as observed information in data analysis. For this reason, we investigate in this section the perfect ORKHS with respect to a family of integral operators.
We first introduce the integral operators to be considered in this section. Let $(\mathcal{X},\mu)$ be a measure space and $\mathcal{Y}$ a Hilbert space. Suppose that $\Lambda$ is an index set and $\{\lambda_{\alpha}:
\ \alpha\in \Lambda\}$ is a family of mappings from $\mathcal{X}$ to $\mathcal{B}(\mathcal{Y},\mathcal{Y})$. Associated with these mappings, we define the family $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of integral operators for a function $f$ from $\mathcal{X}$ to $\mathcal{Y}$ by $$\label{integral-operator}
L_{\alpha}(f):=\int_{\mathcal{X}}\lambda_{\alpha}(x)f(x)
d\mu(x),\ \alpha\in\Lambda.$$
We now describe a construction of perfect ORKHSs with respect to $\mathcal{L}$ defined by following Theorem \[L\_RKHS\_Thm\]. To this end, we suppose that $W$ is a Hilbert space and a mapping $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ satisfies the density condition . The perfect ORKHSs will be constructed through a subspace of $W$. For each $\alpha\in\Lambda$ and each $\xi\in\mathcal{Y}$, we introduce a function from $\mathcal{X}$ to $W$ by letting $$\label{psi}
\psi_{\alpha,\xi}(x):=\Phi(x)\lambda_{\alpha}(x)^{*}\xi,
\ x\in\mathcal{X}.$$ If the functions $\psi_{\alpha,\xi}$, $\alpha\in\Lambda$, $\xi\in\mathcal{Y}$, are all Bochner integrable, then we define a closed subspace of $W$ by $$\label{W}
\widetilde{W}:=\overline{\span}\left\{\int_{\mathcal{X}}
\psi_{\alpha,\xi}(x)d\mu(x):
\ \alpha\in\Lambda,\xi\in\mathcal{Y}\right\}.$$ We let $\mathcal{P}_{\widetilde{W}}$ denote the orthogonal projection from $W$ to $\widetilde{W}$. We then define the mapping $\widetilde{\Phi}$ by $$\widetilde{\Phi}(x):=\mathcal{P}_{\widetilde{W}}\Phi(x),
\ \ x\in \mathcal{X}.$$ The next lemma establishes the density condition involving the mapping $\widetilde{\Phi}:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},\widetilde{W})$ in the Hilbert space $\widetilde{W}$.
\[Projection\] Let $W$ be a Hilbert space and $\Phi:\mathcal{X}\rightarrow\mathcal{B}(\mathcal{Y},W)$ a mapping satisfying the density condition . If $\psi_{\alpha,\xi}$, $\alpha\in\Lambda$, $\xi\in\mathcal{Y}$, defined by , are all Bochner integrable and $\widetilde{W}$ and $\widetilde{\Phi}$ are defined as above, then there holds the density condition $$\label{density_phi1}
\overline{\span}\{\widetilde{\Phi}(x)\xi:
x\in \mathcal{X},\xi\in\mathcal{Y}\}=\widetilde{W}.$$
By the definition of $\widetilde{\Phi}$, we have for each $u\in \widetilde{W}$ that $$\langle\widetilde{\Phi}(x)\xi,u\rangle_{\widetilde{W}}
=\langle\Phi(x)\xi,u\rangle_{W},
\ x\in \mathcal{X},\xi\in\mathcal{Y}.$$ This together with the density condition of $\Phi$ leads to (\[density\_phi1\]).
We describe in the next theorem the construction of the perfect ORKHS with respect to $\mathcal{L}$.
\[integral-perfectORKHS\] Let $W$ be a Hilbert space and $\Phi:\mathcal{X}\rightarrow\mathcal{B}(\mathcal{Y},W)$ a mapping satisfying the density condition . If a collection of mappings $\lambda_{\alpha}:\mathcal{X}\to
\mathcal{B}(\mathcal{Y},\mathcal{Y})$, $\alpha\in \Lambda$ satisfies $$\label{condition}
\int_{\mathcal{X}}\|\Phi(x)\|_{\mathcal{B}(\mathcal{Y},W)}
\|\lambda_{\alpha}(x)\|_{\mathcal{B}(\mathcal{Y},\mathcal{Y})}
d\mu(x)<\infty,\ \ \mbox{for all}\ \ \alpha\in\Lambda,$$ then the space $$\label{ORKHS-integral-operator}
\widetilde{\mathcal{H}}:=\left\{\Phi(\cdot)^{*}u:
u\in\widetilde{W}\right\}$$ with the inner product $$\label{inner-product-integral-operator}
\langle\Phi(\cdot)^{*}u, \Phi(\cdot)^{*}v
\rangle_{\widetilde{\mathcal{H}}}
:=\langle u,v\rangle_{\widetilde{W}},$$ is a perfect ORKHS with respect to $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ defined as in and its operator reproducing kernel $K$ has the form $$\label{L_reproducing_kernel_integral}
K(\alpha)\xi=\Phi(\cdot)^{*}\int_{\mathcal{X}}
\psi_{\alpha,\xi}(x)d\mu(x),\ \alpha\in\Lambda,
\ \xi\in \mathcal{Y}.$$
We prove this theorem by employing Theorem \[L\_RKHS\_Thm\]. Specifically, we verify the hypothesis of Theorem \[L\_RKHS\_Thm\]. For this purpose, we introduce a mapping $\Psi:\Lambda\rightarrow \mathcal{B}(\mathcal{Y},\widetilde{W})$ which satisfies the density condition with $W$ being replaced by $\widetilde{W}$. It follows for each $\alpha\in\Lambda$ and each $\xi\in\mathcal{Y}$ that $$\label{Bochner-integral1}
\int_{\mathcal{X}}\|\psi_{\alpha,\xi}\|_{W}d\mu(x)
\leq\|\xi\|_{\mathcal{Y}}\int_{\mathcal{X}}\|\Phi(x)
\|_{\mathcal{B}(\mathcal{Y},W)}
\|\lambda_{\alpha}(x)\|_{\mathcal{B}(\mathcal{Y},
\mathcal{Y})}d\mu(x).$$ This together with (\[condition\]) leads to $\psi_{\alpha,\xi}$ is Bochner integrable. This allows us to define the operator $\Psi:\Lambda\rightarrow
\mathcal{L}(\mathcal{Y},\widetilde{W})$ by $$\Psi(\alpha)\xi:=\int_{\mathcal{X}}\psi_{\alpha,\xi}(x)
d\mu(x),\ \ \alpha\in\Lambda,\ \ \xi\in\mathcal{Y}.$$ By the definition of $\psi_{\alpha,\xi}$, the linearity of $\Psi(\alpha)$ is clear. Combining equation (\[Bochner-integral1\]) with the fact $$\|\Psi(\alpha)\xi\|_{\widetilde{W}}\leq\int_{\mathcal{X}}
\|\psi_{\alpha,\xi}\|_{W}d\mu(x),$$ we conclude that $\Psi(\alpha)\in\mathcal{B}(\mathcal{Y},\widetilde{W})$. Observing from the definition of $\Psi$, we obtain easily its density condition. By Lemma \[Projection\], the hypotheses of Theorem \[L\_RKHS\_Thm\] are satisfied. Consequently, by identifying for each $\alpha\in\Lambda$, $$L_{\alpha}(\widetilde{\Phi}(\cdot)^{*}u)
=\Psi(\alpha)^{*}u,\ u\in \widetilde{W},$$ we conclude that the space $\widetilde{\mathcal{H}}:=\left\{\widetilde{\Phi}
(\cdot)^{*}u:u\in\widetilde{W}\right\}$ with the inner product $\langle\widetilde{\Phi}(\cdot)^{*}u, \widetilde{\Phi}(\cdot)^{*}v\rangle_{\widetilde{\mathcal{H}}}
:=\langle u,v\rangle_{\widetilde{W}}$ is a perfect ORKHS with respect to $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ and that the corresponding operator reproducing kernel has the form $K(\alpha)=\widetilde{\Phi}(\cdot)^{*}
\Psi(\alpha), \alpha\in\Lambda$.
We next show that the perfect ORKHS $\widetilde{\mathcal{H}}$ has the form and the operator reproducing kernel $K$ has the form (\[L\_reproducing\_kernel\_integral\]). Noting that $$\widetilde{\Phi}(x)^{*}
=\Phi(x)^{*}\mathcal{P}_{\widetilde{W}}^{*}
=\Phi(x)^{*}, \ \ \mbox{for all} \ \ x\in \mathcal{X},$$ we can represent $\widetilde{\mathcal{H}}$ as in (\[ORKHS-integral-operator\]) and $K$ as $K(\alpha)=\Phi(\cdot)^{*}\Psi(\alpha),\alpha\in\Lambda$. This together with the definition of $\Psi$ leads to the form (\[L\_reproducing\_kernel\_integral\]). It suffices to rewrite $L_{\alpha},\alpha\in\Lambda,$ in the form (\[integral-operator\]). It follows for each $\alpha\in\Lambda$ and each $f=\Phi(\cdot)^{*}u\in \widetilde{\mathcal{H}}$ that $$\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\langle\Psi(\alpha)^{*}u,\xi\rangle_{\mathcal{Y}}
=\langle u,\Psi(\alpha)\xi\rangle_{W},
\ \xi\in\mathcal{Y}.$$ Hence, by the definition of $\Psi$ and $\psi_{\alpha,\xi}$ we have that $$\begin{aligned}
\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}}
=\int_{\mathcal{X}}\langle u,
\Phi(x)\lambda_{\alpha}(x)^{*}\xi\rangle_{W}d\mu(x)
=\left\langle\int_{\mathcal{X}}\lambda_{\alpha}(x)f(x)d\mu(x),
\xi\right\rangle_{\mathcal{Y}},\ \xi\in\mathcal{Y},\end{aligned}$$ which yields that $L_{\alpha}$ has the form (\[integral-operator\]) and thus, completes the proof.
Associated with the Hilbert space $W$ and the mapping $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ in Theorem \[integral-perfectORKHS\], the space $\mathcal{H}:=\left\{\Phi(\cdot)^{*}u: u\in W\right\}$, with the inner product $\langle\Phi(\cdot)^{*}u, \Phi(\cdot)^{*}v\rangle_{\mathcal{H}}
:=\langle u,v\rangle_{W}$, is an RKHS on $\mathcal{X}$. The perfect ORKHS $\widetilde{\mathcal{H}} $ with the form (\[ORKHS-integral-operator\]) is a closed subspace of $\mathcal{H}$. Observing from the representation of $\widetilde{\mathcal{H}} $, we obtain the following result.
\[density-W\] Let $W$ be a Hilbert space and $\Phi:\mathcal{X}\rightarrow \mathcal{B}(\mathcal{Y},W)$ a mapping satisfying the density condition . If a collection of mappings $\lambda_{\alpha}:\mathcal{X}\to
\mathcal{B}(\mathcal{Y},\mathcal{Y})$, $\alpha\in \Lambda,$ satisfies and $\widetilde{W}$ is defined by , then the RKHS $\mathcal{H}:=\left\{\Phi(\cdot)^{*}u:u\in W\right\}$ with the inner product $\langle\Phi(\cdot)^{*}u,
\Phi(\cdot)^{*}v\rangle_{\mathcal{H}}
:=\langle u,v\rangle_{W}$, is a perfect ORKHS with respect to $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ defined as in if and only if there holds $\widetilde{W}=W.$
It follows from Theorem \[integral-perfectORKHS\] that the closed subspace $\widetilde{\mathcal{H}}$, defined by (\[ORKHS-integral-operator\]), of $\mathcal{H}$ is a perfect ORKHS with respect to $\mathcal{L}$. If $\widetilde{W}=W,$ then we have $\mathcal{H}=\widetilde{\mathcal{H}}$, which shows that $\mathcal{H}$ is just the perfect ORKHS. Conversely, we suppose that $\mathcal{H}$ is a perfect ORKHS with respect to $\mathcal{L}$. For each $u\in W$, we set $$\label{u-psi}
h_{\alpha,\xi}(u):=\langle u,\int_{\mathcal{X}}
\psi_{\alpha,\xi}(x)d\mu(x)\rangle_{W},
\ \alpha\in\Lambda,\xi\in\mathcal{Y}.$$ It suffices to verify that if $u\in W$ satisfies $h_{\alpha,\xi}(u)=0$ for any $\alpha\in\Lambda,
\xi\in\mathcal{Y}$, then $u=0$. By the definition of $\psi_{\alpha,\xi}$, we have for each $\alpha\in\Lambda,\xi\in\mathcal{Y}$ that $$h_{\alpha,\xi}(u)=\int_{\mathcal{X}}\langle u,\Phi(x)\lambda_{\alpha}(x)^{*}\xi\rangle_{W}d\mu(x)
=\langle\int_{\mathcal{X}}\lambda_{\alpha}(x)
\Phi(x)^{*}ud\mu(x),\xi\rangle_{\mathcal{Y}}.$$ This together with the definition of $L_{\alpha},\alpha\in\Lambda,$ leads to $h_{\alpha,\xi}(u)=\langle L_{\alpha}(\Phi(x)^{*}u)
,\xi\rangle_{\mathcal{Y}}.$ Since the norm of $\mathcal{H}$ is compatible with $\mathcal{L}$, we have that $h_{\alpha,\xi}(u)=0$ holds for any $\alpha\in\Lambda,\xi\in\mathcal{Y}$ if and only if $\|\Phi(x)^{*}u\|_{\mathcal{H}}=0$, which is equivalent to $u=0$.
We now turn to presenting two specific examples of perfect ORKHSs with respect to the family of integral operators defined by (\[integral-operator\]). These perfect ORKHSs come from widely used RKHSs.
The first perfect ORKHS is constructed based on the translation invariant RKHSs. Let $\mathcal{Y}$ be a Hilbert space. A reproducing kernel $\mathcal{K}:
\mathbb{R}^d\times\mathbb{R}^d\to
\mathcal{B}(\mathcal{Y},\mathcal{Y})$ is said to be translation invariant if for all $a\in\mathbb{R}^d$, $\mathcal{K}(x-a,y-a)=\mathcal{K}(x,y)$, for $x,y\in\mathbb{R}^d$. In the scalar-valued case, a well-known characterization of translation invariant kernels was established in [@B59], which states that every continuous translation invariant kernel on $\mathbb{R}^d$ must be the Fourier transform of a finite nonnegative Borel measure on $\mathbb{R}^d$, and vice versa. The characterization was generalized to the operator-valued case in [@F]. We consider a matrix-valued translation invariant reproducing kernel, for which the positive semi-definite matrix-valued Borel measure of bounded variation on $\mathbb{R}^d$ has the Radon-Nikodym property with respect to the Lebesgue measure. Specifically, we denote by $\mathbf{M}_n$ the space of $n\times n$ matrices endowed with the spectral norm and $\mathbf{M}_n^{+}$ its subspace of positive semi-definite matrices. Let $\varphi$ be a Bochner integrable function from $\mathbb{R}^d$ to $\mathbf{M}_n^{+}$. The translation invariant kernel $\mathcal{K}:\mathbb{R}^d\times\mathbb{R}^d\to
\mathbf{M}_n$ to be considered has the special form $$\label{translation_invariant_rk_phi}
\mathcal{K}(x,y)=\int_{\mathbb{R}^d}e^{i(x-y,t)}\varphi(t)dt,
\ \ x,y\in\mathbb{R}^d.$$ We shall make use of Theorem \[integral-perfectORKHS\] to construct a perfect ORKHS, which is a closed subspace of the translation invariant RKHS of $\mathcal{K}$. To this end, we first present the feature space and the feature map of $\mathcal{K}$. Associated with the $\mathbf{M}_n^{+}$-valued Bochner integrable function $\varphi$ on $\mathbb{R}^d$, we denote by $L^2_{\varphi}(\mathbb{R}^d,\mathbb{C}^n)$ the space of $\mathbb{C}^n$-valued Lebesgue measurable functions $f$ on $\mathbb{R}^d$ satisfying $\varphi^{1/2}f$ belongs to $L^2(\mathbb{R}^d,\mathbb{C}^n)$. The space $L^2_{\varphi}(\mathbb{R}^d,\mathbb{C}^n)$ is a Hilbert space endowed with the inner product $$\langle f,g\rangle_{L^2_{\varphi}(\mathbb{R}^d,\mathbb{C}^n)}
:=\int_{\mathbb{R}^d}\langle\varphi(t)f(t),
g(t)\rangle_{\mathbb{C}^n}dt,
\ \ f,g\in L^2_{\varphi}(\mathbb{R}^d,\mathbb{C}^n).$$ We choose $W:=L^2_{\varphi}(\mathbb{R}^d,\mathbb{C}^n)$ and $\Phi(x):=e^{i(x,\cdot)},\ x\in\mathbb{R}^d$. We need to verify that $\Phi(x), x\in\mathbb{R}^d,$ belong to $\mathcal{B}(\mathbb{C}^n,W)$ and satisfy the density condition with $\mathcal{Y}=\mathbb{C}^n$. It follows for each $x\in\mathbb{R}^d$ and each $\xi\in\mathbb{C}^n$ that $$\|\Phi(x)\xi\|^2_{W}
=\int_{\mathbb{R}^d}\langle\varphi(t)e^{i(x,t)}\xi,
e^{i(x,t)}\xi\rangle_{\mathbb{C}^n}dt
=\int_{\mathbb{R}^d}\langle\varphi(t)\xi,
\xi\rangle_{\mathbb{C}^n}dt
\leq\|\xi\|^2_{\mathbb{C}^n}\int_{\mathbb{R}^d}
\|\varphi(t)\|_{\mathbf{M}_n}dt.$$ This together with the Bochner integrability of $\varphi$ yields that $\Phi(x)\xi\in W$ and $\Phi(x)$ is continuous from $\mathbb{C}^n$ to $W$ with the norm satisfying $$\label{phinorm}
\|\Phi(x)\|_{\mathcal{B}(\mathbb{C}^n,W)}
\leq\|\varphi\|_{L^1(\mathbb{R}^d,\mathbf{M}_n)}^{1/2}.$$ Moreover, it is clear that there holds the density condition (\[density\_phi\]) with $\mathcal{Y}=\mathbb{C}^n$. To describe the integral operators in this case, we introduce a sequence $u_{\alpha}=[u_{\alpha,j}:j\in\mathbb{N}_n],
\ \alpha\in\Lambda,$ of $\mathbb{C}^n$-valued Bochner integrable functions on $\mathbb{R}^d$ and choose $$\label{lambda-alpha}
\lambda_{\alpha}(x):={\rm diag}(\overline{u_{\alpha,j}
(x)}:j\in\mathbb{N}_{n}),\ x\in\mathbb{R}^d,\ \alpha\in\Lambda.$$ Then the integral operator (\[integral-operator\]) reduces to $$\label{integral-operator1}
L_{\alpha}(f):=\int_{\mathbb{R}^d}f(x)\circ
\overline{u_{\alpha}(x)}dx,\ \ \alpha\in\Lambda.$$ Here, $u\circ v$ denotes the Hadamard product of two $\mathbb{C}^{n}$-valued functions $u$ and $v$, which has been defined in (\[example\_integration\_functional\]). Associated with the Bochner integrable functions $\varphi:\mathbb{R}^d\to \mathbf{M}_n^{+}$ and $u_{\alpha}:\mathbb{R}^d\to\mathbb{C}^n,\ \alpha\in\Lambda,$ we define the inner product space $$\label{example-ORKHS1}
\mathcal{H}:=\left\{(\varphi(\cdot)^{*}u)^{\wedge}:
u\in\overline{\span}\left\{\check{u}_{\alpha}\circ\xi:
\ \alpha\in\Lambda,\xi\in\mathbb{C}^n\right\}\right\},$$ with the inner product $$\label{example-inner-product}
\langle f,g \rangle_{\mathcal{H}}
:=\int_{\mathbb{R}^d}\langle\varphi(t)u(t),
v(t)\rangle_{\mathbb{C}^n}dt$$ for $f=(\varphi(\cdot)^{*}u)^{\wedge}$ and $g=(\varphi(\cdot)^{*}v)^{\wedge}$. The closure in is taken in $W$.
As a consequence of Theorem \[integral-perfectORKHS\], we get in the next theorem a construction of perfect ORKHSs with respect to a family of integral operators defined as in (\[integral-operator1\]).
\[translation\_invariant\_ORKHS\] If $\varphi$ is a Bochner integrable function from $\mathbb{R}^d$ to $\mathbf{M}_n^{+}$ and $u_{\alpha}, \ \alpha\in\Lambda,$ is a family of $\mathbb{C}^n$-valued Bochner integrable functions, then the space $\mathcal{H}$ defined by with the inner product is a perfect ORKHS with respect to $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ defined as in and the operator reproducing kernel is determined by $K(\alpha)\xi=(2\pi)^d[\varphi(\cdot)^{*}
(\check{u}_{\alpha}\circ\xi)]^{\wedge}$.
In order to use the construction in Theorem \[integral-perfectORKHS\], we first verify condition (\[condition\]) with $\lambda_{\alpha}(x),
\ \alpha\in\Lambda,$ being defined by . According to inequality (\[phinorm\]), we have for each $\alpha\in\Lambda$ that $$\label{condition-verify}
\int_{\mathbb{R}^d}\|\Phi(x)\|_{\mathcal{B}(\mathbb{C}^n,W)}
\|\lambda_{\alpha}(x)\|_{\mathbf{M}_n}dx
\leq\|\varphi\|_{L^1(\mathbb{R}^d,\mathbf{M}_n)}^{1/2}
\int_{\mathbb{R}^d}\|\lambda_{\alpha}(x)\|_{\mathbf{M}_n}dx.$$ By the definition of $\lambda_{\alpha}$, we get $$\int_{\mathbb{R}^d}\|\lambda_{\alpha}(x)\|_{\mathbf{M}_n}dx
\leq\|u_{\alpha}\|_{L^1(\mathbb{R}^d,\mathbb{C}^n)}.$$ Substituting the above inequality into (\[condition-verify\]) and by the Bochner integrability of $u_{\alpha}$, we have that $$\int_{\mathbb{R}^d}\|\Phi(x)\|_{\mathcal{B}(\mathbb{C}^n,W)}
\|\lambda_{\alpha}(x)\|_{\mathbf{M}_n}dx
\leq\|\varphi\|_{L^1(\mathbb{R}^d,\mathbf{M}_n)}^{1/2}
\|u_{\alpha}\|_{L^1(\mathbb{R}^d,\mathbb{C}^n)}
<\infty.$$ Hence, by Theorem \[integral-perfectORKHS\] we obtain a perfect ORKHS $\widetilde{\mathcal{H}}$ with the form (\[ORKHS-integral-operator\]) and the corresponding operator reproducing kernel $K$ with the form (\[L\_reproducing\_kernel\_integral\]).
We next turn to representing $\widetilde{\mathcal{H}}$ and $K$ by making use of the functions $\varphi$ and $u_{\alpha}, \alpha\in \Lambda$. It follows for each $\alpha\in\Lambda$ and each $\xi\in\mathbb{C}^n$ that $$\int_{\mathbb{R}^d}\Phi(x)\lambda_{\alpha}(x)^{*}\xi dx
=\int_{\mathbb{R}^d}\Phi(x)(u_{\alpha}(x)\circ\xi)dx
=(2\pi)^d\check{u}_{\alpha}\circ\xi.$$ Then the closed subspace $\widetilde{W}$ in (\[ORKHS-integral-operator\]) of $W$ can be represented as $$\label{concrete-W}
\widetilde{W}=\overline{\span}\left\{\check{u}_{\alpha}
\circ\xi:\ \alpha\in\Lambda, \xi\in\mathbb{C}^n
\right\}.$$ There holds for each $x\in\mathbb{R}^d$ and each $u\in \widetilde{W}$ that $$\langle\Phi(x)\xi,u\rangle_{\widetilde{W}}
=\int_{\mathbb{R}^d}\langle\varphi(t)e^{i(x,t)}\xi,
u(t)\rangle_{\mathbb{C}^n}dt
=\left\langle\xi,\int_{\mathbb{R}^d}e^{-i(x,t)}
\varphi(t)^{*}u(t)dt\right\rangle_{\mathbb{C}^n},
\ \xi\in\mathbb{C}^n,$$ which leads to $$\Phi(x)^{*}u
=\int_{\mathbb{R}^d}e^{-i(x,t)}\varphi(t)^{*}u(t) dt
=(\varphi(\cdot)^{*}u)^{\wedge}(x).$$ Substituting the representation (\[concrete-W\]) of $\widetilde{W}$ and the above equation into (\[ORKHS-integral-operator\]), we have that $\widetilde{\mathcal{H}}=\mathcal{H}$ and the inner product of $\widetilde{\mathcal{H}}$ can be represented as in (\[example-inner-product\]). Moreover, the operator reproducing kernel $K$ can be represented as $$K(\alpha)\xi:=\Phi(\cdot)^{*}
\int_{\mathbb{R}^d}\Phi(x)\lambda_{\alpha}(x)^{*}\xi dx
=(2\pi)^d[\varphi(\cdot)^{*}(\check{u}_{\alpha}\circ\xi)
]^{\wedge},\ \alpha\in\Lambda,\ \xi\in \mathbb{C}^n.$$
The widely used Gaussian kernel on $\mathbb{R}^d$ is a translation invariant kernel with $\varphi$ being the Gaussian function. A matrix-valued Gaussian kernel on $\mathbb{R}^d$ discussed in [@MP05] is determined by $$\label{matrix-valued-Gaussian-kernel}
\mathcal{K}(x,y):=\sum_{j\in\mathbb{N}_m}e^{-\sigma_j\|x-y\|^2}
\mathbf{A}_j,$$ where $\sigma_j\geq0,\ j\in\mathbb{N}_m$ and $\{\mathbf{A}_j:j\in\mathbb{N}_m\}$ is a sequence of real matrices in $\mathbf{M}_n^{+}$. Here, we denote by $\|\cdot\|$ the norm on $\mathbb{R}^d$ induced by the standard inner product. By letting $$\varphi(t):=\sum_{j\in\mathbb{N}_m}
\frac{1}{(2\sigma_j)^{d/2}}e^{-\frac{1}{4\sigma_j}\|t\|^2}
\mathbf{A}_j,$$ we can represent $\mathcal{K}$ in the form (\[translation\_invariant\_rk\_phi\]). Combining Corollary \[density-W\] with Theorem \[translation\_invariant\_ORKHS\], we conclude that the RKHS of the Gaussian kernel (\[matrix-valued-Gaussian-kernel\]) is a perfect ORKHS with respect to $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ defined as in in terms of a family of $\mathbb{C}^n$-valued Bochner integrable functions $u_{\alpha}, \alpha\in\Lambda, $ if the linear span of $\left\{\check{u}_{\alpha}\circ\xi:
\ \alpha\in\Lambda,\xi\in\mathbb{C}^n\right\}$ is dense in $L^2_{\varphi}(\mathbb{R}^d,\mathbb{C}^n)$. Accordingly, the operator reproducing kernel is given by $$(K(\alpha)\xi)(x)=\overline{L_{\alpha}(\mathcal{K}(x,\cdot)
\overline{\xi})}.$$
We next present a perfect FRKHS, which is a closed subspace of a shift-invariant space. To this end, we first recall the shift-invariant spaces, which have been extensively studied in the areas of wavelet theory [@D; @GLT], approximation theory [@DDR; @J] and sampling [@AG; @AST; @H07]. Given $\phi\in L^2(\mathbb{R}^d)$ that satisfies $$\label{riesz_basis}
m\leq\sum_{j\in\mathbb{Z}^d}|\hat{\phi}(\xi+2\pi j)|^2
\leq M,\ \ \mbox{for almost every} \ \xi,$$ for some $m,M>0$, we construct the shift-invariant space by $$V^2(\phi):=\left\{\sum_{k\in\mathbb{Z}^d}c_k\phi(\cdot-k):
\ \{c_k:k\in\mathbb{Z}^d\}\in l^2(\mathbb{Z}^d)\right\}.$$ Here, $l^2(\mathbb{Z}^d)$ is the Hilbert space of square-summable sequences on $\mathbb{Z}^d$. The function $\phi$ is called the generator of the shift-invariant space $V^2(\phi)$. Clearly, $V^2(\phi)$ is a closed subspace of $L^2(\mathbb{R}^d)$ and condition (\[riesz\_basis\]) ensures that the sequence $\phi(\cdot-k),\ k\in\mathbb{Z}^d,$ constitutes a Riesz basis of $V^2(\phi)$. To guarantee that $V^2(\phi)$ is an RKHS, additional conditions need to be imposed on the generator $\phi$. Following [@AG], we first introduce a weight function $\omega$ on $\mathbb{R}^d$, which is assumed to be continuous, symmetric and positive, and to satisfy the condition $0<\omega(x+y)\leq\omega(x)\omega(y)$, for any $x,y\in\mathbb{R}^d$. Moreover, we assume that $\omega$ satisfies the growth condition $$\sum_{n=1}^{\infty}\frac{\log \omega(nk)}{n^2}<\infty,
\ \ \mbox{for all}\ \ k\in\mathbb{Z}^d.$$ We also recall the Wiener amalgam spaces $W(L_w^p),
1\leq p\leq\infty$. For $1\leq p<\infty$, a measurable function $f$ is said to belong to $W(L_w^p)$, if it satisfies $$\|f\|_{W(L_w^p)}^p:=\sum_{k\in\mathbb{Z}^d}
{\rm ess \ sup}\{|f(x+k)|^p\omega(k)^p:x\in[0,1]^d\}
<\infty.$$ For $p=\infty$, a measurable function $f$ is said to belong to $W(L_w^{\infty})$ if it satisfies $$\|f\|_{W(L_w^{\infty})}:=\sup_{k\in\mathbb{Z}^d}
\{{\rm ess \ sup}\{|f(x+k)|\omega(k):x\in[0,1]^d\}\}
<\infty.$$ We denote by $W_0(L_w^p)$ the subspace of continuous functions in $W(L_w^p)$. It has been proved in [@AG] that if $\phi\in W_0(L_w^1)$, the space $V^2(\phi)$ is an RKHS with the reproducing kernel $$\label{shift_invariant_rk}
\mathcal{K}(x,y)=\sum_{k\in\mathbb{Z}^d}\overline{\phi(x-k)}
\widetilde{\phi}(y-k),\ \ x,y\in\mathbb{R}^d,$$ where $\widetilde{\phi}\in V^2(\phi)$ satisfies the condition that the sequence $\widetilde{\phi}(\cdot-k)$, $k\in\mathbb{Z}^d$, is the dual Riesz basis of $\phi(\cdot-k)$, $k\in\mathbb{Z}^d.$ To see the reproducing property, for each $$f:=\sum_{j\in\mathbb{Z}^d}c_j\phi(\cdot-j)\in V^2(\phi),$$ we note that $$\langle f,\mathcal{K}(x,\cdot)\rangle_{L^2(\mathbb{R}^d)}
=\sum_{j\in\mathbb{Z}^d}\sum_{k\in\mathbb{Z}^d}
c_j\phi(x-k)\langle\phi(\cdot-j),\widetilde{\phi}(\cdot-k)
\rangle_{L^2(\mathbb{R}^d)}.$$ Due to the biorthogonality relation $$\langle\phi(\cdot-j),\widetilde{\phi}(\cdot-k)
\rangle_{L^2(\mathbb{R}^d)}=\delta_{j,k},
\ j,k\in\mathbb{Z}^d,$$ the above equation reduces to $$\langle f,\mathcal{K}(x,\cdot)\rangle_{L^2(\mathbb{R}^d)}
=\sum_{j\in\mathbb{Z}^d}c_j\phi(x-j)=f(x).$$ In order to use the construction described in Theorem \[integral-perfectORKHS\], we choose a natural feature map representation for the reproducing kernel $\mathcal{K}$. That is, $W:=V^2(\phi)$ and $\Phi(x):=\mathcal{K}(x,\cdot),
\ x\in\mathbb{R}^d$. Moreover, we introduce a family of measurable functions $u_{\alpha}, \alpha\in\Lambda$, on $\mathbb{R}^d$ satisfying for each $\alpha\in\Lambda$, $$\label{condition_shift_invariant}
\int_{\mathbb{R}^d}|u_{\alpha}(x)|
\left(\sum_{k\in\mathbb{Z}^d}|\phi(x-k)|^2\right)^{1/2}dx
<\infty.$$ By choosing $\lambda_{\alpha}(x):=\overline{u_{\alpha}(x)}$, the integral operator in (\[integral-operator\]) reduces to $$\label{integral-operator2}
L_{\alpha}(f):=\int_{\mathbb{R}^d}
f(x)\overline{u_{\alpha}(x)}dx,
\ \ \alpha\in\Lambda.$$ Associated with $u_{\alpha}, \alpha\in\Lambda$, we define the closed subspace of $V^2(\phi)$ as $$\label{phi_H_K_L}
\mathcal{H}:=\overline{\span}\left\{\sum_{k\in\mathbb{Z}^d}
\left(\int_{\mathbb{R}^d}u_{\alpha}(t)\overline{\phi(t-k)}dt
\right)\tilde{\phi}(\cdot-k):\alpha\in\Lambda\right\}.$$
We construct in the next theorem a perfect FRKHS with respect to a family of integral functionals defined as in (\[integral-operator2\]).
If $\phi\in W_0(L_w^1)$ satisfies and a family of measurable functions $u_{\alpha}, \alpha\in\Lambda,$ on $\mathbb{R}^d$ satisfies , then the closed subspace $\mathcal{H}$ defined by of $V^2(\phi)$ is a perfect FRKHS with respect to a family $\mathcal{F}:=\{\mathcal{L}_{\alpha}: \alpha\in\Lambda\}$ of integral functionals defined as in and the corresponding functional reproducing kernel is determined by $$\label{phi_K_L}
K(\alpha)(x):=\sum_{k\in\mathbb{Z}^d}
\left(\int_{\mathbb{R}^d}u_{\alpha}(t)\overline{\phi(t-k)}dt
\right)\tilde{\phi}(x-k),
\ \alpha\in\Lambda,\ x\in\mathbb{R}^d.$$
To apply Theorem \[integral-perfectORKHS\] in the case that $\mathcal{Y}=\mathbb{C}$, we need to verify for each $\alpha\in\Lambda$ that $u_{\alpha}\Phi$, as a $W$-valued function, is Bochner integrable. By the reproducing property, we obtain for any $x\in\mathbb{R}^d$ that $$\|\Phi(x)\|_W^2=\mathcal{K}(x,x)=\sum_{k\in\mathbb{Z}^d}
\overline{\phi(x-k)}\widetilde{\phi}(x-k),$$ which leads to for each $\alpha\in\Lambda$, $$\label{equality1}
\int_{\mathbb{R}^d}\|u_{\alpha}(x)\Phi(x)\|_Wdx
=\int_{\mathbb{R}^d}|u_{\alpha}(x)|
\left(\sum_{k\in\mathbb{Z}^d}\overline{\phi(x-k)}
\widetilde{\phi}(x-k)\right)^{1/2}dx.$$ For each $x\in\mathbb{R}^d$, we introduce two sequences $a:=\{a_k:k\in\mathbb{Z}^d\}$ and $\tilde{a}:=\{\tilde{a}_k:k\in\mathbb{Z}^d\}$ with $a_k:=\phi(x+k)$ and $\tilde{a}_k:=\tilde{\phi}(x+k)$. Note that the dual generator $\tilde{\phi}\in V^2(\phi)$ has the expansion $\tilde{\phi}=\sum_{k\in\mathbb{Z}^d}b_k\phi(\cdot-k),$ where the coefficients $b_k, k\in\mathbb{Z}^d,$ are the Fourier coefficients of the $2\pi$-periodic function $$b(\xi)=\left(\sum_{j\in\mathbb{Z}^d}|\hat{\phi}
(\xi+2\pi j)|^2\right)^{-1},\ \xi\in [-\pi,\pi]^d.$$ Set $b:=\{b_k:k\in\mathbb{Z}^d\}$. It follows for each $k\in\mathbb{Z}^d$ that $$\tilde{a}_k
=\sum_{l\in\mathbb{Z}^d}b_l\phi(x+k-l)
=\sum_{l\in\mathbb{Z}^d}b_la_{k-l}
=(a\ast b)_k.$$ Since $\phi\in W_0(L_w^1)$, we get that $a\in l^2(\mathbb{Z}^d)$. By Lemmas 2.8 and 2.11 in [@AG], we also have that $b\in l^1(\mathbb{Z}^d):=\left\{\{c_k:k\in\mathbb{Z}^d\}:
\sum_{k\in\mathbb{Z}^d}|c_k|<+\infty\right\}$. Hence, the above equation leads to $\tilde{a}\in l^2(\mathbb{Z}^d)$ and $$\|\tilde{a}\|_{l^2(\mathbb{Z}^d)}\leq
\|a\|_{l^2(\mathbb{Z}^d)}\|b\|_{l^1(\mathbb{Z}^d)}.$$ Then by (\[equality1\]), we obtain that $$\int_{\mathbb{R}^d}\|u_{\alpha}(x)\Phi(x)\|_Wdx
\leq\|b\|_{l^1(\mathbb{Z}^d)}^{1/2}
\int_{\mathbb{R}^d}|u_{\alpha}(x)|\left(\sum_{k\in
\mathbb{Z}^d}|\phi(x-k)|^2\right)^{1/2}dx
<+\infty,$$ which together with shows that the function $u_{\alpha}\Phi$ is Bochner integrable.
By Theorem \[integral-perfectORKHS\] in the case that $\mathcal{Y}=\mathbb{C}$, we get the perfect FRKHS as a closed subspace of $V^2(\phi)$ $$\widetilde{\mathcal{H}}
:=\overline{\span}\left\{\int_{\mathbb{R}^d}
u_{\alpha}(x)\Phi(x)dx:\alpha\in\Lambda\right\}$$ with the functional reproducing kernel $$K(\alpha):=\int_{\mathbb{R}^d}u_{\alpha}(x)\Phi(x)dx.$$ According to (\[shift\_invariant\_rk\]) we get $\widetilde{\mathcal{H}}=\mathcal{H}$ and $K$ represented as in (\[phi\_K\_L\]).
To end this example, we shall give a sufficient and necessary condition which ensures that the FRKHS $\mathcal{H}$ appearing in (\[phi\_H\_K\_L\]) is identical to $V^2(\phi)$, that is, $$\label{density_V2}
\overline{\span}\left\{\sum_{k\in\mathbb{Z}^d}
\left(\int_{\mathbb{R}^d}u_{\alpha}(t)\overline{\phi(t-k)}dt
\right)\tilde{\phi}(\cdot-k):\alpha\in\Lambda\right\}
=V^2(\phi).$$ For this purpose, we introduce a sequence of functions by letting for each $\alpha\in\Lambda$ $$g_\alpha(\xi):=\sum_{l\in\mathbb{Z}^d}\widehat{u_{\alpha}}
(\xi+2l\pi)\overline{\hat{\phi}(\xi+2l\pi)},
\ \xi\in[-\pi,\pi]^d.$$
If $\phi\in W_0(L_w^1)$ satisfies and for each $\alpha\in \Lambda$, $u_{\alpha}\in L^2(\mathbb{R}^d)$ satisfies , then the density condition holds if and only if there holds $$\label{density_V2_1}
\overline{\span}\{g_\alpha:\ \alpha\in\Lambda\}
=L^2([-\pi,\pi]^d).$$
By the definition of $g_\alpha, \alpha\in\Lambda,$ we have for each $\alpha\in\Lambda$ and $k\in\mathbb{Z}^d$ that $$\begin{aligned}
\int_{\mathbb{R}^d}u_{\alpha}(x)\overline{\phi(x-k)}dx
&=&\int_{\mathbb{R}^d}\widehat{u_{\alpha}}(\xi)
\overline{\hat{\phi}(\xi)}e^{i(k,\xi)}d\xi\\
&=&\sum_{l\in\mathbb{Z}^d}\int_{[-\pi,\pi]^d}
\widehat{u_{\alpha}}(\xi+2l\pi)
\overline{\hat{\phi}(\xi+2l\pi)}e^{i(k,\xi)}d\xi\\
&=&\int_{[-\pi,\pi]^d}g_\alpha(\xi)e^{i(k,\xi)}d\xi
=c_k(g_\alpha),\end{aligned}$$ where we denote by $c_k(h)$ the $k$-th Fourier coefficients of $h\in L^2([-\pi,\pi]^d)$. According to the representation of the functions in $V^2(\phi)$, we note that the density condition (\[density\_V2\]) holds if and only if there holds $$\overline{\span}\{\{c_k(g_\alpha):k\in\mathbb{Z}^d\}:
\alpha\in\Lambda\}=l^2(\mathbb{Z}^d),$$ which is equivalent to (\[density\_V2\_1\]).
A functional reproducing kernel $K$ for a perfect FRKHS $\mathcal{H}$ can be represented by the classical reproducing kernel $\mathcal{K}$ for $\mathcal{H}$ by . Especially with respect to the the family $\mathcal{F}$ of the integral functionals $$\label{integralfun}
L_{\alpha}(f):=\int_{\mathcal{X}}f(x)\overline{u_{\alpha}(x)}
d\mu(x),\ \alpha\in\Lambda,$$ the functional reproducing kernel $K$ has the form $$\label{phi_suffi_nessi}
K(\alpha)(x):=\int_{\mathcal{X}}\mathcal{K}(t,x)
u_{\alpha}(t)d\mu(t).$$ To end this section, we shall show that under a somewhat stronger hypothesis, the operator $K$ with the form (\[phi\_suffi\_nessi\]) is a functional reproducing kernel if and only if $\mathcal{K}$ is a classical reproducing kernel on $\mathcal{X}$.
\[sufficient\_necessary\_1\] Let $\mathcal{X}$ be a compact set, $\mu$ a finite Borel measure on $\mathcal{X}$ and $\mathcal{K}:\mathcal{X}
\times\mathcal{X}\rightarrow \mathbb{C}$ a continuous function. If $\{u_{\alpha}:\alpha\in\Lambda\}$ satisfies for each $\alpha\in \Lambda$, $u_{\alpha}\in L^1(\mathcal{X},\mu)$ and $\overline{\span}\{u_{\alpha}:\alpha\in\Lambda\}
=L^1(\mathcal{X},\mu)$, then the operator $K$ defined by is a functional reproducing kernel with respect to a family $\mathcal{F}$ of linear functionals $L_{\alpha}, \alpha\in\Lambda$ defined by if and only if $\mathcal{K}$ is a reproducing kernel on $\mathcal{X}$.
We suppose that $\mathcal{K}$ is a reproducing kernel on $\mathcal{X}$ and $\mathcal{H}$ is the RKHS of $\mathcal{K}$. To show that $K$, defined by (\[phi\_suffi\_nessi\]), is a functional reproducing kernel, we first introduce the feature space $W:=\mathcal{H}$ and the feature map $\Phi(x):=\mathcal{K}(x,\cdot), x\in\mathcal{X}$ of $\mathcal{K}$ and set $\lambda_{\alpha}(x):=\overline{u_{\alpha}(x)}, x\in\mathcal{X},\alpha\in\Lambda$. We then verify the assumption (\[condition\]) in Theorem \[integral-perfectORKHS\]. Based on the above notations, we have for each $\alpha\in\Lambda$ that $$\int_{\mathcal{X}}|\lambda_{\alpha}(x)|
\|\Phi(x)\|_{\mathcal{H}}d\mu(x)
=\int_{\mathcal{X}}|u_{\alpha}(x)|
\sqrt{\mathcal{K}(x,x)}d\mu(x)
\leq\max_{x\in \mathcal{X}}\{\sqrt{\mathcal{K}(x,x)}\}
\int_{\mathcal{X}}|u_{\alpha}(x)|d\mu(x)
<+\infty.$$ Hence, by Theorem \[integral-perfectORKHS\], we get a functional reproducing kernel $$K(\alpha)(x)=\left\langle\int_{\mathcal{X}}u_{\alpha}(t)
\Phi(t)d\mu(t),\Phi(x)\right\rangle_{W},
\ \alpha\in\Lambda, \ x\in \mathcal{X},$$ with respect to a family $\mathcal{F}$ of linear functionals $L_{\alpha}, \alpha\in\Lambda$ defined by . By the definition of $\Phi$ and the reproducing property of $\mathcal{K}$, we conclude that $K$ has the form (\[phi\_suffi\_nessi\]).
Conversely, we suppose that $K$ with the form (\[phi\_suffi\_nessi\]) is a functional reproducing kernel with respect to $\mathcal{F}$ and show that $\mathcal{K}$ is a classical reproducing kernel on $\mathcal{X}$. We note from [@B41] that $\mathcal{K}$ is a classical reproducing kernel on $\mathcal{X}$ if and only if there holds for all $g\in L^1(\mathcal{X},\mu)$, $$\label{ifonlyif}
\int_{\mathcal{X}}\int_{\mathcal{X}}
g(s)\mathcal{K}(s,t)\overline{g(t)}d\mu(s)d\mu(t)
\geq 0.$$ Hence, it suffices to verify that (\[ifonlyif\]) holds true for all $g\in L^1(\mathcal{X},\mu)$. Observing from the definition of the functional reproducing kernel and the integral functionals, we have for any $u=\sum_{j\in\mathbb{N}_n}c_ju_{\alpha_j}
\in\span\{u_{\alpha}:\alpha\in \Lambda\}$ that $$\int_{\mathcal{X}}\int_{\mathcal{X}}
u(s)\mathcal{K}(s,t)\overline{u(t)}d\mu(s)d\mu(t)
=\sum_{j\in\mathbb{N}_n}\sum_{k\in\mathbb{N}_n}
c_j\overline{c_k}L_{\alpha_k}(K(\alpha_j)).$$ Since $K$ is a functional reproducing kernel with respect to $\mathcal{F}$, we get by the above equation that $$\int_{\mathcal{X}}\int_{\mathcal{X}}
u(s)\mathcal{K}(s,t)\overline{u(t)}d\mu(s)d\mu(t)
\geq 0$$ for any $u\in\span\{u_{\alpha}:\alpha\in\Lambda\}$. By the density of $u_{\alpha},\alpha\in\Lambda,$ in $L^1(\mathcal{X},\mu)$, there exists for each $g\in L^1(\mathcal{X},\mu)$ a sequence $u_n\in \span\{u_{\alpha}:\alpha\in\Lambda\}$ such that $$\lim_{n\rightarrow \infty}
\|u_n-g\|_{L^1(\mathcal{X},\mu)}=0.$$ By the triangular inequality there holds $$\begin{aligned}
&&\left|\int_{\mathcal{X}}\int_{\mathcal{X}}
\mathcal{K}(s,t)(g(s)\overline{g(t)}-u_n(s)
\overline{u_n(t)})d\mu(s)d\mu(t)\right|\\
&\leq&\max_{s,t\in X}\{|\mathcal{K}(s,t)|\}
\left(\|g\|_{L^1(\mathcal{X},\mu)}+
\|u_n\|_{L^1(\mathcal{X},\mu)}\right)
\|u_n-g\|_{L^1(\mathcal{X},\mu)},\end{aligned}$$ which implies that $$\int_{\mathcal{X}}\int_{\mathcal{X}}
g(s)\mathcal{K}(s,t)\overline{g(t)}d\mu(s)d\mu(t)
=\lim_{n\rightarrow\infty}\int_{\mathcal{X}}
\int_{\mathcal{X}}u_n(s)\mathcal{K}(s,t)
\overline{u_n(t)}d\mu(s)d\mu(t).$$ This together with the inequalities $$\int_{\mathcal{X}}\int_{\mathcal{X}}
u_n(s)\mathcal{K}(s,t)\overline{u_n(t)}d\mu(s)d\mu(t)
\geq 0, \ \ \mbox{for all} \ \ n,$$ ensures that (\[ifonlyif\]) holds for each $g\in L^1(\mathcal{X},\mu)$. Therefore, we have that $\mathcal{K}$ is a reproducing kernel on $\mathcal{X}$.
Sampling and Reconstruction in FRKHSs
=====================================
We describe in this section a reconstruction of an element in an FRKHS from its functional values. This study is motivated by the average sampling, which aims at recovering a function from its local average values represented as integral functional values. As pointed out in the classical sampling theory in RKHSs, the local average functionals used in the reconstruction should be continuous to ensure the stability of the sampling process. In practice, only finite local average functionals are used to establish a reconstruction of a function. However, to obtain a more precise approximation of the target function, more local average functionals are demanded. Hence, the average sampling should be considered in a space which ensures the continuity of the local average functionals with respect to all the points in the domain. Such a space is just an FRKHS with respect to the family of all the local average functionals. We establish the sampling theorem in the context of general FRKHSs and present explicit examples concerning the Paley-Wiener spaces. The average sampling has been widely studied in classical RKHSs, such as the Paley-Wiener spaces and the shift-invariant spaces without the availability of FRKHSs [@A; @S; @SZ]. We point out that these spaces are FRKHSs with respect to the local average functionals and the existing sampling theorems are special cases of the main results to be presented in this section.
The general sampling and reconstruction problem in an ORKHS may be reformulated as it in a corresponding FRKHS. Let $\mathcal{H}$ be an FRKHS with respect to a family $\mathcal{F}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear functionals on $\mathcal{H}$. We shall consider the complete reconstruction of an element $f\in\mathcal{H}$ from its sampled data $\{L_{\alpha_j}(f):j\in\mathbb{J}\}$, where $\mathbb{J}$ is a countable index set and $\{\alpha_j:j\in\mathbb{J}\}\subseteq \Lambda$. The general sampling and reconstruction problem in an ORKHS $\mathcal{H}$ with respect to a family $\mathcal{L}$ of linear operators $L_{\alpha},\ \alpha\in\Lambda,$ from $\mathcal{H}$ to $\mathcal{Y}$, where the sampled data have the form $\{\langle L_{\alpha_j}(f),\xi_j\rangle_{\mathcal{Y}}:
j\in\mathbb{J}\},$ with $\{\alpha_j:j\in\mathbb{J}\}\subseteq\Lambda$ and $\{\xi_j:j\in\mathbb{J}\}\subseteq\mathcal{Y}$, can be reformulated as the sampling and reconstruction problem in an FRKHS. In fact, if we introduce the family $\mathcal{F}$ of linear functionals $\widetilde{L}_{(\alpha,\xi)},\ (\alpha,\xi)
\in\Lambda\times\mathcal{Y},$ as $$\widetilde{L}_{(\alpha,\xi)}(f)
=\langle L_{\alpha}(f),\xi\rangle_{\mathcal{Y}},
\ (\alpha,\xi)\in\Lambda\times\mathcal{Y},$$ space $\mathcal{H}$ is also an FRKHS with respect to $\mathcal{F}$. From this observation, the above sampling problem in an ORKHS can also be interpreted as reconstructing an element $f$ in an FRKHS $\mathcal{H}$ from its functionals values $\widetilde{L}_{(\alpha_j,\xi_j)}(f), \ j\in\mathbb{J}$. Hence, we shall consider sampling and reconstruction in FRKHSs only.
Complete reconstruction of a function requires the availability of a frame or a Riesz basis of a Hilbert space. We review the concept of frames and Riesz bases of a Hilbert space according to [@D; @DS]. Let $\mathcal{H}$ be a separable Hilbert space with the inner product $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ and let $\mathbb{J}$ be a countable index set. A sequence $\{f_j: j\in\mathbb{J}\}\subset\mathcal{H}$ is called a frame of $\mathcal{H}$ if there exist constants $0<A\leq B<+\infty$ such that $$A\|f\|_{\mathcal{H}}
\leq \left(\sum_{j\in\mathbb{J}}
|\langle f,f_j\rangle_{\mathcal{H}}|^2\right)^{1/2}
\leq B\|f\|_{\mathcal{H}},
\ \mbox{for all}\ f\in\mathcal{H}.$$ For a frame $\{f_j: j\in\mathbb{J}\}$ of $\mathcal{H}$, there exists a dual frame $\{g_j: j\in\mathbb{J}\}\subset\mathcal{H}$, for which $$\label{reconstruction-formula-H}
f=\sum_{j\in\mathbb{J}}\langle f,
g_j\rangle_{\mathcal{H}}f_j,
\ \mbox{for each}\ f\in\mathcal{H}.$$ A frame $\{f_j: j\in\mathbb{J}\}$ is a Riesz basis of $\mathcal{H}$ if it is minimal, that is, $$f_j\notin\overline{\span}\{f_k:k\in\mathbb{J},k\neq j\},
\ \mbox{for each}\ j\in\mathbb{J}.$$ If $\{f_j: j\in\mathbb{J}\}$ is a Riesz basis of $\mathcal{H}$, there exists a unique sequence $\{g_j: j\in\mathbb{J}\}\subset \mathcal{H}$ satisfying (\[reconstruction-formula-H\]) and we call it the dual Riesz basis of the original Riesz basis. Moreover, the dual Riesz basis $\{g_j: j\in\mathbb{J}\}$ satisfies the biorthogonal condition that $\langle f_j,g_k\rangle_{\mathcal{H}}=\delta_{jk}$, for all $j,k\in\mathbb{J}$, where $\delta_{j,k}$ is the Kronecker delta notation.
We now consider reconstructing an element in an FRKHS in terms of its sampled functional values. Let $\mathcal{H}$ be an FRKHS with respect to a family $\mathcal{F}$ of linear functionals $L_{\alpha},\ \alpha\in \Lambda$ and $K$ the functional reproducing kernel for $\mathcal{H}$. As far as sampling is concerned, we assume that there exists a countable sequence $\alpha_j\in \Lambda, j\in\mathbb{J}$, such that $\mathbb{K}:=\{K(\alpha_j):\ j\in\mathbb{J}\}$ constitutes a frame for $\mathcal{H}$. We need to construct a dual frame of $\mathbb{K}$. To this end, we define the frame operator $T: \mathcal{H}\rightarrow \mathcal{H}$ for each $f\in \mathcal{H}$ by $$\label{frame_operator}
Tf:=\sum_{j\in\mathbb{J}}\langle f,
K(\alpha_j)\rangle_{\mathcal{H}} K(\alpha_j).$$ It is well-known that operator $T$ is bounded, invertible, self-adjoint and strictly positive on $\mathcal{H}$ and the sequence $\{G_j: j\in\mathbb{J}\}$ with $G_j:=T^{-1}(K(\alpha_j))$ is the dual frame of $\mathbb{K}$. We shall show that the dual frame $\{G_j: j\in\mathbb{J}\}$ is also induced from a functional reproducing kernel. For this purpose, we define operator $\widetilde{K}$ from $\Lambda$ to $\mathcal{H}$ by $$\label{dualkernel}
\widetilde{K}(\alpha):=T^{-1}(K(\alpha)),
\ \mbox{for each} \ \alpha\in \Lambda$$ and introduce the space $\widetilde{\mathcal{H}}:=\{f:f\in \mathcal{H}\}$ with the bilinear mapping $\langle \cdot,\cdot\rangle_{\widetilde{\mathcal{H}}}$ defined by $$\label{dualinner_product}
\langle f,g\rangle_{\widetilde{\mathcal{H}}}
:=\langle Tf,g\rangle_{\mathcal{H}},
\ f,g\in\widetilde{\mathcal{H}}.$$ Since operator $T$ is self-adjoint and strictly positive, we conclude that $\widetilde{\mathcal{H}}$ is an inner product space endowed with the inner product (\[dualinner\_product\]). The spaces $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ have the same elements although their norms are different. The next lemma reveals the equivalence of their norms.
\[norm-equivalence\] Let $\mathcal{H}$ be an FRKHS with respect to a family $\mathcal{F}$ of linear functionals $L_{\alpha},\ \alpha\in \Lambda$ and $K$ the functional reproducing kernel for $\mathcal{H}$. If for a countable sequence $\alpha_j$, $j\in\mathbb{J}$, in $\Lambda$, the set $\mathbb{K}$ constitutes a frame for $\mathcal{H}$ and $T$ is the frame operator, then the norms of $\mathcal{H}$ and $\widetilde{\mathcal{H}}$, defined as above, are equivalent.
It is known [@D] that there exist $A,B>0$ such that for any $f\in \mathcal{H}$, $$A\|f\|_{\mathcal{H}}^2
\leq\langle Tf,f\rangle_{\mathcal{H}}
\leq B\|f\|_{\mathcal{H}}^2.$$ This together with the fact $\langle f,f\rangle_{\widetilde{\mathcal{H}}}
=\langle Tf,f\rangle_{\mathcal{H}}$ yields the norm equivalence of $\mathcal{H}$ and $\widetilde{\mathcal{H}}$.
It follows from Lemma \[norm-equivalence\] that $\widetilde{\mathcal{H}}$ is a Hilbert space since $\mathcal{H}$ is a Hilbert space. The following proposition shows that $\widetilde{\mathcal{H}}$ is also an FRKHS with respect to $\mathcal{F}$ and has $\widetilde{K}$ as its functional reproducing kernel.
\[dualLkernel\] Let $\mathcal{H}$ be an FRKHS with respect to a family $\mathcal{F}$ of linear functionals $L_{\alpha}, \alpha\in \Lambda,$ and $K$ the functional reproducing kernel for $\mathcal{H}$. If for a countable sequence $\alpha_j$, $j\in\mathbb{J}$, in $\Lambda$, the set $\mathbb{K}$ constitutes a frame for $\mathcal{H}$ and $T$ is the frame operator, then space $\widetilde{\mathcal{H}}$ is also an FRKHS with respect to $\mathcal{F}$ and $\widetilde{K}$ is the corresponding functional reproducing kernel for $\widetilde{\mathcal{H}}$.
By the norm equivalence of $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ established in Lemma \[norm-equivalence\], we can show that the norm of $\widetilde{\mathcal{H}}$ is compatible with $\mathcal{F}$ since so is the norm of $\mathcal{H}$. Combining the norm equivalence with the continuity of the functionals $L_{\alpha}, \alpha\in\Lambda,$ on $\mathcal{H}$, we also obtain that these functionals are continuous on $\widetilde{\mathcal{H}}$. The definition of the FRKHS ensures that $\widetilde{\mathcal{H}}$ is an FRKHS with respect to $\mathcal{F}$.
It remains to show that the operator $\widetilde{K}$ reproduces the functionals $L_{\alpha}(f)$, for $\alpha\in\Lambda$. By (\[dualinner\_product\]), the self-adjoint property of $T$ and the reproducing property of the kernel $K$, we get for each $f\in \widetilde{\mathcal{H}}$ that $$\langle f,\widetilde{K}(\alpha) \rangle_{\widetilde{\mathcal{H}}}
=\langle Tf,\widetilde{K}(\alpha)\rangle_{\mathcal{H}}
=\langle f,T\widetilde{K}(\alpha)\rangle_{\mathcal{H}}
=\langle f,K(\alpha)\rangle_{\mathcal{H}}=L_{\alpha}(f),
\ \ \mbox{for all}\ \ \alpha\in\Lambda.$$ Hence, $\widetilde{K}$ is the corresponding functional reproducing kernel for $\widetilde{\mathcal{H}}$.
It is clear that $G_j=\widetilde{K}(\alpha_j)$, $j\in\mathbb{J}.$ That is, $\widetilde{\mathbb{K}}:=\{\widetilde{K}(\alpha_j):
\ j\in\mathbb{J}\}$ is the dual frame of $\mathbb{K}$ in $\mathcal{H}$. We shall call operator $\widetilde{K}$ defined by (\[dualkernel\]) the dual functional reproducing kernel of $K$ with respect to the set $\{\alpha_j: j\in\mathbb{J}\}$.
With the help of the dual functional reproducing kernel, we establish the complete reconstruction formula of an element in an FRKHS from its functional values.
\[complete-reconstruction-formula\] Let $\mathcal{H}$ be an FRKHS with respect to the family $\mathcal{F}$ of linear functionals $L_{\alpha}$, $\alpha\in \Lambda$ and $K$ the functional reproducing kernel for $\mathcal{H}$. If for a countable sequence $\alpha_j$, $j\in\mathbb{J}$, in $\Lambda$, the set $\mathbb{K}$ constitutes a frame for $\mathcal{H}$ and $\widetilde{K}$ is the dual functional reproducing kernel of $K$ with respect to the set $\{\alpha_j: j\in\mathbb{J}\}$, then $\widetilde{\mathbb{K}}$ is a dual frame of $\mathbb{K}$ in $\mathcal{H}$ and there holds for each $f\in\mathcal{H}$, $$\begin{aligned}
\label{reconstruction}
f=\sum_{j\in\mathbb{J}}L_{\alpha_j}(f)\widetilde{K}(\alpha_j),\end{aligned}$$ where the reconstruction formula holds also in $\widetilde{\mathcal{H}}$. Moreover, if $\mathbb{K}$ is a Riesz basis for $\mathcal{H}$, then $\widetilde{\mathbb{K}}$ is an orthonormal basis for the FRKHS $\widetilde{\mathcal{H}}$ of $\widetilde{K}$.
It follows from the definition of the dual functional reproducing kernel that $\widetilde{\mathbb{K}}$ is a dual frame of $\mathbb{K}$ in $\mathcal{H}$. Then, by (\[reconstruction-formula-H\]) we have that $$f=\sum_{j\in\mathbb{J}}\langle f,K(\alpha_j)
\rangle_{\mathcal{H}}\widetilde{K}(\alpha_j).$$ This together with the reproducing property $L_{\alpha_j}(f)=\langle f,K(\alpha_j)
\rangle_{\mathcal{H}}$, $j\in\mathbb{J},$ leads to the reconstruction formula (\[reconstruction\]). Lemma \[norm-equivalence\] guarantees that the norms of $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ are equivalent. Moreover, the two spaces have the same elements. Therefore, the reconstruction formula holds also in $\widetilde{\mathcal{H}}$.
In the case when $\mathbb{K}$ is a Riesz basis for $\mathcal{H}$, we verify the orthonormality of $\widetilde{\mathbb{K}}$ by using the biorthogonality of Reisz basis $\mathbb{K}$ and its dual $\widetilde{\mathbb{K}}$. Indeed, for each $j,k\in\mathbb{J}$, we have that $$\langle\widetilde{K}(\alpha_j),
\widetilde{K}(\alpha_k)\rangle_{\widetilde{\mathcal{H}}}
=\langle T\widetilde{K}(\alpha_j),
\widetilde{K}(\alpha_k)\rangle_{\mathcal{H}}
=\langle K(\alpha_j), \widetilde{K}(\alpha_k)\rangle_{\mathcal{H}}
=\delta_{j,k}.$$
According to Theorem \[complete-reconstruction-formula\], we have to find the frame in the form $\{K(\alpha_j): j\in\mathbb{J}\}$ for an FRKHS in order to establish the complete reconstruction formula. It is helpful to have a characterization of such a frame in terms of the feature map representation (\[FRKHS\_feature\_representation\_KL1\]) of the corresponding functional reproducing kernel.
\[feature\_Riesz\] Let $\Phi$ be a map from $\Lambda$ into a Hilbert space $W$ satisfying $\overline{\span}\{\Phi(\alpha):\alpha\in\Lambda\}=W.$ If $K$ is a functional reproducing kernel for an FRKHS $\mathcal{H}$, defined by , then for a countable sequence $\alpha_j, j\in\mathbb{J},$ in $\Lambda$, the set $\mathbb{K}$ constitutes a frame for $\mathcal{H}$ if and only if $\{\Phi(\alpha_j): j\in\mathbb{J}\}$ forms a frame of $W$.
We prove this theorem by establishing an isometric isomorphism between $\mathcal{H}$ and $W$. To this end, we let $$\mathcal{K}(\alpha,\beta):=L_{\beta}(K(\alpha)),
\ \ \alpha,\beta\in \Lambda.$$ By Corollary \[FRKHS\_isometric-isomorphism\], we have that $f\rightarrow L_{(\cdot)}(f)$ is an isometric isomorphism between the FRKHS $\mathcal{H}$ of $K$ and the RKHS $\widetilde{\mathcal{H}}$ of $\mathcal{K}$. Equation shows that $\Phi$ is the feature map of the kernel $\mathcal{K}$ and $W$ is the corresponding feature space. We then see that $\langle u,\Phi(\cdot)\rangle_W\rightarrow u$ is an isometric isomorphism from $\widetilde{\mathcal{H}}$ to $W$. We define the operator $T:\mathcal{H}\rightarrow W$ as follows: For each $f\in\mathcal{H}$, $Tf\in W$ satisfies $$\langle Tf, \Phi(\beta)\rangle_W=L_{\beta}(f),
\ \ \mbox{for all}\ \ \beta\in\Lambda.$$ It follows that $T$ is an isometric isomorphism from $\mathcal{H}$ to $W$. By noting that $T(K(\alpha))=\Phi(\alpha)$, we obtain immediately the desired result of this theorem.
In general, the reconstruction formula (\[reconstruction\]) holds only in the norm of an FRKHS. However, when an FRKHS is perfect, the series in (\[reconstruction\]) converges both pointwise and uniformly on any set where the reproducing kernel is bounded. This is due to the point-evaluation functionals are also continuous, when an FRKHS is perfect. Motivated by this observation, it is of practical importance to consider sampling and reconstruction in a perfect FRKHS. In this case, the frame $\{K(\alpha_j): j\in\mathbb{J}\}$ has the following characterization.
\[feature\_Riesz1\] Let $W$ be a Hilbert space, $\Phi:\mathcal{X}\rightarrow W$ satisfy $\overline{\span}\{\Phi(x):x\in \mathcal{X}\}=W$ and $\Psi:\Lambda\rightarrow W$ satisfy $\overline{\span}\{\Psi(\alpha):\alpha\in \Lambda\}=W$. If $K$ is a functional reproducing kernel for a perfect FRKHS $\mathcal{H}$, defined as in , then for a countable sequence $\alpha_j$, $j\in\mathbb{J}$, in $\Lambda$, the set $\mathbb{K}$ constitutes a frame for $\mathcal{H}$ if and only if $\{\Psi(\alpha_j): j\in\mathbb{J}\}$ forms a frame for $W$.
In order to apply Theorem \[feature\_Riesz\], we need to represent the functional reproducing kernel $K$ as in . It is known that the linear functionals on the perfect FRKHS $\mathcal{H}$ are determined by $$L_{\alpha}(\langle u,\Phi(\cdot)\rangle_W)
:=\langle u,\Psi(\alpha)\rangle_W,
\ u\in W,\ \alpha\in\Lambda.$$ By the representation of $K$, we have that $$L_{\beta}(K(\alpha))
=\langle\Psi(\alpha), \Psi(\beta)\rangle_W,
\ \ \mbox{for all}\ \ \alpha,\beta\in\Lambda.$$ That is, the mapping $\Psi: \Lambda\rightarrow W$ is the feature map of $K$, defined as in . Noting that $$\overline{\span}\{\Psi(\alpha):\alpha \in\Lambda\}=W,$$ the desired result of this theorem follows directly from Theorem \[feature\_Riesz\].
Based upon the discussion above, we conclude that the frame $\{K(\alpha_j):j\in\mathbb{J}\}$ for an FRKHS $\mathcal{H}$ is important for establishing the complete reconstruction formula of functions in $\mathcal{H}$. These desirable frames can be obtained by characterizing the frame $\{\Phi(\alpha_j):j\in\mathbb{J}\}$ for the feature space $W$. In the light of this point, we shall consider the complete reconstruction of functions in two concrete FRKHSs.
Below, we consider reconstructing a vector-valued function $f:\mathbb{R}\rightarrow\mathbb{C}^n$ from its sampled data $$\langle f(x_j),\xi_j\rangle_{\mathbb{C}^n},
\ (x_j,\xi_j)\in \mathbb{R}\times \mathbb{C}^n,
\ j\in\mathbb{Z}.$$ The functions to be reconstructed are taken from the Paley-Wiener space of $\mathbb{C}^n$-valued functions on $\mathbb{R}$ $$\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n)
:=\left\{f\in L^2(\mathbb{R},\mathbb{C}^{n}):
\ {\rm supp}\hat f\subseteq [-\pi,\pi]\right\}$$ with the inner product $$\langle f,g\rangle_{\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n)}
:=\sum_{j\in\mathbb{N}_n}\langle f_j,g_j
\rangle_{L^{2}(\mathbb{R})}, \ \ \mbox{for}\ f:=[f_j:j\in\mathbb{N}_n], \ g:=[g_j:j\in\mathbb{N}_n].$$ Set $\Lambda:=\mathbb{R}\times \mathbb{C}^n$ and for each $(x,\xi)\in\Lambda,$ introduce a functional on $\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n)$ as $$L_{(x,\xi)}(f):=\langle f(x),\xi\rangle_{\mathbb{C}^n},
\ f\in\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n).$$ It was pointed out that the space $\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n)$ is an FRKHS with respect to the family $\mathcal{F}$ of linear functionals $L_{(x,\xi)}, (x,\xi)\in\Lambda$. The functional reproducing kernel $K$ for $\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n)$ is given by $$K(x,\xi):={\rm diag} (\mathcal{K}(x,\cdot):j\in\mathbb{N}_n)\xi,
\ (x,\xi)\in \Lambda,$$ where $\mathcal{K}$ is the sinc kernel on $\mathbb{R}$ defined by $$\label{sincR}
\mathcal{K}(x,y):=\frac{\sin \pi(x-y)}{\pi(x-y)},
\ x,y\in\mathbb{R}.$$ Set $W:=L^2([-\pi,\pi],\mathbb{C}^n)$ and define $\Phi:\Lambda\rightarrow W$ by $$\Phi(x,\xi):=\frac{1}{\sqrt{2\pi}}e^{ix(\cdot)}\xi,
\ \ (x,\xi)\in \Lambda.$$ It follows from $$\mathcal{K}(x,y)
=\left\langle\frac{1}{\sqrt{2\pi}}e^{ix(\cdot)},
\frac{1}{\sqrt{2\pi}}e^{iy(\cdot)}\right
\rangle_{L^2([-\pi,\pi])},
\ \ x,y\in\mathbb{R},$$ that for all $(x,\xi),(y,\eta)\in \Lambda$, $$\begin{aligned}
L_{(y,\eta)}(K(x,\xi))
=\langle K(x,\xi)(y),\eta\rangle_{\mathbb{C}^n}
=\sum_{j\in\mathbb{N}_n}\left\langle\frac{1}{\sqrt{2\pi}}
e^{ix(\cdot)}\xi_j,\frac{1}{\sqrt{2\pi}}
e^{iy(\cdot)}\eta_j\right\rangle_{_{L^2([-\pi,\pi])}}.\end{aligned}$$ This leads to $$L_{(y,\eta)}(K(x,\xi))
=\langle\Phi(x,\xi), \Phi(y,\eta)\rangle_{W},
\ \ \mbox{for all} \ \ (x,\xi),(y,\eta)\in\Lambda.$$ Thus, we conclude that $W$ and $\Phi$ are the feature space and the feature map of the functional reproducing kernel $K$, respectively. Moreover, the density condition $\overline{\span}\{\Phi(x,\xi):\ (x,\xi)\in\Lambda\}=W$ is satisfied.
We now turn to considering reconstructing a function $f\in\mathcal{B}_{\pi}(\mathbb{R}, \mathbb{C}^n)$ from its functional values $L_{(x_j,\xi_j)}(f),
(x_j,\xi_j)\in\Lambda, j\in\mathbb{Z}$. This will be done by finding the sampling set $\{(x_j,\xi_j):j \in\mathbb{Z}\}\subseteq\Lambda$ such that $\{K(x_j,\xi_j): j\in\mathbb{J}\}$ forms a frame for $\mathcal{B}_{\pi}(\mathbb{R}, \mathbb{C}^n)$. By Theorem \[feature\_Riesz\] such a frame can be characterized by the frame for $W$ with the form $\Phi(x_j,\xi_j), j\in\mathbb{Z}$. We shall give a positive example by choosing the sampling set $\{(x_j,\xi_j):j\in\mathbb{Z}\}$ as follows. We present each $j\in\mathbb{Z}$ as $j:=nm+l$, where $m\in\mathbb{Z}$ and $l\in\mathbb{Z}_n:=\{0,1,\dots,n-1\}$. We then choose the sequence $x_j, j\in\mathbb{Z},$ in $\mathbb{R}$ such that for each $l\in\mathbb{Z}_n$, $\{\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l}(\cdot)}:m\in\mathbb{Z}\}$ is a frame for $L^2([-\pi,\pi])$. We introduce $n$ vectors $\eta_{l}:=[\eta_{l,k}:k\in\mathbb{Z}_n], l\in\mathbb{Z}_n,$ satisfying that the matrix $\mathbf{U}:=[\eta_l:l\in\mathbb{Z}_n]$ is unitary. We then choose the sequence $\xi_j,j\in\mathbb{Z},$ in $\mathbb{C}^n$ as $\xi_{nm+l}:=\eta_{l}$ for all $l\in\mathbb{Z}_n$ and all $m\in\mathbb{Z}$. Based on the above notations, we show below that the sequence $\Phi(x_j,\xi_j), j\in\mathbb{Z},$ constitutes a frame for $W$.
\[frame-vector\] If the sequence $\{(x_j,\xi_j):j\in\mathbb{Z}\}
\subseteq\Lambda$ is defined as above, then $\Phi(x_j,\xi_j), j\in\mathbb{Z},$ constitutes a frame for $W$. Moreover, if for each $l\in\mathbb{Z}_n$, the sequence $\{\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l}(\cdot)}:
m\in\mathbb{Z}\}$ is a Riesz basis for $L^2([-\pi,\pi])$, then $\Phi(x_j,\xi_j), j\in\mathbb{Z},$ forms a Riesz basis for $W$.
By the definition of frames, it suffices to verify that there exist positive constants $0<A\leq B<+\infty$ such that for all $u\in W$, $$\label{equivalence_A_B}
A\|u\|_{W}\leq \left(\sum_{j\in\mathbb{Z}}|
\langle u,\Phi(x_j,\xi_j)\rangle_{W}|^2\right)^{1/2}
\leq B\|u\|_{W},$$ which implies $\Phi(x_j,\xi_j),j\in\mathbb{Z}$, is a frame for $W$. According to the description of the sequence $\{(x_j,\xi_j): j\in\mathbb{Z}\}\subseteq\Lambda$ we have for any $u:=[u_k:k\in\mathbb{Z}_n]\in W$ that $$\begin{aligned}
\label{frame1}
\sum_{j\in\mathbb{Z}}\left|\langle u,
\Phi(x_j,\xi_j)\rangle_{W}\right|^2
=\sum_{l\in\mathbb{Z}_n}\sum_{m\in\mathbb{Z}}
\left|\left<\sum_{k\in\mathbb{Z}_n}\overline{\eta}_{l,k}u_k,
\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l}(\cdot)}
\right>_{L^2([-\pi,\pi])}\right|^2.\end{aligned}$$ For each $l\in\mathbb{Z}_n$, set $\displaystyle{v_l:=\sum_{k\in\mathbb{Z}_n}
\overline{\eta}_{l,k}u_k}$. Since $\{\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l}(\cdot)}:
m\in\mathbb{Z}\}$ is a frame for $L^2([-\pi,\pi])$, there exist positive constants $0<A_l\leq B_l<+\infty$ such that $$A_l^2\|v_l\|_{L^2([-\pi,\pi])}^2
\leq\sum_{m\in\mathbb{Z}}\left|\left< v_l,
\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l}(\cdot)}
\right>_{L^2([-\pi,\pi])}\right|^2
\leq B_l^2\|v_l\|_{L^2([-\pi,\pi])}^2.$$ Substituting the above estimates for all $l\in\mathbb{Z}_n$ into (\[frame1\]), we get that $$\begin{aligned}
\label{frame2}
A^2\sum_{l\in\mathbb{Z}_d}\|v_l\|_{L^2([-\pi,\pi])}^2
\leq\sum_{j\in\mathbb{Z}}\left|\left< u,
\Phi(x_j,\xi_j)\right>_{W}\right|^2\leq B^2
\sum_{l\in\mathbb{Z}_d}\|v_l\|_{L^2([-\pi,\pi])}^2,\end{aligned}$$ where $A:=\min\{A_l:l\in\mathbb{Z}_n\}$ and $B:=\max\{B_l:l\in\mathbb{Z}_n\}$. Note that there holds $$\begin{aligned}
\label{gn}
\sum_{l\in\mathbb{Z}_n}\|v_l\|_{L^2([-\pi,\pi])}^2
=\sum_{k\in\mathbb{Z}_n}\sum_{j\in\mathbb{Z}_n}
\left(\sum_{l\in\mathbb{Z}_n}
\overline{\eta}_{l,k}\eta_{l,j}\right)
\langle u_k, u_j\rangle_{L^2([-\pi,\pi])}.\end{aligned}$$ Equation (\[gn\]) with the unitary assumption on $\mathbf{U}$ leads to $$\sum_{l\in\mathbb{Z}_n}\|v_l\|_{L^2([-\pi,\pi])}^2
=\sum_{k\in\mathbb{Z}_n}\|u_k\|_{L^2([-\pi,\pi])}^2
=\|u\|^2_{W}.$$ Hence, by (\[frame2\]) we get the equivalent property .
We next prove that $\Phi(x_j,\xi_j),j\in\mathbb{Z},$ is also a Riesz basis for $W$ based on the hypothesis that for each $l\in\mathbb{Z}_n$, $\{\frac{1}{\sqrt{2\pi}}
e^{ix_{nm+l}(\cdot)}:m\in\mathbb{Z}\}$ is a Riesz basis for $L^2([-\pi,\pi])$. It suffices to show that if there exist $\{c_j:j\in\mathbb{Z}\}\in l^2(\mathbb{Z})$ such that $\sum_{j\in\mathbb{Z}}c_j\Phi(x_j,\xi_j)=0$, then $c_j=0,j\in\mathbb{Z}.$ It follows from $\sum_{j\in\mathbb{Z}}c_j\Phi(x_j,\xi_j)=0$ that for all $j'\in\mathbb{Z}$, $$\begin{aligned}
\label{Riesz1}
\sum_{j\in\mathbb{Z}}c_j\langle\Phi(x_j,\xi_j),
\Phi(x_{j'},\xi_{j'})\rangle_{W}=0.\end{aligned}$$ There holds for any $j:=nm+l$ and any $j':=nm'+l'$ that $$\langle\Phi(x_j,\xi_j),\Phi(x_{j'},\xi_{j'})\rangle_{W}
=\sum_{k\in\mathbb{Z}_n}\overline{\eta}_{l,k}\eta_{l',k}
\left\langle\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l}(\cdot)},
\frac{1}{\sqrt{2\pi}}e^{ix_{nm'+l'}(\cdot)}
\right\rangle_{L^2([-\pi,\pi])}.$$ Again, using the unitary property of ${\bf U}$, we obtain that $$\begin{aligned}
\label{Riesz2}
\langle\Phi(x_j,\xi_j),\Phi(x_{j'},\xi_{j'})\rangle_{W}
=\delta_{l,l'}\left\langle\frac{1}{\sqrt{2\pi}}
e^{ix_{nm+l}(\cdot)}, \frac{1}{\sqrt{2\pi}}
e^{ix_{nm'+l'}(\cdot)}\right\rangle_{L^2([-\pi,\pi])}.\end{aligned}$$ Substituting (\[Riesz2\]) into (\[Riesz1\]), we obtain for any $m'\in\mathbb{Z}$ and any $l'\in\mathbb{Z}_n$ that $$\begin{aligned}
\left\langle\sum_{m\in\mathbb{Z}}c_{nm+l'}
\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l'}(\cdot)},
\frac{1}{\sqrt{2\pi}}e^{ix_{nm'+l'}(\cdot)}
\right\rangle_{L^2([-\pi,\pi])}=0.\end{aligned}$$ This together with the hypothesis that $\{\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l'}(\cdot)}:
m\in \mathbb{Z}\}$ is a Riesz basis for $L^2([-\pi,\pi])$ leads to $$\frac{1}{\sqrt{2\pi}}\sum_{m\in\mathbb{Z}}
c_{nm+l'}e^{ix_{nm+l'}(\cdot)}=0.$$ Since for any $l'\in\mathbb{Z}_N$, $\{\frac{1}{\sqrt{2\pi}}e^{ix_{nm+l'}(\cdot)}:
m\in\mathbb{Z}\}$ is a Riesz basis for $L^2([-\pi,\pi])$, we conclude that $c_{nm+l'}=0$, for all $m\in\mathbb{Z}$ and $l'\in\mathbb{Z}_n,$ which completes the proof.
Sampling sets $\{(x_j,\xi_j): j\in\mathbb{Z}\}$ such that $\Phi(x_j,\xi_j),j\in\mathbb{Z},$ constitute a Riesz basis for $W$ were characterized in [@AI95; @AI08] in terms of the generating matrix function. The authors considered the matrix Sturm-Liouville problems and applied the boundary control theory for hyperbolic dynamical systems in producing a wide class of matrix functions generating sampling sets. Such sampling sets can provide us the desired frames for $\mathcal{B}_{\pi}(\mathbb{R}, \mathbb{C}^n)$ having the form $\{K(x_j,\xi_j): j\in\mathbb{Z}\}$. By constructing the frame operator (\[frame\_operator\]) and obtaining the dual functional reproducing kernel $\widetilde{K}$ by (\[dualkernel\]), we can build the reconstruction formula of $f\in\mathcal{B}_{\pi}(\mathbb{R},\mathbb{C}^n)$ as $$f=\sum_{j\in\mathbb{Z}}\langle f(x_j),
\xi_j\rangle_{\mathbb{C}^n}\widetilde{K}(x_j,\xi_j).$$
We next consider recovering a scalar-valued function $f$ on $\mathbb{R}$ from the sampled data determined by integral functionals. As a special case, the local average sampling is devoted to reconstructing a function $f$ from its local average values in the form $$\label{averagefunctional}
L_{x_j}(f):=\int_{\mathbb{R}}f(t)u_{x_j}(t)dt,
\ x_j\in\mathbb{R},\ j\in\mathbb{Z},$$ where $u_x$, $x\in\mathbb{R}$, are nonnegative functions that satisfy $$\label{average-functions}
\int_{\mathbb{R}}u_x(t)dt=1\ \mbox{and}\ \
\supp u_x\subseteq [x-\delta, x+\delta],$$ with $\delta$ being a positive constant. We call $u_x$, $x\in\mathbb{R}$, the average functions. We shall restrict ourselves to investigating the local average sampling in the framework of FRKHSs. The FRKHS to be considered is the Paley-Wiener space of scalar-valued functions on $\mathbb{R}$ $$\mathcal{B}_{\pi}:=\{f\in L^2(\mathbb{R}):
{\rm supp}\hat f\subseteq [-\pi,\pi]\}.$$ The space $\mathcal{B}_{\pi}$ is a standard RKHS, in which the ideal sampling and the local average sampling are usually considered. The sinc kernel for $\mathcal{B}_{\pi}$, defined by (\[sincR\]), is a translation invariant reproducing kernel defined by (\[translation\_invariant\_rk\_phi\]) with $\varphi:=\frac{1}{2\pi}\chi_{[-\pi,\pi]}$. As pointed out in the last section, the space $W:=L^2([-\pi,\pi])$ and the mapping $\Phi:\mathbb{R}\rightarrow W$, defined by $\Phi(x):=\frac{1}{\sqrt{2\pi}}e^{ix(\cdot)}$, are the feature space and the feature map of the sinc kernel, respectively. It is clear that the function $\varphi$ and the average functions $u_{x},\ x\in\mathbb{R},$ are all integrable. Hence, by Theorem \[translation\_invariant\_ORKHS\] we obtain that the closed subspace $$\mathcal{H}:=\left\{f\in \mathcal{B}_{\pi}:
\check{f}\in\overline{\span}\left\{u_{x}^{\vee}
\chi_{[-\pi,\pi]}:\ x\in\mathbb{R}\right\}\right\}$$ of $\mathcal{B}_{\pi}$ is a perfect FRKHS with respect to the family $\mathcal{F}$ of the local average functionals $L_{x},x\in\mathbb{R}$. The closure in the representation of $\mathcal{H}$ is taken in $W$. Moreover, the functional reproducing kernel is given by $$K(x)(y)=[u_{x}^{\vee}\chi_{[-\pi,\pi]}]^{\wedge}(y),
\ x,y\in\mathbb{R}.$$ For each $x\in\mathbb{R}$, set $$\Psi(x):=\sqrt{2\pi}u_{x}^{\vee}\chi_{[-\pi,\pi]}.$$ Then $K$ can be represented as in .
To relate our new theory to the existing average sampling theorems in the literature, we consider recovering functions in $\mathcal{B}_{\pi}$, rather than the closed subspace $\mathcal{H}$, from the local average values. Hence, we recall Corollary \[density-W\], which states that $\mathcal{B}_{\pi}$ itself is a perfect FRKHS if and only if there holds the density $\overline{\span}\{\Psi(x):x\in\mathbb{R}\}=W$. We restrict ourselves to this special case. Since $\mathcal{B}_{\pi}$ is a perfect FRKHS, with the help of Theorem \[feature\_Riesz1\] we shall try to construct the sampling set $\{x_j:j\in\mathbb{Z}\}$ such that the sequence $\Psi(x_j), j\in\mathbb{Z},$ constitutes a frame for $W$. The next theorem gives a sufficient condition for $\Psi(x_j), j\in\mathbb{Z},$ to be a frame for $W$. For any $u\in W$, we let $R_{u}$ and $I_{u}$ denote its real part and imaginary part, respectively.
\[frame-Paley\] Let $\delta$ be a positive constant. Suppose that the sequence of nonnegative functions $u_x,x\in\mathbb{R},$ satisfy condition . If there exist positive constants $A,B$ depending only on the sampling set $\{x_j:j\in\mathbb{Z}\}$ and $\delta$ such that for any sequence $t_j, j\in\mathbb{Z},$ with $t_j\in [x_j-\delta,x_j+\delta]$, $\{e^{it_j(\cdot)}:j\in\mathbb{Z}\}$ forms a frame for $W$ with the frame bounds $A,B$, then $\{\Psi(x_j): j\in\mathbb{Z}\}$ constitutes a frame for $W$.
To show that $\Psi(x_j), j\in\mathbb{Z},$ constitutes a frame for $W,$ we need to estimate the quantities $Q_j(u):=\langle u,\Psi(x_j)\rangle_{W}$, $j\in\mathbb{Z},$ for any $u\in W$. It follows for any $u\in W$ that $$\begin{aligned}
\label{I}
|Q_j(u)|^2
=\left|\int_{-\pi}^{\pi}u(t)\left(\frac{1}{2\pi}
\int_{\mathbb{R}}u_{x_j}(s)e^{-ist}ds\right)dt\right|^2
=\frac{1}{(2\pi)^2}\left|\int_{\mathbb{R}}
\langle u, e^{is(\cdot)}\rangle_{W}u_{x_j}(s)ds\right|^2.\end{aligned}$$ We decompose $u$ into the odd and the even parts. That is, $u=u_{o}+u_{e}$, where $$u_{o}(x)=\frac{u(x)-u(-x)}{2},\ u_{e}(x)=\frac{u(x)+u(-x)}{2},\ x\in[-\pi,\pi].$$ Substituting the decomposition $u=u_{o}+u_{e}$ into , we get that $$\label{I1}
|Q_j(u)|^2
=\frac{1}{(2\pi)^2}\Big{|}\int_{\mathbb{R}}F_u(s)u_{x_j}(s)ds
+i\int_{\mathbb{R}}G_u(s)u_{x_j}(s)ds\Big{|}^2,$$ where $$F_u(s):=\langle R_{u_e}+iI_{u_o},
e^{is(\cdot)}\rangle_{W} \ \mbox{and}\
G_u(s):=\langle I_{u_e}-iR_{u_o}, e^{is(\cdot)}\rangle_{W}, \ \ s\in\mathbb{R}.$$ Note that $F_u$ and $G_u$ are both real and continuous functions. By the mean value theorem and the hypothesis on $u_x,x\in\mathbb{R}$, there exist $s_j,\tilde{s}_j \in [x_j-\delta,x_j+\delta]$ such that $$\int_{\mathbb{R}}F_u(s)u_{x_j}(s)ds=F_u(s_j)\ \mbox{and}\
\int_{\mathbb{R}}G_u(s)u_{x_j}(s)ds=G_u(\tilde{s}_j).$$ Substituting the above equations into (\[I1\]), we obtain that $$\begin{aligned}
\label{I2}
|Q_j(u)|^2=\frac{1}{(2\pi)^2}[F_u^2(s_j)+G_u^2(\tilde{s}_j)].\end{aligned}$$ By the hypothesis of this theorem, both $\{e^{is_j(\cdot)}:j\in\mathbb{Z}\}$ and $\{e^{i\tilde{s}_j(\cdot)}:j\in\mathbb{Z}\}$ are frames for $W$ with the frame bounds $A,B$. Hence, by the definition of $F_u$ and $G_u$, we get that $$A^2\|R_{u_e}+iI_{u_o}\|_{W}^2
\leq\sum_{j\in\mathbb{Z}}F_u^2(s_j)
\leq B^2\|R_{u_e}+iI_{u_o}\|_{W}^2$$ and $$A^2\|I_{u_e}-iR_{u_o}\|_{W}^2
\leq\sum_{j\in\mathbb{Z}}G_u^2(\tilde{s}_j)
\leq B^2\|I_{u_e}-iR_{u_o}\|_{W}^2.$$ By (\[I2\]) and noting that $$\|u\|_{W}^2=\|R_{u_e}+iI_{u_o}\|_{W}^2+\|I_{u_e}-iR_{u_o}\|_{W}^2,$$ we conclude that $$\frac{1}{2\pi}A\|u\|_{W}
\leq\left(\sum_{j\in\mathbb{Z}}|Q_j(u)|^2\right)^{1/2}
\leq\frac{1}{2\pi}B\|u\|_{W},$$ which completes the proof.
Observing from Theorem \[frame-Paley\], we have that the frames for $W$ with the form $\{e^{it_j(\cdot)}: j\in\mathbb{Z}\}$ is important for the study of the frames with the form $\{\Psi(x_j): j\in\mathbb{Z}\}$. For a complete characterization of the former, one can see [@C; @Y]. By making use of the sampling set $\{x_j:j\in\mathbb{Z}\}$, we obtain the frame $K(x_j),\ j\in\mathbb{Z},$ for $\mathcal{B}_{\pi}$. Accordingly, we get the complete reconstruction formula for $f\in\mathcal{B}_{\pi}$ as $$f=\sum_{j\in\mathbb{Z}}\left(\int_{\mathbb{R}}f(t)u_{x_j}(t)
dt\right)\widetilde{K}(x_j),$$ where $\widetilde{K}$ is the dual functional reproducing kernel of $K$ with respect to the set $\{x_j:j\in\mathbb{Z}\}$.
As applications of Theorem \[frame-Paley\], we present two specific examples for choosing the sampling set $\{x_j:j\in\mathbb{Z}\}$ and the average functions $u_{x},x\in\mathbb{R}$.
[Let $x_j=j, j\in\mathbb{Z},$ and $0<\delta<1/4$. The well-known Kadec’s $1/4$-theorem shows that for any sequence $\{t_j:j\in\mathbb{Z}\}$ satisfying $t_j\in[x_j-\delta,x_j+\delta], j\in\mathbb{Z},$ the sequence $\{e^{it_j(\cdot)}:j\in\mathbb{Z}\}$ forms a frame for $L^2([-\pi,\pi])$ with the bounds $$2\pi(\cos(\delta \pi)-\sin(\delta\pi))^2\ \mbox{and}\ 2\pi(2-\cos(\delta \pi)+\sin(\delta\pi))^2.$$ That is, the hypothesis of Theorem \[frame-Paley\] is satisfied. Hence, we have by Theorem \[frame-Paley\] that $\{\Psi(x_j): j\in\mathbb{Z}\}$ constitutes a frame for $W$ and then we can establish the complete reconstruction formula of functions in $\mathcal{B}_{\pi}$ as in Theorem \[complete-reconstruction-formula\]. In fact, the local average sampling theorem in this case was established in [@SZ] for $\mathcal{B}_{\pi}$ without the concept of FRKHSs. A generalized Kadec’s $1/4$-theorem can help us extend this result. Specifically, we suppose that the sampling set $\{x_j:j\in\mathbb{Z}\}$ satisfies that $\{e^{ix_j(\cdot)}: j\in\mathbb{Z}\}$ is a frame for $W$ with bounds $A,B$ and $0<\delta<1/4$ such that $$\label{Kadec}
1-\cos(\delta\pi)+\sin(\delta\pi)<\sqrt{\frac{A}{B}}.$$ The generalized Kadec’s $1/4$-theorem states that for any $\{t_j:j\in\mathbb{Z}\}$ with $t_j\in[x_j-\delta,x_j+\delta], j\in\mathbb{Z},$ $\{e^{it_j(\cdot)}:j\in\mathbb{Z}\}$ forms a frame for $W$ with bounds $$A\left(1-\sqrt{\frac{B}{A}}(1-\cos(\delta\pi)
-\sin(\delta\pi))\right)^2\ \mbox{and}\
B(2-\cos(\delta \pi)+\sin(\delta\pi))^2.$$ Hence, the sampling set $\{x_j:\ j\in\mathbb{Z}\}$ and $\delta$ in this case also satisfy the hypothesis of Theorem \[frame-Paley\] and thus, $\{\Psi(x_j): j\in\mathbb{Z}\}$ constitutes a frame for $W$.]{}
[We review an example considered in [@SZ] and reinterpret it by Theorem \[frame-Paley\]. Specifically, we suppose that $\alpha, L$ are positive constants and $0<\epsilon<1$. It is known [@DS] that for any sampling set $\{x_j:\ j\in\mathbb{Z}\}$ such that $|x_j-x_k|\geq\alpha, j\neq k$ and $|x_j-j\epsilon|\leq L$, the sequence $\{e^{ix_j(\cdot)}:j\in\mathbb{Z}\}$ forms a frame for $W$ with bounds $A, B$ depending only on $\alpha, L, \epsilon.$ If we choose the sampling set $\{x_j:j\in\mathbb{Z}\}$ satisfying the above condition with $\alpha, L, \epsilon$ and $0<\delta<\alpha/2$, then we can verify that for any sequence $t_j, j\in\mathbb{Z},$ with $t_j\in [x_j-\delta,x_j+\delta]$, there holds $$|t_j-t_k|\geq\alpha-2\delta, j\neq k,\ |t_j-j\epsilon|
\leq L+\delta.$$ Hence, we conclude that $\{e^{it_j(\cdot)}: j\in\mathbb{Z}\}$ forms a frame for $W$ with the bounds $A,B$ depending only on $\alpha, L, \epsilon$ and $\delta$. That is, the hypothesis of Theorem \[frame-Paley\] is satisfied. Hence, we get that $\{\Psi(x_j):j\in\mathbb{Z}\}$ constitutes a frame for $W$. The corresponding complete reconstruction formula of functions in $\mathcal{B}_{\pi}$ can be built as in Theorem \[complete-reconstruction-formula\]. Such a local average sampling theorem was also established in [@SZ].]{}
To end the discussion about the local average sampling in the Paley-Wiener space $\mathcal{B}_{\pi}$, we consider a special case that $u_x(t)=u(t-x),\ t\in\mathbb{R}$, with $u\in L^1(\mathbb{R})$ and present a sufficient condition for $\{\Psi(x_j): j\in\mathbb{J}\}$ to be a frame for $W$.
Let $u\in L^1(\mathbb{R})$ satisfy $|\check{u}(t)|\geq c$ for a positive constant $c$ and any $t\in[-\pi,\pi]$ and $u_x:=u(\cdot-x),\ x\in\mathbb{R}$. If $\{e^{ix_j(\cdot)}:
j\in\mathbb{Z}\}$ is a frame for $W$, then $\{\Psi(x_j):
j\in\mathbb{Z}\}$ also constitutes a frame for $W$.
It follows for any $x\in\mathbb{R}$ that $u_{x}^{\vee}=e^{ix(\cdot)}\check{u}$. Then we have for any $w\in W$ that $$\sum_{j\in\mathbb{Z}}|\langle w,\Psi(x_j)\rangle_{W}|^2
=\sum_{j\in\mathbb{Z}}|\langle w \overline{\check{u}}
\chi_{[-\pi,\pi]},e^{ix_j(\cdot)}\rangle_{W}|^2.$$ Since $\{e^{ix_j(\cdot)}:j\in\mathbb{Z}\}$ is a frame for $W$, there exist $0<A\leq B<\infty$ such that $$\label{equality11}
A^2\|w \overline{\check{u}}\chi_{[-\pi,\pi]}\|_{W}^2
\leq \sum_{j\in\mathbb{Z}}|\langle w,
\Psi(x_j)\rangle_{W}|^2
\leq B^2\|w \overline{\check{u}}\chi_{[-\pi,\pi]}\|_{W}^2.$$ By the assumptions on function $u$, we have that $$c\|w\|_{W}
\leq \|w\overline{\check{u}}\chi_{[-\pi,\pi]}\|_{W}
\leq \frac{1}{2\pi}\|u\|_{L^1(\mathbb{R})}\|w\|_{W}.$$ Substituting the above estimate into (\[equality11\]), we obtain for any $w\in W$ that $$A'\|w\|_{W}
\leq \left(\sum_{j\in\mathbb{Z}}|\langle w,\Psi(x_j)
\rangle_{W}|^2\right)^{1/2}
\leq B'\|w\|_{W},$$ with $A':=Ac$ and $B':=\frac{B}{2\pi}\|u\|_{L^1(\mathbb{R})}$. That is, the sequence $\{\Psi(x_j): j\in\mathbb{Z}\}$ constitutes a frame for $W$.
To close this section, we remark that taking into account the continuity of the sampling process, FRKHSs are the right spaces for reconstruction using functional-valued data. The complete reconstruction formula in such spaces can be obtained by constructing the frames in terms of the corresponding functional reproducing kernels. Characterizing the desired frames by features provides us a convenient approach for finding them. The specific examples discussed in this section include not only the existing sampling theorems concerning the functional-valued data but also new ones. Hence, we point out that FRKHSs indeed provide an ideal framework for systematically study sampling and reconstruction with respect to functional-valued data.
Regularized Learning in ORKHSs
==============================
Regularization, a widely used approach in solving an ill-posed problem of learning the unknown target function from finite samples, has been a focus of attention in the field of machine learning [@CS; @MP05; @SS; @SC; @V]. We study in this section the regularized learning from finite operator-valued data in the framework of ORKHSs. The resulting representer theorem shows that as far as the operator-valued data are concerned, learning in ORKHSs has the advantages over RKHSs.
We begin with recalling the regularized learning schemes for learning a function in an RKHS from finite point-evaluation data. Suppose that $\mathcal{X}$ is the input space and the output space $\mathcal{Y}$ is a Hilbert space. Set $\mathcal{Y}^m:=\mathcal{Y}\times\cdots\times\mathcal{Y}$ ($m$ times). Let $Q:\mathcal{Y}^m\times\mathcal{Y}^m
\to\mathbb{R}_{+}:=[0,+\infty]$ be a prescribed loss function, $\phi:\mathbb{R}_{+}\to\mathbb{R}_{+}$ a regularizer and $\lambda$ a positive regularization parameter. The regularized learning of a function in a $\mathcal{Y}$-valued RKHS $\mathcal{H}$ on $\mathcal{X}$ with the reproducing kernel $\mathcal{K}$ from its finite samples $\{(x_j,\xi_j):j\in\mathbb{N}_m\}
\subset\mathcal{X}\times\mathcal{Y}$ may be formulated as the following minimization problem: $$\label{regularization}
\inf_{f\in\mathcal{H}}\left\{Q(f(\mathbf{x}), \mathbf{\xi})+\lambda\phi(\|f\|_{\mathcal{H}})\right\},$$ where $\mathbf{x}:=[x_j:j\in\mathbb{N}_m], f(\mathbf{x}):=[f(x_j):j\in\mathbb{N}_m]$ and $\mathbf{\xi}:=[\xi_j:j\in\mathbb{N}_m]$. Many popular learning schemes such as regularization networks and support vector machines correspond to the minimization problem (\[regularization\]) with different choices of $Q$ and $\phi$. Specifically, set $\phi(t):=t^2,
\ t\in\mathbb{R}_{+}$. If the loss function $Q$ is chosen as $$Q(f(\mathbf{x}), \mathbf{\xi})
:=\sum_{j\in\mathbb{N}_m}\|f(x_j)-\xi_j\|_{\mathcal{Y}}^2,$$ the corresponding algorithm is called the regularization networks. We next choose the loss function as $$Q(f(\mathbf{x}), \mathbf{\xi}):=\sum_{j\in\mathbb{N}_m}
\max(\|f(x_j)-\xi_j\|_{\mathcal{Y}}-\varepsilon,0),$$ where $\varepsilon$ is a positive constant. In this case, the algorithm coincides with the support vector machine regression.
The success of the regularized learning lies on the well-known representer theorem. This remarkable result shows that for certain choice of loss functions and regularizers the solution to the minimization problem (\[regularization\]) can be represented by a linear combination of the kernel sections $\mathcal{K}(x_j,\cdot),\ j\in\mathbb{N}_m$. Then the original minimization problem in a potentially infinite-dimensional Hilbert space can be converted into one about finitely many coefficients emerging in the linear combination of the kernel sections. Moreover, since the reproducing kernel is used to measure the similarity between inputs, the representer theorem provides us with a reasonable output by making use of the input similarities.
The representer theorem for the regularization networks in the scalar case dates from [@KW] and was generalized for non-quadratic loss functions and nondecreasing regularizers [@AMP; @CO; @SHS]. As the multi-task learning and learning in Banach spaces received considerable attention recently, the representer theorems in vector-valued RKHSs and RKBSs were also established [@MP05; @ZXZ; @ZZ12; @ZZ13]. We state in the following theorem a general result for the solution to the minimization problem (\[regularization\]). To this end, we suppose that the loss function $Q$ is continuous with respect to each of its first $m$ variables under the weak topology on $\mathcal{Y}$ and convex on $\mathcal{Y}^m$. Moreover, the regularizer $\phi$ is assumed to be continuous, nondecreasing, strictly convex and satisfy $\displaystyle{\lim_{t\to+\infty}\phi(t)=+\infty}$.
\[representer-theorem\] Let $\mathcal{H}$ be a $\mathcal{Y}$-valued RKHS on $\mathcal{X}$ with the reproducing kernel $\mathcal{K}$. If $Q$ and $\phi$ satisfy the above hypothesis then there exists a unique minimizer $f_0$ of and it has the form $$\label{representer-theorem-formula}
f_0=\sum_{j\in\mathbb{N}_m}\mathcal{K}(x_j,\cdot)\eta_j$$ for some sequence $\eta_j,j\in\mathbb{N}_m,$ in $\mathcal{Y}$.
Substituting the representation (\[representer-theorem-formula\]) of the minimizer into (\[regularization\]), the original minimization problem in an RKHS is converted into the minimization problem about the parameters $\eta_j,j\in\mathbb{N}_m.$ Especially in some occasions, with the help of the representer theorem, we can convert the minimization problem into a system of equations about the parameters. For the regularization networks, if $\mathcal{K}(x_j,\cdot),j\in\mathbb{N}_m,$ are linearly independent in $\mathcal{H}$, then the parameters $\eta_j,j\in\mathbb{N}_m,$ in (\[representer-theorem-formula\]) satisfy the linear equations $$\label{linearsystem}
\sum_{k\in\mathbb{N}_m}\mathcal{K}(x_k,x_j)\eta_k
+\lambda\eta_j=\xi_j,\ j\in\mathbb{N}_m.$$
As the point-evaluation data are not readily available in practical applications, it is desired to study the regularized learning from finite samples which are in general the linear functional values or linear operator values. Most of the past work mainly focused on such learning problems in RKHSs and RKBSs, where the point-evaluation functionals are continuous [@V; @YC; @ZZ12]. A basic assumption of these work is that the finite linear functionals are continuous on a prescribed space, which guarantees the stability of the finite sampling process. Under this hypothesis the representer theorem was obtained. Specifically, in RKHSs, one can represent the solution to the regularization problem by a linear combination of the dual elements of the given linear functionals.
In general, learning a function in a usual RKHS from its non-point-evaluation data is not appropriate. Two reasons account for this consideration. First of all, although the representer theorem still exists for the regularized learning from functional-valued data in an RKHS, it represents the solution by making use of the dual elements of the continuous linear functionals other than the reproducing kernel. This will bring difficulties to the computation of the solution. Secondly, when only finite linear functionals values are used to learn a target element, the underlying RKHS may ensure the continuity of the finite linear functionals. However, to improve the accuracy of approximation, more functional values should be used in the regularized learning scheme. An RKHS may not have the ability to guarantee the continuity of a family of linear functionals. The disadvantages of RKHSs lead us to consider regularized learning from operator-valued data in the ORKHSs setting.
Suppose that $\mathcal{H}$ is an ORKHS with respect to the set $\mathcal{L}:=\{L_{\alpha}:\alpha\in \Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$. Let $\Lambda_m:=[\alpha_j:j\in\mathbb{N}_m]\in \Lambda^m$, $\mathbf{\xi}:=[\xi_j:j\in\mathbb{N}_m]\in\mathcal{Y}^m$ and for each $f\in\mathcal{H}$ set $L_{\Lambda_m}(f):=[L_{\alpha_j}(f): j\in\mathbb{N}_m]$. We consider the regularized learning algorithm with respect to the values of linear operators as $$\label{general-regularization}
\inf_{f\in\mathcal{H}}\left\{Q(L_{\Lambda_m}(f), \mathbf{\xi})+\lambda\phi(\|f\|_{\mathcal{H}})\right\},$$ where $Q,\phi$ and $\lambda$ are defined as in (\[regularization\]). The following theorem gives the existence and the uniqueness of the minimizer of (\[general-regularization\]) and the corresponding representer theorem. We note that the theorem can be proved by the similar arguments to Theorem \[representer-theorem\]. However, we will give an alternative proof by making use of the isometric isomorphism between $\mathcal{H}$ and a vector-valued RKHS.
\[representer\_theorem\_ORKHS\] Suppose that $\mathcal{H}$ is an ORKHS with respect to the set $\mathcal{L}:=\{L_{\alpha}:\alpha\in \Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ and $K$ is the operator reproducing kernel for $\mathcal{H}$. If the loss function $Q$ and the regularizer $\phi$ satisfy the hypothesis in Theorem \[representer-theorem\] then there exists a unique minimizer $f_0$ of and there exist $\eta_j\in\mathcal{Y},\ j\in\mathbb{N}_m,$ such that $$\label{representer_theorem_ORKHS-formula}
f_0=\sum_{j\in\mathbb{N}_m}K(\alpha_j)\eta_j.$$
We introduce the operator $\mathcal{K}:\Lambda\times \Lambda\rightarrow \mathcal{B}(\mathcal{Y},\mathcal{Y})$ by $$\label{kernel_representation1}
\mathcal{K}(\alpha,\beta)\xi:=L_{\beta}(K(\alpha)\xi),
\ \alpha,\beta\in \Lambda,\xi\in\mathcal{Y}.$$ Then by Theorem \[isometric-isomorphism1\] we have that $\mathcal{K}$ is a reproducing kernel on $\Lambda$ and its vector-valued RKHS $\mathcal{H}_{\mathcal{K}}$, composed by functions with the form $\tilde{f}(\alpha):=
L_{\alpha}(f), \alpha\in\Lambda, f\in\mathcal{H},$ is isometrically isomorphic to $\mathcal{H}.$ For each $\tilde{f}\in\mathcal{H}_{\mathcal{K}}$ set $\tilde{f}(\Lambda_m):=[L_{\alpha_j}(f):j\in\mathbb{N}_m]$. Due to the isometrically isomorphic between $\mathcal{H}$ and $\mathcal{H}_{\mathcal{K}}$ we have that $f_0$ is the minimizer of (\[general-regularization\]) if and only if $\tilde{f}_0$ with $\tilde{f}_0(\alpha):=L_{\alpha}(f_0),\ \alpha\in\Lambda$, is the minimizer of $$\inf_{f\in\mathcal{H}_{\mathcal{K}}}
\left\{Q(\tilde{f}(\Lambda_m),\mathbf{\xi})
+\lambda\phi(\|f\|_{\mathcal{H}_{K}})\right\}.$$ By Theorem \[representer-theorem\] we get that the unique minimizer $\tilde{f}_0$ of the above optimization problem has the form $$\widetilde{f}_0=\sum_{j\in\mathbb{N}_m}
\mathcal{K}(\alpha_j,\cdot)\eta_j$$ for some $\eta_j\in\mathcal{Y}, j\in\mathbb{N}_m.$ Hence, we conclude that there exists a unique minimizer of . Moreover, since there holds for each $\beta\in\Lambda$, $$\widetilde{f}_0(\beta)
=\sum_{j\in\mathbb{N}_m}\mathcal{K}(\alpha_j,\beta)\eta_j
=\sum_{j\in\mathbb{N}_m}L_{\beta}(K(\alpha_j)\eta_j)
=L_{\beta}\left(\sum_{j\in\mathbb{N}_m}
K(\alpha_j)\eta_j\right),$$ we get the representation of $f_0$ as in (\[representer\_theorem\_ORKHS-formula\]).
As a special case, we consider the regularization networks (\[general-regularization\]), where the regularizer $\phi$ is chosen as $\phi(t):=t^2,\ t\in\mathbb{R}_{+}$ and the loss function $Q$ is chosen as $$Q(L_{\Lambda_m}(f), \mathbf{\xi})
:=\sum_{j\in\mathbb{N}_m}\|L_{\alpha_j}(f)
-\xi_j\|_{\mathcal{Y}}^2.$$ It is clear that the loss function $Q$ and the regularizer $\phi$ satisfies the hypothesis in Theorem \[representer\_theorem\_ORKHS\]. We then conclude by Theorem \[representer\_theorem\_ORKHS\] that there exists a unique minimizer $f_0$ of the regularization networks. Let $\mathcal{K}$ be the reproducing kernel defined by (\[kernel\_representation1\]). Observing from the proof of Theorem \[representer\_theorem\_ORKHS\], we have that if $K(\alpha_j), j\in\mathbb{N}_m$ are linearly independent in $\mathcal{H}$, then the parameters $\eta_j,j\in\mathbb{N}_m$, in (\[representer-theorem-formula\]) satisfy the linear equations $$\sum_{k\in\mathbb{N}_m}\mathcal{K}(\alpha_k,\alpha_j)\eta_k
+\lambda\eta_j=\xi_j,\ j\in\mathbb{N}_m.$$ By the definition of $\mathcal{K}$, the above equations are equivalent to $$\label{linearsystem1}
\sum_{k\in\mathbb{N}_m}L_{\alpha_j}(K(\alpha_k)\eta_k)
+\lambda\eta_j=\xi_j,\ j\in\mathbb{N}_m.$$
Observing from representation (\[representer\_theorem\_ORKHS-formula\]), we note that for the regularized learning from operator-valued data in an ORKHS, the target element can be obtained by the operator reproducing kernel in a desired manner. That is, it can be viewed as a linear combination of the kernel sections. As shown in the system of equations (\[linearsystem1\]), the finite coefficients in the representation (\[representer\_theorem\_ORKHS-formula\]) can be obtained by solving a minimization problem in a finite-dimensional space, which reduces to a linear system in some special cases. Furthermore, the desired result holds for all the finite set of linear operators in the family $\mathcal{L}:=\{L_{\alpha}:\alpha\in \Lambda\}$. All these facts show that ORKHSs and operator reproducing kernels provide a right framework for investigating learning from operator-valued data.
Stability of Numerical Reconstruction Algorithms
================================================
When we use a numerical algorithm to reconstruct an element from non-point-evaluation data (its operator values), stability of the algorithm is crucial. In this section, we study stability of a numerical reconstruction algorithm using operator values in an ORKHS. As will be pointed out, the continuity of linear operators, used to obtain the non-point-evaluation data, on an ORKHS is necessary for the algorithm to be stable.
Numerical reconstruction algorithms are often established for understanding a target element. There are two stages in the reconstruction. The first one is sampling used to obtain a finite set of functional-valued or operator-valued data processed digitally on a computer. The second one is the reconstruction of an approximation of the target element from the resulting sampled data. With the number of the sampled data increasing, the numerical reconstruction algorithm is expected to provide a more and more accurate approximation for the target element. As an admissible numerical reconstruction algorithm, it needs to be stable. Such a property will ensure that the effect of the noise, emerging in both sampling and reconstruction, are not amplified through the underlying numerical reconstruction algorithm.
We now define the notion of the stability. Suppose that $\mathcal{H}$ is an ORKHS with respect to a family $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$ and $\widetilde{\Lambda}:=\{\alpha_j:j\in\mathbb{J}\}$ is a subset of $\Lambda$ with $\mathbb{J}$ being a countable index set. With respect to each finite set $\mathbb{J}_m$ of $m$ elements in $\mathbb{J}$, we set $\widetilde{\Lambda}_m:=\{\alpha_j:j\in\mathbb{J}_m\}$ and define the sampling operator $\mathcal{I}_{\widetilde{\Lambda}_m}:\mathcal{H}\to \mathcal{Y}^m$ by $$\label{Sampling-operator-finite}
\mathcal{I}_{\widetilde{\Lambda}_m}(f)
:=[L_{\alpha_j}(f):j\in\mathbb{J}_m].$$ We denote by $\mathcal{A}_m$ the reconstruction operator from $\mathcal{Y}^m$ to $\mathcal{H}$. The sampling operator and the reconstruction operator together describe a numerical reconstruction algorithm. Namely, $\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))$ describes a numerical algorithm for constructing an approximation of $f$. We say the numerical reconstruction algorithm is stable if there exists a positive constant $c$ such that for all $f\in\mathcal{H}$, all $m\in\mathbb{N}$ and all finite subset $\widetilde{\Lambda}_m$ of $\widetilde{\Lambda}$, $$\|\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
\|_{\mathcal{H}}\leq c\|f\|_{\mathcal{H}}.$$
We present in the following theorem a sufficient condition for the stability of a numerical reconstruction algorithm. To this end, we define the Hilbert space of square-summable sequence in $\mathcal{Y}$ on $\mathbb{J}$ by $$l^2(\mathbb{J},\mathcal{Y})
:=\left\{\{\xi_j:j\in\mathbb{J}\}:\sum_{j\in\mathbb{J}}
\|\xi_j\|^2_{\mathcal{Y}}<+\infty\right\}.$$ With respect to $\widetilde{\Lambda}$ we define the sampling operator $\mathcal{I}_{\widetilde{\Lambda}}:\mathcal{H}
\rightarrow l^2(\mathbb{J},\mathcal{Y})$ for $f\in\mathcal{H}$ by $\mathcal{I}_{\widetilde{\Lambda}}(f)
:=\{L_{\alpha_j}(f):j\in\mathbb{J}\}$.
\[stable\] Let $\mathcal{H}$ be an ORKHS with respect to a family $\mathcal{L}:=\{L_{\alpha}:\alpha\in\Lambda\}$ of linear operators from $\mathcal{H}$ to a Hilbert space $\mathcal{Y}$. Let $\widetilde{\Lambda}:=\{\alpha_j:j\in\mathbb{J}\}$ be a countable subset of $\Lambda$. Suppose that the operator $\mathcal{I}_{\widetilde{\Lambda}}$ is continuous on $\mathcal{H}$ and the operators $\mathcal{I}_{\widetilde{\Lambda}_m}$, $\mathcal{A}_m$ are defined as above. If there exist a positive constant $c$ such that for all $f\in\mathcal{H}$, all $m\in\mathbb{N}$ and all finite subset $\widetilde{\Lambda}_m$ of $\widetilde{\Lambda}$, $$\|\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
\|_{\mathcal{H}}\leq c
\|\mathcal{I}_{\widetilde{\Lambda}_m}(f)\|_{\mathcal{Y}^m},$$ then the numerical reconstruction algorithm is stable.
It follows from the continuity of operator $\mathcal{I}_{\widetilde{\Lambda}}$ that there exists a positive constant $c'$ such that for all $f\in\mathcal{H}$, $$\label{Sampling-operator-continuity}
\|\mathcal{I}_{\widetilde{\Lambda}}(f)
\|_{l^2(\mathbb{J},\mathcal{Y})}
\leq c'\|f\|_{\mathcal{H}}.$$ By the definition of sampling operators and the norm of the space $l^2(\mathbb{J},\mathcal{Y})$, we get for all $f\in\mathcal{H}$, all $m\in\mathbb{N}$ and all finite subset $\widetilde{\Lambda}_m$ that $$\|\mathcal{I}_{\widetilde{\Lambda}_m}(f)
\|_{\mathcal{Y}^m}=\left(\sum_{j\in\mathbb{J}_m}
\|L_{\alpha_j}(f)\|_{\mathcal{Y}}^2\right)^{1/2}
\leq \|\mathcal{I}_{\widetilde{\Lambda}}(f)
\|_{l^2(\mathbb{J},\mathcal{Y})}.$$ This combined with (\[Sampling-operator-continuity\]) leads to the estimate $$\|\mathcal{I}_{\widetilde{\Lambda}_m}(f)
\|_{\mathcal{Y}^m}\leq c'\|f\|_{\mathcal{H}}.$$ Substituting the above inequality into the inequality in the assumption, we have for all $f\in\mathcal{H}$, all $m\in\mathbb{N}$ and all finite subset $\widetilde{\Lambda}_m$ that $$\|\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
\|_{\mathcal{H}}\leq cc'\|f\|_{\mathcal{H}}.$$ This together with the definition of the stability of the numerical reconstruction algorithm proves the desired result.
According to Theorem \[stable\], to ensure the stability of the numerical reconstruction algorithm, we need both the family $\{\mathcal{I}_{\widetilde{\Lambda}_m}:
\widetilde{\Lambda}_m\subseteq\widetilde{\Lambda}\}$ of sampling operators and the family $\{\mathcal{A}_m: m\in\mathbb{N}\}$ of reconstruction operators to be uniformly bounded. We note that the continuity of the sampling operators on an ORKHS is necessary for the uniform boundedness of the family of the sampling operators. For the reconstruction operators, we shall show below that two classes of commonly used operators are uniformly bounded.
The first numerical reconstruction algorithm that we consider is the truncated reconstruction of elements in an FRKHS from a countable set of functional-valued data discussed in section 6. Let $\mathcal{H}$ be an FRKHS with respect to a family $\mathcal{L}$ of linear functionals $\{L_{\alpha}:\alpha\in\Lambda\}$ and $K$ the functional reproducing kernel for $\mathcal{H}$. Suppose that $\widetilde{\Lambda}:=\{\alpha_j: j\in\mathbb{J}\}$ is a countable set of $\Lambda$ such that $\{K(\alpha_j): j\in\mathbb{J}\}$ constitutes a Riesz basis for $\mathcal{H}$. We denote by $\widetilde{K}$ the dual functional reproducing kernel of $K$ with respect to the set $\{\alpha_j: j\in\mathbb{J}\}$. Since only finite sampled data are used in practice, we consider the reconstruction operator $\mathcal{A}_m$ from $\mathbb{C}^m$ to $\mathcal{H}$ defined by $$\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
:=\sum_{j\in\mathbb{J}_m}L_{\alpha_j}(f)
\widetilde{K}(\alpha_j),$$ with $\mathbb{J}_m$ being a finite subset of $\mathbb{J}$ and $\widetilde{\Lambda}_m:=\{\alpha_j:j\in\mathbb{J}_m\}$. The stability of the resulting numerical reconstruction algorithm for this case is established below.
Let $\mathcal{H}$ be an FRKHS with respect to the family $\mathcal{F}$ of linear functionals $L_{\alpha}$, $\alpha\in\Lambda$ and $K$ the functional reproducing kernel for $\mathcal{H}$. If for a countable subset $\widetilde{\Lambda}:=\{\alpha_j: j\in\mathbb{J}\}$ of $\Lambda$, the set $\{K(\alpha_j): j\in\mathbb{J}\}$ constitutes a Riesz basis for $\mathcal{H}$, then the numerical reconstruction algorithm defined as above is stable.
We prove this result by employing Theorem \[stable\] with $\mathcal{Y}:=\mathbb{C}$. To this end, we verify the hypothesis of Theorem \[stable\] for this special case. Since $\{K(\alpha_j): j\in\mathbb{J}\}$ constitutes a Riesz basis for $\mathcal{H}$, there exist positive constants $A$ and $B$ such that $$\label{inequalityAB}
A\|f\|_{\mathcal{H}}\leq \left(\sum_{j\in\mathbb{J}}
|\langle f,K(\alpha_j)\rangle_{\mathcal{H}}|^2
\right)^{1/2}\leq B\|f\|_{\mathcal{H}},
\ \mbox{for all}\ f\in\mathcal{H}.$$ By the first inequality in and the definition of $\mathcal{A}_m$, we have that $$\|\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
\|_{\mathcal{H}}\leq \frac{1}{A}
\left(\sum_{k\in\mathbb{J}}
\sum_{j\in\mathbb{J}_m}|L_{\alpha_j}(f)|^2
\left|\left< \widetilde{K}(\alpha_j),K(\alpha_k)
\right>_{\mathcal{H}}\right|^2\right)^{1/2}.$$ Using the biorthogonal condition of the sequence pair $\{\widetilde{K}(\alpha_j), K(\alpha_j): j\in\mathbb{J}\}$ in the right hand side of the inequality above yields that for all $f\in\mathcal{H}$, all $m\in\mathbb{N}$ and all finite subset $\widetilde{\Lambda}_m$, $$\|\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
\|_{\mathcal{H}}\leq \frac{1}{A} \left(\sum_{j\in\mathbb{J}_m}|L_{\alpha_j}(f)|^2\right)^{1/2}
=\frac{1}{A}\|\mathcal{I}_{\widetilde{\Lambda}_m}(f)
\|_{\mathbb{C}^m}.$$ Employing the reproducing property of $K$ in the second inequality in (\[inequalityAB\]), we get that $$\|\mathcal{I}_{\widetilde{\Lambda}}(f)
\|_{l^2(\mathbb{J}, \mathbb{C})}\leq B\|f\|_{\mathcal{H}},
\ \mbox{for all}\ f\in\mathcal{H},$$ yielding the continuity of operator $\mathcal{I}_{\widetilde{\Lambda}}$. By Theorem \[stable\], we conclude the desired result.
We next consider the regularization network, discussed in section 7, which infers an approximation of a target element by solving the optimization problem $$\label{regularization-network}
\inf_{g\in\mathcal{H}}\left\{\sum_{j\in\mathbb{N}_m}
\|L_{\alpha_j}(g)-\xi_j\|_{\mathcal{Y}}^2
+\lambda\|g\|_{\mathcal{H}}^2\right\}.$$ Suppose that $\mathcal{H}$ is an ORKHS with respect to a family $\mathcal{L}$ of linear operators $\{L_{\alpha}:\alpha\in\Lambda\}$ from $\mathcal{H}$ to $\mathcal{Y}$ and $K$ is the operator reproducing kernel for $\mathcal{H}$. Let $\widetilde{\Lambda}:=\{\alpha_j:j\in\mathbb{J}\}$ be a countable subset of $\Lambda$. For each finite subset $\widetilde{\Lambda}_m:=\{\alpha_j:j\in\mathbb{J}_m\}$, the sampling operator $\mathcal{I}_{\widetilde{\Lambda}_m}:\mathcal{H}
\rightarrow \mathcal{Y}^m$ is defined as in (\[Sampling-operator-finite\]). For each $f\in\mathcal{H}$, we let $f_{\widetilde{\Lambda}_m}$ be the unique minimizer of (\[regularization-network\]) with $[\xi_j:j\in\mathbb{J}_m]
:=\mathcal{I}_{\widetilde{\Lambda}_m}(f).$ Accordingly, the reconstruction operator $\mathcal{A}_m:\mathcal{Y}^m\rightarrow\mathcal{H}$ is defined by $$\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
:=f_{\widetilde{\Lambda}_m}.$$
The next result concerns the stability of the corresponding numerical reconstruction algorithm. To this end, we introduce another operator by considering the following regularization problem $$\label{Tikhonov-regularization}
\inf_{g\in\mathcal{H}}\left\{\sum_{j\in\mathbb{J}}
\|L_{\alpha_j}(g)-\xi_j\|^2_{\mathcal{Y}}+
\lambda\|g\|_{\mathcal{H}}^2\right\}.$$ It follows from theory of the Tikhonov regularization [@K] that if the operator $\mathcal{I}_{\widetilde{\Lambda}}:\mathcal{H}
\rightarrow l^2(\mathbb{J},\mathcal{Y})$ is continuous on $\mathcal{H}$, there exists for any $\xi:=\{\xi_j:j\in\mathbb{J}\}\in l^2(\mathbb{J},\mathcal{Y})$ a unique minimizer $g_0$ of the optimization problem (\[Tikhonov-regularization\]). Furthermore, the minimizer $g_0$ is given by the unique solution of the equation $\lambda g_0+\mathcal{I}_{\widetilde{\Lambda}}^*
\mathcal{I}_{\widetilde{\Lambda}}(g_0)=
\mathcal{I}_{\widetilde{\Lambda}}^*(\xi)$. We define the operator $\mathcal{A}_{\widetilde{\Lambda}}:
l^2(\mathbb{J},\mathcal{Y})\rightarrow\mathcal{H}$ by $$\label{Tikhonov-regularization-operator}
\mathcal{A}_{\widetilde{\Lambda}}
:=(\lambda \mathcal{I}+\mathcal{I}_{\widetilde{\Lambda}}^*
\mathcal{I}_{\widetilde{\Lambda}})^{-1}
\mathcal{I}_{\widetilde{\Lambda}}^*,$$ where $\mathcal{I}$ denotes the identity operator on $\mathcal{H}$.
Suppose that $\mathcal{H}$ is an ORKHS with respect to the set $\mathcal{L}:=\{L_{\alpha}:\alpha\in \Lambda\}$ of linear operators from $\mathcal{H}$ to $\mathcal{Y}$ and $K$ is the operator reproducing kernel for $\mathcal{H}$. If the countable subset $\widetilde{\Lambda}:=\{\alpha_j:j\in\mathbb{J}\}$ of $\Lambda$ is chosen to satisfy the condition that there exists a positive constant $c$ such that for all $\xi:=\{\xi_j:j\in\mathbb{J}\}\in l^2(\mathbb{J},\mathcal{Y})$ with $\|\xi\|_{l^2(\mathbb{J},\mathcal{Y})}=1$, $$\label{inequality-c}
\left\|\sum_{j\in\mathbb{J}}K(\alpha_j)\xi_j\right
\|_{\mathcal{H}}\leq c,$$ then the corresponding numerical reconstruction algorithm defined as above is stable.
Again, we prove this result by employing Theorem \[stable\]. It follows for each $f\in\mathcal{H}$ that $$\|\mathcal{I}_{\widetilde{\Lambda}}(f)\|_{l^2(\mathbb{J}, \mathcal{Y})}=\sup_{\|\xi\|_{l^2(\mathbb{J}, \mathcal{Y})}=1}\left|\langle
\mathcal{I}_{\widetilde{\Lambda}}(f),
\xi\rangle_{l^2(\mathbb{J},\mathcal{Y})}\right|
=\sup_{\|\xi\|_{l^2(\mathbb{J}, \mathcal{Y})}=1}\left|\sum_{j\in\mathbb{J}}\langle
L_{\alpha_j}(f),\xi_j\rangle_{\mathcal{Y}}\right|.$$ Using the reproducing property of $K$, we get that $$\|\mathcal{I}_{\widetilde{\Lambda}}(f)\|_{l^2(\mathbb{J}, \mathcal{Y})}=\sup_{\|\xi\|_{l^2(\mathbb{J}, \mathcal{Y})}=1}\left|\langle f,
\sum_{j\in\mathbb{J}}K(\alpha_j)\xi_j
\rangle_{\mathcal{H}}\right|.$$ This together with inequality (\[inequality-c\]) implies that for all $f\in\mathcal{H}$, $$\|\mathcal{I}_{\widetilde{\Lambda}}(f)\|_{l^2(\mathbb{J}, \mathcal{Y})}\leq c\|f\|_{\mathcal{H}}.$$ That is, the operator $\mathcal{I}_{\widetilde{\Lambda}}$ is continuous on $\mathcal{H}$.
It follows for each $g\in\mathcal{H}$ that $$\langle\lambda g+\mathcal{I}_{\widetilde{\Lambda}}^*
\mathcal{I}_{\widetilde{\Lambda}}(g),
g\rangle_{\mathcal{H}}=\lambda\|g\|_{\mathcal{H}}^2+
\|\mathcal{I}_{\widetilde{\Lambda}}(g)\|_{\mathcal{H}}^2
\geq\lambda\|g\|_{\mathcal{H}}^2.$$ By the Lax-Milgram theorem [@Yo], we have that $(\lambda \mathcal{I}+\mathcal{I}_{\widetilde{\Lambda}}^*
\mathcal{I}_{\widetilde{\Lambda}})^{-1}$ is continuous on $\mathcal{H}$. According to the definition of $\mathcal{A}_{\widetilde{\Lambda}}$, there exists a positive constant $c'$ such that for all $f\in\mathcal{H}$, $$\label{Tikhonov-regularization-inequality}
\|\mathcal{A}_{\widetilde{\Lambda}}
(\mathcal{I}_{\widetilde{\Lambda}}(f))
\|_{\mathcal{H}}\leq c'
\|\mathcal{I}_{\widetilde{\Lambda}}(f)
\|_{l^2(\mathbb{J},\mathcal{Y})}.$$ For each finite subset $\widetilde{\Lambda}_m$ and each $f\in\mathcal{H}$, we choose $\tilde{f}\in\mathcal{H}$ to satisfy that $L_{\alpha_j}(\tilde{f})=L_{\alpha_j}(f)$ for $j\in\mathbb{J}_m$ and $L_{\alpha_j}(\tilde{f})=0$ for $j\notin\mathbb{J}_m$. By the definition of $\mathcal{A}_m$ and $\mathcal{A}_{\widetilde{\Lambda}}$, we get that $\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
=\mathcal{A}_{\widetilde{\Lambda}}
(\mathcal{I}_{\widetilde{\Lambda}}(\tilde{f})).$ This combined with (\[Tikhonov-regularization-inequality\]) leads to that there holds for all $f\in\mathcal{H}$, all $m\in\mathbb{N}$ and all finite subset $\widetilde{\Lambda}_m$, $$\|\mathcal{A}_m(\mathcal{I}_{\widetilde{\Lambda}_m}(f))
\|_{\mathcal{H}}=\|\mathcal{A}_{\widetilde{\Lambda}}
(\mathcal{I}_{\widetilde{\Lambda}}(\tilde{f}))
\|_{\mathcal{H}}\leq c'
\|\mathcal{I}_{\widetilde{\Lambda}}(\tilde{f})
\|_{l^2(\mathbb{J},\mathcal{Y})}=c'
\|\mathcal{I}_{\widetilde{\Lambda}_m}(f)
\|_{\mathcal{Y}^m}.$$ Hence, the stability of the corresponding numerical reconstruction algorithm follows directly from Theorem \[stable\].
Conclusion
==========
We have introduced the notion of the ORKHS and established the bijective correspondence between an ORKHS and its operator reproducing kernel. Particular attention has been paid to the perfect ORKHS, which relates to two families of continuous linear operators with one being the family of the point-evaluation operators. The characterization and the specific examples for the perfect ORKHSs with respect to the integral operators have been provided. To demonstrate the usefulness of ORKHSs, we have discussed in the ORKHSs setting the sampling theorems and the regularized learning schemes from non-point-evaluation data. We have considered stability of numerical reconstruction algorithms in ORKHSs. This work provides an appropriate mathematical foundation for kernel methods when non-point-evaluation data are used.
[**Acknowledgment:**]{} This research is supported in part by the United States National Science Foundation under grant DMS-152233, by Guangdong Provincial Government of China through the “Computational Science Innovative Research Team” program and by the Natural Science Foundation of China under grants 11471013 and 11301208.
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[^1]: School of Mathematics, Jilin University, Changchun 130012, P. R. China. E-mail address: [*[email protected]*]{}.
[^2]: Corresponding author. Guangdong Provincial Key Lab of Computational Science, School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510275, P. R. China and Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA. E-mail address: [*[email protected]*]{}
|
A conformally invariant generalization of string theory to higher-dimensional objects
=====================================================================================
The remarkable achievements of string and superstring theories are well known: evaluation of the space-time dimension, fixing a particular gauge group, inclusion of gravity into a unified scheme etc. [@grin]. These achievements stimulate an interest in studies of geometric objects of higher dimension, such as membranes or p-branes. It is known, however, that [@ket] in standard membrane theories the absence of conformal invariance precludes the usage of string-theoretical methods. For instance, the requirement that conformal invariance should be preserved at the quantum level leads, in string theory, to fixing the space-time dimension [@brn]. There are also other arguments [@vit] in favour of the requirement that a physical field theory should be conformally invariant, at least at the classical level.
On this basis, we have previouly suggested a conformally invariant generalization of string theory to higher-dimensional objects [@zar]. This paper, aimed at further realization of this approach, is devoted to obtaining and investigation of Hamiltonian equations and constraint equations of the theory under consideration. This idea was originally suggested as a quantum theory by analogy with string theory. However, a further analysis has shown the necessity of an initial classical analysis of this theory. It has turned out that even the classical level of the theory contains results of interest related to gravitation theory and p-brane theory. The action that serves as a basis for the suggested theory, being a generalization of string and p-brane theory, is simultaneously a certain generalization of Einstein’s general relativity. We suggest that general relativity should be considered as a special case of a conformally invariant sigma model, appearing as a result of conformal symmetry violation.
This paper is devoted to foundation and analysis of the above ideas in the classical case.
The recent development of multidimensional theories have been, to a large extent, related to the so-called branes. In this theory \[6–9\], the observable Universe is considered as a surface (brane) embedded in a higher-dimensional space-time. It is hoped that this approach can lead to a success in solving the fundamental problem of the hierarchy of physical coupling constants and the cosmological constant problem. The hierarchy problem lies in the existence of a huge difference between the elecroweak energy scale of about 1 TeV and the gravitational energy scale of the order of $10^{19}$ GeV. Besides, the energy density related to the cosmological constant should be about 120 orders of magnitude smaller than the possible energy density values for the known models of quantum theory of the weak and strong interactions.
In the theory suggested, the gravitational constant is related to the dynamic characteristics of the model, and it is obtained in multidimensional space-time due to localization of solutions to nonlinear equations, by analogy with the Higgs effect in gauge field theory.
The Lagrangian approach
-----------------------
The action for a membrane (or p-brane) does not admit conformal transformations, and these models do not possess a natural candidate for the role of an anomalous symmetry like conformal symmetry in string theory. To circumvent this difficulty without abandoning the string-theoretical ideology, we suggest the following generalization of string theory: \[ns1\] S = { - (\_ X,\^ X ) + (X,X) + (X,X)\^ } d\^[p+1]{} , where we use the notations: (X,X) X\^[A]{}X\^[B]{}\_[AB]{}, (\_[X]{},\^[X]{}) \_X\^[A]{}\_X\^[B]{}g\^\_[AB]{}, (p+1)/(p-1). In the action (\[ns1\]), the functions $X^{A}=X^{A}(\sigma^{\mu})$, with $A,\,B = 1,\ 2, \ldots , D;\quad \mu,\nu=0,1,\ldots,p$, map the $n = p+1$-dimensional manifold $\Pi$, described by the metric $g_{\mu\nu}$, into $D$-dimensional space-time $M$ with the metric $\eta_{AB}$, where the space $M$ is determined by the Minkowski metric with the signature $(-,+,\ldots,+)$. However, it turns out that, in a detailed study, it is more convenient to leave the signature of $M$ arbitrary. The flat space signature is here understood as the set of signs of the elements along the main diagonal ($+1$ and $-1$) of the metric matrix. The quantity $\stackrel{n}{R}$ is the scalar curvature of the manifold $\Pi$, the operator $\nabla_{\nu}$ means a covariant derivative in the manifold $\Pi$, where the Christoffel symbols are connected with the metric in the standard manner. We will assume that the space $\Pi$ is parametrized by the coordinates $\sigma^{\mu}$, where $\sigma^{0}= t$ is the temporal coordinate while the components $\sigma^i$ $(i=1,2,\ldots,p)$ describe a certain $p$-dimensional object, to be designated as $\Gamma$. The quantities $w$, ${\tilde\xi}$ and $\Lambda$ are constants. The models like that with the action (\[ns1\]) are also often called nonlinear $\sigma$ models.
The action (\[ns1\]) is conformally invariant if \[2\] = - . This invariance is expressed in the fact that the equations obtained by varying the action (\[ns1\]) with respect to the fields $\hat g$ and $\hat X$ are invariant under the local Weyl scale changes \[cp\] g\_ \^[2]{} g\_, X\^[A]{} \^[4p]{}X\^[A]{}, for an arbitraryf $\phi = \phi(\sigma^{\mu})$.
After varying the action (\[ns1\]), the field equations for $\hat X$ and $\hat g$ have the following form: \[Y\] Y\^[A]{}X\^[A]{} + 2 X\^[A]{} + 2 (X,X)\^[- 1]{} X\^[A]{} = 0,\
\[T\] T\_ T\^[1]{}\_ + \[ - \_ + g\_ + \_ \_ - g\_ \](XX) = 0, where \[T1\] T\^[1]{}\_= \[(\_X, \_X) + L\_1 g\_ \] are the terms appearing due to variation of the Lagrangian density \[L\_1\] L\_1 = - g\^ (\_ X, \_ X) + (X, X)\^.
If the action is supplemented by Lagrange functions of other matter fields, then (\[T\]) is replaced by the equation \[T3\] T\_ + T\^[e]{}\_=0, where $T^{e}_{\alpha\beta}$ is the energy-momentum tensor of the other fields. In case $(X,X) = \const$, as follows from (\[T3\]) and (\[T\]), the equations are similar to Einstein’s, with the canonical energy-momentum tensor $T^{1}_{\alpha \beta}$ and the effective gravitational constant \[kap\] G\_e = -, w= . Let us point out the important fact that for strings ($p = 1$) the general solution to (\[T\]) has the form \[gtti\] Bg\_ = (\_ X, \_ X), , = , where $B$ is an arbitrary function. Thus the original metric $g_{\mu \nu}$ is connected by a conformal transformation with the induced metric $(\nabla_{\mu} X, \nabla_{\nu} X) $. Unfortunately, in the general case $p > 1 $, for (\[Y\])–(\[T\]), the solution (\[gtti\]) is not a general solution. The problem of connection between the metric of the manifold $\Pi_g$ with the metric induced by the solutions $X^A = X^A (\sigma^{\mu})$, as well as that of a physical interpretation of this connection, have not been solved for an arbitrary dimension.
In what follows, we will consider some special solutions to (\[Y\])–(\[T3\]), being of interest for physics.
Hamiltonian formalism
---------------------
To pass over to the Hamiltonian formalism, we make, in the action (\[ns1\]), a $(p + 1)$-partition. Employing the results of Refs.[@tor; @tut], we introduce the parameters $N$ and $N_{i}$, the “lapse” or “shift” functions, and the metric functions of the $p$-dimensional geometry $h_{i j}$, where $i = \overline{1,p}$: g\_[00]{} = N\_[s]{} N\^[s]{} - N\^2,g\_[0 i]{} = N\_[i]{}, g\_[i j]{}= h\_[i j]{}, = N. Then, taking into account the results of Ref.[@tor], we can present the scalar curvature in the form \[RN\] = R - (K)\^2 + (K\^2) - \_\[N (n\^K + \_ n\^)\]. Here $R$ is the scalar curvature calculated for the metric $h_{ij}$, $a^{\alpha} = \nabla_{\beta} n^{\alpha} n^\beta$ is the $(p+1)$-dimensional acceleration of an observer moving along a timelike normal $\vec n$ to consecutive sections. The space-time $\Pi$ is assumed to be foliated into a one-parameter family of spacelike hypersurfaces with the parameter $t$. The quantity $\hat K$ is the extrinsic curvature tensor of the spacelike sections: \[naa\] n\_ = {- N, 0},(n n)= - 1,\
\[spp\] K = h\^[i j]{} K\_[ij]{},K\^2 = K\_[ij]{} K\^[ij]{}, K\_[ij]{} = (D\_i N\_J + D\_j N\_i - \_t h\_[ij]{}), where $D_i$ is a covariant derivative calculated with the metric $h_{ij}$.
Let us now pass over to a description in terms of the phase-space variables, i.e., the generalized coordinates and momenta: $$\nq\,
\{ q_I \} = \{ X^A, h_{lk}, N, N_i \},\quad
\{ p_I \} = \{ P_A, \Pi^{lk}, p_N, p_{N_i} \},$$ where \[PI\] p\_I = , where $L$ is the Lagrangian corresponding to the action (\[ns1\]). We obtain as a result: P = \[X - N\^i D\_i X + 4 X N \], \^[lk]{} = , the remaining momenta are zero. In the conformal transformations (\[cp\]), the phase variables are transformed as follows: h\_[lk]{} \^[2 ]{} h\_[lk]{} ,X \^[4p ]{} X ,N \^ N , \[pl\] N\_i \^ N\_i, P \^[-4p ]{}P , \^[lk]{}\^[-2]{} \^[lk]{}.
As follows from (\[pl\]), there is a constraint between $P$ and $\Pi^{lk}$. This constraint may be written as \[M1\] M 2 + 4 p (P,X) = 0. Integrating by parts and rejecting terms with a full divergence, one can write the action (1) in terms of the canonical variables as \[S\] S = d\^[p+1]{} , where H\_0 \[ () - (SP )\^2\] + P\^2 \[H0\] + , \[Hl\] H\^l ( P, D\^l X ) - 2 D\_s \^[sl]{} , \[Qi\] Q\^i = \[ (X, X) D\^i N - N D\^i (X, X) \] + 2 N\_k \^[ki]{} + . Here we have used the notations \[E\] E = \_t (X, X) - N\^s D\_s (X,X) , $\Delta = h^{lk} D_l D_k$ and $(D_s X,D^s X) = (D_i X, D_j X) h^{ij}$. The function $\lambda_M$ is arbitrary. This is related to the impossibility of resolving the velocity $\lambda_E = \d_t (X,X)$ in terms of the momenta. It can be shown that $$p_E \equiv \frac{\delta L}{\delta \lambda_E} = \frac{M}{ 8 \xi p(X,X)}.$$ The constraint (\[M1\]) has appeared because of the invariance of the theory with respect to(\[pl\]). To take this fact into account explicitly, let us transform the integrand in (\[S\]) according to (\[pl\]) for $\phi = \psi$ and introduce the field $\psi$. Then the expression (\[S\]) takes the form \[S1\] S = d\^[p+1]{}. That is, we could use, instead of $\lambda_{M}$, the field $\psi (\sigma)$. The corresponding momentum is $p_{\psi} = M$. The divergence term in (\[S1\]) does not affect the equations of motion but affects the boundary conditions. After the transformations (\[pl\]), the quantities $Q^i$ turn into $\tilde Q^i$, where \[Q1\] Q\^i = Q\^i + (X, X) . The latter expression implies that, taking into account the boundary effects, we shall obtain certain boundary conditions applied to the function $\psi$, violating the invariance of the theory with respect to (\[pl\]). It is probably reasonable to omit the divergence term (\[Q1\]) from the action, replacing the original Lagrangian density with $L + \d_i Q^i$. An argument in favour of such a replacement is that for $(X, X) = \const$ and $\ X^i = \const$, the action acquires the Einstein form. In the construction of the Hamiltonian formalism for Einstein’s theory, such terms are omitted [@tut].
The conditions that the primary constraints are conserved in time, \[Phi1\] \^[(1)]{}\_[I]{}: p\_ { p\_N, p\_[N\_[i]{}]{}} = 0, p\_ - M = 0, with the Hamiltonian constructed in the standard way [@tut], \[H\] H = \_i Q\^i + N H\_0 + N\_s H\^s + \_0 M and the extended Hamiltonian $$H^1 = H + \lambda_I \Phi^{(1)}_{I},$$ do not allow one to determine the functions $\lambda_{I}$, but there emerge secondary constraints: \[Phi2\] \^[(2)]{}\_I : H\^[v]{} { H\_0, H\^l, M } = 0.
Consider the conservation conditions for the constraints (\[Phi2\]). To do so, it is necessary to calculate the Poisson brackets: $$[ \Phi_K, \Phi_J ] \equiv \ \frac{\delta \Phi_K}{\delta q_I}
\frac{\delta \Phi_J}{\delta p^I} \ - \ \frac{\delta \Phi_K}{\delta p^I}
\frac{\delta \Phi_J}{\delta q_I} .$$ After cumbersome calculations, it can be shown that if the appearing divergence terms, leading to surface integrals, vanish, then the constraint conservation conditions do not allow determining the functions $\lambda_I$ and do not lead to new constraints. All constraints are thus first-class constraints.
The equations of motion $\dot q = [q, H]$ have the following form: \[x1\] X = P + N\^s D\_s X + 4p\_M X, \[x2\] h\_[lk]{} = + D\_[(l]{}N\_[k)]{} + 2\_M h\_[lk]{}, P = X + \[ R N - 2N\] X \[x3\] + ( N X + D\^s N D\_s X ) + (X, X)\^[- 1]{} X + D\_s(N\^s P) - 4 p \_M P,\
\^[lk]{} = - + h\^[lk]{}{ \[( ) - ()\^2\] + P\^2 } + \[ N D\^l D\^k (X, X) - N R\^[lk]{} + (X, X) D\^l D\^k N - h\^[lk]{} ( (X,X)N + D\_s(X, X) D\^s N ) \] \[plk\] + (D\^l X, D\^k X) - c\^[lk]{} + P N\^[(l]{} D\^[k)]{} X - 2\_M \^[lk]{} - h\^[lk]{} H\_0, where $$c^{lk} = \sqrt{h} D_s\biggl(\frac{1}{\sqrt h} ( N^{(l}\Pi^{k)s}
- N^s \Pi^{lk} \ )\biggr)$$ In what follows, we will put the constant $w$ equal to unity. If, instead of the indefinite coefficient $\lambda_M$, we introduce the field $ \psi$, we should make the following substitution in the equations of motion: \_M = \_0 - + \_s N\^s, where $\lambda_0$ is an arbitrary function of the phase variables. This function may be chosen to be equal to zero, which simply re-defines the function $\psi$. Then, using the substitutions h\_[lk]{} = \^[- 2]{} |[h]{}\_[lk]{} , X = \^[- 4p ]{} |X, N= \^[- ]{} |N , \[pk\] N\_i = \^[- ]{} |N\_i, P = \^[4p ]{} |P, \^[lk]{} = \^[2]{} |\^[lk]{}, one can exclude the field $\psi$ from (\[x1\])–(\[plk\]) and pass over to the conformally invariant canonical variables $\{ \bar {q_I}, \bar {p^I} \}$, which is equivalent to putting $\lambda_M = 0$ in the equations. However, for studying different gauge conditions, it is more convenient to preserve the arbitrariness in choosing the function $\lambda$. To impose the canonical gauge, it is necessary to impose $2p+4$ supplementary conditions, according to the number of first-class constraints. We will consider as such constraints the class of additional conditions $\Phi_G$ of the form N = N,N\_l = N\_l,= 0, \_ = 0,F = 0, where $\chi_\mu$ are $p + 1$ functions of the phase variables $h_{lk}$ and $X^A$, while $F$ is a function of the phase variables $h_{lk}, \ X^A, \ P,
\ \Pi^{lk}$. These functions are chosen in such a way that $\det [\hat
\Phi, \hat \Phi] \not = 0$, where $\hat \Phi = \{\Phi, \Phi_G \}$. Let us denote $H^{v} = \{H_0, H^l,M\}$ and $\chi_u=\{\chi_\mu, F\}$, then, in the case under consideration, = ()\^2 ( \[, p\_\])\^2 . A gauge, related to a choice of the function $F$, violates the conformal symmetry and determines a “representative” from each class of conformally equivalent metrics. To reach comprehension of the different kinds of gauge conditions, let us consider some consequences of the constraint equations (\[Phi2\]) and the equations of motion (\[x1\]), (\[plk\]). Let us define the conformally invariant tensor $\Theta_{lk}$ which is traceless on the surface of the constraints: \_[lk]{} 2 N \[Th\] + 2 + N + H\_0. Then, using (\[x1\]) and (\[plk\]) as well as the definitions (\[H0\]), (\[Hl\]) and (\[M1\]), it is easy to prove the following identity: \[TH\] \_t - D\_s =F\^l\_k + \^l\_k + N\^[(l]{} H\^[s)]{} h\_[sk]{}-N\^[s]{} H\_[s]{}, where $$F^l_k = \Pi^{l}_{s} D_{k} N^{s} -\Pi^{s}_{k} D_{s} N^{l}.$$ The latter equation is equivalent to (\[plk\]) provided the conditions (\[x1\])–(\[x3\]) hold.
Induced gravity
===============
Let us call the “partial embedding” condition the choice of the supplementary conditions $\Phi_G$ obtained from the requirement \[gi\] h\_[lk]{} = B (D\_l X, D\_k X), l,k = , where $B$ is a certain function. Thus the metric $h_{lk}$, entering into the original action, is connected with the induced metric $(\nabla_{l} X, \nabla_{k}X)$ by a conformal mapping.
In the string case, the general solution to the constraing equations has the form \[gj\] Bg\_ = (\_ X, \_ X), , = , where $B$ is an arbitrary function. This solution follows from (\[Th\]) and (\[TH\]) if one puts the momenta $\Pi^{lk}$ and the parameter $\xi$ equal to zero. For an arbitrary dimension, (\[gj\]) (for $B=1$) determines the condition of full embedding of the mainfold $\Pi$ into the space-time $M$. This equation, written in the $p+1$-partition formalism, is equivalent to (\[gi\]) and the equations \[kal\] (X, X) B(N\_sN\^s - N\^2), (D\_l X, X) B N\_l, B (D\_l X, D\^l X)/p =1.
Thus we can consider two kinds of solutions corresponding to “partial” or full embedding. In the first case, the validity of (\[gi\]) (for $B=1$), as well as for strings, would permit one to interpret the fields $X^A$ as the conventional coordinates of a $d$-object $\Gamma$ in the space-time $M$. In other words, this means that, from each class $\Gamma$ of conformally equivalent manifolds, it is possible to choose at least one “representative” $\Gamma_0$, such that the functions $X^A$ perform embedding of $\Gamma_0$ into the surrounding space $M$. Here, conformally equivalent manifolds are understood as manifolds whose metrics are connected with the reparametrization invariance and the conformal invariance (\[pl\]).
An invalidity of the relations (\[kal\]), if (\[gi\]) is valid, leads to some difficulties in the physical interpretation. If, by analogy with string theory, the space-time $M$ is considered as physical space-time, then the coincidence between the original metric $h_{lk}$ and the induced metric $(D_l X, D_k X)$ makes the theory transparent, making it possible to interpret the solutions $X^A = X^A (t, \sigma)$ as an embedding of a $p$-dimensional object $\Gamma$ into the physical space-time $M$. However, a non-coincidence between the “lapse” or “shift” functions of the original manifold $\Pi$ and the $p+1$-dimensional “world history” mainfold of the object $\Gamma$ poses a question on the physical meaning of the original functions $N$ and $N_i$. To answer this question, one can try to invoke the ideas of the Kaluza-Klein theory. We, however, put forward a conjecture according to which it is possible, at the expense of a choice of the corresponding reference frame and conformal gauge, and maybe also the dimension $D$, to achieve the validity of the conditions of full embedding of the whole manifold $\Pi$ into the space-time $M$. (\[Th\]) and (\[TH\]) determine $\frac{p(p+1)}{2} - 1=k_p$ equations, while the number of arbitrary functions, determining the constraints $\Phi_G$, is equal to $2p + 4$. It is necessary to specify $p+2$ functions $N,N_l,\lambda_M$. Besides, according to the number of first-class conditions, we shoud impose $p+2$ supplementary conditions. One can try to impose the latter relations by requiring the $p+2$ conditions (\[kal\]) to be valid. As follows from (\[Th\]) and (\[TH\]), to fulfil the “partial embedding” conditions (\[gi\]), the following equations should hold: \_t - D\_s- F\^l\_k \[kali\] = { N + }. If (\[kali\]) holds, then (\[gi\]) follows from (\[Th\]) and (\[TH\]) The $p+2$ functions $N,\ N_l,\ \lambda_M$ should be chosen in such a way that, due to this choice, (\[kali\]) hold. The number of these functions for the dimensions $p=2$ and $p=3$ is 4 and 5, respectively, while the number of equations (\[kali\]) (the number $k_p$) is 2 and 5, respectively. This simple counting of the degrees of freedom shows that, in the cases of interest $p = 2$ and $p = 3$, it is possible to choose a full embedding gauge.
In the most general case, it can be proved that, to satisfy the full embedding conditions (\[gj\]) (for $B=1$), it is necessary that the following equations hold: \[ind\] \[ + (X, X)\^[-1]{}\] \_(X,X) = 0, where $\mu = \overline{0, p}$. The simplest proof can be performed with the aid of the generally covariant equations (\[Y\]). They are equivalent to the Hamiltonian equations (\[x1\])–(\[x3\]). Acting with the covariant derivative $\nabla_{\gamma}$ on the relation (\[gj\]) and contracting different pairs of indices, we obtain the equations (\_ X, X) + (\^ X, \_ \_ X) = \_B, 2 (\^ X, \_ \_ X) =\_B (p+1), where $\mu, \nu, \gamma = \overline{0, p}$.
From the latter equations combined with (\[Y\]), we obtain (\[ind\]). In the derivation, we taking into account the covariant constancy of the metric tensor $g_{\mu \nu}$ and that $B=1$. Thus, as follows from (\[ind\]), we can use two kinds of supplementary gauge conditions agreeing with the full embedding condition: \[oda1\] (1) F(X, X) - C =0, C = ; \[oda2\] (2) F + (X,X)\^[-1]{}= 0. As follows from (\[T\]) and (\[oda1\]), in the first case the constraint equations are similar to Einstein’s equations with the canonical energy-momentum tensor (\[T1\]) and the effective gravitational constant (\[kap\]). For the second case, one cannot exclude solutions in which $G_e$ is variable and coordinate-dependent.
Model theory
------------
The equations obtained, like Einstein’s equations, are strongly nonlinear and cannot be solved in a general form. However, the existing additional conformal symmetry simplifies the search for solutions of these equations. In this section, we simplify the equations obtained by restricting the class of metrics under consideration. Paying more attention to the dimension $n=4$, let us consider a model problem with the metric tensor $h_{lk}$ chosen in the form \[mod\] h\_[lk]{} = b\^2 \_[lk]{}, where $b^2 \ = \ b^2(t, \sigma)$ is an arbitrary function and $\omega_{lk}$ is some fixed metric. It will be essential for what follows that the functions $\omega_{lk}$ are time-independent: $\dot \omega_{lk} = 0$. In the two-dimensional case, the following relations always hold: \[rij\] \_[lk]{} - \_[lk]{}=0. The index $\omega$ means that the corresponding quantities are calculated for the metric coefficients $\omega_{lk}$. For instance, $\omega_{lk}$ may be chosen to be the metric of a constant-curvature space. Then, \_[lk]{} = k\_0 (p - 1)\_[lk]{} = -8k\_0 p \_[lk]{}, $k_0 = \{0, 1, -1\}$ for surfaces of zero, positive and negatice curvature, respectively.
Here and henceforth, we leave the dimension arbitrary, considering simultaneously two-dimensional $\Gamma$ objects and objects of an arbitrary dimension, however, for the latter we restrict ourselves to spaces which are conformal to constant-curvature spaces.
The “conformal time” gauge
--------------------------
Let us impose the following conditions on the “lapse” and “shift” functions: \[n1\] N\^2 = b\^2, N\_i = 0. After taking the trace of (\[x2\]), it follows: \_M = = - \_[lk]{} = ()h\_[lk]{}. With the constraint $M=0$, we obtain \_[lk]{} = -2(P,X)h\_[lk]{}.
Consider the gauge condition (\[oda1\]) \[C\] (X,X)- C = 0, C = = 0.
Then, substituting $X = g b^{4\xi p}$, from (\[x1\]) and (\[x3\]) we obtain the equations for the field $g^A$ components: \[gx\] g - g -2g = 2C\^[-1]{} b\^2g .
The constraint equations $H_0=0$ and $H_l=0$ may be brought to the form \[xxx\] (X, X) = -(4p \_M)\^2 C + b\^2 (2C R + 2C\^ - Bd), \[xx2\] (X,D\_l X) = 16 \^2 p C D\_l (). (\[kali\]) are reduced to the following ones: \[rsii\] 8 p C (- D\_l D\_k + D\_l D\_k - (- + D\_s D\^s ) ) + C ( - ) = 0. Using (\[xxx\]), (\[xx2\]) and the consequence of (\[C\]) $$( \dot X, \dot X) = - ( \ddot X, X),$$ we obtain an equation for finding the conformal factor: \[psii\] + 4p \^2 - b\^2 (R + 2 - 2D\_s D\^s -2 (p+1) q ) = 0.
The scalar curvature may be expressed in terms of the function $\psi$: \[RR\] R = b\^[-2]{} + 8 p \[ - 2 - (p-2)\_s \^s \].
If one requires that the first and the second relations of the condition (\[kal\]) hold, this leads to the equations \[goi12\] - 8p\^2 \^2 + b\^2 (R + 16 p q) = 0, \[goj1\] D\_l = - D\_l , where $$q =\frac{1}{4\xi C} \biggl(B + \Lambda C^\rho \frac{1}{4\xi p}\biggr).$$ It can be shown that (\[psii\]) is a differential consequence of (\[goi12\]) and (\[goj1\]). Then (\[psii\]) can be brought to the form p + b\^2( -2 q + - D\_s D\^s ) = 0, or, in terms of the metric $\omega$, \[gpsi\] p + -2q b\^2 + +8pD\_[s]{} D\^[s]{} = 0, Thus the function $\psi$ is found by solving (\[goi12\]), (\[goj1\]) and (\[rsii\]).
Then the functions $X^A$ are determined by (\[gx\]). If we write (\[gx\]) directly in terms of the variables $X^A$, we obtain a linear equation with respect to $X^A$. This equation, with (\[psii\]), may be written as \[gxx\] X - 8p X - X + 8 p D\_[s]{} X D\^[s]{} - b\^2 X= 0. Solutions of the latter equations should satisfy the remaining constraint equations which have the form \[xxxx\] (X, X) = - b\^2, (X, X) = C, \[xx2x\] (X,D\_l X) = 0. (D\_k X, D\_l X) = h\_[kl]{}.
Let us present some special solutions to the equations obtained for the case of conformally flat manifolds. Let us first consider a flat $p$-dimensional model: $$\stackrel{\omega}{R} =0.$$ Let the metric matrix $(\omega_{lk})$ be a unit matrix. A solution to (\[goi12\]), (\[goj1\]) and (\[rsii\]) has the form \[bab\] b\^[-2]{} \^[2]{}= \^2, where $$r^2 = \sum_{i=1}^{p} (\sigma^{i})^2,$$ $u_0,\ m,\ n_i,\ l_0$ being integration constants satisfying the condition $m^2 +2 u_0 l_0 - n_i n^i = {2q}/{p}$. Here $t$ is the time coordinate, and the $p$-dimensional coordinates may be interpreted as the conventional Cartesian coordinates. It can be verified by a direct inspection that the functions $f_0=a_0 \dot \psi$ and $f_l= a_l D_l \psi$ (where $a_0,
a_l = \const$) are special solutions to (\[gx\]). Using this, let us build solutions which also satisfy (\[xxxx\])–(\[xx2x\]). Probably, there can be many such solutions. But we will here seek solutions with a minimal set of fields $X^A $. With this approach, we consider solutions which describe an embedding of the manifold $\Pi$ into the 5-dimensional space $M$. Calculations show that the solutions linear in the functions $f_0$ and $f_l$ satisfy (\[xxxx\])–(\[xx2x\]) with the following values of the constants: \[cosm\] q = C\^ = , \[con1\] a\_0\^2 = a\_l\^2 = =C c\_0, where $c_0=2 l_0/u_0$. Without losing generality in the solution (\[bab\]), we put $m = n_i =0$, which simply corresponds to a parallel transport. Then the scale factor (\[bab\]) is rewritten in the form \[bab1\] b\^[2]{} = . ˆ§ (\[con1\]) it follows that all solutions split into two types: (1) C > 0, c\_0 > 0; (2) C < 0, c\_0 < 0. To embed the manifold $\Pi$ into $M$, it turns out to be convenient (see [@sing]) that, for the second type of solutions, the metric signature in $M$ be $(-,+,+,+,-)$. For the first type it should be $(-,+,+,+,+)$. If we define $|c_0| = g_0^2$, then the solution have the following form: \[zac\] X\^0 t , X\^l \^l , X\^4 for the first type and \[otc\] X\^0 t , X\^l \^l , X\^4 for the second type.
To study the global properties of the manifold, let us study its boundaries. For the second type of solutions, consider a range $W_o$ specified by the following constraints on the coordinate variables: \[cor1\] - |g\_0| (t[+]{}r) |g\_0|, -|g\_0| (t[-]{}r) |g\_0|. Let us introduce the new coordinate $(\eta, \chi)$ instead of $(t, r)$: \[cor2\] t +r g\_0 ((+ )), t - r g\_0 ((- )). In the new coordinates, using a conformal transformation, the metric may be brought to a form exactly coinciding with the open anti-de Sitter space metric \[met\] d s\^2 = a\^2 () \[ d \^2 - (d)\^2 - K ()d \^2\], where $$a^2 (\eta) = \frac{C}{\cosh^2 {\eta}}, \qquad K (\chi) = \sinh^2 {\chi},$$ and $d \Omega^2$ is the metric form of a $(p-1)$-sphere of unit radius, expressed in spherical coordinates.
In a similar way, for the first type of solutions, the whole range $W_cl :
-\infty < t < +\infty,\ -\infty < r < +\infty$ may be mapped into a part of the compact (closed) de Sitter space.
To this end, we introduce new coordinates by the relations \[corz2\] t +r g\_0 ((+\_0 + )), t - r g\_0 ((+\_0- )), with $\eta_0 = \const$. The metric has the form (\[met\]), where \[zakrit\] a\^2 () = , K () = \^2,
The functions $X^A$ are scalars with respect to the above coordinate transformations and may be rewritten in the new coordinates. For instance, for de Sitter space, \[zakritx\] X\^[0]{} = , X\^a = k\^a, where $k^a $ are the embedding functions of a $p$-dimensional sphere. For dimension $p=3$, these functions are k\^1 = , k\^2 = , \[ka\] k\^3 = , k\^4 = . In this case, the metric form is d s\^2 = a\^2 () \[ d \^2 - d \^2 - \^2(d \^2 + \^2 d\^2\], which corresponds to the Robertson-Walker metric describing the Friedmann cosmological models.
The solutions for anti-de Sitter space are obtained from the above equations if one makes there the following substitution: $$\sin \chi \mapsto \sinh \chi,\quad
\cos \chi \mapsto \cosh \chi, \quad
\cos \eta \mapsto \cosh \eta.$$
Induced gravity as a result of a spontaneous violation of the conformal invariance
----------------------------------------------------------------------------------
In addition to considering different gauge conditions, let us note that the field equations and constraint equations may also be studiedly the canonical gauge. To do so, using the substitution $X = b^{4\xi p} g$, without imposing the supplementary condition (\[C\]), one can entirely exclude the field $\lambda_M$ in the Hamiltonian equations if the condition (\[n1\]) is valid. In this case, the following equations are obtained: \[kar\] g = g + 2g + 2g Z\^[- 1]{}, Z (g, g), \[gry\] \[ (D\_l g, D\_k g) - B\_g \_[lk]{}\] + D\_l D\_k Z - Z \_[lk]{} - \[\_[lk]{} - \_[lk]{}\] Z = 0, \[ghy\] (g, g) + B\_g p + 4 Z - 2 Z -2 Z\^ = 0, B\_g \^[lk]{} (D\_l g, D\_k g), \[gly\] D\_l Z + (g, D\_l g) = 0. The latter two equations are equivalent to the constraint equations $H_0$, $H_l$. As their consequence, we obtain an equation for the function $Z$: \[gzy\] Z = (1 - 8 ) Z + 8 Z - Z\^- 4d B\_g. (\[kali\]) has the form D\_l D\_k Z - Z \_[lk]{} -Z = 0,
We will seek special solutions to (\[kar\])–(\[gzy\]) for the dimension $p = 3$, when the metric $\omega_{lk}$ is determined by the 3-dimensional part of the linear element of an open-type Robertson-Walker space-time. We seek solutions in the form \[gnc\] g\_0 = u\_[0]{}(), g\^a = u () k\^a (\^i), a = 1,2,3,4, and, doing so, we do not require that the full embedding conditions (\[kal\]) should hold. Then, for the functions $u_{0}(\eta)$ and $u(\eta)$ we obtain \[gnc1\] u\^2 + 4k u\^2 - 8k(u\^2 - k u\_0\^2)u du - 2H = 0, H = , \[gnc2\] u\_[0]{}\^2 + k u\_0\^2 - 8k(u\^2 -k u\_[0]{}\^2) u\_[0]{} du\_[0]{} + 2H = 0. The last two equation with respect to the variables $u_{0}(\eta)$ and $u(\eta)$) may be considered as a dynamic system with the potential energy $$\nq
U(u, u_0) = -k \Lambda u^2 (u^2 +2 u_{0}^2)+\Lambda u_{0}^4
+k(2 u^2 - u_0^2 /2)$$ and zero total energy. Integrating by parts and summing, we can obtain $$\dot u^2 + \dot u_{0}^2 + 2U = 0.$$ A further study shows that the previously found solution, describing an open de Sitter space, is a stable exceptional solution to (\[gnc1\])–(\[gnc2\]). In the present formulation, the following solution corresponds to the one obtained above: \[gnc3\] u = , u\_0 = , r\^2 = .
Using the terminology of the qualitative theory of differential equations, the singular point $u = 0$, $u_0 = 0$ is unstable. There are no other static points. Meanwhile, the solution (\[gnc3\]), being a separatrix in the phase space of the variables $u_{0}(\eta)$ and $u(\eta)$, minimizes the total energy.
From this point of view, it is of interest to invoke the Higgs mechanism to obtain the constraints (\[gi\]). The fields $X^A$, being coordinates of the space $M$, may play the role of Higgs’ fields in Grand Unification models. On the other hand, from the viewpoint of the Hamiltonian formalism considered above, solutions with a broken symmetry may be treated as a particular choice of the gauge.
The hierarchy of coupling constants
-----------------------------------
As has been already noted above, we here obtain equations similar to the Einstein equations with an effective gravitational constant, see (\[kap\]). Indeed, in case $(X,X)=C$ and if \[gti\] g\_ =(\_ X, \_ X), ,=, (\[T3\]) take the form \[TE3\] G\_ =8 G\_e T\^[e]{}\_ + \_e g\_, where $G_e$ is given by (\[kap\]), while the cosmological constant is \[lamcos\] \_e=- (-1+C\^2). From the solution (\[cosm\]) and (\[lamcos\]), we find that \[coslam\] = , \_e = .
In a closed model, the constant $C$ satisfies the equation \[constc\] -(X\^0)\^2+(X\^1)\^2+(X\^2)\^2+(X\^3)\^2+(X\^4)\^2=C. Then $\sqrt{C}$ characterizes the size of the observed part of the Universe. In the solution (\[zakrit\]) (for $\eta_0=\pi/2$), we pass over to the proper time $t$ and obtain: \[massht\] a(t) (t/), H =(t/). Suppose that the Hubble “constant” $H\sim (3 \cdot 10^{17})^{-1}
c^{-1}$ (in the Planck units, $\hbar = 1$ and $c = 1$) and that our epoch corresponds to the time $t\simeq \sqrt{C}$, then we obtain the calue of $C$: $\sqrt{C}\simeq 7.2 \cdot 10^{27}$ cm $\sim 10^{28}$ cm. Substituting this value into (\[coslam\]), we find $\Lambda_e\sim 10^{-56}\ {\rm cm}^{-2}$, or, the same in energy units, $\Lambda_e \sim 10^{-46}\ GeV^4$. This result confirms the existence of a nonzero cosmological constant $\Lambda$, which is also in agreement with the observational data, see, e.g., Ref.[@starob].
Equating the expression (\[kap\]) to $1/{M_p^2} \sim
10^{-66}\ {\rm cm}^2$, we find that $w_0\sim 4\cdot 10^{-10}cm^4$. The parameter $w_0$ also corresponds to distances of the order $l_w = \sqrt[4]{w_0}\sim 0.05$ mm. To explain the nature of the emerging scale, one can invoke the Randall-Sundrum conjecture [@randal1], where the existence of extra dimensions ($n>4$) is supposed, with a sufficiently small size ($l < 0.2$ mm) for being in agreement with the experimental data.
In conclusion, let us note that if we consider the action obtained from (\[ns1\]) by adding to it the Einstein term $(1/G) R$, which violates the conformal invariance of the equations, then there emerges the effective gravitational “constant” $G_e = wG (w+2\xi G (X, X))^{-1} $ . As is shown in Ref.[@zar1], this leads to an instability of cosmological solutions for $G_e \to 0 $. This result is one of the arguments in favour of consideration of an intially conformally invariant theory of gravity; this invariance will then be probably violated due to quantum effects.
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abstract: 'Considerable progress has been recently made in the theoretical understanding of the colossal magnetoresistance (CMR) effect in manganites. The existence of inhomogeneous states has been shown to be directly related with this phenomenon, both in theoretical studies and experimental investigations. The analysis of simple models with two competing states and a resistor network approximation to calculate conductances has confirmed that CMR effects can be theoretically reproduced using non-uniform clustered states. However, a direct computational study of the CMR effect in realistic models has been difficult since large clusters are needed for this purpose. In this paper, the recently proposed Truncated Polynomial Expansion method (TPEM) for spin-fermion systems is tested using the double-exchange one-band, with finite Hund coupling $J_{\rm H}$, and two-band, with infinite $J_{\rm H}$, models. Two dimensional lattices as large as 48$\times$48 are studied, far larger than those that can be handled with standard exact diagonalization (DIAG) techniques for the fermionic sector. The clean limit (i.e. without quenched disorder) is here analyzed in detail. Phase diagrams are obtained, showing first-order transitions separating ferromagnetic metallic from insulating states. A huge magnetoresistance is found at low temperatures by including small magnetic fields, in excellent agreement with experiments. However, at temperatures above the Curie transition the effect is much smaller confirming that the standard finite-temperature CMR phenomenon cannot be understood using homogeneous states. By comparing results between the two methods, TPEM and DIAG, on small lattices, and by analyzing the systematic behavior with increasing cluster sizes, it is concluded that the TPEM is accurate to handle realistic manganite models on large systems. Our results pave the way to a frontal computational attack of the colossal magnetoresistance phenomenon using double-exchange like models, on large clusters, and including quenched disorder.'
author:
- 'C. Şen'
- 'G. Alvarez'
- 'Y. Motome'
- 'N. Furukawa'
- 'I. A. Sergienko'
- 'T. Schulthess'
- 'A. Moreo'
- 'E. Dagotto'
title: 'Study of the One- and Two-band Models for Colossal Magnetoresistive Manganites Using The Truncated Polynomial Expansion Method'
---
Introduction
============
The study of transition metal oxides (TMOs) has been among the most important areas of investigations in condensed matter physics in the last two decades. The excitement in this context started with the high-temperature superconductors and was later followed by the discovery of the colossal magnetoresistance in manganites,[@tokura; @review] as well as a plethora of equally interesting phenomena in several other oxides. Strong correlations ($i.e.$ large electron-electron or electron-phonon couplings, or both) play a major role in the physics of these materials. In addition, the presence of nearly degenerate states renders some of these oxides highly susceptible to external perturbations. In fact, TMOs appear to share a phenomenology similar to that of complex systems, with nonlinearities and sensitivity to details.[@dagotto-science]
We focus in this work on the manganites, area where in recent years considerable progress has been made, both in theory and experiments.[@review] In the late 90’s, it was predicted that many Mn-oxides should be inhomogeneous at the nanoscale, due to the unveiling of tendencies toward electronic phase separation.[@yunoki] On the experimental front, the evidence for inhomogeneous states was rapidly gathered and it is by now widely accepted, with building blocks typically having small nanoscale sizes.[@cheong] Subsequent theoretical work showed that similar tendencies can also occur after the inclusion of quenched disorder effects – caused for instance by chemical doping– near first-order phase transitions.[@giant] The key influence of quenched disorder was also observed in simulations of the one-band model for manganites including cooperative phonons [@motome; @sen] and also for two bands with Jahn-Teller phonons.[@aliaga] This key role of quenched disorder postulated by theory was observed experimentally using a Mn-oxide compound that can be prepared both in crystal ordered and disordered forms.[@tokura-disorder; @ueda] Remarkably, only the latter presents a CMR effect.
While the presence of quenched disorder was theoretically found to generate metal insulator transitions similar to those found experimentally, the actual existence of large magnetotransport effects is difficult to test in unbiased theoretical studies. Using toy models that have phase competition and first-order transitions, and supplementing the investigations with a random-resistor network approximation, huge magnetoresistances were obtained in resistance vs. temperature profiles in excellent agreement with experiments.[@burgy] However, it is certainly desirable to carry out similar investigations in more realistic models, of the double-exchange variety, and with quantum mechanical procedures to calculate the conductances. Alas, this task is tremendously difficult with standard computational methods that rely on the exact diagonalization of the fermionic sector and the Monte Carlo simulation of the classical $t_{\rm 2g}$ spins.[@review] The effort in this context grows like $N^4$, with $N$ the number of sites, severely limiting the clusters that can be studied. Since theory suggests that physics related with percolative phenomena is anticipated upon the introduction of magnetic fields in nano-clustered states,[@review; @sen] large clusters must be used for accurate simulations. Fortunately, important progress has been made in recent years toward the development of a novel method to carry out these investigations.[@motome99; @furukawa01; @furukawa03; @motome-disorder] The technique, TPEM, has a CPU time that scales like $N$ and, as a consequence, it is a promising technique for these investigations. Previous studies have shown that the $J_{\rm H}$=$\infty$ one-band model with and without quenched disorder can be accurately studied.[@motome99; @furukawa01; @furukawa03; @motome-disorder; @alvarez] However, the method has not been tested yet under some of the more severe circumstances needed for a realistic study, namely the use of two active orbitals per site ($i.e.$ the doubly degenerate $e_{\rm g}$ sector of Mn ions) and/or with a finite Hund coupling.
It is important to remark that there are at least two other independent methods that have been proposed to improve on the exact diagonalization of the fermionic sector approach: (1) The hybrid Monte Carlo technique of Alonso [*et al.*]{},[@alonso1] inspired in lattice gauge theory, that comfortably allows calculations on lattices up to $10^3$ sites in the limit of $J_{\rm H}$=$\infty$ and for one orbital, [@alonso2] and with a linear with $N$ increase in effort, and (2) the method of Kumar and Majumdar [@kumar] that has been already applied to a variety of models reaching 32$\times$32 clusters for one orbital and at $J_{\rm H}$=$\infty$. The scaling of this method is $N^3$. Our choice of the TPEM is motivated by the perception that a linear with $N$ cost is needed to handle the anticipated percolative physics that emerges when quenched disorder and phase competition occurs. Also it seems easier to implement than method (1) where auxiliary fields are needed. Nevertheless, this is not a critique on methods (1) and (2): they should be strongly pursued as well. Only future work can decide which method is the best for the type of problems described here.
It is the main purpose of this paper to present a systematic study of the TPEM applied to models that are widely believed to be realistic for manganite investigations, in the clean limit. The conclusions indicate that the technique works properly for these investigations, opening an avenue of research toward the ultimate goal of conducting a fully realistic simulation of the two-band double-exchange model including quenched disorder. The present results include a detailed analysis of both metallic and insulating phases on lattices as large as 48$\times$48 sites, about 20 times larger in number of sites than it is possible to handle with exact diagonalization techniques. In addition, here it is discussed the existence of a huge magnetoresistance at low temperatures, in agreement with experiments for $\rm(Nd_{\it 1-y} Sm_{\it y})_{1/2} Sr_{1/2} Mn O_3$.[@low-T-CMR] This phenomenon was theoretically studied before by Aliaga [*et al.*]{} [@aliaga] as well, although on much smaller systems. Nevertheless, the conclusions are similar: this somewhat exotic “low temperature” CMR phenomenon can be understood as a natural consequence of the existence of a first-order metal-insulator transition in the phase diagram and, thus, a clean-limit simulation is sufficient for this purpose. However, in our investigations it is also confirmed that these clean limit simulations are $not$ able to generate the standard CMR effect above the Curie temperature using states that are homogeneous. Quenched disorder or strain fields are likely important for this purpose, which will be the subject of near future efforts.
The organization of the paper is simple. The theoretical aspects of the TPEM are briefly reviewed in Section \[sec:tpem\]. The systematics of the TPEM in the case of the one-band model, with emphasis on the dependence with parameters and size effects is discussed in Section \[sec:oneband\]. We also present physical results related with large magnetoresistance effects found at low temperatures, in large clusters. Then, in Section \[sec:twoband\], the emphasis shifts to the two bands model, with an analogous study of TPEM performance and effects of magnetic fields. Conductances are evaluated for both models using the TPEM and reasonable results are observed. Overall, it is concluded that the TPEM is adequate for a frontal future attack of the CMR phenomenon using realistic models and quenched disorder.
Review of the TPEM {#sec:tpem}
==================
For completeness, here a brief review of the TPEM is presented following closely Ref. . Consider a model defined by a general Hamiltonian, $\hat{H}=\sum_{ij\alpha\beta}c^\dagger_{i\alpha} H_{i\alpha,j\beta}(\phi)c_{j\beta}$, where the indices $i$ and $j$ denote a spatial index, while $\alpha$ and $\beta$ are internal degree(s) of freedom, [[*e. g.*]{}]{} spin and/or orbital. As in the case of spin-fermion models, the Hamiltonian matrix depends on the configuration of one or more classical fields, represented by $\phi$. Although no explicit indices will be used, the field(s) $\phi$ are supposed to have a spatial dependence. The partition function for this Hamiltonian is schematically given by: $
Z=\int d\phi \sum_n \langle n | \exp(-\beta\hat{H}(\phi)+\beta\mu\hat{N}) |n\rangle
$ where $|n\rangle$ are the eigenvectors of the one-electron sector, and the $\phi$ integral denotes the integration over all the classical fields. Here $\beta$=$1/(k_{\rm B} T)$ is the inverse temperature. The number of particles (operator $\hat{N}$) is adjusted via the chemical potential $\mu$.
The procedure to calculate observables (energy, density, action, etc) is the following. First, an arbitrary configuration of classical fields $\phi$ is selected as a starting point. The Boltzmann weight or action of that configuration, $S(\phi)$, is calculated by diagonalizing the one-electron sector. Then, a small local change of the field configuration is proposed, so that the new configuration is denoted by $\phi'$ and its action is recalculated to obtain the difference in action $\Delta S = S(\phi')-S(\phi)$. Finally, this new configuration is accepted or rejected based on a Monte Carlo algorithm, such as Metropolis or heat bath, and the cycle starts again. In summary, in the standard algorithm (that we will denote here as DIAG) the observables are calculated using an exact diagonalization of the one-electron sector for every classical field configuration, and Monte Carlo integration for those classical fields.[@review]
The TPEM replaces the diagonalization for a polynomial expansion as discussed below (details can be found in Refs. ). It will be assumed that the Hamiltonian $H(\phi)$ is “normalized”, which simply implies a rescaling in such a way that the normalized Hamiltonian has eigenvalues $\epsilon_v\in[-1,1]$. Simple observables can be divided in two categories: (i) those that do not depend directly on fermionic operators, [[*e. g.*]{}]{} the magnetization, susceptibility and classical spin-spin correlations in the double exchange model, and (ii) those for which a function $F(x)$ exists such that they can be written as: $A(\phi)=\int_{-1}^{1}F(x)D(\phi,x)dx$, where $D(\phi,\epsilon)=\sum_\nu\delta(\epsilon(\phi)-\epsilon_\nu)$, and $\epsilon_\nu$ are the eigenvalues of $H(\phi)$, [[*i. e.*]{}]{} $D(\phi,x)$ is the density of states of the system.
For category (i), the calculation is straightforward and simply involves the average over Monte Carlo configurations. Category (ii) includes the effective action or generalized Boltzmann weight and this quantity is particularly important because it is calculated at every MC step to integrate the classical fields. Therefore, we first briefly review how to deal with this type of observables. As Furukawa [[*et al.*]{}]{} showed, it is convenient to expand the function $F(x)$ in terms of Chebyshev polynomials in the following way: $F(x)=\sum_{m=0}^{\infty}f_mT_m(x)$, where $T_m(x)$ is the $m-$th Chebyshev polynomial evaluated at $x$. Let $\alpha_m=2-\delta_{m,0}$. The coefficients $f_m$ are calculated with the formula: $f_m=\int_{-1}^1 \alpha_m F(x)T_m(x)/(\pi\sqrt{1-x^2})$. The moments of the density of states are defined by: $$\mu_m(\phi)\equiv\sum_{\nu=1}^{N_{dim}}\langle \nu| T_m(H(\phi))|\nu\rangle,
\label{eq:moments}$$ where $N_{dim}$ is the dimension of the one-electron sector. Then, the observable corresponding to the function $F(x)$, can be calculated using $$A(\phi)=\sum_m f_m \mu_m(\phi).
\label{eq:expansion}$$ In practice, the sum in Eq. (\[eq:expansion\]) is performed up to a certain cutoff value $M$, without appreciable loss in accuracy as described in Refs. and , and as extensively tested in the main sections of this paper for realistic manganite Hamiltonians. The calculation of $\mu_m$ is carried out recursively using $|\nu;m\rangle = T_m(H)|\nu\rangle=2H|\nu;m-1\rangle-|\nu;m-2\rangle$. These same vectors are used to calculate the moments. The process involves a sparse matrix-vector product, [[*e. g.*]{}]{} in $T_m(H)|\nu\rangle$, yielding a cost of $O(N^2)$ for each configuration, [[*i. e.*]{}]{} $O(N^3)$ for each Monte Carlo step. This represents an improvement in a factor $N$ compared with DIAG.
In addition, an extra improvement of the method described thus far has been proposed [@furukawa03] based on two truncations: (i) of the sparse matrix-vector product and (ii) of the difference in action for local Monte Carlo updates. The resulting algorithm has an expected CPU time growing like $N$. The first truncation is possible because of the form of the vectors $|\nu;m\rangle$. Indeed, for $m$=$0$ these are simply basis vectors that can be chosen with only one non-zero component. The $m$=$1$ vector is obtained by applying $T_1(H)$=$H$ to the basis vector. Since $H$ is sparse (consider for instance a nearest-neighbor hopping), the vector $|\nu;m=1\rangle$ will have non-zero elements only in the vicinity of $\nu$. For general $m$, the non-zero elements will propagate in what resembles a diffusion process. Note that we only have to keep track of the non-zero indices to perform the sparse matrix-vector product. Since we are only discarding null terms, this truncation does not introduce any approximation. It is possible to go a step further and discard not only zero elements but elements smaller than a certain threshold denoted by $\epsilon_{\rm pr}$. In this case, the results are approximate, but the exact results are recovered in the limit $\epsilon_{\rm pr}\rightarrow0$. In this paper, the dependence of results with various values for this cutoff (and the one described below) is discussed.
The second truncation involves the difference in effective action, which is calculated very frequently in the Monte Carlo integration procedure. The function corresponding to the effective action for a configuration $\phi$ is defined by $F^S(x) = -\log(1+\exp(-\beta(x-\mu)))$ and $S(\phi)$ admits an expansion as in Eq. (\[eq:expansion\]), with coefficients $f^S_m$ corresponding to $F^S(x)$. In practice, only the difference in action, $\Delta S $=$ S(\phi')-S(\phi)$ has to be computed for every change of classical fields. Since this operation requires calculating two sets of moments, for $\phi$ and $\phi'$, the authors of Ref. have also developed a truncation procedure for this trace operation controlled by a small parameter, $\epsilon_{\rm tr}$. This truncation is based on the observation that if $\phi$ and $\phi'$ differ only in very few sites (as is the case with local Monte Carlo updates), then the corresponding moments will differ only for certain indices allowing for a significant reduction of the computational effort. Again, the exact results are recovered for $\epsilon_{\rm tr}\rightarrow0$, so this approximation is controllable. Another key advantage of the TPEM is that it can be parallelized, because the calculation of the moments in Eq. (\[eq:moments\]) for each basis ket $|\nu\rangle$ is independent. Thus, the basis can be partitioned in such a way that each processor calculates the moments corresponding to a portion of the basis. It is important to remark that the data to be moved between different nodes are small compared to calculations in each node: communication among nodes is mainly done here to add up all the moments. The possibility of parallelizing this algorithm can be contrasted with the conventional method where a matrix diagonalization is performed at every Monte Carlo step; in that case the calculations must be serial because it is difficult to make an efficient parallelization of the matrix diagonalization.
RESULTS FOR THE ONE BAND MODEL {#sec:oneband}
==============================
Definition
----------
After introducing the computational method, we will now focus on its performance starting with the one-band model for manganites. Historically, this model was among the first proposed for Mn-oxides and it is still widely used, although it does not incorporate the two orbitals $e_{\rm g}$ of relevance in Mn ions. This more involved two-band version will be studied in the next section.
The one band Hamiltonian used in this study is given by: $$\begin{aligned}
H_{1b}&=&-t\sum_{\langle ij \rangle , \alpha} (c_{i,\alpha}^{\dag}c_{j,\alpha} + \mbox{h.c.})
-J_{\rm H}\sum_{i,\alpha, \beta}c_{i,\alpha}^{\dag} \sigma_{\alpha,\beta}c_{i,\beta} \nonumber
\\ &+&J_{\rm AF}\sum_{\langle ij \rangle}\mathbf{S}_{i}\cdot\mathbf{S}_{j},\end{aligned}$$ where $c_{i,\alpha}^{\dag}$ creates an electron at site $i$ with spin $\alpha$, $\mathbf{\sigma}$ are the Pauli spin matrices, $\mathbf{S}_i$ is the total spin of the $t_{\rm 2g}$ electrons (assumed to be localized and classical), $\langle ij
\rangle$ indicates summing over nearest neighbor sites, $t$ is the nearest neighbor hopping amplitude for the movement of electrons ($t$ also sets the energy unit), $J_{\rm H}>0$ is the Hund coupling, and $J_{\rm AF}>0$ is the antiferromagnetic coupling between the localized spins. The study carried out in this manuscript is based on the $clean$ limit, namely the couplings that appear in the Hamiltonian do not have a site index, which would be necessary if quenched disorder is incorporated. The study of realistic models including disorder will be carried out in a future investigation, since it represents an order of magnitude extra effort due to the average over disorder configurations.
Here and in the rest of the paper, spatial indices will always be denoted without arrows or bold letters independently of the dimension. Also the notation $i+j$ is meant to represent the lattice site given by the vectorial sum of the vectors corresponding to $i$ and $j$, respectively.
TPEM performance
----------------
### Test of the TPEM in Small Systems
As explained before, the TPEM has three controlling parameters: $M$, $\epsilon_{\rm pr}$, and $\epsilon_{\rm tr}$. In the limit when the first parameter runs to infinity, and the other two to zero, the exact results are recovered. For the TPEM to be useful, accurate results must be obtained for values for these parameters that allow for a realistic computational study. In Fig. \[Figure1\](a), the dependence of the zero-momentum spin structure factor, of relevance for ferromagnetism, is shown vs. temperature, using a 12$^2$ cluster and the values of $J_{\rm H}$ and $J_{\rm AF}$ indicated. In this case, it is expected that a FM state will form at low temperatures, as observed numerically. Results for many values of $M$ are shown, at fixed values of $\epsilon_{\rm pr}$, and $\epsilon_{\rm tr}$. Clearly, $M$=10 only captures the low and high temperature limits, but it is not accurate near the critical temperature. The results for $M$=20 are much better, but still there is a visible discrepancy near the region where $S(0,0)$ changes the fastest. However, for $M$=30 and 40, fairly accurate results are obtained. In Fig. \[Figure1\](c), it is shown that even the spin correlations at the largest distances are accurately reproduced with $M$=40 terms in the expansion.
![(color online). Dependence of the TPEM algorithm results on the number of terms M in the expansion. Shown are the spin structure factors at the momenta characteristic of (a) the FM state and (b) the Flux state, normalized to 1 and 0.5, respectively, for the perfect states. Results are obtained on a $12\times12$ lattice and they are compared with the numbers gathered using the exact diagonalization algorithm, all calculated at density $\langle n\rangle=0.5$. (a),(c) correspond to $J_{\rm AF}=0.0$ and (b),(d) to $J_{\rm AF}=0.1$. Measurements were taken every $10$ steps of a MC run of $2000$ total iterations, after discarding $2000$ steps for thermalization. A random starting configuration is used for each $T$. In (a), at the most difficult temperature, $T$=0.06, where critical fluctuations are strong, the result shown was confirmed using several different starting configurations, including ordered ones. The average $S(0,0)$ obtained by this procedure was very similar among the several starts. The shown error bars at this temperature and M=30 mainly arise from the expected critical fluctuations. In (b), a good convergence at $T$=0.04 is only achieved by using 40 moments with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-7}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-5}$ and the result is shown with an orange star just below the exact result. In (c), the TPEM parameters used are $M=30$, $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-5}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-3}$, except at $T$=0.06, where the convergence is achieved by using 40 moments with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-7}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-5}$. In (d), all the results shown were obtained with $M=30$, $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-5}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-3}$, and solid lines represent the DIAG results while the dashed lines are the TPEM results.[]{data-label="Figure1"}](\mypath{merge1.eps}){width="9cm"}
In Fig. \[Figure1\](b), similar results are presented but now for the spin-structure-factor corresponding to the “Flux” phase – nearest-neighbor spins at 90 degrees forming a staggered arrangement of nonzero plaquette fluxes. This phase appears at $n$=0.5 with increasing $J_{\rm AF}$, as reported in previous investigations. [@flux-phase] In this case, $M$=10 and even 20 produces results dramatically different from those of DIAG. However, for $M$=30 discrepancies are observed only in a range of temperatures near the critical transition, with low and high temperatures under control. Finally, $M$=40 leads to very accurate results, as in case (a). Even the spin correlations are under well control for this number of terms in the expansion (see Fig. \[Figure1\](d)). The range $M$$\sim$30-40 appears systematically in our investigations, and it is expected to provide safe values of $M$ for studies of the type of spin-fermion models under investigation in manganites.
![(color online). $\varepsilon$ dependence of the TPEM algorithm results (M=30) for a $12\times12$ lattice, compared with the exact (DIAG) results. Both are obtained at density $\langle n \rangle$=0.5, using (a) $J_{\rm AF}=0.0$ and (b) $J_{\rm AF}=0.1$. Measurements were taken at every $10$ steps of a $2000$ MC steps total run, after $2000$ steps for thermalization. In (b), the convergence at $T$=0.04 is achieved by using 40 moments with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-7}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-5}$, as shown with an orange star just below the exact result. The starting configurations and error bar convention is as in Fig.\[Figure1\].[]{data-label="Figure2"}](\mypath{merge2.eps}){width="9cm"}
In Fig. \[Figure2\], a study of the dependence of results with $\epsilon_{\rm pr}$ and $\epsilon_{\rm tr}$ is presented, working at $M$=30. From Fig. \[Figure2\], clearly there are large $\epsilon$’s that lead to unphysical results, but with decreasing values an accurate evaluation of observables is reached. In this and other investigations, values such as $\epsilon_{\rm pr}$=10$^{-5}$ and $\epsilon_{\rm tr}$=10$^{-3}$ are generally found to be accurate, with only a few exceptions.
### Dependence of Results on Lattice Sizes
An approximate method that depends on some parameters, such as in the case of the TPEM, is practical only if by fixing those parameters on small systems, their values still provide accurate numbers as the lattice sizes increase. A qualitative way to carry out this test is to perform the studies on large clusters and see that all the trends and approximate numbers remain close to those known to be accurately obtained on small systems, or expected from other techniques or physical argumentations. Figure \[Figure3\](a,b) supports the notion that TPEM indeed behaves properly in this respect, namely the range of $M$ and $\epsilon$’s identified in the previous subsection are sufficient to produce qualitatively similar results even when the number of sites grows by a factor 10. In (a), the expected size dependence corresponding to a second order FM transition is found. For a 40$\times$40 cluster, the Curie temperature appears located at $T$$\sim$0.07. In (b), the size dependence is almost negligible. The transition is far sharper for the paramagnetic-flux transition, as already noticed in Fig. \[Figure2\](b). This is an intriguing feature that will be investigated in future work: while the first-order low-temperature metal-insulator transitions are clear and well established in realistic models for manganites, the presence of first-order transitions between ordered and disordered phases varying temperature is far less obvious, and TPEM studies on large lattices can properly address this issue. For our current purposes, here it is sufficient to state that the TPEM appears to behave properly with increasing lattice sizes, both in metallic and insulating regimes.
![(color online). Lattice size dependence of the TPEM algorithm results for the lattices and parameters shown, working at density $\langle n \rangle$=0.5 and using (a) $J_{\rm AF}=0.0$ and (b) $J_{\rm AF}=0.1$. Measurements were taken at every $10$ steps of a $2000$ MC total steps run, after discarding $2000$ MC steps for thermalization. For the 20$\times$20 lattice and larger, the starting configuration used was a perfect FM state. In (b), the starting configuration used is a random one except at $T$=0.04 where the starting configuration is chosen to be a perfect flux state for faster convergence. In addition, for lattices except $40\times40$, the convergence at $T$=0.04 is achieved by using 40 moments with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-7}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-5}$, while for other temperatures $M=30$ with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-5}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-3}$ were sufficient. For the lattice $40\times40$, $M=40$ with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-6}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-4}$ were used for all temperatures.[]{data-label="Figure3"}](\mypath{merge3.eps}){width="9cm"}
At this point a clarification is important. In principle, two-dimensional systems should not show true critical temperatures due to the Mermin-Wagner theorem. However, it is well known that in systems where the Mermin-Wagner theorem applies, such as the two-dimensional NN Heisenberg model, the antiferromagnetic spin correlations exponentially diverge with decreasing temperature. An exponential behavior, defines via the exponent a temperature scale $T^*$ below which the correlations are much larger than any lattice size that can be practically studied numerically. This may seem like a problem, but it is not: very large correlation lengths also render the system [*very susceptible to small perturbations*]{}. In particular, we have shown that tiny deviations from the fully symmetric Heisenberg model, such as introducing Ising anisotropies, stabilize $T^*$ into a true critical temperature. In fact, simulations performed with Ising anisotropies typically reveal no important differences with the results obtained with fully vector models on finite systems. Small couplings in the third direction play a similar role. As a consequence, for all practical purposes the critical behavior observed in the present studies describes properly the expected physics of manganite models, which are always embedded in three dimensional environments, and that have small anisotropies. A final note on this subject: The CE phase of manganites can show a finite-temperature transition even in two dimensions, since the order parameter for charge order can be Ising type.
The CPU time needed to obtain the results shown in this subsection follows the expected trends reported in previous investigations (see Table \[tab:comp2D\]). In particular, the TPEM time needed for a 32$\times$32 cluster is comparable to the DIAG time on a 12$\times$12 cluster, a very encouraging result. Of course, this comparison will be even more favorable to the TPEM with increasing number of CPU’s for parallelization.
$L$ Algorithm \# of CPUs CpuTime(h)
------ ----------- ------------ ------------ --
$12$ DIAG $1$ $19.48$
$12$ TPEM $2$ $5.46$
$20$ TPEM $2$ $18.08$
$32$ TPEM $8$ $25.92$
: \[tab:comp2D\] Comparison of the CPU times for the algorithms indicated, using an Intel Pentium 4 (clock speed 3.06Gz) computer. Shown are results for different square lattices of size $L\times L$, assuming 2000 MC steps for thermalization, and 2000 MC steps for measurements (taken every 10 MC steps). Since the TPEM can be parallelized, some results were obtained using more than one CPU, as indicated.
![(color online). Spin structure factor vs. $T$ for $J_{\rm AF}$=$0.05$ using the lattice sizes and parameters shown in the figure.[]{data-label="Figure48"}](\mypath{merge48.eps}){width="9cm"}
Phase Diagram
-------------
Using the TPEM, the phase diagram of the one-band model for manganites at $\langle n \rangle$=0.5 was obtained (see Fig. \[Figure4\]). The transition between the FM and Flux states at low temperatures is of first order. In fact, in the absence of quenched disorder the zero temperature result can be obtained by using the perfect classical spin configurations for both the FM and Flux states, and calculating their energy vs $J_{\rm AF}$ (not shown). By this procedure the zero-temperature critical $J_{\rm AF}$ was found to be close to 0.03. Raising the temperature, this transition line is not vertical, but has a tilting. Figure \[Figure4\] shows that the estimated critical temperatures do not present severe size effects, and the TPEM can be comfortably used at least up to 40$\times$40 clusters. The presence of a first-order transition in the competition between the FM and Flux phases is in qualitative agreement with several previous investigations that have shown similar trends both for the one and two bands models, at any electronic density.[@review] This transition is expected to be severely affected by the influence of quenched disorder, and this issue will be investigated in the near future.
![(color online). Phase diagram at $\langle n \rangle$=0.5, varying temperature and $J_{\rm AF}$. Results are shown for a $12\times12$ lattice using both DIAG and TPEM techniques, and for larger lattices using TPEM, as indicated. The origin of the tilting of the first-order low-temperature FM-Flux line is explained in the text.[]{data-label="Figure4"}](\mypath{phasefinal.eps}){width="8cm"}
The critical temperatures in Fig. \[Figure4\] were estimated from the behavior of the spin structure factors at the two momenta of relevance for the FM and Flux phases, as shown in Fig. \[Figure5\].
![(color online). Examples of the criteria used in the calculation of the critical temperatures in Fig. \[Figure4\]. (a) and (c): Spin structure factors at the momenta of relevance vs. temperatures, for many values of $J_{\rm AF}$, as indicated, using the DIAG technique on a $12\times12$ cluster. (b) and (d): Same as in (a) and (c) except the technique used here is the TPEM. Shown are some of the results for $12\times12$ and $20\times20$, as indicated in the figure.[]{data-label="Figure5"}](\mypath{merge4.eps}){width="9cm"}
Density of States
-----------------
In this section, it is shown that the density-of-states (DOS) can be reproduced properly by the TPEM. This is nontrivial, since it may be suspected that a method based on an expansion of the DOS may have problems in an insulating phase due to the rapid changes in the DOS near the gap. To our knowledge, this is the first time that the TPEM is applied to an insulator. The discussion in this subsection shows that the technique works satisfactorily.
In general, the density-of-states for a configuration of classical fields $\phi$ is given by $$N_{\phi}(\omega)=\sum_{\lambda}\delta(\omega^{\prime}-\epsilon_{\lambda}),$$ with $\omega^{\prime}=(\omega-b)/a$, and where $a$ and $b$ are the parameters that normalize the Hamiltonian in such a way that the new eigenvalues are in the interval $[-1,1]$. These constants are given by $a$=$(E_{\rm max} - E_{\rm min})/2$ and $b$=$(E_{\rm max} + E_{\rm min})/2$, where $E_{\rm max}$ and $E_{\rm min}$ are the maximum and minimum eigenvalues of the Hamiltonian. Then, following the discussion of Section \[sec:tpem\], the corresponding function $F(x)$ for the density-of-states in the expression $$A(\phi)=\int_{-1}^{1}F(x)D(\phi,x)dx$$ is the $\delta$-function $$F(x)=\delta(\omega^{\prime}-x).$$ In the expansion $F(x)$=$\sum_{m=0}^{\infty}f_{m}T_{m}(x)$, by using the expression for the coefficients $f_m$=$\int_{-1}^{1}\alpha_{m}F(x)T(x)/(\pi\sqrt{1-x^{2}})$, the final result for the density-of-states becomes $$N_{\phi}(\omega)=\frac{\sum_{m}\alpha_{m}T_{m}(\omega^{\prime})\mu_{m}(\phi)}{\pi\sqrt{1-\omega^{'2}}}.$$ This sum is truncated to a certain cutoff $M$ and such an abrupt truncation results in unwanted Gibbs oscillations, as shown, e.g., in Fig. \[dosfigure\] for $M$=30. This problem may be avoided by multiplying the moments by dumping factors if needed.[@gonzalo-thomas]
![(color online). (a) Density-of-states calculated for a perfect Flux state using both the DIAG and TPEM methods for a lattice of size $12\times12$. In order to get the DOS accurately, even removing the Gibbs oscillations, one needs larger number of moments than what is usually required for other observables. (b) Monte Carlo results for the density-of-states obtained from simulations performed on a 40$\times$40 lattice. In this case the last configuration of the MC run has been used to calculate the density-of-states at $T$=$0.02$.[]{data-label="dosfigure"}](\mypath{clean.dos2.eps}){width="7cm"}
The results for the density-of-states of the Flux phase are shown in Fig. \[dosfigure\]. Clearly, even at $M$=30 there is a very good agreement between the DOS calculated exactly and with TPEM (with the exception of the in-gap Gibbs oscillations). Increasing $M$ further, even this effect disappears. Using a 40$\times$40 lattice, the results are almost the same as those observed on the smaller system. It is concluded that the TPEM can produce the DOS of insulating states accurately, and the method can be used to study phase competition between metals and insulators.
Conductances: Comparison TPEM vs. DIAG, and Results with Increasing Lattice Sizes
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To compare theory with experiments, it is crucial to evaluate the conductance of the cluster under study. Its temperature and magnetic field dependence will clarify whether the double-exchange models for manganites contain the essence of the CMR phenomenon. The conductance calculation here follows the steps previously extensively discussed by Vergés [*et al.*]{}, and it basically relies on the Landauer formalism that links conductance with transmission. We refer the readers to original references for more details (see for instance Ref. ). In Fig. \[Figure6\], the conductance and its inverse (resistance) are shown as a function of temperature for the model on a 12$\times$12 cluster that can be solved both with DIAG and TPEM. The agreement between the results obtained with both techniques is excellent, at the two values of $J_{\rm AF}$ shown (one in the FM and the other in the Flux phase). Thus, the conductance calculation does not present an obstacle in the use of the TPEM .
![(a) Conductance and (b) resistance (1/conductance) vs. temperature for a $12\times12$ lattice, calculated with both DIAG and TPEM algorithms showing that the results agree. The couplings used are $J_{\rm AF}$=$0.0$ and $J_{\rm AF}$=$0.1$ as indicated, and the density is $\langle n \rangle$=0.5. The convergence at $T$=0.04 is achieved by using 40 moments with $\varepsilon_{\mbox{\scriptsize{{pr}}}}$=$10^{-7}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}$=$10^{-5}$, as discussed in previous figures captions.[]{data-label="Figure6"}](\mypath{merge5.eps}){width="9cm"}
With increasing lattice size, the TPEM conductance behaves smoothly and the finite-size effects are small (see Fig. \[Figure7\]), with the only exception of the insulating Flux phase regime at low temperatures where the 12$\times$12 cluster results appear appreciably different from those on larger systems. Considering the small value of the conductance in this insulating regime and the subsequent convergence of the Flux-phase resistance between the 20$\times$20 and 32$\times$32 clusters, this appears to be only a minor issue.
![(color online). (a) Conductance and (b) resistance (1/conductance) vs. temperature calculated with the TPEM algorithm at $J_{\rm AF}=0.0$ and $J_{\rm AF}=0.1$, $\langle n \rangle$=0.5, and for the cluster sizes shown in the figure. The figure shows that the conductance does $not$ suffer from strong size effects. The convergence at $T$=0.04 for the $12\times12$ lattice was achieved by using 40 moments with $\varepsilon_{\mbox{\scriptsize{{pr}}}}=10^{-7}$ and $\varepsilon_{\mbox{\scriptsize{{tr}}}}=10^{-5}$, as discussed elsewhere.[]{data-label="Figure7"}](\mypath{merge6.eps}){width="9cm"}
Influence of Magnetic Fields in the Clean Limit and Partial Conclusions
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As discussed in the introduction, it is important to investigate if the models studied here, in the clean limit, present a large magnetoresistance effect. Previous studies by Aliaga [*et al.*]{} [@aliaga] on 4$\times$4 clusters, suggested that the “low temperature” large magnetoresistance experimentally observed in some manganites [@low-T-CMR] can be explained by double-exchange models in the clean limit. This result is important and deserves to be confirmed using larger clusters. Here, the case of the one-band model is analyzed, with results shown in Fig. \[resmagneticfigure\] (two bands will be studied later in this paper). The value of $J_{\rm AF}$ was chosen to be on the insulating side (Flux phase) of the phase diagram Fig. \[Figure4\], but close to the first-order transition separating the metal from the insulator. The application of ‘small’ magnetic fields favors the FM state over the Flux state and that manifests as a sharp transition from the metal to the insulator, for values of the magnetic field that appear abnormally small in the natural units of the problem. Thus, this model presents a huge negative magnetoresistance, an encouraging result that shows theory is in the right track to understand manganites. The effect shown in Fig. \[resmagneticfigure\] is caused by the proximity in energy of two states with quite different properties, i.e. there is a [*hidden small energy scale*]{} in the problem.
However, note that the standard large finite-temperature magnetoresistance traditionally studied in Mn-oxides cannot be understood with clean limit models, as shown in Fig. \[resmagneticfigure\]: the zero magnetic-field resistivity does not have the large peak near Curie temperatures characteristic of CMR manganites. Future work using the TPEM will analyze whether this more traditional CMR effect can be obtained including quenched disorder.
![(color online). (a) Conductance vs. temperature for lattices of sizes $20\times20$ and $32\times32$, with and without magnetic fields. (b) Resistance vs. temperature calculated by taking the inverse of the conductance in (a). Close to the first-order transition in the phase diagram, a small magnetic field can destabilize the insulating Flux state into a metallic FM state. For each temperature, 1000 thermalization and 2000 measurements MC steps were used, with actual measurements taken at every 10 steps.[]{data-label="resmagneticfigure"}](\mypath{resh.eps}){width="7cm"}
0.4cm
Overall, it can be safely concluded that the study of the one-band model with the TPEM has proven that the technique works properly, and that the FM vs. Flux competition occurs via a first-order metal-insulator transition. This regime is ideal for the analysis of the influence of quenched disorder in future calculations.
RESULTS FOR THE TWO-BAND MODEL {#sec:twoband}
==============================
Definition
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In this effort, the two-band model for manganites was also investigated using the TPEM. The two bands arise from the $e_{\rm g}$ bands that are active at the Mn ions in Mn-oxides, as extensively discussed before.[@review] The overall conclusion of this section is that the TPEM is also a good approximation to carry out computational studies, conclusion similar to that reached for only one active band. The Hamiltonian for this model is[@review]
$$\begin{aligned}
H_{2b}&=&\sum_{\gamma,\gamma',i,\alpha}t^\alpha_{\gamma\gamma'}
{\mathcal S}(\theta_i,\phi_i,\theta_{i+\alpha},\phi_{i+\alpha})
c^\dagger_{i,\gamma}c_{i+\alpha,\gamma'} \nonumber \\ &+&
\lambda\sum_{i}(Q_{1i} \rho_i + Q_{2i} \tau_{xi} + Q_{3i} \tau_{zi}) \nonumber \\&+&
\sum_{i}\sum_{\alpha=1}^{\alpha=3} D_\alpha Q_{\alpha i}^2,
\label{eq:hamtwobands}\end{aligned}$$
where the factor that renormalizes the hopping in the $J_{\rm H}$=$\infty$ limit is $$\begin{aligned}
{\mathcal S}(\theta_i,\phi_i,\theta_{j},\phi_{j})&=&\cos(\frac{\theta_{i}}{2})\cos(\frac{\theta_{j}}{2})
\nonumber \\&+&\sin(\frac{\theta_{i}}{2})\sin(\frac{\theta_{j}}{2})e^{-i (\theta_{i}-\theta_{j})}.\end{aligned}$$ The parameters $t^\alpha_{\gamma\gamma'}$ are the hopping amplitudes between the orbitals $\gamma$ and $\gamma'$ in the direction $\alpha$. In this paper, we restrict ourselves to two dimensions, such that $t^{x}_{aa}=-\sqrt{3}t^{x}_{ab}=
-\sqrt{3}t^{x}_{ba}=3t^{x}_{bb}=1$, and $t^{y}_{aa}=\sqrt{3}t^{y}_{ab}=\sqrt{3}t^{y}_{ba}
=3t^{y}_{bb}=1$. $Q_{1i}$, $Q_{2i}$ and $Q_{3i}$ are normal modes of vibration that can be expressed in terms of the oxygen coordinate $u_{i,\alpha}$ as: $$\begin{aligned}
Q_{1i}&=&\frac {1}{\sqrt{3}} [(u_{i,z}-u_{i-z,z}) + (u_{i,x}-u_{i-x,x}) \nonumber \\ &+&
(u_{i,y}- u_{i-y,y})], \nonumber \\
Q_{2i}&=&\frac {1}{\sqrt{2}} (u_{i,x}-u_{i-x,x}), \nonumber \\
Q_{3i}&=&\frac {2}{\sqrt{6}} (u_{i,z}-u_{i-z,z})
-\frac{1}{\sqrt{6}} (u_{i,x}-u_{i-x,x}) \nonumber \\&-&\frac{1}{\sqrt{6}} (u_{i,y}- u_{i-y,y}). \nonumber\end{aligned}$$ Also, $\tau_{xi}=c^{\dagger}_{ia}c_{ib}+c^{\dagger}_{ib}c_{ia}$, $\tau_{zi}=c^{\dagger}_{ia}c_{ia}-c^{\dagger}_{ib}c_{ib}$, and $\rho_{i}=c^{\dagger}_{ia}c_{ia}+c^{\dagger}_{ib}c_{ib}$. The constant $\lambda$ is the electron-phonon coupling related to the Jahn-Teller distortion of the MnO$_6$ octahedron.[@tokura; @review] Regarding the phononic stiffness, and in units of $t^{x}_{aa}=1$, the $D_{\alpha}$ parameters are $D_1=1$ and $D_2=D_3=0.5$, as discussed in previous literature.[@aliaga] The rest of the notation is standard. In our effort here, the emphasis is on the case $\lambda$=0 believed to be of sufficient relevance to deserve a special study since it already contains[@aliaga] a competition between FM metallic and CE insulating states at $\langle n \rangle$=0.5. Thus, this is an excellent testing ground for the TPEM, particularly having in mind the next challenge involving a TPEM study in the presence of quenched disorder. However, briefly some results at nonzero $\lambda$ will also be shown.
TPEM Performance
----------------
### Test of the TPEM in Small Systems
As in the case of the one-band model, the analysis starts here by comparing DIAG and TPEM results on small systems. Figure \[Figure8\] contains the magnetization (in absolute value, and coming from the classical spins) vs. temperature. The results in (a) and (c) were obtained at $J_{\rm AF}$=0.0, $\lambda$=0, and $\langle n \rangle$=0.5, a regime known to develop ferromagnetism at low temperatures.[@aliaga] Indeed, both methods show a nonzero value for the magnetization. The dependence with the TPEM parameters indicates that a $M$ of approximately 30 or higher is sufficient to get accurate results. This is a conclusion that also appears in (b) where the case of a CE state is studied, which is stabilized with increasing $J_{\rm AF}$. Regarding the other TPEM parameters, (d) shows that $\epsilon_{\rm pr}$=10$^{-5}$, as used for the one-band case, leads to accurate results. Overall, it seems that the same set of parameters deduced from the one-band model investigations can also be used for two bands, an interesting simplifying result.
![(color online). Convergence of the structure factors, varying the parameters of the TPEM algorithm: (a) Magnetization $|M|$=$\sqrt{S(0,0)}$ vs. $T/t$ at $J_{\rm AF}$=$0.0$ using the DIAG method and TPEM with $\epsilon_{\rm pr}$=$10^{-5}$, $\epsilon_{\rm tr}$=$10^{-6}$, and the values of $M$ indicated. (b) Order parameter associated with the CE phase $|O_{\rm CE}|$=$\sqrt{S(\pi,0)}$ vs. $T/t$ at $J_{\rm AF}$=$0.2$ using the DIAG method and TPEM with $\epsilon_{\rm pr}$=$10^{-5}$, $\epsilon_{\rm tr}$=$10^{-6}$, and the values of $M$ indicated. (c) Magnetization $|M|$=$\sqrt{S(0,0)}$ vs. $T/t$ at $J_{\rm AF}= 0.0$ using the DIAG method and TPEM with $M$=$20$ and $\epsilon_{\rm tr}=10^{-6}$, varying $\epsilon_{\rm pr}$=$\epsilon$ as indicated. (d) Order parameter of the CE phase $|O_{\rm CE}|$=$\sqrt{S(\pi,0)}$ vs. $T/t$ at $J_{\rm AF}= 0.2$ using the DIAG method and TPEM with $M$=$20$, $\epsilon_{\rm tr}$=$10^{-6}$, varying $\epsilon_{\rm pr}$=$\epsilon$ as indicated. All calculations were done on a $12\times12$ lattice, using $1000$ Monte Carlo steps for thermalization and $1000$ for measurements.[]{data-label="Figure8"}](\mypath{figurefakenumber1.eps}){width="9cm"}
### Dependence of Results on Lattice Sizes
Figure \[Figure9\] illustrates the dependence of results on lattice sizes. The systematic behavior is similar to that observed in the case of the one-band model at the same density $\langle n \rangle$=0.5. In (a) results for the magnetization vs. temperature indicate the existence of a FM state at low temperatures, as well as small finite-size effects when the 20$\times$20 and 32$\times$32 clusters are compared. Even less pronounced size effects are found in the CE regime, increasing $J_{\rm AF}$ as shown in (b). There is no indication that the TPEM deteriorates with increasing lattice size, providing hope that this method will be strong enough to handle the introduction of quenched disorder in future studies.
![(color online). Lattice size dependence of the square root of the structure factors at the momenta characteristic of (a) a FM state, ${\mathbf k}$=$(0,0)$ and $J_{\rm AF}$=$0.0$, and (b) a CE phase, ${\mathbf k}$=$(\pi,0)$ and $J_{\rm AF}$=$0.2$. Results were obtained with the TPEM with $M$=$20$, $\epsilon_{\rm pr}$=$10^{-5}$, $\epsilon_{\rm tr}$=$10^{-6}$. In addition, for (a) the DIAG method was also used on a $8\times8$ lattice as indicated. In the simulation, $1000$ MC steps were used for thermalizations and $1000$ steps for measurements. []{data-label="Figure9"}](\mypath{figurefakenumber2.eps}){width="9cm"}
Phase Diagram
-------------
### Results without Phonons
To further test the TPEM, the phase diagram of the two-band model at $\lambda$=0 and $\langle n \rangle$=0.5 was obtained. Also at very low temperature, the energy was found as a function of $J_{\rm AF}$. The results are in Fig. \[Figure10\]. Part (a) shows an excellent agreement among the several lattice sizes studied here. The abrupt change in the slope of the curve near $J_{\rm AF}$=0.15 indicates a first-order transition, similar to that found in the one-band case and in previous literature. In (b), the full phase diagram is obtained. There are clear qualitative similarities with the results presented before by Aliaga [*et al.*]{} using a 4$\times$4 cluster.[@aliaga] In particular, the curve defining the CE phase at low temperature has a positive slope rather than being vertical as in other cases. The fact that the TPEM gives results in excellent agreement with DIAG but on substantially larger systems is very encouraging and establishes this technique as a key method for a frontal attack to the CMR problem using realistic models and quenched disorder.
![(color online). (a) Total Energy vs. $J_{\rm AF}$ at low temperature ($T$=$0.01t$) for different lattices as indicated. For the $12\times12$ lattice the DIAG method was used, while for the others the TPEM was employed with $M$=$20$, $\epsilon_{\rm pr}$=$10^{-5}$, $\epsilon_{\rm tr}$=$10^{-6}$. In the simulation, $1000$ MC steps were used for thermalizations and $1000$ steps for measurements. (b) Phase diagram of Hamiltonian Eq. (\[eq:hamtwobands\]) varying temperature and $J_{\rm AF}$ ($\lambda$=0). The three magnetically different regions: FM, PM, and CE are indicated. The phase diagram was calculated for different lattices as shown. For $12\times12$ the DIAG method was used and for the others the TPEM with $M$=$20$, $\epsilon_{\rm pr}$=$10^{-5}$, $\epsilon_{\rm tr}$=$10^{-6}$. The critical temperatures were obtained from the calculation of structure factors, as shown in Fig. \[Figure9\].[]{data-label="Figure10"}](\mypath{figurefakenumber3.eps}){width="9cm"}
For completeness, CPU times for the case of the two-band model are provided in Table \[Table2\]. The CPU time per site does not change dramatically with $N$, close to the expected theoretical estimation for the TPEM. Clearly, lattices well beyond 32$\times$32 can be handled with this technique.
$L\times L$ CPU Time(s) CPU Time/N
---------------- ------------- ------------
$12 \times 12$ 124 $0.86$
$20 \times 20$ 642 $1.61$
$24 \times 24$ 992 $1.72$
$32 \times 32$ 2451 $2.39$
: \[Table2\] CPU Times of the TPEM in seconds per 5 Monte Carlo steps for Hamiltonian Eq. (\[eq:hamtwobands\]) with $J_{\rm AF}$=$0$ and inverse temperature $\beta=50$, and the lattices shown. The third column is the ratio of CPU time per lattice site. The computer used was an AMD Opteron(tm) 244, 1.8GHz with 1MB cache. The TPEM parameters were $M$=20, $\epsilon_{\rm pr}$=$10^{-5}$, and $\epsilon_{\rm tr}$=$10^{-6}$.
### Influence of Phonons
As explained in the introduction, the general scenario proposed for manganites does not depend on particular details of the competing phases, but in the competition itself.[@review] Thus, to the extent that the FM metallic and CE insulating phases are found in competition, the value of electron-phonon coupling $\lambda$ is not of crucial relevance. We believe that $\lambda$ is likely small in practice, since recent experiments are not finding evidence of a robust charge checkerboard (see, for example, Ref. ) and, in addition, theoretical studies have shown that a large $\lambda$ renders the FM state also insulating.[@aliaga] Nevertheless, to confirm that the results are not severely affected by switching on $\lambda$, in Fig. \[Figure11\] the phase diagram for $\lambda$=0.5 is presented on a lattice substantially larger than used in previous investigations.[@aliaga] Comparing Figs. \[Figure10\] and \[Figure11\], clearly both cases lead to very similar phase diagrams. Since removing the phononic degree of freedom speeds up the simulations, these results suggest that the future effort in this context could focus in the $\lambda$=0 case and still expect to find realistic conclusions.
![Phase diagram of Hamiltonian Eq. (\[eq:hamtwobands\]) varying temperature and $J_{\rm AF}$ for $\lambda$=$0.5$ and the harmonic parameters discussed in the text, i.e., with the inclusion of phonons. The critical temperatures were obtained from the calculation of structure factors on a $12\times12$ lattice with the DIAG method. Note the similarity of this phase diagram with the result obtained at $\lambda$= 0.0, suggesting that to simulate the competition between FM metallic and CE insulating regimes the presence of a robust electron-phonon coupling is not necessary.[]{data-label="Figure11"}](\mypath{figurefakenumber5.eps}){width="9cm"}
Density of States
-----------------
As in the case of the one-band model, we also tested whether the TPEM technique can reproduce the DOS of the two-band model in the regime where the system is insulating (CE phase). The results are in Fig. \[FigureDOS\]. Part (a) shows a comparison between DIAG and TPEM on a 12$\times$12 cluster. The agreement is excellent for the case of $M$=100 (shown), and fairly acceptable for smaller values of $M$. For larger lattices that can only be studied with TPEM (part (b)), the results are also in good agreement with expectations. Then, no problems have been detected in calculating the DOS using the TPEM technique in the regime where the model is in an insulating state.
![ (color online) (a) DOS of a perfect CE phase (single spin configuration) obtained on a 12$\times$12 lattice calculated with the DIAG method (solid black line) and with the TPEM (dashed red line) using $M$=$100$. The chemical potential lies in the left gap (arrows) indicating that the system is an insulator. (b) DOS of the system with $J_{\rm AF}$=$0.2$ (CE-phase ground state) on a 20$\times$20 lattice calculated with the TPEM using $M$=$100$, as described in the text. The location of the chemical potential is indicated by the vertical dashed line. []{data-label="FigureDOS"}](\mypath{figurefnumberadd8.eps}){width="9cm"}
Conductances: Comparison TPEM vs. DIAG, and Results for Increasing Lattice Sizes
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As in the case of the one-band model, the final test for the two-band case is the calculation of the conductance. Results are shown in Fig. \[Figure12\]. Even using clusters much larger than can be handled with DIAG, the behavior of the conductance is properly captured by TPEM. There are no hidden subtleties involved in this estimation of transport properties, opening the way toward many future applications of this technique.
![(color online) [*Upper Panel:* ]{} Conductance vs. temperature for $J_{\rm {AF}}=0.0$ (ferromagnet) and $J_{\rm {AF}}=0.2$ (CE phase), different lattices sizes and algorithms, as indicated. [*Lower Panel:*]{} Logarithm of the resistivity vs. temperature for the same parameters as before.[]{data-label="Figure12"}](\mypath{figurefakenumber6.eps}){width="9cm"}
Influence of Magnetic Fields in the Clean Limit
-----------------------------------------------
Similarly as for the case of the one-band model, we have also studied the influence of a magnetic field in the case of the two-band model. The value of $J_{\rm AF}$ was chosen to be 0.175, namely on the CE side but close to the first-order metal-insulator transition. As shown in Fig. \[FigureMAG\], the application of a relatively small field – in the natural units of the problem – leads to a drastic change in the resistivity at low temperatures. In this regime, the insulator is transformed into a metal (negative magnetoresistance). As remarked before, this is in agreement with previous studies carried out by Aliaga [*et al.*]{} [@aliaga], showing that the “low temperature” large magnetoresistance materials[@low-T-CMR] can be understood by double-exchange models in the clean limit. However, as in the case of one-band models studied before, the finite-temperature CMR effect in Mn-oxides cannot be understood with clean limit models, as shown in Fig. \[FigureMAG\], since the zero magnetic-field resistivity does not have the large peak characteristic of manganites. Future work with TPEM will address this issue including quenched disorder.
![(color online) Effect of a small magnetic field on the resistivity of the CE phase, for couplings in the vicinity of the first order transition metal-insulator. Shown is the resistivity vs. $T$ at $J_{\rm AF}$=$0.175$, $\lambda$=0, on a 20$\times$20 lattice for $B$=$0.1t$ and without field ($B$=$0$) for comparison. Note the large change in the resistivity at low temperature, compatible with a colossal magnetoresistance. The TPEM was used with $M$=$30$, $\epsilon_{\rm pr}$=$10^{-5}$ and $\epsilon_{\rm tr}$=$10^{-6}$. 1,000 Monte Carlo steps were done for thermalization and an additional 100 more for measurements. []{data-label="FigureMAG"}](\mypath{figurefnumberadd9.eps}){width="9cm"}
CONCLUSIONS
===========
In this paper, it has been shown that the computationally intensive exact-diagonalization algorithm for the study of CMR-manganite models can in practice be replaced by the novel Truncated Polynomial Expansion Method (TPEM) without any substantial loss in accuracy. The DIAG, although being exact, does not permit simulations of large clusters owing to the fact that the computational cost grows like the $4^{th}$ power of the system size, $N$. On the other hand, the newly developed TPEM algorithm reduces the computational complexity to $\mathcal{O}(N)$, thereby allowing for simulations of larger systems.
For both the one- and two-band double exchange model with interacting Mn spins, we have compared systematically the results calculated with both the diagonalization method and the TPEM algorithm. As the spin-spin coupling $J_{\rm AF}$ is varied, the one-band model reveals a low-temperature first-order phase transition between conducting FM and insulating Flux states in the vicinity of $J_{\rm AF}$=$0.045$. For two bands, a similar first-order transition separates FM metallic and CE insulating phases. Our calculations presented here included a systematic study of the performance of the TPEM algorithm varying its parameters $M$, $\epsilon_{\rm pr}$ and $\epsilon_{\rm tr}$, already defined in the introductory sections. It was shown that the results of the TPEM algorithm converge to those of the DIAG algorithm with increasing $M$, and for $M \ge 30$ TPEM results are sufficiently accurate to obtain phase diagrams with small error bars. This number appears to be fairly stable under changes in the model, couplings, and for different phases, and lattice sizes. Also nothing indicates that working at density different from 0.5, the focus of the current effort, will spoil the TPEM performance. \[However, it is advisable to be particularly cautious near critical temperatures, where in some cases we found the need to increase $M$ to 40\]. Similar systematic results were presented for the $\epsilon$’s. Overall, the general process of fixing TPEM parameters on small systems by comparing with DIAG and then using the same parameters on larger lattices appears reliable, and this method will be applied to other systems in the near future.
Parallelization of the TPEM algorithm makes it possible to study large clusters. Taking advantage of this possibility, in the absence of quenched disorder, we have made calculations on lattices of up to $40\times40$ sites even for the case of a finite Hund coupling in the one-band model. But previous calculations in the limit of $J_{\rm H}$=$\infty$ used up to 8000 sites,[@heisenberg] thus even accounting for the factors of 2 involved in comparing finite and infinite $J_{\rm H}$ there is still room for further improvement. We have also checked that calculations of conductances, crucial to predict transport properties, also can be carried out smoothly with the TPEM, and in addition the size effects are in general small. While obviously the increase in lattice size allows for more accurate determinations of critical temperatures, even more importantly these large lattices will be crucial for the next big step in large-scale manganite simulations which is the introduction of quenched disorder. This disorder is expected to lead to a percolative-like picture that causes the CMR phenomenon. Any percolative effect requires large systems, and having access to clusters substantially larger than those studied with DIAG is a key necessary condition to unveil the conceptual reason behind the CMR phenomenon.
Interesting physical results were also here reported. This includes the one-band phase diagram with a metal-insulator transition. But the main result in this context is the presence of an enormous magnetoresistance effect at low temperatures, even in the clean limit studied here. This effect was already anticipated by Aliaga [*et al.*]{} in their pioneering work on this subject on small systems.[@aliaga] The survival of the effect on the large clusters reachable by the TPEM, as described in this manuscript, shows that [*some forms of CMR found experimentally can already be accurately reproduced using realistic models*]{}, providing further support that a theoretical solution of the CMR puzzle is within reach. But for the most common form of CMR in Mn-oxides at temperatures close to the Curie temperature, studies with quenched disorder will likely be needed.
Summarizing, we here reported a successful implementation of the TPEM for the study of double-exchange-like models for manganites. The technique has a CPU time that grows linearly with the number of sites, and in addition it is parallelizable. Thus, the main result is that a novel technique has been identified and tested that can led to a frontal attack of the most interesting problem in manganites: the analysis of large magnetoresistance effects in the presence of quenched disorder, when phases compete via a first-order transition in the clean limit. This “holy grail” of simulations will be the focus of our effort in the near future. It will demand at least an order of magnitude more effort than in the present manuscript, but this can be alleviated by increasing the number of nodes available for the simulations and we are already working on this aspect. The large-scale computational facilities at Oak Ridge National Laboratory will play a key role in reaching this ambitious goal.
ACKNOWLEDGMENTS
===============
Most of the computational work in this effort was performed at the supercomputing facilities of the Center for Computational Science at the Oak Ridge National Laboratory (ORNL), managed by UT-Battelle, LLC, for the U.S. Department of Energy under Contract DE-AC05-000R22725. This work is supported by the LDRD program at ORNL and by the NSF grant DMR-0443144. We also acknowledge the help of J. A. Vergés in the study of conductances. This research used the SPF computer program and software toolkit developed at ORNL (http://mri-fre.ornl.gov/spf).
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J. A. Verges, Comput. Phys. Commun. [**118**]{}, 71 (1999), and references therein.
G. Subias, J. Garcia, J. Blasco, M. Grazia Proietti, H. Renevier, and M.C. Sanchez, Phys. Rev. Lett. [**93**]{}, 156408 (2004), and references therein.
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|
---
abstract: 'In this paper, we derive a more precise version of the Strong Pair Correlation Conjecture on the zeros of the Riemann zeta function under Riemann Hypothesis and Twin Prime Conjecture.'
author:
- Tsz Ho Chan
title: More precise Pair Correlation Conjecture
---
Introduction
============
In the early 1970s, H. Montgomery studied the distribution of the difference $\gamma - {\gamma}'$ between the imaginary parts of the non-trivial zeros of the Riemann zeta function. Let $$\label{basic}
F(x,T) = \mathop{\sum_{0 \leq \gamma \leq T}}_{0 \leq {\gamma}' \leq T}
x^{i(\gamma - {\gamma}')} w(\gamma - {\gamma}')
\mbox{ and }
w(u) = {4 \over 4 + u^2}.$$ Assuming Riemann Hypothesis, he proved in \[\[M\]\] that, as $T \rightarrow
\infty$, $$F(x,T) \sim {T \over 2 \pi} \log{x} + {T \over 2 \pi x^2} (\log{T})^2$$ for $1 \leq x \leq T$ (actually he only proved for $1 \leq x \leq o(T)$ and the full range was done by Goldston \[\[G\]\]). He conjectured that $$F(x,T) \sim {T \over 2 \pi} \log{T}$$ for $T \leq x$ which is known as the Strong Pair Correlation Conjecture. From this, one has the (Weak) Pair Correlation Conjecture: $$\mathop{\sum_{0 < \gamma, \gamma' \leq T}}_{0 < \gamma - \gamma' \leq {2 \pi
\alpha \over \log{T}}} 1 \sim {T\log{T} \over 2 \pi} \int_{0}^{\alpha} 1 -
\Bigl({\sin{\pi u} \over \pi u}\Bigr)^2 du.$$
In \[\[C1\]\], the author proved that, assuming Riemann Hypothesis, for any $\epsilon > 0$, $$\label{0.1}
\begin{split}
F(x,T) =& {1 \over 2\pi}T\log{x} + {1 \over x^2}\Bigl[{T \over 2\pi}(\log{T
\over 2\pi})^2 - 2{T \over 2\pi}\log{T \over 2\pi} \Bigr] \\
&+ O(x\log{x}) + O\Bigl({T \over x^{1/2-\epsilon}}\Bigr)
\end{split}$$ for $1 \leq x \leq {T \over \log{T}}$. This gives a more precise formula for $F(x,T)$ in the range $1 \leq x \leq {T \over \log{T}}$. Meanwhile, in \[\[C2\]\], the author derived a more precise Strong Pair Correlation Conjecture: For every fixed $\epsilon > 0$ and $A \geq 1+\epsilon$, $$\label{0.2}
F(x,T) = {T \over 2 \pi}\log{T \over 2\pi}- {T \over 2\pi}+ O(T^{1-\epsilon_1})$$ holds uniformly for $T^{1+\epsilon} \leq x \leq T^A$ with some $\epsilon_1 > 0$. It would be interesting to know how $F(x,T)$ changes from (\[0.1\]) to (\[0.2\]) when $x$ is close to $T$. We have the following
\[theorem1\] Assume Riemann Hypothesis and Twin Prime Conjecture. For any small $\epsilon >
0$ and any integer $M > 2$, $$\begin{aligned}
F(x,T) &=& {T \over 2\pi}\log{x} - {4x \over 3\pi}\int_{0}^{T/x} {\sin{v} \over
v} dv + {x^2 \over \pi T} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h) \over h^2}
\Bigr) \Bigl(1 - \cos{T \over x}\Bigr) \\
&-& {x \over 2\pi}\int_{1}^{\infty} {\sin{Ty \over x} \over y^2} dy + \Bigl({B
\over 2} + {11 \over 12}\Bigr) {x \over \pi} \int_{1}^{\infty} {\sin{Ty \over x}
\over y^4} dy - {4T \over \pi} \int_{1}^{\infty} {f(y) \over y^2} {\sin{{T \over
x}y} \over {T \over x}y} dy \\
&+& {2T \over \pi} \int_{1}^{\infty} {\int_{1}^{y} f(u) du \over y^3} {\sin{{T
\over x}y} \over {T \over x}y} dy + {6T \over \pi} \int_{1}^{\infty} y \int_{y}^
{\infty} {f(u) \over u^4} du {\sin{{T \over x}y} \over {T \over x}y} dy \\
&+& O\Bigl({x^{1+6\epsilon} \over T}\Bigr) + O(x^{{1/2}+7\epsilon}) + O
\Bigl({T \over (\log{T})^{M-2}}\Bigr).\end{aligned}$$ for ${T \over (\log{T})^M} \leq x \leq T^{2-\epsilon}$. $B = -C_0 - \log{2\pi}$ and $C_0$ is Euler’s constant $0.5772156649..$. $H^{*}$, ${\mathfrak S}(h)$ and $f(u)$ are defined as in the next section. The implicit constants may depend on $\epsilon$ and $M$.
As corollaries of Theorem \[theorem1\], we have
\[corollary1\] Assume Riemann Hypothesis and Twin Prime Conjecture. For any integer $M >2$, $$F(x,T) = {T \over 2\pi}\log{x} + O(x) + O_{M}\Bigl({T \over (\log{T}
)^{M-2}}\Bigr)$$ for ${T \over (\log{T})^M} \leq x \leq T$.
\[corollary2\] Assume Riemann Hypothesis and Twin Prime Conjecture. For any small $\epsilon >
0$ and any integer $M > 2$, $$F(x,T) = {T \over 2\pi} \log{T \over 2\pi} - {T \over 2 \pi} + O_{\epsilon}
\Bigl(T \bigl({T \over x}\bigr)^{{1/2}-\epsilon}\Bigr) + O_{\epsilon, M}
\Bigl({T \over (\log{T})^{M-2}} \Bigr)$$ for $T \leq x \leq T^{2-29\epsilon}$.
Preparations
============
We mentioned Twin Prime Conjecture in the previous section. The form needed is the following: For any $\epsilon > 0$, $$\sum_{n=1}^{N} \Lambda(n) \Lambda(n+d) = {\mathfrak S}(d) N + O(N^{1/2+
\epsilon})$$ uniformly in $|d| \leq N$. $\Lambda(n)$ is the von Mangoldt lambda function. ${\mathfrak S}(d) = 2\prod_{p>2}\bigl(1-{1 \over (p-1)^2}\bigr) \prod_{p|d, p>2}
{p-1 \over p-2}$ if $d$ is even, and ${\mathfrak S}(d) = 0$ if $d$ is odd. We also need a lemma concerning ${\mathfrak S}(d)$.
\[lemma2.1\] For any $\epsilon > 0$, $$\sum_{k=1}^{h} (h-k) {\mathfrak S}(k) = {1 \over 2} h^2 - {1 \over 2} h\log{h}
+ Ah + O(h^{{1/2}+\epsilon})$$ where $A = {1 \over 2} (1 - C_0 -\log{2\pi})$ and $C_0$ is Euler’s constant.
Proof: This is a theorem in Montgomery and Soundararajan \[\[MS\]\]. Borrowing from \[\[GGOS\]\], $$S_{\alpha}(y) := \sum_{h \leq y} {\mathfrak S}(h) h^{\alpha} - {y^{\alpha +1}
\over \alpha +1} \mbox{ for } \alpha \geq 0,$$ and $$T_{\alpha}(y) := \sum_{h>y} {{\mathfrak S}(h) \over h^{\alpha}} \mbox{ for }
\alpha>1.$$ Then from \[\[FG\]\], $$\label{1}
S_0(y) = -{1 \over 2}\log{y} + O((\log{y})^{2/3}).$$ Suppose $S_0(y) = -{1 \over 2}\log{y} + \epsilon(y)$. By partial summation, $$\label{2}
S_{\alpha}(y) = -{y^{\alpha} \over 2\alpha} + \epsilon(y) y^{\alpha} - \alpha
\int_{1}^{y} \epsilon(u) u^{\alpha -1} du + \Bigl({1 \over 2\alpha} + {\alpha
\over \alpha+1}\Bigr),$$ and $$\label{3}
T_{\alpha}(y) = {1 \over (\alpha -1)y^{\alpha -1}} - {\epsilon(y) \over
y^{\alpha}} - {1 \over 2\alpha y^{\alpha}} + \alpha \int_{y}^{\infty}
{\epsilon(u) \over u^{\alpha +1}} du.$$
\[lemma2.2\] For any $\epsilon > 0$, $$\int_{1}^{y} \epsilon(u) du = {B \over 2}y + O(y^{{1/2}+\epsilon})$$ where $B = -C_0 - \log{2\pi}$ as in the previous section.
Proof: By Lemma \[lemma2.1\], $$\begin{aligned}
\int_{1}^{y} \epsilon(u) du &=& \int_{1}^{y} \Bigl(\sum_{h \leq u} {\mathfrak S}
(h) -u+{1 \over 2} \log{u} \Bigr) du \\
&=&\sum_{h \leq y} (y-h) {\mathfrak S}(h) - {1 \over 2}y^2 + {1 \over 2}y\log{y}
- {1 \over 2}y + 1 \\
&=&Ay - {1 \over 2}y + O(y^{{1/2}+\epsilon}) \\
&=& {B \over 2} y + O(y^{{1/2}+\epsilon}).\end{aligned}$$ Now, let us define $$f(y) := \int_{1}^{y} \epsilon(u) - {B \over 2} du.$$ By integration by parts and Lemma \[lemma2.2\], one has $$\label{4}
\int_{1}^{y} \epsilon(u)u du = {B \over 4}y^2 + yf(y) - \int_{1}^{y} f(u) du -
{B \over 4} = {B \over 4}y^2 + O(y^{{3/2}+\epsilon}),$$ and $$\label{5}
\int_{y}^{\infty} {\epsilon(u) \over u^3} du = {B \over 4 y^2} - {f(y) \over
y^3} + 3 \int_{y}^{\infty} {f(u) \over u^4} du = {B \over 4 y^2} + O(y^{-{5/2}+
\epsilon}).$$ Next, we are going to define a smooth weight $\Psi_U (t)$. Fix a small positive real number $\epsilon$ and let $K$ be a large integer depending on $\epsilon$. Let $M$ be an integer greater than $2$ and $U = (\log{T})^M$. We want $\Psi_U
(t)$ to have support in $[-1/U, 1+1/U]$, $0 \leq \Psi_U (t) \leq 1$, $\Psi_U (t)
= 1$ for $1/U \leq t \leq 1-1/U$, and $\Psi_{U}^{(j)} (t) \ll U^j$ for $j = 1,2,
...,K$.
Let $\Delta = 1 / (2^K U)$. We define a sequence of functions as follow (which is Vinogradov’s construction) : $$\begin{aligned}
\chi_0 (t) &=& \left\{ \begin{array}{ll}
1, & \mbox{if $0 \leq t \leq 1$},\\
0, & \mbox{else}.\end{array} \right. \\
\chi_i (t) &=& {1 \over 2 \Delta} \int_{-\Delta}^{\Delta} \chi_{i-1} (t+x) dx
\mbox{ for } i=1,2,...,K+1.\end{aligned}$$
Clearly, $0 \leq \chi_i (t) \leq 1$ for $1 \leq i \leq K+1$. One can easily check by induction that $\chi_i (t) =1$ for $2^{i-1} \Delta \leq t \leq 1 -
2^{i-1} \Delta$, and $\chi_i (t)=0$ for $t < -2^{i-1} \Delta$ or $t>1+2^{i-1}
\Delta$ for $i=1,2,...,K+1$.
\[lemma2.3\] $\chi_{i}^{(j)} (t)$ exist and are continuous, and $\chi_{i}^{(j)} (t) \leq
\Delta^{-j}$ for $0 \leq j \leq i-1$ and $2 \leq i \leq K+1$.
Proof: Induction on $i$. First note that $\chi_1(t)$ is continuous because $$\begin{aligned}
|\chi_1(t+\delta) - \chi_1(t)| &=& \Big| {1 \over 2\Delta} \int_{-\Delta}^
{\Delta} \chi_0(t+\delta+x) dx - {1 \over 2\Delta} \int_{-\Delta}^{\Delta}
\chi_0(t+x) dx \Big| \\
&=& \Big| {1 \over 2\Delta} \int_{-\Delta+\delta}^{\Delta+\delta} \chi_0(t+x)
dx - {1 \over 2\Delta} \int_{-\Delta}^{\Delta} \chi_0(t+x) dx \Big| \\
&=& \Big| {1 \over 2\Delta} \int_{\Delta}^{\Delta+\delta} \chi_0(t+x) dx -
{1 \over 2\Delta} \int_{-\Delta}^{-\Delta+\delta} \chi_0(t+x) dx \Big| \\
&\leq& {\delta \over \Delta}.\end{aligned}$$ Similarly, $$\begin{aligned}
{\chi_2(t+h) - \chi_2(t) \over h} &=& {1 \over h} \Bigl[ {1 \over 2\Delta}
\int_{\Delta}^{\Delta+h} \chi_1(t+x) dx - {1 \over 2\Delta} \int_{-\Delta}^
{-\Delta+h} \chi_1(t+x) dx \Bigr] \\
&=& {1 \over 2\Delta} [ \chi_1(t+\Delta+\xi_1) - \chi_1(t-\Delta+\xi_2) ]\end{aligned}$$ for some $0 \leq \xi_1, \xi_2 \leq h$ by mean-value theorem. So $\chi_{2}'(t)$ exists and equals to ${1 \over 2\Delta} [ \chi_1(t+\Delta) -
\chi_1(t-\Delta) ]$ which is continuous and $\leq {1 \over \Delta}$. Assume that $\chi_{i}^{(j)} (t)$ are continuous and satisfy $\chi_{i}^{(j)} \ll \Delta^{-j}$ for some $2 \leq i \leq K$ and all $0 \leq j
\leq i-1$. Now, for $0 \leq j \leq i-1$, $\chi_{i+1}^{(j)} (t) = {1 \over
2 \Delta} \int_{-\Delta}^{\Delta} \chi_{i}^{(j)} (t+x) dx \leq \Delta^{-j}$ by induction hypothesis. For $j=i$, $$\begin{aligned}
\chi_{i+1}^{(i)} (t) &=& \lim_{h \rightarrow 0} {\chi_{i+1}^{(i-1)} (t+h) -
\chi_{i+1}^{(i-1)} (t) \over h} \\
&=& \lim_{h \rightarrow 0} {1 \over h} \Bigl[ {1 \over 2\Delta} \int_{\Delta}^
{\Delta+h} \chi_{i}^{(i-1)} (t+x) dx - {1 \over 2\Delta} \int_{-\Delta}^
{-\Delta+h} \chi_{i}^{(i-1)} (t+x) dx \Bigr] \\
&=& {1 \over 2\Delta} [\chi_{i}^{(i-1)} (t+\Delta) - \chi_{i}^{(i-1)}
(t-\Delta)]\end{aligned}$$ which is continuous and $\leq \Delta^{-i}$ by induction hypothesis.
\[lemma2.4\] $\hat{\chi}_{0} (y) = e^{\pi i y} {\sin{\pi y} \over \pi y}$ and $\hat{\chi}_
{i+1} (y) = \hat{\chi}_{i} (y) {\sin{2 \pi \Delta y} \over 2 \pi \Delta y}$ for $0 \leq i \leq K$. Here $\hat{f}(y)$ denotes the inverse Fourier transform of $f(t)$, $\hat{f}(y) = \int_{-\infty}^{\infty} f(t) e(yt) dt$.
Note: We use inverse Fourier transform so that the notation matches with \[\[GG\]\] and \[\[GGOS\]\].
Proof: $\hat{\chi}_{0} (y) = \int_{0}^{1} e(yt) dt = {e^{2\pi i y} - 1 \over
2\pi i y} = e^{\pi i y} {\sin{\pi y} \over \pi y}$. $$\begin{aligned}
\hat{\chi}_{i+1} (y) &=& \int_{-\infty}^{\infty} \chi_{i+1} (t) e(yt) dt \\
&=& {1 \over 2\Delta} \int_{-\Delta}^{\Delta} \int_{-\infty}^{\infty} \chi_i
(t+x) e(yt) dt dx \\
&=& {1 \over 2\Delta} \int_{-\Delta}^{\Delta} \hat{\chi}_{i} (y) e(-yx) dx \\
&=& {\hat{\chi}_{i} (y) \over 2\Delta} {e(-y\Delta) - e(y\Delta) \over -2\pi i
y} = \hat{\chi}_{i} (y) {\sin{2\pi \Delta y} \over 2 \pi \Delta y}.\end{aligned}$$ Now we take $\Psi_{U} (t) = \chi_{K+1} (t)$, then $\Psi_U (t)$ has the required properties by the above discussion and Lemma \[lemma2.3\]. From Lemma \[lemma2.4\], we know that $\hat{\Psi}_{U} (y) = e^{\pi i y}
{\sin{\pi y} \over \pi y} ({\sin{2 \pi \Delta y} \over 2 \pi \Delta y})^{K+1}$. It follows that $$\label{452}
\begin{split}
Re \hat{\Psi}_{U}(y) &= {\sin{2\pi y} \over 2\pi y} \Bigl({\sin{2\pi \Delta y}
\over 2\pi \Delta y}\Bigr)^{K+1}, \\
\hat{\Psi}_{U} (y) &\ll y^{-K} \mbox{ for } y \gg T^{\epsilon}, \\
\mbox{ and } \hat{\Psi}_{U} (Ty) &\ll T^{-K \epsilon} \mbox{ for } y \gg
\tau^{-1} \mbox{ where } \tau = T^{1-\epsilon}.
\end{split}$$ These are similar to ($18$) and ($19$) in \[\[GG\]\]. Also, by Lemma \[lemma2.3\], it follows from the discussion in \[\[GG\]\] that $$\hat{\Psi}_{U} (y), \hat{\Psi}'_{U} (y) \ll \mbox{min}\Bigl(1, ({U \over 2
\pi y})^K \Bigr)$$ which is ($17$) in \[\[GG\]\]. Consequently, the results in \[\[GG\]\] are true with our choice of $\Psi_U (t)$. Moreover, if one follows their arguments carefully, one has their Corollaries $1$ $\&$ $2$ (except that the error term may need to be modified by a factor of $N^{\epsilon}$) and Theorem $3$ as long as $\tau = T^{1-\epsilon} \leq x$. We shall need the following lemmas concerning our weight function $\Psi_U (t)$. Here we assume $T \Delta \leq x$.
\[lemma2.5\] For any integer $n \geq 1$, $$\int_{1}^{\infty} {1 \over y^n} Re \hat{\Psi}_U \Bigl({Ty \over 2\pi x}\Bigr)
dy = {x \over T} \int_{1}^{\infty} {\sin{Ty \over x} \over y^{n+1}} dy +
O\Bigl(K \Delta \log{1 \over \Delta}\Bigr).$$
Proof: By a change of variable $v = {Ty \over x}$ and (\[452\]), the left hand side $$\begin{aligned}
&=& \Bigl({T \over x}\Bigr)^{n-1} \int_{T/x}^{\infty} {1 \over v^n} {\sin{v}
\over v} \Bigl({\sin{\Delta v} \over \Delta v}\Bigr)^{K+1} dv \\
&=& \Bigl({T \over x}\Bigr)^{n-1} \int_{T/x}^{1/\Delta} {\sin{v} \over v^{n+1}}
(1+O(K \Delta^2 v^2)) dv + O\Bigl(\Bigl({T \over x}\Bigr)^{n-1}
\int_{1/\Delta}^{\infty} {1 \over v^{n+1}}dv\Bigr) \\
&=& \Bigl({T \over x}\Bigr)^{n-1} \int_{T/x}^{1/\Delta} {\sin{v} \over v^{n+1}}
dv + O\Bigl(\Bigl({T \over x}\Bigr)^{n-1} K \Delta^2 \int_{T/x}^{1/\Delta} {1
\over v^{n-1}}dv\Bigr) + O\Bigl(\Bigl({T \over x}\Bigr)^{n-1} \Delta^n\Bigr) \\
&=& \Bigl({T \over x}\Bigr)^{n-1} \int_{T/x}^{\infty} {\sin{v} \over v^{n+1}}
dv + O\Bigl(K\Delta \log{1 \over \Delta}\Bigr) \\
&=& {x \over T} \int_{1}^{\infty} {\sin{Ty \over x} \over y^{n+1}} dy +
O\Bigl(K \Delta \log{1 \over \Delta}\Bigr)\end{aligned}$$ because $T\Delta \leq x$. Note that the error term comes from the case $n=2$. If $n \not= 2$, then we can replace the error term by $O(K \Delta)$.
\[lemma2.6\] If $F(y) \ll y^{-{3/2} + \epsilon}$ for $y \geq 1$, then $$\int_{1}^{\infty} F(y) Re \hat{\Psi}_U \Bigl({Ty \over 2 \pi x}\Bigr) dy =
\int_{1}^{\infty} F(y) {\sin{{T \over x}y} \over {T \over x}y} dy + O(K \Delta)
.$$
Proof: By a change of variables $v={Ty \over x}$ and (\[452\]), the left hand side $$\begin{aligned}
&=&{x \over T}\int_{T/x}^{\infty} F({x \over T}v) {\sin{v} \over v}
\Bigl({\sin{\Delta v} \over \Delta v}\Bigr)^{K+1} dv \\
&=&{x \over T}\int_{T/x}^{1/\Delta} F({x \over T}v) {\sin{v} \over v} \bigl(1 +
O(K \Delta^2 v^2)\bigr) dv + O\Bigl({x \over T \Delta^{K+1}} \int_{1/\Delta}^
{\infty} {|F({x \over T}v)| \over v^{K+2}} dv \Bigr) \\
&=&{x \over T}\int_{T/x}^{1/\Delta} F({x \over T}v) {\sin{v} \over v} dv +
O\Bigl(K \bigl({T \over x}\bigr)^{{1/2}-\epsilon} \Delta^{{3/2}-\epsilon}\Bigr)
\\
&=&{x \over T}\int_{T/x}^{\infty} F({x \over T}v) {\sin{v} \over v} dv +
O\Bigl({x \over T} \int_{1/\Delta}^{\infty} {|F({x \over T}v)| \over v}dv \Bigr)
+ O(K \Delta) \\
&=&\int_{1}^{\infty} F(y) {\sin{{T \over x}y} \over {T \over x}y} dy + O(K
\Delta).\end{aligned}$$ Finally, we need the following
\[lemma2.7\] Assume Riemann Hypothesis. For any $\epsilon > 0$, $$\sum_{n \leq x} \Lambda(n)^2 n = {1 \over 2}x^2\log{x} - {1\over 4} x^2 +
O({x^{3/2+\epsilon}})$$ $$\sum_{n >x} {\Lambda(n)^2 \over n^3} = {1 \over 2}{\log{x} \over x^2} + {1
\over 4}{1 \over x^2} + O({1 \over x^{5/2-\epsilon}})$$ where the implicit constants may depend on $\epsilon$.
Proof: By partial summation and the form of prime number theorem under Riemann Hypothesis.
Proof of main results
=====================
Throughout this section, we assume $\tau = T^{1-\epsilon} \leq {T \over
(\log{T})^M} \leq x$, $U = (\log{T})^M$ for $M>2$, $H^{*}=\tau^{-2} x^{2/(1-
\epsilon)}$, and $\Psi_{U}(t)$ is defined as in the previous section. Keep in mind the $\epsilon$ and $M$ dependency in the error terms.
Proof of Theorem \[theorem1\]: Our method is that of Goldston and Gonek \[\[GG\]\]. Let $s = \sigma + it$, $$A(s) := \sum_{n \leq x} {\Lambda(n) \over n^s} \mbox{ and } A^{*}(s) :=
\sum_{n>x} {\Lambda(n) \over n^s}.$$ Assume Riemann Hypothesis, it follows from Theorem $3.1$ of \[\[C1\]\] with slight modification that $$F(x,T) =$$ $${1 \over 2\pi} \int_{0}^{T} \Big| {1 \over x} \Bigl( A(-{1 \over 2}
+it) - \int_{1}^{x} u^{1/2-it} du \Bigr) + x \Bigl( A^{*}({3 \over 2}+it) -
\int_{x}^{\infty} u^{-3/2-it} du \Bigr) \Big|^2 dt$$ $$+ O((\log{T})^3).$$ Inserting $\Psi_{U} (t/T)$ into the integral and extending the range of integration to the whole real line, we can get $$\label{4.6.1}
F(x,T) = {1 \over 2 \pi x^2} I_1 (x, T) + {x^2 \over 2 \pi} I_2 (x, T) + O
\Bigl({T(\log{T})^2 \over U}\Bigr) + O \Bigl({x^{1+6\epsilon} \over T}\Bigr)$$ where $$I_1(x,T) = \int_{-\infty}^{\infty} \Psi_{U}\Bigl({t \over T}\Bigr) \Big| A(-{1
\over 2}+it) - \int_{1}^{x} u^{1/2-it} du \Big|^2 dt,$$ and $$I_2(x,T) = \int_{-\infty}^{\infty} \Psi_{U}\Bigl({t \over T}\Bigr) \Big| A^{*}
({3 \over 2}+it) - \int_{x}^{\infty} u^{-3/2-it} du \Big|^2 dt.$$ This is essentially by Lemma $1$ of \[\[GGOS\]\] with modification that $V=-{T
\over U}$ and $T-{T \over U}$, and $W = {2T \over U}$. Riemann Hypothesis is assumed here so that the contribution from the cross term is estimated via Theorem $3$ of \[\[GG\]\].
Now, we assume the Twin Prime Conjecture in the previous section. By Corollary $1$ of \[\[GG\]\] (see also the calculations at the end of \[\[GG\]\] and \[\[GGOS\]\]) and Lemma \[lemma2.7\], one has, $$\begin{aligned}
I_1(x,T) &=& \hat{\Psi}_{U}(0) T \sum_{n \leq x} \Lambda^2 (n) n \\
& &+ 4 \pi \Bigl({T \over 2 \pi}\Bigr)^3 \int_{T/2\pi x}^{\infty} \Bigl(
\sum_{h \leq 2 \pi x v /T} {\mathfrak S} (h) h^2 \Bigr) Re \hat{\Psi}_{U}(v)
{dv \over v^3} \\
& &- 4\pi \Bigl({T \over 2 \pi}\Bigr)^3 \int_{T/2\pi \tau x}^{\infty} \Bigl(
\int_{0}^{2\pi x v/T} u^2 du \Bigr) Re \hat{\Psi}_{U}(v) {dv \over v^3} \\
& &+ O\Bigl({x^{3+6\epsilon} \over T}\Bigr) + O(x^{{5/2}+7\epsilon}) \\
&=&{1 \over 2}Tx^2 \log{x} - {1 \over 4}Tx^2 \\
& &+4 \pi \Bigl({T \over 2 \pi}\Bigr)^3 \int_{T/2\pi x}^{\infty} \Bigl(
\sum_{h \leq 2 \pi x v /T} {\mathfrak S} (h) h^2 - \int_{0}^{2\pi x v/T} u^2 du
\Bigr) Re \hat{\Psi}_{U}(v) {dv \over v^3} \\
& &- {4\pi \over 3} x^3 \int_{T/2\pi\tau x}^{T/2\pi x} Re \hat{\Psi}_U (v) dv
+ O\Bigl({x^{3+6\epsilon} \over T}\Bigr) + O(x^{{5/2}+7\epsilon}) \\
&=&{1 \over 2}Tx^2 \log{x} - {1 \over 4}Tx^2 \\
& &+4 \pi \Bigl({T \over 2 \pi}\Bigr)^3 \int_{T/2\pi x}^{\infty} \Bigl(
\sum_{h \leq 2 \pi x v /T} {\mathfrak S} (h) h^2 - \int_{0}^{2\pi x v/T} u^2 du
\Bigr) Re \hat{\Psi}_{U}(v) {dv \over v^3} \\
& &-{2 \over 3} x^3 \int_{0}^{T/x} {\sin{v} \over v} dv + O\Bigl({K T x^2 \over
(\log{T})^M}\Bigr) + O\Bigl({x^{3+6\epsilon} \over T}\Bigr)
+ O(x^{{5/2}+7\epsilon}) \end{aligned}$$ because, from (\[452\]), $$\begin{aligned}
\int_{T/2\pi\tau x}^{T/2\pi x} Re \hat{\Psi}_U (v) dv &=& \int_{0}^{T/2\pi x}
{\sin{2\pi v} \over 2\pi v} \Bigl({\sin{2\pi \Delta v} \over 2\pi \Delta
v}\Bigr)^{K+1} dv + O\Bigl({T \over \tau x}\Bigr) \\
&=&{1 \over 2\pi} \int_{0}^{T/x} {\sin{u} \over u} (1 + O(K \Delta^2 u^2)) du
+ O\Bigl({T \over \tau x}\Bigr) \\
&=&{1 \over 2\pi} \int_{0}^{T/x} {\sin{u} \over u} du + O\Bigl({K \Delta^2 T^2
\over x^2}\Bigr) + O\Bigl({T \over \tau x}\Bigr).\end{aligned}$$
Similarly, by Corollary $2$ of \[\[GG\]\] and Lemma \[lemma2.7\], $$\begin{aligned}
I_2(x,T) &=& \hat{\Psi}_{U}(0) T \sum_{x<n} {\Lambda^2(n) \over n^3} \\
& &+{8 \pi^2 \over T} \int_{0}^{T/2 \pi x} \Bigl(\sum_{1 \leq h \leq H^{*}}
{{\mathfrak S}(h) \over h^2} \Bigr) Re \hat{\Psi}_{U} (v) v dv \\
& &+{8 \pi^2 \over T} \int_{T/2\pi x}^{T H^{*}/2 \pi x} \Bigl(\sum_{2\pi xv/T <
h \leq H^{*}} {{\mathfrak S}(h) \over h^2} \Bigr) Re \hat{\Psi}_{U} (v) v dv \\
& &-{8 \pi^2 \over T} \int_{0}^{T H^{*}/2\pi x} \Bigl(\int_{2\pi xv/T}^{H^{*}}
u^{-2} du\Bigr) Re \hat{\Psi}_{U} (v) v dv \\
& &+O(T^{-1} x^{-1+6\epsilon}) + O(x^{-{3/2} + 6\epsilon}) + O(T^{1-{\epsilon/2}
} x^{-2}) \\
&=&{T \log{x}\over 2x^2} + {1 \over 4} {T \over x^2} \\
& &+ {8 \pi^2 \over T} \int_{0}^{T H^{*}/2\pi x} \Bigl(\sum_{2\pi xv/T <
h \leq H^{*}} {{\mathfrak S}(h) \over h^2} - \int_{2\pi xv/T}^{H^{*}} {du \over
u^2} \Bigr) Re \hat{\Psi}_{U} (v) v dv \\
& & + O\Bigl({x^{-1+6\epsilon} \over T}\Bigr) + O(x^{-{3/2}+ 6\epsilon}).\end{aligned}$$ Therefore, by a change of variable $y={2\pi xv \over T}$ and putting back to (\[4.6.1\]), we have $$\begin{aligned}
F(x,T) &=& {T \over 2\pi}\log{x} + {T \over \pi}\int_{1}^{\infty} \Bigl(
\sum_{h \leq y} {\mathfrak S}(h)h^2 - {y^3 \over 3} \Bigr) Re \hat{\Psi}_{U}
\Bigl({Ty \over 2\pi x}\Bigr) {dy \over y^3} \\
& &+{T \over \pi}\int_{1}^{H^{*}} \Bigl(\sum_{y<h \leq H^{*}} {{\mathfrak S}(h)
\over h^2} - \int_{y}^{H^{*}} {du \over u^2} \Bigr) Re \hat{\Psi}_{U}
\Bigl({Ty \over 2\pi x}\Bigr) y dy \\
& & + {T \over \pi} \int_{0}^{1} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h)
\over h^2} - \int_{y}^{H^{*}} {du \over u^2} \Bigr) Re \hat{\Psi}_{U}
\Bigl({Ty \over 2\pi x}\Bigr) y dy \\
& &-{x \over 3\pi} \int_{0}^{T/x} {\sin{v} \over v} dv + O\Bigl({KT \over
(\log{T})^{M-2}}\Bigr) + O\Bigl({x^{1+6\epsilon} \over T}\Bigr) + O(x^{{1/2}+
7\epsilon})\end{aligned}$$ $$\label{4.6.main}
= {T \over 2\pi}\log{x} + {T \over \pi}I_1 + {T \over \pi}I_2 -{4x \over 3\pi}
\int_{0}^{T/x} {\sin{v} \over v} dv +$$ $${x^2 \over \pi T} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h) \over h^2}\Bigr)
\Bigl(1 - \cos{T \over x}\Bigr)+ O\Bigl({KT \over (\log{T})^{M-2}}\Bigr) +
O\Bigl({x^{1+6\epsilon} \over T}\Bigr) + O(x^{{1/2}+7\epsilon}),$$ where $I_1$ and $I_2$ are the first and second integral respectively. This is because $$\begin{aligned}
& &{T \over \pi} \int_{0}^{1} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h)
\over h^2} - \int_{y}^{H^{*}} {du \over u^2} \Bigr) Re \hat{\Psi}_{U}
\Bigl({Ty \over 2\pi x}\Bigr) y dy \\
&=&{4\pi x^2 \over T} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h) \over h^2}
\Bigr) \int_{0}^{T/2\pi x} Re \hat{\Psi}_U (v) vdv - {x \over \pi} \int_{0}^
{T/x} {\sin{u} \over u} du + O\Bigl({KT \over (\log{T})^M}\Bigr) \\
&=&{x^2 \over \pi T} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h) \over h^2}
\Bigr) \int_{0}^{T/x} \sin{u} \bigl(1+ O(K \Delta^2 u^2)\bigr) du \\
& &- {x \over \pi} \int_{0}^{T/x} {\sin{u} \over u} du + O\Bigl({KT \over
(\log{T})^M}\Bigr) \\
&=&{x^2 \over \pi T} \Bigl(\sum_{h \leq H^{*}} {{\mathfrak S}(h) \over h^2}
\Bigr) \Bigl(1 - \cos{T \over x}\Bigr) - {x \over \pi} \int_{0}^{T/x} {\sin{u}
\over u} du + O\Bigl({KT \over (\log{T})^M}\Bigr)\end{aligned}$$ by a similar calculation as before and $T\Delta \leq x$. With the notation of $S_{\alpha}(y)$ and $T_{\alpha}(y)$, $$\begin{aligned}
I_1 &=& \int_{1}^{\infty} S_2(y) Re \hat{\Psi}_{U} \Bigl({Ty \over 2\pi
x}\Bigr) {dy \over y^3} \\
&=&\int_{1}^{\infty} \Bigl[ {-1 \over 4y} + {\epsilon(y) \over y} - {2
\int_{1}^{y} \epsilon(u)u du \over y^3} \Bigr] Re \hat{\Psi}_{U} \Bigl(
{Ty \over 2\pi x}\Bigr) dy \\
& &- 2\int_{1}^{\infty} {f(y) \over y^2} Re \hat{\Psi}_{U} \Bigl({Ty \over 2\pi
x}\Bigr) dy + 2\int_{1}^{\infty} {\int_{1}^{y} f(u) du \over y^3} Re \hat{\Psi}_
{U} \Bigl({Ty \over 2\pi x}\Bigr) dy \\
& &+ \Bigl({B \over 2} + {11 \over 12}\Bigr) {x \over T} \int_{1}^{\infty}
{\sin{Ty \over x} \over y^4} dy + O(K \Delta) \\
&=&\int_{1}^{\infty} \Bigl[ {-1 \over 4y} + {\epsilon(y) \over y} - {2
\int_{1}^{y} \epsilon(u)u du \over y^3} \Bigr] Re \hat{\Psi}_{U} \Bigl(
{Ty \over 2\pi x}\Bigr) dy \\
& &- 2\int_{1}^{\infty} {f(y) \over y^2} {\sin{{T \over x}y} \over
{T \over x}y} dy + 2\int_{1}^{\infty} {\int_{1}^{y} f(u) du \over y^3} {\sin{{T
\over x}y} \over {T \over x}y} dy \\
& &+ \Bigl({B \over 2} + {11 \over 12}\Bigr) {x \over T} \int_{1}^{\infty}
{\sin{Ty \over x} \over y^4} dy + O(K \Delta)\end{aligned}$$ by (\[2\]), (\[4\]), Lemma \[lemma2.5\] and Lemma \[lemma2.6\]. As for $I_2$, note that by (\[1\]) and (\[3\]), $$\label{4.6.7}
T_2(z) = {1 \over z} + O\Bigl({(\log{z})^{2/3} \over z^2}\Bigr),$$ and $$\begin{aligned}
\sum_{y<h \leq H^{*}} {{\mathfrak S}(h) \over h^2} - \int_{y}^{H^{*}} {du \over
u^2}
&=&T_2(y)-T_2(H^{*}) -{1 \over y} -{1 \over H^{*}} \\
&=&T_2(y)-{1 \over y} +O\Bigl({(\log{H^{*}})^{2/3} \over (H^{*})^2} \Bigr).\end{aligned}$$ Therefore, $$\begin{aligned}
I_2 &=& \int_{1}^{H^{*}} \Bigl(T_2(y) - {1 \over y} + O\Bigl({(\log{H^{*}})^
{2/3} \over (H^{*})^2} \Bigr) \Bigr) Re \hat{\Psi}_{U} \Bigl({Ty \over 2\pi x}
\Bigr) y dy \\
&=&\int_{1}^{H^{*}}\Bigl(T_2(y) - {1 \over y}\Bigr) Re \hat{\Psi}_{U} \Bigl({Ty
\over 2\pi x}\Bigr) y dy \\
& &+ O\Bigl({(\log{H^{*}})^{2/3} x^2 \over (H^{*})^2 T^2} \int_{T/2 \pi x}^{T H
^{*} /2 \pi x} |\hat{\Psi}_{U}(v)| vdv \Bigr) \\
&=&\int_{1}^{\infty}\Bigl(T_2(y) - {1 \over y}\Bigr)Re \hat{\Psi}_{U} \Bigl({Ty
\over 2\pi x}\Bigr) y dy + O\Bigl({1 \over T^{\epsilon}}\Bigr)\end{aligned}$$ because of (\[4.6.7\]) and the formula of $\hat{\Psi}_{U}(y)$ in the previous section that the integral $\int_{H^{*}}^{\infty} \ll {x (\log{H^{*}})^{2/3}
\over T H^{*}} \ll {1 \over T^{\epsilon}}$ by the definition of $H^{*}$ (similar estimation for the error term). Applying (\[3\]), (\[5\]) and Lemma \[lemma2.6\], $$\begin{aligned}
I_2 &=& \int_{1}^{\infty} \Bigl[{-1 \over 4y} - {\epsilon(y) \over y} + 2y
\int_{y}^{\infty} {\epsilon(u) \over u^3} du \Bigr] Re \hat{\Psi}_{U} \Bigl({Ty
\over 2\pi x}\Bigr) dy + O\Bigl({1 \over T^{\epsilon}}\Bigr) \\
&=&\int_{1}^{\infty} \Bigl[{-1 \over 4y} - {\epsilon(y) \over y} + {B \over 2y}
\Bigr] Re \hat{\Psi}_{U} \Bigl({Ty \over 2\pi x}\Bigr) dy -2\int_{1}^{\infty}
{f(y) \over y^2} Re \hat{\Psi}_{U} \Bigl({Ty \over 2\pi x}\Bigr) dy \\
& & + 6\int_{1}^{\infty} y \int_{y}^{\infty} {f(u) \over u^4} du Re \hat
{\Psi}_{U} \Bigl({Ty \over 2\pi x}\Bigr) dy + O\Bigl({1 \over T^{\epsilon}}
\Bigr) \\
&=&\int_{1}^{\infty} \Bigl[{-1 \over 4y} - {\epsilon(y) \over y} + {B \over 2y}
\Bigr] Re \hat{\Psi}_{U} \Bigl({Ty \over 2\pi x}\Bigr) dy \\
& &-2\int_{1}^{\infty} {f(y) \over y^2} {\sin{{T \over x}y} \over {T \over x}y}
dy + 6\int_{1}^{\infty} y \int_{y}^{\infty} {f(u) \over u^4} du {\sin{{T \over
x}y} \over {T \over x}y} dy + O(K\Delta).\end{aligned}$$ Consequently, with miraculous cancellations, one has $$\begin{aligned}
I_1 + I_2 &=& -{x \over 2T} \int_{1}^{\infty} {\sin{Ty \over x} \over y^2} dy +
\Bigl({B \over 2} + {11 \over 12}\Bigr) {x \over T} \int_{1}^{\infty} {\sin{Ty
\over x} \over y^4} dy \\
& &-4\int_{1}^{\infty} {f(y) \over y^2} {\sin{{T \over x}y} \over {T \over x}y}
dy +2\int_{1}^{\infty} {\int_{1}^{y} f(u) du \over y^3} {\sin{{T \over x}y}
\over {T \over x}y} dy \\
& &+6\int_{1}^{\infty} y \int_{y}^{\infty} {f(u) \over u^4} du {\sin{{T \over x}
y} \over {T \over x}y} dy + O(K \Delta)\end{aligned}$$ by Lemma \[lemma2.5\] again. Putting this back to (\[4.6.main\]), we have Theorem \[theorem1\].
Proof of Corollary \[corollary1\]: This follows from Theorem \[theorem1\] straighforwardly as $x \leq T$ and $f(u) \ll u^{1/2+\epsilon}$ by Lemma \[lemma2.2\]. Note that the error term is better than (\[0.1\]) for $x$ in the given range.
Before proving Corollary \[corollary2\], we need the following lemmas.
\[4lemma7.1\] $$\int_{1}^{\infty} {\sin{ax} \over x^{2n}} dx = {a^{2n-1} \over (2n-1)!}
\Bigl[\sum_{k=1}^{2n-1} {(2n-k-1)! \over a^{2n-k}} \sin{\bigl(a+(k-1){\pi \over
2}\bigr)} + (-1)^n ci(a) \Bigr]$$ where $ci(x) = -\int_{x}^{\infty} {\cos{t} \over t}dt = C_0 + \log{x} +
\int_{0}^{x} {\cos{t}-1 \over t} dt$ and $C_0$ is Euler’s constant.
Proof: This is formula $3.761(3)$ on P.430 of \[\[GR\]\] which can be proved by integration by parts inductively.
\[4lemma7.2\] If $F(y) \ll y^{-{3/2}+\epsilon}$ for $y \geq 1$, then for $T \leq x$, $$\int_{1}^{\infty} F(y) {\sin{{T \over x}y} \over {T \over x}y} dy =
\int_{1}^{\infty} F(y) dy + O\Bigl(\bigl({T \over x}\bigr)^{{1/2}-\epsilon}
\Bigr).$$
Proof: Since $T \leq x$, the left hand side $$\begin{aligned}
&=& \int_{1}^{x/T} F(y) \Bigl(1 + O\Bigl(\bigl({T \over x}\bigr)^2 y^2 \Bigr)
\Bigr) dy + O\Bigl(\int_{x/T}^{\infty} {|F(y)| \over {T \over x}y} dy \Bigr) \\
&=& \int_{1}^{x/T} F(y) dy + O\Bigl(\bigl({T \over x}\bigr)^{{1/2}-\epsilon}
\Bigr) \\
&=& \int_{1}^{\infty} F(y) dy + O\Bigl(\int_{x/T}^{\infty} |F(y)| dy \Bigr) +
O\Bigl(\bigl({T \over x}\bigr)^{{1/2}-\epsilon}\Bigr) \\
&=& \int_{1}^{\infty} F(y) dy + O\Bigl(\bigl({T \over x}\bigr)^{{1/2}-\epsilon}
\Bigr).\end{aligned}$$
\[4lemma7.3\] $$\sum_{h \leq H^{*}} {{\mathfrak S}(h) \over h^2} = {7 \over 4}+ {B \over 2}+
6\int_{1}^{\infty} {f(u) \over u^4} du + O\Bigl({1 \over H^{*}}\Bigr)$$ where $B = -C_0-\log{2\pi}$ and $C_0$ is Euler’s constant again.
Proof: First, from (\[1\]), $$\begin{aligned}
\sum_{h > H^{*}} {{\mathfrak S}(h) \over h^2} &=& \int_{H^{*}}^{\infty} {1 \over
u^2} d(S_0(u) + u) \\
&\ll& {1 \over H^{*}} + \int_{H^{*}}^{\infty} {\log{u} \over u^3} du \ll {1
\over H^{*}}\end{aligned}$$ which accounts for the error term. It remains to see that $$\begin{aligned}
\sum_{h=1}^{\infty} {{\mathfrak S}(h) \over h^2} &=& \int_{1}^{\infty} {1 \over
u^2} d(S_0(u)+u) \\
&=&2 \int_{1}^{\infty} {S_0(u)+u \over u^3} du \\
&=&2 \int_{1}^{\infty} {u - {1 \over 2}\log{u} + \epsilon(u) \over u^3} du \\
&=&2 - {1 \over 4} + 2 \int_{1}^{\infty} {\epsilon(u) \over u^3} du \\
&=&{7 \over 4} + {B \over 2} + 2 \int_{1}^{\infty} {\epsilon(u) - {B \over 2}
\over u^3} du \\
&=&{7 \over 4} + {B \over 2} + 2 \int_{1}^{\infty} {1 \over u^3} df(u) \\
&=&{7 \over 4} + {B \over 2} + 6\int_{1}^{\infty} {f(u) \over u^4} du\end{aligned}$$ by integration by parts and the definitions of $\epsilon(u)$ and $f(u)$.
Proof of Corollary \[corollary2\]: First observe that when $x$ is in the required range, the error terms in Theorem \[theorem1\] is $O_{\epsilon, M}
\Bigl({T \over (\log{T})^{M-2}}\Bigr)$. Rewrite Theorem \[theorem1\] as $$F(x,T)= {T \over 2\pi} \log{x} - T_1 + T_2 - T_3 + T_4 - T_5 + T_6 + T_7 +
\mbox{error}.$$ Then, by Lemma \[4lemma7.1\], Lemma \[4lemma7.2\] and Lemma \[4lemma7.3\], $$\begin{aligned}
T_1 &=& {4x \over 3\pi}\int_{0}^{T/x} 1+O(v^2) dv = {4T \over 3\pi} + O\Bigl({
T^3 \over x^2}\Bigr), \\
T_2 &=& {x^2 \over \pi T}\Bigl({7 \over 4} + {B \over 2} + 6\int_{1}^{\infty}
{f(u) \over u^4} du + O ({1 \over H^{*}})\Bigr) \Bigl[{1 \over 2}\bigl({T \over
x}\bigr)^2 + O\Bigl(\bigl({T \over x}\bigr)^4\Bigr) \Bigr] \\
&=&{T \over 2\pi} \Bigl({7 \over 4} + {B \over 2} + 6\int_{1}^{\infty}
{f(u) \over u^4} du \Bigr) + O(T^{1-2\epsilon}) + O\Bigl(T \bigl({T \over
x}\bigr)^2 \Bigr), \\
T_3 &=& {T \over 2\pi} \Bigl[{1 \over {T/x}}\sin{T \over x} - ci({T \over x})
\Bigr] \\
&=& -{T \over 2\pi}\log{T \over x} - {C_0 T \over 2\pi} + {T \over 2\pi} +
O\Bigl(T({T \over x})\Bigr), \\
T_4 &=& \Bigl({B \over 2}+{11 \over 12}\Bigr) {x \over 6 \pi} \Bigl({T \over x}
\Bigr)^3 \Bigl[2\bigl({x \over T}\bigr)^3 \sin{T \over x} + \bigl({x \over T}
\bigr)^2 \sin{({T \over x} + {\pi \over 2})} \\
& &+ \bigl({x \over T}\bigr)\sin{({T \over x} + \pi)} + ci({T \over x})
\Bigr] \\
&=& \Bigl({B \over 2}+{11 \over 12}\Bigr) {T \over 2\pi} + O\Bigl(T \bigl({T
\over x}\bigr)\Bigr),\end{aligned}$$ $$\begin{aligned}
T_5 &=& {4T \over \pi} \int_{1}^{\infty} {f(y) \over y^2} dy + O\Bigl( T
\bigl({T \over x}\bigr)^{{1/2}-\epsilon} \Bigr), \\
T_6 &=& {2T \over \pi} \int_{1}^{\infty} {\int_{1}^{y} f(u) du \over y^3} dy +
O\Bigl( T\bigl({T \over x}\bigr)^{{1/2}-\epsilon} \Bigr) \\
&=& {T \over \pi} \int_{1}^{\infty} {f(y) \over y^2} dy + O\Bigl( T\bigl({T
\over x}\bigr)^{{1/2}-\epsilon} \Bigr), \\
T_7 &=& {6T \over \pi} \int_{1}^{\infty} y \int_{y}^{\infty} {f(u) \over u^4}
du dy + O\Bigl( T\bigl({T \over x}\bigr)^{{1/2}-\epsilon} \Bigr) \\
&=&-{3T \over \pi} \int_{1}^{\infty} {f(u) \over u^4} du + {3T \over \pi}
\int_{1}^{\infty} {f(y) \over y^2} dy + O\Bigl( T\bigl({T \over x}\bigr)^{{1/2}
-\epsilon} \Bigr).\end{aligned}$$ Combining these, we get $$\begin{aligned}
F(x,T) &=& {T \over 2\pi}\log{T} + {T \over 2\pi} \Bigl[-{8 \over 3} + {7 \over
4} + {B \over 2} + C_0 -1 + {B \over 2} + {11 \over 12}\Bigr] \\
& & + O\Bigl( T\bigl({T \over x}\bigr)^{{1/2}-\epsilon} \Bigr) + O\Bigl({T \over
(\log{T})^{M-2}} \Bigr) \\
&=& {T \over 2\pi}\log{T} + {T \over 2\pi} [-1 -\log{2\pi}] + O\Bigl( T\bigl({T
\over x}\bigr)^{{1/2}-\epsilon} \Bigr) + O\Bigl({T \over (\log{T})^{M-2}} \Bigr)\end{aligned}$$ which gives the corollary.
Conclusion
==========
Based on (\[0.1\]), (\[0.2\]), Corollary \[corollary1\] and Corollary \[corollary2\], we propose the following more precise Strong Pair Correlation Conjecture: For any small $\epsilon > 0$ and any large $A > 1$, $$F(x,T) = \left\{ \begin{array}{ll}
{T \over 2\pi}\log{x} + {1 \over x^2}\Bigl[{T \over 2\pi}(\log{T
\over 2\pi})^2 - 2{T \over 2\pi}\log{T \over 2\pi} \Bigr] \\
+ O(x) + O({T \over x^{{1/2}-\epsilon}}), & \raisebox{1ex}{if $1 \leq x \leq T$,
}\\
\raisebox{-1ex}{${T \over 2 \pi}\log{T \over 2\pi} - {T \over 2\pi} + O(T({T
\over x})^{{1/2}-\epsilon})$,} & \raisebox{-1ex}{if $T \leq x \leq T^{1+
\epsilon}$,}\\
\raisebox{-1ex}{${T \over 2 \pi}\log{T \over 2\pi} - {T \over 2\pi} + O(T^{1-
\epsilon_1})$,} &
\raisebox{-1ex}{if $T^{1+\epsilon} \leq x \leq T^A$.}
\end{array} \right.$$ where $\epsilon_1 > 0$ may depend on $\epsilon$, and the implicit constants may depend on $\epsilon$ and $M$.
[99]{}
\[C1\] T.H. Chan, [*On a conjecture of Liu and Ye*]{}, submitted.
\[C2\] T.H. Chan, [*More precise pair correlation of zeros and primes in short intervals*]{}, in preparation.
\[FG\] J.B. Friedlander and D.A. Goldston, [*Some singular series averages and the distribution of Goldbach numbers in short intervals*]{}, Illinois J. Math. [**39**]{} (1995), 158-180.
\[G\] D.A. Goldston, [*Large Differences between Consecutive Prime Numbers*]{}, Thesis, U. of Calif., Berkeley, 1981.
\[GG\] D.A. Goldston and S.M. Gonek, [*Mean value theorems for long Dirichlet polynomials and tails of Dirichlet series*]{}, Acta Arith. (2) [**84**]{} (1998), 155-192.
\[GGOS\] D.A. Goldston, S.M. Gonek, A.E. Ozluk and C. Synder, [*On the pair correlation of zeros of the Riemann zeta-function*]{}, Proc. London Math. Soc. (3) [**80**]{} (2000), 31-49.
\[GR\] I.S. Gradshteyn and I.M. Ryzhik, [*Table of Integrals, Series and Products*]{}, Academic Press, sixth edition, 2000.
\[M\] H.L. Montgomery, [*The pair correlation of zeros of the zeta function*]{}, Analytic Number Theory (St. Louis Univ., 1972), Proc. Sympos. Pure Math. [**24**]{}, Amer. Math. Soc., Providence, 1973, pp. 181-193.
\[MS\] H.L. Montgomery and K. Soundararajan, [*Beyond pair correlation*]{}, Preprint.
|
---
abstract: 'We report Spitzer Space Telescope observations during predicted transits of the exoplanet Proxima Centauri b. As the nearest terrestrial habitable-zone planet we will ever discover, any potential transit of Proxima b would place strong constraints on its radius, bulk density, and atmosphere. Subsequent transmission spectroscopy and secondary-eclipse measurements could then probe the atmospheric chemistry, physical processes, and orbit, including a search for biosignatures. However, our photometric results rule out planetary transits at the 200 ppm level at 4.5 , yielding a 3$\sigma$ upper radius limit of 0.4 (Earth radii). Previous claims of possible transits from optical ground- and space-based photometry were likely correlated noise in the data from Proxima Centauri’s frequent flaring. Follow-up observations should focus on planetary radio emission, phase curves, and direct imaging. Our study indicates dramatically reduced stellar activity at near-to-mid infrared wavelengths, compared to the optical. Proxima b is an ideal target for space-based infrared telescopes, if their instruments can be configured to handle Proxima’s brightness.'
author:
- |
James S. Jenkins$^{1,2,\ast}$, Joseph Harrington$^{3}$, Ryan C. Challener$^{3}$, Nicolás T. Kurtovic$^{1}$, Ricardo Ramirez$^{1}$, Jose Pe[ñ]{}a$^{1}$, Kathleen J. McIntyre$^{3}$, Michael D. Himes$^{3}$, Eloy Rodríguez$^{4}$, Guillem Anglada-Escudé$^{5}$, Stefan Dreizler$^{6}$, Aviv Ofir$^{7}$, Pablo A. Pe[ñ]{}a Rojas$^{1}$, Ignasi Ribas$^{8,9}$, Patricio Rojo$^{1}$, David Kipping$^{10}$, R. Paul Butler$^{11}$, Pedro J. Amado$^{4}$, Cristina Rodríguez-López$^{4}$, Eliza M.-R. Kempton$^{12,13}$, Enric Palle$^{14,15}$, Felipe Murgas$^{14,15}$\
$^1$Departamento de Astronomía, Universidad de Chile, Camino El Observatorio 1515, Las Condes, Santiago, Chile\
$^2$Centro de Astrofísica y Tecnologías Afines (CATA), Casilla 36-D, Santiago, Chile\
$^3$Planetary Sciences Group, Department of Physics, University of Central Florida, Orlando, Florida, USA\
$^4$Instituto de Astrofísica de Andalucía (IAA, CSIC) Glorieta de la Astronomía, s/n E-18008 Granada, Spain\
$^5$School of Physics and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK\
$^6$Institut für Astrophysik, Georg-August-Universität Göttingen Friedrich-Hund-Platz 1, 37077 Göttingen, Germany\
$^7$Department of Earth and Planetary Sciences, Weizmann Institute of Science, 234 Herzl Street, Rehovot 76100, Israel\
$^8$Institut de Ciéncies de l’Espai (ICE, CSIC), C/Can Magrans s/n, Campus UAB, 08193 Bellaterra, Spain\
$^9$Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain\
$^{10}$Department of Astronomy, Columbia University, 550 W 120th Street, New York NY 10027\
$^{11}$Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road NW, Washington D.C. USA 20015-1305\
$^{12}$Department of Physics, Grinnell College, 1116 8th Ave., Grinnell, IA 50112, USA\
$^{13}$Department of Astronomy, University of Maryland, College Park, MD 20742, USA\
$^{14}$Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain\
$^{15}$Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain
bibliography:
- 'refs.bib'
date: 'Submitted 2019 March 28; Accepted 2019 April 30'
title: Proxima Centauri b is not a transiting exoplanet
---
\[firstpage\]
stars: planetary systems; stars: planetary systems: formation; stars: activity
Introduction
============
The search for the nearest small planets has accelerated in recent years with the development of purpose-built instrumentation [e.g., @mayor03; @crane06; @cosentino12; @pepe13; @QuirrenbachEtal2018spieCARMENES amongst others]. Some highlights include the multi-planet systems orbiting the nearby stars HD 69830 [@lovis06], HD 10180 [@lovis11; @tuomi12], HD 40307 [@tuomi13], or 61 Virginis [@vogt10]. The value of these nearby planetary systems significantly increases when the planets are found to transit their host stars, like those orbiting HD 219134 [see for example, @vogt15; @motalebi15; @gillon17b], 55 Cancri e [@winn11], or the recent Transiting Exoplanet Survey Satellite discovery of $\Pi$ Mensae c [@huang18].
M-dwarf stars are particularly fruitful targets. Their occurrence rate for planets with masses below 10 (Earth masses) is at least one per star, with a habitable zone (HZ) occurrence rate in the mass range 3–10 of 0.21$^{+0.03}_{-0.05}$ planets per star [@tuomi14]. Within this population are dense multi-planet systems like the seven planet candidates orbiting GJ 667C [@AngladaEscudeEtal2013aaGJ667C] and the four planets around GJ 876 [@rivera10; @jenkins14]. Furthermore, [@RibasEtal2018natBarnardb] detected a small planetary candidate orbiting Barnard’s star.
M dwarfs host some spectacular transiting systems, particularly for the low-mass population, both inside and outside of the HZ. The super-Earth GJ 1214 b transits its host star [@charbonneau09], allowing studies of its atmospheric composition [e.g., @bean10; @rackham17]. The star LHS 1140 hosts two transiting planets [@dittmann17; @ment18]. But, the standard-bearer in this class is the TRAPPIST-1 system, with seven small transiting exoplanets, at least three of which are in the HZ [@gillon17a].
Proxima Centauri b [@AngladaEscudeEtal2016natProxCenb hereafter Proxima b] provides potentially the best possible opportunity for exoplanet characterization. Not only does the planet orbit our nearest stellar neighbour, but it has an equilibrium temperature and minimum mass similar to the Earth, meaning it could be rocky and have liquid surface water. If it transits, characterisation of its atmosphere and surface would be possible. [@KippingEtal2017ajNoProxMOST] used optical photometry from the Microvariability and Oscillations of STars (MOST) Space Telescope. Although they reported a candidate signature matching the planet’s expected properties, they could not confirm the feature. [@LiEtal2017rnaasProxCandidate] also reported a possible optical transit of Proxima b using data taken with the 30 cm telescope at Las Campanas, but again without confirmation. Using the Bright Star Survey Telescope in Antartica, [@LiuEtal2018ajProxPrelim] report a number of transit-like events that could be ascribed to Proxima b, and assuming large transit timing variations (TTV), they could phase with the event reported in [@KippingEtal2017ajNoProxMOST]. On the contrary, [@blank18] were unable to confirm any of the previously reported transit events using optical data spanning 11 years, albeit with heterogeneous and non-continuous data sets. They do, however, confirm the impact of high stellar activity on optical light curves.
Observations at longer wavelengths mitigate the effects of stellar activity, increasing precision. In $\S$ \[observations\] we discuss an observing campaign with the Spitzer Space Telescope at 4.5 to search for the Proxima b’s transits. We then present new constraints on the mass provided by the latest radial-velocity (RV) data in $\S$ \[rvanalysis\], both for Proxima b and any planets interior to its orbit. Finally, we summarise our findings in $\S$ \[conclusions\].
Photometric Observations and Analysis {#observations}
=====================================
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In November 2016, we observed Proxima Centauri for over 48 hours in the 4.5 band of the InfraRed Array Camera (IRAC, [@FazioEtal2004apjIRAC]) on the Spitzer Space Telescope [@WernerEtal2004apjsSpitzer], with target reacquisition roughly every 16 hours. We centered the observation on the predicted transit time of 2,457,708.02 $\pm$ 0.33 BJD, calculated from the original orbital solution for Proxima b published in @AngladaEscudeEtal2016natProxCenb. The subarray mode frame time of 0.02 s resulted in $\sim600,000$ individual frames (Spitzer Proposal ID 13155, PI: James Jenkins). Due to on-board data storage limits, there are short gaps between successive sets of 64 subarray images. The IRAC heater was off for the duration of the stare.
We used Basic Calibrated Data frames from Spitzer pipeline version S19.2.0. We performed twice-iterated 4$\sigma$ bad pixel rejection at every pixel position within each 64-frame set of subarray images to mask cosmic ray hits, and combined these with masks supplied by Spitzer.
Two groups within our team, RC and JH at UCF and NT, RR, and JJ at U. de Chile, analyzed all the data with completely independent codes, obtaining closely similar results. The UCF group used its Photometry for Orbits, Eclipses, and Transits pipeline (POET; [@StevensonEtal2012apjBLISS; @CubillosEtal2014apjTrES1b]), while the Chilean group wrote a new code, in consultation with the UCF group, but not sharing code in either direction. The codes performed centering, aperture photometry, and light-curve modeling.
We considered Gaussian, center-of-light, and least-asymmetry [@LustEtal2014paspLeastAsym] centering, as well as fixed and (at UCF only) variable-aperture [@LewisEtal2013apjHATP2b] photometry. We selected the optimal centering and photometry method by minimizing the standard deviation of normalized residuals (SDNR) and the binned-$\sigma$ $\chi^2$ ($\chi^2_{\rm bin}$, [@DemingEtal2015apjPLD]) of the model. This second metric looks for a broad-bandwidth solution by comparing a curve of SDNR vs. bin size to the expected inverse square root. We find that **$\chi^2_{\rm bin}$** more sucessfully selects against correlated noise, so we present the results using that selection criterion. The raw photometry and the position of the target on the detector relative to pixel center are shown in Figure \[fig:rawlc\].
To remove IRAC’s intrapixel sensitivity variations, we applied both BiLinearly Interpolated Subpixel Sensitivity mapping (BLISS; [@StevensonEtal2012apjBLISS]) and Pixel-Level Decorrelation (PLD; [@DemingEtal2015apjPLD]), using independent codes we each developed. In brief, BLISS iteratively computes a subpixel-resolution sensitivity grid from the light curve. We account for other effects (transit features, non-flat baselines, etc.) in other model components, all of which fit simultaneously. The flux is then:
$$\label{eqn:bliss}
F = F_s Tr(t) M(x,y) R(t),$$
where $F_s$ is the stellar flux, $Tr$ is a transit model [e.g., @MandelAgolEtal2002apjlTransitShape; @RappaportEtal2014apjKOI-2700b], $M$ is the subpixel sensitivity grid, and $R$ is the non-flat baseline (typically linear or quadratic). PLD corrects the same effect by noting that motion of the target on the detector will be correlated with individual pixel flux values, and models the light curve as a weighted sum of the brightest pixels, after normalisation:
$$F = F_s \left(\sum_{i=1}^{n} c_i \hat{P_i} + Tr(t) + R(t)\right),
\label{eqn:pld}$$
where $i$ denotes each of $n$ pixels, $c_i$ are the weights, and $\hat{P_i}$ are normalized pixel values. Both methods have been used extensively to correct Spitzer photometry [e.g., @StevensonEtal2012apjBLISS; @BlecicEtal2014apjWASP43; @CubillosEtal2014apjTrES1b; @DemingEtal2015apjPLD; @BuhlerEtal2016apjHATP13b; @WongEtal2016apjWASP19HATP7].
We fit each model to every combination of centering method, photometry method, and aperture size. We used fixed apertures with 1.5–4.0 pixel radii in 0.25 pixel increments, and variable apertures with radii from $\sqrt{N}$ to $\sqrt{N}$+2.0 pixels in 0.25 pixel increments. $N$ is the “noise pixel” parameter ([@LewisEtal2013apjHATP2b]; Spitzer IRAC handbook), defined as
$$\label{eqn:noisepix}
N = \frac{\left(\sum I(i)\right)^2}{\sum I(i)^2}$$
where $I(i)$ is the intensity of pixel $i$, and all pixels within the centering aperture are considered. We used a 17x17 pixel box, centered on the pixel containing Proxima Centauri, for centering. We take the combination of centering and photometry that result in the lowest $\chi^2_{\rm bin}$, Gaussian centering with a fixed 2.0 pixel radius aperture, as the best.
POET finds the best-fitting model using least squares. Since we find that the Spitzer pipeline tends to overestimate uncertainties, we rescale our photometric uncertainties such that the reduced $\chi^2$ of the best fit is 1. For fits with a BLISS map, we set the $x$ and $y$ widths of the subpixel grid equal to the root-mean-square of the point-to-point variation in the $x$ and $y$ positions found from centering. We also require that each subpixel bin contain at least 4 frames.
We then explore the parameter space using Multi-Core Markov-Chain Monte Carlo (MC$^3$, [@CubillosEtal2017ajMC3]), a Markov-Chain Monte Carlo (MCMC) wrapper, to determine accurate parameter uncertainties. Our Markov chains use DEMCzs, or “snooker”, a form of differential evolution Markov Chain, to explore the parameter space efficiently [@terBraak2006scDEMC; @terBraakVrugt2008scDEMCzs]. We run sufficent iterations for all parameters to pass the Gelman & Rubin convergence test within 1% of unity [@GelmanRubin1992stscConvergence].
With this analysis, only a single asymmetric transit-like feature appears, towards the end of the time series (see Fig. \[fig:lc\] upper panel). At $\sim$0.3% max depth below the continuum, it is smaller than the 0.5% transit depth that we predict for Proxima b using the parameters determined from the RV modeling effort. However, if we consider variable-aperture photometry (which is not preferred by our noise-minimization metrics, as this method significantly increases the white noise in the light curve), the feature disappears completely (see Fig. \[fig:lc\] lower panel). The asymmetric transit-like feature corresponds to a telescope vibration that smears the point-spread function, frequently associated with the “noise pixel” parameter. The strength of the vibrational effect depends upon the noise-minimization metric (i.e., the choice of centering and photometry techniques) as well as the decorrelation model (BLISS, PLD, etc.). Further study of this effect, including detection and mitigation techniques, will appear in a forthcoming paper (Challener et al. 2019, in prep).
Proxima flares over 60 times per day [@DavenportEtal2016apjlProxFlares], giving an optical light curve stability at the 0.5–1% level. We find an SDNR of 7527 and 9300 ppm for the fixed and variable aperture cases, respectively, in this infrared filter. When binned over a typical $\sim 2$ hour transit, these SDNR drop to 170 and 222 ppm. Taking these as uncertainties and the stellar radius to be 0.154 , we rule out transiting objects with radii $>$0.43 at the 3$\sigma$ level of confidence, using the more-conservative variable-aperture photometry. Previously detected features in optical light curves for this star (e.g., [@KippingEtal2017ajNoProxMOST; @LiEtal2017rnaasProxCandidate; @LiuEtal2018ajProxPrelim]) are not due to Proxima b. They may be residual correlated noise from the stellar activity.
{width="18cm"}
Spectroscopic Observations and Analysis {#rvanalysis}
=======================================
### The Ultraviolet and Visual Echelle Spectrograph (UVES)
The RV data from the Ultraviolet and Visual Echelle Spectrograph (UVES) and High Accuracy Radial velocity Planet Searcher (HARPS; see [@AngladaEscudeEtal2016natProxCenb] for details) that were used to discover Proxima b, along with new HARPS data observed as part of the Red Dots[^1] campaign, allowed further confirmation of the existence of Proxima b, along with improved upper limits on the mass of any additional body orbiting Proxima with an orbit interior to that of Proxima b. The 77 observations from UVES span a baseline of over seven years, with RVs acquired from Julian Date 2,451,634.731 to 2,454,189.714 at signal-to-noise ratios over 100 at 5500Å, necessary to measure optical RVs at the 1 level, and generally taken at low observing cadence (for instance, no measurements were acquired on successive nights). The resolving power of the UVES spectra was $R=\lambda/\Delta\lambda$ = 100,000 – 120,000, where $\lambda$ is wavelength, through application of image slicer \#3, which redistributes the light from the $1''$ opening along the 0.3$''$ slit. An iodine cell placed in the optical light path before entrance into the echelle spectrograph had an operational temperature of 70$^\circ$ C. It imprinted a dense forest of molecular iodine lines on the stellar spectra between $\sim$5000$\--$6200Å. More details on the observational strategy for the UVES spectra can be found in @kurster03, @endl08, @zechmeister09, and @AngladaEscudeEtal2016natProxCenb.
The treatment of the spectra follow the classical reduction steps for such data [see @baranne96; @jenkins17], including debiasing, cosmic ray removal, echelle order location, flatfielding, scattered-light removal, spectral extraction, spectral deblazing, and wavelength calibration. For UVES, the resulting spectrum, when compared against a previously measured Fourier transform spectrometer (FTS) iodine spectrum at much higher resolution, enables modeling of the iodine spectrum. However, this requires a prior template observation of the star without the iodine cell and without the spectrograph’s point-spread function included. The mathematical form of the process is
$$F_o(\lambda) = c [F_t(\lambda) F_i(\lambda + \Delta\lambda)] \ast PSF,
\label{eqn:iod}$$
where $F_o$ is the observed spectrum, $F_t$ is the observed stellar template without iodine, $F_i$ is the FTS iodine spectrum, $\Delta\lambda$ is the subsequent shift in wavelength between the iodine model and the FTS observation, and $c$ is a normalisation constant. Forward modeling of the iodine spectrum, the stellar spectrum (Proxima), and the instrumental response can yield precise measurements of the star’s RV signature. The final UVES internal precision using this technique is $\sim$0.6 m/s.
### High Accuracy Radial velocity Planet Searcher (HARPS)
The HARPS data cover Julian Dates from 2,457,406.870 to 2,458,027.479, giving rise to 115 measurements after cleaning outliers. Contrary to the UVES observing cadence, the HARPS data were acquired at very high observing frequency, with multiple observations being taken on individual nights, and many covering successive nights.
HARPS uses a different observing methodology, but there are many similarities in the processing. For initial reduction we used the HARPS Data Reduction Software [@MayorEtal2003] to perform the steps outlined above for UVES, extracting calibrated spectra from the raw images. To calculate the RVs, we used the HARPS-TERRA package [@AngladaEscudeEtal2012]. Without a gas cell in front of the spectrograph, the calculation can be performed directly by comparing the observed spectra. TERRA first shifts and combines all high S/N observed spectra. Their mean shift is the reference velocity for the rest of the observations. Precise velocities come from a Gaussian fit to the minimum of the cross-correlation function between the template and individual observations. We correct this for internal wavelength drift, measured by a ThAr gas lamp spectrum taken at the same time as the observations. Internal velocity precision for Proxima is $\sim$0.9 m/s.
Radial Velocity Constraints {#rvs}
---------------------------
In order to analyse the latest RV data for this work, which had the aim of confirming the existence of Proxima b and searching for additional companions on orbits interior to that of the planet, we employed the Exoplanet Mcmc Parallel tEmpering Radial VelOcity fitteR (EMPEROR) code [@PenaEtal2019]. The algorithm uses MCMC to explore the posterior parameter space, along with Bayesian statistics to determine if any signal exists. In this work we employed EMPEROR with a first-order moving-average (MA), correlated-noise model, to smooth out the high-frequency noise that tends to dominate RV measurements. No linear correlation terms were included, therefore we did not model any impact from stellar activity that is tracked by measured indices drawn from the stellar spectra themselves [for example, see @diaz18], beyond the MA model, since when included they were mostly found to be statistically similar to zero. The model ($m(t)$) we employ as function of time for a given planet ($k$) and dataset ($d$) is described by
$$m(t) = \sum_{i=1}^{k} \sum_{j=1}^{d} [K_{i,j}(cos(\omega_{i,j} + T_{i,j}(t)) + e_{i,j}cos\omega_{i,j})]
+ \dot{\gamma} + \sigma_{jit,j} + MA_{j}~~~~~
\label{eq:rvmod}$$
where $K$ is the semiamplitude of the planet model, $e$ is the eccentricity, $\omega$ is longitude of periastron, $T$ corresponds to the time of periastron passage, $\dot{\gamma}$ is the linear trend added to the model, $\sigma_{jit}$ is the excess jitter noise, and $MA$ is the moving average model.
The full timeseries was modeled as four separate datasets simultaneously, those data coming from four separate programs, three using HARPS and one using UVES (see Table \[tab:data\] for a brief summary). The pre-2016 HARPS data, HARPS Pale Red Dot (PRD), and UVES data were discussed in [@AngladaEscudeEtal2016natProxCenb] and the HARPS Red Dots (RD) data, which was an extension and expansion of the PRD program and followed the observing procedure set out there, can be found at the website[^2]. The simultaneous modeling contained five Keplerian parameters to model the planet, along with independent data offsets, MA coefficients, and excess noise (jitter) parameters, and finally a linear trend (Equation \[eq:rvmod\]). The MA modeling finds all four datasets have correlation coefficients that are statistically significantly different from zero at around the 3$\sigma$ level of confidence (see Table \[tab:rvs\]). Therefore, both the high cadence and low cadence data require a correlated noise model to extract the most RV information from the negative effects of the noise. It also appears that the jitter level slightly decreases between the PRD timeseries and the RD data, possibly due to a decrease in the activity state of Proxima. The pre-2016 and UVES jitter values are significantly larger, likely due to the lower precision of these datasets compared to the post-HARPS upgrade observations. No signficant linear trend was found in the full timeseries, however this does not rule out longer period companions as we split the data up into individual runs and we also nightly binned any data that were observed on the same night. Such data handling would ultimately disfavour long period signals in the data, which is fine for these purposes since we were looking to constrain any planets with orbital periods less than that of Proxima b.
Data set Instrument No. Observations Baseline Cadence
---------- ------------ ------------------ ----------------------------- ---------
UVES UVES 77 31/03/2000 $\--$ 30/03/2007 Low
pre-2016 HARPS 63 27/05/2004 $\--$ 23/03/2013 Low
PRD HARPS 53 19/01/2016 $\--$ 01/04/2016 High
RD HARPS 62 01/06/2017 $\--$ 30/09/2017 High
EMPEROR provides excellent constraints on the orbital characteristics of Proxima b, particularly refining the orbital period of 11.1855 days to a level better than $\pm$2.3 minutes (see Table \[tab:rvs\]). The eccentricity is also better constrained, with a 3$\sigma$ upper limit of 0.29, a movement towards zero of 0.06 compared with the value published in [@AngladaEscudeEtal2016natProxCenb]. Further limits on the eccentricity are warranted, since lower values of eccentricity require the planet’s orbit to be tidally locked to the star, providing additional constraints on the habitability of Proxima b [see for example, @ribas16]. These results highlight that the latest HARPS data are in excellent agreement with the previous data (see Fig. \[fig:rvs\]) and strongly confirm the existence of Proxima b. We find that at orbital periods shorter than that of Proxima b, the radial-velocity precision we can reach is $\sim$0.5 m/s, mainly coming from the high-cadence datasets, placing an upper limit of $\sim$0.5 on any possible inner Proxima c. The semiamplitude of the signal we find here is also in excellent agreement with that already found for Proxima b, with a difference of only 0.06 between the model published in @AngladaEscudeEtal2016natProxCenb and the model found here; although the uncertainties we find are almost half those found previously.
Orbital Parameter Model Value
-------------------------------------- -------------------------------
Amplitude \[\] $1.32_{-0.14}^{+0.12}$
Period \[d\] $11.1855_{-0.0014}^{+0.0016}$
Phase \[rads\] $3.44_{-1.79}^{+0.62}$
Longitude \[rads\] $4.40_{-3.47}^{+0.84}$
Eccentricity $0.08_{-0.06}^{+0.07}$
$\dot{\gamma}$ \[ms$^{-1}$d$^{-1}$\] $-0.0005_{-0.0002}^{+0.0003}$
$\sigma_{jit,pre-2016}$ \[\] $1.84_{-0.11}^{+0.09}$
$\gamma_{pre-2016}$ \[\] $1.23_{-0.73}^{+0.67}$
$\phi_{pre-2016}$ $0.63_{-0.11}^{+0.16}$
$\tau_{pre-2016}$ \[d\] $7.57_{-3.10}^{+1.52}$
$\sigma_{jit,PRD}$ \[\] $1.44_{-0.20}^{+0.10}$
$\gamma_{PRD}$ \[\] $1.98_{-0.92}^{+1.08}$
$\phi_{PRD}$ $0.38_{-0.11}^{+0.17}$
$\tau_{PRD}$ \[d\] $7.86_{-4.46}^{+0.97}$
$\sigma_{jit,RD}$ \[\] $1.14_{-0.10}^{+0.13}$
$\gamma_{RD}$ \[\] $2.10_{-0.90}^{+1.21}$
$\phi_{RD}$ $0.50_{-0.20}^{+0.16}$
$\tau_{RD}$ \[d\] $6.91_{-2.58}^{+2.33}$
$\sigma_{jit,UVES}$ \[\] $1.91_{-0.11}^{+0.09}$
$\gamma_{UVES}$ \[\] $-0.20_{-0.32}^{+0.27}$
$\phi_{UVES}$ $0.62_{-0.17}^{+0.19}$
$\tau_{UVES}$ \[d\] $4.54_{-1.63}^{+2.35}$
: Orbital constraints and nuisance parameters for the Proxima b model from the EMPEROR analysis of RV data.[]{data-label="tab:rvs"}
pre-2016 - parameters for data taken with HARPS prior to the PRD program PRD - parameters for the Pale Red Dot program RD - parameters for the Red Dots program
![[**Top:**]{} RV measurements of Proxima Centauri from HARPS prior to the Pale Red Dot program (blue), HARPS Pale Red Dot (green), HARPS Red Dots (red), and UVES (yellow), phase-folded to the planet’s orbital period. The black line is the best-fit Keplerian model. [**Bottom:**]{} Residuals to the fit.[]{data-label="fig:rvs"}](Fig3.pdf){width="10cm"}
Summary {#conclusions}
=======
We have addressed the recent claims of transit-like events in optical photometry arising from the habitable-zone terrestrial planet Proxima b. We observed the system with the Spitzer Space Telescope for $\sim$48 hours at 4.5 . The observations covered the 99% probability window predicted for the transit using the published RV model in @AngladaEscudeEtal2016natProxCenb. The limits on this window were drawn from the posterior density distribution of the model, assuming 99% uncertainty limits on the model parameters like period and eccentricity.
Our observations and BLISS analysis allowed us to reach an unbinned photometric precision of 7500 ppm, with a 2 hr (rough transit duration) binned precision of 200 ppm. No transit-like event could be attributed to the passage of Proxima b in front of its star. The previously witnessed transit-like events may result from residual correlated noise arising from the star’s complex and frequent flaring and activity patterns. Our photometric precision places a 3$\sigma$ upper limit on the size of a transiting Proxima b of 0.4 . This corresponds to an implausible minimum density of $\sim$112 g cm$^{-3}$.
We performed a short radial-velocity experiment to search for additional small planets interior to the orbit of Proxima b, whilst constraining better Proxima b’s orbital characteristics. Beyond the data published in @AngladaEscudeEtal2016natProxCenb, we also included newly observed HARPS data from the Red Dots program. After fitting for the orbit of Proxima b, the residuals reveal no inner planet down to the 0.5 m/s level, which relates to planets with minimum masses of 0.5 . The orbital period and eccentricity of Proxima b’s orbital solution were also better constrained in this process, with a precision in period of better than $\pm$30 s found, and a 3$\sigma$ upper limit on the eccentricity of 0.29.
Finally, we did witness a transit-like event at the 0.3% depth level and with an asymmetric morphology. However, we found we could remove the feature completely from the time series by using variable-radius photometry apertures. A study of this and similar features in additional Spitzer data of Proxima and beyond, as well as detection and treatment methods, will be published in a future paper (Challener et al. 2019, in prep).
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank the Spitzer Science Center staff for making these observations possible. This work is based on observations made with the [*Spitzer Space Telescope*]{}, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. We also thank the anonymous referee for their efficient and detailed review. The authors acknowledge support from the following: CATA-Basal/Chile PB06 Conicyt and Fondecyt/Chile project \#1161218 (JSJ). CONICYT Chile through CONICYT-PFCHA/Doctorado Nacional/2017-21171752 (JP). Spanish MINECO programs AYA2016-79245-C03-03-P, ESP2017-87676-C05-02-R (ER), ESP2016-80435-C2-2-R (EP) and through the “Centre of Excellence Severo Ochoa” award SEV-2017-0709 (PJA, CRL and ER). STFC Consolidated Grant ST/P000592/1 (GAE). NASA Planetary Atmospheres Program grant NNX12AI69G, NASA Astrophysics Data Analysis Program grant NNX13AF38G. Spanish Ministry of Science, Innovation and Universities and the Fondo Europeo de Desarrollo Regional (FEDER) through grant ESP2016-80435-C2-1-R (IR). We thank contributors to SciPy, Matplotlib, and the Python Programming Language; the free and open-source community; and the NASA Astrophysics Data System for software and services. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programmes 096.C-0082, 191.C-0505, and 099.C-0880.
\[lastpage\]
[^1]: https://reddots.space/
[^2]: https://reddots.space/
|
\#1[[$\backslash$\#1]{}]{} Metal-insulator transitions (MITs) remain a fascinating and only incompletely understood phenomenon[@R]. Conceptually, one distinguishes between Anderson transitions in models of noninteracting electrons, and Mott-Hubbard transitions of clean, interacting electrons. At the former, the electronic charge diffusivity $D$ is driven to zero by quenched, or frozen-in, disorder, while the thermodynamic properties do not show critical behavior. At the latter, the thermodynamic density susceptibility $\partial n/\partial\mu$ vanishes due to electron-electron interaction effects. In either case, the conductivity $\sigma = (\partial n/\partial\mu)D$ vanishes at the MIT. In many real systems both quenched disorder and interactions are present, which makes a theoretical understanding of the resulting Anderson-Mott transition very difficult. One particular complication is provided by the presence of magnetic local moments (LMs) in such systems. There is much experimental evidence for LMs[@LMs], and their formation has been studied theoretically[@Milovanovic], but no existing theory can describe their interplay with the transport properties near the MIT[@R]. Another complication is the possible presence of [*annealed*]{} disorder, which is in thermal equilibrium with the rest of the system and hence involves disorder averaging of the partition function. This is in contrast to quenched disorder that requires an averaging of the free energy, which is usually done by means of the replica trick[@Grinstein].
In this Letter we make two contributions to the MIT problem. (1) We show that annealed disorder leads to a MIT that belongs to none of the previously studied classes. It is driven by a vanishing $\partial n/\partial\mu$ and thus resembles a Mott-Hubbard transition, even if no correlation effects are explicitly considered. (2) We propose a mechanism by which additional annealed disorder is generically self-generated in quenched disordered systems, and we argue that a type of LMs can be described in terms of it. We further develop a method for incorporating these ‘annealed LMs’ into a transport theory.
Let us start by considering Wegner’s nonlinear sigma-model (NL$\sigma$M)[@Wegner] for noninteracting electrons with nonmagnetic quenched disorder. The action reads $${\cal A} = \frac{-1}{2G}\int\! d{\bf x}\,\tr \left(\nabla
Q({\bf x})\right)^2 + 2H\int\! d{\bf x}\,\tr \left(\Omega\,Q({\bf x})\right).
\label{eq:1}$$ Here $Q({\bf x})$ is a matrix field that comprises two fermionic degrees of freedom. Accordingly, $Q$ carries two Matsubara frequency indices $n$ and $m$, and two replica indices $\alpha$ and $\beta$ to deal with the quenched disorder. The matrix elements $Q_{nm}^{\alpha\beta}$ are spin-quaternion valued to allow for particle-hole and spin degrees of freedom. It is convenient to expand them in a basis $\tau_r\otimes s_i$ ($r,i=0,1,2,3$) where $\tau_0=s_0$ is the $2\times 2$ unit matrix, and $\tau_{1,2,3} = - s_{1,2,3} = -i\sigma_{1,2,3}$, with $\sigma_j$ the Pauli matrices. For simplicity, we will ignore the particle-particle or Cooper channel, which amounts to dropping $\tau_1$ and $\tau_2$ from the spin-quaternion basis[@R]. $Q$ is subject to the constraints $Q^2({\bf x}) \equiv 1$, and $\tr Q({\bf x}) \equiv 0$. $\Omega_{nm}^{\alpha\beta} = \delta_{nm}\delta_{\alpha\beta}\Omega_n\,
(\tau_0\otimes s_0)$ is a frequency matrix with $\Omega_n = 2\pi Tn$ a bosonic Matsubara frequency and $T$ the temperature. $G$ is a measure of the disorder that is proportional to the bare resistivity, and the frequency coupling $H$ is proportional to the bare density of states at the Fermi level. $\tr$ denotes a trace over all discrete degrees of freedom that are not shown explicitly.
The properties of this model are well known[@Wegner; @ELK; @R]. The bare action describes diffusive electrons, with $D=1/GH$ the diffusion coefficient. Under renormalization $D$ decreases with increasing disorder until a MIT is reached at a critical disorder value. The critical behavior is known in an $\epsilon$-expansion about the lower critical dimension $d=2$. In the absence of the Cooper channel, the MIT appears only at two-loop order at a critical disorder strength of $O(\sqrt\epsilon)$. $H$, which determines the specific heat coefficient, the spin susceptibility, and $\partial n/\partial\mu$, is uncritical, which makes this MIT an Anderson transition.
Now we add magnetic annealed disorder to the model. Since our general results are independent of its origin, we first proceed without specifying it. Annealed disorder implies that the $Q$ in the resulting terms all carry the same replica index[@Grinstein]; otherwise, the functional form of the resulting additional terms in the action can be taken from Ref. . We obtain two additional terms, viz.
\[eqs:2\] $$\begin{aligned}
\Delta{\cal A}^{(1)}&=&\frac{TM_1}{8}\sum_{\alpha}\int d{\bf x}\sum_{j=1}^{3}
\left[\tr\left(\left(\tau_3\otimes s_j\right)\,Q^{\alpha\alpha}({\bf x})
\right)^2\right.
\nonumber\\
&&\qquad\qquad\qquad - \left. \tr\left(Q^{\alpha\alpha}({\bf x})\right)^2
\right]\quad,
\label{eq:2a}\end{aligned}$$ and $\Delta{\cal A}^{(2)} = \Delta{\cal A}^{(2,s)} + \Delta{\cal A}^{(2,t)}$, where $$\begin{aligned}
\Delta{\cal A}^{(2,j)}&=&\frac{TM_2^j}{8}
\sum_{\alpha\neq\beta}
\sum_{nm} \sum_{r=0,3} \int\! d{\bf x}\,\left[\tr (\tau_r\otimes s_i)\,
Q_{nm}^{\alpha\beta}({\bf x})\right]
\nonumber\\
&&\times \left[\tr (\tau_r^{\dagger}
\otimes s_i)\,Q_{mn}^{\beta\alpha}({\bf x})\right]\quad,
\label{eq:2b}\end{aligned}$$
with $j=s$ for $i=0$ (spin-singlet), and $j=t$ for $i=1,2,3$ (spin-triplet). $\Delta{\cal A}^{(2)}$ arises from the need to absorb the scattering rate due to the annealed disorder in $G$. The coupling constants $M_1$, $M_2^s$, and $M_2^t$ are related to the strength of the magnetic annealed disorder. The factor of $T$ appears naturally in front of any annealed disorder term, a crucial point that we will come back to later.
The action ${\cal A} + \Delta{\cal A}^{(1)}
+ \Delta{\cal A}^{(2)}$ can be analyzed by standard means. Note that the mass terms in Eqs. (\[eqs:2\]) are proportional to temperature, making them quite different from conventional masses due to quenched disorder. In many respects, they are similar to electron-electron interaction terms in a Q-field theory formalism[@us_fermions]. We denote the renormalized coupling constants that correspond to $G$, $H$, $M_1$, and $M_2^{s,t}$ by $g$, $h$, $m_1$, and $m_2^{s,t}$, and define $\delta_{1,2}^{s,t} = m_{1,2}^{s,t}/h$. The renormalization group (RG) flow equations to one-loop order are
\[eqs:3\] $$\begin{aligned}
\frac{dg}{dl}&=&-\epsilon g + g^2(\delta_2^s + 3\delta_2^t - 3\delta_1)\quad,
\label{eq:3a}
\\
\frac{dh}{dl}&=&-hg(\delta_2^s + 3\delta_2^t - 3\delta_1)\quad,
\label{eq:3b}
\\
\frac{d\delta_1}{dl}&=&-g\left[-4\delta_1^2 + \delta_1(\delta_2^s + 3\delta_2^t)
+ (\delta_2^s - \delta_2^t)^2\right]\quad,
\label{eq:3c}
\\
\frac{d\delta_2^s}{dl}&=&g\left[3\delta_1^2
+ 3\delta_1(\delta_2^s - 2\delta_2^t)
- 3\delta_2^t(\delta_2^s - \delta_2^t)\right]\quad,
\label{eq:3d}
\\
\frac{d\delta_2^t}{dl}&=&g\left[3\delta_1^2 - \delta_1(2\delta_2^s + \delta_2^t)
-(\delta_2^s)^2 \right.
\nonumber\\
&&\qquad\qquad\qquad-\left. 2(\delta_2^t)^2 + 3\delta_2^s\delta_2^t\right]\quad,
\label{eq:3e}\end{aligned}$$
where $l=\ln b$ with $b$ the RG length scale factor. Besides unstable fixed points (FPs), there is a line of critical fixed points (FPs) $(g^*,h^*,\delta_1^*,\delta_2^{s*},
\delta_2^{t*}) = (\epsilon/4\delta_2^*,0,0,\delta_2^*,\delta_2^*)$ that correspond to an MIT (all of these FPs belong to the same universality class). Linearization about any of these FPs yields one relevant eigenvalue $\lambda_g = \epsilon + O(\epsilon^2)$ that determines the correlation length exponent $\nu = 1/\lambda_g$, one marginal eigenvalue that corresponds to moving along the line of FPs, and two irrelevant eigenvalues equal to $-\epsilon + O(\epsilon^2)$. The anomalous dimension of $h$ is $\kappa = -\epsilon + O(\epsilon^2)$. In addition, the critical behavior of the single-particle density of states (DOS), $N$, at the Fermi level can be obtained from the wavefunction renormalization. Choosing the critical exponent of the DOS, $\beta$, the correlation length exponent $\nu$, and the dynamical critical exponent $z = d + \kappa$ as independent exponents, we find $$\nu = 1/\epsilon + O(1)\ ,\ \beta = \epsilon + O(\epsilon^2)\ ,\
z = 2 + O(\epsilon^2)\ .
\label{eq:4}$$ For the conductivity exponent we find $s=\nu\epsilon = 1 + O(\epsilon)$, and $\partial n/\partial\mu$, the spin susceptibility $\chi_s$, and the specific heat coefficient $\gamma = C_V/T$, which we collectively denote by $\chi$, all vanish with a critical exponent determined by $\kappa$. The diffusion coefficient, on the other hand, has no anomalous dimension and thus is uncritical to one-loop order, as can be seen from Eqs. (\[eq:3a\], \[eq:3b\]). With $t$ the dimensionless distance from the critical point at $T=0$, and $E$ the energy, we can summarize the critical behavior of these quantities by the homogeneity laws
\[eqs:5\] $$\begin{aligned}
\chi(t,T)&=&b^{\kappa} \chi(tb^{1/\nu},Tb^z)\quad,
\label{eq:5a}\\
N(t,T,E)&=&b^{-\beta/\nu} N(tb^{1/\nu},Tb^z,Eb^z)\quad,
\label{eq:5b}\\
\sigma(t,T)&=&b^{-s/\nu} \sigma(tb^{1/\nu},Tb^z)\quad,
\label{eq:5c}\\
D(t,T)&=&b^{-(s/\nu+\kappa)} D(tb^{1/\nu},Tb^z)\quad.
\label{eq:5d}\end{aligned}$$
We conclude that the MIT is driven by the vanishing of $\partial n/\partial\mu$, and therefore is qualitatively different from the localization transition that is found in the absence of annealed disorder. Indeed, putting $M_1 = M_2^s = M_2^t = 0$ we find that all thermodynamic anomalies disappear, as does the one-loop correction to $g$. At two-loop order, one finds instead a MIT of Anderson type[@R].
We now turn to a specific realization, via local magnetic moments, of the annealed disorder that leads to the striking effects discussed above. To explain the salient points, it is easiest to initially consider a simpler field theory than the $Q$-matrix theory studied above, and adapt a classical line of reasoning from Ref. to quantum field theories. Accordingly, we consider a scalar quantum field $\phi({\bf x},\tau)$ and an action $$S[\phi] = \int dx\,\left(\phi\partial_{\tau}\phi
- {\cal H}[\phi,\nabla\phi]\right)\quad,
\label{eq:6}$$ Here $x=({\bf x},\tau)$ comprises position ${\bf x}$ and imaginary time $\tau$, $\int dx \equiv \int d{\bf x}\int d\tau$, ${\cal H}$ is a Hamiltonian density, and we use units such that $\hbar=k_{\rm B}=1$. We will assume that $S$ describes a phase transition from a disordered to an ordered phase, and will use a magnetic language, referring to $\langle\phi\rangle$ as ‘magnetization’. Suppose that ${\cal H}$ contains quenched disorder of random-mass type, and that we are in the nonmagnetic phase, $\langle\phi\rangle=0$. The key idea is to [*not*]{} integrate out the quenched disorder as a first step, as one does in a conventional treatment[@Grinstein], but rather to work with a particular disorder realization. Due to the quenched disorder there will be regions in space that energetically favor local order, $\langle\phi\rangle\neq 0$, even though there is no global order. Deep inside the disordered phase these regions will be rare, but in an arbitrarily large system we will find arbitrarily large rare regions with a finite probability. The action $S$ will then have static saddle-point solutions $\Phi({\bf x})$ that have a nonvanishing value of the magnetization only in the rare regions. Let there be $N$ such rare regions and associated local blobs of magnetization or LMs. Then we can actually construct $2^N$ such saddle points, which differ only by the way the sign of the magnetization is distributed among the LMs. Since the LMs are far apart, the energy differences between these $2^N$ saddle points will be small. In expanding about the saddle points, we therefore have no reason to prefer one of them over any of the others. Furthermore, since the LMs are self-generated by the system, albeit in response to the quenched potential, we assume that they are in thermal equilibrium with all other degrees of freedom as well as with each other. To calculate the partition function $Z$ it is therefore necessary to take into account fluctuations in the vicinity of each of the $2^N$ saddle points[@barrier_footnote]: $$Z \approx \sum_{a=1}^{2^N} \int_{<} D[\varphi]\
\exp \left(-S[\Phi^{(a)} + \varphi]\right)\quad.
\label{eq:7}$$ Here $\int_{<} D[\varphi]$ denotes an integration over small fluctuations $\varphi$ in the vicinity of each of the saddle points. Notice that this restriction to small fluctuations is necessary in order to avoid double counting. Conversely, if we could perform the integral over the fluctuations exactly, then it would be sufficient to expand about one of the saddle points. In practice, however, one is restricted to a perturbative evaluation of the functional integral, and Eq. (\[eq:7\]) is a good approximation[@nonperturbative_footnote].
We now consider the thermodynamic limit. Then the discrete set of $2^N$ saddle points turns into a saddle-point manifold ${\cal M}(\Phi)$ that needs to be integrated over. Splitting off the saddle-point part of the action, $S[\phi] = S[\Phi] + \Delta S[\Phi,\varphi]$, we have
\[eqs:8\] $$Z = \int D[\Phi]\ P[\Phi]\int D[\varphi]\
\exp\left(-\Delta S[\Phi,\varphi]\right)\quad,
\label{eq:8a}$$ with the probability distribution $P$ given by $$P[\Phi] = {\cal S}(\Phi)\ \exp\left(-\frac{1}{T}\int d{\bf x}\
{\cal H}[\Phi,\nabla\Phi]\right)\quad.
\label{eq:8b}$$
Here ${\cal S}$ denotes the support of the saddle-point manifold ${\cal M}$. Notice the factor of $1/T$ in the exponent, which results from the static nature of the saddle points[@sign_footnote].
In general it is not possible to determine $P[\Phi]$ explicitly. However, if we perform the $\Phi$ integration by means of a cumulant expansion, the most relevant term in the effective action will be the one that results from the term quadratic in ${\cal H}[\Phi]$ and the linear coupling between $\Phi$ and $\varphi^2$ in $\Delta S$. To obtain the most relevant term in the effective theory for the fluctuations $\varphi$, we thus can write, with $w>0$ a number[@sign_footnote], $$\begin{aligned}
Z&\approx&\int D[\varphi]\ e^{-S[\varphi]}\int D[\Phi]\
\exp\left(\frac{-1}{wT}\int d{\bf x}\, \Phi^2({\bf x})\right)
\nonumber\\
&&\times \exp\left({\int dx\ \Phi({\bf x})\,
\varphi^2(x)}\right)\quad.
\label{eq:9}\end{aligned}$$
Equation (\[eq:9\]) is the partition function one would obtain by expanding perturbatively about just one of the saddle points, with static, annealed disorder appearing in addition to the quenched disorder still contained in $S[\varphi]$. The annealed disorder is governed by a Gaussian distribution whose variance is proportional to $T$. This property reflects the fact that the annealed disorder, as classical degrees of freedom in equilibrium with the rest of the system, must come with a Boltzmann weight, and it is the reason for the factors of $T$ in Eqs. (\[eqs:2\]).
Let us now explain how these arguments can be applied to the $Q$-field theory of interacting electrons to arrive at the action, Eqs. (\[eq:1\], \[eqs:2\]). The magnetization is proportional to the expectation value $\langle\tr (\tau_3\otimes s_i)\,Q({\bf x})
\rangle$ [@us_fermions], and in the presence of quenched disorder that favors the formation of magnetic LMs, the exact fermionic theory that underlies the NL$\sigma$M[@us_fermions] allows for saddle-point solutions where these components of $Q$ are locally nonzero and play the role of the field $\Phi$ above. This is in addition to a globally nonzero $\langle\tr (\tau_0\otimes s_0)\,Q({\bf x})
\rangle$ which reflects a nonvanishing DOS. By following the above reasoning for a scalar field, and going through the derivation of the sigma-model again, one obtains Eqs. (\[eq:1\],\[eqs:2\]).
We conclude with several remarks. First, we emphasize that we have studied a simplified model, neglecting both the Cooper channel and the electron-electron interaction. The latter point requires some clarification. In order to generate the annealed disorder from LMs, some interaction is necessary, (1) for local magnetic order to develop, and (2) in order for our canonical averaging over the saddle points to make physical sense. A truly noninteracting system would not sample all of these field configurations. Put differently, interactions make the energy barriers between the saddle points, which are infinite in a noninteracting system, finite and thus allow for an equilibration of the saddle-point degrees of freedom[@barrier_footnote; @nonperturbative_footnote]. We have simplified our model by assuming points (1) and (2) above to be the [*only*]{} effect of the interactions. Of course, if the annealed disorder were due to some other mechanism, then our results would also apply to strictly noninteracting electrons. Clearly, one can study generalizations of our model. In addition to adding an explicit interaction term, one can restore the Cooper channel, which will make the FP we found compete with the ordinary localization FP that also occurs at one-loop order. In systems with time reversal symmetry, one then expects the MIT studied here to get preempted by a localization transition if the bare dimensionless mass $M_2/H$ is smaller than a number of $O(1)$. It would also be interesting to consider the present model to 2-loop order to see whether the diffusion coefficient will still not be renormalized (apart from the ‘diffuson’ localization contributions that will appear at that order), and whether the line of FPs gives way to a more conventional FP structure. These questions will be considered in the future.
Second, we point out that the strong effects of annealed disorder we found are characteristic of quantum statistical mechanics. In a classical scalar field theory, the leading term in the action generated upon integrating out annealed disorder is of the form (see Eq. (\[eq:9\])) $-\int d{\bf x}\,\varphi^4({\bf x})$. It thus has the same form as the ordinary $\varphi^4$-term and is in general not very interesting (although it can lead, e.g., to a first order phase transition). In a quantum system, on the other hand, integrating out the annealed disorder yields $-\int d{\bf x}\int d\tau\,d\tau'\,\varphi^2({\bf x},\tau)\,
\varphi^2({\bf x},\tau')$, which has a different time structure than the usual $\varphi^4$ term. It is the extra time integral that makes the annealed disorder term more relevant than in the classical case.
Third, we come back to the fact that the variance of the Gaussian distribution for the annealed disorder is linear in $T$. If one used a Gaussian distribution with a temperature independent width, one would encounter factors of $1/T$ in perturbation theory that force one to scale the annealed disorder strength with $T$ to obtain a meaningful theory. Annealed disorder with an unbounded distribution and a finite variance at $T=0$ is unphysical, since it allows the system to lower its energy arbitrarily far by digging itself a deeper and deeper trough. The necessity of the factor of $T$ was realized in Ref. , but its origin was not recognized[@1/T_footnote].
Finally, let us explain why annealed disorder leads to a critical $\partial n/\partial\mu$, while quenched disorder without electron-electron interactions does not. To see this, we realize that annealed disorder essentially means potential troughs that are somewhat flexible, i.e. they adjust in response to the electrons. Let the sytem be in equilibrium at some value of the chemical potential $\mu$, and change $\mu$ slightly. Then the flexible potential will adjust, and as a result fewer electrons will have to flow out of or into the grand canonical reservoir than would be the case in the absence of annealed disorder. This explains why there is a correction to $\partial n/\partial\mu$ in perturbation theory. Furthermore, the diffusive dynamics of the electrons lead to this correction being a frequency-momentum integral over diffusion propagators, which is logarithmically singular in $2$-$d$. In $d=2+\epsilon$ this leads to a critical $\partial n/\partial\mu$, as it happens with other quantities that are singular in perturbation theory in $2$-$d$. This is the only known mechanism for a critical $\partial n/\partial\mu$ at a MIT in low-dimensional systems[@high-d]. The recent observation of a critical $\partial n/\partial\mu$ at a $2$-$d$ MIT[@Jiang] is therefore very interesting in this context, even though our current theory does not describe a MIT in $d=2$.
This work was initiated at the Aspen Center for Physics, and supported by the NSF under grant Nos. DMR-98-70597 and DMR–96–32978, and by the DFG under grant No. SFB 393/C2.
-4mm
For a review, see, e.g., D. Belitz and T.R. Kirkpatrick, Rev. Mod. Phys. [**66**]{}, 261 (1994). M.A. Paalanen, S. Sachdev, R.N. Bhatt, and A.E. Ruckenstein, Phys. Rev. Lett. [**57**]{}, 2061 (1986); M.A. Paalanen, J.E. Graebner, R.N. Bhatt, and S. Sachdev, Phys. Rev. Lett. [**61**]{}, 597 (1988); Y. Ootuka and N. Matsunaga, J. Phys. Soc. Japan [**59**]{}, 1801 (1990). M. Milovanovic, S. Sachdev, and R.N. Bhatt, Phys. Rev. Lett. [**63**]{}, 82 (1989), and references therein. See, e.g., G. Grinstein, in [*Fundamental Problems in Statistical Mechanics VI*]{}, E.G.D. Cohen (ed.), North Holland (Amsterdam, 1985). F. Wegner, Z. Phys. B [**35**]{}, 207 (1979). We use the fermionic formulation of the model given in Ref. . See also Ref. . K.B. Efetov, A.I. Larkin, and D.E. Khmelnitskii, Zh. Eksp. Teor. Fiz. [**79**]{}, 1120 (1980) \[Sov. Phys. JETP [**52**]{}, 568 (1980)\]. D. Belitz and T.R. Kirkpatrick, Phys. Rev. B [**56**]{}, 6513 (1997); D. Belitz, T.R. Kirkpatrick, and F. Evers, Phys. Rev. B [**58**]{}, 9710 (1998). Viktor Dotsenko, A.B. Harris, D. Sherrington, and R.B. Stinchcombe, J. Phys. A [**28**]{}, 3093 (1995). Elsewhere we have argued that for noninteracting LMs, the typical barriers between different saddle point states diverge in the bulk limit[@rr_magnets]. We expect that one effect of interactions between the LMs is to make the barriers finite, so that the averaging discussed here applies. R. Narayanan, T. Vojta, D. Belitz, and T.R. Kirkpatrick, Phys, Rev. B [**60**]{}, 10150 (1999). The basic assumption underlying these arguments is that typical pairs of saddle-point configurations are separated by finite barriers[@barrier_footnote], but not perturbatively accessible from one another. This amounts to an assumption about a separation of time scales that we expect to be true in some, but not all, systems. If one uses a simple saddle-point theory of noninteracting LMs for $\Phi$ in the Hamiltonian density in Eq. (\[eq:8b\]), then the coefficient of $1/T$ is found to be positive, indicating that the LM state is a thermodynamic equilibrium state. We expect a more realistic treatment, including LM interactions, to weaken the effects of these LMs and make the state thermodynamically unstable, changing the sign of the $1/T$ term to the one used in Eqs. (\[eq:8b\],\[eq:9\]), which physically makes these LMs [*dynamic*]{} rare fluctuations. Developing the theory without the explicit factor of $T$ in Eqs. (\[eqs:2\]) leads to Eqs. (\[eqs:3\]) with an extra factor of $1/T$ on the righ-hand side. The only physical fixed point then requires the masses to scale with $T$. The factor of $T$ in Eqs. (\[eqs:2\]) is thus unavoidable on technical grounds, even if its presence on physical grounds is not realized. In high dimensions an order parameter theory of the Anderson-Mott transition yields a critical $\partial n/\partial\mu$, T.R. Kirkpatrick and D. Belitz, Phys. Rev. Lett. [**73**]{}, 862 (1994), while in $d=2+\epsilon$, $\partial n/\partial\mu$ is uncritical[@R]. S.C. Dultz and H.W. Jiang, cond-mat/9909314.
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abstract: 'We show the existence of a deformation process of hypersurfaces from a product space $\m_1\times\r$ into another product space $\m_2\times\r$ such that the relation of the principal curvatures of the deformed hypersurfaces can be controlled in terms of the sectional curvatures or Ricci curvatures of $\m_1$ and $\m_2$. In this way, we obtain barriers which are used for proving existence or non existence of hypersurfaces with prescribed curvatures in a general product space $\mm$.'
---
\[section\]
------------------------------------------------------------------------
[**Geometric barriers for the existence of hypersurfaces\
with prescribed curvatures in M$^n\times$R.**]{}\
------------------------------------------------------------------------
\
José A. Gálvez$^{a\ }$[^1], Victorino Lozano$^b$
$\mbox{}^a$ Departamento de Geometría y Topología, Universidad de Granada, E-18071 Granada, Spain; e-mail: [email protected]
$\mbox{}^b$ Departamento de Matemática Aplicada, U.N.E.D. E-13600 Alcázar de San Juan, Ciudad Real, Spain; e-mail: [email protected]
Mathematics Subject Classification: 58J05, 53A10.\
Introduction.
=============
Our main objective is to describe a simple method for obtaining barriers in a product space $\m^n\times\r$. To that end, we consider a hypersurface $S$ in a product space $\m_1\times\r$ and obtain a new hypersurface $S^\ast$ in a different product space $\m_2\times\r$ such that the principal curvatures of $S$ and $S^\ast$ can be related in terms of the sectional curvatures or Ricci curvatures of $\m_1$ and $\m_2$.
The previous method has a special interest when $S$ is a hypersurface with constant mean curvature $H$, or in general with constant r-mean curvature $H_r$, and $\m_1$ has constant sectional curvature $c$. In such a case, we obtain barriers for the existence of hypersurfaces with constant r-mean curvature $H_r$ in a general $\m_2\times\r$ if the sectional (or Ricci) curvature of $\m_2$ is bounded from above or below by $c$.
Thus, we will generalize different known results in the homogeneous spaces $\h^n\times\r$, $\r^{n+1}$ or $\s^n\times\r$ to general product spaces $\m^n\times\r$. For the case $n=2$ the analogous method was described in [@GL].
The paper is organized as follows. Section \[s2\] is devoted to the calculus of the principal curvatures of certain graphs in a general product $\m^n\times\r$ in terms of Jacobi fields on $\m^n$ (Lemma \[l1\]). In Section \[s3\] we describe in detail our method for obtaining barriers in a product space $\m^n\times\r$, and obtain two comparison results. The first one relates the principal curvatures of two graphs $S$ in $\m_1\times\r$ and $S^{\ast}$ in $\m_2\times\r$ with the sectional curvatures of $\m_1$ and $\m_2$ (Theorem \[t1\]). The second comparison result relates the mean curvatures of $S$ and $S^{\ast}$ with the Ricci curvatures of $\m_1$ and $\m_2$ (Theorem \[t2\]).
In Section 4 we show different examples of how these barriers can be used. Thus, in Theorem \[t3\] we prove that given a closed geodesic ball $B_r\subseteq \m^n$ of radius $r$ then there is an explicit constant $H_0$, which only depends on the radius $r$ and the minimum of its Ricci curvature, such that there exists no vertical graph over $B_r$ in $\m^n \times \r$ with minimum of its mean curvature greater than or equal to $H_0$. This generalizes a previous result by Espinar and Rosenberg in [@ER] for $n=2$. In Theorem \[t4\] we obtain an analogous result for Gauss-Kronecker curvature, or in general for r-mean curvature, depending on the sectional curvatures of the closed geodesic ball.
Moreover, in Theorem \[t5\], we prove that, under certain restrictions on the ambient space $\m^n \times \r$, for every properly embedded hypersurface $\Sigma \subset \m^n \times \r$ with mean curvature $H \geq H_0 >0$, its mean convex component cannot contain a certain geodesic ball of radius $r$, where $r$ only depends on $H_0$ and the infimum of the Ricci curvature of $\m$. In particular, this shows the non existence of entire horizontal graphs over a Hadamard manifold for certain values of the mean curvature (Corollary \[c1\]).
In Theorem \[t7\] we also prove the existence of vertical graphs in $\m^n \times \r$ with boundary on a horizontal slice and constant mean curvature $H_0$ for any $H_0\in[0,(n-1)/n]$, when $\m^n$ is a Hadamard manifold with sectional curvature pinched between $-c^2$ and $-1$. In fact, we show that any compact hypersurface with constant mean curvature $H_0$ and the same boundary must be the previous graph or its reflection with respect to the slice. This generalizes previous results in $\h^n\times\r$ (see [@NSST] and [@BE2]).
Finally, in Theorem \[t8\] we give a result of existence for vertical graphs with positive constant Gauss-Kronecker curvature in $\m^n \times \r$, which solves the Dirichlet problem for the associated Monge-Ampère equation with zero boundary values.
The principal curvatures of the graph. {#s2}
======================================
Let ${\cal H}$ be an (n-1)-dimensional manifold and $(\m,g)$ be an $n$-dimensional Riemannian manifold. Consider two smooth maps $i:{\cal H}{\longrightarrow}\m$ (non necessarily an immersion) and $n:{\cal H}{\longrightarrow}T\m$ such that $n(x)\in T_{i(x)}\m$ is a unit vector with $g(di_x(v),n(x))=0$ for all $v\in T_x{\cal H}$. Here, for instance, $T_p\m$ denotes the tangent space to $\m$ at the point $p\in\m$.
Let $I$ be an open real interval such that 0 is in its closure $\bar{I}$, and assume that the map $$\label{eliminado}
\varphi(x,t)=\exp_{i(x)}(t\ n(x)),\qquad (x,t)\in {\cal H}\times \bar{I},$$ is smooth and $\varphi_{|{\cal H}\times I}$ a global diffeomorphism onto its image; where $\exp$ denotes the exponential map in $\m$.
Observe that $\varphi_{|{\cal H}\times I}$ can be seen as a certain parametrization of an open set of $\m$, and the parameter $t$ can be considered as a distance function to $i(x)$.
The polar geodesic parameters at a point $p\in\m$ are examples of the previous situation. For that, one can consider ${\cal H}$ as the unit sphere of $T_p\m$, $i$ as the constant map $i(x)=p$ and $n(x)=x$.
Now, let us consider the product space $\mm$ with the standard product metric and let us call $h$ to the parameter in $\r$. Let $\psi(x,t)$ be the graph given by the height function $f(t)$ which only depends on the distance function, that is, the graph in $\mm$ parameterized as $$\label{psi}
\psi(x,t)=(\exp_{i(x)}(t\ n(x)),f(t))=(\varphi(x,t),f(t)).$$
Then, one has $$\begin{array}{l}
\overline{\partial}_{x_i}=\partial_{x_i},\qquad i=1,\ldots,n-1\\
\overline{\partial}_t=\partial_t+f'(t)\partial_h,
\end{array}$$ where $x=(x_1,\ldots,x_{n-1})$ are local coordinates in ${\cal H}$. Here, for instance, $\partial_{x_i}$ denotes the vector field $\frac{\partial\ }{\partial x_i}$ in $\mm$ and $\overline{\partial}_{x_i}$ the corresponding vector field in the graph.
If $\langle,\rangle=g+dh^2$ stands for the product metric in $\mm$, then from the Gauss lemma we obtain $$\langle\overline{\partial}_{x_i},\overline{\partial}_{t}\rangle=g(\partial_{x_i},\partial_{t})=0.$$
Hence, the pointing upwards unit normal of the graph is $$N=\frac{1}{\sqrt{1+f'(t)^2}}(-f'(t)\partial_t+\partial_h).$$
So, if we denote by $\overline{\nabla}$ the Levi-Civita connection in $\mm$, it is easy to see that $$-\overline{\nabla}_{\overline{\partial}_{t}}N=\frac{f''(t)}{(1+f'(t)^2)^{3/2}}\ \overline{\partial}_{t}.$$ In particular, $\overline{\partial}_{t}$ is a principal direction with associated principal curvature $$k_n=\frac{f''(t)}{(1+f'(t)^2)^{3/2}}.$$ Observe that this principal curvature does not depend on either ${\cal H}$, or $\m$, or its metric.
In order to compute the rest of principal curvatures of the graph we will focus on the directions which are orthogonal to $\overline{\partial}_{t}$, that is, the ones generated by $\overline{\partial}_{x_i}$.
Let $\gamma(t)=\varphi(x_0,t)$ be a geodesic in $\m$ and $J(t)$ a Jacobi field along $\gamma(t)$ with $g(J(t),\gamma'(t))=0$. If we denote by $\nabla$ the Levi-Civita connection in $\m$ then the second fundamental form of the graph satisfies $$II(J,J)=\langle-\overline{\nabla}_J N,J\rangle=g(-\nabla_J\left(\frac{-f'(t)}{\sqrt{1+f'(t)^2}}\partial_t\right),J)=\frac{f'(t)}{\sqrt{1+f'(t)^2}}\ g(\nabla_J\partial_t,J).$$
On the other hand, since $J$ is a Jacobi field then $$\frac{D^2J}{dt^2}+R(J,\gamma')\gamma'=0,$$ where, as usual, we use the notation $\frac{DJ}{dt}$ for $\nabla_{\gamma'(t)}J$ and $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$.
Moreover, since $\frac{DJ}{dt}=\nabla_J\partial_t$ because $J$ is a Jacobi field, we obtain that $$\begin{aligned}
g(\nabla_J\partial_t,J )\left|_{t_0}\right.&=&g( J(0),\frac{DJ}{dt}(0))+\int_0^{t_0}\frac{d\ }{dt}g(\frac{DJ}{dt},J )\ dt\\
&=&g( J(0),\frac{DJ}{dt}(0) )+\int_0^{t_0}\left(\left|\frac{DJ}{dt}\right|^2-g(R(J,\gamma')\gamma',J)\right)\ dt.\end{aligned}$$
Observe that since $\partial_t(x,0)=n(x)$ we have that if $i$ is an immersion then $$\label{e1}
g( J(0),\frac{DJ}{dt}(0) )=g( \nabla_J\partial_t,J )(0)=-II_{\cal H}(J(0),J(0)),$$ where $II_{\cal H}$ will denote the second fundamental form of $i:{\cal H}{\longrightarrow}\m$ in the direction of $n$. On the other hand, the amount $g( J(0),\frac{DJ}{dt}(0) )$ vanishes if $i$ is constant (as in the polar geodesic coordinates).
Given a vector field $V(t)$ along $\gamma_{|[0,t_0]}$, with $g(V,\gamma'(t))=0$, we will define the [*index form*]{} of $V$ as $$\label{e2}
{\cal I}_{(t_0,{\cal H})}(V,V)=g(V(0),\frac{DV}{dt}(0) )+\int_0^{t_0}\left(\left|\frac{DV}{dt}\right|^2-g(R(V,\gamma')\gamma',V)\right)\ dt.$$
With all of this, we obtain
\[l1\] In the previous conditions the graph $\psi(x,t)$ given by (\[psi\]) has a principal curvature $k_n$, with principal direction $\overline{\partial}_t$, given by $$k_n=\frac{f''(t)}{(1+f'(t)^2)^{3/2}}.$$ Moreover, the second fundamental form of the graph at $\psi(x_0,t_0)$ for a tangent vector $v_0$ perpendicular to $\overline{\partial}_t$ can be computed as follows: Take a perpendicular Jacobi field $J(t)$ along the geodesic $\gamma(t)=\varphi (x_0,t)$ with $J(t_0)=v_0$ then the second fundamental form is given by $$\label{e3}
II(v_0,v_0)=II(J(t_0),J(t_0))=\frac{f'(t_0)}{\sqrt{1+f'(t_0)^2}}\ {\cal I}_{(t_0,{\cal H})}(J,J).$$
Comparison results. {#s3}
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The above lemma will help us to compare the principal curvatures of two graphs $\psi_j(x,t)$, $j=1,2$, with the same height function in two different product spaces $\m_j\times\r$. For that, we need to relate the index forms in the manifolds $\m_1$ and $\m_2$.
Thus, let $(\m_1,g_1)$, $(\m_2,g_2)$ be two Riemannian manifolds with $dim(\m_1)\leq dim(\m_2)$ and ${\cal H}_j$, $j=1,2,$ two smooth manifolds with $dim({\cal H}_j)= dim(\m_j)-1$. Consider smooth maps $i_j:{\cal H}_j{\longrightarrow}\m_j$ and $n_j:{\cal H}_j{\longrightarrow}T\m_j$ such that $n_j(x)\in T_{i_j(x)}\m_j$ is a unit vector with $g(d(i_j)_x(v),n_j(x))=0$ for all $v\in T_x {\cal H}_j$.
Moreover, assume that the maps $$\varphi_j(x,t)=\exp_{i_j(x)}(t\ n_j(x)),\qquad (x,t)\in {\cal H}_j\times \bar{I},$$ are smooth and ${\varphi_j}_{|{\cal H}_j\times I}$ a global diffeomorphism onto its image, where $I$ is an open real interval such that 0 is in its closure $\bar{I}$. Then, we consider the two graphs $$\label{psi2}
\psi_j(x,t)=(\exp_{i_j(x)}(t\ n_j(x)),f(t))=(\varphi_j(x,t),f(t)),$$ for the same height function $f(t)$ with $f'(t)\geq0$.
In order to compare the second fundamental forms of both graphs, let $\overline{\i}:{\cal H}_1{\longrightarrow}{\cal H}_2$ be an immersion, $x_0\in {\cal H}_1$ and $\gamma_j:[0,t_0]{\longrightarrow}\m_j$ be the geodesics $$\gamma_1(t)=\varphi_1(x_0,t),\qquad \gamma_2(t)=\varphi_2(\overline{\i}(x_0),t).$$
\[t1\] In the previous conditions assume $K^1_{\gamma_1(t)}(\pi_1)\leq K^2_{\gamma_2(t)}(\pi_2)$, $t\in[0,t_0]$, for each planes $\pi_j$, where $K^j_{\gamma_j(t)}(\pi_j)$ is the sectional curvature of a plane $\pi_j$ in $\m_j$ containing $\gamma'_j(t)$. If:
1. $i_j$ are constant, or
2. $i_j$ are immersions and $II_{{\cal H}_1}(v,v)\leq II_{{\cal H}_2}(w,w)$ for all $v\in di_1(T_{x_0}{\cal H}_1), w\in di_2(T_{\overline{\i}(x_0)}{\cal H}_2)$ with $|v|=|w|$,
then the second fundamental forms of the graphs satisfy $$II_1(V,V)\geq II_2(W,W)$$ for all tangent vectors $V,W$ such that $|V|=|W|$ and $\langle V,\gamma'_1(t_0)\rangle=0=\langle W,\gamma'_2(t_0)\rangle$.
In particular, every principal curvature of the graph $\psi_1$ at $\varphi_1(x_0,t_0)$ is greater than or equal to every principal curvature of the graph $\psi_2$ at $\varphi_2(\overline{\i}(x_0),t_0)$.
Let $V\in T_{\gamma_1(t_0)}\m_1$ and $W\in T_{\gamma_2(t_0)}\m_2$ with $$|V|=1=|W|\quad \text{ and }\quad g_1(V,\gamma_1'(t_0))=0=g_2(W,\gamma_2'(t_0)).$$ As above, $V$ and $W$ are identified as tangent vectors to the graphs at the points $\psi_j(\gamma_j(t_0),f(t_0))$, $j=1,2$, respectively.
There exist unique perpendicular Jacobi fields $J_j:[0,t_0]{\longrightarrow}T\m_j$, $j=1,2$, along the geodesics $\gamma_1$ and $\gamma_2$ respectively, such that $J_1(t_0)=V$, $J_2(t_0)=W$ with the additional property: $J_j(0)=0$ if $i_j$ is constant, or $-dn(J_j(0))+\frac{DJ_j}{dt}(0)$ is proportional to $\gamma_j'(0)$ if $i_j$ is an immersion (see, for instance, [@Wa]).
Let $\{e_k(t)\}_{k=1}^m$ be an orthonormal basis of parallel vector fields along $\gamma_1$, which are perpendicular to $\gamma_1'(t)$, with $m=dim(\m_1)-1$, and such that $e_1(t_0)=J_1(t_0)$. In a similar way, let $\{b_k(t)\}_{k=1}^n$ be an orthonormal basis of parallel vector fields along $\gamma_2$, which are perpendicular to $\gamma_2'(t)$, with $n=dim(\m_2)-1$, and such that $b_1(t_0)=J_2(t_0).$
We define the functions $a_k(t)$ as the ones given by the equality $$J_1(t)=\sum_{k=1}^m a_k(t) e_k(t).$$ And consider a new vector field along $\gamma_2(t)$ given as $$W(t)=\sum_{k=1}^m a_k(t) b_k(t).$$
Since $W(t_0)=J_2(t_0)$ the minimizing property of the Jacobi fields (see [@Wa]) gives us $${\cal I}_{(t_0,{\cal H}_2)}(J_2,J_2)\leq {\cal I}_{(t_0,{\cal H}_2)}(W,W).$$ Hence, $$II_{2}(J_2(t_0),J_2(t_0))=\frac{f'(t_0)}{\sqrt{1+f'(t_0)^2}}\ {\cal I}_{(t_0,{\cal H}_2)}(J_2,J_2)\leq \frac{f'(t_0)}{\sqrt{1+f'(t_0)^2}}\ {\cal I}_{(t_0,{\cal H}_2)}(W,W).$$
Since $|J_1(t)|=|W(t)|$, $|\frac{DJ_1}{dt}(t)|=|\frac{DW}{dt}(t)|$ and $II_{{\cal H}_1}(J_1(0),J_1(0))\leq II_{{\cal H}_2}(W(0),W(0))$ when $i_j$ is an immersion, then we obtain from (\[e1\]), (\[e2\]), (\[e3\]) and the previous inequality that $$II_{2}(J_2(t_0),J_2(t_0))\leq II_{1}(J_1(t_0),J_1(t_0)),$$ as we wanted to show.
A different proof of this result was given in [@GL] when $dim(\m_1)=dim(\m_2)=2$ as well as many applications.
Theorem \[t1\] gives us a criterium for comparing all the principal curvatures of a graph at a point with all the principal curvatures of another graph at the corresponding point. Now, we look for some weaker conditions in order to compare the mean curvature of both graphs.
\[t2\] In the previous conditions assume $dim(\m_1)=dim(\m_2)$ and the metric of $\m_1$ can be written as $$\label{model}
g_1=dt^2+G(t)g_0,$$ where $g_0$ is the (n-1)-dimensional metric of a space form. If $Ric^1(\gamma_1'(t))\leq Ric^2(\gamma_2'(t))$, where $Ric^j(\gamma_j'(t))$ denotes the Ricci curvature in the direction of the unit vector $\gamma_j'(t)$, and
1. $i_j$ are constant, or
2. $i_j:{\cal H}_j{\longrightarrow}\m_j$ are immersions and their mean curvatures $H_{{\cal H}_j}$ satisfy $H_{{\cal H}_1}(x_0)\leq H_{{\cal H}_2}(\overline{\i}(x_0))$,
then the mean curvatures $H_j$ of the graphs in $\m_j\times\r$ satisfy $$H_1(\gamma_1(t_0))\geq H_2(\gamma_2(t_0)).$$
In order to compare the mean curvatures of the graphs given by (\[psi2\]), we use the trace of the second fundamental forms at points $\psi_1(x_0, t_0)$ and $\psi_2(\overline{\i}(x_0), t_0)$. For this, from (\[e3\]) and the fact that the function $f(t)$ is increasing, it is sufficient to compare the corresponding sums of the index forms.\
Let $\{e_k(t)\}_{k=1}^{n-1}$ be an orthonormal basis of parallel fields along $\gamma_1(t)$, orthogonal to $\gamma'_1(t)$, and let $\{b_k(t)\}_{k=1}^{n-1}$ be an orthonormal basis of parallel fields along $\gamma_2(t)$, orthogonal to $\gamma'_2(t)$ with $t \in [0, t_0]$. Then, there exist unique perpendicular Jacobi fields $J_k^j:[0,t_0]\longrightarrow TM_j$, with $j=1,2$ and $k=1,...,n-1$ along $\gamma_1(t)$ and $\gamma_2(t)$ respectively, such that
1. $J_k^1(t_0)=e_k(t_0), \quad J_k^2(t_0)=b_k(t_0), \quad k=1,...,n-1.$
2. $J_k^j(0)=0 $ if $i_j$ are constant maps, and so $$\sum_{k=1}^{n-1} g_j(J_k^j(0),\frac{DJ_k^j}{dt}(0))=0, \quad j=1,2; \quad \mbox{or}$$ $-dn(J_k^j(0))+\frac{DJ_k^j}{dt}(0)$ is proportional to $\gamma'_j(0) $ if $i_j$ are immersions (see, for instance, [@Wa]), and so $$\sum_{k=1}^{n-1}g_1(J_k^1(0),\frac{DJ_k^1}{dt}(0))=-(n-1)\,H_{\mathcal{H}_1}(x_0),$$ and $$\sum_{k=1}^{n-1}g_2(J_k^2(0),\frac{DJ_k^2}{dt}(0))=-(n-1)\,H_{\mathcal{H}_2}(\overline{\i}(x_0)).$$
From (\[model\]) the previous Jacobi fields $J_k^1$ in $\m_1$ satisfy $$|J_k^1(t)|=|J_i^1(t)| \quad \mbox{and} \quad |\frac{DJ_k^1}{dt}(t)|=|\frac{DJ_i^1}{dt}(t)| ,\qquad \mbox{with} \quad i, k = 1,...,n-1.$$
Now, we define the functions $a_{ik}$ as the ones given by the equalities $$J_i^1(t)=\sum_{k=1}^{n-1} a_{ik}(t) \,e_k(t), \quad i=1,...,n-1.$$ Consider the new vector fields $W_i(t)$ along $\gamma_2(t)$ given by $$W_i(t)=\sum_{k=1}^{n-1} a_{ik}(t) \,b_k(t), \quad i=1,...,n-1.$$ In these conditions, $a_{ik}(t_0)=\delta_{ik}$ and $W_i(t_0)=J_i^2(t_0)$. Moreover, by construction, $$|J_i^1(t)|=|W_i(t)| \quad \mbox{and} \quad |\frac{DJ_i^1}{dt}(t)|=|\frac{DW_i}{dt}(t)|.$$
Observe now that if $i_j$ are constant maps $$\sum_{i=1}^{n-1} g(J_i^2(0),\frac{DJ_i^2}{dt}(0))= \sum_{i=1}^{n-1} g(W_i(0),\frac{DW_i}{dt}(0))= \sum_{i=1}^{n-1} g(J_i^1(0),\frac{DJ_i^1}{dt}(0)) = 0.$$ On the other hand, if the maps $i_j$ are immersions, then $$\sum_{k=1}^{n-1}g(J_k^2(0),\frac{DJ_k^2}{dt}(0))=-(n-1)\,H_{\mathcal{H}_2}(\overline{\i}(x_0)) \leq -(n-1)\,H_{\mathcal{H}_1}(x_0) = \sum_{k=1}^{n-1}g(J_k^1(0),\frac{DJ_k^1}{dt}(0)).$$
In addition, the fields ${W_i(t)}$ are also orthogonal on $[0,t_0]$ by construction, and $|W_i(t)|=|W_1(t)|$, $i=1,\ldots,n-1$. Hence $$-\sum_{i=1}^{n-1}g_2(R(W_i,\gamma_2')\gamma_2',W_i) = -(n-1)
|W_1|^2\,Ric^2(\gamma_2'(t))\leq$$ $$\leq -(n-1)|J_1|^2\,Ric^1(\gamma_1'(t)) =-\sum_{i=1}^{n-1}g_1(R(J_i^1,\gamma_1')\gamma_1',J_i^1),$$
In these conditions, and by the minimizing property of the Jacobi fields, it is obtained
$$\sum_{i=1}^{n-1} {\cal I}_{(t_0,{\mathcal{H}_2})}(J_i^2(t_0),J_i^2(t_0))=$$ $$=\sum_{i=1}^{n-1} g_2(J_i^2(0),\frac{DJ_i^2}{dt}(0))+\int_0^{t_0}\left(\sum_{i=1}^{n-1}\left|\frac{DJ_i^2}{dt}\right|^2-\sum_{i=1}^{n-1}g_2(R(J_i^2,\gamma_2')\gamma_2',J_i^2)\right)\ dt$$ $$\leq \sum_{i=1}^{n-1} g_2(W_i(0),\frac{DW_i}{dt}(0))+\int_0^{t_0}\left(\sum_{i=1}^{n-1}\left|\frac{DW_i}{dt}\right|^2-\sum_{i=1}^{n-1}g_2(R(W_i,\gamma_2')\gamma_2',W_i)\right)\ dt$$ $$\leq \sum_{i=1}^{n-1} g_1(J_i^1(0),\frac{DJ_i^1}{dt}(0))+\int_0^{t_0}\left(\sum_{i=1}^{n-1}\left|\frac{DJ_i^1}{dt}\right|^2-\sum_{i=1}^{n-1}g_1(R(J_i^1,\gamma_1')\gamma_1',J_i^1)\right)\ dt$$ $$=\sum_{i=1}^{n-1} {\cal I}_{(t_0,{\mathcal{H}_1})}(J_i^1(t_0),J_i^1(t_0))$$ as we wanted to show.
Observe that a manifold $\m_1$ whose metric is described by (\[model\]) in geodesic polar coordinates is classically known as a [*model manifold*]{} (see [@GW]).
Existence of barriers in $\m^n \times \r$. {#s4}
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Our comparison results will allow us to extend some results only known for $\m^n\times\r$ when $\m^n$ is a space form to general ambient spaces $\m^n\times\r$. Thus, in this section we will follow our approach in [@GL] for obtaining some existence and non existence results for hypersurfaces in $\m^n\times\r$.
From now on, we denote by $\m^n (c)$ the complete simply connected n-dimensional space form of constant curvature $c$, that is a hyperbolic space if $c<0$, the Euclidean space if $c=0$ or a sphere if $c>0$. Let $s_{c,n}= \frac{n-1}{n} \sqrt{-c}$ be the infimum of the mean curvature of the topological spheres of constant mean curvature in $\m^n(c) \times \r$ when $c<0$. Also, for each $H_0>0$ ($H_0 > s_{c,n}$ if $c<0$) we denote by $r_{c,n}(H_0)$ the radius of the topological sphere of constant mean curvature $H_0$ in $\m^n(c) \times \r$, and for each $K_0>0$ we will denote by $r_{c,n}^*(K_0)$ the radius of the topological sphere of constant Gauss-Kronecker curvature $K_0>0$ in the same ambient space.
Let us start with a topological sphere $S$ of constant mean curvature $H_0$ in $\m^n (c)\times \r$ (see, for instance, [@AR; @AEG; @BE2; @HH; @PR]). Observe that $S$ is unique up to isometries of the ambient space and only exists for $H_0>s_{c,n}$ if $c<0$. Moreover, $S$ is rotational with respect to a vertical axis and symmetric with respect to a horizontal slice. In particular, $S$ is a bigraph over a geodesic ball of $\m^n (c)$ of radius $r_{c,n}(H_0)>0$.
Thus, let $p\in \m^n (c)$ and $(x,t)$ be geodesic polar coordinates around $p$. Since $S$ is a rotational surface, the lower part of $S$ can be considered as a graph over the geodesic ball centered at $p$ and radius $r_{c,n}(H_0)$, with height function $h(t)$ which only depends on the distance function $t$ to the point $p$. Moreover, $h(t)$ is strictly increasing. Hence, this part of the hypersurface $S$ of constant mean curvature can be described as $$\psi_1(x,t)=(x,t,h(t)) \in \m^n (c) \times \r.$$ Note that, for convenience, we have deleted the parametrization $\varphi$ (given by (\[eliminado\])) in the previous expression.
Now, given an n-dimensional Riemannian manifold $\m^n$ and geodesic polar coordinates $(x,t)$ around a point $q\in\m^n$, which are well defined for $0<t\leq r_{c,n}(H_0)$, we can consider the new immersion $$\psi_2(x,t)=(x,t,h(t))\in\m^n \times \r.$$
Applying the same process for the upper part of $S$, we obtain a sphere $S^{\ast}$ in $\m^n \times \r$ which is a bigraph over the geodesic ball of radius $r_{c,n}(H_0)$ centered at $q$.
We remark that $S^{\ast}$ is symmetric with respect to a horizontal slice as $S$, and any vertical translation of $S^{\ast}$ is congruent to $S^{\ast}$. However, $S^{\ast}$ depends strongly on the point $q\in\m^n$, i. e. if we start with another point $\widetilde{q}\in\m^n$ and obtain a new surface $\widetilde{S^{\ast}}$ following the same process then $S^{\ast}$ and $\widetilde{S^{\ast}}$ are not isometric in general. If the Ricci curvature in the radial directions on $B_r(q) \subset \m$ are greater than or equal to $c$, for all the geodesics $\gamma(t)$ in $\m$ emanating from $q$, using Theorem 2, we have that the mean curvature of $S^{\ast}$ satisfies $H(S^{\ast}) \leq H_0$. With all of this we obtain
\[t3\] Let $B_r$ be a closed geodesic ball of radius $r>0$ in an n-dimensional Riemannian manifold $\m^n$, and $c$ the minimum of the Ricci curvature in the radial directions of unit vectors $\gamma'(t)$ on $B_r$, for all the geodesics $\gamma(t)$ in $\m^n$ emanating from the center of $B_r$. Consider $H_0>0$ such that $r_{c,n}(H_0) = r$. Then, there is no vertical graph over $B_r$ with minimum of its mean curvature satisfying $min(H)\geq H_0$.
Assume $\Sigma$ is a graph over $B_r$ with $min(H)\geq H_0$ for a unit normal $N$. Without loss of generality, we assume that the unit normal $N$ points upwards.
Let $q\in\m^n$ be the center of the geodesic disk $B_r$ and consider the sphere $S^{\ast}$ centered at $q$ previously obtained, which has mean curvature smaller than or equal to $H_0$ for its inner normal.
Move the sphere $S^{\ast}$ up until $\Sigma$ is below $S^{\ast}$, and go down until $S^{\ast}$ intersects $\Sigma$ for the first time. Then, the classical maximum principle for mean curvature asserts that both surfaces must agree locally. In particular, $\Sigma$ and $S^{\ast}$ have constant mean curvature $H_0$ and $\Sigma$ agrees with the lower hemisphere of $S^{\ast}$. However, this is a contradiction because $S^{\ast}$ is not a strict graph over the boundary of $B_r$ since its unit normal is horizontal at those points.
\[t4\] Let $B_r$ be a closed geodesic ball of radius $r>0$ in an n-dimensional Riemannian manifold $\m^n$, $c:= min \{ K_p (\pi) : \partial_t \in \pi, p \in B_r \}$ be the minimum of the radial sectional curvatures on $B_r$, and $K_0>0$ such that $r_{c,n}^*(K_0) = r$. Then, there is no vertical graph over $B_r$ with minimum of its Gauss-Kronecker curvature satisfying $min(K)\geq K_0$ and a point with definite second fundamental form.
The proof follows the same process that in Theorem \[t3\], taking now a sphere with constant Gauss-Kronecker curvature in $\m^n(c) \times \r$ (see, for instance, [@EGR; @ES]), and using Theorem \[t1\]. The requirement of the graph of having a point with definite second fundamental form is now needed for using the maximum principle.
It should be observed that a similar result to Theorem \[t4\] is possible for any r-mean curvature $H_r$, with $2\leq r\leq n$, and not only for the Gauss-Kronecker curvature $H_n$.
\[t5\] Let $\m^n$ be a complete, simply connected Riemannian manifold with injectivity radius $i>0$ and $c \in \r$ the infimum of its Ricci curvature on $\m^n$. Consider a properly embedded hypersurface $\Sigma$ in $\m^n \times \r$ with mean curvature $H \geq H_0 >0$, ($H_0 > s_{c,n}$ if $c<0$). If $r_{c,n}(H_0)<i$ then the mean convex component of $\Sigma$ cannot contain a closed geodesic ball in $\m^n \times \r$ of radius greater than or equal to the extrinsic semi-diameter of a sphere with constant mean curvature $H_0$ in $\m^n(c) \times \r$.
The proof is a consequence of the maximum principle and follows the same process that [@GL Theorem2], using now Theorem \[t2\].
Also observe that a weaker version of Theorem \[t5\] is possible for the r-mean curvatures $H_r$, $2\leq r\leq n$, using Theorem \[t1\].
Let $\m^n$ be a Hadamard manifold, that is, a complete simply connected Riemannian manifold with non-positive sectional curvature. Since its injectivity radius is $i=\infty$, we obtain as a consequence of the previous result:
\[c1\] Let $H_0>0$ and $\m^n$ be a Hadamard manifold with infimum of its Ricci curvature $c>-\infty$. Then, there exists no entire horizontal graph in $\m^n \times \r$ with mean curvature $H \geq H_0 > s_{c,n}$.
Let us denote by $S_{\frac{n-1}{n}}$ the simply connected rotational entire vertical graph with constant mean curvature $H=\frac{n-1}{n}$ in $\h^n \times \r$. This graph has been described in [@BE2]. Again, as a consequence of our comparison results, if we consider the corresponding entire vertical graph $S_{\frac{n-1}{n}}^{\ast}$ in $\m^n\times\r$, we have:
\[c2\] Let $\m^n$ be an n-dimensional Hadamard manifold with Ricci curvature smaller than or equal to $-1$. Assume $\Sigma$ is an immersed hypersurface in $\m^n \times \r$ with mean curvature $H \leq \frac{n-1}{n}$ and cylindrically bounded vertical ends. Then $\Sigma$ must have more than one end.
We obtain now a generalization to $\m^n\times\r$ of a theorem proven in [@BE2] for the product space $\h^n\times\r$.
\[t7\] Let $\m^n$ be an n-dimensional Hadamard manifold with sectional curvature pinched between $-c^2$ and $-1$, for a constant $c \geq 1$. Let $\Omega$ be a bounded domain in $\m^n \times \{0\}$, with boundary given by a compact embedded hypersurface $\Gamma$. Assume all the principal curvatures of $\Gamma$ are greater than $c$, then for any $H_0 \in [0, \frac{n-1}{n}]$ there exists a graph $h$ over $\Omega$ with constant mean curvature $H_0$ and zero boundary data.
Moreover, if $\Sigma$ is a compact hypersurface immersed in $\m^n \times \r$ with boundary $\Gamma$ and constant mean curvature $H_0$ then, up to a symmetry with respect to $\m^n \times \{0\}$, $\Sigma$ agrees with the previous graph.
Observe that $\Omega$ must be a convex bounded domain in $\m^n$ and homeomorphic to a ball, and $\Gamma$ must be homeomorphic to a sphere (see [@Alex]).
Let $m_0$ be the minimum of the principal curvatures of $\Gamma$. Since $m_0>c$ we can take a radius $R_0$ big enough such that for every $R>R_0$ the geodesic spheres of radius $R$ in the hyperbolic space of seccional curvature $-c^2$ have principal curvatures smaller than $m_0$. As the sectional curvature of $\m^n$ is bigger than or equal to $-c^2$, the geodesic spheres in $\m^n$ of radius $R\geq R_0$ have principal curvatures smaller than $m_0$.
Let $p\in\Gamma$ and $\gamma_p(t)$ be the geodesic in $\m^n$ starting at $p$ with initial speed given by the unit normal to $\Gamma$ pointing to $\Omega$. It is clear that the geodesic sphere $S_p(R)\subseteq\m^n$ centered at $\gamma_p(R)$ and radius $R$ is tangent to $\Gamma$ at $p$. Moreover, if $R\geq R_0$ the open geodesic ball bounded by $S_p(R)$ contains a punctured neighborhood of $p\in\Gamma$ because the principal curvatures of $S_p(R)$ are bigger than the principal curvatures of $\Gamma$ at $p$ for the same interior unit normal.
Let $S_0\subseteq\m^n$ be a geodesic sphere such that $\Gamma$ is contained in the geodesic ball bounded by $S_0$, and the distance from $S_0$ to $\Gamma$ is greater than or equal to $R_0$. Then, consider the map $G:\Gamma\rightarrow S_0$ defined in the following way: given $p\in\Gamma$ the point $G(p)$ is given by the intersection of the geodesic $\gamma_p(t)$ for $t\geq 0$ with $S_0$. It is well known that the previous intersection is given by a unique point due to the convexity of the geodesic spheres (see, for instance, [@Alex]).
Thus, if we denote by $S_p$ the geodesic sphere centered at $G(p)$ passing across $p\in\Gamma$, then we have shown that $S_p$ is tangent to $\Gamma$ at $p$ and a punctured neighborhood of $p$ in $\Gamma$ is contained in the open geodesic ball bounded by $S_p$. In fact, we assert
[*Claim:*]{} For every $p\in\Gamma$ the closed geodesic ball bounded by $S_p$ contains to $\Gamma$.
Observe that if $G:\Gamma\rightarrow S_0$ is injective then the Claim would be proven. Indeed, if there existed $p_1\in \Gamma$ such that $\Gamma\not\subseteq S_{p_1}$ then there would be a point $p_2\neq p_1$ such that $d(p_2,G(p_1))\geq d(p,G(p_1))$ for all $p\in\Gamma$. Thus, the geodesic sphere centered at $G(p_1)$ passing across $p_2$ is tangent to $\Gamma$ and contains $\Gamma$ in its interior, and so $G(p_2)=G(p_1)$.
Hence, assume there exist two points $p_1,p_2\in\Gamma$ such that $G(p_1)=G(p_2)$. In such a case, we have shown that $p_1$ and $p_2$ are two strict local maxima for the distance function $\varrho(p)$ from $p\in\Gamma$ to the fixed point $G(p_1)=G(p_2)$. Now, we distinguish two cases depending on the dimension of $\Gamma$:
1. If dim($\Gamma$)$\geq2$ then we can use the mountain pass lemma for the function $\varrho$ and there must exist a third point $p_3$ which is a saddle point for $\varrho$. Thus, the geodesic sphere $\widetilde{S}_{p_3}$ centered at $G(p_1)$ passing across $p_3$ is tangent to $\Gamma$. Therefore, depending on the orientation of the unit normal to $\Gamma$, we have that $\widetilde{S}_{p_3}=S_{p_3}$ or $\widetilde{S}_{p_3}\cap S_{p_3}=\{p_3\}$. But, this contradicts that $p_3$ is a saddle point, because a punctured neighborhood of $p_3$ is contained in the interior of the geodesic ball bounded by $S_{p_3}$.
2. If dim($\Gamma$)$=1$ then we can consider the two closed arcs $\Gamma_1$ and $\Gamma_2$ of $\Gamma$ joining $p_1$ and $p_2$. Since $p_1$ and $p_2$ are strict local maxima for the function $\varrho$, there must exist $p_3\in\Gamma_1$ and $p_4\in\Gamma_2$ different from $p_1$ and $p_2$ which are local minima for $\varrho$. Assume $\varrho(p_3)\leq \varrho(p_4)$, then from the convexity of $\overline{\Omega}$ the geodesic arc $\Lambda$ joining $p_3$ and $p_4$ is contained in $\overline{\Omega}$. But, $\Lambda\backslash\{p_3,p_4\}$ is contained in the open geodesic ball centered at $G(p_1)$ and radius $\varrho(p_4)$ from the convexity of the geodesic ball. This contradicts that $p_4$ is a minimum for $\varrho$.
Once the Claim is proven, consider a compact hypersurface $\Sigma$ immersed in $\m^n \times \r$ with boundary $\Gamma$ and constant mean curvature $H_0$.
Let $S$ be the rotational entire graph with constant mean curvature $H_0$ in $\h^n\times\r$ for its unit normal pointing upwards. Consider a point $p\in\Gamma\subseteq\m^n$ and the associated entire graph $S^{\ast}\subseteq\m^n\times\r$ when we use geodesic polar coordinates at $G(p)\in\m^n$. Up to a vertical translation we can assume that $S^{\ast}\cap\m^n\times\{0\}$ is the geodesic sphere centered at $G(p)$ and containing to $p$ in $\m^n\times\{0\}$. Thus, from the previous Claim, $\Gamma\times\{0\}$ is contained in the closed mean convex component of $S^{\ast}$, and $(p,0)\in S^{\ast}$.
Let us also denote by $\overline{S^{\ast}}$ the reflection of $S^{\ast}$ with respect to the horizontal slice $\m^n\times\{0\}$. The entire graphs $S^{\ast}$ and $\overline{S^{\ast}}$ are congruent, and from Theorem \[t1\] we obtain that they have mean curvature $H\geq H_0$ for its unit normal pointing to the mean convex component.
As $\Sigma$ is compact we can move vertically $S^{\ast}$ in such a way that $\Sigma$ is completely contained in the mean convex component of $S^{\ast}$. Now, from the maximum principle, if we move back $S^{\ast}$ then the surfaces $\Sigma$ and $S^{\ast}$ do not intersect until $S^{\ast}$ is in its initial position. The same is true for $\overline{S^{\ast}}$.
Therefore, for every $p\in\Gamma$ the hypersurface $\Sigma$ is contained in the compact domain determined by the intersection of the mean convex components of $S^{\ast}$ and $\overline{S^{\ast}}$. In particular, the interior of $\Sigma$ is contained in the solid cylinder $\Omega\times\r$, and if $\Sigma$ was given by the graph of a function $h$ then its height is bounded a priori and so is its gradient at the boundary $\Gamma$.
Now we can prove that there exists a graph $h$ over $\Omega$ with constant mean curvature $H_0 \in [0, \frac{n-1}{n}]$ and zero boundary data. That is, we want to solve the following Dirichlet problem $$\left\{\begin{array}{ll}
{\displaystyle div\left(\frac{\nabla h}{\sqrt{1+|\nabla h|^2}}\right)=n\,H_0,}& \qquad\text{in }\Omega\\[2mm]
h=0&\qquad\text{on }\Gamma
\end{array}\right.$$ where the divergence and gradient $\nabla h$ are taken with respect to the metric on $\m^n$ (see [@Sp]).
We have proven the existence of height estimates and gradient estimates at the boundary. Hence, from [@Sp], we also have global gradient estimates, and the existence of $h$ follows from the classical elliptic theory (see [@GT] and [@Sp]).
Finally, we want to show that if $\Sigma$ is a compact hypersurface in $\m^n\times\r$ with constant mean curvature $H_0$ and boundary $\Gamma$, then it is the graph of the previous function $h$ or $-h$.
First, let us observe that $\Sigma$ is a vertical graph. In fact, we have shown that $\Sigma$ is contained in the cylinder $\overline{\Omega}\times\r$ and $\Sigma$ has no interior point in $\Gamma\times\r$. Thus, we can use the maximum principle with respect to horizontal slices from the highest point of $\Sigma$ to the lowest point of $\Sigma$, which proves that $\Sigma$ is a graph.
Moreover, let $\Sigma_0$ be the graph of $h$ or $-h$ which points in the same direction (upwards or downwards) as $\Sigma$. Moving $\Sigma$ vertically upwards until $\Sigma$ and $\Sigma_0$ are disjoint, and coming down again we observe that, from the maximum principle, $\Sigma$ cannot touch $\Sigma_0$ till the boundaries agree. Hence, $\Sigma$ is above $\Sigma_0$. Repeating the same process, but moving now $\Sigma$ vertically downwards, one has $\Sigma$ is below $\Sigma_0$. Therefore, $\Sigma$ and $\Sigma_0$ agree, as we wanted to show.
It is an interesting open question if, under the previous conditions on $\m^n$, there exists an entire vertical graph over $\m^n$ for every constant mean curvature $H_0\in(0,(n-1)/n]$.
\[t8\] Let $B_r$ be a closed geodesic ball of radius $r>0$ in $\m^n$, $c:= max \{ K_p (\pi) : \partial_t \in \pi, p \in B_r \}$ be the maximum of the radial sectional curvatures on $B_r$, and $K_0>0$ such that $r_{c,n}^*(K_0) = r$. Then, there exists a strictly convex graph $h_K$ over $B_r$ of constant Gauss-Kronecker curvature $K>0$ in $\m^n \times \r$ y $h_K |_{\partial B_r}=0$, for any $K<K_0$.
Let us consider a sphere $S$ in $\m^n(c) \times \r$ with positive constant Gauss-Kronecker curvature $K<K_0$. From Theorem \[t1\], the corresponding sphere $S^{\ast}$ in $\m^n\times\r$, using polar coordinates at the center of $B_r$, has Gauss-Kronecker curvature greater than or equal to $K$, and so it is a subsolution for the existence of the graph we are looking for. Thus, from [@Gu] (see also [@Sp]) there exists a strictly convex graph of constant Gauss-Kronecker curvature $K$ and zero boundary data.
[99999]{} U. Abresch, H. Rosenberg, *A Hopf differential for constant mean curvature surfaces in $\s^2 \times \r$ and $\h^2 \times \r$*. Acta Math. **193** (2004), nº 2, 141–174. J.A. Aledo, J.M. Espinar, J.A. Gálvez, *Height estimates for surfaces with positive constant mean curvature in $\m \times \r$*. Illinois J. Math. **52** (2008), nº 1, 203–211. S. Alexander, [*Locally convex hypersurfaces of negatively curved spaces*]{}. Proc. Amer. Math. Soc. [**64**]{} (1977), no. 2, 321–325. P. Bérard, R. Sa Earp. *Examples of H-hypersurfaces in $\h^n \times \r$ and geometric applications*. Mat. Contemp. [**34**]{} (2008), 19–51. M.F. Elbert, R. Sa Earp, [*Constructions of $H_r$-hypersurfaces, barriers and Alexandrov Theorem in $\h^n\times\r$*]{}, arxiv.org/pdf/1402.6886.pdf; preprint (2014). J.M. Espinar, J.A. Gálvez, H. Rosenberg, *Complete surfaces with positive extrinsic curvature in product spaces*. Comment. Math. Helv. **84** (2009), nº 2, 351–386. J.M. Espinar, H. Rosenberg, *Complete constant mean curvature surfaces and Berstein type teorems in $\m^2 \times \r$*. J. Diff. Geometry **82** (2009), nº 3, 611–628. J.A. Gálvez, V. Lozano, *Existence of barriers for surfaces with prescribed curvatures in $\m^2 \times \r$*. J. Diff. Equations **255** (2013), nº 7, 1828–1838. D. Gilbarg, N.S. Trudinger, *Elliptic partial differential equations of second order.* Springer–Verlag, Berlin, second edition, 1983. R.E. Green, H. Wu. [*Function Theory on Manifolds which possess a pole*]{}. Springer-Verlag, Lectures notes in mathematics 699, Berlin-Heidelberg-New York 1979. B. Guan, *The Dirichlet problem for Monge-Ampère equations in nonconvex domains and spacelike hypersurfaces of constant Gauss curvature*. Trans. Amer. Math. Soc. **350** (1998), nº 12, 4955-4971. W.Y. Hsiang y W.T. Hsiang, *On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact simmetric spaces*. Invent. Math. **98** (1998), nº 1, 39–58. B. Nelli, R. Sa Earp, W. Santos, E. Toubiana, *Uniqueness of H-surfaces in $\h^2 \times \r$, $|H| \leq 1/2$, with boundary one or two parallel horizontal circles*. Ann. Global Anal. Geom. **33** (2008), nº 4, 307–321. R. Pedrosa, M. Ritoré, *Isoperimetric domains in the Riemannian product of a circle with a simple connected space form and applications to free boundary problems*. Indiana Univ. Math. J. **48** (1999), nº 4, 1357–1394. J. Spruck, *Interior gradient estimates and existence theorems for constant mean curvature graphs in $\m^n\times\r$*. Pure Appl. Math. Q. **3** (2007), nº 3, 785–800. F.W. Warner, [*Extensions of the Rauch comparison theorem to submanifolds*]{}, Trans. Am. Math. Soc. [**122**]{} (1966) 341–356.
[^1]: The first author is partially supported by MICINN-FEDER, Grant No. MTM2010-19821 and by Junta de Andalucía Grant No. FQM325 and P09-FQM-5088.
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---
abstract: 'This paper investigates the conditions for producing rapid variations of solar energetic particle (SEP) intensity commonly known as “dropouts”. In particular, we use numerical model simulations based on solving the focused transport equation in the 3-dimensional Parker interplanetary magnetic field to put constraints on the properties of particle transport coefficients both in the direction perpendicular and parallel to the magnetic field. Our calculations of the temporal intensity profile of 0.5 and 5 MeV protons at the Earth show that the perpendicular diffusion must be small enough while the parallel mean free path should be long in order to reproduce the phenomenon of SEP dropouts. When the parallel mean free path is a fraction of 1 AU and the observer is located at $1$ AU, the perpendicular to parallel diffusion ratio must be below $10^{-5}$, if we want to see the particle flux dropping by at least several times within three hours. When the observer is located at a larger solar radial distance, the perpendicular to parallel diffusion ratio for reproducing the dropouts should be even lower than that in the case of 1 AU distance. A shorter parallel mean free path or a larger radial distance from the source to observer will cause the particles to arrive later, making the effect of perpendicular diffusion more prominent and SEP dropouts disappear. All these effects require that the magnetic turbulence that resonates with the particles must be low everywhere in the inner heliosphere.'
author:
- 'Y. Wang, G. Qin, M. Zhang and S. Dalla'
title: A numerical simulation of solar energetic particle dropouts during impulsive events
---
INTRODUCTION
============
Solar Energetic Particles (SEPs) encounter small-scale irregularities during transport in the large scale interplanetary magnetic field. The particles are scattered by the irregularities whose scales are comparable to the particles’ gyro radius. The parallel diffusion is produced by the pitch angle scattering, while the perpendicular diffusion is caused by crossing the local field line or following magnetic field lines randomly walking in space. Low-rigidity particles tend to follow more tightly along individual field lines, whereas high-rigidity particles can cross local field lines more easily. As a result, the perpendicular diffusion of lower rigidity particles is generally smaller than that of high-rigidity particles.
The diffusion coefficients of SEPs depend on the magnetic turbulence in the solar wind. Jokipii (1966) was the first to use the Quasi-Linear Theory (QLT) to calculate particle diffusion coefficients from magnetic turbulence spectrum. But later it was found that the observed particle mean free paths are usually much larger than the QLT results derived from a slab magnetic turbulence [@Palmer82]. According to [@matthaeus1990evidence], the slab model is not a good approximation to describe the Interplanetary Magnetic Field (IMF) turbulence because there is also a stronger two-dimensional (2D) component. @BieberEA94 showed that with a ratio of turbulence energy between slab and 2D components, $E^{slab}:E^{2D}=20:80$, the QLT was able to derive a parallel mean free path much better in agreement with observations. However, the perpendicular diffusion remained a puzzle for many years. It is shown that the particles’ perpendicular diffusion model of Field Line Random Walk (FLRW) based on QLT has difficulty to describe spacecraft observations and numerical simulations. Recently, the Non-Linear Guiding Center Theory (NLGC) [@Matthaeus2003ApJ...590L..53M] was developed to describe the perpendicular diffusion in magnetic turbulence. The perpendicular diffusion coefficient from the NLGC theory agrees quite satisfactorily with numerical simulations of particle transport in typical solar wind conditions.
Observations by the ACE and Wind spacecraft show that there are rapid temporal structures in the time profiles of $ \sim 20 $ keV nucleon$^{-1}$ to $ \sim 5 $ MeV nucleon$^{-1}$ ions during impulsive SEP events. The phenomenon is commonly known as “dropouts” or “cutoffs” in some cases. In the dropouts, the particle intensities exhibit short time scale (about several hours) variations, whereas, the cutoffs are referred to as some special dropouts in which the intensities suddenly decrease without recovery. They do not seem to be associated with visible local magnetic field changes [@mazur2000interplanetary; @Gosling2004Correlated; @chollet2008multispacecraft; @Droge2010]. Contrarily to the previous studies, by performing a detailed analysis of magnetic field topology during SEP events, [@trenchi2013solar; @trenchi2013observations] identified magnetic structures associated with SEP dropouts. found that SEP dropouts are generally associated with magnetic boundaries which represent the borders between adjacent magnetic flux tubes while [@trenchi2013observations], using the Grad-Shafranov reconstruction, identified flux ropes or current sheet associated with SEP maxima. The dropouts and cutoffs can be interpreted as a result of magnetic field lines that connect or disconnect the observer alternatively to the SEP source on the Sun. However, with perpendicular diffusion, particles can cross the field lines as they propagate in the interplanetary space. A strong enough perpendicular diffusion can efficiently diminish longitudinal gradients of fluxes. The dropouts and cutoffs provide a good chance for us to estimate the level of perpendicular diffusion in the interplanetary space.
In an effort to interpret the SEP dropouts, [@giacalone2000small] did a simulation of test particle trajectory in a model with random-walking magnetic field lines using Newton-Lorentz equation to study SEPs dropouts. It was found that the phenomenon is consistent with random-walking magnetic field lines. In addition, it was found from their simulations that, particle perpendicular diffusion relative to the Parker spiral due to the field line random walk can be significant, and the ratio of perpendicular diffusion to the parallel one relative to the Parker spiral can be as large as $2\%$. However, their perpendicular diffusion coefficients relative to the background magnetic field (instead of Parker spiral) could still be very small, so dropouts can be obtained. Recently, using the same technique as in [@giacalone2000small], [@Guo2013Small] found that in some condition, dropouts can be reproduced in the foot-point random motion model, but no dropout is seen in the slab + 2D model. In [@Droge2010], the large scale magnetic field is assumed to be a Parker spiral, and the observer is located at $1$ AU equatorial plane. At the start the observer is connected to the source region, and leaves the region after some time due to the effect of co-rotation. Based on a numerical solution of the focused transport equation, they found that in order to reproduce cutoffs, the ${\kappa _ \bot }/{\kappa _\parallel }$ should be as small as a few times of ${10^{ - 5}}$. In the turbulence view, some other mechanisms [@Ruffolo2003Trapping; @Chuychai2005Suppressed; @Chuychai2007Trapping; @Kaghashvili2006Propagation; @Seripienlert2010Dropouts] are also proposed to interpret the dropout phenomenon.
In this work, we use a Fokker-Planck focused transport equation to calculate the transport of SEPs in three-dimensional Parker interplanetary magnetic field. We intend to put constraints on the conditions of the perpendicular and parallel diffusion coefficients for observing the SEP dropouts and cutoffs. In SECTION 2 we describe our SEP transport model. In SECTION 3 simulation results are presented. In SECTION 4 the simulation results are discussed, and conclusions based on our simulations are made.
MODEL
=====
Our model is based on solving a three-dimensional focused transport equation following the same method in our previous research [e.g., @Qin2006JGRA..11108101Q; @Zhang2009ApJ...692..109Z; @Wang2012apj]. The transport equation of SEPs can be written as [@Skilling1971ApJ...170..265S; @Schlickeiser2002cra..book.....S; @Qin2006JGRA..11108101Q; @Zhang2009ApJ...692..109Z; @Droge2010; @HeEA11; @Wang2012apj; @zuo2013acceleration; @qin2013transport] $$\begin{aligned}
\frac{{\partial f}}{{\partial t}} = \nabla\cdot\left( \bm
{\kappa_\bot}
\cdot\nabla f\right)- \left(v\mu \bm{\mathop b\limits^ \wedge}
+ \bm{V}^{sw}\right)
\cdot \nabla f + \frac{\partial }{{\partial \mu }}\left(D_
{\mu \mu }
\frac{{\partial f}}{{\partial \mu }}\right) \nonumber \\
+ p\left[ {\frac{{1 - \mu ^2 }}{2}\left( {\nabla \cdot \bm{V}^
{sw} -
\bm{\mathop b\limits^ \wedge \mathop b\limits^ \wedge } :\nabla
\bm{V}^{sw} } \right) +
\mu ^2 \bm{\mathop b\limits^ \wedge \mathop b\limits^ \wedge} :
\nabla \bm{V}^{sw} }
\right]\frac{{\partial f}}{{\partial p}} \nonumber \\
- \frac{{1 - \mu ^2 }}{2}\left[ { - \frac{v}{L} + \mu \left
( {\nabla \cdot
\bm{V}^{sw} - 3\bm{\mathop b\limits^ \wedge \mathop b\limits^
\wedge} :\nabla \bm{V}^{sw} }
\right)} \right]\frac{{\partial f}}{{\partial \mu }},\label{dfdt}\end{aligned}$$ where $f(\bm{x},\mu,p,t)$ is the gyrophase-averaged particle distribution function as a function of time $t$, position in a non-rotating heliographic coordinate system $\bm{x}$, particle momentum $p$ and pitch angle cosine $\mu$ in the plasma reference frame. In the equation, $v$ is the particle speed, $\bm{\mathop b\limits^ \wedge}$ is a unit vector along the local magnetic field; $\bm{V}^{sw}=V^{sw}\bm{\mathop r\limits^ \wedge}$ is the solar wind velocity in the radial direction; and $L$ is the magnetic focusing length given by $L=\left(\bm{\mathop b\limits^ \wedge}\cdot\nabla\\{ln} B_0
\right)^{-1}$ with $B_0$ being the magnitude of the background IMF. This equation includes many important particle transport effects such as particle streaming along field line, adiabatic cooling, magnetic focusing, and the diffusion coefficients parallel and perpendicular to the IMF. It is noted that for low-energy SEPs propagating in inner heliosphere, the drift effects can be neglected. Here, we use the Parker field model for the IMF, and the solar wind speed is $400$ km/s.
The parallel particle mean free path $\lambda _\parallel$ is related to the particle pitch angle diffusion $D_{\mu\mu}$ through [@Jokipii1966ApJ...146..480J; @Earl74] $$\lambda _\parallel = \frac{{3v}}{8}\int_{ - 1}^{ + 1}
{\frac{{\left(1 - \mu ^2 \right)^2 }}{{D_{\mu \mu } }}d\mu
}\label{lambda_parallel_1},$$ and parallel diffusion coefficient $\kappa_\parallel$ can be written as $\kappa_\parallel=v\lambda_\parallel/3$.
We choose to use a pitch angle diffusion coefficient from [@Beeck1986ApJ...311..437B; @QinEA05; @Qin2006JGRA..11108101Q] $${D_{\mu \mu }}(\mu ) = {D_0}v{R^{s - 2}}\left({\mu ^{s - 1}} + h\right)\left(1 -
{\mu ^2}\right)$$ where the constant ${D_0}$ is adopted from $${D_0} = {\left( {\frac{{\delta {B_{slab}}}}{{{B_0}}}} \right)^2}\frac{{\pi (s - 1)}}{{4s}}{k_{\min }}$$ here $ \delta {B_{slab}} $ is the magnitude of slab turbulence, $ {k_{\min }} $ is the lower limit of wave number of the inertial range in the slab turbulence power spectrum, $ R = pc/\left( {\left| q \right|{B_0}} \right) $ is the maximum particle Larmor radius, $q$ is the particle charge, and $s = 5/3$ is the Kolmogorov spectral index of the magnetic field turbulence in the inertial range. The constant $h$ comes from the non-linear effect of magnetic turbulence on the pitch angle diffusion at $\mu=0$ [@QinAShalchi09; @QinAShalchi14]. In following simulations, we set $ h=0.01$, and ${k_{\min }} =1/l_{slab}$, where, $l_{slab}$ is the slab turbulence correlation length. In this formula, we assume that ${\left( {\delta {B_{slab}}} \right)^2}/{\left( {{B_0}}
\right)^2} \cdot {k_{\min }} = {A_1}$. Different parallel particle mean free path values can be obtained by altering the parameter $A_1$.
The perpendicular diffusion coefficient is taken from the NLGC theory [@Matthaeus2003ApJ...590L..53M] with the following analytical approximation [@ShalchiEA04; @ShalchiLi2010] $${\bm{\kappa} _ \bot } = \frac{1}{3}v{\left[ {{{\left( {\frac{{\delta {B_{2D}}}}{{{B_0}}}} \right)}^2}\sqrt {3\pi } \frac{{s - 1}}{{2s}}\frac{{\Gamma \left( {\frac{s}{2} + 1} \right)}}{{\Gamma \left( {\frac{s}{2} + \frac{1}{2}} \right)}}{l_{2D}}} \right]^{2/3}}{\lambda _\parallel }^{1/3}\left( {{\bf{I}} - \bm{ \mathop b\limits^ \wedge} \bm{ \mathop b\limits^ \wedge} } \right)$$ where $B_{2D}$ and $ l_{2D} $ are the magnitude and the correlation length of 2D component of magnetic turbulence, respectively. $ \Gamma $ is the gamma function. Here for simplicity, $\bm{\kappa _\bot}$ is assumed to be independent of $\mu$, since particle pitch angle diffusion usually is much faster than the perpendicular diffusion, so the particle sense the effect of perpendicular diffusion averaged over all pitch angles. $\bm{{\rm I}}$ is a unit tensor. In our simulations, we set ${\left( {\delta {B_{2D}}} \right)^2}/{\left( { {B_{0}}} \right)^2} \cdot{l_{2D}}= A_2$, and $s = 5/3$. As a result, the value of perpendicular diffusion coefficient can be altered by changing the parameter $A_2$, and parallel diffusion coefficient.
The particle source on the solar wind source surface covers certain ranges of longitudinal and latitudinal $S_{long}\times S_{lat}$. The spatial distribution of the SEP source is shown in Figure \[source\]. The source region is divided into small cells with the same longitude and latitude intervals. The regions filled with ions are labeled as “1”, and the regions devoid of ions are labeled as “0”. The size of every cell is set as $1.5^\circ$ in both latitudinal and longitudinal direction. This setup of source region is to mimic the effect of braided magnetic field lines due to random walk of foot-point in the low corona. The size of the cell is equivalent to the typical size of supergranular motion. Without perpendicular diffusion, the particles propagate outward from the source to the interplanetary space only along the field lines. In this case, only the magnetic flux tubes connected to the regions labelled as “1” in the phase of source particle injection are filled with particles, and the rest of tubes are devoid of particles. As the magnetic flux tubes past by an observer at 1 AU, the observer can see alternating switch-ons and switch-offs of SEPs. However, with perpendicular diffusion, particles can cross the field lines when they propagate in the interplanetary space. In this case, the longitudinal gradients in the particle intensities at different locations of interplanetary space will be reduced.
We use boundary values to model the particles’ injection from the source. The source rotates with the Sun, and the boundary condition is chosen as the following form $${f_b}(z \le 0.05AU,{E_k},\theta ,\varphi ,t) = \frac{a}{t} \cdot \frac{{E_k^{ - \gamma }}}{{{p^2}}} \cdot \exp \left( { - \frac{{{t_c}}}{t} - \frac{t}{{{t_l}}}} \right) \cdot \xi,$$ $$\xi\left( {\theta ,\varphi } \right) = \left\{ {\begin{array}{*{20}{l}}
{{e^{( - a\phi /\phi_0 )}} {\kern 15pt}\textrm{in source region 1}}\\
{0 {\kern 15pt}\textrm{otherwise}}
\end{array}} \right.\nonumber \label{SEPsource}$$ where the particles are injected from the SEP source near the Sun. $\xi $ indicates the spatial scale of every cell. $E_k$ is the particles energy. We set a typical value of $\gamma=-3$ for the spectral index of source particles. Because of adiabatic energy loss, those particles observed at $1$ AU have less energy than their initial energy at the source. In our simulations, energy of particles at source are just a few times larger than that of particles at $1$ AU. Time constants $t_c=0.48$ hour and $t_l=1.24$ hours indicate the rise and decay time scales, respectively. The injection time scales are used to model an impulsive SEP event. $\phi$ is the angle distance from the center of the cell and where the particles are injected. $\phi_0$ and $a$ are the constant. $\phi_0$ is set to be $0.75^\circ$ (the half width of each cell), but $a$ is allowed to change according to several different scenarios. The inner boundary is 0.05 AU and the outer boundary is 50 AU. The transport equation (\[dfdt\]) is solved by a time-backward Markov stochastic process method [@Zhang1999ApJ...513..409Z; @Qin2006JGRA..11108101Q]. The transport equation can be reformulated to stochastic differential equations, so it can be solved by a Monte-Carlo simulation of Markov stochastic process, and the SEP distribution function can be derived. In this method, we trace virtual particles from the observation point back to the injection time from the SEP source. More details of the technique can be found in those references.
RESULTS
=======
${\kappa _ \bot }/{\kappa _\parallel }$ Ratio
----------------------------------------------
Figure \[compose\_15\_and\_20\_dBToB\_03\] shows the interplanetary magnetic field in the ecliptic in the left panel, and the omni-directional fluxes for $500$ keV protons which are detected at $1$ AU in the right panel. In the left panel, the grey region indicates that the field lines are connected to the source. The dropouts and cutoffs are interpreted as the magnetic flux tubes which are alternately filled with and devoid of ions pass the spacecraft. In our model, the source rotates with the Sun. As a result, the magnetic flux tube which connects with the source also rotates with the source. The observer is located at $1$ AU ($x=0, y=1$) in the equatorial plane as indicated by the black circle. The magnetic flux tubes rotate with the Sun, and the angular speed is $0.55^\circ$ per hour. According to the typical size of supergranulation, the size of every cell is set as $1.5^\circ$ in both latitudinal and longitudinal direction. An observer in the ecliptic traverses a cell in nearly $2.7$ hours. Due to the connection of the magnetic flux tubes, the observer’s field lines can connect to different regions which are alternately filled with and devoid of ions. In the right panel, the source parameter $a$ is set as $0$, so the source intensity is uniform in every cell. The source width is $S_{long}=S_{lat}=18^\circ$. When the particles are injected, the observer at $1$ AU is magnetically connected with the first boundary of the source region. This same magnetic connection is also verified for the other simulations, except the last one when the observer is located at larger distance. In all the cases, the parallel mean free paths are the same (${\lambda _\parallel =0.087}$ AU), but the perpendicular diffusion coefficients, and subsequently the ratios of perpendicular diffusion coefficients to parallel ones, are set to several different values. With perpendicular diffusion, particles can be detected even if the observer is not connected directly to region 1 by field lines. The observer detects enhancements of particles starting at nearly $0.3$ day after the particles are injected on the Sun. With a larger perpendicular diffusion, the onset time of flux changes to an earlier time. The onset time of flux is the earliest and the latest in the cases of ${\kappa_\perp}/{\kappa _\parallel } = 1
\times 10^{-4} $ and ${\kappa _ \bot }/{\kappa _\parallel } = 0 $, respectively. During the time interval from $0.55$ to $0.65$ day, the observer’s field lines are connected to region 0. Without perpendicular diffusion, the observer can not detect energetic particles, so the flux suddenly drops to zero. As the ${\kappa _ \bot }/{\kappa_\parallel } $ increases from $0$ to $1\times 10^{-4} $, the variation of flux becomes increasingly smaller during the interval from $0.55$ to $0.65$ day. Especially in the case of ${\kappa_\bot}/{\kappa_\parallel}=1\times 10^{-4}$, there is essentially no difference in the flux between the time intervals when the observer’s field lines are not connected to the source. Later on, when the observer is connected to the region $0$ again during the time interval from $0.8$ to $0.9$ day, the fluxes behave similarly to those in the interval from $0.55$ to $0.65$ day. After $1.25$ day, the observer is completely disconnected from the source region. The flux decreases very quickly in the case of ${\kappa_\bot}/{\kappa_\parallel } = 1 \times 10^{-5}$, but the decreases slow down as the perpendicular diffusion coefficient increases. The time between neighboring valleys and peaks is less than $3$ hours. Let us define a ratio $R_i=f_{p,i}/f_{v,i}$, where $f_{p,i}$ and $f_{v,i}$ are the ith peak value and valley value of the flux, respectively. If the ratio $R_i$ is larger than $2$, we count it as a dropout. Since the ratios $R_i$ in each of the valleys are similar, we only use the ratio $R_2$ to identify the dropouts of the fluxes. When ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $0$, $1 \times 10^{-5}$, and $5 \times 10^{-5}$, $R_2$ is approximately equal to $ + \infty$, $30.4$, and $2.2$, respectively. We find that the dropouts can be reproduced only in the cases with ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 5 \times 10^{-5}$. In any case, if a dropout is reproduced, a cutoff, the step-like intensities decrease without recovery, is also reproduced. In this sense, dropouts and cutoffs are the same phenomenon, with the only difference being whether or not the flux is recovered by a follow-on connection to the particle source.
Parallel Mean Free Path
------------------------
The level of parallel mean free path affects the speed of particle propagation from the Sun to the Earth. Figure \[compose\_15\_and\_20\_dBToB\_1\] is similar to Figure \[compose\_15\_and\_20\_dBToB\_03\] but with different turbulence parameters $A_1$ or parallel mean free paths. The left panel corresponds to ${\lambda _\parallel } = 0.0{\rm{26}}$ AU, and the right panel is corresponding to ${\lambda _\parallel } = 0.{\rm{5}}$ AU. Due to the smaller parallel mean free path in the left panel of Figure \[compose\_15\_and\_20\_dBToB\_1\], the onset of the fluxes shifts to a later time than that in Figure \[compose\_15\_and\_20\_dBToB\_03\]. In the right panel, where the mean free path is larger than that in Figure \[compose\_15\_and\_20\_dBToB\_03\], the onset is earlier. In the left panel, the observer only detects two dropouts, then its foot-point goes away becomes disconnected from the source region. However, due to a larger parallel mean free path, the observer detects four dropouts in the right panel. When ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $1 \times 10^{-5}$ and $5 \times 10^{-5}$, $R_2$ is approximately equal to $560$ and $3.7$ in the left panel, and is approximately equal to $5.5$ and $1.5$ in the right panel, respectively. The dropouts requires ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 5\times 10^{-5}$ in the case of ${\lambda _\parallel } = 0.0{\rm{26}}$ AU, and requires ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 1\times 10^{-5}$ in the case of ${\lambda _\parallel } = 0.{\rm{5}}$ AU. Otherwise, the dropouts disappear. The results show that the upper limit of ${\kappa _ \bot }/{\kappa _\parallel }$ in dropouts changes little with different ${\lambda _\parallel}$. The relation of the appearance of dropout and parallel mean free path at given ${\kappa _ \bot }/{\kappa
_\parallel }$ ratio can be understood as follows. When the mean free path increase, it takes a shorter time for the particles to propagate from the Sun to the Earth, during which the particle can not diffuse across magnetic field lines too much even with a large perpendicular diffusion coefficient. Therefore, it is the ratio of ${\kappa _ \bot }/{\kappa _\parallel }$ that determines whether a dropout of particle intensity is observed at the distance of 1 AU.
Spatial Variation of SEP Source
-------------------------------
The left panel of Figure \[different\_source\_width\_dBToB\_03\_087AU\] is the same as Figure \[compose\_15\_and\_20\_dBToB\_03\]. In the right panel, the source region is set to $S_{long}=S_{lat}=15^\circ$ which is narrower than that in the left panel. Due to a narrower source, the observer only encounters two dropouts instead of three. Other than this, the fluxes in the right panel of Figure \[different\_source\_width\_dBToB\_03\_087AU\] show behaviors similar to those in left panel.
Figure \[different\_source\_spatial\_variation\_dBToB\_03\_087AU\] shows $500$ keV proton fluxes with different spatial distribution of source. The four panels (a), (b), (c), and (d) correspond to $\kappa_\perp/\kappa _\parallel=0$, $\kappa_\perp/\kappa _\parallel=1 \times 10^{-5}$, $\kappa_\perp/\kappa _\parallel=5 \times 10^{-5}$, and $\kappa_\perp/\kappa _\parallel=1 \times 10^{-4}$, respectively. In every panel, the source parameter $a$ varies from $0$ to $12$. With a larger parameter $a$, the source intensity decreases more quickly towards the flank of each cell. In the panel (a), the ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $0$. No particle is detected by the observer when the field line is disconnected from the source. In the panel (b), the ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $1 \times 10^{-5}$. Due to the variation of source intensity, the fluxes observed at 1 AU drop much more as $a$ increases. When $a$ increases from $0$ to $12$, $R_2$ also increase from $30.4$ to $1962$. In the panel (c), the ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $5 \times 10^{-5}$. The fluxes observed at 1 AU drop much slower as $a$ increases than that in the panel (b). When $a$ is set as $0$, $3$, $6$, and $12$, $R_2$ is approximately equal to $2.2$, $2.7$, $3.3$, and $3.6$. In the panel (d), the ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $1 \times 10^{-4}$. There is no significant difference in the fluxes observed at 1 AU as $a$ increases. Based on the results in the four panels, we find that the dropouts can be detected in the cases with a slightly higher ratio of ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 5\times 10^{-5}$, if the source distribution becomes narrower.
Energy Dependence of Dropouts and Cutoffs
-----------------------------------------
The left panel of Figure \[different\_energy\_channel\_source\_constant\] is the same as Figure \[compose\_15\_and\_20\_dBToB\_03\]. The right panel of Figure \[different\_energy\_channel\_source\_constant\] shows the omni-directional flux for 5 MeV protons detected at 1 AU in the ecliptic. The parallel mean free paths remain the same (${\lambda _\parallel } = 0.{\rm{13}}$ AU at $1$ AU) in all cases, but the perpendicular diffusion coefficient is set to several different values. The source width in the two panels are set as $S_{long}=S_{lat}=18^\circ$. The source parameter $a$ is set as $0$ in the two panels. Comparing with the left panel, the onset time of flux is earlier because of the higher energy and the larger parallel mean free path in the right panel. In the right panel, when ${\kappa _ \bot }/{\kappa _\parallel }$ is set as $0$, $1 \times 10^{-5}$, and $5\times 10^{-5}$, $R_2$ is approximately equal to $+\infty$, $10$, and $1.7$, respectively. The dropouts can be reproduced in the cases of ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 1 \times {10^{ - 5}}$ in the right panel. As a comparison, in the left panel, dropouts can be reproduced in the cases of ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 5 \times {10^{ - 5}}$.
Observer at A Larger Radial Distance
------------------------------------
Results for an observer at $3$ AU in the ecliptic are shown in the Figure \[different\_location\_1AU\_and\_3AU\_dBToB\_03\_087AU\]. The left panel illustrates the interplanetary magnetic field lines. Due to a larger radial distance, the particles spend more time propagating from the source to the observer than in the case of $1$ AU. In order to detect the particles, the foot-point of observer is set as west $40^\circ$ to the boundary of source at the beginning of the simulation. In the right panel, the onset of the fluxes shifts to a later time than that in Figure \[compose\_15\_and\_20\_dBToB\_03\]. As the source rotates with the Sun, the observer encounters five dropouts, which is more than seen in Figure \[compose\_15\_and\_20\_dBToB\_03\], because the observer is located at the boundary of the source at the initial time in Figure \[compose\_15\_and\_20\_dBToB\_03\], and particles spend some time propagating from source to $1$ AU. As a results, the observer missed two dropouts in Figure \[compose\_15\_and\_20\_dBToB\_03\]. In the right panel of Figure \[different\_location\_1AU\_and\_3AU\_dBToB\_03\_087AU\], the dropouts can be detected in the cases of ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 1 \times {10^{ - 5}}$. In the case of ${\kappa _ \bot }/{\kappa _\parallel } = 5 \times {10^{ - 5}}$, the dropout is absent in Figure \[different\_location\_1AU\_and\_3AU\_dBToB\_03\_087AU\], but it appears in Figure \[compose\_15\_and\_20\_dBToB\_03\]. The reason is that it takes a longer time for the particles to propagate from the Sun to the observer, and for perpendicular diffusion to be effective, when the solar radial distance increases.
DISCUSSIONS AND CONCLUSIONS
===========================
By numerically solving the Fokker-Planck focused transport equation for $500$ keV and $5$ MeV protons, we have investigated the effect of the perpendicular diffusion coefficients on the dropouts and cutoffs when an observer is located at $1$ AU or $3$ AU in the ecliptic. SEPs are injected from a source near the Sun, and the source rotates with the Sun. The dropouts and cutoffs are caused by the magnetic flux tubes which are alternately filled with and devoid of ions past the spacecraft. In this paper, all the times between neighbouring valleys and peaks are less than $3$ hours, and the ratio $R_2$ between the second peak value and the second valley value are used to identify the dropouts. The dropout is defined to be present when $R_2$ is more than a significant factor, which is set to be 2. We list the values of $R_2$ in the cases of different magnetic field turbulence intensities in Table \[diffusionCoefficients\].
Our simulations are performed for several different parallel mean free paths (${\lambda _\parallel } = $ $0.5$ AU, $0.087$ AU, $0.026$ AU at $1$ AU) with different assumption for the ratios of perpendicular diffusion coefficient to the parallel one. With a larger parallel mean free path, the onset time of SEP flux appears earlier, and more dropouts can be detected. Meanwhile, the flux increases more quickly, and the peak time is also earlier. This feature is closely related to the pitch angle distribution of particles arriving at the observer. Since the particles encounter fewer scatterings when they propagate in the interplanetary space with a larger parallel mean free path. Therefore, the distribution of pitch angle would be anisotropic for a longer time in this case. In order to reproduce the dropouts and cutoffs at 1 AU, the perpendicular diffusion has to be small: ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 5 \times {10^{ - 5}}$ when observer is located at $1$ AU, while ${\kappa _ \bot }/{\kappa _\parallel } \lesssim 1 \times {10^{ - 5}}$ when observer is located at $3$ AU. In any case when the dropouts are reproduced, cutoffs can also be reproduced when the observer’s flux tubes completely move out of the source region. If the observer is located at a larger radial distance (eg, $3$ AU in our simulation), it takes a longer time for the particles to propagate from the Sun to the observer, and perpendicular diffusion has more time to be effective. In order to reproduce the dropout at several AU, the ratio of ${\kappa _ \bot }/{\kappa _\parallel }$ should be lower than that in the cases of $1$ AU. As a result, our simulation also predicts that the dropout may disappear at larger radial distances, which can be checked by analysing data from Ulysses or other spacecraft at large distance.
With a wider source, the observer can detect more dropouts. Other than this, the fluxes with a wider source show behaviors similar to those with a narrower source. As $a$ changes from $0$ to $12$, in the case of ${\kappa_\bot}/{\kappa_\parallel}=1\times{10^{-5}}$, the ratio $R_i$ is much larger in the case of $a=12$ than that in the case of $a=0$. However, in the cases of ${\kappa_\bot}/{\kappa_\parallel}=1\times{10^{-4}}$ and ${\kappa_\bot}/{\kappa_\parallel}=5\times{10^{-5}}$, the ratio $R_i$ does not change significantly as $a$ changes from $0$ to $12$.
For $5$ MeV protons, the case of ${\lambda _\parallel } = 0.13$ AU has been analysed. Due to the higher particle speed and a typically larger parallel mean free path, the onset time appears earlier than for $500$ keV protons. As a result, more dropouts can be detected. In order to reproduce the dropouts and cutoffs, the ratio of the perpendicular diffusion coefficient to the parallel one should be smaller than $10^{-5}$, which is a little lower than that in the cases for $500$ keV protons.
Different ${\kappa _ \bot }$ and ${\kappa _\parallel } $ are obtained by altering the parameters $A_1$ and $A_2$ in our simulations, respectively, where ${A_1}={\left({\delta{B_{slab}}}\right)^2}/({{{B_0}}^2}\cdot l_{slab})$, and ${A_2}={\left({\delta{B_{2D}}}\right)^2}/{\left({{B_{0}}}\right)^2}\cdot{l_{2D}}$. In Table \[diffusionCoefficients\], we list all coefficients in the diffusion formulae which are used in our simulations with $\lambda_\parallel$ equal to $0.5$ AU, $0.087$ AU, $0.026$ AU for $500$ keV protons and $0.13$ AU for $5$ MeV protons at $1$ AU. In this table, we assume a slab turbulence correlation length $l_{slab}=0.03$ AU, and 2D correlation length $l_{2D}=0.003$ AU. As we can see, the ${\left({\delta{B_{slab}}/{B_0}}\right)^2}$ is much larger than ${\left({\delta{B_{2D}}/{B_0}}\right)^2}$ in all cases. This result is consistent with the observation [@Tan2014Correlation]. However, we should note that the exact values of ${\left({\delta {B_{slab}}/{B_0}}\right)^2}$, ${\left({\delta{B_{2D}}/{B_0}}\right)^2}$, $l_{slab}$ and $l_{2D}$ cannot be well determined. For example, we can assume slab turbulence correlation length $l_{slab}=0.003$ AU instead, and ${\lambda _\parallel } = 0.5$ AU, $0.087$ AU, and $0.026$ AU for $500$ keV protons given ${\left({\delta{B_{slab}}/{B_0}}\right)^2}=0.05$, $0.3$, and $1$, respectively. In our results we need a very small $\delta B_{2D}/\delta B_{slab}$ to get a small perpendicular diffusion coefficients from the NLGC theory. However, the NLGC results with a small $\delta B_{2D}/\delta B_{slab}$ are much larger than simulation results [e.g., @Qin07]. Although $\delta B_{2D}/\delta B_{slab}$ must be small to get the small perpendicular diffusion coefficients, the actual value of $\delta B_{2D}/\delta B_{slab}$ needed according to simulations [@Qin07] is not as extremely small as that shown in Table \[diffusionCoefficients\] from NLGC theory.
In [@Droge2010], the cutoffs can be reproduced for a ratio of ${\kappa _ \bot }/{\kappa _\parallel }$ a few times $10^{-5}$. This ratio is similar to what we deduced from our simulations. The basic difference between this simulation and the one in [@Droge2010] is that we reproduced the dropouts and cutoffs simultaneously, while their simulation only reproduced the cutoffs. We believe that the cutoffs are only a special type of dropout in which the intensity suddenly decreases without recovery. In [@giacalone2000small] and [@Guo2013Small], perpendicular diffusion coefficients relative to the background magnetic field (instead of Parker spiral) needed to be very small for reproducing the dropouts. That is consistent with our results. It should be noted that in our model, the large scale magnetic field is assumed to be a Parker spiral so that the Fokker-Planck focused transport equation can be solved efficiently with our stochastic method. In reality, the magnetic field lines with randomly walking foot-point do not have an azimuthal symmetry. However the large-scale geometry of interplanetary magnetic field and the behaviours of particle transport in it are not much different from those with a Parker magnetic field. The only difference is a slight shift of SEP source location relative to the magnetic field line passing through the observation at 1 AU.
Some special sets of turbulence parameters are needed in our simulations to produce small perpendicular diffusion coefficients in order to produce the dropouts and cutoffs. For example, turbulence dominated by a slab component leads to very small perpendicular diffusion coefficients. In the future, it will be interesting for us to study solar wind turbulence geometry from spacecraft observations when SEP dropouts and cutoffs occur.
The authors thank the anonymous referee for valuable comments. Y. W. benefited from the discussions with Fan Guo. We are partly supported by grants NNSFC 41304135, NNSFC 41374177, and NNSFC 41125016, the CMA grant GYHY201106011, and the Specialized Research Fund for State Key Laboratories of China. The computations were performed by Numerical Forecast Modeling R&D and VR System of State Key Laboratory of Space Weather and Special HPC work station of Chinese Meridian Project. M.Z. was supported in part by NSF under Grant AGS-1156056 and by NASA under Grant NNX08AP91G. SD acknowledges funding from the UK Science and Technology Facilities Council (STFC) (grant ST/J001341/1) and the International Space Science Institute.
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[|l|l|l|l|l|l|l|]{}
${\lambda _\parallel }$ $(AU)$& $l_{slab}$ $(AU)$ & $l_{2D}$ $(AU)$ & ${\left( {\delta {B}/{B_0}} \right)^2}$ & ${\kappa_ \bot }/{\kappa _\parallel }$ & ${\left( {\delta {B_{2D}}/{B_{slab}}} \right)^2}$ & ${R_2}^c$\
& & & & 0 & 0 & $ + \infty $\
& & & & $1 \times 10^{-5} $ & $1 \times 10^{-5}$ & $ 5.5$\
& & & & $5 \times 10^{-5} $ & $9 \times 10^{-5}$ & $ 1.5$\
& & & & $1 \times 10^{-4} $ & $3 \times 10^{-4}$ & $ 1.5$\
& & & & 0 & 0 & $ + \infty $\
& & & & $1 \times 10^{-5} $ & $2 \times 10^{-6}$ & $ 30.4$\
& & & & $5 \times 10^{-5} $ & $2 \times 10^{-5}$ & $ 2.2 $\
& & & & $1 \times 10^{-4} $ & $5 \times 10^{-5}$ & $ 1.1$\
& & & & 0 & 0 & $ + \infty $\
& & & & $1 \times 10^{-5} $ & $4 \times 10^{-7}$ & $ 560$\
& & & & $5 \times 10^{-5} $ & $5 \times 10^{-6}$ & $ 3.7$\
& & & & $1 \times 10^{-4} $ & $1 \times 10^{-5}$ & $ 1.4$\
& & & & 0 & 0 & $ + \infty $\
& & & & $1 \times 10^{-5} $ & $2 \times 10^{-6}$ & $ 10$\
& & & & $5 \times 10^{-5} $ & $2 \times 10^{-5}$ & $ 1.7$\
& & & & $1 \times 10^{-4} $ & $7 \times 10^{-5}$ & $ 1.3$\
|
---
abstract: 'In this paper, we propose a new joint dictionary learning method for example-based image super-resolution (SR), using sparse representation. The low-resolution (LR) dictionary is trained from a set of LR sample image patches. Using the sparse representation coefficients of these LR patches over the LR dictionary, the high-resolution (HR) dictionary is trained by minimizing the reconstruction error of HR sample patches. The error criterion used here is the mean square error. In this way we guarantee that the HR patches have the same sparse representation over HR dictionary as the LR patches over the LR dictionary, and at the same time, these sparse representations can well reconstruct the HR patches. Simulation results show the effectiveness of our method compared to the state-of-art SR algorithms.'
address: |
$^1$Department of Electrical Engineering, Sharif University of Technology, Iran.\
$^2$Department of Electrical Engineering, Ferdowsi University of Mashhad, Iran
bibliography:
- 'strings.bib'
- 'refs.bib'
title: 'JOINT DICTIONARY LEARNING FOR EXAMPLE-BASED IMAGE SUPER-RESOLUTION'
---
Image super-resolution, sparse representation, dictionary learning
Introduction {#sec:intro}
============
Super-resolution is the problem of reconstructing a high resolution[^1] image from one or several low resolution images [@park2003super]. It has many potential applications like enhancing the image quality of low-cost imaging sensors (e.g., cell phone cameras) and increasing the resolution of standard definition (SD) movies to display them on high definition (HD) TVs, to name a few.
Prior to SR methods, the usual way to increase resolution of images was to use simple interpolation-based methods such as bilinear, bicubic and more recently the resampling method described in [@zhang2006edge] among many others. However all these methods suffer from blurring high-frequency details of the image especially for large upscaling factors (the amount by which the resolution of image is increased in each dimension). Thus, over the last few years, a large number of SR algorithms have been proposed [@milanfar2010super]. These methods can be classified into two categories: multi-image SR, and single-image SR.
Since the seminal work by Tsai and Huang [@tsai1984multiframe] in 1984, many multi-image SR techniques were proposed [@elad1997restoration; @elad1999super; @capel2001super; @farsiu2004fast]. In the conventional SR problem, multiple images of the same scene with subpixel motion are required to generate the HR image. However the performance of these SR methods are only acceptable for small upscaling factors (usually smaller than 2). As the upscaling factor increases, the SR problem becomes severely ill-conditioned and a large number of LR images are needed to recover the HR image with acceptable quality.
To address this problem, example-based SR techniques were developed which require only a single LR image as input [@freeman2002example]. In these methods, an external training database is used to learn the correspondence between manifolds of LR and HR image patches. In some approaches, instead of using an external database, the patches extracted from the LR image itself across different resolutions are used [@glasner2009super]. In [@freeman2002example] Freeman *et al.* used a Markov network model for super-resolution. Inspired by the ideas in locally linear embedding (LLE) [@roweis2000nonlinear], the authors of [@chang2004super] used the similarity between manifolds of HR patches and LR patches to estimate HR image patches. Motivated by results of compressive sensing [@candes2006compressive], Yang *et al.* in [@yang2008image] and [@yang2010image] used sparse representation for SR. In [@yang2012coupled] they introduced coupled dictionary training in which the sparse representation of LR image patches better reconstructs the HR patches.
Recently, joint and coupled learning methods are utilized for efficient modeling of correlated sparsity structures [@7301775; @yang2010image]. However joint learning methods and the coupled learning methods proposed in [@yang2008image; @yang2010image; @7533090] still does not guarantee that the sparse representation of HR image patches over the HR dictionary is the same as the sparse representation of LR patches over LR dictionary. To address this problem, in this paper we propose a direct way to train the dictionaries that enforces the same sparse representation for LR and HR patches. Moreover since the HR dictionary is trained by minimizing the final error in reconstruction of HR patches, the reconstruction error in our method is smaller.
The rest of this paper is organized as follows. In section 2, Yang’s method for super-resolution via sparse representation is reviewed. In section 3, a flaw in Yang’s method is discussed, and our method to solve this problem is presented. Finally, section 4 is devoted to simulation results.
review of super-resolution via sparse representation {#sec:review}
====================================================
In SR via sparse representation we are given two sets of training data: a set of LR image patches, and a set of corresponding HR image patches. In other words, in the training data we have pairs of LR and HR image patches. The goal of SR is to use this database to increase the resolution of a given LR image.
Let $\{{\boldsymbol}{y}_i\}_{i=1}^{N}$ be the set of LR patches (each patch is arranged into a column vector ${\boldsymbol}{y}_i$) and $\{{\boldsymbol}{x}_i\}_{i=1}^N$ be the set of corresponding HR patches. In SR using sparse representation, the problem is to train two dictionaries ${\boldsymbol}{D}_l$ and ${\boldsymbol}{D}_h$ for the set of LR patches (or a feature of these patches) and HR patches respectively, such that for any LR patch ${\boldsymbol}{y}_i$, its sparse representation ${\boldsymbol}{w}_i$ over ${\boldsymbol}{D}_l$, reconstructs the corresponding HR patch ${\boldsymbol}{x}_i$ using ${\boldsymbol}{D}_h$: ${\boldsymbol}{x}_i\approx{\boldsymbol}{D}_h{\boldsymbol}{w}_i$ [@yang2010image]. Towards this end, first the dictionary learning problem is briefly reviewed in section \[ssec:dic\]. Then the dictionary learning method for SR proposed in [@yang2010image] is studied in section \[sec:yangdic\]. Finally in section \[sec:SR\], it is shown how these trained dictionaries can be used to perform SR on a LR image.
Dictionary learning {#ssec:dic}
-------------------
Given a set of signals $\{{\boldsymbol}{x}_i\}_{i=1}^N$, dictionary learning is the problem of finding a wide matrix ${\boldsymbol}{D}$ over which the signals have sparse representation [@elad2010sparse]. This problem is highly related to subspace identification [@rahmani2015innovation]. However, sparsity helps us to turn the subspace recovery to a well-defined problem. This approach has attracted lot of attentions in the last decade and found diverse applications [@minaee2015screen; @abavisani2015robust; @joneidi2015eigen]. If we denote the sparse representation of ${\boldsymbol}{x}_i$ over ${\boldsymbol}{D}$ by ${\boldsymbol}{w}_i$, the dictionary learning problem can be formulated as $$\min_{{\boldsymbol}{D},\{{\boldsymbol}{w}_i\}_{i=1}^N}{\sum_{i=1}^N{\|{\boldsymbol}{w}_i\|_0}}\quad s.t. \|{\boldsymbol}{x}_i-{\boldsymbol}{D}{\boldsymbol}{w}_i\|_2^2\leq\epsilon , i=1,...,N \label{diclearn}$$ in which the $\|\cdot\|_0$ is the $l_0$-norm which is the number of nonzero components of a vector and $\epsilon$ is a small constant which determines the maximum tolerable error in sparse representations. Replacing the $l_0$-norm by $l_1$-norm, Yang *et al.* in [@yang2010image] used the following formulation for sparse coding instead of $$\min_{{\boldsymbol}{D},\{{\boldsymbol}{w}_i\}_{i=1}^N}{\sum_{i=1}^N{\|{\boldsymbol}{x}_i-{\boldsymbol}{D}{\boldsymbol}{w}_i\|_2^2}+\lambda\sum_{i=1}^N{\|{\boldsymbol}{w}_i\|_1}}.\label{yangdiclearn}$$ By defining ${\boldsymbol}{X}\triangleq [{\boldsymbol}{x}_1 {\ } \cdots{\ } {\boldsymbol}{x}_N]$ and ${\boldsymbol}{W}\triangleq[{\boldsymbol}{w}_1 {\ } \cdots{\ } {\boldsymbol}{w}_N]$, it can be rewritten in matrix form as $$\min_{{\boldsymbol}{D},{\boldsymbol}{W}}{\|{\boldsymbol}{X}-{\boldsymbol}{D}{\boldsymbol}{W}\|_F^2+\lambda{\|{\boldsymbol}{W}\|_1}},\label{yangdiclearn2}$$ in which $\|\cdot\|_F$ stands for the Frobenius norm. and are not equivalent, but closely related. can be interpreted as minimizing the representation error of signals over the dictionary, while forcing these representations to be sparse by adding a $l_1$-regularization to the error. Therefore $\lambda$ can be used as a parameter that balances the sparsity and the error; a larger $\lambda$ results in sparser representations with larger errors.
Dictionary learning for SR {#sec:yangdic}
--------------------------
Given the sets of LR and HR training patches, $\{{\boldsymbol}{y}_i\}_{i=1}^{N}$ and $\{{\boldsymbol}{x}_i\}_{i=1}^{N}$, by defining ${\boldsymbol}{Y}\triangleq[{\boldsymbol}{y}_1 {\ } \cdots{\ } {\boldsymbol}{y}_N]$ , and having in mind, Yang *et al.* in [@yang2010image] proposed the following joint dictionary learning to ensure that the sparse representation of LR patches over ${\boldsymbol}{D}_l$ is the same as sparse representation of HR image patches over ${\boldsymbol}{D}_h$: $$\min_{{\boldsymbol}{D}_l,{\boldsymbol}{D}_h,{\boldsymbol}{W}}{{\|{\boldsymbol}{Y}-{\boldsymbol}{D}_l{\boldsymbol}{W}\|_F^2}+{\|{\boldsymbol}{X}-{\boldsymbol}{D}_h{\boldsymbol}{W}\|_F^2}+\lambda{\|{\boldsymbol}{W}\|_1}}\label{yanggg}$$ The key point here is that they have used the same matrix ${\boldsymbol}{W}$ for sparse representation of both LR and HR patches to make sure that their representation is the same over the dictionaries ${\boldsymbol}{D}_l$ and ${\boldsymbol}{D}_h$. If we define the concatenated space of HR and LR patches: $${\boldsymbol}{Z}=\left[\begin{matrix}
{\boldsymbol}{Y} \\
{\boldsymbol}{X}
\end{matrix}\right],\quad {\boldsymbol}{D}=\left[\begin{matrix}
{\boldsymbol}{D}_l \\
{\boldsymbol}{D}_h
\end{matrix}\right]$$ then joint dictionary training can also be written equivalently as $$\min_{{\boldsymbol}{D},{\boldsymbol}{W}}{\|{\boldsymbol}{Z}-{\boldsymbol}{D}{\boldsymbol}{W}\|_F^2+\lambda{\|{\boldsymbol}{W}\|_1}}.\label{con}$$ This formulation is clearly the same as . In other words, in the concatenated space, joint dictionary learning is the same as conventional dictionary learning, and any dictionary learning algorithm can be used for joint dictionary learning.
Super-Resolution {#sec:SR}
----------------
After training the two dictionaries ${\boldsymbol}{D}_l$ and ${\boldsymbol}{D}_h$, the input LR image can be super-resolved using the following steps:
1. The input LR image is divided into a set of overlapping LR patches: $\{{\boldsymbol}{y}_i^{LR}\}_{i=1}^M$.
2. From each image patch ${\boldsymbol}{y}_i^{LR}$, subtract its mean, $\eta_i$, $${\hat{{\boldsymbol}{y}}_i^{LR}}\triangleq{\boldsymbol}{y}_i^{LR}-\eta_i,$$ and find its sparse representation over ${\boldsymbol}{D}_l$ $${\boldsymbol}{w}_i=\arg \min_{{\boldsymbol}{\alpha}_i}{\|\hat{{\boldsymbol}{y}}_i^{LR}-{\boldsymbol}{D}_l{\boldsymbol}{\alpha}_i\|_2^2+\lambda\|{\boldsymbol}{\alpha}_i\|_1}.$$
3. Using the sparse representation of each LR patch and its mean, the corresponding HR patch is estimated by $$\hat{{\boldsymbol}{x}}_i^{HR}={\boldsymbol}{D}_h\cdot {\boldsymbol}{w}_i,\quad {{\boldsymbol}{x}}_i^{HR}=\hat{{\boldsymbol}{x}}_i^{HR}+\eta_i.$$
4. Combining the estimated HR image patches, the output HR image is generated.
our proposed method {#ours}
===================
Our method for SR is to improve the dictionary learning part of Yang’s method, described in section \[sec:yangdic\]. Having the dictionaries trained, the rest of the method is the same as what described in section \[sec:SR\].
As mentioned earlier, in SR the dictionaries should be trained in a way that the sparse representation of each LR patch well reconstructs the corresponding HR patch. The Yang’s method uses to accomplish this. It uses the same sparse representation matrix ${\boldsymbol}{W}$ for both LR and HR patches to ensure that each LR and HR patch, both have the same sparse representation. However as can be seen from , this joint dictionary learning is only optimal in the concatenated space of LR and HR patches, but if we look at the space of LR and HR patches separately, we may find a sparser representation for some patches than the sparse representation found in the concatenated space.
To address this problem, note first that the SR method described in section \[sec:SR\] consists of two distinct operations: finding the sparse representation of the LR patch, and the reconstruction of the HR patch. Then, we note that the first operation uses only ${\boldsymbol}{D}_l$, and the second operation uses only ${\boldsymbol}{D}_h$. Therefore, instead of training the dictionaries jointly as in , we propose to train ${\boldsymbol}{D}_l$ for LR patches solely, and then to train the HR dictionary by minimizing the reconstruction error when sparse representation of LR patches are used.
Mathematically, we propose to train the LR dictionary as $$\min_{{\boldsymbol}{D}_l,{\boldsymbol}{W}}{\|{\boldsymbol}{Y}-{\boldsymbol}{D}_l{\boldsymbol}{W}\|_F^2+\lambda{\|{\boldsymbol}{W}\|_1}},\label{mylr}$$ which is a conventional dictionary learning problem. After training the LR dictionary, for each LR patch, its sparse representation ${\boldsymbol}{w}_i$ is found over ${\boldsymbol}{D}_l$ (note that this step is already done during the dictionary training in ) $${\boldsymbol}{W}={\mathop\textrm{argmin}}_{{\boldsymbol}{V}}{\|{\boldsymbol}{Y}-{\boldsymbol}{D}_l{\boldsymbol}{V}\|_F^2+\lambda{\|{\boldsymbol}{V}\|_1}}.$$ Using the sparse representation of LR patches ${\boldsymbol}{W}$, the HR dictionary ${\boldsymbol}{D}_h$ is found such that the reconstruction error of the corresponding HR patches are minimized, that is, $${\boldsymbol}{D}_h={\mathop\textrm{argmin}}_{{\boldsymbol}{D}_h}{\|{\boldsymbol}{X}-{\boldsymbol}{D}_h{\boldsymbol}{W}\|_F^2}.\label{HRtrain}$$ This is an unconstrained quadratic optimization problem which has the following closed-form solution: $${\boldsymbol}{D}_h={\boldsymbol}{X}{\boldsymbol}{W}\left({\boldsymbol}{WW}^T\right)^{-1}={\boldsymbol}{X}{\boldsymbol}{W}^\dagger$$ in which $(\cdot)^T$ and $(\cdot)^\dagger$ represent transpose and pseudo-inverse of a matrix, respectively.
Note that unlike Yang’s method, in the proposed method ${\boldsymbol}{D}_h$ is not trained in a way that explicitly enforces the sparsity of representation of HR patches over it, rather it is trained to minimize the final reconstruction error.
simulation results {#sec:sim}
==================
![Results of Lena image magnified by a factor of 2 using: (b) Bicubic interpolation, (c) Yang’s method, (d) our proposed method. The original image is also given in (a) for comparison. []{data-label="fig:res"}](lena_gnd){width="4cm"}
\(a) Original
![Results of Lena image magnified by a factor of 2 using: (b) Bicubic interpolation, (c) Yang’s method, (d) our proposed method. The original image is also given in (a) for comparison. []{data-label="fig:res"}](lena-bicubic){width="4cm"}
\(b) Bicubic interpolation
![Results of Lena image magnified by a factor of 2 using: (b) Bicubic interpolation, (c) Yang’s method, (d) our proposed method. The original image is also given in (a) for comparison. []{data-label="fig:res"}](lena-yang-12){width="4cm"}
\(c) Yang’s method
![Results of Lena image magnified by a factor of 2 using: (b) Bicubic interpolation, (c) Yang’s method, (d) our proposed method. The original image is also given in (a) for comparison. []{data-label="fig:res"}](man-lena-217final){width="4cm"}
\(d) Proposed method
In this section we compare the performance of our method with Yang’s method. The error criteria used here are [Peak Signal to Noise Ratio]{} (PSNR) and Structural SIMilarity (SSIM) index [@wang2004image]. PSNR criterion is defined as $$\textrm{PSNR}=20\log_{10}\left(\frac{255}{\sqrt{MSE}}\right)\label{psnr},$$ where MSE is mean square error given by $$\textrm{MSE}=\frac{\|{\boldsymbol}{I}_{SR}-{\boldsymbol}{I}_g\|_F^2}{mn}\label{mse},$$ in which ${\boldsymbol}{I}_g$ is the original distortion-free image, and ${\boldsymbol}{I}_{SR}$ is the super-resolved image derived from the SR algorithm, and $m$ and $n$ are dimensions of the image in pixels.
For the definition of SSIM refer to [@wang2004image]. From these definitions it is clear that higher PSNR means less mean square error, however it does not necessarily mean a better image quality when perceived by human eye. Many other error criteria have been proposed to solve this problem of PSNR. SSIM is one of these error criteria. But still PSNR is widely used because of its simple mathematical form. This can be seen in where MSE is to train the HR dictionary, but in order to show the effectiveness of our method, here we use both SSIM and PSNR to compare images produced by our method with Yang’s method.
To make a fair comparison, the same set of 80000 training data patches sampled randomly from natural images is used to train dictionaries for both Yang’s and our method. The size of LR patches is $5\times5$ and they are magnified by a factor of 2, i.e. the size of generated HR image patches is $10\times10$. The LR patches extracted from the input image have a 4 pixel overlap. Dictionary size is fixed at 1024, and $\lambda=0.15$ is used for both methods as in [@yang2010image].
In Fig. \[fig:res\], simulation results of Yang’s method and proposed method on Lena image can be seen. The original image and the image magnified using bicubic interpolation are also given as references. The PSNRs of these images are $32.79$dB, $34.73$dB and $34.86$dB for bicubic interpolation, Yang’s method and ours respectively. It is clear that the quality of images magnified by SR is much better than the image magnified by bicubic interpolation and the details are more visible, which has resulted in sharper images. But the difference between image (c) and image (d) is not noticeable visually, although the PSNR of image (d) which is super-resolved by our method is about $0.1$dB higher.
[cc|\*[3]{}[c|]{}]{} & &[Bicubic]{} & [ Yang ]{} &[proposed]{}\
&PSNR& 32.79& 34.73& **[34.86]{}\
&SSIM& 0.9012& 0.9268& **[0.9283]{}\
&PSNR& 26.50 & 27.77& **[27.89]{}\
&SSIM& 0.8334 & 0.8737& **[0.8762]{}\
&PSNR& 24.66& 25.30& **[25.39]{}\
&SSIM& 0.9529& 0.9872& **[0.9873]{}\
&PSNR& 27.93 & **[28.61]{}& 28.59\
&SSIM& 0.9609 & **[0.9852]{}& **[0.9852]{}\
&PSNR& 30.51& 33.24 & **[33.36]{}\
& SSIM&0.9230& 0.9526 & **[0.9538]{}\
&PSNR& 28.47 & 29.93 & **[30.02]{}\
&SSIM& 0.9143&0.9451&**[0.9462]{}\
**************************
In Table \[tab\] the PSNRs and SSIMs of some images produced by our method is compared with those of Yang’s method and bicubic interpolation. Almost all of the images recovered by our method have higher PSNRs than images recovered by Yang’s method. The average PSNRs given in the last row show that our method performs slightly better than Yang’s method on average.
The SSIMs in Table \[tab\] also confirm that our method is performing better than Yang’s method. The images super-resolved by the proposed method have on average a higher SSIM than images recovered by Yang’s method. Since SSIM is much more consistent with the image quality as it is perceived by human eye compared to PSNR, higher SSIM of images recovered by our method suggests that they also have better visual quality.
Conclusion and Future Works {#sec:conclusion}
===========================
In this paper, we presented a new dictionary learning algorithm for example-based SR. The dictionaries were trained from a set of sample LR and HR image patches in order to minimize the final reconstruction error. Simulation results on real images showed the effectiveness of our algorithm in super-resolving images with less error compared to Yang’s method. The average PSNR and average SSIM of images produced by our method were higher than images recovered by Yang’s method. In future, we can extend this work by training the HR dictionary using a better error criterion instead of PSNR. One of the advantages of our method is that training of ${\boldsymbol}{D}_h$ is separated from ${\boldsymbol}{D}_l$ in and . We can use another error criterion that better represents the image quality like SSIM in without making the training of ${\boldsymbol}{D}_l$ more complex. Changing the error criterion in each of Yang’s methods will make the optimizations in their algorithms much more complex.
[^1]: In this article by resolution we mean spatial resolution.
|
---
abstract: |
Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process.
Given a fixed sigma-finite measure $\mu$, we consider positive definite kernels defined on the subset of the sigma algebra having finite $\mu$ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when $\mu$ is assumed non-atomic. By contrast, when $\mu$ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their Itô calculus and diffusion. Examples include fractional Brownian motion, and time-change processes.
address:
- '(Palle E.T. Jorgensen) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A. '
- '(Feng Tian) Department of Mathematics, Hampton University, Hampton, VA 23668, U.S.A.'
author:
- Palle Jorgensen and Feng Tian
bibliography:
- 'ref.bib'
title: 'On reproducing kernels, and analysis of measures'
---
Introduction
============
A reproducing kernel Hilbert space (RKHS) is a Hilbert space $\mathscr{H}$ of functions (defined on a prescribed set) in which point-evaluation is a continuous linear functional; so continuity is required to hold with respect to the norm in $\mathscr{H}$. These Hilbert spaces (RKHS) have a host of applications, including to complex analysis, to harmonic analysis, and to quantum mechanics.
A fundamental theorem of Aronszajn yields an explicit correspondence between positive definite kernels on the one hand and RKHSs on the other. Now every positive definite kernel is also the covariance kernel of a Gaussian process; a fact which is a point of departure in our present analysis. Given a positive definite kernel, we shall explore its use in the analysis of the associated Gaussian process; and vice versa.
This point of view is especially fruitful when one is dealing with problems from stochastic analysis. Even restricting to stochastic analysis, we have the exciting area of applications to statistical learning theory [@MR2327597; @MR3236858]. The RKHSs are useful in statistical learning theory on account of a powerful representer theorem: It states that every function in an RKHS that minimizes an associated empirical risk-function can be written as a generalized linear combination of samplings of the kernel function; i.e., samples evaluated at prescribed training points. Hence, it is a popular tool for empirical risk minimization problems, as it adapts perfectly to a host of infinite dimensional optimization problems.
Analysis with the use of reproducing kernel Hilbert space (RKHS) has found diverse applications in many areas. However, presently we shall focus on applications to probability theory; applications to such important and related topics as metric entropy computations, to small deviation problems for Gaussian processes, and to i.i.d. series representations for general classes of Gaussian processes. We refer to a detailed discussion of these items below, with citations.
Recall that a *reproducing kernel Hilbert space* (RKHS) is a Hilbert space $\mathscr{H}$ of functions, say $f$, on a fixed set $X$ such that every linear functional (induced by $x\in X$), $$E_{x}\left(f\right):=f\left(x\right),\quad f\in\mathscr{H}.\label{eq:A1}$$ is continuous in the norm of $\mathscr{H}$.
Hence, by Riesz’ representation theorem, there is a corresponding $h_{x}\in\mathscr{H}$ such that $$E_{x}f=\left\langle f,h_{x}\right\rangle _{\mathscr{H}}\label{eq:I2}$$ where $\left\langle \cdot,\cdot\right\rangle _{\mathscr{H}}$ denotes the inner product in $\mathscr{H}$. Setting $$K\left(x,y\right)=\left\langle h_{y},h_{x}\right\rangle _{\mathscr{H}},\quad\left(x,y\right)\in X\times X$$ we get a *positive definite* (p.d.) kernel, i.e., $\forall n\in\mathbb{N}$, $\forall\left\{ \alpha_{i}\right\} _{1}^{n}$, $\forall\left\{ x_{i}\right\} _{1}^{n}$, $\alpha_{i}\in\mathbb{C}$, $x_{i}\in X$, we have $$\sum_{i}\sum_{j}\alpha_{i}\overline{\alpha}_{j}K\left(x_{i},x_{j}\right)\geq0.\label{eq:I3}$$
Conversely, if $K$ is given p.d., i.e., satisfying (\[eq:I3\]), then by [@MR0051437], there is a RKHS such that (\[eq:I2\]) holds.
Given $K$ p.d., we may take $\mathscr{H}\left(K\right)$ to be the completion of $$\psi=\sum_{i}\alpha_{i}K\left(\cdot,x_{i}\right)\label{eq:I4}$$ in the norm $$\left\Vert \psi\right\Vert _{\mathscr{H}\left(K\right)}^{2}=\sum_{i}\sum_{j}\alpha_{i}\overline{\alpha}_{j}K\left(x_{i},x_{j}\right),$$ but quotiented out by those functions $\psi$ in (\[eq:I4\]) with $\left\Vert \psi\right\Vert _{\mathscr{H}\left(K\right)}^{2}=0$. (In fact, by below, $\left\Vert \psi\right\Vert _{\mathscr{H}\left(K\right)}=0$ implies that $\psi\left(x\right)=0$, for all $x\in X$.)
A key fact which we shall be using throughout the paper is the following:
\[lem:I1\]Let $K$ be a positive definite kernel on $X\times X$, and let $\mathscr{H}\left(K\right)$ be the corresponding RKHS.
Then a function $f$ on $X$ is in $\mathscr{H}\left(K\right)$ there is a finite constant $C=C_{f}$, depending on $f$, such that $\forall n\in\mathbb{N}$, $\forall\left\{ x_{i}\right\} _{1}^{n}$, $\left\{ \alpha_{i}\right\} _{1}^{n}$, $x_{i}\in X$, $\alpha_{i}\in\mathbb{C}$, we have: $$\left|\sum\nolimits _{i=1}^{n}\alpha_{i}f\left(x_{i}\right)\right|^{2}\leq C\sum\nolimits _{i}\sum\nolimits _{j}\alpha_{i}\overline{\alpha}_{j}K\left(x_{i},x_{j}\right).\label{eq:I6}$$
One direction in the proof is immediate from the following observation regarding the norm $\left\Vert \cdot\right\Vert _{\mathscr{H}\left(K\right)}$ in the RKHS $\mathscr{H}\left(K\right)$. Here a reproducing kernel $K$ is fixed: For all finite sums, $\alpha_{i}\in\mathbb{C}$, $x_{i}\in X$, $1\leq i\leq N$, we then have: $$\left\Vert \sum\nolimits _{i=1}^{N}\alpha_{i}K\left(x_{i},\cdot\right)\right\Vert _{\mathscr{H}\left(X\right)}^{2}=\sum\nolimits _{i=1}^{N}\sum\nolimits _{j=1}^{N}\alpha_{i}\overline{\alpha_{j}}K\left(x_{i},x_{j}\right),$$ i.e., the RKHS in (\[eq:I6\]).
Now assume a function $f$ on $X$ is given to satisfy (\[eq:I6\]). Then define a linear functional $T_{f}$ on $\mathscr{H}\left(K\right)$, as follows: First define it on the above finite linear combinations (recall dense in $\mathscr{H}\left(K\right)$): $$T_{f}\left(\sum\nolimits _{i=1}^{N}\alpha_{i}K\left(x_{i},\cdot\right)\right)=\sum\nolimits _{i=1}^{N}\alpha_{i}f\left(x_{i}\right).$$ The assumption (\[eq:I6\]) simply amounts to the following *a priori* estimate: $$\left|T_{f}\left(\psi\right)\right|^{2}\leq C\left\Vert \psi\right\Vert _{\mathscr{H}\left(K\right)}^{2}\label{eq:A7}$$ where $\psi$ has the form of (\[eq:I4\]). Since, by (\[eq:A7\]), $T_{f}$ defines a bounded linear functional on a dense subspace in $\mathscr{H}\left(K\right)$, it extends by limits (in the $\mathscr{H}\left(K\right)$-norm) to $\mathscr{H}\left(K\right)$. So by Riesz’ lemma (for Hilbert spaces) applied to $\mathscr{H}\left(K\right)$, we get the stated inner-product representation $$T_{f}\left(\psi\right)=\left\langle F,\psi\right\rangle _{\mathscr{H}\left(K\right)}$$ for a unique $F\in\mathscr{H}\left(K\right)$. Using again the reproducing property (\[eq:A1\])-(\[eq:I2\]) for $\left\langle \cdot,\cdot\right\rangle _{\mathscr{H}\left(K\right)}$ (inner product), we conclude that $F=f$ holds (pointwise identity) for the two functions; hence $f\in\mathscr{H}\left(K\right)$.
Our present core theme, is motivated by, and makes direct connections to, a number of areas in probability theory. For the benefit of readers, we add below some hints to a number of such important and related topics, metric entropy, small deviation problems for Gaussian processes, and series representations of Gaussian processes. Of special note are the following three:
**1.** Metric entropy of the unit ball of the RKHS of Gaussian measures/processes. We refer to the fundamental papers by Dudley and Sudakov [@MR0220340; @MR0247034]. For a reformulation of their results in functional-analytic terms see [@MR695735].
**2.** Small ball problems for Gaussian measures on Banach spaces/small deviation problems for Gaussian processes. Kuelbs and Li achieved a breakthrough in this area [@MR1237989]. Further relevant contributions (including also fractional Brownian motion) can be found e.g. in [@MR1733160] and [@MR1724026].
**3.** Series representations of Gaussian processes (similar to Karhunen-Loéve expansions). See e.g., [@MR1935652].
\[sec:SigA\]Sigma-algebras and RKHSs of signed measures
=======================================================
Now our present focus will be a class of p.d. kernels, defined on subsets of a fixed $\sigma$-algebra. Specifically, if $\left(M,\mathscr{B},\mu\right)$ is a $\sigma$-finite measure space, we set $X=\mathscr{B}_{fin}$; see (\[eq:S1\]) below.
Consider a measure space $\left(M,\mathscr{B},\mu\right)$ where $\mathscr{B}$ is a sigma-algebra of subsets in $M$, and $\mu$ is a $\sigma$-finite measure on $\mathscr{B}$. Set $$\mathscr{B}_{fin}=\left\{ A\in\mathscr{B}\mid\mu\left(A\right)<\infty\right\} .\label{eq:S1}$$ Let $\mathscr{H}$ be a Hilbert space having the following property: $$\left\{ \chi_{A}\mid A\in\mathscr{B}_{fin}\right\} \subset\mathscr{H},\label{eq:S2}$$ where $\chi_{A}$ denotes the indicator function for the set $A$.
We shall restrict the discussion to real valued functions, and the extension to the complex case is straightforward. The latter can be found in a number of treatments, for example Peres et al. [@MR2552864].
\[thm:S2\]Let $\beta$ be a function, $\mathscr{B}_{fin}\times\mathscr{B}_{fin}\longrightarrow\mathbb{R}$. Then TFAE:
1. \[enu:S1\]$\beta$ is positive definite, i.e., $\forall n\in\mathbb{N}$, $\forall\left\{ \alpha_{i}\right\} _{1}^{n}$, $\forall\left\{ A_{i}\right\} _{1}^{n}$, $\alpha_{i}\in\mathbb{R}$, $A_{i}\in\mathscr{B}_{fin}$, we have $$\sum_{1}^{n}\sum_{1}^{n}\alpha_{i}\alpha_{j}\beta\left(A_{i},A_{j}\right)\geq0.$$
2. \[enu:S2\]There is a Hilbert space $\mathscr{H}$ which satisfies (\[eq:S2\]); and also $$\beta\left(A,B\right)=\left\langle \chi_{A},\chi_{B}\right\rangle _{\mathscr{H}},\quad\forall\left(A,B\right)\in\mathscr{B}_{fin}\times\mathscr{B}_{fin}.\label{eq:S4}$$
3. \[enu:S3\]There is a Hilbert space $\mathscr{H}$ which satisfies (\[eq:S2\]); and also a linear mapping: $$\mathscr{H}\ni f\longmapsto\mu_{f}\in\left(\begin{matrix}\text{signed finitely additive}\\
\text{measures on \ensuremath{\left(M,\mathscr{B}\right)}}
\end{matrix}\right)\label{eq:S5}$$ with $$\mu_{f}\left(A\right)=\left\langle \chi_{A},f\right\rangle _{\mathscr{H}},\quad\forall A\in\mathscr{B}_{fin}.\label{eq:S6}$$
We shall divide up the reasoning in the implications: (\[enu:S1\]) $\Rightarrow$ (\[enu:S2\]) $\Rightarrow$ (\[enu:S3\]) $\Rightarrow$ (\[enu:S1\]). The characterization in (\[eq:S5\]) of the elements in the RKHS $\mathscr{H}$ is based on an application of , combined with the detailed reasoning below.
**Case (\[enu:S1\]) $\Rightarrow$ (\[enu:S2\]).** Given a function $\beta$ as in (\[enu:S1\]), we know that, by [@MR0051437], there is an associated reproducing kernel Hilbert space $\mathscr{H}\left(\beta\right)$. The vectors in $\mathscr{H}\left(\beta\right)$ are obtained by the quotient and completion procedures applied to the functions $$\mathscr{B}_{fin}\ni B\longmapsto\beta\left(A,B\right)\in\mathbb{R}\label{eq:S7}$$ defined for every $A\in\mathscr{B}_{fin}$. Moreover, the inner product in $\mathscr{H}\left(\beta\right)$, satisfies $$\left\langle \beta\left(A_{1},\cdot\right),\beta\left(A_{2},\cdot\right)\right\rangle _{\mathscr{H}\left(\beta\right)}=\beta\left(A_{1},A_{2}\right).$$
Now let $$\mathscr{H}=\big(span\left\{ \chi_{A}\mid A\in\mathscr{B}_{fin}\right\} \big)^{\sim}$$ with $\left(\cdots\right)^{\sim}$ denoting the Hilbert completion: $$\begin{aligned}
\left\Vert \sum\nolimits _{i}\alpha_{i}\chi_{A_{i}}\right\Vert _{\mathscr{H}}^{2} & = & \sum\nolimits _{i}\sum\nolimits _{j}\alpha_{i}\alpha_{j}\beta\left(A_{i},A_{j}\right)\\
& \underset{\text{see \ensuremath{\left(\ref{eq:S7}\right)}}}{=} & \left\Vert \sum\nolimits _{i=1}^{n}\alpha_{i}\beta\left(A_{i},\cdot\right)\right\Vert _{\mathscr{H}\left(\beta\right)}^{2}.\end{aligned}$$ It is then immediate from this that the Hilbert space $\mathscr{H}$ satisfies the conditions stated in (\[enu:S2\]) of the theorem.
**Case (\[enu:S2\]) $\Rightarrow$ (\[enu:S3\]).** Let $\mathscr{H}$ satisfy the conditions in (\[enu:S2\]); and for $f\in\mathscr{H}$, let $\mu_{f}$ be as in (\[eq:S5\]). We must show that if $n\in\mathbb{N}$, $\left\{ A_{i}\right\} _{1}^{n}$, $A_{i}\in\mathscr{B}_{fin}$, satisfy $A_{i}\cap A_{j}=\emptyset$, $i\neq j$, then $$\mu_{f}\left(\cup_{1}^{n}A_{i}\right)=\sum\nolimits _{1}^{n}\mu_{f}\left(A_{i}\right).\label{eq:S10}$$ But $$\begin{aligned}
\text{LHS}_{\left(\ref{eq:S10}\right)} & \underset{\text{by \ensuremath{\left(\ref{eq:S6}\right)}}}{=} & \left\langle \chi_{\cup_{i=1}^{n}A_{i}},f\right\rangle _{\mathscr{H}}\\
& = & \sum_{i}\left\langle \chi_{A_{i}},f\right\rangle _{\mathscr{H}}=\sum_{i}\mu_{f}\left(A_{i}\right)=\text{RHS}_{\left(\ref{eq:S10}\right)}.\end{aligned}$$ The remaining assertions in (\[enu:S3\]) are clear.
**Case (\[enu:S3\]) $\Rightarrow$ (\[enu:S1\]).** This step is immediate from (\[eq:S4\]).
Let $\left(M,\mathscr{B},\mu\right)$ be a $\sigma$-finite measure space. As in , we specify a pair $\left(\beta,\mathscr{H}\right)$ where $\beta$ is defined on $\mathscr{B}_{fin}\times B_{fin}$, and $\mathscr{H}$ is a Hilbert space subject to condition (\[eq:S2\]). For $f\in\mathscr{H}$, set $$\mu_{f}\left(A\right)=\left\langle \chi_{A},f\right\rangle _{\mathscr{H}},\quad A\in\mathscr{B}_{fin}.$$ Then $\mu_{f}\in\mathscr{H}\left(\beta\right)\left(=\text{the RKHS of \ensuremath{\beta}.}\right)$ Moreover, $$\left\Vert \mu_{f}\right\Vert _{\mathscr{H}\left(\beta\right)}\leq\left\Vert f\right\Vert _{\mathscr{H}}.\label{eq:I12}$$
This will be a direct application of , but now applied to $X=\mathscr{B}_{fin}$. Hence we must show that, $\forall n\in\mathbb{N}$, $\left\{ A_{i}\right\} _{1}^{n}$, $\left\{ \alpha_{i}\right\} _{1}^{n}$, $A_{i}\in\mathscr{B}_{fin}$, $\alpha_{i}\in\mathbb{R}$, the estimate (\[eq:I6\]) holds, and with a finite constant $C_{f}$.
In fact, we may take $C_{f}=\left\Vert f\right\Vert _{\mathscr{H}}^{2}$, so $\left\Vert \mu_{f}\right\Vert _{\mathscr{H}\left(\beta\right)}\leq\left\Vert f\right\Vert _{\mathscr{H}}$ as claimed. Specifically, $$\begin{aligned}
\left|\sum\nolimits _{i=1}^{n}\alpha_{i}\mu_{f}\left(A_{i}\right)\right|^{2} & = & \left|\sum\nolimits _{i=1}^{n}\alpha_{i}\left\langle \chi_{A_{i}},f\right\rangle _{\mathscr{H}}\right|^{2}\\
& = & \left|\left\langle \sum\nolimits _{i=1}^{n}\alpha_{i}\chi_{A_{i}},f\right\rangle _{\mathscr{H}}\right|^{2}\\
& \underset{{\scriptscriptstyle \text{by Schwarz}}}{\leq} & \left\Vert \sum\nolimits _{i=1}^{n}\alpha_{i}\chi_{A_{i}}\right\Vert _{\mathscr{H}}^{2}\left\Vert f\right\Vert _{\mathscr{H}}^{2}\\
& \underset{{\scriptscriptstyle \text{by \ensuremath{\left(\ref{eq:S4}\right)}}}}{=} & \left\Vert f\right\Vert _{\mathscr{H}}^{2}\sum\nolimits _{i}\sum\nolimits _{j}\alpha_{i}\alpha_{j}\beta\left(A_{i},A_{j}\right),\end{aligned}$$ which is the desired conclusion.
Our present focus is on the case when the prescribed $\sigma$-finite measure $\mu$ is non-atomic. But the atomic case is also important, for example in interpolation theory in the form of Shannon, see e.g., [@MR0442564].
Consider, for example, the case $X=\mathbb{R}$, and $$K\left(x,y\right)=\frac{\sin\pi\left(x-y\right)}{\pi\left(x-y\right)},\label{eq:I7}$$ defined for $\left(x,y\right)\in\mathbb{R}\times\mathbb{R}$. In this case, the RKHS $\mathscr{H}\left(K\right)$ is familiar: It may be realized as functions $f$ on $\mathbb{R}$, such that the Fourier transform $$\hat{f}\left(\xi\right)=\int_{\mathbb{R}}e^{-i2\pi x\xi}f\left(x\right)dx\label{eq:I8}$$ is well defined, *and* supported in the compact interval $\left[-\frac{1}{2},\frac{1}{2}\right]$, frequency band, with $\left\Vert f\right\Vert _{\mathscr{H}\left(K\right)}^{2}=\int_{-\frac{1}{2}}^{\frac{1}{2}}|\hat{f}\left(\xi\right)|^{2}d\xi.$
Set $\mu=\sum_{n\in\mathbb{Z}}\delta_{n}$ (the Dirac-comb). Then Shannon’s theorem states that $$l^{2}\left(\mathbb{Z}\right)\ni\left(\alpha_{n}\right)_{n\in\mathbb{Z}}\xrightarrow{\quad T\quad}\mathscr{H}\left(K\right),$$ given by $$\big(T\left(\left(\alpha_{n}\right)\big)\right)\left(x\right)=\sum_{n\in\mathbb{Z}}\alpha_{n}\,\frac{\sin\pi\left(x-n\right)}{\pi\left(x-n\right)}$$ is isometric, mapping $l^{2}$ onto $\mathscr{H}\left(K\right)$. Its adjoint operator $$T^{*}:\mathscr{H}\left(K\right)\longrightarrow l^{2}\left(\mathbb{Z}\right)$$ is $$\left(T^{*}f\right)_{n}=f\left(n\right),\quad n\in\mathbb{Z}.\label{eq:I11}$$ Compare (\[eq:I11\]) with (\[eq:T6\]) below in a much wider context.
The RKHS for the kernel (\[eq:I7\]) $\mathscr{H}\left(K\right)$ is called the Paley-Wiener space. Functions in $\mathscr{H}\left(K\right)$ also go by the name, band-limited signals. We refer to (\[eq:I11\]) as (Shannon) sampling. It states that functions (continuous time-signals) $f$ from $\mathscr{H}\left(K\right)$ may be reconstructed perfectly from their discrete $\mathbb{Z}$ samples.
\[sec:SA\]The sigma-additive property
=====================================
The sigma-additive property alluded to here is not a minor technical point. Indeed, one of the basic problems related to the propositional calculus and the foundations of quantum mechanics is the description of probability measures (called states in quantum physical terminology) on the set of experimentally verifiable propositions. In the quantum setting, the set of propositions is then realized as an orthomodular partially ordered set, where the order is induced by a relation of implication, called a quantum logic. Now quantum-observables are generally non-commuting, and the precise question is in fact formulated for states (measures) on $C^{*}$-algebras; i.e., normalized positive linear functionals (see e.g., [@MR3642406]).
The classical Gleason theorem (see [@MR0096113]) is the assertion that a state on the $C^{*}$-algebra $\mathscr{B}\left(\mathscr{H}\right)$ of all bounded operators on a Hilbert space is uniquely described by the values it takes on orthogonal projections, assuming the dimension of the Hilbert space $\mathscr{H}$ is not 2. The precise result entails extension of finitely additive measures to sigma-additive counterparts, i.e., when we have additivity on countable unions of disjoint sets from the underlying sigma-algebra.
We now turn to the question of when the finitely additive measures $\mu_{f}$ are in fact $\sigma$-additive. (See , part (\[enu:S3\]).)
Given $\left(M,\mathscr{B},\mu\right)$ as above, we shall set $$\mathscr{D}_{fin}\left(\mu\right)=span\left\{ \chi_{A}\mid A\in\mathscr{B}_{fin}\right\} .$$ Recall that $\mathscr{D}_{fin}\left(\mu\right)$ is automatically a dense subspace in $L^{2}\left(\mu\right)$.
\[thm:T1\]Let $\mathscr{B}_{fin}$ be as specified in (\[eq:S1\]) with a fixed $\sigma$-finite measure space $\left(M,\mathscr{B},\mu\right)$. Let $\beta$ be given, assumed positive definite on $\mathscr{B}_{fin}\times\mathscr{B}_{fin}$, and let $\mathscr{H}$ be a Hilbert space which satisfies conditions (\[eq:S2\]) and (\[eq:S4\]).
Then there is a dense subspace $\mathscr{H}_{\mu}\subset\mathscr{H}$ such that the signed measures $$\left\{ \mu_{f}\mid f\in\mathscr{H}_{\mu}\right\}$$ are $\sigma$-additive if and only if the following implication holds: $$\begin{aligned}
{1}
\left.\begin{matrix}\left(\alpha\right) & & \left\{ \varphi_{n}\right\} _{n\in\mathbb{N}},\:\varphi_{n}\in\mathscr{D}_{fin}\left(\mu\right),\:\left\Vert \varphi_{n}\right\Vert _{L^{2}\left(\mu\right)}\xrightarrow[\;n\rightarrow\infty\;]{}0\\
\left(\beta\right) & & f\in\mathscr{H},\;\left\Vert \varphi_{n}-f\right\Vert _{\mathscr{H}}\xrightarrow[\;n\rightarrow\infty\;]{}0
\end{matrix}\right\} \Longrightarrow & f=0,\end{aligned}$$ i.e., if a vector $f\in\mathscr{H}$ satisfies $\left(\alpha\right)$ and $\left(\beta\right)$, it must be the null vector in $\mathscr{H}$.
Note that, because of assumptions (\[eq:S2\]) and (\[eq:S4\]), we get a natural inclusion mapping, denoted $T$, $$L^{2}\left(\mu\right)\xrightarrow{\quad T\quad}\mathscr{H}\label{eq:T3}$$ with dense domain $\mathscr{D}_{fin}\left(\mu\right)$ in $L^{2}\left(\mu\right)$. Recall, if $A\in\mathscr{B}_{fin}$, then the indicator function $\chi_{A}$ is assumed to be in $\mathscr{H}$.
With these assumptions, we see that the implication in the statement of the theorem simply states that $T$ is closable when viewed as a densely defined operator as in (\[eq:T3\]).
By a general theorem (see e.g., [@MR3642406]), $T$ is closable if and only if the domain $dom\left(T^{*}\right)$ of its adjoint $T^{*}$ is dense in $\mathscr{H}$.
We have that a vector $f$ in $\mathscr{H}$ is in $dom\left(T^{*}\right)$ if and only if $\exists C_{f}<\infty$ such that $$\left|\left\langle T\varphi,f\right\rangle _{\mathscr{H}}\right|\leq C_{f}\left\Vert \varphi\right\Vert _{L^{2}\left(\mu\right)}$$ holds for all $\varphi\in\mathscr{D}_{fin}\left(\mu\right)$. Also note that, if $\varphi=\chi_{A}$, $A\in\mathscr{B}_{fin}$, then $$\left\langle T\varphi,f\right\rangle _{\mathscr{H}}=\mu_{f}\left(A\right);$$ and so if $f\in dom\left(T^{*}\right)$, then $$\begin{aligned}
\mu_{f}\left(A\right) & =\big\langle\chi_{A},\underset{{\scriptscriptstyle \in L^{2}\left(\mu\right)}}{\underbrace{T^{*}f}}\big\rangle_{L^{2}\left(\mu\right)}=\int_{A}\left(T^{*}f\right)d\mu,\quad\forall A\in\mathscr{B}_{fin}.\label{eq:T6}\end{aligned}$$ Note, by definition, $T^{*}f\in L^{2}\left(\mu\right)$. Indeed, the converse holds as well. Since the right-hand side in (\[eq:T6\]) is clearly $\sigma$-additive, one implication holds. Moreover, the other implication follows from general facts about $L^{2}\left(M,\mathscr{B},\mu\right)$ valid for any $\sigma$-finite measure $\mu$ on $\left(M,\mathscr{B}\right)$.
\[cor:T2\]Let $\left(\beta,\mathscr{H}\right)$ be as in the statement of , and let $T$ be the closable inclusion $L^{2}\left(\mu\right)\xrightarrow{\;T\;}\mathscr{H}$. Then for $f\in dom\left(T^{*}\right)$, dense in $\mathscr{H}$, the corresponding signed measure $\mu_{f}$ is absolutely continuous w.r.t. $\mu$ with Radon-Nikodym derivative $$\frac{d\mu_{f}}{d\mu}=T^{*}f.\label{eq:T7}$$
Let $\left(M,\mathscr{B},\mu\right)$ be a $\sigma$-finite measure space, and on $\mathscr{B}_{fin}\times\mathscr{B}_{fin}$ define $$\beta_{\mu}\left(A,B\right):=\mu\left(A\cap B\right),\quad\forall A,B\in\mathscr{B}_{fin}.\label{eq:T8}$$ Let $\mathscr{H}\left(\beta_{\mu}\right)=\text{RKHS}(\beta_{\mu})$, i.e., the reproducing kernel Hilbert space associated with the p.d. function $\beta_{\mu}$. Then $\mathscr{H}\left(\beta_{\mu}\right)$ consists of all signed measures $m$ of the form $$m\left(A\right)=\int_{A}\varphi\,d\mu,\quad\varphi\in L^{2}\left(\mu\right);\label{eq:T9}$$ and when (\[eq:T9\]) holds, $$\left\Vert m\right\Vert _{\mathscr{H}\left(\beta_{\mu}\right)}^{2}=\int_{M}\left|\varphi\right|^{2}d\mu.$$
When $\beta_{\mu}$ is specified as in (\[eq:T8\]), then one checks immediately that the inclusion operator $T:L^{2}\left(\mu\right)\longrightarrow\mathscr{H}\left(\beta_{\mu}\right)$ is isometric, and maps onto $\mathscr{H}\left(\beta_{\mu}\right)$. Indeed, for finite linear combinations $\sum_{i=1}^{n}\alpha_{i}\chi_{A_{i}}$ as above, we have $$\begin{aligned}
\left\Vert \sum\nolimits _{i}\alpha_{i}\chi_{A_{i}}\right\Vert _{L^{2}\left(\mu\right)}^{2} & =\sum\nolimits _{i}\sum\nolimits _{j}\alpha_{i}\alpha_{j}\mu\left(A_{i}\cap A_{j}\right)\\
& =\left\Vert \sum\nolimits _{i}\alpha_{i}\beta_{\mu}\left(A_{i},\cdot\right)\right\Vert _{\mathscr{H}\left(\beta_{\mu}\right)}^{2},\end{aligned}$$ so $T$ is isometric and onto.
\[sec:GF\]Gaussian Fields
=========================
Let $\left(M,\mathscr{B},\mu\right)$ be a $\sigma$-finite measure space. By a *Gaussian field* based on $\left(M,\mathscr{B},\mu\right)$, we mean a probability space $\left(\Omega,\mathscr{C},\mathbb{P}^{\left(\mu\right)}\right)$, depending on $\mu$, such that $\mathscr{C}$ is a $\sigma$-algebra of subsets of $\Omega$, and $\mathbb{P}^{\left(\mu\right)}$ is a probability measure on $\left(\Omega,\mathscr{C}\right)$. For every $A\in\mathscr{B}_{fin}$, it is assumed that $X_{A}^{\left(\mu\right)}$ is in $L^{2}\left(\Omega,\mathscr{C},\mathbb{P}^{\left(\mu\right)}\right)$; and in addition, $$X_{A}^{\left(\mu\right)}\sim N\left(0,\mu\left(A\right)\right),$$ i.e., the distribution of $X_{A}^{\left(\mu\right)}$, computed for $\mathbb{P}^{\left(\mu\right)}$ is the standard Gaussian with variance $\mu\left(A\right)$.
Finally, set $\mathbb{E}_{\mu}\left(\cdot\right)=\int_{\Omega}\left(\cdot\right)d\mathbb{P}^{\left(\mu\right)}$; then it is required that $$\mathbb{E}_{\mu}\left(X_{A}^{\left(\mu\right)}X_{B}^{\left(\mu\right)}\right)=\mu\left(A\cap B\right),\quad\forall A,B\in\mathscr{B}_{fin}.\label{eq:G2}$$ For a background reference on probability spaces, see e.g., [@MR1278486].
\[prop:G1\]Given $\left(M,\mathscr{B},\mu\right)$, $\sigma$-finite, then there is an associated Gaussian field $\{X_{A}^{\left(\mu\right)}\}_{A\in\mathscr{B}_{fin}}$ satisfying $$\mathbb{E}\left(X_{A}^{\left(\mu\right)}X_{B}^{\left(\mu\right)}\right)=\mu\left(A\cap B\right),$$ for all $A,B\in\mathscr{B}_{fin}$.
For all $n\in\mathbb{N}$, $\left\{ A_{i}\right\} _{1}^{n}$, $A_{i}\in\mathscr{B}_{fin}$, let $g^{\left(A_{i}\right)}$ be the Gaussian distribution on $\mathbb{R}^{n}$, with mean zero, and covariance matrix $$\big[\mu\left(A_{i}\cap A_{j}\right)\big]_{i,j=1}^{n}.\label{eq:G3}$$ By Kolmogorov’s theorem [@MR0032961; @MR0150810; @MR0279844; @MR562914; @MR3272038; @MR3642406], there is a unique probability measure $\mathbb{P}^{\left(\mu\right)}$ on the infinite Cartesian product $$\Omega=\dot{\mathbb{R}}^{\mathscr{B}_{fin}}\label{eq:G4}$$ such that $$\mathbb{E}_{\mu}\left(\cdot\cdot\mid\left\{ A_{1},\cdots,A_{n}\right\} \right)=g^{\left(A_{i}\right)}.$$ For $\omega\in\Omega=\dot{\mathbb{R}}^{\mathscr{B}_{fin}}$, set $$X_{A}^{\left(\mu\right)}\left(\omega\right)=\omega\left(A\right),\quad A\in\mathscr{B}_{fin}.$$ For the $\sigma$-algebra $\mathscr{C}$ of subsets in $\Omega$, we take the cylinder $\sigma$-algebra, which is generated by $$\left\{ \omega\in\Omega\mid a_{i}<\omega\left(A_{i}\right)<b_{i}\right\} ,$$ with $\left\{ A_{i}\right\} _{1}^{n}\subset\mathscr{B}_{fin}$, and open intervals $\left(a_{i},b_{i}\right)$; see .
![\[fig:cyl\]A cylinder set in $\Omega$.](cyl){width="60.00000%"}
\[cor:G2\]Let $\left(M,\mathscr{B},\mu\right)$ be given, $\sigma$-finite, and let $X^{\left(\mu\right)}$ be an associated Gaussian field; see , and (\[eq:G2\]).
Let $\mathscr{D}_{fin}\left(\mu\right)=span\left\{ \chi_{A}\mid A\in\mathscr{B}_{fin}\right\} $; then $$\mathscr{D}_{fin}\left(\mu\right)\ni\sum_{i}\alpha_{i}\chi_{A_{i}}\longmapsto\sum_{i}\alpha_{i}X_{A_{i}}^{\left(\mu\right)}$$ extends by closure to an isometry of $L^{2}\left(\mu\right)$ into $L^{2}\left(\Omega,\mathbb{P}^{\left(\mu\right)}\right)$, called the generalized Itô-Wiener integral.
We have for all linear combinations as above, $$\begin{aligned}
\left\Vert \sum\nolimits _{i}\alpha_{i}X_{A_{i}}^{\left(\mu\right)}\right\Vert _{L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})}^{2} & = & \sum\nolimits _{i}\sum\nolimits _{j}\alpha_{i}\alpha_{j}\mathbb{E}_{\mu}\left(X_{A_{i}}^{\left(\mu\right)}X_{A_{j}}^{\left(\mu\right)}\right)\\
& \underset{\text{by \ensuremath{\left(\ref{eq:G2}\right)}}}{=} & \sum\nolimits _{i}\sum\nolimits _{j}\alpha_{i}\alpha_{j}\mu\left(A_{i}\cap A_{j}\right)\\
& = & \left\Vert \sum\nolimits _{i}\alpha_{i}\chi_{A_{i}}\right\Vert _{L^{2}\left(\mu\right)}^{2}\end{aligned}$$ which is the desired isometry. Hence $$T_{\mu}:\underset{\varphi}{\underbrace{\sum\nolimits _{i}\alpha_{i}\chi_{A_{i}}}}\longrightarrow\sum\nolimits _{i}\alpha_{i}X_{A_{i}}^{\left(\mu\right)}$$ extends by closure to an isometry $$T_{\mu}\left(\varphi\right):=X_{\varphi}^{\left(\mu\right)},\label{eq:G10}$$ i.e., $$\mathbb{E}_{\mu}\left(\left|X_{\varphi}^{\left(\mu\right)}\right|^{2}\right)=\int_{M}\left|\varphi\right|^{2}d\mu,\quad\text{and}\quad\mathbb{E}_{\mu}\left(X_{\varphi_{1}}^{\left(\mu\right)}X_{\varphi_{2}}^{\left(\mu\right)}\right)=\int_{M}\varphi_{1}\varphi_{2}\,d\mu$$ hold for all $\varphi_{1},\varphi_{2}\in L^{2}\left(\mu\right)$. Moreover, $X_{\varphi}^{\left(\mu\right)}\sim N\big(0,\left\Vert \varphi\right\Vert _{L^{2}\left(\mu\right)}^{2}\big)$ as stated.
\[cor:G3\]Let $\left(M,\mathscr{B},\mu\right)$ be as above, i.e., $\mu$ is assumed $\sigma$-finite. Suppose, in addition, that $\mu$ is non-atomic; then the quadratic variation of the Gaussian process $X^{\left(\mu\right)}$ coincides with the measure $\mu$ itself.
Consider $B\in\mathscr{B}_{fin}$, and consider all partitions $PAR\left(B\right)$ of the set $B$, i.e., $$\pi=\left\{ \left(A_{i}\right)\right\}$$ specified as follows: $A_{i}\in\mathscr{B}_{fin}$, $A_{i}\cap A_{j}=\emptyset$ if $i\neq j$, and $\cup_{i}A_{i}=B$.
We consider the limit over the net of such partitions.We show that $$\mathbb{E}_{\mu}\left(\left|\mu\left(B\right)-\sum\nolimits _{i}(X_{A_{i}}^{\left(\mu\right)})^{2}\right|^{2}\right)\longrightarrow0\label{eq:G12}$$ as $\pi\rightarrow0$, i.e., $\max_{i}\mu\left(A_{i}\right)\rightarrow0$, for $\pi=\left(A_{i}\right)\in PAR\left(B\right)$.
Since, for $\pi=\left(A_{i}\right)\in PAR\left(B\right)$, we have $\sum_{i}\mu\left(A_{i}\right)=\mu\left(B\right)$, to prove (\[eq:G12\]), we need only consider the individual terms; $i$ fixed: $$\begin{aligned}
& & \mathbb{E}_{\mu}\left(\left|\mu\left(A_{i}\right)-(X_{A_{i}}^{\left(\mu\right)})^{2}\right|^{2}\right)\\
& = & \mu\left(A_{i}\right)^{2}-2\mu\left(A_{i}\right)\mathbb{E}_{\mu}\left((X_{A_{i}}^{\left(\mu\right)})^{2}\right)+\mathbb{E}_{\mu}\left((X_{A_{i}}^{\left(\mu\right)})^{4}\right).\end{aligned}$$ But $$\mathbb{E}_{\mu}\left((X_{A_{i}}^{\left(\mu\right)})^{2}\right)=\mu\left(A_{i}\right),\quad\text{and}\quad\mathbb{E}_{\mu}\left((X_{A_{i}}^{\left(\mu\right)})^{4}\right)=3\mu\left(A_{i}\right)^{2};$$ and so $$\mathbb{E}_{\mu}\left(\left|\mu\left(A_{i}\right)-(X_{A_{i}}^{\left(\mu\right)})^{2}\right|^{2}\right)=2\mu\left(A_{i}\right)^{2}.$$ Now, for $\pi=\left(A_{i}\right)\in PAR\left(B\right)$, we have: $$\sum_{i}\mu\left(A_{i}\right)^{2}\leq\underset{\rightarrow0}{\underbrace{\left(\max_{i}\mu\left(A_{i}\right)\right)}}\mu\left(B\right)\quad\text{as \ensuremath{\pi\rightarrow0;}}$$ and the desired conclusion (\[eq:G12\]) follows.
By general theory, fixing a non-atomic measure space $\left(\mathscr{B},\mu\right)$, then the set $\pi$ of all $\left(\mathscr{B},\mu\right)$-partitions (see above) can be given an obvious structure of refinement. This in turn yields a corresponding net, and net-convergence refers limit over this net, as the refinement mesh tends to zero. Specifically, as $\max_{i}\mu\left(A_{i}\right)\rightarrow0$.
Let $\mu$ and $\nu$ be two positive $\sigma$-finite measures on a fixed measure space $\left(M,\mathscr{B}\right)$; see for the detailed setting. Let $X^{\left(\mu\right)}$ and $X^{\left(\nu\right)}$ be the corresponding Gaussian fields. Consider nets of partitions $\pi=\left\{ \left(A_{i}\right)\right\} $ from $\left(M,\mathscr{B}\right)$.
1. If $B\in\mathscr{B}$, then the limit $$\lim_{\stackrel{\pi\rightarrow0}{{\scriptscriptstyle \pi\in PAR\left(B\right)}}}\sum_{i}X_{A_{i}}^{\left(\mu\right)}X_{A_{i}}^{\left(\nu\right)}$$ exists; and it defines a signed measure, denoted $\langle X^{\left(\mu\right)},X^{\left(\nu\right)}\rangle$, satisfying $$\langle X^{\left(\mu\right)},X^{\left(\nu\right)}\rangle=\frac{1}{2}\left(\langle X^{\left(\mu\right)}\rangle+\langle X^{\left(\nu\right)}\rangle-\langle X^{\left(\mu\right)}-X^{\left(\nu\right)}\rangle\right).$$
2. If $\lambda$ is a positive measure on $\left(M,\mathscr{B}\right)$ satisfying $\mu\ll\lambda$, and $\nu\ll\lambda$, with respective Radon-Nikodym derivatives $d\mu/d\lambda$ and $d\nu/d\lambda$, then $$\langle X^{\left(\mu\right)},X^{\left(\nu\right)}\rangle=\sqrt{\frac{d\mu}{d\lambda}\frac{d\nu}{d\lambda}}\,d\lambda,\label{eq:G15}$$ where the representation in (\[eq:G15\]) is of the choice of measures $\lambda$ subject to: $\mu\ll\lambda$, $\nu\ll\lambda$.
The details follow those in the proof of above; and we also make use of the theory of sigma-Hilbert spaces (universal Hilbert spaces); see e.g., [@MR0282379; @zbMATH06897817; @2018arXiv180506063J].
Let $\left(M,\mathscr{B},\mu\right)$, $X^{\left(\mu\right)}$, and $T_{\mu}:L^{2}\left(\mu\right)\longrightarrow L^{2}(\mathbb{P}^{\left(\mu\right)})$ be as in , then the adjoint $$T_{\mu}^{*}:L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})\longrightarrow L^{2}\left(M,\mu\right)$$ is specified as follows:
Let $n\in\mathbb{N}$, and let $p\left(x_{1},x_{2},\cdots,x_{n}\right)$ be a polynomial on $\mathbb{R}^{n}$. For $$F:=p\left(X_{\varphi_{1}}^{\left(\mu\right)},\cdots,X_{\varphi_{n}}^{\left(\mu\right)}\right),\quad\left\{ \varphi_{i}\right\} _{1}^{n},\:\varphi_{i}\in L^{2}\left(\mu\right);\label{eq:G16}$$ set $$D\left(F\right):=\sum_{i=1}^{n}\frac{\partial p}{\partial x_{i}}\left(X_{\varphi_{1}}^{\left(\mu\right)},\cdots,X_{\varphi_{n}}^{\left(\mu\right)}\right)\otimes\varphi_{i}.\label{eq:G17}$$ Then we get the adjoint $T_{\mu}^{*}$ of the isometry $T_{\mu}$ expressed as: $$T_{\mu}^{*}\left(F\right)=\sum_{i=1}^{n}\mathbb{E}_{\mu}\left(\frac{\partial p}{\partial x_{i}}\left(X_{\varphi_{1}}^{\left(\mu\right)},\cdots,X_{\varphi_{n}}^{\left(\mu\right)}\right)\right)\varphi_{i}.\label{eq:G18}$$ (Note that the right-hand side in (\[eq:G18\]) is in $L^{2}\left(\mu\right)$.)
Recall that $$T_{\mu}\psi:=X_{\psi}^{\left(\mu\right)}:L^{2}\left(\mu\right)\longrightarrow L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})$$ as in (\[eq:G10\]), and $$X_{\psi}^{\left(\mu\right)}=\int_{M}\psi\,dX^{\left(\mu\right)}\label{eq:G19}$$ is the stochastic integral, where $dX^{\left(\mu\right)}$ denotes the Itô-Wiener integral.
The arguments combine the results in the present section, and standard facts regarding the Malliavin derivative. (See, e.g., [@MR3630401; @MR1952822; @MR2382071; @MR3501849].) Recall that the operator $$D:L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})\longrightarrow L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})\otimes L^{2}\left(\mu\right)$$ from (\[eq:G17\]) is the Malliavin derivative corresponding to the Gaussian field (\[eq:G19\]); see also .
In the arguments below, we restrict consideration to the case of real valued functions. We shall also make use of the known fact that the space of functions $F$ in (\[eq:G16\]) is dense in $L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})$ as $n\in\mathbb{N}$, polynomials $p\left(x_{1},\cdots,x_{n}\right)$, and $\left\{ \varphi_{i}\right\} _{1}^{n}$ vary, $\varphi_{i}\in L^{2}\left(\mu\right)$.
The key step in the verification of the formula (\[eq:G18\]) for $T^{*}$, form $L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})$ onto $L^{2}\left(\mu\right)$, is the following assertion: Let $F$ and $X_{\psi}^{\left(\mu\right)}$, $\psi\in L^{2}\left(\mu\right)$, be as stated; then $$\begin{aligned}
\left\langle F,X_{\psi}^{\left(\mu\right)}\right\rangle _{L^{2}(\Omega,\mathbb{P}^{\left(\mu\right)})} & =\mathbb{E}_{\mu}\left(FX_{\psi}^{\left(\mu\right)}\right)\nonumber \\
& =\sum_{i=1}^{n}\mathbb{E}_{\mu}\left(\frac{\partial p}{\partial x_{i}}\left(X_{\varphi_{1}}^{\left(\mu\right)},\cdots,X_{\varphi_{n}}^{\left(\mu\right)}\right)\right)\left\langle \varphi_{i},\psi\right\rangle _{L^{2}\left(\mu\right)}.\label{eq:G421}\end{aligned}$$ But (\[eq:G421\]) in turn follows from the basic formula for the finite-dimensional Gaussian distributions $g^{\left(n\right)}\left(x\right)$ in above. We have: $$\begin{aligned}
& & \int_{\mathbb{R}^{n}}\frac{\partial p}{\partial x_{i}}\left(x_{1},\cdots,x_{n}\right)g^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right)d^{\left(n\right)}x\\
& = & \int_{\mathbb{R}^{n}}x_{i}p\left(x_{1},\cdots,x_{n}\right)g^{\left(n\right)}\left(x_{1},\cdots,x_{n}\right)d^{\left(n\right)}x\end{aligned}$$ where $d^{\left(n\right)}x=dx_{1}dx_{2}\cdots dx_{n}$ is the standard Lebesgue measure on $\mathbb{R}^{n}$.
The general case is as follows: Set $C=\left[\mu\left(A_{i}\cap A_{j}\right)\right]_{i,j}$, the covariance matrix from (\[eq:G3\]), and $$g\left(x\right):=g^{\left(A_{i}\right)}\left(x\right)=\left(\det C\right)^{-n/2}e^{-\frac{1}{2}\left\langle x,C^{-1}x\right\rangle _{\mathbb{R}^{n}}};$$ then $$\begin{aligned}
& & \mathbb{E}_{\mu}\left(\sum\nolimits _{i}\frac{\partial p}{\partial x_{i}}\left\langle \varphi_{i},\psi\right\rangle _{L^{2}\left(\mu\right)}\right)\\
& = & \int_{\mathbb{R}^{n}}\sum\nolimits _{i}\frac{\partial p}{\partial x_{i}}\left(x\right)g\left(x\right)\left\langle \varphi_{i},\psi\right\rangle _{L^{2}\left(\mu\right)}dx^{\left(n\right)}\\
& = & \int_{\mathbb{R}^{n}}p\left(x\right)\left(\sum\nolimits _{i,j}C_{ij}^{-1}x_{j}\right)g\left(x\right)\left\langle \varphi_{i},\psi\right\rangle _{L^{2}\left(\mu\right)}dx^{\left(n\right)}\\
& = & \mathbb{E}_{\mu}\left(p\,T_{\mu}\left(\sum\nolimits _{i,j}C_{ij}^{-1}\varphi_{j}\left\langle \varphi_{i},\psi\right\rangle _{L^{2}\left(\mu\right)}\right)\right),\end{aligned}$$ where $$\psi\longmapsto\sum_{i,j}C_{ij}^{-1}\varphi_{j}\left\langle \varphi_{i},\psi\right\rangle _{L^{2}\left(\mu\right)}$$ is the projection from $\psi$ onto $span\left\{ \varphi_{i}\right\} $.
Recall the correspondence $\left(p,\varphi_{1},\cdots,\varphi_{n}\right)\longleftrightarrow F$ in (\[eq:G16\]), where $p=p\left(x_{1},\cdots,x_{n}\right)$, $x=\left(x_{1},\cdots,x_{n}\right)\in\mathbb{R}^{n}$. The random variable $F$ has the Wiener-chaos representation in (\[eq:G16\]).
Let $\left(M,\mathscr{B},\mu\right)$ be a $\sigma$-finite measure, and let $\{X_{\varphi}^{\left(\mu\right)}\mid\varphi\in L^{2}\left(\mu\right)\}$ be the corresponding Gaussian field. We then have the following covariance relations for $(X_{\varphi}^{\left(\mu\right)})^{m}$ corresponding to the even and odd values of $m\in\mathbb{N}$: $$\begin{aligned}
\mathbb{E}_{\mu}\left(\left(X_{\varphi}^{\left(\mu\right)}\right)^{2n}X_{\psi}^{\left(\mu\right)}\right) & =0,\quad\forall\varphi,\psi\in L^{2}\left(\mu\right);\;\text{and}\\
\mathbb{E}_{\mu}\left(\left(X_{\varphi}^{\left(\mu\right)}\right)^{2n+1}X_{\psi}^{\left(\mu\right)}\right) & =\left\langle \varphi,\psi\right\rangle _{L^{2}\left(\mu\right)}\left\Vert \varphi\right\Vert _{L^{2}\left(\mu\right)}^{2n}\left(2n+1\right)!!\end{aligned}$$ where $$\left(2n+1\right)!!=\left(2n+1\right)\left(2n-1\right)\cdots5\cdot3=\frac{\left(2\left(n+1\right)\right)!}{2^{n+1}\left(n+1\right)!}.$$
This is immediate from (\[eq:G421\]), and an induction argument. Take $n=1$, and $p\left(x\right)=x^{m}$; starting with $$\mathbb{E}_{\mu}\left(\left(X_{\varphi}^{\left(\mu\right)}\right)^{2}X_{\psi}^{\left(\mu\right)}\right)=2\mathbb{E}_{\mu}\left(X_{\varphi}^{\left(\mu\right)}\right)\left\langle \varphi,\psi\right\rangle _{L^{2}\left(\mu\right)}=0$$ and $$\mathbb{E}_{\mu}\left(\left(X_{\varphi}^{\left(\mu\right)}\right)^{3}X_{\psi}^{\left(\mu\right)}\right)=3\underset{{\scriptscriptstyle \left\Vert \varphi\right\Vert _{L^{2}\left(\mu\right)}^{2}}}{\underbrace{\mathbb{E}_{\mu}\left(\left(X_{\varphi}^{\left(\mu\right)}\right)^{2}\right)}}\left\langle \varphi,\psi\right\rangle _{L^{2}\left(\mu\right)}.$$
Itô calculus
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In this section we discuss properties of the Gaussian process corresponding to the Hilbert space factorizations from the setting in .
The initial setting is a fixed $\sigma$-finite measure space $\left(M,\mathscr{B},\mu\right)$ with corresponding $$\mathscr{B}_{fin}=\left\{ A\in\mathscr{B}\mid\mu\left(A\right)<\infty\right\} .\label{eq:GI1}$$ As in , we shall study positive definite (p.d.) functions $\beta$ $$\mathscr{B}_{fin}\times\mathscr{B}_{fin}\xrightarrow{\quad\beta\quad}\mathbb{R};$$ i.e., it is assumed that $\forall n\in\mathbb{N}$, $\forall\left\{ c_{i}\right\} _{1}^{n}$, $\left\{ A_{i}\right\} _{1}^{n}$, $c_{i}\in\mathbb{R}$, $A_{i}\in\mathscr{B}_{fin}$, we have $$\sum_{i}\sum_{j}c_{i}c_{j}\beta\left(A_{i},A_{j}\right)\geq0.\label{eq:GI3}$$ Then let $X=X^{\left(\beta\right)}$ be the Gaussian process with $$\left\{ \begin{split} & \mathbb{E}\left(X_{A}\right)=0,\;\text{and}\\
& \mathbb{E}\left(X_{A}X_{B}\right)=\beta\left(A,B\right),\;\text{for }\text{\ensuremath{\forall A,B\in\mathscr{B}_{fin}}}.
\end{split}
\right.\label{eq:GI25}$$
\[thm:G7\]Let $\left(M,\mathscr{B},\mu\right)$ be as above, and let $\beta$ be a corresponding p.d. function, i.e., we have (\[eq:GI1\])–(\[eq:GI3\]) satisfied.
Now suppose there is a Hilbert space $\mathscr{H}$ such that the conditions in are satisfied.
Then the Gaussian process $X=X^{\left(\beta\right)}$ admits an Itô-integral representation: Let $X^{\left(\mu\right)}$ denote the Gaussian field from and . Then there is a function $l$, as follows: $$\left\{ \begin{split}\mathscr{B}_{fin} & \xrightarrow{\quad l\quad}L^{2}\left(M,\mu\right)\\
\overset{\rotatebox{90}{\text{\ensuremath{\in}}}}{A} & \xmapsto{\quad\phantom{l}\quad}\overset{\rotatebox{90}{\text{\ensuremath{\in}}}}{l_{A}}
\end{split}
\right.$$ such that $$X_{A}=\int_{M}l_{A}\left(x\right)dX_{x}^{\left(\mu\right)},\quad\forall A\in\mathscr{B}_{fin};\label{eq:GI27}$$ where (\[eq:GI27\]) is the Itô-integral from .
We shall first need a lemma which may be of independent interest.
\[lem:G8\]With the conditions on $\left(\beta,\mu\right)$ as in the statement of and , we get existence of an $L^{2}\left(\mu\right)$-factorization for the initially given p.d. function $\beta$ see (\[eq:GI1\])–(\[eq:GI3\]). Specifically, $\beta$ admits a representation: $$\beta\left(A,B\right)=\int_{M}l_{A}\left(x\right)l_{B}\left(x\right)d\mu\left(x\right),\quad\forall A,B\in\mathscr{B}_{fin}\label{eq:GI28}$$ with $l_{A}\in L^{2}\left(\mu\right)$, $\forall A\in\mathscr{B}_{fin}$.
An application of yields a closed linear operator $T$ from $L^{2}\left(\mu\right)$ into $\mathscr{H}$, having $\mathscr{D}_{fin}\left(\mu\right)\subset L^{2}\left(\mu\right)$ as dense domain. Moreover, we have: $$\begin{aligned}
\beta\left(A,B\right) & \underset{\text{by \ensuremath{\left(\ref{eq:S4}\right)}}}{=} & \left\langle \chi_{A},\chi_{B}\right\rangle _{\mathscr{H}}\\
& \underset{\text{by \ensuremath{\left(\ref{eq:T3}\right)}}}{=} & \left\langle T\left(\chi_{A}\right),T\left(\chi_{B}\right)\right\rangle _{\mathscr{H}}\\
& = & \left\langle T^{*}T\chi_{A},\chi_{B}\right\rangle _{L^{2}\left(\mu\right)}\\
& \underset{{\scriptscriptstyle \stackrel{\text{\text{since \ensuremath{T^{*}T} is}}}{\text{selfadjoint}}}}{=} & \left\langle \left(\left(T^{*}T\right)^{\frac{1}{2}}\right)^{2}\chi_{A},\chi_{B}\right\rangle _{L^{2}\left(\mu\right)}\\
& = & \left\langle \left(T^{*}T\right)^{\frac{1}{2}}\chi_{A},\left(T^{*}T\right)^{\frac{1}{2}}\chi_{B}\right\rangle _{L^{2}\left(\mu\right)}.\end{aligned}$$ Now setting, $$l_{A}:=\left(T^{*}T\right)^{\frac{1}{2}}\chi_{A},\quad A\in\mathscr{B}_{fin},\label{eq:GI29}$$ the desired conclusion (\[eq:GI28\]) follows.
Let $\left(\beta,\mu\right)$ be as in the statement of , and let $\left\{ l_{A}\right\} _{A\in\mathscr{B}_{fin}}$ be the $L^{2}\left(\mu\right)$-function in (\[eq:GI29\]). We see that the factorization (\[eq:GI28\]) is valid.
Hence, by , the corresponding Itô-integral (\[eq:GI27\]) is well defined; and the resulting Gaussian process $X_{A}:=\int_{M}l_{A}\left(x\right)dX_{x}^{\left(\mu\right)}$ is a Gaussian field with $\mathbb{E}\left(X_{A}\right)=0$. Hence we only need to verify the convariance condition in (\[eq:GI25\]) above:
Let $A,B\in\mathscr{B}_{fin}$, and compute: $$\begin{aligned}
\mathbb{E}\left(X_{A}X_{B}\right) & = & \mathbb{E}\left[\left(\int_{M}l_{A}\left(x\right)dX_{x}^{\left(\mu\right)}\right)\left(\int_{M}l_{B}\left(x\right)dX_{x}^{\left(\mu\right)}\right)\right]\\
& \underset{{\scriptscriptstyle \text{by Cor. \ref{cor:G3}}}}{=} & \int_{M}l_{A}\left(x\right)l_{B}\left(x\right)d\mu\left(x\right)\quad\left(\mu=QV\left(X^{\left(\mu\right)}\right)\right)\\
& \underset{{\scriptscriptstyle {\scriptscriptstyle \stackrel{\text{by Lem. \ref{lem:G8},}}{\text{see \ensuremath{\left(\ref{eq:GI28}\right)}}}}}}{=} & \beta\left(A,B\right);\end{aligned}$$ and the proof is completed.
\[rem:GI9\]As an application of , consider the case of $\left(\mathbb{R},\mathscr{B},\lambda_{1}\right)$ (so $\mu=\lambda_{1}$), i.e., standard Lebesgue measure on $\mathbb{R}$, with $\mathscr{B}$ denoting the standard Borel-sigma-algebra. We shall discuss fractional Brownian motion with Hurst parameter $H$ (see [@MR0242239; @MR672910; @MR2123205; @MR2178502; @MR2793121; @MR2966130]).
Recall, on $[0,\infty)$, fractional Brownian motion $\{X_{t}^{\left(H\right)}\}_{t\in[0,\infty)}$, $0<H<1$, fixed, may be normalized as follows: $X_{0}^{\left(H\right)}=0$, $$\begin{aligned}
\mathbb{E}\left(X_{t}^{\left(H\right)}\right) & =0,\quad\text{and}\\
\mathbb{E}\left(X_{s}^{\left(H\right)}X_{t}^{\left(H\right)}\right) & =\frac{1}{2}\left(s^{2H}+t^{2H}-\left|s-t\right|^{2H}\right),\quad\forall s,t\in[0,\infty).\label{eq:GI31}\end{aligned}$$ The corresponding process induced by $\mathscr{B}_{fin}$ is $$X_{\left[0,t\right]}^{\left(H\right)}:=X_{t}^{\left(H\right)};\label{eq:GI32}$$ and we shall adapt (\[eq:GI32\]) as an identification. The following spectral representation is known: Set, for $\lambda\in\mathbb{R}$, $$d\mu^{\left(H\right)}\left(\lambda\right)=\frac{\sin\left(\pi H\right)\Gamma\left(1+2H\right)}{2\pi}\left|\lambda\right|^{1-2H}d\lambda;\label{eq:G33}$$ then $$\mathbb{E}\left(X_{s}^{\left(H\right)}X_{t}^{\left(H\right)}\right)=\int_{\mathbb{R}}\frac{\left(e^{i\lambda s}-1\right)\left(e^{-i\lambda t}-1\right)}{\lambda^{2}}d\mu^{\left(H\right)}\left(\lambda\right).\label{eq:G34}$$
A choice of factorization for the kernel $K^{\left(H\right)}\left(s,t\right)$ in (\[eq:GI31\]) is then as follows: $$\begin{aligned}
K^{\left(H\right)}\left(s,t\right) & =\frac{1}{2}\left(s^{2H}+t^{2H}-\left|s-t\right|^{2H}\right)\\
& =\int_{\mathbb{R}}l_{s}\left(x\right)l_{t}\left(x\right)dx\quad\left(s,t\in[0,\infty)\right)\end{aligned}$$ with $$\begin{aligned}
l_{t}\left(x\right) & =\frac{1}{\Gamma\left(H+\frac{1}{2}\right)}\Big(\chi_{(-\infty,0]}\left(x\right)\left(\left(t-x\right)^{H-\frac{1}{2}}-\left(-x\right)^{H-\frac{1}{2}}\right)\nonumber \\
& \qquad+\chi_{\left[0,t\right]}\left(x\right)\left(t-x\right)^{H-\frac{1}{2}}\Big),\quad x\in\mathbb{R},\:t\in[0,\infty).\label{eq:G35}\end{aligned}$$
Application to fractional Brownian motion
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Fix $H$, $0<H<1$, the Hurst parameter, and let $\{X_{t}^{\left(H\right)}\}_{t\in[0,\infty)}$ be fractional Brownian motion (fBM), see . Then the special case $H=\frac{1}{2}$ corresponds to standard Brownian motion (BM). we shall write $X_{t}^{(\nicefrac{1}{2})}=W_{t}$; where “$W$” is for Wiener. Now $\left\{ W_{t}\right\} _{t\in[0,\infty)}$ is a martingale; and standard Brownian motion has *independent increments*, by contrast to the case when $H\neq\frac{1}{2}$, i.e., fBM.
**(i) Itô-integral representation for $X_{t}^{\left(H\right)}$ when $H\neq\frac{1}{2}$.**
We now combine , (\[eq:G35\]) and (\[eq:GI27\]) to conclude that $X_{t}^{\left(H\right)}$ has the following Itô-integral representation:
Let $\{l_{t}^{\left(H\right)}\}_{t\in[0,\infty)}$ be the integral kernel from (\[eq:G35\]). Note, it depends on the value of $H$, but we shall fix $H$, $H\neq\frac{1}{2}$. Then $$X_{t}^{\left(H\right)}=\int_{\mathbb{R}}l_{t}^{\left(H\right)}\left(x\right)dW_{x};\label{eq:GI36}$$ where $\text{RHS}_{\left(\ref{eq:GI36}\right)}$ is the Itô-integral introduced in in the more general setting of $X^{\left(\mu\right)}$. Here, $\mu=\lambda_{1}=dx$ is standard Lebesgue measure; and $QV\left(W_{x}\right)=dx$; see .
**(ii) Filtrations.**
Returning to the probability space $\left(\Omega,\mathscr{C}\right)$ for $\left\{ W_{t}\right\} _{t\in[0,\infty)}$; see , and let $\mathscr{B}$ be the standard Borel $\sigma$-algebra of subsets of $\mathbb{R}$. For $A\in\mathscr{B}$, we denote by $\mathscr{F}\left(A\right):=$ the sub $\sigma$-algebra of the cylinder $\sigma$-algebra in $\Omega$ (see (\[eq:G4\])) generated by the random variables $W_{B}$, as $B$ in $\mathscr{B}$ varies over subsets $B\subseteq A$. Let $l_{t}^{\left(\pm\right)}\left(x\right)$ denote the two separate terms on $\text{RHS}_{\left(\ref{eq:G35}\right)}$, i.e., $$l_{t}^{\left(-\right)}\left(x\right)=\chi_{(-\infty,0]}\left(x\right)\left(\left(t-x\right)^{H-\frac{1}{2}}-\left(-x\right)^{H-\frac{1}{2}}\right)\Big/\Gamma\left(H+\tfrac{1}{2}\right)$$ and $$l_{t}^{\left(+\right)}\left(x\right)=\chi_{\left[0,t\right]}\left(x\right)\left(t-x\right)^{H-\frac{1}{2}}\Big/\Gamma\left(H+\tfrac{1}{2}\right).$$ Then there are two components (of fractional Brownian motion): $$X_{t}^{\left(-\right)}=\int_{-\infty}^{0}l_{t}^{\left(-\right)}\left(x\right)dW_{x},$$ and $$X_{t}^{\left(+\right)}=\int_{0}^{t}l_{t}^{\left(+\right)}\left(x\right)dW_{x};$$ where $H\neq\frac{1}{2}$ is fixed; (supposed in the notation.)
The two processes $(X_{t}^{\left(-\right)})$ and $(X_{t}^{\left(+\right)})$ are independent, and $$X_{t}=X_{t}^{\left(H\right)}=X_{t}^{\left(-\right)}\oplus X_{t}^{\left(+\right)}$$ with $$\mathbb{E}\left(X_{s}X_{t}\right)=\mathbb{E}\left(X_{s}^{\left(-\right)}X_{t}^{\left(-\right)}\right)+\mathbb{E}\left(X_{s}^{\left(+\right)}X_{t}^{\left(+\right)}\right),\quad\forall s,t\in[0,\infty).$$
These processes $(X_{t}^{\left(\pm\right)})$ result from the initial fBM $X_{t}$ (\[eq:GI36\]) itself, as conditional Gaussian processes as follows: $$\mathbb{E}\left(X_{t}^{\left(+\right)}\mid\mathscr{F}\left((-\infty,0]\right)\right)=0;\label{eq:G39}$$ and $$\mathbb{E}\left(X_{t}\mid\mathscr{F}\left((-\infty,0]\right)\right)=X_{t}^{\left(-\right)}\label{eq:G38}$$ and $$\mathbb{E}\left(X_{t}\mid\mathscr{F}\left(\left[0,t\right]\right)\right)=X_{t}^{\left(+\right)}.\label{eq:G41}$$ So $X_{t}^{\left(-\right)}$ in (\[eq:G38\]) is the backward process, while $X_{t}^{\left(+\right)}$ is the corresponding forward process.
Fix $H$ (Hurst parameter) as above, and consider the fractional Brownian motion $X_{t}^{\left(H\right)}$, and its forward part $X_{t}^{\left(+\right)}:=(X_{t}^{\left(H\right)})^{+}$ given in (\[eq:G41\]). Then $(X_{t}^{\left(H\right)})^{+}$ is a , i.e., if $0<s<t$, then $$\mathbb{E}\left(X_{t}^{\left(+\right)}\mid\mathscr{F}\left(\left[0,s\right]\right)\right)=X_{s}^{\left(+\right)}.\label{eq:GS42}$$
$$\begin{aligned}
\text{LHS}_{\left(\ref{eq:GS42}\right)} & \underset{{\scriptscriptstyle \text{by \ensuremath{\left(\ref{eq:G41}\right)}}}}{=} & \mathbb{E}\left(\mathbb{E}\left(X_{t}^{\left(H\right)}\mid\mathscr{F}\left(\left[0,t\right]\right)\right)\mid\mathscr{F}\left(\left[0,s\right]\right)\right)\\
& = & \mathbb{E}\left(X_{t}^{\left(H\right)}\mid\mathscr{F}\left(\left[0,s\right]\right)\right)\\
& = & \mathbb{E}\left(\left(X_{t}^{\left(H\right)}-X_{s}^{\left(H\right)}\right)+X_{s}^{\left(H\right)}\mid\mathscr{F}\left(\left[0,s\right]\right)\right)\\
& \underset{{\scriptscriptstyle \text{by \ensuremath{\left(\ref{eq:G39}\right)}}}}{=} & \mathbb{E}\left(X_{s}^{\left(H\right)}\mid\mathscr{F}\left(\left[0,s\right]\right)\right)\underset{{\scriptscriptstyle \text{by \ensuremath{\left(\ref{eq:G41}\right)}}}}{=}X_{s}^{\left(+\right)}.\end{aligned}$$
We stress that the proofs of these properties of fBM, (with $H\neq\frac{1}{2}$) follow essentially from our conclusions in , as well as Corollaries \[cor:G2\] and \[cor:G3\].
**The spectral representation.**
The formula (\[eq:G34\]) is a *spectral representation* in following sense: The choice of $d\mu^{\left(H\right)}$ in (\[eq:G33\]) yields the following generalized Paley-Wiener space (compare (\[eq:I7\])–(\[eq:I8\]) above):
Let $\mathscr{H}(\mu^{\left(H\right)})$ denote the Hilbert space of functions $f$ on $\mathbb{R}$ such that the Fourier transform $\widehat{f}$ is well defined and is in $L^{2}(\mu^{\left(H\right)})$. Then set $$\left\Vert f\right\Vert _{\mathscr{H}(\mu^{\left(H\right)})}^{2}=\Vert\widehat{f}\Vert_{L^{2}(\mu^{\left(H\right)})}^{2}=\int_{\mathbb{R}}|\widehat{f}\left(\lambda\right)|^{2}d\mu^{\left(H\right)}\left(\lambda\right).\label{eq:G40}$$ For $f\in\mathscr{H}(\mu^{\left(H\right)})$, consider the Itô-integral, $$X^{\left(H\right)}\left(f\right):=\int f\left(t\right)dX_{t}^{\left(H\right)}.$$ Then it follows from (\[eq:G34\]), and Theorems \[thm:T1\] and \[thm:G7\] that $$\mathbb{E}\left(\left|X^{\left(H\right)}\left(f\right)\right|^{2}\right)=\left\Vert f\right\Vert _{\mathscr{H}(\mu^{\left(H\right)})}^{2}.$$ In particular, $$\mathbb{E}\left(\left|X^{\left(H\right)}\left(f\left(\cdot+t\right)\right)\right|^{2}\right)=\mathbb{E}\left(\left|X^{\left(H\right)}\left(f\right)\right|^{2}\right).$$ This follows since the RHS in (\[eq:G40\]) is translation invariant, i.e., we have: $$\widehat{f\left(\cdot+t\right)}\left(\lambda\right)=e^{i\lambda t}\widehat{f}\left(\lambda\right).$$
A Karhunen-Loève representation
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The Karhunen-Loève (KL) theorem is usually stated for the special case of positive definite kernels $K$ which are also continuous (typically on a bounded interval), so called Mercer-kernels. The starting point is then an application of the spectral theorem to the corresponding selfadjoint integral operators, $T_{K}$ in $L^{2}$ of the interval. Mercers theorem states that if $K$ is Mercer, then the integral operator $T_{K}$ is trace-class. A Karhunen-Loève representation for a stochastic process (with specified covariance kernel $K$) is a generalized infinite linear combination, or orthogonal expansion, for the random process, analogous to a Fourier series representation for (deterministic) functions on a bounded interval; see e.g., [@MR0008270; @MR2268393]. The KL representation we give below is much more general, and it applies to the most general positive definite kernel, and makes essential use of our RKHS theorem ( below). In our KL-theorem, we also make precise the random i.i.d $N(0,1)$-terms inside the KL-expansion; see (\[eq:G36\]).
\[cor:G7\]Let $\left(M,\mathscr{B},\mu\right)$ be a $\sigma$-finite measure space, and let $\{X_{A}^{\left(\mu\right)}\}_{A\in\mathscr{B}_{fin}}$ be the associated Gaussian field (see and .) Let $\left\{ \varphi_{k}\right\} _{k\in\mathbb{N}}$ be an orthonormal basis (ONB) in $L^{2}\left(\mu\right)$, and set $$Z_{k}:=X_{\varphi_{k}}^{\left(\mu\right)}=\int_{M}\varphi_{k}\,dX^{\left(\mu\right)}.\label{eq:G36}$$
1. Then $\left\{ Z_{k}\right\} _{k\in\mathbb{N}}$ is an i.i.d. $N\left(0,1\right)$-system (i.e., a system of independent, identically distributed standard Gaussians.)
2. Moreover, $X^{\left(\mu\right)}$ admits the following Karhunen-Loève representation ($A\in\mathscr{B}_{fin}$): $$X_{A}^{\left(\mu\right)}\left(\cdot\right)=\sum_{k\in\mathbb{N}}\left(\int_{A}\varphi_{k}\,d\mu\right)Z_{k}\left(\cdot\right),\label{eq:G23}$$ and, more generally, for $\psi\in L^{2}\left(\mu\right)$, $$X_{\psi}^{\left(\mu\right)}\left(\cdot\right)=\sum_{k\in\mathbb{N}}\left\langle \psi,\varphi_{k}\right\rangle _{L^{2}\left(\mu\right)}Z_{k}\left(\cdot\right).$$
3. In particular, $X^{\left(\mu\right)}$ admits a realization on the infinite product space $\Omega=\mathbb{R}^{\mathbb{N}}$, equipped with the usual cylinder $\sigma$-algebra, and the infinite-product measure $$\mathbb{P}:=\vartimes_{\mathbb{N}}g_{1}=g_{1}\times g_{1}\times\cdots,\label{eq:G25}$$ where $g_{1}\left(x\right)=\frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}=$ the $N\left(0,1\right)$-distribution. (We compute the expectation $\mathbb{E}$ with respect to $\mathbb{P}$, the infinite product measure $\mathbb{P}$ in (\[eq:G25\]))
When the system $\left\{ Z_{k}\right\} _{k\in\mathbb{N}}$ is specified as in (\[eq:G36\]), it follows from standard Gaussian theory (see e.g., [@MR2362796; @MR2446590; @MR2670567; @MR2793121; @MR2966130; @MR3402823; @MR3687240] and the papers cited there) that it is an i.i.d. $N\left(0,1\right)$-system.
For $A,B\in\mathscr{B}_{fin}$, $$\begin{aligned}
\mathbb{E}\left(X_{A}^{\left(\mu\right)}X_{B}^{\left(\mu\right)}\right) & = & \sum_{k\in\mathbb{N}}\sum_{l\in\mathbb{N}}\int_{A}\varphi_{k}\,d\mu\int_{B}\varphi_{l}\,d\mu\,\underset{{\scriptscriptstyle \delta_{k,l}}}{\underbrace{\mathbb{E}\left(Z_{k}Z_{l}\right)}}\\
& = & \sum_{k\in\mathbb{N}}\int_{A}\varphi_{k}\,d\mu\int_{B}\varphi_{k}\,d\mu\\
& \underset{{\scriptscriptstyle \text{by Parseval}}}{=} & \left\langle \chi_{A},\chi_{B}\right\rangle _{L^{2}\left(\mu\right)}=\mu\left(A\cap B\right).\end{aligned}$$ Since the representation in (\[eq:G23\]) yields a Gaussian process with mean zero, it is determined by its covariance kernel, and the result follows.
In this section, we have addressed some questions that are naturally implied by our present setting, but we wish to stress that there is a vast literature in the general area of the subject, and dealing with a variety of different important issues for Gaussian fields. Below we cite a few papers, and readers may also want to consult papers cited there: [@MR2677883; @MR2805533; @MR3266271; @MR3803148; @MR1952822; @MR2382071; @MR3501849; @MR2322706].
Measures on $\left(I,\mathscr{B}\right)$ when $I$ is an interval
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We consider the spaces consisting of the measure spaces when $I$ is an interval (taking $I=\left[0,1\right]$ for specificity); and where $\mathscr{B}$ is the standard Borel $\sigma$-algebra of subsets in $I$.
In this case, our results above, especially , take the following form:
Let $\mu$ be a $\sigma$-finite measure on $\left(M,\mathscr{B}\right)$, and $\beta_{\mu}\left(A,B\right)=\mu\left(A\cap B\right)$ the p.d. function from (\[eq:T8\]). Let $\mathscr{H}\left(\beta_{\mu}\right)$ be the corresponding RKHS.
1. Then $\mathscr{H}\left(\beta_{\mu}\right)$ consists of all functions $F$ on $\left[0,1\right]$, such that $F\left(0\right)=0$, and $$\sup_{0\leq a<b\leq1}\frac{\left|F\left(b\right)-F\left(a\right)\right|}{\mu\left(\left[a,b\right]\right)}<\infty,\label{eq:M1}$$ supremum over all intervals contained in $\left[0,1\right]$.
2. If $dF/d\mu$ denotes the Radon-Nikodym derivative corresponding to (\[eq:M1\]), then the $\mathscr{H}\left(\beta_{\mu}\right)$-norm is as follows:
First, $dF/d\mu\in L^{2}\left(\mu\right)$, and $$\left\Vert F\right\Vert _{\mathscr{H}\left(\beta_{\mu}\right)}^{2}=\int_{0}^{1}\left|\frac{dF}{d\mu}\right|^{2}d\mu.\label{eq:M2}$$
The idea is essentially contained in the considerations above from . Indeed, if $F$ is as specified in (\[eq:M1\]) & (\[eq:M2\]), set for all $A\in\mathscr{B}$, $$\mu_{F}\left(A\right)=\int_{A}\frac{dF}{d\mu}d\mu.\label{eq:M3}$$ Then the Radon-Nikodym derivative $d\mu_{F}/d\mu$ in (\[eq:T7\]) satisfies $d\mu_{F}/d\mu=dF/d\mu$ (see (\[eq:M2\])–(\[eq:M3\])).
Moreover, $$L^{2}\left(\mu\right)\ni\varphi\longmapsto\underset{F_{\varphi}\left(\cdot\right)\in\mathscr{H}\left(\beta_{\mu}\right)}{\underbrace{\int_{A}\varphi\,d\mu=F_{\varphi}\left(A\right)}}$$ defines an isometry, mapping onto $\mathscr{H}\left(\beta_{\mu}\right)$.
If, for example, $\mu=\mu_{3}$ is the middle-third Cantor measure, then the Devil’s Staircase function (see ) is $$F\left(x\right)=\mu_{3}\left(\left[0,x\right]\right).$$ It is in $\mathscr{H}\left(\beta_{\mu}\right)$, and $$\frac{dF}{d\mu_{3}}=\chi_{\left[0,1\right]}.\label{eq:M6}$$ Note that it is important that the Radon-Nikodym derivative in (\[eq:M6\]) is with respect to the Cantor measure $\mu_{3}$. If, for example, $\lambda$ denotes the Lebesgue measure on $\left[0,1\right]$, then $dF/d\lambda=0$.
For graphical illustration of these functions, see Figures \[fig:DS\]–\[fig:cum\] below.
![\[fig:DS\]The middle-third Cantor set.](c3){width="40.00000%"}
**Time-change**
While there is earlier work in the literature, dealing with time-change in Gaussian processes, see e.g., [@MR2484103; @MR3363697]; our aim here is to illustrate the use of our results in Sections \[sec:SA\] and \[sec:GF\] as they apply to the change of the time-variable in a Gaussian process. To make our point, we have found it sufficient to derive the relevant properties for time-change for time in a half-line.
[>p[0.45]{}>p[0.45]{}]{} ![\[fig:cum\]The two *cumulative distributions*, with support sets $[0,1]$ and $C_{\nicefrac{1}{3}}$. ](dev0 "fig:"){width="30.00000%"} & ![\[fig:cum\]The two *cumulative distributions*, with support sets $[0,1]$ and $C_{\nicefrac{1}{3}}$. ](dev1 "fig:"){width="30.00000%"}[\
]{}[ $F_{\lambda}\left(x\right)=\lambda\left(\left[0,x\right]\right)$; points of increase = the support of the normalized $\lambda$, so the interval $[0,1]$. ]{} & [ $F_{\nicefrac{1}{3}}\left(x\right)=\mu_{3}\left(\left[0,x\right]\right)$; points of increase = the support of $\mu_{3}$, so the middle third Cantor set $C_{\nicefrac{1}{3}}$ (the Devil’s staircase).]{}[\
]{}
\[prop:M3\]Let $J=[0,\infty)$ denote the positive half-line, and let $\left\{ B_{t}\right\} _{t\in J}$ be the standard Brownian motion, i.e., $B_{t}\sim N\left(0,t\right)$, and $$\mathbb{E}\left(B_{s}B_{t}\right)=s\wedge t,\quad\forall s,t\in J\label{eq:M7}$$ where $s\wedge t=\min\left(s,t\right)$. Let $h:J\rightarrow J$ be a monotone (increasing) function such that $h\left(0\right)=0$, and set $X=X^{\left(h\right)}$ given by $$X_{t}:=B_{h\left(t\right)},\quad t\in J.\label{eq:M8}$$
1. Then $X_{t}$ is the Gaussian process determined by the following induced covariance kernel: $$\mathbb{E}\left(X_{s}X_{t}\right)=h\left(s\wedge t\right)$$
2. \[enu:M32\]The for $\{X_{t}^{\left(h\right)}\}_{t\in J}$ is $$d\mu\left(t\right)=h'\left(t\right)dt,\label{eq:M10}$$ where $dt$ is the usual Lebesgue measure on $J$.
(Recall that, since $h\left(s\right)\leq h\left(t\right)$ for all $s,t$, $s\leq t$; it follows, by Lebesgue’s theorem, that $h$ is differentiable almost everywhere on $J$ with respect to $dt$.)
Note that, if $h\left(t\right)=t^{2}$, then $\mathbb{E}\left(X_{s}X_{t}\right)=\left(s\wedge t\right)^{2}$; see for an illustration.
Since $h$ is monotone (increasing) and $h\left(0\right)=0$, we get $$h\left(s\right)\wedge h\left(t\right)=h\left(s\wedge t\right),\quad\forall s,t\in J\label{eq:M11}$$ and so the covariance kernel satisfies: $$\begin{aligned}
\mathbb{E}\left(X_{s}X_{t}\right) & = & \mathbb{E}\left(B_{h\left(s\right)}B_{h\left(t\right)}\right)\\
& \underset{{\scriptscriptstyle \text{by \ensuremath{\left(\ref{eq:M7}\right)}}}}{=} & h\left(s\right)\wedge h\left(t\right)\underset{{\scriptscriptstyle \text{by \ensuremath{\left(\ref{eq:M11}\right)}}}}{=}h\left(s\wedge t\right)\\
& = & \int_{0}^{s\wedge t}h'\left(x\right)dx=\mu\left(s\wedge t\right)\\
& = & \mu\left(\left[0,s\right]\cap\left[0,t\right]\right)\end{aligned}$$ where $\mu$ is the measure given in (\[eq:M10\]).
It now follows from that then $\mu$ is indeed the quadratic variation measure for $\{X_{t}^{\left(h\right)}\}_{t\in J}$, as asserted.
\[cor:M5\]Let $h:J\rightarrow J$ be as in , i.e., $h\left(0\right)=0$, $h\left(s\right)\leq h\left(t\right)$, for $s\leq t$; and, as in (\[eq:M8\]), consider: $$X_{t}=B_{h\left(t\right)},\quad t\in J.$$ Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be given, assumed twice differentiable. Then the Itô-integral formula for $f\left(X_{t}\right)$ is as follows: For $t>0$, we have: $$f\left(X_{t}\right)=\int_{0}^{t}f'\left(X_{s}\right)dX_{s}+\frac{1}{2}\int_{0}^{t}f''\left(X_{s}\right)h'\left(s\right)ds.\label{eq:M13}$$
The result is immediate from Itô’s lemma applied to the quadratic variation term on the right-hand side in (\[eq:M13\]).
Recall, we proved in (\[enu:M32\]), eq. (\[eq:M10\]) that the quadratic variation of a Gaussian process with covariance measure $\mu$ is $\mu$ itself. Hence (\[eq:M13\]) follows from a direct application to $d\mu\left(s\right)=h'\left(s\right)ds$, where $ds$ is standard Lebesgue measure on the interval $J$.
Let $h:J\rightarrow J$, $h\left(0\right)=0$, $h$ monotone be as specified as , and let $X_{t}=B_{h\left(t\right)}$ be the corresponding time-change process.
Set $d\mu=dh=\left(\text{the Stieltjes measure}\right)=h'\left(t\right)dt$; see (\[eq:M13\]). For $\left(t,x\right)\in J\times J$, and $f\in L^{2}\left(\mu\right)$, let $$u\left(t,x\right)=\mathbb{E}_{X_{0}=x}\left(f\left(X_{t}\right)\right).\label{eq:M14}$$ Then $u$ satisfies the following diffusion equation $$\frac{\partial}{\partial t}u\left(t,x\right)=\frac{1}{2}h'\left(t\right)\frac{\partial^{2}}{\partial x^{2}}u\left(t,x\right),\label{eq:M15}$$ with boundary condition $$u\left(t,\cdot\right)\big|_{t=0}=f\left(\cdot\right).$$
The assertion follows from an application of the conditional expectation $\mathbb{E}_{X_{0}=x}$ to both sides in (\[eq:M13\]). Since the expectation of the first of the two terms on the right-hand side in (\[eq:M13\]) vanishes, we get from the definition (\[eq:M14\]) that: $$u\left(t,x\right)=\frac{1}{2}\int_{0}^{t}\mathbb{E}_{X_{0}=x}\left(f''\left(X_{s}\right)\right)h'\left(s\right)ds,$$ and so $$\frac{\partial}{\partial t}u\left(t,x\right)=\frac{1}{2}h'\left(t\right)\frac{\partial^{2}}{\partial x^{2}}u\left(t,x\right)$$ as claimed in (\[eq:M15\]). The remaining conclusions in the corollary are immediate.
Let $0<H<1$ be fixed, and set $$h\left(t\right):=t^{2H},\quad t\in J.$$ Then the corresponding process $$X_{t}^{\left(H\right)}:=B_{t^{2H}},\quad t\in J,$$ is a time-changed process, as discussed in . We have $$\mathbb{E}\left((X_{t}^{\left(H\right)})^{2}\right)=t^{2H}.$$
Now this is the same variance as the fractional Brownian motion $Y_{t}^{\left(H\right)}$; but we stress that (when $H$ is fixed, $H\neq1/2$), then the two Gaussian processes $X_{t}^{\left(H\right)}$ (time-change), and $Y_{t}^{\left(H\right)}$ (fractional Brownian motion with Hurst parameter $H$), are different. (See .)
$H=1/3$ $H=2/3$
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![\[fig:fBM\]Fractional Brownian motion. The top two figures are sample paths of fractional Brownian motion, while the bottom two are the corresponding processes resulting from time change in the standard Brownian motion.](fbm1a "fig:"){width="40.00000%"} ![\[fig:fBM\]Fractional Brownian motion. The top two figures are sample paths of fractional Brownian motion, while the bottom two are the corresponding processes resulting from time change in the standard Brownian motion.](fbm2a "fig:"){width="40.00000%"}
$Y_{t}^{\left(H\right)}$ $Y_{t}^{\left(H\right)}$
![\[fig:fBM\]Fractional Brownian motion. The top two figures are sample paths of fractional Brownian motion, while the bottom two are the corresponding processes resulting from time change in the standard Brownian motion.](fbm1b "fig:"){width="40.00000%"} ![\[fig:fBM\]Fractional Brownian motion. The top two figures are sample paths of fractional Brownian motion, while the bottom two are the corresponding processes resulting from time change in the standard Brownian motion.](fbm2b "fig:"){width="40.00000%"}
$X_{t}^{\left(H\right)}=B_{t^{2H}}$ $X_{t}^{\left(H\right)}=B_{t^{2H}}$
The reason is that the two covariance kernels are difference. Indeed, when $H\neq1/2$, $$\underset{{\scriptscriptstyle \mathbb{E}\left(X_{s}^{\left(H\right)}X_{t}^{\left(H\right)}\right)}}{\underbrace{\left(s\wedge t\right)^{2H}}}\neq\underset{{\scriptscriptstyle \mathbb{E}\left(Y_{s}^{\left(H\right)}Y_{t}^{\left(H\right)}\right)}}{\underbrace{\tfrac{1}{2}(s^{2H}+t^{2H}-\left|s-t\right|^{2H})}};\label{eq:M16}$$ i.e., the two functions from (\[eq:M16\]) are different on $J\times J$.
For general facts on fractional Brownian motion, and Hurst parameter, see e.g., [@MR2793121] and [@MR1921068].
Laplacians
==========
The purpose of the present section is to show that there is an important class of Laplace operators, and associated energy Hilbert spaces $\mathscr{H}$, which satisfies the conditions in our results from Sections \[sec:SigA\] and \[sec:SA\] above. Starting with a fixed sigma-finite measure $\mu$, the setting from sect \[sec:SA\] entails pairs $(\beta,\mathscr{H})$, subject to conditions (\[eq:S2\]) and (\[eq:S4\]), which admit a certain spectral theory. With the condition in , we showed that there are then induced sigma-finite measures $\mu_{f}$, indexed by $f$ in a dense subspace in $\mathscr{H}$. The key consideration implied by this is a closable, densely defined, operator $T$ from $L^{2}(\mu)$ into $\mathscr{H}$. The induced measures $\mu_{f}$ are then indexed by $f$ in $dom(T^{*})$, the dense domain of the adjoint operator $T^{*}$. If $\mathscr{H}$ is one of the energy Hilbert spaces, then $T^{*}$ will be an associated Laplacian; see details in .
Now the Laplacians we introduce include variants from both discrete network analysis, and more classical Laplacians from harmonic analysis. As well as more abstract Laplacians arising in potential theory. There is a third reason for the relevance of such new classes of Laplace-operators: Each one of these Laplacians corresponds to a reversible Markov process (and vice versa.) The latter interconnection will be addressed at the end of section, but the more detailed implications, following from it, will be postponed to future papers. As for the research literature, it is fair to say that papers on reversible Markov processed far outnumber those dealing with generalized Laplacians.
Let $\left(M,\mathscr{B},\mu\right)$ be a fixed $\sigma$-finite positive measure, and let $\rho$ be a *symmetric* positive measure on the product space $\left(M\times M,\mathscr{B}_{2}\right)$ where $\mathscr{B}_{2}$ denotes the product $\sigma$-algebra on $M\times M$, i.e., the $\sigma$-algebra of subsets of $M\times M$ generated by the cylinder sets $$\left\{ A\times B\mid A,B\in\mathscr{B}\right\} .$$
We assume that $\rho$ admits a disintegration with $\mu$ as marginal measure: $$d\rho\left(x,y\right)=\rho^{\left(x\right)}\left(dy\right)d\mu\left(x\right);\label{eq:L2}$$ equivalently, $$\rho\left(A\times B\right)=\int_{A}\rho^{\left(x\right)}\left(B\right)d\mu\left(x\right),\label{eq:L3}$$ $\forall A,B\in\mathscr{B}$. Note that since $\rho$ is symmetric, we also have a field of measures $\rho^{\left(y\right)}\left(dx\right)$ such that $$\rho\left(A\times B\right)=\int_{B}\rho^{\left(y\right)}\left(A\right)d\mu\left(y\right).\label{eq:L4}$$ For the theory of disintegration of measures, we refer to [@MR3441734; @zbMATH06897817] and the papers cited there.
Let $\pi_{i}$, $i=1,2$, denote the coordinate projections $$\pi_{1}\left(x,y\right)=x,\quad\text{and}\quad\pi_{2}\left(x,y\right)=y,$$ for $\left(x,y\right)\in M\times M$. Then from the assumptions above, we get $$\mu=\rho\circ\pi_{1}^{-1}=\rho\circ\pi_{2}^{-1}.$$
We shall finally assume that $$\rho\left(A\times M\right)<\infty,\quad\forall A\in\mathscr{B}_{fin}\label{eq:L5}$$ where $\mathscr{B}_{fin}=\left\{ A\in\mathscr{B}\mid\mu\left(A\right)<\infty\right\} $; and we set $$c\left(x\right)=\rho^{\left(x\right)}\left(M\right),\quad x\in M,\label{eq:L6}$$ where the measures $\rho^{\left(x\right)}$ are the slice measures from the disintegration formula (\[eq:L2\]), or equivalently (\[eq:L3\]). We note that assumption (\[eq:L5\]) may be relaxed. For the results proved below, it will be enough to assume only that the function $c(x)$ defined by the RHS in (\[eq:L6\]) be finite for almost all $x$, so for a.a. $x$ with respect to the measure $\mu$. See (\[eq:L3\]).
We shall need the measure $\nu$, given by $$d\nu\left(x\right)=c\left(x\right)d\mu\left(x\right).\label{eq:L7}$$
Given a pair $\left(\mu,\rho\right)$, as above, set $$\left(Rf\right)\left(x\right)=\int_{M}f\left(y\right)\rho^{\left(x\right)}\left(dy\right),$$ defined on all measurable functions $f$ on $\left(M,\mathscr{B}\right)$.
The associated *Laplacian* (*Laplace operator*) is as follows: $$\begin{aligned}
\left(\Delta f\right)\left(x\right) & =\int_{M}\left(f\left(x\right)-f\left(y\right)\right)\rho^{\left(x\right)}\left(dy\right)\nonumber \\
& =c\left(x\right)f\left(x\right)-\left(Rf\right)\left(x\right).\label{eq:L9}\end{aligned}$$
\[def:T1\]Let $\left(\mu,\rho\right)$ be as above, and let $\mathscr{E}$ be the associated energy Hilbert space consisting of measurable functions $f$ on $\left(M,\mathscr{B}\right)$ such that $$\left\Vert f\right\Vert _{\mathscr{E}}^{2}=\frac{1}{2}\iint_{M\times M}\left|f\left(x\right)-f\left(y\right)\right|^{2}d\rho\left(x,y\right)<\infty;\label{eq:L10}$$ modulo functions $f$ s.t. $\text{RHS}_{\left(\ref{eq:L10}\right)}=0$.
\[lem:L2\]Let a fixed pair $\left(\mu,\rho\right)$ be as above; and let $\nu$ be the induced measure on $\left(M,\mathscr{B}\right)$ given by (\[eq:L7\]).
1. Then condition (\[eq:S2\]) is satisfied for $\mathscr{H}=\mathscr{E}$ (the energy Hilbert space), and with $$\left\langle f,g\right\rangle _{\mathscr{E}}=\frac{1}{2}\iint_{M\times M}\left(f\left(x\right)-f\left(y\right)\right)\left(g\left(x\right)-g\left(y\right)\right)d\rho\left(x,y\right),\label{eq:L11}$$ we have, for $A,B\in\mathscr{B}_{fin}$: $$\left\langle \chi_{A},\chi_{B}\right\rangle _{\mathscr{E}}=\nu\left(A\cap B\right)-\rho\left(A\times B\right),$$ and for $A=B$, $$\left\Vert \chi_{A}\right\Vert _{\mathscr{E}}^{2}=\nu\left(A\right)-\rho\left(A\times A\right).$$
2. If $\varphi\in\mathscr{D}_{fin}\left(\mu\right)$, and $f\in\mathscr{E}$, then $$\left\langle \varphi,f\right\rangle _{\mathscr{E}}=\int_{M}\varphi\left(x\right)\left(\Delta f\right)\left(x\right)d\mu\left(x\right).\label{eq:T14}$$
Most of the assertions follow by direct computation, using the results in Sections \[sec:SigA\]–\[sec:SA\] above; see also [@MR2159774; @MR3441734; @MR3701541; @MR3630401; @MR3796644], and the papers cited there.
Let the pair $\left(\mu,\rho\right)$ be as stated in , and let $\nu$ be the measure $d\nu\left(x\right)=c\left(x\right)d\mu\left(x\right)$ where $c\left(x\right)=\rho^{\left(x\right)}\left(M\right)$ as in (\[eq:L7\]). On $\mathscr{B}_{fin}\times\mathscr{B}_{fin}$, set $$\beta\left(A,B\right)=\nu\left(A\cap B\right)-\rho\left(A\times B\right).$$
Then $\beta$ is positive definite, and the corresponding RKHS $\mathscr{H}\left(\beta\right)$ naturally and isometrically, embeds as a closed subspace in the energy Hilbert space $\mathscr{E}$ from (\[eq:L11\]).
\[prop:L4\]Let $\left(\mu,\rho\right)$ be as above, we denote by $T$ the inclusion identification $$\begin{aligned}
\mathscr{D}_{fin}\left(\mu\right)\subset L^{2}\left(\mu\right) & \xrightarrow{\quad T\quad} & \mathscr{E}\\
\left(\varphi\in L^{2}\left(\mu\right)\right) & \xmapsto{\quad\phantom{T}\quad} & \left(\varphi\in\mathscr{E}\right).\nonumber \end{aligned}$$
1. Then $T$ is closable with respect to the respective inner products in $L^{2}\left(\mu\right)$ and $\mathscr{E}$; see (\[eq:L11\]).
Moreover, for $f\in dom\left(T^{*}\right)\left(\subseteq_{dense}\mathscr{E}\right)$ we have $$T^{*}f=\Delta f$$ where $\Delta$ is the Laplacian in (\[eq:L9\]).
2. For $f\in dom\left(T^{*}\right)$, the induced measure $\mu_{f}$ from , satisfies $$\mu_{f}\left(A\right)=\int_{A}\left(\Delta f\right)d\mu,\quad\forall A\in\mathscr{B}_{fin}.$$
The details are essentially contained in the above.
It is convenient to derive the closability of $T$ as a consequence of the following symmetry property:
For operators $T$ and $T^{*}$, $L^{2}\left(\mu\right)\xrightarrow{\;T\;}\mathscr{E}$, and $\mathscr{E}\xrightarrow{\;T^{*}\;}L^{2}\left(\mu\right)$, we consider the following dense subspaces, respectively: $$\begin{aligned}
{1}
& \mathscr{D}_{fin}\left(\mu\right)\subset L^{2}\left(\mu\right),\;\text{dense w.r.t. the \ensuremath{L^{2}\left(\mu\right)}-norm; and}\label{eq:T18}\\
& \big\{ f\in\mathscr{E}\mid\Delta f\in L^{2}\left(\mu\right)\big\}\subset\mathscr{E},\;\text{dense w.r.t. the \ensuremath{\mathscr{E}}-norm \ensuremath{\left(\ref{eq:L10}\right)}.}\label{eq:T19}\end{aligned}$$ Then a direct verification, using , (\[eq:L10\])–(\[eq:T14\]), yields: $$\langle\underset{{\scriptscriptstyle =\varphi}}{\underbrace{T\varphi}},f\rangle_{\mathscr{E}}=\left\langle \varphi,\Delta f\right\rangle _{L^{2}\left(\mu\right)}$$ for all $\varphi\in dom\left(T\right)$ (see (\[eq:T18\])), and all $f\in dom\left(T^{*}\right)$ (see (\[eq:T19\])).
Equivalently, $$\left\langle \varphi,f\right\rangle _{\mathscr{E}}=\int_{M}\varphi\:\left(\Delta f\right)d\mu,\label{eq:T21}$$ for functions $\varphi$ and $f$ in the respective domains. But we already established (\[eq:T21\]) in above; see (\[eq:T14\]). Now the conclusions in the Proposition follow.
**Discrete time reversible Markov processes**
Let $\left(M,\mathscr{B}\right)$ be a measure space. A Markov process with state space $M$ is a stochastic process $\left\{ X_{n}\right\} _{n\in\mathbb{N}_{0}}$ having the property that, for all $n,k\in\mathbb{N}_{0}$, $$Prob\left(X_{n+k}\in A\mid X_{1},\cdots,X_{n}\right)=Prob\left(X_{n+k}\mid X_{n}\right)$$ holds for all $A\in\mathscr{B}$. A Markov process is determined by its transition probabilities $$P_{n}\left(x,A\right)=Prob\left(X_{n}\in A\mid X_{0}=x\right),$$ indexed by $x\in M$, and $A\in\mathscr{B}$.
It is known and easy to see that, if $\left\{ X_{n}\right\} _{n\in\mathbb{N}_{0}}$ is a Markov process, then $$P_{n+k}\left(x,A\right)=\int_{M}P_{n}\left(x,dy\right)P_{k}\left(y,A\right);$$ and so, in particular, we have: $$P_{n}\left(x,A\right)=\int_{y_{1}}\int_{y_{2}}\cdots\int_{y_{n-1}}P\left(x,dy_{1}\right)P\left(y_{1},dy_{2}\right)\cdots P\left(y_{n-1},A\right),$$ for $x\in M$, $A\in\mathscr{B}$.
\[def:L5\]Let $\mu$ be a $\sigma$-finite measure on $\left(M,\mathscr{B}\right)$. We say that a Markov process is reversible iff there is a positive measurable function $c$ on $M$ such that, for all $A,B\in\mathscr{B}$, we have: $$\int_{A}c\left(x\right)P\left(x,B\right)d\mu\left(x\right)=\int_{B}c\left(y\right)P\left(y,A\right)d\mu\left(y\right).$$
\[prop:T6\]Let $\left(M,\mathscr{B},\mu\right)$ be as usual, and let $\left(P\left(x,\cdot\right)\right)$ be the generating transition system for a Markov process. Then this Markov process $\left\{ X_{n}\right\} _{n\in\mathbb{N}_{0}}$ is reversible if and only if there is a positive measurable function $c$ on $M$ such that the assignment $\rho$: $$\rho\left(A\times B\right)=\int_{A}c\left(x\right)P\left(x,B\right)d\mu\left(x\right),\quad A,B\in\mathscr{B},$$ extends to a sigma-additive positive measure on the product $\sigma$-algebra $\mathscr{B}_{2}$, i.e., the $\sigma$-algebra on $M\times M$ generated by product sets $\left\{ A\times B\mid A,B\in\mathscr{B}\right\} $.
The conclusion follows from the considerations above, and the remaining details are left to the reader.
\[cor:T7\]Let $\left(\mu,\rho\right)$ be a pair of measures, $\mu$ on $\left(M,\mathscr{B}\right)$, $\rho$ on $\left(M\times M,\mathscr{B}_{2}\right)$ satisfying the conditions in (\[eq:L3\])–(\[eq:L4\]), and let $c$ be the function from (\[eq:L6\]), then $$P\left(x,A\right):=\frac{1}{c\left(x\right)}\rho^{\left(x\right)}\left(A\right)$$ defines a reversible Markov process.
For measurable function $f$ on $\left(M,\mathscr{B}\right)$, i.e., $f:M\rightarrow\mathbb{R}$, set $$\left(Pf\right)\left(x\right)=\int_{M}f\left(y\right)P\left(x,dy\right).$$ Then the path space measure for the associated Markov-process $\left\{ X_{n}\right\} _{n\in\mathbb{N}_{0}}$ is determined by its conditional expectations evaluated on cylinder functions: $$\begin{aligned}
& & \mathbb{E}_{X_{0}=x}\left[f_{0}\left(X_{0}\right)f_{1}\left(X_{1}\right)f_{2}\left(X_{2}\right)\cdots f_{n}\left(X_{n}\right)\right]\\
& = & f_{0}\left(x\right)P\left(f_{1}P\left(f_{2}\left(\cdots P\left(f_{n-1}P\left(f_{n}\right)\right)\right)\right)\cdots\right)\left(x\right).\end{aligned}$$ The result is now immediate from .
Let the pair $\left(\mu,\rho\right)$ be as above, and as in . Let $\left\{ X_{n}\right\} _{n\in\mathbb{N}_{0}}$ be the corresponding reversible Markov process; see .
1. Then, for measurable functions $f$ on $\left(M,\mathscr{B}\right)$, we have the following variance formula: $$VAR_{X_{0}=x}\left(f\left(X_{1}\right)\right)=\int_{M}\left|f\left(y\right)-P\left(f\right)\left(x\right)\right|^{2}P\left(x,dy\right)$$
2. Set $d\nu=c\left(x\right)d\mu\left(x\right)$, and let $\mathscr{E}$ denote the energy Hilbert space from . Then a measurable function $f$ on $\left(M,\mathscr{B}\right)$ is in $\mathscr{E}$ iff $f-P\left(f\right)\in L^{2}\left(\nu\right)$, and $VAR_{x}\left(f\left(X_{1}\right)\right)\in L^{1}\left(\nu\right)$. In this case, $$\left\Vert f\right\Vert _{\mathscr{E}}^{2}=\frac{1}{2}\left[\int_{M}\left|f-Pf\right|^{2}d\nu+\int_{M}VAR_{x}\left(f\left(X_{1}\right)\right)d\nu\left(x\right)\right].$$
Immediate from the details in and .
In the last section we pointed out the connection between reversible Markov processes, and the Laplace operators, the energy Hilbert space, and our results in Sections \[sec:SigA\] and \[sec:SA\]. However we have postponed applications to reversible Markov processes to future papers. For earlier papers regarding Laplace operators and associated energy Hilbert space, see eg., [@MR3096586]. The literature on reversible Markov processes is vast; see e.g., [@MR2584746; @MR3638040; @MR3441734; @MR3530319].
The present work was started during the NSF CBMS Conference, Harmonic Analysis: Smooth and Non-Smooth, held at the Iowa State University, June 4–8, 2018, where the first named author gave 10 lectures. We thank the NSF for funding, the organizers, especially Prof Eric Weber; as well as the CBMS participants, especially Profs Daniel Alpay, and Sergii Bezuglyi, for many fruitful discussions on the present topic, and for many suggestions. We are extremely grateful to a referee who offered a number of excellent suggestions, helped us broaden the list of pointers to additional applications of our RKHS analysis; applications to yet more areas of probability theory, and stochastic analysis. And finally, spotted places where corrections were needed. We followed all suggestions. Indeed, his/her kind help and suggestions much improved our paper.
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abstract: 'Searches for gravitational wave signals which do not have a precise model describing the shape of their waveforms are often performed using power detectors based on a quadratic form of the data. A new, optimal method of generalizing these power detectors so that they operate coherently over a network of interferometers is presented. Such a mode of operation is useful in obtaining better detection efficiencies, and better estimates of the position of the source of the gravitational wave signal. Numerical simulations based on a realistic, computationally efficient hierarchical implementation of the method are used to characterize its efficiency, for detection and for position estimation. The method is shown to be more efficient at detecting signals than an incoherent approach based on coincidences between lists of events. It is also shown to be capable of locating the position of the source.'
author:
- Julien Sylvestre
title: 'Optimal generalization of power filters for gravitational wave bursts, from single to multiple detectors'
---
Introduction {#intro}
============
Six kilometer-scale laser interferometers designed to observe gravitational waves (GW) with unprecedented sensitivities should complete or approach the end of their commissioning in the year 2003. Three are operated in North America by the LIGO Laboratory [@LIGO], two in Europe by the Virgo [@Virgo] and the GEO600 [@GEO] projects, on one in Asia by the TAMA300 [@TAMA] project. A collaborative analysis of the data collected by these instruments provides the best prospects for detecting and analyzing GW events of astronomical origin.
The focus of this article will be on “bursts” of gravitational radiation, loosely defined as GW of duration of the order of a few seconds at most, and present in a frequency range overlapping at least partially with the bandwidth of the interferometers (10 Hz $\alt f \alt$ 1 kHz). Other types of signals that will not be discussed here include continuous GW from rotating neutron stars, and a stochastic background of GW of cosmological origin.
Arguments based on the astrophysics and on the dynamics of the sources of GW bursts show that the detection of these signals will be challenging, as the signals will be deeply buried in the instrumental noise [@Thorne300yrs]. Consequently, a significant research effort is currently on-going to develop and study efficient algorithms for the detection and the characterization of the elusive GW signals. An important fraction of the literature on the subject concerns signals with a precisely known form [@matched]. The knowledge of the signal allows the construction of a phase coherent filter (the Wiener or matched filter) which is known to be optimal for signal detection. Only the coalescence of compact binaries and possibly the ringdown of excited black holes should be detectable using matched filtering. For the particular case of compact binaries, it is known that a coherent analysis using data from all the interferometers of the international network will improve the detection prospects noticeably [@Pai; @Finn], although the computational cost of such an analysis might be prohibitive [@Pai2]. In addition, it was shown in [@Sylvestre] that the use of the Advanced LIGO detectors and of the Virgo interferometer cooperatively might allow the localization of the GW source with enough accuracy to permit its observation with electromagnetic instruments, thus complementing with information about the thermodynamics of the source the information on its dynamics provided by the GW.
The remainder of this article will be concerned only with GW signals that are not known with enough precision to allow matched filtering. The algorithms that have been proposed in the literature to detect these signals fall into two general categories: time-domain filters, and power detectors. Time-domain filters [@timeDomain] rely on the development of a small bank of linear filters which are expected to cover relatively well the space of possible GW signals. They offer the advantages of speed, simplicity, and possibly ease of interpretation, but might lack the robustness and efficacy of power detectors. Only the latter will be discussed here. The power detectors threshold on some non-linear measure of the data, often constructed from a time-frequency representation of the signal [@tfplane; @tfplane1; @epower; @vicere; @tfclusters; @wnoise]. They have been shown to be optimal for the detection of signals with especially poor waveform descriptions.
All power detectors were explicitly designed and implemented to process data from the different interferometers of the world-wide network independently. Under this mode of operation, it is expected that event lists are generated individually from the data stream provided by each interferometer, and are later compared to form coincidences based on temporal, frequency, or more general information. This [*incoherent*]{} approach should not yield the maximum efficacy, in part because GW bursts in individual interferometers have to be rather loud to register with the power detector and to give accurate estimates of their start time, duration, frequency band, amplitude, etc., all of which might be needed by the coincidence gate. The alternative is to combine all data streams first, and then run a burst detector on the [*synthetic*]{} data stream so produced. I implement this [*coherent*]{} approach as a generalization of the power detectors developed to date for single interferometers, by calculating the optimal way to combine any number of interferometer data streams into a single time-series, such that when this time-series is fed to a single interferometer power detector, a larger signal-to-noise ratio is obtained than for any other combination of the data streams. This brings to the already implemented and well-characterized power detectors the benefits of a network coherent analysis, which include improved sensitivity, and the ability to precisely locate the source position on the sky.
Summary of Results
------------------
The coherent power filter ([CPF]{}) algorithm presented in this paper involves the following steps:
1. A point on a grid in parameter space is chosen, where the parameters are the source angular position, and two numbers describing the plus and cross polarizations of the GW signal. One of these numbers is the ratio of the power in the cross polarization to the power in the plus polarization, and the other is the amount of overlap between the two polarizations, approximately measured by integrating the product of the two waveforms over time.
2. Given the source position and the network topology, the data streams from all interferometers are time-shifted to align the GW signals to a common origin in time.
3. Every data stream is multiplied by a scale factor, and all the data streams are added together to form a synthetic time-series. The choice of the scale factors depends on the network topology and on the four parameters chosen in Step 1.
4. The synthetic time-series is processed by a power detector, and the power measurement is recorded.
5. If all points of the grid in parameter space have been visited, the algorithm exits. If the maximum of the power measurements exceeds a certain threshold, a detection is announced, and the parameters that gave the largest power measurement are returned as an estimate of the source parameters.
6. Back to Step 1.
A more detailed discussion of this algorithm is presented in section \[Algorithm\].
The scale factors in Step 3 are chosen so that the value of the signal-to-noise ratio (SNR) is maximized. It is quite significant that only two parameters in addition to the source position are required to perform this maximization. Geometrically, this can be understood by realizing that the signals in all interferometers, after being properly time-shifted, are linear combinations of the two polarization waveforms, and therefore lie in a hyperplane spanned by these two polarization waveforms. Consequently, a knowledge of the ratio of the lengths of the two polarization waveforms and of their angle with respect to each other, together with a knowledge of the beam-patterns of all interferometers, is sufficient to determine the signals in all interferometers, up to an overall scale factor, and up to the orientation of the hyperplane. However, neither of these two pieces of information are needed to calculate how to linearly combine the signals to get maximum power in the synthetic time-series.
The validity of the geometrical picture and the actual conclusion that only four parameters are required to perform the SNR maximization depends critically on the right choice of the position dependent time-shifts. As shown in section \[Algorithm\], in the specific case where the cross-correlation function of the plus and the cross polarizations does not have an extremum at zero time lag, there are no formal guarantees that the [CPF]{} algorithm will converge to the right source parameters. Physically, this results from possible interactions (or cross-terms) between the plus and the cross polarizations which cannot be properly handled by the coherent algorithm. This does not affect the detection performances of [CPF]{}, but in some cases is significant for source position estimations. As discussed in section \[Algorithm\], a number of canonical sources do not satisfy this condition exactly, so that a careful study of the position systematic errors is needed. For the difficult case where the two polarizations are long monochromatic signals with a phase difference of a quarter of a cycle, it is shown in section \[systematics\] that a correction for this systematic error can be implemented such that for $\sim 25\%$ of the sky the systematic error is negligible, while that for about $50\%$ of the sky it is too large to allow any position estimation. This correction procedure only requires the additional knowledge of a quantity which is closely related to the characteristic frequency of the signal.
The performances of the [CPF]{} algorithm are explored empirically in section \[simulations\] through numerical simulations. All the experiments are limited to the three interferometer network (the HLV network) consisting of the LIGO interferometer near Hanford, Washington, the LIGO interferometer near Livingston, Louisiana, and the Virgo interferometer in Italy. The signal is short (1/16 s) and narrow band (25 Hz), and is assumed to originate from a position along the northern hemisphere normal to the HLV plane. All experiments are performed with the [TFCLUSTERS]{} [@tfclusters] algorithm as the single interferometer power detector. A realistic, computationally efficient hierarchical implementation of the [CPF]{} algorithm is shown to offer better detection performances than a incoherent approach which uses only coincidences between events generated by independent [TFCLUSTERS]{} operating on the three interferometers. It is also shown that the [CPF]{} algorithm can be used to estimate the position of the source of GW. When the GW signal has four times more power in its plus polarization than in its cross polarization, roughly one quarter of all trials lead to a position estimate that is within one degree from the true source position, for signals with reasonable amplitudes. The ability to pinpoint the source location is debilitated by the misalignment of Virgo with respect to the LIGO detectors, the reduction of the signal-to-noise ratio, and the reduction of the ratio of the power in the plus and in the cross polarizations.
Notation
========
Let bold characters denote time-series; whether these time-series are continuous in time or discretely sampled will be immaterial in the following discussion. It is assumed that a GW is observed with a network of $N$ independent detectors. Calibrated data corresponding to measurements of the GW strain in all detectors are denoted ${\bm{y}}_i$, $i=1,2, ... , N$. The noises ${\bm{n}}_i$ are assumed to be additive, so $${\bm{y}}_i = F^+_i(\theta, \phi, \psi) T[\Delta_i(\theta, \phi)] {\bm{s}}_+ + F^\times_i(\theta, \phi, \psi) T[\Delta_i(\theta, \phi)] {\bm{s}}_\times + {\bm{n}}_i, \label{eq:sigmodel}$$ where ${\bm{s}}_+$ and ${\bm{s}}_\times$ are the two polarizations of the GW signal, $F^+_i$, $F^\times_i$ are the beam-pattern functions[@Thorne300yrs] of the $i^{\rm th}$ detector, and $T(\Delta)$ denotes the time-shift operator; for time-series with continuous time, for instance, $T(\Delta){\bm{x}}(t) = {\bm{x}}(t - \Delta)$. The beam-pattern functions depend on the two angles describing the source position (the right ascension and the declination, denoted $\theta$ and $\phi$ respectively), and on the polarization angle $\psi$. The time-shift at the $i^{\rm th}$ detector, denoted $\Delta_i$, is the same for the two polarizations, and depends only on the source position on the sky. The frame in which ${\bm{s}}_+$ and ${\bm{s}}_\times$ are defined is irrelevant since the waveforms are not assumed to be known a priori; a rotation of that frame is equivalent to a change in ${\bm{s}}_+$, ${\bm{s}}_\times$ and $\psi$. The parameters $\theta, \phi, \psi, {\bm{s}}_+$, and ${\bm{s}}_\times$, and the derived quantities $F^+_i(\theta, \phi, \psi)$, $F^\times_i(\theta, \phi, \psi)$, and $\Delta_i(\theta, \phi)$, will be used below to describe the parameters of a real source which is assumed to be present in the data, and which we are trying to detect.
The scalar product between two time-series is denoted ${\bm{x}}\cdot{\bm{y}}$. For time-series with continuous time, it is defined as $${\bm{x}}\cdot{\bm{y}} = \int_{-\infty}^\infty\int_{-\infty}^\infty {\bm{x}}(t_x) Q(t_x, t_y) {\bm{y}}(t_y) dt_x dt_y, \label{eq:dotproduct}$$ and similarly for time-series with discrete time. The kernel $Q$ can be viewed as a filter applied to the time-series in order to detect more efficiently a particular signal or to modify the character of the noise, for instance. The square of the norm of a time-series, also called its power, is denoted $|{\bm{x}}|^2$, and is defined by $|{\bm{x}}|^2 = {\bm{x}} \cdot {\bm{x}}$.
The noise in each of the $N$ interferometers is only assumed to be wide-sense stationary [@WSS], i.e., it does not have to be Gaussian or white. The noises can always be made zero mean and independent by linear filtering [@KL], so $E[{\bm{y}}_i \cdot T(\Delta){\bm{y}}_j] = R_i(\Delta)$ if $i=j$ and is zero otherwise, for $R_i(\Delta)$ the autocorrelation of the noise, and $E[\cdot]$ denoting the expectation value of its argument. As usual, the Fourier transform of the noise autocorrelation function is the noise power spectral density.
Algorithm {#Algorithm}
=========
The [*synthetic response*]{} of the network is denoted ${\bm{Y}}$, and is a simple linear combination of the time-shifted individual detector responses: $${\bm{Y}} = \sum_{i=1}^N a_i T(\delta_i) {\bm{y}}_i, \label{eq:syntheticresponse}$$ for some set of real coefficients $a_i$ and time shifts $\delta_i$, $i=1, ..., N$, which are arranged in two vectors, ${\overrightarrow{a}}$ and ${\overrightarrow{\delta}}$, respectively. Note that the $\delta_i$ are the trial time delays used in the data analysis, and the algorithm defined below is used to estimate these delays so that they are close from the real delays $\Delta_i(\theta, \phi)$ corresponding to a source located at position $(\theta, \phi)$. The [*network power*]{} $\hat{P}$ is the estimate of the power in the GW signal, according to our norm definition, i.e., $\hat{P} = |{\bm{Y}}|^2$.
The motivation behind this particular design is that it is the simplest generalization of the numerous power detectors for single interferometers described in the literature, which in many cases have already been implemented and characterized. In practical terms, a software code can be designed to compute the synthetic response for a network of interferometers, and these data can be fed to a power detector, as if they were data from a single interferometer, in order to measure the network power. The kernel of the dot product used for the computation of the network power is then determined by the single interferometer power detector used to process the synthetic response. Different power detectors are efficient for detecting different types of signals, so this generality of the synthetic response approach is very economical in terms of code development. Some single interferometer power detectors provide a non-linear measure of the power; this is not a serious limitation given the algorithm structure defined above, for the power measurements are all very nearly linear for detectable signals.
Let $\delta_{ij} = \delta_i - \delta_j + \Delta_i - \Delta_j$ denotes the error on the estimated time-of-flight between detectors $i$ and $j$. The network power can be expanded as $$\hat{P} = \zeta + \eta,$$ where the signal term is given by $$\begin{aligned}
\zeta = \sum_{i,j=1}^N a_i a_j [ F^+_i F^+_j R_{++}(\delta_{ij}) + F^+_i F^\times_j R_{+\times}(\delta_{ij}) + \nonumber \\
F^\times_i F^+_j R_{\times+}(\delta_{ij}) + F^\times_i F^\times_j R_{\times\times}(\delta_{ij})], \label{eq:zeta}\end{aligned}$$ and where the noise term is given by $$\eta = \sum_{i,j=1}^N a_i a_j [ T(\delta_i + \Delta_i)(F^+_i {\bm{s}}_+ + F^\times_i {\bm{s}}_\times) \cdot T(\delta_j + \Delta_j) {\bm{n}}_j + T(\delta_i + \Delta_i) {\bm{n}}_i \cdot T(\delta_j + \Delta_j) {\bm{n}}_j ].$$ The signal correlation functions are given by $$R_{ij}(t_i - t_j) = T(t_i) {\bm{s}}_i \cdot T(t_j) {\bm{s}}_j,$$ for $i,j = +$ or $\times$.
The signal-to-noise ratio $\rho$ is defined as $$\rho^2 = \frac{\zeta}{E[\eta]},$$ where the expectation of the noise can be rewritten as $$E[\eta] = \sum_{i = 1}^N a_i^2 \sigma^2_i,$$ where the noise variance is $\sigma^2_i = R_i(0)$. For fixed noises and signals, the signal-to-noise ratio depends only on ${\overrightarrow{a}}$ and ${\overrightarrow{\delta}}$. It can therefore be maximized for a given choice of the source parameters ($\theta, \phi, \psi, {\bm{s}}_+, {\bm{s}}_\times$) by varying ${\overrightarrow{a}}$ and ${\overrightarrow{\delta}}$. Let ${\overrightarrow{a}}_m(\theta', \phi', \psi', {\bm{s}}_+', {\bm{s}}_\times')$ and ${\overrightarrow{\delta}}_m(\theta', \phi', \psi', {\bm{s}}_+', {\bm{s}}_\times')$ denote those values of ${\overrightarrow{a}}$ and ${\overrightarrow{\delta}}$ which maximize $\rho^2$ for some source parameters identified by primes to differentiate them from the true source parameters. The following algorithm is then defined:
1. Pick a set of trial source parameters $\theta', \phi', \psi', {\bm{s}}_+', {\bm{s}}_\times'$.
2. Compute ${\overrightarrow{a}}_m(\theta', \phi', \psi', {\bm{s}}_+', {\bm{s}}_\times')$ and ${\overrightarrow{\delta}}_m(\theta', \phi', \psi', {\bm{s}}_+', {\bm{s}}_\times')$.
3. Form the synthetic response ${\bm{Y}}$ from ${\overrightarrow{a}}_m$ and ${\overrightarrow{\delta}}_m$.
4. Estimate $\hat{P}({\bm{Y}})$ using a single detector algorithm.
5. Retain the source parameters $\theta', \phi', \psi', {\bm{s}}_+', {\bm{s}}_\times'$ if $[\hat{P}({\bm{Y}}) - E[\eta]]/E[\eta]$ is the largest to date.
6. Go back to Step 1.
The expectation of $[\hat{P}({\bm{Y}}) - E[\eta]]/E[\eta]$ is just $\rho^2$, so on average this algorithm will converge to the true parameters of the source.
The maximization problem for $\rho^2$ can be recast as the maximization of $\zeta$ subjected to the constraint that $E[\eta]$ is constant. The normal equations are $$\begin{aligned}
\sum_{j=1, j\neq i}^N a_j [ F^+_i F^+_j R_{++}'(\delta_{ij}) + F^+_i F^\times_j R_{+\times}'(\delta_{ij}) + \nonumber \\
F^\times_i F^+_j R_{\times+}'(\delta_{ij}) + F^\times_i F^\times_j R_{\times\times}'(\delta_{ij}) - \nonumber \\
F^+_i F^+_j R_{++}'(\delta_{ji}) - F^\times_i F^+_j R_{+\times}'(\delta_{ji}) - \nonumber \\
F^+_i F^\times_j R_{\times+}'(\delta_{ji}) - F^\times_i F^\times_j R_{\times\times}'(\delta_{ji}) ] = 0 \label{eq:normdelta}\\
\lambda a_i \sigma_i^2 + a_i [F^+_i F^+_i {\bm{s}}_+ \cdot {\bm{s}}_+ + (F^+_i F^\times_i + F^\times_i F^+_i) {\bm{s}}_+ \cdot {\bm{s}}_\times + F^\times_i F^\times_i {\bm{s}}_\times \cdot {\bm{s}}_\times] + \nonumber \\
\sum_{j=1, j\neq i}^N a_j [ F^+_i F^+_j R_{++}(\delta_{ij}) + F^+_i F^\times_j R_{+\times}(\delta_{ij}) + \nonumber \\
F^\times_i F^+_j R_{\times+}(\delta_{ij}) + F^\times_i F^\times_j R_{\times\times}(\delta_{ij}) ] = 0, \label{eq:normalpha} \end{aligned}$$ where $\lambda$ is the Lagrange parameter for the constraint, $R_{ij}'(x) = dR_{ij}(x)/dx$, and $i = 1, 2, 3$.
Eq. (\[eq:normdelta\]) can be simplified using the identity $R_{ij}(x) = R_{ji}(-x)$: $$\sum_{j=1, j\neq i}^N a_j [ F^+_i F^+_j R_{++}'(\delta_{ij}) + F^+_i F^\times_j R_{+\times}'(\delta_{ij}) + \nonumber \\
F^\times_i F^+_j R_{\times+}'(\delta_{ij}) + F^\times_i F^\times_j R_{\times\times}'(\delta_{ij})] = 0. \label{eq:normdeltaI}$$ If $R_{+\times}(x)$ has an extremum at $x=0$, i.e. if $R_{+\times}'(x)|_{x=0}=0$, then a solution to Eq. (\[eq:normdeltaI\]) is $\delta_i = -\Delta_i$ for $i=1,...,N$, since $R_{++}(x)$ and $R_{\times\times}(x)$ are maximal at $x=0$. Back into Eq. (\[eq:normalpha\]), this solution gives $$\begin{aligned}
\lambda a_i \sigma_i^2 + \sum_{j=1}^N a_j [F^+_i F^+_j {\bm{s}}_+ \cdot {\bm{s}}_+ + (F^+_i F^\times_j + F^\times_i F^+_j) {\bm{s}}_+ \cdot {\bm{s}}_\times + F^\times_i F^\times_j {\bm{s}}_\times \cdot {\bm{s}}_\times] = 0. \label{eq:normalphaII}\end{aligned}$$ The choice of ${\overrightarrow{a}}$ to maximize the signal-to-noise ratio then depends on the angles $\theta, \phi, \psi$ as before, but now only on the two numbers $|{\bm{s}}_+| / |{\bm{s}}_\times|$ and ${\bm{s}}_+ \cdot {\bm{s}}_\times / |{\bm{s}}_+||{\bm{s}}_\times|$ instead of the full waveforms for the two polarizations. These numbers are denoted $\Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$, respectively. Eq. (\[eq:normalphaII\]) is an eigenvalue problem (with the Lagrange parameter $\lambda$ playing the role of the eigenvalue, and the weight vector ${\overrightarrow{a}}$ that of the eigenvector), and is straightforward to solve numerically. The matrix of the eigenvalue problem is hermitian, so all eigenvalues are real; the eigenvector which maximizes Eq. (\[eq:zeta\]) with $\delta_{ij}=0$ is picked to form the synthetic response.
To recapitulate, if $R_{+\times}$ has an extremum at zero, a synthetic waveform can be constructed from the data of all interferometers such that this waveform is optimal for its processing by power detectors, and the maximization of the detection statistic for parameter estimation can be performed over only four parameters: $\theta, \phi, \Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$. I do not include $\psi$ in this list of parameters because it is completely degenerated with $\Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$. In particular, changing the basis where the two polarizations are defined by a rotation (i.e., a redefinition of the polarization angle $\psi$) does not change the value of $R_{+\times}'(x)|_{x=0}$. For the reminder of this paper, I will take $\psi \equiv 0$; this fixes the definition of ${\bm{s}}_+$ and ${\bm{s}}_\times$ with respect to the frame of reference of the detectors network. The [CPF]{} algorithm is then defined as followed:
1. Pick a set of trial source parameters $\theta'$, $\phi'$, $\Lambda'_{+/\times}$, and $\Lambda'_{+\cdot\times}$.
2. Compute ${\overrightarrow{a}}$ from Eq.(\[eq:normalphaII\]), and set ${\overrightarrow{\delta}}=0$.
3. Form the synthetic response ${\bm{Y}}$ using Eq.(\[eq:syntheticresponse\]).
4. Use a single interferometer power detector to calculate $\hat{P} = |{\bm{Y}}|^2$.
5. Retain the source parameters $\theta'$, $\phi'$, $\Lambda'_{+/\times}$, and $\Lambda'_{+\cdot\times}$ if $\hat{P}$ is the largest to date.
6. Go back to Step 1.
If the signal is linearly polarized, it can be written as ${\bm{s}}_+ = {\bm{s}} \cos 2\psi$ and ${\bm{s}}_\times = -{\bm{s}} \sin 2\psi$, for some polarization angle $\psi$ and some waveform ${\bm{s}}$. The eigenmatrix in Eq. (\[eq:normalphaII\]) is then of rank one, and consequently only has a single non-trivial eigenvalue. The corresponding eigenvector is given by $a_i = [F^+_i(\theta, \phi, 0) \cos 2\psi - F^\times_i(\theta, \phi, 0) \sin 2\psi] / \sigma_i^2 = F^+_i(\theta, \phi, \psi) / \sigma_i^2$ for $i=1,2,3$. Hence, when the GW signal is linearly polarized, the signals from the different interferometers are optimally combined by weighting them with the ratio of their noise variance to the value of the beam pattern functions, weighted appropriately by the polarization angle. The signals from the different interferometers of the network are therefore emphasized linearly in the observed power of the GW signal, and inversely in the power of the noise.
Another interesting subcase involves [*directed*]{} searches: in that case, the position $(\theta,\phi)$ of a potential GW source is known precisely ($\delta_{ij} = 0$), and the goal is to be maximally sensitive to gravitational radiation from that source. For $\hat{a}_i = a_i \sigma_i$, the maximization of the signal-to-noise ratio can be rewritten as the maximization of the quadratic form $$\rho^2 = \sum_{i,j = 1}^N \hat{a}_i \hat{a}_j M_{ij}$$ subjected to $$\sum_{i=1}^N \hat{a}_i^2 = 1,$$ where the matrix $M$ has elements $M_{ij}$ given by $$M_{ij} = \frac{|{\bm{s}}_+| |{\bm{s}}_\times|}{\sigma_i \sigma_j}\left[F^+_i F^+_j \Lambda_{+/\times} + (F^+_i F^\times_j + F^\times_i F^+_j) \Lambda_{+\cdot\times} + \frac{F^\times_i F^\times_j}{\Lambda_{+/\times}} \right].$$ It is well known from Rayleigh’s Principle [@noble] that the maximum value of $\rho^2$ is given by the largest eigenvalue of the matrix $M$. This maximum signal-to-noise ratio can be compared to the signal-to-noise ratio that can be obtained using the best interferometer in the network, which is typical of the signal-to-noise ratio of an incoherent search. It is given by $$\rho^2_{\rm best} = \max_{i=1...N} M_{ii}.$$ Considering the HLV network simplified so that all interferometers have the same noise level ($\sigma_i^2 = $constant), and fixing the beam-pattern functions by selecting a source along the northern hemisphere normal of the HLV plane, the ratio $\rho / \rho_{\rm best}$ varies between 1.03 and 1.57, depending on the value of $\Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$. Figure \[fig:snrratios\] shows that variation of the ratio $\rho / \rho_{\rm best}$ with $\Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$: when the signal has a large degree of linear polarization, the improvement is fairly large, with $\rho / \rho_{\rm best} \sim 1.4$. When both polarizations roughly contain the same power, the improvement can be large ($\agt 1.5$) or very small ($\sim 1$), depending on the structure of the signal (i.e., on $\Lambda_{+\cdot\times}$). If we add a detector at TAMA’s location to the HLV network, with noise similar to the noise of the other interferometers, the ratio $\rho / \rho_{\rm best}$ varies between 1.15 and 1.79 for a source at the same location as before. The four interferometer network works better than the three interferometer HLV network, although the improvement is somewhat limited by the unavoidable misalignment between instruments located on different continents.
[![The ratio $\rho / \rho_{\rm best}$ as a function of $\Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$, for a source along the northern hemisphere normal of the HLV plane.[]{data-label="fig:snrratios"}](fig1.eps "fig:"){height="3in"}]{}
If $R_{+\times}$ does not have an extremum at zero, the normal equations are not necessarily satisfied at $\delta_i = -\Delta_i$. They become non-linear and rather complex to solve, but, more importantly, a full knowledge of the waveforms for the two polarizations is then necessary to obtain a solution. Physically, this is a result of the fact that the two polarizations may interfere together constructively when shifted by a non-zero lag; this lag gets added to the estimated time delay between the interferometers in the network, and a systematic error in the source position estimate appears. It is important to realize that this systematic error does not significantly reduce the detection capabilities of [CPF]{}, but only its positioning ability. Essentially, the convergence to a secondary maximum provides an alternative way to cross the detection threshold when the global maximum of the signal power is plagued by noise, such that the measured power at its position is not a global maximum of the measured network power. It is nevertheless interesting to estimate the size of the position systematic error when the [CPF]{} algorithm is used on a signal for which $R_{+\times}'(x)|_{x=0} = 0$ does not hold, i.e. when Eq. (\[eq:normalphaII\]) is imposed as a solution. This is the subject of the following subsection.
Before this question is addressed, however, it is worth examining the physical meaning of the condition $R_{+\times}'(x)|_{x=0}=0$. As it was pointed out by [@ThorneTh Eqns. 2.30d, 4.3, and 5.18a], the GW strain for slow-motion, weak gravity sources radiating mostly due to variations of their mass moments (by contrast to current moments) can be written as $$h^{TT}(t) \propto \sum_{l=2}^\infty \sum_{m=-l}^l (\nabla \nabla Y^{lm})^{\rm STT} \frac{d^l}{dt^l} \int \rho(t-r/c) Y^{lm*} r^{2+l} dr d\Omega, \label{eq:hsph}$$ where STT means “symmetric transverse-traceless”, $Y^{lm}$ are the spherical harmonics, and $\rho$ is the mass density of the source. A number of possible GW sources consist in anisotropic mass distributions that are rapidly rotating about a well-defined axis, and consequently radiate principally with $l=2$, $m=2$ (e.g., binaries, bar or fragmentation instabilities, longest live mode of a perturbed Kerr black hole, as noted by [@KM2002]). The imaginary part of $Y^{22}$ is rotated by $\pi/4$ about the polar axis (the rotation axis) with respect to its imaginary part. This results in the mass distribution being sampled at any given time according to two spatial patterns rotated by $\pi/4$ with respect to each other. Since the GW are emitted at twice the rotation frequency of the source, this rotation angle produces a $\pi/2$ phase difference between the real and imaginary parts of the time dependent integral in Eq. (\[eq:hsph\]). As noted by [@KM2002], the dependence of the polarization on the inclination angle comes from the pure-spin tensor harmonics $(\nabla \nabla Y^{lm})^{\rm STT}$, which for $l=2$ can be found in [@Mathews]. For $m=2$, $$(\nabla \nabla Y^{22})^{\rm STT} \propto (1 + \cos^2 \iota) e_+ + 2i \cos \iota \; e_\times,$$ where $\iota$ is the inclination angle of the rotation axis with respect to the line of sight ($\iota = 0$ along the polar axis), and where $e_+$ and $e_\times$ are the unit linear-polarization tensors for the GW. Consequently, the plus and cross polarization waveforms can be written as $$\begin{aligned}
h_+ \propto (1 + \cos^2 \iota) \cos \Phi(t) \label{eq:pluspolarizations} \\
h_\times \propto 2 \cos \iota \sin \Phi(t), \label{eq:crosspolarizations}\end{aligned}$$ where $\Phi(t)$ is some phase function.
By definition, equations (\[eq:pluspolarizations\]) and (\[eq:crosspolarizations\]) give $$R_{+\times}(\tau) \propto \cos \iota (1 + \cos^2 \iota) \int_{-\infty}^\infty \int_{-\infty}^\infty dt_1 dt_2 Q(t_1, t_2) \cos \Phi(t_1) \sin \Phi(t_2 - \tau),$$ so that $$R_{+\times}'(x)\left|_{x=0}\right. \propto - \cos \iota (1 + \cos^2 \iota) \int_{-\infty}^\infty \int_{-\infty}^\infty dt_1 dt_2 Q(t_1, t_2) \Phi'(t_2) \cos \Phi(t_1) \cos \Phi(t_2).$$ In general, the phase $\Phi(t)$ will have at least a linear component \[$\Phi(t) \sim \omega t$\], so that $$R_{+\times}'(x)\left|_{x=0}\right. \propto - \cos \iota (1 + \cos^2 \iota) \omega.$$ As the system’s polar axis becomes aligned with the line-of-sight, the assumption that $R_{+\times}'(x)|_{x=0}=0$ becomes progressively worst, and significant systematic position errors might appear, and will have to be accounted for, as discussed in the next section. Some signals might also satisfy the condition $R_{+\times}'(x)|_{x=0}=0$. This is the case, for instance, if $\Phi(t)$ is an even function of $t$.
Sources which radiate predominantly in a $l=2$, $m=1$ mode will show a similar correlation between their plus and cross polarizations. In that case, the real and imaginary parts of $Y^{21}$ differ by a $\pi/2$ rotation about the polar axis, but have a $m=1$ symmetry, so that the waves are radiated at the spin frequency, and consequently the phase shift between the plus and cross polarizations is again $\pi/2$. It might be that the dominating population of sources to be observed will not be dominated by rotation about a principal axis; as a result, the correlation between the plus and cross polarizations might be quite arbitrary. In an axisymmetric core collapse, for instance, the $l=2$, $m=0$ mode dominates [@Muller]. The angular response is $(\nabla \nabla Y^{20})^{\rm STT} \propto \sin^2 \iota \; e_+$, so the waves are linearly polarized.
Systematic Position Errors {#systematics}
--------------------------
Eq. (\[eq:normdeltaI\]) can be used to check the error on $\theta$ and $\phi$ by solving it for ${\overrightarrow{\delta}}$, with ${\overrightarrow{a}}$ obtained from the [CPF]{} algorithm, i.e., from the solution of Eq. (\[eq:normalphaII\]). For every trial choice of $(\theta', \phi', \Lambda'_{+/\times}, \Lambda'_{+\cdot\times})$, the algorithm returns a set of weights ${\overrightarrow{a}}$ (by definition of [CPF]{}, ${\overrightarrow{\delta}} = 0$). However, the maximum of the signal-to-noise ratio occurs when the normal equations are satisfied; assuming that we let $\Lambda_{+/\times}$ and $\Lambda_{+\cdot\times}$ be varied freely in the maximization of $\hat{P}$, the values of $\theta$ and $\phi$ returned by this maximization will be those that solve Eq. (\[eq:normdeltaI\]).
Consider the following parameterization of the signal cross-correlation functions: $$\begin{aligned}
R'_{++}(t) = R'_{\times\times}(t) = -\omega_1^2 t + O(t^3) \\
R'_{+\times}(t) = \omega_0 - \omega_2^3 t^2 + O(t^4) \\
R'_{\times+}(t) = -\omega_0 + \omega_2^3 t^2 + O(t^4),\end{aligned}$$ for some parameters $\omega_0$, $\omega_1$, and $\omega_2$. If ${\bm{s}}_+$ and ${\bm{s}}_\times$ were long, nearly monochromatic signals of angular frequency $\omega$, with the same amplitude but a phase difference of $\pi/2$, for instance, the parameters could be chosen to be $\omega_0 = \omega_1 = \omega_2 \equiv \omega$. In order to get an analytical solution, I linearize Eq. (\[eq:normdeltaI\]) to obtain the first order equation $$\sum_{j=1, j\neq i}^N a_j [ F_{ij} \omega_1^2 \delta^{(1)}_{ij} - \omega_0 G_{ij} ] = 0,$$ where $\delta^{(1)}_{ij}$ represent first order errors between the true time delays and the delays returned by the algorithm, and where $F_{ij} = F^+_i F^\times_j + F^\times_i F^+_j$ and $G_{ij} = F^+_i F^\times_j - F^\times_i F^+_j$. The solutions to this linear system of equations are degenerate; for $N=3$, they are $$\frac{\omega_1^2}{\omega_0} \delta^{(1)}_{13} = -\frac{a_1 F_{12} G_{13} + a_2 F_{12} G_{23} + a_2 F_{23} G_{12} + a_3 F_{23} G_{13}}{a_1 F_{12} F_{13} + a_2 F_{12} F_{23} + a_3 F_{13} F_{23}} \label{eq:delta13}$$ and $$\frac{\omega_1^2}{\omega_0} \delta^{(1)}_{23} = -\frac{a_1 F_{12} G_{13} + a_2 F_{12} G_{23} - a_1 F_{13} G_{12} + a_3 F_{13} G_{23}}{a_1 F_{12} F_{13} + a_2 F_{12} F_{23} + a_3 F_{13} F_{23}} \label{eq:delta23}.$$
The times-of-flight between two pairs of detectors are sufficient to triangulate the position of the source on the sky and to obtain its position $(\theta, \phi)$, up to a reflection with respect to the plane containing the three detectors. The magnitude $l^{(1)}$ of the systematic error due to $R_{+\times}'(x)|_{x=0} \neq 0$ is given by the arclength of the portion of the great circle connecting the true source position and the position obtained by triangulation from $\delta^{(1)}_{13}$ and $\delta^{(1)}_{23}$ in Eqs. (\[eq:delta13\]) and (\[eq:delta23\]). To give a representative example, $l^{(1)}$ is computed for the long monochromatic signal described above, for the HLV network with $\sigma^2_i = 1$, i.e. under the simplifying assumption of instruments with identical noises at all sites. Only a very small fraction of the sky (about 0.7%) has a negligible systematic error \[$l^{(1)} (2\pi \times 40 \; {\rm Hz}/\omega) < 0.01$ rad\]. It is therefore plain that position estimations will be grossly off target if the assumption that $R_{+\times}'(x)|_{x=0} = 0$ is wrongly made.
When $R_{+\times}'(x)|_{x=0} \neq 0$, more information about the waveforms is required to estimate the source position. One possible approach is to use the [CPF]{} algorithm, which assumes $R_{+\times}'(x)|_{x=0} = 0$, and then to correct the source position estimate using Eqs. (\[eq:delta13\]) and (\[eq:delta23\]). This requires only a knowledge of the slope of $R_{+\times}$ with respect to the slope of $R_{++}$ at zero lag (i.e., $\omega_1^2/\omega_0$), which is closely related to the characteristic frequency of the waveforms. A map from estimated position (with systematic error) to actual position must be constructed for every choice of $\omega_1^2/\omega_0$. The remaining systematic error is given by the higher order terms not included in the correction. For $\delta^{(2)}_{ij}$ the second order errors on the time delays, the second order equation derived from the linearization of Eq. (\[eq:normdeltaI\]) is $$\sum_{j=1, j\neq i}^N a_j [ F_{ij} \omega_1^2 \delta^{(2)}_{ij} - \omega_2^3 G_{ij} ( \delta^{(1)2}_{ij} + 2 \delta^{(1)}_{ij} \delta^{(2)}_{ij} ) ] = 0.$$ This linear system of equations can be solved to obtain: $$\begin{aligned}
\frac{-\delta^{(2)}_{13} D}{\delta^{(1)}_{13}} = a_1 \delta^{(1)}_{23} [ 2 a_2 \delta^{(1)}_{12} F_{12} G_{12} + a_3 \delta^{(1)}_{13} G_{13} ( F_{12} - 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{12} G_{12} ) ] + a_3 \{ a_3 \delta^{(1)}_{12} \delta^{(1)}_{13} G_{13} [ F_{23} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{23} G_{23} ] + \nonumber \\
a_2 [ \delta^{(1)2}_{23} F_{12} G_{23} + \delta^{(1)}_{12} G_{12} ( \delta^{(1)}_{12} F_{23} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{23} (\delta^{(1)}_{12} + \delta^{(1)}_{23} ) G_{23} ) ] \} \end{aligned}$$ and $$\begin{aligned}
\frac{-\delta^{(2)}_{23} D}{\delta^{(1)}_{23}} = a_3 \delta^{(1)}_{23} G_{23} [ a_2 \delta^{(1)}_{13} ( F_{12} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{12} G_{12} ) + a_3 \delta^{(1)}_{12} ( F_{13} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{13} G_{13} ) ] + a_1 \{ 2 a_2 \delta^{(1)}_{12} \delta^{(1)}_{13} F_{12} G_{12} + \nonumber \\
a_3 [ \delta^{(1)2}_{13} F_{12} G_{13} + \delta^{(1)}_{12} G_{12} ( \delta^{(1)}_{12} F_{13} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{13} (\delta^{(1)}_{12} - \delta^{(1)}_{13}) G_{13} ) ] \},\end{aligned}$$ where $$\begin{aligned}
D = a_3 \frac{\omega_1^2}{\omega_2^3} \{ a_1 \delta^{(1)}_{23} ( F_{12} - 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{12} G_{12} ) ( F_{13} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{13} G_{13} ) + \nonumber \\
(F_{23} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{23} G_{23})[a_2 \delta^{(1)}_{13} (F_{12} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{12} G_{12}) + a_3 \delta^{(1)}_{12}(F_{13} + 2\frac{\omega_2^3}{\omega_1^2} \delta^{(1)}_{13} G_{13}) ] \}.\end{aligned}$$
For $l^{(2)}$ the arclength of the portion of the great circle connecting the true source position and the position obtained by triangulation using $\delta^{(2)}_{ij}$, i.e., for $l^{(2)}$ the systematic error after correction using the first order expansion, and for the same example as above (with $\omega = 80\pi$ rad/s), one finds that 23.2% of the sky has a negligible systematic error ($l^{(2)} < 0.01$ rad). Figure \[fig:syserr\] shows the fraction of the sky with a systematic error smaller than a certain value, for $l^{(1)}$ and $l^{(2)}$. It should be noted that for only about 50% of the sky is it meaningful to use an expansion in the small parameters $\delta^{(1)}_{ij}$ and $\delta^{(2)}_{ij}$. All in all, these numbers show that it is possible to use the [CPF]{} algorithm for any signals, at the cost of limiting the sky coverage to $\sim 25\%$, and of requiring one additional piece of information about the signal, the value of the ratio $\omega_1^2/\omega_0$, which describes the behavior of the signal cross-correlation functions near zero lag. Shorter signals, or signals with less overlap between their two polarizations, are likely to offer smaller systematic errors on position estimates, so that the correction suggested above becomes unnecessary.
[![The fraction of the sky with a systematic error smaller than the value of the position error plotted on the horizontal axis, for a long monochromatic signal of angular frequency $\omega = 80\pi$ rad/s, with two polarizations of the same amplitude, but at a phase offset of $\pi/2$. The rightmost curve is for $l^{(1)}$, the leftmost one for $l^{(2)}$.[]{data-label="fig:syserr"}](fig2.eps "fig:"){height="3in"}]{}
Numerical Simulations {#simulations}
=====================
All the numerical simulations were performed using the [CPF]{} implementation for the LIGO Data Analysis System (LDAS), which uses the [TFCLUSTERS]{} algorithm [@tfclusters] to measure the signal power in a given time-series. The HLV network was analyzed, except that the noises of all three interferometers were assumed identical, for simplicity. For every realization of the simulation procedure, a 10 s long segment of white noise of unit variance, sampled at 16384 Hz, was generated independently for all three interferometers. The use of white instead of colored noise is not an important loss of generality, since [TFCLUSTERS]{} is quite robust with respect to the presence of correlations in the background noise.
Simultaneously, two 1/8 s long segments of unit variance white noise were generated, and filtered by a 6$^{\rm th}$ order elliptical bandpass filter with 3 dB cutoff frequencies at 125 Hz and 150 Hz, so that the amount of power outside this band was negligible after filtering. These two segments were then truncated to their central 1/16 s long portion, and were used as the plus and cross polarizations of the gravitational wave signal incident on the network of detectors. By construction, the average value of $\Lambda_{+/\times}$ was 1, and that of $\Lambda_{+\cdot\times}$ was zero. In some simulations, the plus polarization waveform was multiplied by $\sqrt{2}$ and the cross polarization waveform was divided by $\sqrt{2}$, so that on average $\Lambda_{+/\times}$ was equal to 2.
All gravitational wave signals were injected from the position corresponding to the northern hemisphere normal of the HLV plane, so that they arrived in phase at the three interferometers. The two polarizations were combined at each interferometer using the beam-pattern functions $F^+_i$ and $F^\times_i$, and were added directly to the background white noise.
A scale factor $A$ was multiplying the GW signal injected in the simulated data for all interferometers. If the frequency band occupied by the signal, its duration, and its arrival time had all been known exactly, it would have been possible to filter the data optimally in that band. In that case, the signal-to-noise ratio $\rho_{\rm opt}$ for a certain scale factor $A$ would have been given by $$\rho_{\rm opt}(A) = A \sqrt{\sum_{i=1}^3 (F^{+ 2}_i + F^{\times 2}_i)}. \label{eq:snropt}$$ Of course, it cannot be assumed that this information is available, but I will nevertheless use Eq.(\[eq:snropt\]) as a measure of the strength of the injected signals, instead of using $A$ which is less intuitive. Numerically, $\rho_{\rm opt}(A) = 1.34 A$ for a source at the northern hemisphere normal of the HLV plane.
The simulations were performed using a realistic hierarchical implementation of [CPF]{}. The simulated data from the three interferometers were first processed separately by [TFCLUSTERS]{} to produce three lists of events. In terms of the notation developed in [@tfclusters], the settings of [TFCLUSTERS]{} were $\alpha = 0$, $\sigma=5$, and ${\bm{\delta}} = [0,0,0,0,0,0,2,3,4,4]$. Only the frequencies below 1024 Hz were considered, and the time resolution of the time-frequency decomposition was $T=1/8$ s. The number of events in this first stage was controlled by the black pixel probability threshold, $p_0$ (note that larger values of $p_0$ give [*larger*]{} false alarm rates).
The events produced by [TFCLUSTERS]{} are in the form of rectangles in the time-frequency plane, with information about the power present in each pixel of these rectangles. For a given cluster identified by [TFCLUSTERS]{}, the rectangle is defined as the smallest rectangle containing all the pixels of the cluster. A triple-coincidence condition was therefore applied as in [@THESE]: to be coincident, the time-frequency rectangles corresponding to the events from all three interferometers had to be overlapping. This coincidence condition selects events that are close in time and in frequency, and can be understood as a standard time and frequency coincidence gate with varying windows that are fixed by the events under consideration.
All the coincident events that were present in a given 10 s segment were then considered in turn. Their start time, duration, central frequency and bandwidth were estimated from the smallest rectangle in the time-frequency plane that could contain the union of the rectangles from the individual events. The [CPF]{} algorithm was then ran on the data, once for each coincident event. The implementation employed used [TFCLUSTERS]{} to process the synthetic data, and the measure of the power was the sum of the power in all the pixels identified by [TFCLUSTERS]{} that were inside the time-frequency rectangle identified from the triple coincidence. The parameters for [TFCLUSTERS]{} were the same as those mentioned above, except for the black pixel probability, which was set to a value $p_1$. The threshold defined by $p_1$ was chosen so that only loud enough signals were detected by [TFCLUSTERS]{}, and their estimated power was linearly related to their actual power.
The power measured by [TFCLUSTERS]{} was maximized over the source position (two angles) and over the parameter $\Lambda_{+\cdot\times}$. It was assumed that the value of $\Lambda_{+/\times}$ was known beforehand, in order to keep the size of the parameter space small enough for simulations. In a first time, the sky was covered by picking 100 points uniformly distributed in the range $[0,2\pi[$ for the right ascension, and 100 points in $[-1,1[$, uniform in the sine of the declination angle. In addition, 10 points were used to cover the range $[-1,1[$ uniformly for the parameter $\Lambda_{+\cdot\times}$. Consequently, ${\tt TFCLUSTERS}$ was ran $10^5$ times on every 10 s long simulation. Including the overhead from the LDAS system, this part of the search ran in $\sim 225$ s on 31, 2 GHz Pentium IV computers, with 512 Mb of RAM.
If none of the triple coincidence events registered above the $p_1$ threshold of [TFCLUSTERS]{} when analyzed by [CPF]{}, a non-detection was reported and the analysis was stopped. Otherwise, a detection was announced, and [CPF]{} produced a scan of the parameter space for every triple coincidence event above threshold. The triple coincidence event with the largest maximum power was then selected as a possible GW candidate, and was analyzed in more details. The point in parameter space where the power was maximum defined the parameters for a second run of [CPF]{}, used to obtain a refined position estimate not limited by the coarseness of the grid covering the parameter space. The value of $\Lambda_{+\cdot\times}$ was fixed to the value estimated in the first run, and a square search window of size 0.2 rad in right ascension and in declination, with 50 steps in both angles, was centered on the value of the position obtained in the first run. The position with the maximum power in this second run was taken as the final estimate of the source position.
Detection
---------
For the sole purpose of detecting the presence of a signal in the data, only the first run of [CPF]{} was required. Data were ran through [TFCLUSTERS]{} separately, the events were combined in the triple coincidence gate, and the data were fetch through [CPF]{} for the parameters defined by each triple coincidence event. If at least one of the triple coincidence event lead to a detection by [CPF]{}, the 10 s long time interval under scrutiny was assumed to contain a signal. By design, this hierarchical scheme required a fairly permissive threshold in the first stage where [TFCLUSTERS]{} was ran independently on every interferometer, so that a given signal was very likely to make it to the second stage where [CPF]{} was ran. The limit on this threshold was determined by the availability of computational resources, and by the confusion that resulted from the proliferation of events at low threshold. Most of the rejection of accidental coincidences occurred at the second stage, where [CPF]{} was operated with a reasonably strict threshold.
A numerical experiment was performed by running this simulation a large number of times, with and without signal injection. When injected, the signal had $\rho_{\rm opt} = 13.4$ and $\Lambda_{+/\times} = 1$. The thresholds were chosen to be $p_0 = 0.14$ and $p_1 = 0.012$. In the first stage (triple coincidence), the probability to detect a signal was $P_D = 0.92 \pm 0.01$, and the probability of a false alarm when no signal was present was $P_F = 0.62 \pm 0.02$. In the second stage ([CPF]{}), it was measured that $(P_D, P_F) = (0.86 \pm 0.01, 0.10 \pm 0.01)$. Overall, it was measured with both staged combined that $(P_D, P_F) = (0.79 \pm 0.02, 0.064 \pm 0.009)$. Table \[tab:detprob\] gives the detailed results from the simulations. The errors quoted here come from 68.3% confidence intervals (“$1\sigma$”) for a Bernoulli process, built using the prescription of [@FC]. The 6.4% probability of false alarm is larger than typical values for GW searches, as it would give a false alarm rate around 7 mHz. It was chosen, however, in order to provide enough detections for small errors on the measured probabilities. In a more realistic setting, $p_0$ would be similar to the value used here, while $p_1$ might be smaller by one or two orders of magnitude. It should be noted that the threshold settings were found by trial and error; there is an infinity of points along a curve in the $p_0,p_1$-plane that give a 6.4% probability of false alarm, and a choice different than the one above may give a larger probability of detection.
----------------------- ------------------- ------------------------------
no signal injected signal
injected with $\rho_{\rm opt} = 13.4$
total number 981 730
detected by 611 669
triple coincidence \[0.606,0.640\] \[0.904,0.927\]
undetected by triple 370 61
coincidence \[0.361,0.394\] \[0.0726,0.0962\]
detected by [CPF]{} 62 578
\[0.0549,0.0731\] \[0.775,0.808\]
undetected by [CPF]{} 549 91
\[0.542,0.576\] \[0.112,0.139\]
----------------------- ------------------- ------------------------------
: Details of the simulations to measure the efficacy of the hierarchical implementation of the [CPF]{}. Numbers in brackets show the 68.3% (“$1\sigma$”) confidence interval for the fraction of the number of trials to the total number of trials.[]{data-label="tab:detprob"}
Nevertheless, it seems that the choice I made for the $p_0$ and $p_1$ thresholds is sufficient to show the superiority of the coherent approach over the incoherent one for detection. I repeated the experiment above, but using only the first stage to detect events. A detection was announced when at least one triple coincidence was observed between the outputs of the three [TFCLUSTERS]{} runs on the interferometers’ data. In that case, a threshold of $p_0 = 0.11$ gave $(P_D, P_F) = (0.70 \pm 0.02, 0.07 \pm 0.01)$, in an experiment with 742 trials for the measurement of $P_D$ (519 detections), and 742 trials for $P_F$ (48 detections). At a similar false alarm probability, the probability of detection is significantly smaller in the incoherent case than in the coherent case. For the particular signal and false alarm probability under consideration, the signal-to-noise ratio would have to be increased to $\rho_{\rm opt} = 16.8$ in order for the incoherent approach to be as efficient as the coherent one. For a homogeneous distribution of sources in space, this corresponds to a factor of $\sim 2$ improvement in detection rate, assuming no significant degradation of the [CPF]{} algorithm performances with respect to the incoherent algorithm as the position of the source is varied away from the northern hemisphere normal to the HLV plane.
The performances of the [CPF]{} search were mostly limited by the quality of the estimation of the time-frequency rectangle containing the burst, in the first stage of the analysis (the triple coincidence). With the signal injection for $\rho_{\rm opt} = 13.4$, the first stage gave an estimated rectangle that overlapped with the one containing the signal in 96% of the 578 cases where the signal was discovered, but only in 3% of the 61 missed detections was this the case. Without signal injection, only 1% of the 611 triple coincidences had time-frequency rectangles overlapping with the signal rectangle. These numbers show that if an oracle were available to provide the rectangle containing the signal without error every time the search is ran, a probability of detection $\agt 97\%$ would be possible for a probability of false alarm $\alt 1\%$. Stated differently, [CPF]{} is extremely efficient at detecting a burst when it receives the right parameters describing that burst; the triple coincidence incoherent search provides many candidates; when one such candidate corresponds to the signal, [CPF]{} picks it out of the others very efficiently. A better approximation to this oracle than the one used here might be to tile the time-frequency plane with a variety of rectangles, and to run [CPF]{} on each rectangle.
Suppose that it is known that the signal has a bandwidth of 25 Hz. One can cover the time-frequency plane with non-overlapping rectangles of duration 1/16 s, and bandwidth of 25 Hz, so that the 10 s long, 1024 Hz bandwidth data segment in my simulations is covered by $\sim 6500$ tiles. The false alarm probability for each run of [CPF]{} must be reduced to $\sim 10^{-5}$ so that the global search has $P_F \sim 7\%$. The efficiency of the search would then approximately be given by the probability of detection of [CPF]{} with an oracle, for a threshold giving $P_F = 10^{-5}$. I have not measured this probability of detection, but it is plausible that it is larger than the 79% efficiency measured for the hierarchical implementation used in the simulations. However, running this search in real time would be prohibitively expensive, as it would require $\sim 10^4$ teraflops of computational power. In the present hierarchical implementation, the first stage took $\sim 30$ gigaflops to run in real time, and the second stage, $\sim 800 (P_F/0.62)$ gigaflops, where $P_F$ is the false alarm probability in the first stage. These numbers should be taken as upper bounds on the required computational power, because the codes were not optimized to minimize overhead, to make an optimal usage of the parallel resources, or to do an optimal scan of the parameter space. Optimized codes should be able to run at least $\sim 5$ times faster.
Position estimation
-------------------
In order to get a precise idea of the magnitude of the position estimation errors, the analysis scheme described above was simplified by removing the first incoherent step involving the three different instances of [TFCLUSTERS]{} running at each site. Instead, the [CPF]{} algorithm was instructed to compute the power according to the output of [TFCLUSTERS]{} in a rectangle of duration 1/8 s located at the right position in the time series, with a lower frequency of 50 Hz and an upper frequency of 150 Hz. The black pixel probability was set to $p_1 = 5\times 10^{-3}$, so that the number of clusters unrelated to the signal and produced only by the noise was small.
Figure \[fig:posU10\] presents a scatter plot of the position estimates obtained for 240 realizations of the simulation, when $\Lambda_{+/\times} = 1$. The estimates tend to cluster along the curve corresponding to the locus of positions having equal delays at the Hanford and Livingston interferometers. This is a direct consequence of the good alignment between these two interferometers, and the relatively poor alignment of the Virgo detector with them. Estimates tend to fall on that curve, but also to cluster at different places along it, where the signals at Hanford and at Livingston are delayed by an integer number of the characteristic periods of the signal with respect to the signal at Virgo.
[![Scatter plot of the estimated position of a source injected along the northern hemisphere normal of the HLV plane, from 240 realizations of the simulation with $\Lambda_{+/\times} = 1$ and $\rho_{\rm opt} = 13.4$. Horizontal axis is right ascension, vertical axis is the sine of the declination. The curves represent loci of equal time delay for the three independent interferometer pairs. The point where they intersect in the upper left corner of the figure is where the signal was injected.[]{data-label="fig:posU10"}](fig3.eps "fig:"){height="3in"}]{}
Figure \[fig:posU210\] shows a similar plot as Fig.\[fig:posU10\], except that $\Lambda_{+/\times} = 2$. This corresponds to a GW signal which has more structure in its polarizations than the one for the case $\Lambda_{+/\times} = 1$, i.e. which is closer to a linearly polarized signal. Since linearly polarized signals are the easiest ones to analyze with a network of interferometers, it is expected that the position estimates will be better. This is indeed was is observed in Fig.\[fig:posU210\]; the estimates still hug the Hanford-Livingston equal-delay curve, but now present less scatter around the points where the signals are in phase with the Virgo signal along that curve.
[![Same as Figure \[fig:posU10\], but with $\Lambda_{+/\times} = 2$.[]{data-label="fig:posU210"}](fig4.eps "fig:"){height="3in"}]{}
The position error can be quantified as in Fig.\[fig:syserr\]: it is taken to be the length of the shortest portion of the great circle joining the estimated position and the true source position or its mirror image with respect to the HLV plane. The error can be defined with respect to the mirror image because there is a natural ambivalence in the estimation of the position when only three detectors are used. Figure \[fig:posErr\] shows the cumulative distribution of the position error from the simulations, for $\Lambda_{+/\times} = 1$ and for $\Lambda_{+/\times} = 2$, and for two different values of the signal-to-noise ratio of the injected signals. As expected, signals with larger values of the signal-to-noise ratio or of $\Lambda_{+/\times}$ lead to smaller position errors. The curves in Fig.\[fig:posErr\] present a number of “steps”, which are produced by the clustering along the Hanford-Livingston equal-delay curve at positions in phase with Virgo.
[![The fraction of all simulations that gave a position error smaller than the value plotted on the horizontal axis. The two continuous lines correspond to $\Lambda_{+/\times} = 1$, and the two dotted lines to $\Lambda_{+/\times} = 2$. In both cases, the rightmost curve is for $\rho_{\rm opt} = 13.4$, and the leftmost one is for $\rho_{\rm opt} = 35.6$. Each curve is built from 240 realizations of the simulation. The error on the curves is estimated to be $\sim 5\%$ of the fraction of trials.[]{data-label="fig:posErr"}](fig5.eps "fig:"){height="3in"}]{}
Roughly 50% of the trials lead to unusable position estimates (errors $\agt$ 10 degrees) when $\Lambda_{+/\times} = 2$, while this number reaches $\sim 80\%$ when $\Lambda_{+/\times} = 1$. However, with $\Lambda_{+/\times} = 2$, approximately 25% of the trials have errors smaller than one degree. Moreover, at least in the regime of signal-to-noise ratios under consideration here, the scaling of the position error with the signal-to-noise ratio is rather weak.
Conclusion
==========
A method was presented for the optimal generalization of the power detectors developed for single interferometers so that they can process coherently data from a network of interferometers. The coherent method, as compared to an incoherent approach where event lists independently generated at all interferometers are searched for coincidences, offers the advantage of better detection efficiency, and the possibility to accurately estimate the position of the source. A few systematic effects affect the performances of the [CPF]{} algorithm for position estimation, including: cross terms between the plus and cross polarization waveforms of the GW signal, lack of differences between the characteristics of the two polarization waveforms, and misalignment of the interferometers of the network.
The three effects are related, and reflect the obvious fact that GW signals incident on a network of misaligned detectors will only show entangled versions of their two polarization waveforms, different in each interferometer. If one of the two polarizations is significantly stronger than the other (i.e., the GW signal has a stronger degree of linear polarization), the problem is drastically simplified. Similarly, aligned interferometers are much less sensitive to this problem. If the two polarization waveforms are fairly coherent with each other and of similar amplitudes, the cross terms between the two polarizations in different interferometers may show significant maxima when the time shifts imposed on the different data streams do not correspond to the differences in time-of-arrival of the GW signals at each interferometer. These maxima cannot be distinguished from the maximum resulting from the product of the waveforms of the same polarization in different interferometers, and systematic position errors may result. It should be noted, however, that the same effect leads to similar position errors when the differences in arrival time of the GW signal in different interferometers are estimated by maximizing the cross-correlation between pairs of interferometers, and these time differences are used to triangulate the source position.
The study of the [CPF]{} algorithm presented here was based on a three interferometer network. However, the structure of the algorithm allows for any number of interferometers. More complex networks will most likely reduce the effects of the systematic errors, by increasing the number of linear combinations of the two polarization waveforms that are being sampled, or, equivalently, by increasing the area of the sky where at least three interferometers are measuring similar combinations of the two polarizations. Numerical simulations on the network formed by the LIGO Hanford, LIGO Livingston, and Virgo interferometers (the HLV network) with a short random signal showed that the [CPF]{} algorithm could be used to accurately measure the position of the source a significant fraction ($\sim 25\%$) of the time, for reasonably strong sources located at the normal of the HLV plane. In about $\sim 90\%$ of the trials, the [CPF]{} algorithm correctly placed the source position at a point where the signals at the Hanford detector and at the Livingston detector were in phase. However, due to its misalignment with respect to the LIGO detectors, the information provided by the Virgo detector was often insufficient to pinpoint correctly the position of the source. The average position error was a rather weak function of the strength of the signal, but a stronger function of the amount of difference in the structure of the two polarization waveforms.
It was also shown that the [CPF]{} algorithm offers better detection efficiencies than its incoherent equivalent, both for directed and for all-sky blind searches, and independently of the systematic errors affecting the position estimations. In the former case, improvements in the detection signal-to-noise ratio of 40% or better are expected, excepted for a few values of the source parameters. In the latter case, a 25% improvement in signal-to-noise ratio was measured for typical source parameters. This improvement for the all-sky blind search comes at the cost of increasing the computational power required to perform the data analysis in real time by a factor of $\sim 5-30$. This may be significant, especially since lengthy simulations probably have to be completed in order to estimate the false alarm and detection statistics of a real search. However, the signals studied in the numerical simulations, which could be viewed as very rough approximations to the signals that could be emitted in the collapse of the core of a star in supernova explosions, would have been detected by the [CPF]{} algorithm at a rate $\sim 2$ times larger than the rate for its incoherent equivalent, assuming a homogeneous spatial distribution of the sources, and that the performances of the [CPF]{} algorithm for a source injected at the normal of the HLV plane are characteristic of its performances at other injection positions.
Finally, an important advantage of the design of the [CPF]{} algorithm is that it inherits the robustness, efficacy, and computational efficiency of the single interferometer power detectors. In other words, the [CPF]{} algorithm should not be more sensitive to non-Gaussian or non-stationary noises than incoherent analyzes, it should be searching more efficiently for GW signals sharing similar morphologies, and, while more computationally intensive, its speed should scales the same way with improvements in the implementation of the single interferometer power detectors, or with computing hardware ameliorations.
This work was supported by the National Science Foundation under cooperative agreement PHY-9210038 and the award PHY-9801158. This document has been assigned LIGO Laboratory document number LIGO-P030007-01-D.
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The Karhunen-Loève expansion can always be used to expand an arbitrary stochastic process, stationary or not, into a series with orthogonal coefficients. Cf. A. Papoulis, [*Probability, Random Variables, and Stochastic Processes*]{} (McGraw-Hill ed., USA, 1991), 3rd edition, p. 413.
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abstract: 'A simple chemical enrichment code is described where the two basic mechanisms driving the evolution of the ages and metallicities of the stellar populations are the star formation efficiency and the fraction of gas ejected from the galaxy. Using the observed Tully-Fisher relation in different passbands as a constraint, it is found that a steep correlation between the maximum disk rotational velocity ($v_{\rm ROT}$) and star formation efficiency (${C_{\rm eff}}$) must exist — ${C_{\rm eff}}\propto v_{\rm ROT}^4$ — either for a linear or a quadratic Schmidt law. Outflows do not play a major role. This result is in contrast with what we have found for early-type systems, where the Faber-Jackson constraint in different bands allows a significant range of outflows and requires a large star formation efficiency regardless of galaxy mass. The extremely low efficiencies found at low masses translate into a large spread in the distribution of stellar ages in these systems, as well as a large gas mass fraction independently of the star formation law. The model predictions are consistent with the star formation rates in low-mass local galaxies. However, our predictions for gas mass are in apparent conflict with the estimates of atomic hydrogen content observed through the flux of the 21cm line of . The presence of large masses of cold molecular hydrogen — especially in systems with low mass and metallicity — is predicted, up to ratios M(H$_2$)/M(HI)$\sim 4$, in agreement with a recent tentative detection of warm H$_2$. The redshift evolution of disk galaxies is explored, showing that a significant change in the [*slope*]{} of the Tully-Fisher relation ($L\propto v_{\rm ROT}^\gamma$) is expected because of the different age distributions of the stellar components in high and low-mass disk galaxies. The slope measured in the rest frame $B, K$-bands is found to change from $\gamma_B \sim 3, \gamma_K \sim 4$ at $z=0$ up to $\sim 4.5, 5$ at $z\sim 1$, with a slight dependence on formation redshift.'
author:
- 'Ignacio Ferreras & Joseph Silk'
title: |
A backwards approach to the formation of disk galaxies\
I. Stellar and gas content
---
Introduction
============
The process of galaxy formation can be divided into two distinct and equally important histories. On the one hand, the dynamical history of galaxies is crucial in understanding the Hubble sequence (or other morphological classifications) that we see now as well as its evolution with redshift. On the other hand, the star formation history allows us to trace the baryonic matter, its conversion from gas to stars and the feedback from stellar evolution in the processing of subsequent generations of stars. Disk galaxies are assumed to be the building blocks from which the rest of the galactic “zoo” we see around us has been assembled. The standard picture of disk galaxy formation involves the infall and cooling of primordial gas onto the centers of dark matter halos and its later conversion into stars (White & Rees 1978). A multitude of models have been published, describing this process in varying degrees of detail, following either a dynamical approach, where gravity plays the dominant role in the properties of galaxies, or a spectrophotometric approach, where the key issue is the highly complex and so far unsolved problem of star formation. The former approach usually over-simplifies the luminous properties of galaxies by assuming a fixed value of the mass-to-light ratio, tying the properties of the observed parts of galaxies to the behavior of the much larger halos in which the luminous matter is embedded. On the other hand, models based on the evolution of the stellar populations lack the dynamical information needed in order to quantify the merger tree. Semi-analytic models (e.g. Kauffmann et al. 1999; Baugh et al. 1998) attempt to fill in the gap between these two approaches at the expense of a large and complicated set of parameters.
We follow a spectrophotometric approach, defining a model that traces both the evolution of gas and metal content in a simple star formation scenario reduced to infall of primordial gas that fuels star formation via a simple power law dependence on gas content. The model allows for outflows of gas driven out of the galaxy, whose cause could either be stellar (feedback from supernova-driven winds) or dynamical (e.g. merging or harassment). In this paper we study the evolution of the gas and stellar content in disk galaxies, ignoring size evolution. In a future paper (Ferreras & Silk, in preparation) we use the same model to explore the evolution of sizes and surface brightnesses along with the associated effects on the selection function of current surveys at moderate and high redshift.
The first part of this paper (§§2,3,4) describes the model as well as the technique used in order to constrain the volume of parameter space sampled. In §5 we discuss the star formation efficiencies predicted by the model. §6 explores the issue of gas content in disk galaxies and speculates on the idea that a large fraction of gas could be found in the form of cold H$_2$. §7 explains the redshift evolution predicted by our model for the star formation rates as well as for the Tully-Fisher relation. Finally §8 is a brief summary of our results with a discussion of the most interesting issues, regarding disk galaxy formation, raised by our work.
Modelling star formation in disk galaxies
=========================================
We follow a chemical enrichment code similar to the one described in more detail in Ferreras & Silk (2000a,b), where the formation and evolution of the stellar populations is parametrized by a range of star formation efficiencies (${C_{\rm eff}}$) and outflows (${B_{\rm out}}$). The model only traces the cold gas component
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that gives rise to star formation, with instantaneous mixing of the ejected gas from stars and the cold gas component being assumed. The gas mass density ($\mu_g$) and metallicity ($Z_g$) obey a simple set of chemical enrichment equations following the formalism of Tinsley (1980): $$\frac{d\mu_g}{dt}=-\psi(t)+(1-{B_{\rm out}})E(t)+f(t),$$ where $\psi(t)$ is the star formation rate, $f(t)$ the gas infall rate, $E(t)$ the gas mass ejected from stars of all masses, and ${B_{\rm out}}$ is the fraction of gas ejected from the galaxy. Since this paper treats galaxies as unresolved objects, and we do not assume a threshold in density for the star formation rate, we can always interchange mass densities and total masses. The evolution of the metallicity follows a similar equation: $$\frac{d(Z_g\mu_g)}{dt}=-Z_g(t)\psi(t) + Z_ff(t) + (1-{B_{\rm out}})E_Z(t),$$ where $Z_f$ is the metallicity of the infalling gas (assumed to be primordial in this paper), and $E_Z(t)$ is the mass in metals ejected from stars. Infall of primordial gas fuels star formation via a Schmidt-type law (Schmidt 1959): $$\psi(t) = {C_{\rm eff}}\mu_g^n(t);$$ where ${C_{\rm eff}}$ is the star formation efficiency parameter. We assume either a linear ($n=1$) or a quadratic law ($n=2$) as extreme examples. The power law index that best fits observational data ($n=1.4$) lies between these two values (Kennicutt 1998a). In the instantaneous recycling approximation — for which the stellar lifetimes are assumed to be either zero or infinity, depending on whether the stellar mass is greater or less than some mass threshold ($M_0$) — the star formation rate can be solved analytically (e.g. Tinsley 1980). Taking $n=1$: $$\psi(t)={C_{\rm eff}}\int_0^t ds f(t-s)\exp(-s/\tau_g),$$ $$\tau_g^{-1} = {C_{\rm eff}}\big[ 1-(1-{B_{\rm out}})R\big];$$ where $R$ is the returned mass fraction from stars with masses $M>M_0$. Hence, the inverse of ${C_{\rm eff}}$ is roughly the time (in Gyr) needed to process gas into stars. The comparison between the efficiency defined here (${C_{\rm eff}}$) and observed efficiencies — such as the parameter $A$ defined in Kennicutt (1998b) with respect to surface densities rather than total masses, i.e. $A=\Sigma_\psi /\Sigma^n_{\rm gas}$ — is straightforward without a star formation threshold, but gets rather complicated with such a threshold, which causes ${C_{\rm eff}}$ to lie systematically below the observed efficiencies by a factor given by the ratio between the surface of the disk with density above threshold with respect to the total area. If we assume a universal star formation law, one would expect a systematic offset between observed star formation efficiencies and model one-zone efficiencies such as our ${C_{\rm eff}}$.
The infall rate that fuels star formation is assumed to have a Gaussian profile peaked at an epoch described by a formation redshift parameter ($z_F$), with two different timescales for ages earlier or later than the time corresponding to that formation redshift, namely: $$f(t)\propto \left\{
\begin{array}{lr}
\exp [-(t-t(z_F))^2/2\tau_1^2] & t < t(z_F)\\
\exp [-(t-t(z_F))^2/2\tau_2^2] & t > t(z_F).\\
\end{array}
\right.$$
For $t<t(z_F)$ we assume a short infall timescale — $\tau_1=0.5$ Gyr — which results in a prompt enrichment of the interstellar medium and thereby circumvents the G dwarf problem. Any model of chemical enrichment must avoid the over-production of low metallicity stars by assuming either a non-monotonically decreasing star formation rate (as in this paper) or initially non-primordial gas (pre-enrichment). For $t>t(z_F)$ the infall timescale ($\tau_2$) is a rough label for disk type: early-type disks will be associated with short timescales ($\tau_2\sim 1-2$ Gyr), whereas late-type disks, with more extended star formation histories, correspond to larger values of $\tau_2$. The Initial Mass Function (IMF) used is a hybrid between a Salpeter (1955) and a Scalo (1986) IMF. The high-mass end follows a Salpeter power law, with upper-mass cutoff at $60 M_\odot$, whereas the low-mass end — truncated at $0.1 M_\odot$ — follows the Scalo IMF. Stellar lifetimes follow a broken power law fit to the data from Tinsley (1980) and Schaller et al. (1992). We refer the reader to Ferreras & Silk (2000a,b) for more details about the chemical enrichment code.
The outflow parameter (${B_{\rm out}}$) is used to quantify the amount of gas ejected out of stars and that leaves the galaxy and does not contribute to the processing of the next generation of stars. This mechanism plays an important role in the mass-metallicity correlation as originally suggested by Larson (1974): massive galaxies are expected to retain most of the gas, thereby increasing the average metallicity, whereas the shallow gravitational wells of low-mass galaxies cannot prevent enriched gas from being ejected from the galaxy, resulting in lower average metallicities. It is a well-known fact that the color range found in early-type systems is mostly driven by metallicity
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(e.g. Kodama et al. 1998), and can be explained by a correlation between the outflow parameter (${B_{\rm out}}$) and the total mass of the system (Ferreras & Silk 2000b).
The chemical enrichment process in galaxies is hence described in this model with a set of five parameters: $(\tau_1,\tau_2,z_F,{C_{\rm eff}},{B_{\rm out}})$. Only the early-infall timescale — $\tau_1$ — is fixed to 0.5 Gyr throughout, whereas the other four parameters are allowed to vary over a large range of values. Every set of parameters chosen describes a star formation history ($\psi(t)$, $Z_g(t)$) which can be used to convolve simple stellar populations in order to find the composite spectral energy distribution of a galaxy whose formation process corresponds to that choice of parameters. We use the latest population synthesis models from Bruzual & Charlot (in preparation), which span the range of metallicities $\frac{1}{50} \leq Z/Z_\odot \leq \frac{5}{2}$ and include all phases of stellar evolution, from the zero-age main sequence to supernova explosions for progenitors more massive than $8 M_\odot$, or to the end of the white dwarf cooling sequence for less massive progenitors. The uncertainties present in population synthesis models (Charlot, Worthey & Bressan 1996) complicate the determination of absolute values for parameters such as stellar ages or star formation efficiencies. This paper aims at showing [*relative*]{} values of efficiencies or outflow parameters between galaxies of different masses.
The predicted spectral energy distribution obtained for each point scanned in this large volume of parameter space is matched against observed data so that a map of the parameter range compatible with observations can be drawn. Here we use the Tully-Fisher relation of disk galaxies in different passbands as a constraint, shown in Figure 1. The data involve multiband $\{B, R, I, K^\prime\}$ photometry of a sample of disk galaxies in the Ursa Major cluster (Verheijen 1997). Section 4 describes the comparison in more detail.
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Outflows versus Star Formation Efficiencies
===========================================
In order to visualize the role of every parameter used in the chemical enrichment model presented here, we show in Figures 2 and 3 the metallicity and age distribution, respectively, for a range of outflows (${B_{\rm out}}$); star formation efficiencies (${C_{\rm eff}}$) and infall parameters ($\tau_2$, $z_F$). The fiducial set chosen is (${B_{\rm out}}$,${C_{\rm eff}}$,$\tau_2$,$z_F$)= (0,1,1 Gyr,2). In each panel, one of the parameters is allowed to vary, while keeping the others fixed. Notice the effect of the early-infall timescale ($\tau_1$) on the metallicity distribution, which avoids the over-production of low metallicity stars. If we had set the timescale $\tau_1$ to zero, the metallicity histogram would have been a monotonically decreasing function, peaked at zero metallicity. Closed-box models with constant star formation efficiencies and a standard initial mass function always yield monotonically decreasing star formation rates that over-produce low metallicity stars which contradict observations, both in our local stellar census (Rocha-Pinto & Maciel 1996) as well as in the integrated photometry of bulge-dominated galaxies (Worthey, Dorman & Jones 1996). Models with pre-enriched gas fuelling star formation generate similar distributions although shifted towards higher abundances. The photometric predictions of pre-enrichment models can be made compatible with the observations. However, the metallicity distribution — a monotonically decreasing function with $Z$ with the maximum at the pre-enriched metallicity — is in disagreement with the observed metallicities of stars in the Milky Way (e.g. Rocha-Pinto & Maciel 1996). However, see Rich (1990) and Ibata & Gilmore (1995), who find an agreement between the prediction of a closed-box model and the distribution in the galactic bulge.
The top left panels of Figures 2 and 3 show that a range in outflows — with all the other parameters fixed — does not generate significantly different age distributions. The net effect of a change in the amount of gas ejected from the galaxy is a shift
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in the average metallicity. Hence, the assumption that an outflow range is the only mechanism driving the luminosity sequence of galaxies results in a pure metallicity sequence. On the other hand, a range of star formation efficiencies generates distributions with different ages and metallicities. The efficiency parameter sets the clock rate at which stars are being formed from the infalling gas (equations 4 and 5). Hence, systems with lower efficiencies generate a larger age spread, and lower average metallicities. Finally, the parameters controlling infall ($\tau_2$ and $z_F$) simply shift the age distributions without a significant change in the metallicities, as expected, since ${B_{\rm out}}$ and ${C_{\rm eff}}$ are the only parameters driving the chemical enrichment process: it is possible to re-define the time variable as $s\equiv (t-t(z_F))/\tau_2$ and solve similar chemical enrichment equations to (1) and (2) with respect to variable $s$, without any significant dependence on the infall parameters ($\tau_2$ and $z_F$).
A comparison of the age and metallicity histograms with the observed mass-metallicity correlation found in disk galaxies (Zaritsky, Kennicutt & Huchra 1994) suggests that either outflows and/or star formation efficiencies must be strongly correlated with galaxy mass. However, one can not determine [*a priori*]{} whether the observed color range found in disks is purely driven by outflows (i.e. a metallicity sequence) or by efficiency (i.e. a mixed age $+$ metallicity sequence).
The Tully-Fisher relation as a constraint
=========================================
The work described here follows a “backwards” approach in that we apply correlations between observables in local galaxies as a constraint and use the chemical enrichment code in order
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to evolve the system backwards in time. For disk galaxies the obvious constraint to be used is the Tully-Fisher relation (Tully & Fisher 1977): this shows a tight correlation between the disk rotation velocity — which is a measure of the ratio between the mass and the size of the galaxy — and the absolute luminosity. The scatter of this correlation is rather small ($\sim 0.3$ mag) which allows it to be used as a distance indicator with uncertainties as low as 10%, notwithstanding unknown systematic effects (e.g. Jacoby et al. 1992). The small scatter hides a fundamental connection between the dynamical evolution of disk galaxies and their star formation histories. The analogous correlation for bulges or early-type systems — the Faber-Jackson relation (Faber & Jackson 1976) between luminosity and velocity dispersion of the spheroid — has a larger scatter, a clear sign of a more complex dynamical evolution.
The conventional scenario for disk formation via infall of gas onto the centers of dark matter halos gives a qualitative idea of the trend followed by the Tully-Fisher relation: massive disks (which have faster rotation velocities) will create more stars, thereby having higher luminosities than low-mass systems. However, a more detailed analysis of the Tully-Fisher relation — including disk sizes which of course affect the measured rotation velocity — shows that the interpretation of the slope and zero point of the Tully-Fisher relation is more complicated. Many dynamical models assume constant mass-to-light ratios ($M/L$), an incorrect assumption unless only stellar populations of the same ages dominate the light from all disks. The Tully-Fisher relation observed through near-infrared passbands minimizes the effect of the chemical enrichment history, and so $K$-band is the preferred filter in order to use the correlation as a distance indicator. Rather than exploring a suitable way of finding a “distilled” Tully-Fisher relation that is independent of the formation process, we will use the correlation in different filters in order to constrain the star formation history of disk galaxies. There are many samples of disk galaxies for which rotation curves have been observed using different techniques such as optical emission lines (Mathewson & Ford 1996) or the 21 cm line of atomic hydrogen (Giovanelli et al. 1997). We use the sample of disk galaxies in the Ursa Major cluster (Tully et al. 1996; Verheijen 1997) imaged in several filters: $B$, $R$, $I$ and $K^\prime$. The rotation curves were determined with the 21 cm line of H I.
For a choice of parameters describing the star formation history: (${B_{\rm out}}$, ${C_{\rm eff}}$, $\tau_2$, $z_F$), we solve equations (1) and (2) describing the evolution of the gas mass and metallicity and use this to convolve simple stellar populations from the latest models of Bruzual & Charlot (in preparation). We explore four different formation redshifts ($z_F=\{1, 2, 3, 5\}$) and four different infall timescales ($\tau_2 = \{1, 2, 4, 8\}$ Gyr), corresponding to a range of galaxy types: from early-type disks such as S0, Sa — for $\tau_2\sim 1-2$ Gyr — to late-type systems such as Sd for longer timescales. For every pair of values describing infall ($\{\tau_2, z_F\}$) we scan a grid of $128\times 128$ star formation efficiencies and outflow fractions over the following range:
$-3 \leq \log{C_{\rm eff}}\leq +1$\
$0 \leq {B_{\rm out}}\leq 0.7$.
For every choice of parameters, the $I-K^\prime$ color of the un-normalized composite spectral energy distribution is used in order to find the absolute luminosity using the color-magnitude relation of the sample of Verheijen (1997): $$M_K^\prime = -\frac{(I-K^\prime) + 1.417}{0.1304},$$ which is shown as a solid line in the inset of Figure 1. The color-magnitude relation found by de Grijs & Peletier (1999) from a subsample of the Surface Photometry Catalogue of ESO-Uppsala galaxies is shown as a dashed line, in remarkable agreement with the fit from the sample of UMa galaxies. In fact, the universality of the color-magnitude relation in disk galaxies is explored by de Grijs & Peletier (1999) as a plausible distance indicator, with accuracies deemed to be better than 25 %.
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Band ($X$) $\alpha_X$ $\beta_X$ $\sigma_X$
------------ ------------ ----------- ------------
$B$ $-6.387$ $-3.842$ $0.370$
$R$ $-7.356$ $-2.284$ $0.348$
$I$ $-8.020$ $-1.061$ $0.394$
$K^\prime$ $-9.368$ $+0.849$ $0.323$
Once the spectral energy distribution is normalized, we can compare the predicted Tully-Fisher relations in different wavebands with the observations. Table 1 shows the result of fitting the photometry of disk galaxies in the Ursa Major cluster (Verheijen 1997) to a Tully-Fisher relation in different filters ($X=\{B, R, I, K^\prime\}$): $$M_X^{\rm TF} = \alpha_X\log v_{\rm ROT} + \beta_X.$$ We perform a two-stage linear fit with a first estimate of the slope and zero point by applying absolute deviations. This first estimate is used to remove points which lie more than one standard deviation off the linear fit. A least squares fit is subsequently applied to the remaining points. This technique prevents outliers from contributing significantly to the fit. These fits allow us to estimate how well different star formation histories — described by the parameters — can account for the observed Tully-Fisher relations in different passbands. We define a $\chi^2$ by: $$\chi^2\equiv\Sigma_{X=\{B, R, I, K^\prime\}}
\frac{(M_X-M_X^{\rm TF})^2}{\sigma_X^2}.$$
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Figures 4 and 5 show the range of star formation efficiencies and outflow fractions, respectively, allowed by the observed photometry. This range is shown in four panels corresponding to four different infall timescales (i.e. four different galaxy types). Each panel gives the result for three different formation redshifts (thick lines) using a linear Schmidt law ($n=1$ in equation 3). The result for a quadratic law ($n=2$) is shown for $z_F=2$ as a thin solid line. The 90% confidence level of the $\chi^2$ fit for $z_F=2$ and $n=1$ is shown as a shaded area in Figure 4 and as error bars in Figure 5.
How efficient is Star Formation ?
=================================
Figures 4 and 5 show that a very large range of star formation efficiencies is needed in order to explain the observations, whereas outflows seem to be unimportant in disk galaxies. Over a velocity range of $\Delta\log v_{\rm ROT}=0.6$ dex we find a range of star formation efficiencies corresponding to gas-to-star processing timescales that span over 2 decades between very small disks ($v_{\rm ROT}\sim 100$ km s$^{-1}$) and massive ones ($v_{\rm ROT}\sim 400$ km s$^{-1}$) for various infall parameters. This result is very robust given the assumptions and the model presented here, as well as the 90% confidence level of our $\chi^2$ fit. While outflows could play a role, these cannot be very important. The error bars shown in Figure 5 show that outflows in low-mass systems cannot be larger than 20%. Furthermore, a constant value ${B_{\rm out}}\sim 0$, regardless of $v_{\rm ROT}$, is also consistent with the data. This is also in agreement with the negligible effect of supernova-driven winds in disks as found in hydrodynamic simulations (MacLow & Ferrara 1999). The correlation between ${C_{\rm eff}}$ and $v_{\rm ROT}$ does not change significantly with respect to morphological type (roughly described by our infall timescale $\tau_2$). This is in agreement with Young et al. (1996) who find no variation in the global ratio $L(H\alpha)/M(H_2)$ for spiral galaxies with types Sa-Sc.
The range of efficiencies is consistent with the observations of Kennicutt (1998b), who defines an efficiency with respect to surface densities: $A\equiv \Sigma_\psi / \Sigma^n_{\rm gas}$. The data fit a power law index $n=1.4\pm 0.15$, with the efficiency ranging over a factor of $100$. However, the observed efficiencies are systematically one order of magnitude higher than our ${C_{\rm eff}}$, with the lowest value around $A\sim 0.1$ (thereby the gas consumption timescales $A^{-1}{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13C$}}}10$ Gyr are significantly shorter than ${C_{\rm eff}}^{-1}$). There are various reasons for such a discrepancy between $A$ and ${C_{\rm eff}}$. We believe the most important one to be a systematic underestimate of the amount of molecular hydrogen, which we discuss in more detail in the next sections. Furthermore, the star formation rates obtained from $H\alpha$ measurements are corrected for extinction by a factor of 2.8 using a comparison between $H\alpha$ fluxes and radio fluxes due to free-free emission at 3 and 6 cm (Kennicutt 1983). However, the radio emission is assumed to be purely thermal. A non-thermal contribution in the radio fluxes would imply a systematic overestimate of the star formation rates, with a corresponding increase of $A$ with respect to ${C_{\rm eff}}$. Additionally, a universal star formation threshold in the surface gas density would result in lower integrated efficiencies — such as the parameter ${C_{\rm eff}}$ — compared to surface-based measurements such as $A$, with the correction factor roughly amounting to the ratio of the disk area over which the surface mass density is above threshold with respect to the total area of the galaxy. A linear Schmidt law and a surface density threshold for the star formation ($\Sigma_{th}$) would imply: $$\psi(t)={C_{\rm eff}}M_g(t)=A\int d^2r\Sigma_g(r,t)\Theta\Big(
\Sigma_g(r,t)-\Sigma_{th}\Big) < AM_g(t),$$ where $\Theta(x)$ is the step function. For an exponential density profile with scale-length $h$, i.e. $\Sigma(r)=\Sigma_0\exp (-r/h)$ we find: $${C_{\rm eff}}= A\Big( 1-e^{-x_0}\Big) < A,$$ such that $\Sigma(r=x_0h)=\Sigma_{th}$. Hence, a threshold will cause a systematic offset towards lower values for the integrated star formation efficiencies. A delay in the galaxy formation process can also yield higher star formation efficiencies. The dot-dash line in the bottom-left panel of Figure 4 gives the star formation efficiencies predicted for $z_F=0.3$, which are about one order of magnitude higher than those for models with $z_F\geq 1$. However, this implies the bulk of the stellar populations should have ages ${\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13C$}}}2.5$ Gyr, which leads to a drastic change of the luminosity function of disk galaxies at moderate redshift that is not seen (e.g. Lilly et al. 1995).
The large range of star formation efficiencies shown in Figure 4 is in remarkable contrast with the region of parameter space allowed by the color-magnitude constraint in early-type systems. The photometry of spheroids is compatible with either a large range of star formation efficiencies [*and/or*]{} a wide range of gas ejected in outflows (Ferreras & Silk 2000b). In fact, the lack of evolution of the slope of the color-magnitude relation of early-type galaxies with redshift suggests the mass sequence of early-type galaxies is a pure metallicity sequence, where the stellar populations have roughly similar ages regardless of galaxy mass so that the color range observed is explained by a mass-metallicity correlation. Such a correlation, without a similar mass-age connection, can only be explained by assuming outflows are very important in low-mass systems. In order to test our model, we applied the same chemical enrichment code and fitting technique to the Faber-Jackson relation in early-type systems. We used the $U$, $V$, $J$ and $K$ photometry of elliptical galaxies in the Coma cluster from Bower, Lucey & Ellis (1992). The best fits are shown in Figure 6 for the star formation efficiency ([*top*]{}) and outflow fraction ([*bottom*]{}), taking a linear Schmidt law and three different formation redshifts: $z_F=\{1, 2, 5\}$. We chose a short infall timescale: $\tau_2=0.25$ Gyr, although larger values were also used, leading to similar results for $\tau_2=0.5$ and $1$ Gyr. Longer infall timescales were incompatible with the photometric data. The shaded area represents the 90% confidence level of the $\chi^2$ fit in the $z_F=2$ case. In contrast to disk galaxies (Figures 4 and 5), early-type systems require very high star formation efficiencies, and are compatible with a significant correlation between galaxy mass and fraction of gas ejected in outflows. This is in agreement with the enhanced star formation efficiencies observed in mergers (Young et al. 1996), which are believed to be associated with the process of spheroid formation. A plausible scenario to reconcile these two very different trends is that disk galaxies are dynamically “ordered” systems, where gas ejection can only take place through feedback from supernovae, whereas early-type galaxies (or disk bulges) have a more complicated dynamical history, which can allow for a significant amount of gas (and metals) being ejected from the galaxy, as expected in dwarf galaxy mergers (Gnedin 1998). Furthermore, this process should be correlated with the mass of the galaxy (i.e. more gas being ejected in events involving less massive mergers).
Of gas and stars
================
The large range of star formation efficiencies required by our model — shown in Figure 4 — implies that there is a large difference in gas content with respect to galaxy luminosity. As a rough approximation, the inverse of the star formation efficiency gives the timescale for the processing of gas into stars (in Gyr). We can see in the figure that in the interval $2.1<\log v_{\rm ROT}<2.6$ — which corresponds to a range of $4.5$ magnitudes in absolute $K$-band luminosity — ${C_{\rm eff}}$ spans two orders of magnitude. This means it takes $100$ times longer to
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process stars in the faintest disk galaxies. Furthermore, the gas mass fraction: $f_g\equiv M_g/(M_g+M_s)$, where $M_g$ and $M_s$ are the mass in gas and stars, respectively, will also vary over a very large range. As a simple example, we can write the gas mass fraction in a closed-box model (using $f(t)=\delta(t)$ in equation 4) as: $$f_g(t) = \exp(-t/\tau_g),$$ with the star-processing timescale ($\tau_g$) defined in (5). Massive systems have efficiencies close to ${C_{\rm eff}}\sim 1$, i.e. $\tau_g\sim 1$ Gyr, which means the gas mass fraction should be very low in disk galaxies at $z=0$. However, low-mass galaxies are predicted to have $\tau_g\sim 100$ Gyr, which boosts the gas mass fraction in local galaxies up to 90%. Hence, galaxies with low rotation velocities should have large amounts of gas. This issue is hard to circumvent by redefining the model: the index of the Schmidt power law used in the correlation between gas mass and star formation rates does not change the range of efficiencies. We also considered several other models with a time-varying star formation efficiency but the final results were very similar to those presented here.
Figure 7 checks the consistency of our model with estimates of star formation in nearby spiral galaxies using the flux of the H$\alpha$ emission line as the observable. The data points are from Kennicutt (1983, squares; 1994, triangles). The top panel plots the prediction of the absolute star formation rate (in $M_\odot {\rm yr}^{-1}$) as a function of $B$-band absolute luminosity for three different infall timescales: $\tau_2=\{1, 4,$ and $8\}$ Gyr, using a linear Schmidt law. The thin solid line is the prediction for $\tau_2=4$ Gyr for a
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quadratic law. A typical error bar from the data is also shown. The star formation rates obtained from these measurements can have uncertainties mainly due to variable extinction up to $\pm 50$%. The model predictions show that even though there is not much gas in larger mass systems, their high star formation efficiencies account for an increase in the SFR with luminosity.
Table 2 gives the fits to the star formation rate as a function of stellar mass ($\psi\propto M_s^b$) predicted by the model, for various infall parameters. The power law index lies in the range $b\sim 0.7-0.8$ and as expected, it varies more with formation redshift for shorter infall timescales. The middle panel of Figure 7 shows the specific star formation rate weighted by stellar mass ($\psi/M_s$), which decreases with increasing mass since $b<1$. In order to transform the absolute luminosities from Kennicutt (1983; 1994) to stellar masses, we assume a large range in mass-to-light ratios $-0.15<\log(M/L_B)<+0.15$ in order to account for different age distributions or initial mass functions of the stellar component. The error bars shown give the range of specific star formation rates depending on the mass-to-light ratio chosen. Finally, the bottom panel plots the specific star formation rate weighted by total (baryonic) matter, i.e. stars and gas. In this case only Kennicutt (1994) gives gas masses. Both atomic and molecular hydrogen are considered and helium is included by correcting the final mass by a factor of $1.4$. The data is consistent with the model predictions, except for the two faintest data points, which appear to be in disagreement with such a high gas content. One explanation for this is that the molecular hydrogen measured has been underestimated. Molecular hydrogen is usually measured indirectly, by assuming a constant H$_2$/CO ratio. Systems with lower metallicities than the galaxies used to calibrate this ratio should have larger H$_2$ masses than the estimates according to this technique (Combes 1999). The bottom panel of Figure 8 also shows the results from Young & Knezek (1989) regarding the dependence of $M(H_2)/M(HI)$ with morphology, where the detection of H$_2$ is based on CO observations. The observed trend, which finds more molecular hydrogen in earlier types (which have higher metallicities), is a reflection of the strong dependence of the H$_2$/CO ratio with metal abundance.
In order to explore this point further, we used the data presented by McGaugh et al (2000) who motivated the existence of a baryon-dominated Tully-Fisher relation, i.e. a tight correlation between total stellar$+$gas mass and rotational velocity. They inferred the stellar masses from photometric data in optical ($I$) and near infrared ($H$ and $K^\prime$) bands, by assuming a fixed mass-to-light ratio (however, see Bell & de Jong 2001). Near infrared observations are usually good stellar mass observables since $M/L_K$ does not change much over a large range of ages. For consistency, we decided to use the Tully-Fisher relation as a constraint so that the stellar masses were corrected to abide by the linear fits shown in Table 1. The gas-to-stellar ratios are shown in Figure 8 along with model predictions for three formation redshifts for a linear Schmidt law and also for a prediction using a quadratic law. The gas fractions predicted for the quadratic Schmidt law are higher since such a model requires even lower star formation efficiencies at the faint end (Figure 4).
The figure also shows the fit-by-eye from Bell & de Jong (2000) who estimated the gas fraction to be: $f_g=M_g/(M_g+M_s)=0.8 + 0.14(M_K-20)$ in a sample of low-inclination spiral galaxies. The fractions were computed from $K$-band luminosities (in order to find stellar masses) and fluxes corrected to include helium. They also included the contribution from molecular hydrogen by using the ratio of molecular to atomic gas masses as a function of morphological type (Young & Knezek 1989). Our models are thereby consistent with the data for a linear Schmidt law. However, the large scatter observed could be associated with the presence of undetected molecular hydrogen. The bottom panel shows the ratio between molecular and atomic hydrogen predicted if we take the models at face value. It is interesting to notice that a larger amount of undetected H$_2$ is expected for fainter galaxies, in agreement with the fact that fainter galaxies have low efficiencies which translate into lower average metallicities (cf fig. 2) so that the H$_2$/CO ratio used to trace H$_2$ could be in significant disagreement with the calibrations (Combes 2000).
The ratios shown in the figure are still consistent with the total amount of matter to be expected from the rotation curves of spirals. Combes (1999) showed that using a ratio of $M(H_2)/M(HI)\sim 6.2$, the gas and stellar content in NGC 1560 can account for the rotation curve out to $8$ kpc. Furthermore, observations by Valentijn & Van der Werf (1999) of the lowest pure rotational lines of H$_2$ in NGC 891 out to $11$ kpc, using the Short-Wavelength Spectrometer on board the Infrared Space Observatory, found large amounts of H$_2$ at temperatures $T\sim 80-90$ K, outweighing by a factor $5-15$. More data is clearly needed to draw a definitive conclusion about the presence of a high mass in molecular hydrogen. Nevertheless, the agreement between these observations and our model predic-
+3.3truein
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tions — arising from the need for very low star formation efficiencies in low-mass galaxies — motivates further exploration of this problem, especially given the impact that such an issue has on galaxy formation as well as on our understanding of the properties of dark matter.
We want to emphasize that a possible weakness in the model presented here is the [*absolute*]{} estimate of star formation efficiencies. One should take the values of ${C_{\rm eff}}$ as lower bounds for a given galaxy if the results are to be compared with local efficiencies defined as $A\sim\Sigma_\psi/\Sigma^n_g$, where $\Sigma_g$ and $\Sigma_\psi$ are the surface densities of gas and star formation rate, respectively. However, even this method of estimating efficiencies is questionable after the discovery of star formation in the extreme outer regions of disk galaxies, which cannot be described by a single-component Schmidt law (Ferguson et al. 1998). The stellar metallicities predicted by our model for rotation velocities corresponding to a Milky Way sized galaxy (which implies $0.1{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13C$}}}{C_{\rm eff}}{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13C$}}}0.2$) are in the right ballpark: $-0.1<[{\rm Fe/H}]<+0.05$, whereas low efficiencies ${C_{\rm eff}}\sim 0.01-0.02$ predicted by the model for a galaxy like the Large Magellanic Cloud ($M_B=-18$) give metallicities ($-1.1<[{\rm Fe/H}]<-0.8$) that are consistent with the observations of the young clusters — with ages $t{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13C$}}}3$ Gyr — with the lowest metallicities (Olszewski, Suntzeff & Mateo 1996). However, notice the large range of metallicities found in old LMC clusters ($-2.1<[{\rm Fe/H}]<-1.4$ for $t\sim 14$ Gyr) which hints at a more complicated star formation history.
+0.1truein
$\tau_2$/Gyr $z_F$ $b$ ${\cal A}$
-------------- ------- --------- ------------
$1$ $1$ $0.665$ $-6.763$
$2$ $0.716$ $-7.311$
$5$ $0.741$ $-7.597$
$4$ $1$ $0.796$ $-7.817$
$2$ $0.795$ $-7.937$
$5$ $0.796$ $-8.018$
$8$ $1$ $0.859$ $-8.311$
$2$ $0.862$ $-8.449$
$5$ $0.862$ $-8.532$
Evolution of the Tully-Fisher relation with redshift
====================================================
The star formation histories given by the best fits to the efficiency and the fraction of gas in outflows shown in Figures 4 and 5 can be used to evolve the systems backwards in time. A major consequence of the large range in star formation efficiencies found with respect to rotational velocity is that low-mass systems will have a broader distribution of stellar ages. This implies a stronger evolution in luminosity with lookback time. Furthermore, a low efficiency will yield a constant and low star formation rate with redshift. On the other hand, massive galaxies will undergo milder changes in absolute luminosity because of their high star formation efficiencies, and this will narrow the expected age distribution (cf. Fig. 3). The star formation rate for massive galaxies, however, experiences a significant increase with redshift. Figure 9 illustrates this point: model predictions of the redshift evolution of the absolute ([*top*]{}) and specific weighted-by-mass ([*bottom*]{}) star formation rates are shown for $\log v_{\rm ROT} = \{ 2.6$ (thick line), $2.4, 2.2,$ and $2.0$ (dashed)$\}$. An increase in the specific star formation rate with decreasing mass is predicted, spanning a factor of $4$ in $\psi/M_s$ between $\log v_{\rm ROT}=2.0$ and $2.6$, i.e. roughly linear with $v_{\rm ROT}$. This difference in $\psi/M_s$ gets narrower at high redshifts, explained by the fact that the increase in the specific SFR in galaxies with a high efficiency will evolve faster with lookback time, whereas low-mass systems — with low values of ${C_{\rm eff}}$ — feature a low and nearly constant SFR.
The estimates of stellar mass and star formation rates of a sample of field galaxies from the Canada France Redshift Survey and from the Hubble Deep Field North and its flanking fields are also shown in Figure 8 (Brinchmann 1999). The sample spans a wide redshift range ($0.2<z<1$). The stellar masses are obtained following $K$-band luminosities, but adding a correction term based on a comparison between multiband optical photometry and population synthesis models for various ages, metallicities and dust extinction values (Brinchmann & Ellis 2000). These authors estimated the uncertainties in the computation of the stellar mass to be $\log\Delta M_s\sim 0.3$ dex. The ongoing star formation rate is computed from the flux of the \[\] emission line. The sample is divided in Figure 8 with respect to stellar mass, so that galaxies with masses larger (smaller) than the median ($\log(M_s/M_\odot)=10.3$) are represented by squares (stars). The bottom panels only show one typical error bar to avoid crowding the figure. Taking the model predictions at face value, one would explain the lack of data points in the bottom right portion of the top panels by a selection bias of the sample which would have to be missing systems with low rotational velocities (or faint absolute luminosities) at high redshift. It is also interesting to notice that the region with $\psi{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13E$}}}5 M_\odot {\rm yr}^{-1}$ is mainly populated by massive galaxies. Brinchmann & Ellis (2000) find a significant increase in the specific star formation rate with redshift, shown in the bottom panels of Figure 9 as big circles with error bars. The points are the values at $\log (M_s/M_\odot )=10$ of the best fit found in the redshift bins represented by the horizontal error bars. The values for higher (lower) mass systems will roughly shift linearly to lower (higher) specific SFRs. Hence, this fits into our model by assuming a correlation between stellar mass and infall parameters, so that galaxies with more extended star formation histories should correspond to low-mass systems and vice versa.
The redshift evolution of the Tully-Fisher relation is another interesting issue which becomes possible to quantify with recent observations of the rotation curves of disk galaxies at moderate and high redshift (Vogt et al. 1997). The power of an analysis including star formation lies in the ability to predict relative changes in luminosity evolution caused by a different age distribution of the stellar populations. In contrast to semi-analytic modellers who are more concerned with the evolution of the zero point of the Tully-Fisher relation, we can “turn the crank” of our model — which is constrained at zero redshift by the local Tully-Fisher relation — backwards in time and find luminosity changes as a function of galaxy mass. Figure 10 shows that even more importantly than the zero point, [*the slope*]{} ($\gamma$, with $L\propto v_{\rm ROT}^\gamma$) of the Tully-Fisher relation is a crucial parameter to explore. The panels show the evolution of several properties in optical ($B$; left) and near infrared ($K$; right) passbands. The bottom panels show the predicted evolution of $\gamma_{B,K}$ for $\tau_2=4$ Gyr and three formation redshifts using a linear Schmidt law, and for $z_F=2$ with a quadratic law (dashed-dotted line). The shaded areas in all panels give the predictions at $z_F=2$ when allowing a wide range of infall timescales ($1 < \tau_2/{\rm Gyr} < 8$). The steepening of the slope is a consequence of the very low star formation efficiencies at the low-mass end, which imply a broader distribution of stellar ages. Models based on simple arguments of structure formation fail to include the effect of the evolution of the stellar populations. Hence, the prediction of Mo, Mao & White (1998), who tie the evolution of the luminous component in disks to the parameters defining the dark matter halo in which they are embedded, uses a scaling argument to find a constant $\gamma =3$, obviously regardless of passband and therefore inconsistent with the observations.
A slope change implies that the measured zero point will depend on the absolute luminosity considered. The top panels plot the luminosity evolution for systems with a rotational velocity of $100$ km s$^{-1}$, whereas the middle panels correspond to massive galaxies, with $\log v_{\rm ROT}=2.6$. As expected, more massive disks undergo a milder luminosity evolution. Recent observations at moderate and high redshift are shown as well. The latest work of Vogt et al. (2000; filled squares) on disks in the Groth Survey Strip, extending over a large redshift range: $0.2<z<1$, shows a very small — if any — evolution in absolute $B$-band luminosity ($\Delta M_B{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13C$}}}-0.2$). The observations of Bershady et al. (1999; hollow squares) at $z\sim 0.4$ hint at a slightly different luminosity evolution depending on rest frame $B-R$ color, which we coarsely map into $\log v_{\rm ROT}$. Hence, low-mass systems are found to have $\Delta M_B=-0.5\pm 0.5$ whereas massive disks appear not to have any significant evolution: $\Delta M_B=0.0\pm 0.5$. This result is in good agreement with the model, which gives a larger luminosity evolution for low-mass disks. The triangle represents the result from Simard & Pritchet (1998), who find very strong evolution ($\Delta M_B\sim -2$) in a sample of strong \[\] emitters at $z\sim 0.35$, in remarkable contrast with the other observations as well as with our model. One could speculate that disks with a star formation rate peaked at $z_F\sim 0.5-1.0$ could in principle explain this last data point, assuming that the selection criterion biased the sample towards systems with a very recent star formation history.
Discussion
==========
In this paper we have presented a phenomenological approach to galaxy formation which reduces the process of star formation to a set of a few parameters controlling the infall of primordial gas, the efficiency of processing gas to form stars, and the fraction of gas and metals that are ejected from the galaxy. These parameters are not set [*a priori*]{}. Instead, they are allowed to vary over a large volume of parameter space. The Tully-Fisher (1977) relation in different wavebands is used as a constraint, in order to find the parameter space compatible with the data. Our results show that gas outflows should not play a major role in explaining the color and luminosity range of disk galaxies, whereas a very large range of star formation efficiencies (${C_{\rm eff}}$) is required. A power law fit to the correlation between ${C_{\rm eff}}$ and rotational velocity ($v_{\rm ROT}$) gives a slope $\Delta\log{C_{\rm eff}}/\Delta\log v_{\rm ROT}\sim 4$ regardless of either the infall parameters chosen or the exponent used in the Schmidt law relating gas density and star formation rates (see Figure 4). For a range of rotational velocities $2.1<\log v_{\rm ROT}<2.6$ — which maps into a $4.5$ mag range in absolute $K$-band luminosity — we therefore predict a star formation efficiency variation of around $2$ orders of magnitude. This corresponds to star processing timescales which are $100$ times longer for low-mass disks compared to the bright ones. In contrast, Bell & Bower (2000) suggest a mass dependency of either infall or outflows in order to explain the observations, although their models did not allow for a wide range of star formation efficiencies. Moreover, Boissier et al. (2001) found no correlation between the star formation efficiency and the galaxy mass, using a chemo-spectrophotometric evolution model calibrated on the Milky Way and extended to a wide range of galaxy masses. However, the observed color range must be accounted for by a large range of formation redshifts, so that massive disks are formed [*earlier*]{} than less massive ones. The predicted large variation in stellar ages as a function
+3.3truein
+0.2truein
of galaxy mass may be in conflict with the redshift evolution of both the luminosity function of disk galaxies and the Tully-Fisher relation. If we impose on our models formation redshifts $z_F{\mathrel{\hbox to 0pt{\lower 3.5pt\hbox{$\mathchar"218$}\hss}
\raise 1.5pt\hbox{$\mathchar"13E$}}}1$ for all galaxy masses, a large range of efficiencies are predicted. Furthermore, a hierarchical scenario — for which small structures collapse first — would increase the slope of the correlation between ${C_{\rm eff}}$ and rotation velocity. The departure of the scaling behavior of the star formation efficiency found in this paper with respect to dynamical back-of-the-envelope estimates frequently used in semi-analytic modelling, i.e. ${C_{\rm eff}}\propto t_{dyn}^{-1}\propto v_{\rm ROT}^3/M$, where $t_{dyn}$ is a characteristic dynamical time-scale for the galaxy (eg. Kauffmann, White & Guiderdoni 1993), shows that the Tully-Fisher relation cannot be fully explained by the properties of the halo, but rather, needs a component incorporating self-regulated star formation (Silk 1997, 2000).
We also applied our model using the Faber-Jackson (1976) correlation in spheroids as a constraint, finding a rather constant and high efficiency and a significant range in gas outflows. This behavior could be explained by the different dynamical history of disks and spheroids. Early-type systems have undergone mergers with equal mass progenitors, which can drive a significant amount of gas out of galaxies. On the other hand, the standard framework for disk galaxy formation assumes smooth infall and cooling of gas on to the centers of dark matter halos. Furthermore, the low outflow fractions found in disks for any galaxy mass enable us to reject supernova-driven winds as an important mechanism for driving gas and metals out of the interstellar medium of galaxies, in agreement with the hydrodynamic simulations of McLow & Ferrara (1999) and the modelling of star formation in dwarf galaxies by Ferrara & Tolstoy (2000).
The large range of star formation efficiencies obtained by our model implies that low-mass galaxies should have a large fraction of baryonic matter in the form of gas. As a sanity check, we compared the predictions for the star formation rate at $z=0$ as a function of stellar mass with Kennicutt (1983; 1994) and found good agreement. A fit to a power law — $\psi\propto M_s^b$ — gives an exponent $b\sim 0.7-0.8$ for a large range of infall parameters (Table 2). Hence, in the tug-of-war between star formation efficiency and galaxy masses, the former wins over, giving higher star formation rates in brighter galaxies. In a speculative mode, we suggest that a large fraction of the considerable amounts of gas predicted for systems with low efficiencies could be in cold molecular hydrogen, a rather elusive component. In the bottom panel of Figure 8 we took our model at face value and compared it with the stellar and gas content (atomic hydrogen plus helium) from the sample compiled by McGaugh et al. (2000), and we inferred a significant increase in the ratio of molecular to atomic hydrogen for less massive galaxies. This can be accounted for if there is a significant offset in galaxies with low metallicities from the usual H$_2$ to CO conversion factor used to detect molecular hydrogen (Combes 2000). Using a direct technique to search for H$_2$, large amounts of not-too-cold molecular gas ($T\sim 80-90$ K) have been detected through the lowest pure rotational transitions in NGC 891 (Valentijn & Van der Werf 1999). Incidentally, an extrapolation of up to $M(H_2)/M(HI)\sim 6.2$ can account for the rotation curves of disks (Combes 1999). This issue has important implications not only for galaxy formation but also for cosmology: the existence of such large amounts of H$_2$ on galaxy scales could favor a parameter space of dark matter with partially suppressed hierarchical galaxy formation on sub-galactic scales, as a consequence of the enhanced free streaming lengths found in scenarios such as warm dark matter (e.g. Colín, Avila-Reese & Valenzuela 2000; Bode, Ostriker & Turok 2000). We remark that the large range in star formation efficiencies should be incorporated into models of galaxy formation. Semi-analytical simulations of galaxy formation which assume a constant star formation efficiency as a function of circular velocity (Lacey & Silk 1991; White & Frenk 1991; Cole et al. 2000; Kauffmann et al. 1999; Somerville & Primack 1999) are far too simplistic to give meaningful predictions of, for example, the cosmic star formation history.
We used our model to infer the redshift evolution of star formation and of the Tully-Fisher relation. Star formation rate predictions are compatible with the data, although the scatter, uncertainties and various biases makes this a rather uncertain endeavor. SFRs are expected to undergo stronger evolution in systems with high rotational velocities, whereas low-mass systems feature a steadier formation rate due to their low efficiencies. The range of specific formation rates with stellar mass is predicted to decrease with redshift. The Tully-Fisher relation is expected to evolve [*in slope*]{}, as well as in zero point, because the age distribution of the stellar population varies with galaxy mass, being more spread out in low-mass disks. Hence, we expect a steepening of the slope. For instance, in rest frame $B$ band the slope changes from $\gamma_B\sim 3$ at zero redshift ($L\propto v_{\rm ROT}^\gamma$) up to $\gamma\sim 4-4.5$ at $z\sim 1$ (Figure 10), with a slight dependence on the formation redshift, giving stronger evolution for low formation redshifts. The zero point should thus be given with respect to some fiducial rotational velocity. We show the evolution of the zero point at low ($\log v_{\rm ROT}=2$) and high mass ($2.6$) is in agreement with the latest observations from Vogt et al. (2000) for high mass systems, as well as with Bershady et al. (1998) both for high and low masses. Notwithstanding the many observational difficulties that complicate accurate estimates of the redshift evolution of $\gamma$, we find the evolution of the slope of the Tully-Fisher relation to be the best observable for exploring the star formation process in disk galaxies, in analogy with the study of the slope of the fundamental plane in early-type systems (Ferreras & Silk 2000b).
Acknowledgments {#acknowledgments .unnumbered}
===============
IF is supported by a grant from the European Community under contract HPMF-CT-1999-00109. We would very much like to thank Jarle Brinchmann for helpful discussions and for making available the data from his PhD thesis. We would also like to thank Eric Bell and Stacy McGaugh for providing their data on disk gas masses. The anonymous referee is gratefully acknowledged for useful comments and suggestions.
Baugh, C. M., Cole, S., Frenk, C. S. & Lacey, C. G. 1998, , 498, 504 Bell, E. F. & Bower, R. G. 2000, , 319, 235 Bell, E. F. & de Jong, R. S. 2000, , 312, 497 Bell, E. F. & de Jong, R. S. 2001, , March 20, astro-ph/0011493 Bershady, M. A., Haynes, M. P., Giovanelli, R. & Andersen, D. R. 1999, in Merritt, D. R., Valluri, M. & Sellwood, J. A., eds, ASP Conf. Ser. 182, Galaxy Dynamics. Astron. Soc. Pac., San Francisco, p. 499 Bode, P., Ostriker, J. P. & Turok, N. 2000, astro-ph/0010389 Boissier, S., Boselli, A., Prantzos, N. & Gavazzi, G. 2001, , 321, 733 Bower, R. G., Lucey, J. R. & Ellis, R. S., 1992, , 254, 589 Brinchmann, J. 1999, PhD thesis, Univ. Cambridge Brinchmann, J. & Ellis, R. S. 2000, , 536, L77 Charlot, S., Worthey, G. & Bressan, A. 1996, , 457, 625 Cole, S., Lacey, C. G., Baugh, C. M. & Frenk, C. S. 2000, , 319, 168 Colín, P., Avila-Reese, V. & Valenzuela, O. 2000, , 542, 622 Combes, F. 2000, astro-ph/0007318 Combes, F. 1999, astro-ph/9910296 de Grijs, R. & Peletier, R. F. 1999, , 310, 157 Faber, S. M. & Jackson, R. 1976, , 204, 668 Ferguson, A. M. N., Wyse, R. .F. G., Gallagher, J. S. & Hunter, D. A. 1998, , 506, L19 Ferrara, A. & Tolstoy, E. 2000, , 313, 291 Ferreras, I. & Silk, J. 2000a, , 532, 193 Ferreras, I. & Silk, J. 2000b, , 316, 786 Giovanelli, R., Haynes, M. P., Herter, T., Vogt, N. P., Wegner, G., Salzer, J. J., Da Costa, L. N. & Freudling, W. 1997, , 113, 22 Gnedin, N. Y. 1998, , 294, 407 Ibata, R. A. & Gilmore, G. F. 1995, , 275, 605 Jacoby, G. H. et al. 1992, , 104, 599 Kauffmann, G., White, S. D. M. & Guiderdoni, G. 1993, , 264, 201 Kauffmann, G., Colberg, J. M., Diaferio, A. & White, S. D. M. 1999, , 303, 188 Kennicutt, R. C. 1983, , 272, 54 Kennicutt, R. C., Tamblyn, P. & Congdon, C. W. 1994, , 435, 22 Kennicutt, R. C. 1998a, , 36, 189 Kennicutt, R. C. 1998b, , 498, 541 Kodama, T., Arimoto, N., Barger, A. J. & Aragón-Salamanca, A. 1998, , 334, 99 Lacey, C. & Silk, J. 1991, , 381, 14 Larson, R. B. 1974, , 169, 229 Lilly, S. J., Tresse, L., Hammer, F., Crampton, D. & Le Fèvre, O. 1995, , 455, 108 McGaugh, S. S., Schombert, J. M., Bothun, G. D. & de Blok, W. J. G. 2000, , 533, L99 MacLow, M.-M. & Ferrara, A. 1999, , 513, 142 Mathewson, D. S. & Ford, V. L. 1996, , 107, 97 Mo, H. J., Mao, S. & White, S. D. M. 1998, , 295, 319 Olszewski, E. W., Suntzeff, N. B. & Mateo, M. 1996, , 34, 511 Rich, R. M. 1990, , 362, 604 Rocha-Pinto, H. J. & Maciel, W. J. 1996, , 279, 447 Salpeter, E. E. 1955, , 121, 161 Scalo, J. M. 1986, fund. Cosm. Phys., 11, 1 Schaller, G., Schaerer, D., Maeder, A. & Meynet, G. 1992, , 96, 269 Schmidt, M. 1959, , 129, 243 Silk, J. 1997, , 481, 703 Silk, J. 2000, , in press, astro-ph/0010624 Simard, L. & Pritchet, C. J. 1998, , 505, 96 Somerville, R. S. & Primack, J. R. 1999, , 310, 1087 Tinsley, B. M. 1980, Fund. Cosm. Phys., 5, 287 Tully, R. B. & Fisher, J. R. 1977, , 54, 661 Tully, R. B., Verheijen, M. A. W., Pierce, M. J., Huang, J.-S. & Wainscoat, R. J. 1996, , 112, 2471 Valentijn, E. A. & van der Werf, P. 1999, , 522, L29 Verheijen, M. A. W. 1997, PhD thesis, Univ. Groningen Vogt, N. P., et al. 1997, , 479, L121 Vogt, N. P. 2000, in Combes, F., Mamon, G. A. & Charmandaris, V., eds, ASP Conf. Ser. 197, Dynamics of Galaxies: from the Early Universe to the Present. Astron. Soc. Pac., San Francisco, p. 435 White, S. D. M. & Rees, M. J. 1978, , 183, 341 White, S. D. M. & Frenk C. S. 1991, , 379, 52 Worthey, G., Dorman, B. & Jones, L. A., 1996, , 112, 948 Young, J. S. & Knezek, P. M. 1989, , 347, L55 Young, J. S., Allen, L., Kenney, J. D. P., Lesser, A. & Rownd, B. 1996, , 112, 1903 Zaritsky, D., Kennicutt, R. C. & Huchra, J. P. 1994, 420, 87
|
---
abstract: 'Voice controlled virtual assistants (VAs) are now available in smartphones, cars, and standalone devices in homes. In most cases, the user needs to first “wake-up” the VA by saying a particular word/phrase every time he or she wants the VA to do something. Eliminating the need for saying the wake-up word for every interaction could improve the user experience. This would require the VA to have the capability to detect the speech that is being directed at it and respond accordingly. In other words, the challenge is to distinguish between system-directed and non-system-directed speech utterances. In this paper, we present a number of neural network architectures for tackling this classification problem based on using only acoustic features. These architectures are based on using convolutional, recurrent and feed-forward layers. In addition, we investigate the use of an attention mechanism applied to the output of the convolutional and the recurrent layers. It is shown that incorporating the proposed attention mechanism into the models always leads to significant improvement in classification accuracy. The best model achieved equal error rates of 16.25% and 15.62% on two distinct realistic datasets.'
address: |
$^1$Nuance Communications\
$^2$McGill University, Canada\
{atta.norouzian, dermot.connolly, daniel.willett}@nuance.com, [email protected]
title: 'Exploring Attention Mechanism for Acoustic-based Classification of Speech Utterances into System-Directed and Non-System-Directed'
---
Human-machine interaction, spoken utterance classification, wake-up word, attention mechanism
Introduction {#sec:intro}
============
Thanks to recent advances in speech recognition and natural language understanding, VAs have become part of our daily lives. The VAs are typically activated by a wake-up word/phrase such as *hi Mercedes*, *hey BMW*, *hey Siri*, *Alexa* or *ok Google*. Eliminating such wake-up words in favor of allowing direct requests for assistance from the VA could significantly improve the user experience. This requires the device to have the capability to detect speech directed at it and ignore human-to-human and background speech. The problem of classifying spoken utterances into system-directed and non-system-directed has previously been investigated within the context of virtual assistants [@mallidi2018device; @reich2011real; @yamagata2009system] and dialogue systems [@wang2013understanding; @dowding2006you].
Both, the spoken words and the way they are spoken provide cues for differentiating between system-directed and non-system-directed speech utterances. Typically, the lexical cues are extracted from a word sequence generated by an automatic speech recognition (ASR) system. The classification can then be performed by applying two language models [@shriberg2012learning; @ravuri2014neural], one for each class, to the hypothesized word sequence to compute a likelihood ratio and choose the class label based on that. Alternatively, the word sequence can be input to a neural network (NN) model to either directly estimate class probabilities [@ravuri2015recurrent] or to generate new features for another model [@mallidi2018device]. The non-lexical acoustic cues can be learned from features corresponding to prosodic structure [@reich2011real; @shriberg2012learning] or the short-time frequency representation of the speech signal [@mallidi2018device; @yamagata2009system]. The frequency-based features are typically extracted using a sliding window of 20-25 ms. One of the challenges involved in using frame-based features for utterance classification is to represent all utterances with a fixed-dimensional vector to train a model regardless of the length of the utterance. Averaging the features over time [@yamagata2009system] or passing the input sequence into a long short-time memory (LSTM) cell and using the last output of the LSTM cell [@mallidi2018device] are examples of how others have dealt with this issue. In this paper, we propose a new technique based on attention to address some shortcomings of the other methods.
Similar to systems developed in [@mallidi2018device; @reich2011real; @yamagata2009system; @shriberg2012learning; @shriberg2013addressee] our plan is to combine information extracted from acoustic features with lexical information for this classification task, however, the focus of this paper is only on acoustic-based classification. The acoustic features explored for this purpose are frame-based log Mel-filterbank coefficients. We favor short-term frame-based acoustic features for this task since they facilitate early detection of user’s intent by gradual application of the trained model on the incoming speech. The proposed classification models are based on deep neural networks (DNN) with combination of convolutional, recurrent and feed-forward layers. In addition, the use of an attention mechanism on top of the convolutional layer as well as the recurrent layer is investigated. This paper is organized as follows. First, an overview of the models developed for this classification problem is presented in \[sec:sys\_overview\]. A number of model architectures proposed for frame-based approach and the architecture of the utterance-based model are described in Section \[sec:models\]. The experimental study containing a description of the evaluation data, the model parameters, and the experimental results is provided in Section \[sec:exp\]. Summary and conclusion are given in Section \[sec:conclusion\].
Overview {#sec:sys_overview}
========
Here, an overview of the two modeling approaches investigated for the system-directed versus non-system-directed classification problem is given. The two approaches are based on using frame-level and utterance-level input features. The models developed based on the frame-level input features are depicted in Figure \[fig:model\_architecture\] on the left and the utterance-based model in shown on the right. Several architectures are explored for the frame-based models as described in Section \[sec:models\] for dealing with the variable length input sequences. The model developed for the fixed-length utterance-level features is comprised of only dense feed-forward layers.
![General architecture of frame-based models **(a)** and the utterance-based model **(b)** developed for the classification task at hand.[]{data-label="fig:model_architecture"}](figures/ICASSP_Fig_model_architecture.png){width="\linewidth"}
Model Architectures {#sec:models}
===================
This section presents a number of model architectures for dealing with the issue of variable-length input feature representation faced in the frame-based approach. Moreover, it describes the input features and the architecture of the utterance-based model in detail.
Frame-based Approach {#sub_sec:frame_based}
--------------------
In this approach the feature vectors input to the models consist of 45 log Mel-filterbank coefficients extracted from 25 ms of acoustic signal with a frame shift of 10 ms. A speech utterance is hence represented by a sequence of feature vectors, $\{\boldsymbol{m}_1^{45},\ldots,\boldsymbol{m}_T^{45}\}$, where ${T}$ is the total number of frames in the utterance. All frame-based models developed here use a two-dimensional convolutional layer as input layer which outputs a set of ${d}$ feature maps denoted by $\{\boldsymbol{E}_1^{j\times l},\ldots,\boldsymbol{E}_d^{j\times l}\}$. The width of the feature maps, ${j}$, is proportional to the acoustic feature vector size (i.e., 45) and their length ${l}$ is proportional to ${T}$. In a realistic scenario, recorded utterances have different lengths which means ${T}$ and consequently ${l}$ vary from one utterance to another. This causes an issue when converting the feature-maps into a vector to pass to feed-forward layers since the input to a feed-forward layer has to have a fixed-dimension for all samples. In the following, three approaches for creating a fixed-length vector from variable-length feature-maps are presented. After creating a fixed-length vector it is input to dense feed-forward layers followed by a softmax layer as shown in Figure \[fig:model\_architecture\].
**Global averaging across time:** A simple way of generating a fixed-length representation is to take the average of each feature-map over its length ${l}$. This will transform every 2-D feature map $\boldsymbol{E}_i^{j\times l}$ to a vector ${\boldsymbol{e}_i^j}$. The resulting vectors, $\{\boldsymbol{e}_1^{j},\ldots,\boldsymbol{e}_d^{j}\}$, are then concatenated and fed to a feed-forward layer as was done in [@shon2018convolutional].
**Using a recurrent layer:** One could obtain a fixed-length vector from variable-length feature maps by using a recurrent neural layer. This is done by first concatenating columns of all feature maps to generate ${l}$ super vectors of dimension ${d \times j}$. Next, the super vectors are fed to a recurrent layer one by one and the last (i.e., ${l}$th) output of the recurrent layer is used for the succeeding layer. In addition, one could use a bi-directional recurrent layer and use the last output vector of forward and backward directions to obtain a richer fixed-length representation. In the model explored here, a bi-directional LSTM layer is used for this purpose and the two resulting vectors from both directions are concatenated and used in the feed-forward layer.
**Using attention mechanism:** Simple averaging of feature maps or passing them through a recurrent layer and using only its last output could result in losing important information. An attention mechanism could retain most of the relevant information while resolving the variable-length issue. The attention mechanism explored here is somewhat different from the traditional encode-decoder based attention introduced in [@bahdanau2014neural]. It is in essence a weighted average of sequence of vectors where the weights are learned through back-propagation. This mechanism was first explored for emotion recognition in [@neumann2017attentive] and is similar to the idea of self-attention in [@vaswani2017attention]. Denoting a sequence of ${l}$ vectors of dimension ${s}$ by the matrix $\boldsymbol{X}^{s \times l}$, attention is computed as $$\begin{split}
\boldsymbol{b}^{l\times 1} &= f(\boldsymbol{w}^{1\times s} \boldsymbol{X}^{s\times l}), \\
\alpha^{i} &= \frac{\exp(b_i)}{\sum_{j=1}^l\exp(b_j)}, \ \ i = 1, \ldots, l, \\
\end{split}
\label{eq:attention}$$ where $\boldsymbol{w}$ is the weight vector learned through back-propagation, ${f}$ is a non-linear function (here $tanh$), $b$ is the attention vector, and $\alpha$ is the normalized attention vector. Applying attention to the input sequence results in a vector known as context vector given by $$\boldsymbol{c}^{s\times 1} = \boldsymbol{X}^{s\times l}\boldsymbol{\alpha}^{l\times 1}.
\label{eq:context}$$ As can be seen in Equation \[eq:context\], the context vector dimension is independent of the length of the input sequence ${l}$. Furthermore, the attention vector ${\boldsymbol{\alpha}}$ helps to put more emphasis on the parts of the input sequence ${\boldsymbol{X}}$ that carry the most relevant information for distinguishing the two classes. The process of computing the attention and applying it to the input sequence is shown in Figure \[fig:my\_label\]. The input sequence in this case could be the flattened feature maps or the output of the recurrent layer.
![Attention mechanism applied to the input sequence ${\boldsymbol{X}}$ to generate the context vector $\boldsymbol{c}$.[]{data-label="fig:my_label"}](figures/ICASSP_Fig_Attention_schema.png){width="0.9\linewidth"}
Utterance-based Approach {#sub_sec:utterance_based}
------------------------
As an alternative to the frame-based approach one could represent every utterance with a fixed-length feature representation prior to any modeling. This can be done by computing some functions over the frame-based features. The feature set used here was developed for INTERSPEECH ComParE emotion recognition sub-challenge [@schuller2013interspeech]. It contains 6373 acoustic-features described in [@weninger2013acoustics]. We used the openSmile toolkit [@eyben2013recent] for extracting these features from the speech utterances in our corpus. Although, this feature set was originally developed for an emotion recognition task, it contains a variety of acoustic-prosodic features (F0, energy, zero crossing, mfccs) many of which are relevant for this classification problem as well. A three-layer dense feed-forward model is trained and evaluated for these features.
{width="0.9\linewidth"}
Experimental study {#sec:exp}
==================
This section provides a description of the two data sets used for training and evaluation of the classifier. Afterwards, the model parameter are given and finally experimental results are presented and analyzed.
Datasets {#sub_sec:dataset}
--------
Two datasets are used for evaluation of the proposed techniques. The first dataset denoted by $\mathcal{D}_1$ contains recordings from a device with virtual assistant. The recordings contain system-directed utterances including questions and commands as well as non-system directed utterances mostly consisting of people dictating phrases. These are actual recordings from multiple users in different environments. The second dataset denoted by $\mathcal{D}_2$ also contains virtual assistant recordings but in addition it includes, background speech, open microphone recordings and some non-speech noise in the non-system-directed subset. The models are only trained on the training subset of $\mathcal{D}_1$ and the dataset $\mathcal{D}_2$ was only used for testing. Table \[tab:dataset\_breakdown\] shows the breakdown of both datasets by class and training/validation/test. Same number of training utterances from both classes where chosen for training the models to prevent them from being biased towards one class.
------------ ---------- ----------------- ------ ------
$\mathcal{D}_2$
(r)[2-4]{} Training Validation Test Test
System 35k 12k 14k 7k
Non-system 35k 5k 4k 127k
------------ ---------- ----------------- ------ ------
: Number of utterances per training, validation and test splits across both classes and datasets $\mathcal{D}_1$ and $\mathcal{D}_2$ rounded to the nearest thousand.[]{data-label="tab:dataset_breakdown"}
Model Parameter {#sub_sec:model_param}
---------------
Prior to training the models, the frame-based and utterance-based feature vectors are normalized to have zero mean and unit standard deviation along each dimension to facilitate model convergence. Moreover, Adam optimizer and early stopping are used for training the models. For the frame-based models, the convolutional layer has a depth of 50 with a kernel height of 20 and width of 9 for the models without the LSTM layer and width of 5 for the models with LSTM layer. A stride of 5 is used along the time access and 3 along the log Mel-filterbank coefficients. The LSTM layer is bi-directional with 128 units in each direction. The three feed-forward layers in frame-based models each have 128 units. The best classification accuracy was obtained from the utterance-based model when three feed-forward layers of 128 units were used. All the models were trained using tensorflow toolkit [@tensorflow2015-whitepaper].
Results {#sub_sec:results}
-------
In this section the classification models described in Section \[sec:sys\_overview\] are evaluated on $\mathcal{D}_1$ and $\mathcal{D}_2$ datasets defined in Section \[sub\_sec:dataset\]. Figure \[fig:eer\] shows the performance of the models in terms of detection error tradeoff (DET) curves. A number of observations can be made from these plots. First, using an LSTM significantly improves the model performance compared to global averaging. Moreover, adding attention to the mix yields an additional boost to the performance with and without the LSTM. Furthermore, having larger improvement with attention on $\mathcal{D}_2$ dataset which was not seen in training suggests that the proposed attention mechanism improves model generalization as well. To have a single point of reference to compare the models, the equal error rate metric which corresponds to equal Type I and Type II errors is measured and shown in Table \[tab:eer\].
$\mathcal{D}_1$ $\mathcal{D}_2$
-------------------- ----------------- -----------------
CNN+Global Average 26.99 39.25
CNN+Attention 26.21 24.90
CNN+LSTM 19.46 18.79
CNN+LSTM+Attention **16.25** **15.62**
: Equal error rates of the four frame-based models measured on $\mathcal{D}_1$ and $\mathcal{D}_2$ test sets.[]{data-label="tab:eer"}
![Attention vector spread across time for a system-directed utterance (top) and a non-system-directed utterance (bottom) from $\mathcal{D}_1$ testset.[]{data-label="fig:attention"}](figures/Fig_ICASSP_Attention.png){width="\linewidth"}
The main question here is what the model is actually learning. This is not easy to answer especially when it comes to neural network models. However, the attention mechanism could help shedding some light on this matter. Aligning the attention vector ${\boldsymbol{\alpha}}$ with the original speech utterance, one could find out where the model is putting the most emphasis. This is done in Figure \[fig:attention\] for two utterances from the two classes. The word sequences associated with the utterances are also shown in the figure to identify possible correlations between the spoken words and where the model is mostly focusing on. The vertical lines correspond to start and end time of the words. The plot shows that for the system-directed utterance the attention is on both “*play*” and “*music*” while for the non-system-directed example the attention is mostly on the words “*that*”, “*fine*”, and “*period*”. It is interesting to note that in the training dataset the word “*period*” is spoken only when the users are dictating a phrase. In other words, this word is a strong indication that the speech utterance is of dictation style which belongs to non-system-directed class. This led us to think that maybe the model is just learning keywords and is not learning any para-linguistic information. To answer this question we looked at a number of system-directed utterances such as “*you didn’t catch that*” and “*one more run after that*” that were not part of the training data and did not contain any word highly correlated with system-directed class. The model classified both of these utterances correctly with high confidence. This suggests that the model is not just learning keywords or para-linguistic information but rather a combination of both. Adding more training data from different domains would make the model less sensitive to words and more sensitive to para-linguistic information.
![Detection error trade-off curves for the utterance-based approach and the proposed frame-based approach measured on $\mathcal{D}_1$ testset.[]{data-label="fig:compare"}](ICASSP_Fig_EER_openSMILE.png){width="\linewidth"}
In Figure \[fig:compare\], the proposed frame-based approach is compared to the utterance-based approach described in Section \[sub\_sec:utterance\_based\] on the $\mathcal{D}_1$ dataset. It should be noted that the utterance-based acoustic-prosodic feature set was designed for emotion recognition and contains several features that may not be relevant for this task. Nevertheless, the gap between the two curves indicates that even without using hand-crafted features and only relying on frame-based log Mel-filterbank features very good classification performance can be achieved with the proposed attention-based modeling technique.
Conclusion {#sec:conclusion}
==========
In this paper, the problem of classifying speech utterances into system-directed and non-system-directed was addressed. A number of neural network architectures based on using convolutional and recurrent layers were investigated. It was shown that having an attention mechanism improves the classification performance whether applied directly to the output of the convolutional layer or to the output of the recurrent layer. The best performing model was built by stacking a convolutional layer, a recurrent layer and three feed-forward layers with attention applied to the output of the recurrent layer. This model achieved an EER rate of ${16.25\%}$ on one test set and ${15.62\%}$ on the second test set. As continuation of this work we are looking into combining direct audio classification with ASR-output based text classification for improved accuracy.
Acknowledgements
================
The authors would like to thank two colleagues from Nuance Communications, Raymond Brueckner for helping out with the openSmile toolkit and Yasser Hifny for very insightful discussions.
|
---
abstract: 'A class of inverse problems for a heat equation with involution perturbation is considered using four different boundary conditions, namely, Dirichlet, Neumann, periodic and anti-periodic boundary conditions. Proved theorems on existence and uniqueness of solutions to these problems are presented. Solutions are obtained in the form of series expansion using a set of appropriate orthogonal basis for each problem. Convergence of the obtained solutions is also discussed.'
address:
- 'Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36, Al-Khodh, PC 123, Muscat, Oman'
- 'LaSIE, Faculté des Sciences et Technologies, Université de La Rochelle, Avenue Michel Crépeau, 17000, La Rochelle, France'
- 'Institute of Mathematics and Mathematical Modeling, 050010, Pushkin st., 125, Almaty, Kazakhstan'
author:
- 'Nasser Al-Salti'
- Mokhtar Kirane
- 'Berikbol T. Torebek'
title: On a class of inverse problems for a heat equation with involution perturbation
---
Inverse Problems, Heat Equation, Involution Perturbation 35R30; 35K05; 39B52
Introduction
============
Differential equations with modified arguments are equations in which the unknown function and its derivatives are evaluated with modifications of time or space variables; such equations are called in general functional differential equations. Among such equations, one can single out, equations with involutions [@Cabada2].
[@Carleman; @Wiener2] A function $\alpha(x)\not\equiv x,$ that maps a set of real numbers, $\Gamma$ onto itself and satisfies on $\Gamma$ the condition $$\alpha\left(\alpha(x)\right)=x,\quad \textrm{or}\quad \alpha^{-1}(x)=\alpha(x)$$ is called an involution on $\Gamma.$
Equations containing involution are equations with an alternating deviation (at $x^* < x$ being equations with advanced, and at $x^*> x$ being equations with delay, where $x^*$ is a fixed point of the mapping $\alpha(x)$).
Equations with involutions have been studied by many researchers, for example, Ashyralyev [@Ashyralyev1; @Ashyralyev2], Babbage [@Babbage], Przewoerska-Rolewicz [@D1; @D2; @D3; @D4; @D5; @D6; @D7], Aftabizadeh and Col. [@Aftabizadeh], Andreev [@Andreev1; @Andreev2], Burlutskayaa and Col. [@Burlutskayaa], Gupta [@Gupta1; @Gupta2; @Gupta3], Kirane [@Kirane2], Watkins [@Watkins], and Wiener [@Viner1; @Viner2; @Wiener1; @Wiener2]. Spectral problems and inverse problems for equations with involutions have received a lot of attention as well, see for example, [@Kirane1; @Kopzhassarova; @Sadybekov; @Sarsenbi1; @Sarsenbi2], and equations with a delay in the space variable have been the subject of many research papers, see for example, [@Aliev; @Rus]. Furthermore, for the equations containing transformation of the spatial variable in the diffusion term , we can cite the talk of Cabada and Tojo [@Cabada1], where they gave an example that describes a concrete situation in physics: Consider a metal wire around a thin sheet of insulating material in a way that some parts overlap some others as shown in Figure \[fig1\].
![An application of heat equation with involution[]{data-label="fig1"}](wires.png "fig:"){width="0.25\linewidth"}
Assuming that the position $y = 0$ is the lowest of the wire, and the insulation goes up to the left at $-Y$ and to the right up to $Y.$ For the proximity of two sections of wires they added the third term with modifications on the spatial variable to the right-hand side of the heat equation with respect to the wire: $$\frac{\partial T}{\partial t}(y,t)=a \frac{\partial^2 T}{\partial y^2}(y,t)+b \frac{\partial^2 T}{\partial y^2}(-y,t).$$ Such equations have also a purely theoretical value. For general facts about partial functional differential equations and for properties of equations with involutions in particular, we refer the reader to the books of Skubachevskii [@Skubachevskii], Wu [@Wu] and Cabada and Tojo [@Cabada2].
In this paper, we consider inverse problems for a heat equation with involution using four different boundary conditions. We seek formal solutions to these problems in a form of series expansions using orthogonal basis obtained by separation of variables and we also examine the convergence of the obtained series solutions. The main results on existence and uniqueness are formulated in four theorems in the last section of this paper along with an illustrating example.
Concerning inverse problems for heat equations, some recent works have been implemented by Kaliev [@Kaliev1; @Kaliev2], Sadybekov [@Orazov1; @Orazov2], Kirane [@Furati; @Kirane3].
Statements of Problems
======================
Consider the heat equation
$$\label{2.1}u_t \left( {x,t} \right) - u_{xx} \left( {x,t} \right) +
\varepsilon u_{xx} \left( { - x,t} \right) = f\left( x \right), \quad
\left( {x,t} \right) \in \Omega,$$
where, $\varepsilon$ is a nonzero real number such that $\left| \varepsilon \right| < 1$ and $\Omega$ is a rectangular domain given by $\Omega = \left\{ {- \pi < x < \pi, \, 0 < t < T} \right\}$. Our aim is to find a regular solution to the following four inverse problems:\
[**IP1: Inverse Problem with Dirichlet Boundary Conditions.**]{}\
Find a pair of functions $u\left( {x,t} \right)$ and $f\left( x \right)$ in the domain $\Omega $ satisfying equation (\[2.1\]) and the conditions $$\label{2.2}
u\left({x,0} \right) = \varphi \left( x \right), \quad u\left( {x,T} \right) =
\psi \left( x \right), \quad x \in \left[ { - \pi ,\pi } \right],$$ and the homogeneous Dirichlet boundary conditions $$\label{2.3}
u\left( { - \pi ,t} \right) = 0, \quad u\left( {\pi ,t} \right) = 0, \quad t \in \left[ {0,T}
\right],$$ where $\varphi \left( x \right)$ and $\psi
\left( x \right)$ are given, sufficiently smooth functions.\
[**IP2: Inverse Problem with Neumann Boundary Conditions.**]{}\
Find a pair of functions $u\left( {x,t} \right)$ and $f\left( x \right)$ in the domain $\Omega $ satisfying equation (\[2.1\]), conditions (\[2.2\]) and the homogeneous Neumann boundary conditions $$\label{2.4}
u_x \left( { - \pi ,t} \right) = 0, \quad u_x \left( {\pi ,t} \right) = 0, \quad t \in \left[ {0,T}\right].$$
[**IP3: Inverse Problem with Periodic Boundary Conditions.**]{}\
Find a pair of functions $u\left( {x,t} \right)$ and $f\left( x \right)$ in the domain $\Omega $ satisfying equation (\[2.1\]), conditions (\[2.2\]) and the periodic boundary conditions $$\label{2.5}
u\left( { - \pi ,t} \right) = u\left(
{\pi ,t} \right), \quad u_x \left( { - \pi ,t} \right) = u_x \left( {\pi
,t} \right), \quad t \in \left[ {0,T} \right].$$
[**IP4: Inverse Problem with Anti-Periodic Boundary Conditions.**]{}\
Find a pair of functions $u\left( {x,t} \right)$ and $f\left( x \right)$ in the domain $\Omega $ satisfying equation (\[2.1\]), conditions (\[2.2\]) and the anti-periodic boundary conditions $$\label{2.6}
u\left( { - \pi ,t} \right) = - u\left(
{\pi ,t} \right), \quad u_x \left( { - \pi ,t} \right) = - u_x \left( {\pi
,t} \right), \quad t \in \left[ {0,T} \right].$$
By a regular solution of problems IP1, IP2, IP3 and IP4, we mean a pair of functions $u\left( {x,t} \right)$ and $f\left( x \right)$ of the class $u\left( {x,t} \right) \in C_{x,t}^{2,1} \left( { \Omega
} \right),$ $f\left( x \right) \in C\left[ { - \pi ,\pi }
\right].$
Solution Method
===============
Here we seek a solution to problems IP1, IP2, IP3 and IP4 in a form of series expansion using a set of functions that form orthogonal basis in $L_2(-\pi, \pi)$. To find the appropriate set of functions for each problem, we shall solve the homogeneous equation corresponding to equation (\[2.1\]) along with the associated boundary conditions using separation of variables.
Spectral Problems
-----------------
Separation of variables leads to the following spectral problems for IP1, IP2, IP3 and IP4, respectively, $$\label{SP IP1}
X''(x)-\epsilon X''(-x)+\lambda X(x)=0,\quad \quad X(-\pi)=X(\pi)=0,$$ $$\label{SP IP2}
X''(x)-\epsilon X''(-x)+\lambda X(x)=0,\quad X'(-\pi)=X'(\pi)=0,$$ $$\label{SP IP3}
X''(x)-\epsilon X''(-x)+\lambda X(x)=0,\quad X(-\pi)=X(\pi), \, X'(-\pi)=X'(\pi),$$ $$\label{SP IP4}
X''(x)-\epsilon X''(-x)+\lambda X(x)=0,\, X(-\pi)=-X(\pi), \, X'(-\pi)=-X'(\pi).$$ The eigenvalue problems (\[SP IP1\]) - (\[SP IP4\]) are self-adjoint and hence they have real eigenvalues and their eigenfunctions form a complete orthogonal basis in $L_2 \left( { - \pi ,\pi } \right)$ [@Naim]. Their eigenvalues are, respectively, given by $$\label{eigenvalue IP1}
\lambda _{1k} = \left( {1 - \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2, \, k \in \mathbb{N}\cup\left\{0\right\}, \quad \lambda _{2k} = \left( {1 + \varepsilon }\right)k^2, \, k \in \mathbb{N}, \tag{7.a}$$ $$\label{eigenvalue IP2}
\lambda _{1k} = \left( {1 - \varepsilon } \right)k^2, \quad \lambda _{2k} = \left( {1 + \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2, \quad k \in \mathbb{N}\cup\left\{0\right\}, \tag{8.a}$$ $$\label{eigenvalue IP3}
\lambda _{1k} = \left( {1 - \varepsilon } \right)k^2, \, k \in \mathbb{N}\cup\left\{0\right\}, \quad \quad \lambda _{2k} = \left( {1 + \varepsilon }\right)k^2, \, k \in \mathbb{N}, \tag{9.a}$$ $$\label{eigenvalue IP4}
\lambda _{1k} = \left( {1 + \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2, \quad \lambda _{2k} = \left( {1 - \varepsilon }\right)\left( {k + \frac{1}{2}} \right)^2, \, k \in \mathbb{N}\cup\left\{0\right\}, \tag{10.a}$$ and the corresponding eigenfunctions are given by $$\label{basis IP1}
X_{1k} = \cos \left( {k + \frac{1}{2}} \right)x,\,\, k \in \mathbb{N}\cup\left\{0\right\}, \quad X_{2k} = \sin kx, \, \, k \in \mathbb{N}, \tag{7.b}$$ $$\label{basis IP2}
X_0=1, \, X_{1k} = \cos kx, \, k \in \mathbb{N}, \, \, \, X_{2k} = \sin \left( {k + \frac{1}{2}} \right)x,\, k \in \mathbb{N}\cup\left\{0\right\},\tag{8.b}$$ $$\label{basis IP3}
X_0=1, \, \, \quad X_{1k} = \cos kx,\,\, \quad X_{2k} = \sin kx, \, \, \quad k \in \mathbb{N}. \tag{9.b}$$ $$\label{basis IP4}
X_{1k} =\sin \left( {k + \frac{1}{2}} \right)x \quad X_{2k} = \cos \left( {k + \frac{1}{2}} \right)x , \, \, \quad k \in \mathbb{N}\cup\left\{0\right\}. \tag{10.b}$$
\[l1\] The systems of functions (\[basis IP1\]) - (\[basis IP4\]) are complete and orthogonal in $L_2\left( { - \pi ,\pi } \right).$
Here we present the proof for the system of functions (\[basis IP1\]). The orthogonality follows from the direct calculations: $$\int_{-\pi}^{\pi} X_{1n}X_{2m}\, dx=0, \quad n \in \mathbb{N}\cup\left\{0\right\}, \, m \in \mathbb{N},$$ and $$\int_{-\pi}^{\pi} X_{in}X_{im}\, dx=0, \quad m \ne n, \, i=1,2.$$ Hence, it only remains to prove the completeness of the system in $L_2(-\pi, \pi)$, i.e., we need to show that if $$\label{complete1}
\int_{-\pi}^{\pi}f(x)\cos\left(k+\frac{1}{2}\right)x \, dx=0, \quad k \in \mathbb{N}\cup\left\{0\right\},$$ and $$\label{complete2}
\int_{-\pi}^{\pi}f(x)\sin kx \,dx=0, \quad k\in N,$$ then $f(x) \equiv 0 $ in $(-\pi, \pi)$. To show this, we are going to use the fact that $\left\{\cos \left(k+\frac{1}{2}\right)x\right\}_{k \in \mathbb{N}\cup\left\{0\right\}}$ and $\left\{\sin kx\right\}_{k\in N}$ are complete in $L_2(0, \pi)$, see [@Moiseev] for example. Now, suppose that the equation (\[complete1\]) holds. We then have $$0=\int_{-\pi}^{\pi}f(x)\cos\left(k+\frac{1}{2}\right)x \,dx=\int_{0}^{\pi}\left(f(x)+f(-x)\right)\cos\left(k+\frac{1}{2}\right)x\,dx.$$ Hence, by the completeness of the system $\left\{\cos \left(k+\frac{1}{2}\right)x\right\}_{k \in \mathbb{N}\cup\left\{0\right\}}$ in $L_2(0, \pi)$, we have $f(x)=-f(-x), \, -\pi<x<\pi$. Similarly, if equation (\[complete2\]) holds, we have $$0=\int_{-\pi}^{\pi}f(x)\sin kx \,dx=\int_{0}^{\pi}\left(f(x)-f(-x)\right)\sin kx \,dx.$$ Then, by the completeness of the system $\left\{\sin kx\right\}_{k\in N}$ in $L_2(0, \pi)$, we have $f(x)=f(-x), \, -\pi<x<\pi.$ Therefore, we must have $f(x) \equiv 0$ in $(-\pi, \pi).$ Completeness and orthogonality of the systems of functions (\[basis IP2\]) - (\[basis IP4\]) can be proved similarly.
Since each one of the systems of eigenfunctions (\[basis IP1\]) - (\[basis IP4\]) is complete and forms a basis in $L_2\left(-\pi, \pi\right)$, the solution pair $u(x,t)$ and $f(x)$ of each inverse problem can be expressed in a form of series expansion using the appropriate set of eigenfunctions.
Existence of Solutions
----------------------
Here, we give a full proof of existence of solution to the Inverse Problem IP1. Existence of solutions to the other three problems can be proved similarly. Using the orthogonal system (\[basis IP1\]), the functions $u\left(
{x,t} \right)$ and $f\left( x \right)$ can be represented as follows $$\label{6.1}
u\left( {x,t} \right) = \sum\limits_{k
= 0}^\infty {u_{1k} \left( t \right)\cos \left( {k + \frac{1}{2}}
\right)x} + \sum\limits_{k = 1}^\infty {u_{2k} \left( t \right)\sin
kx},$$ $$\label{6.2}
f\left( x \right) =
\sum\limits_{k = 0}^\infty {f_{1k} \cos \left( {k + \frac{1}{2}}
\right)x} + \sum\limits_{k = 1}^\infty {f_{2k} \sin kx},$$ where the coefficients $u_{1k} \left( t \right),u_{2k}
\left( t \right), f_{1k} ,f_{2k}$ are unknown. Substituting (\[6.1\]) and (\[6.2\]) into equation (\[2.1\]), we obtain the following equations relating the functions $u_{1k} \left( t \right),$ $u_{2k} \left( t \right)$ and the constants $f_{1k}, f_{2k}$: $$\label{u1k_eqn}
u'_{1k} \left( t \right) + \left({1 - \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 u_{1k}\left( t \right) = f_{1k} ,$$ $$\label{u2k_eqn}
u'_{2k} \left( t \right) + \left( {1 + \varepsilon } \right)k^2 \, u_{2k} \left( t \right) = f_{2k}.$$ Solving these equations we obtain $$u_{1k} \left( t \right) = \frac{{f_{1k} }}{{\left( {1 - \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 }} + C_{1k} e^{ - \left( {1 - \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 t},$$ $$u_{2k} \left( t \right) = \frac{{f_{2k} }}{{\left( {1 + \varepsilon } \right)k^2 }} + C_{2k} e^{ - \left( {1 + \varepsilon } \right)k^2 t},$$ where the unknown constants $C_{1k} ,$ $C_{2k} ,$ $f_{1k},$ $f_{2k}$ are to be determined using the conditions in (\[2.2\]). Let $\varphi
_{ik} ,\psi _{ik} ,i = 1,2$ be the coefficients of the series expansions of $\varphi \left( x \right)$ and $\psi \left( x \right)$, respectively, i.e., $$\varphi _{1k} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi \left( x \right)\cos \left( {k + \frac{1}{2}} \right)x \, dx} , \quad \varphi
_{2k} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi \left( x \right)\sin kx \, dx},$$ $$\psi _{1k} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi \left( x \right)\cos \left( {k + \frac{1}{2}} \right)x \, dx}, \quad \psi _{2k} =
\frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi \left( x \right)\sin kx \, dx}.$$ Then, the two conditions in (\[2.2\]) leads to $$\frac{{f_{1k}
}}{{\left( {1 - \varepsilon } \right)\left( {k + \frac{1}{2}}
\right)^2 }} + C_{1k} = \varphi _{1k}, \quad \frac{{f_{1k} }}{{\left( {1 - \varepsilon } \right)\left( {k +
\frac{1}{2}} \right)^2 }} + C_{1k} e^{ - \left( {1 - \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2 T} = \psi _{1k},$$
$$\frac{{f_{2k}
}}{{\left( {1 + \varepsilon } \right)k^2 }} + C_{2k} = \varphi
_{2k}, \quad \frac{{f_{2k}}}{{\left( {1 +
\varepsilon } \right)k^2 }} + C_{2k} e^{ - \left( {1 + \varepsilon
} \right)k^2 T} = \psi _{2k}.$$ Solving these set of algebraic equations, we get $$C_{1k} = \frac{{\varphi _{1k} - \psi _{1k} }}{{1 - e^{ - \left(
{1 - \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T}
}}, \quad f_{1k} = \left( {1
- \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 \left( \varphi
_{1k} - C_{1k} \right),$$
$$C_{2k} = \frac{{\varphi _{2k} - \psi _{2k} }}{{1 - e^{
- \left( {1 + \varepsilon } \right)k^2 T} }}, \quad \quad \quad f_{2k} = \left( {1 + \varepsilon } \right)k^2 \left(\varphi _{2k} - C_{2k} \right).$$ Now, substituting $u_{1k}\left( t \right),$ $u_{2k} \left( t \right),$ $f_{1k},$ $f_{2k}$ into (\[6.1\]) and (\[6.2\]) we get $$\begin{aligned}
u\left( {x,t} \right) =
\varphi \left( x \right) &+& \sum\limits_{k = 0}^\infty {C_{1k}
\left( {e^{ - \left( {1 - \varepsilon } \right)\left( {k +
\frac{1}{2}} \right)^2 t} - 1} \right)\cos \left( {k +
\frac{1}{2}} \right)x} \\
&+& \sum\limits_{k = 1}^\infty {C_{2k}
\left( {e^{ - \left( {1 + \varepsilon } \right)k^2 t} - 1}
\right)\sin kx},\end{aligned}$$ and $$\begin{aligned}
f\left( x
\right) = -\varphi'' \left( x \right) +\varepsilon\varphi'' \left( -x \right)
&-& \sum\limits_{k = 0}^\infty {\left( 1 - \varepsilon \right) \left( {k + \frac{1}{2}} \right)^2 C_{1k} \cos \left( {k + \frac{1}{2}} \right)x} \\
&-& \sum\limits_{k = 1}^\infty {\left( 1 +\varepsilon \right) k ^2 \, C_{2k} \sin kx} .\end{aligned}$$ Note that for $f\left( x \right) \in C\left[ { - \pi ,\pi }
\right]$, it is required that $\varphi(x) \in C^2\left[ { - \pi ,\pi }\right]$.
Convergence of Series
---------------------
In order to justify that the obtained formal solution is indeed a true solution, we need to show that the series appeared in $u(x,t)$ and $f(x)$ as well as the corresponding series representations of $u_{xx}(x,t)$ and $u_t(x,t)$ converge uniformly in $\Omega$. For this purpose, let $$\varphi ^{\left( i \right)} \left( { - \pi } \right) = \varphi ^{\left( i \right)} \left( \pi \right) = 0, \quad i = 0,2,$$ $$\psi ^{\left( i \right)} \left( { - \pi } \right) = \psi ^{\left( i \right)} \left( \pi \right) = 0, \quad i = 0,2.$$ Hence, on integration by parts, $C_{1k}$ and $C_{2k}$ can now be rewritten as $$C_{1k} = \frac{{\varphi _{2k}^{\left( 3 \right)} - \psi
_{2k}^{\left( 3 \right)} }}{{\left( {1 - e^{ - \left( {1 -
\varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T} }
\right)\left( {k + \frac{1}{2}} \right)^3 }}, \quad C_{2k} = -\frac{{\varphi _{1k}^{\left( 3 \right)} - \psi
_{1k}^{\left( 3 \right)} }}{{\left( {1 - e^{ - \left( {1 +
\varepsilon } \right)k^2 T} } \right)k^3 }}.$$ where, $$\varphi _{1k} ^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\cos kx \, dx} , \quad \varphi
_{2k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\sin \left( {k + \frac{1}{2}} \right) x \, dx},$$ $$\psi _{1k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\cos kx \, dx}, \quad \psi _{2k}^{(3)} =
\frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\sin \left( {k + \frac{1}{2}} \right) x \, dx}.$$ Hence, the series representation of $u\left({x,t} \right)$ and $f(x)$ can be expressed as $$\begin{aligned}
u\left(
{x,t} \right) = \varphi \left( x \right) &+& \sum\limits_{k = 1}^\infty {\frac{{{1-e^{ - \left( {1 + \varepsilon } \right)k^2 t}} }}{{{1 - e^{ - \left( {1 + \varepsilon } \right)k^2 T} } }} \left( \frac{ {\varphi _{1k}^{\left( 3 \right)} - \psi_{1k}^{\left( 3 \right)} }}{k^3} \right) \sin \, kx}\\
&-& \sum\limits_{k = 0}^\infty {\frac{{{1-e^{ - \left( {1 - \varepsilon }\right)\left( {k + \frac{1}{2}} \right)^2 t}} }}{{ {1 - e^{ - \left( {1 -
\varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T} }}}\left( \frac{ {\varphi_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }}{\left( {k + \frac{1}{2}} \right)^3} \right) \cos \left( {k + \frac{1}{2}} \right)x} ,\end{aligned}$$ and
$$\begin{aligned}
f\left( x
\right) = -\varphi'' \left( x \right) &+&\varepsilon\varphi'' \left( -x \right) +\sum\limits_{k = 1}^\infty {\frac{1 + \varepsilon}{k} \left(\frac{\varphi _{1k}^{\left( 3 \right)} - \psi
_{1k}^{\left(3 \right)} }{{{1 - e^{ - \left( {1+ \varepsilon } \right)k^2 T} }}} \right)\sin kx} \\
&-& \sum\limits_{k = 0}^\infty {\frac{\left( {1 - \varepsilon } \right)}{\left( {k + \frac{1}{2}} \right)} \left(\frac{{{\varphi
_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }
}}{{{1 - e^{ - \left( {1 - \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2 T} } }}\right)\cos \left( {k + \frac{1}{2}} \right)x}.\end{aligned}$$
For convergence, we then have the following estimates for $u\left( {x,t} \right)$ and $f \left(x \right)$ $$\left|
{u\left( {x,t} \right)} \right| \le \left| {\varphi \left( x
\right)} \right| +c \sum\limits_{k = 1}^\infty {\frac{{\left|
{\varphi _{1k}^{\left( 3 \right)} } \right| + \left| {\psi
_{1k}^{\left( 3 \right)} } \right|}}{{k^3 }}}+ c\sum\limits_{k = 0}^\infty
{\frac{{\left| {\varphi _{2k}^{\left( 3 \right)} } \right| +
\left| {\psi _{2k}^{\left( 3 \right)} } \right|}}{{\left( {k + \frac{1}{2}} \right)^3 }}}$$ and $$\begin{aligned}
\left| {f\left( x \right)} \right| \le
\left| {\varphi'' \left( x \right)} \right| &+&\left| {\varphi'' \left( -x \right)} \right|+ c\sum\limits_{k =
1}^\infty {\left( \left| {\varphi _{1k}^{\left( 3 \right)} }
\right|^2 + \left| {\psi _{1k}^{\left( 3 \right)} }
\right|^2 + \frac{2}{k^2}\right)} \\
&+& c\sum\limits_{k =
0}^\infty {\left( \left| {\varphi _{2k}^{\left( 3 \right)} }
\right|^2 + \left| {\psi _{2k}^{\left( 3 \right)} }
\right|^2 + \frac{2}{\left( {k + \frac{1}{2}} \right)^2}\right)},\end{aligned}$$ for some positive constant $c$. Here, for the estimate of $f(x)$, we have used the inequality $2ab \le a^2 +b^2$. The convergence of the series in the estimate of $u(x,t)$ is clearly achieved if $\varphi^{(3)}_{ik} ,\psi^{(3)} _{ik} ,i = 1,2$ are finite. This can be ensured by assuming that $\varphi'''(x)$ and $\psi'''(x) \in L_2(-\pi, \pi)$. Furthermore, by Bessel inequality for trigonometric series, the following series converge: $$\sum\limits_{k =1}^\infty {\left| {\varphi _{ik}^{\left(3 \right)} } \right|^2 \le } C\left\| {\varphi''' \left( x \right)}
\right\|_{L_2 \left( { - \pi ,\pi } \right)}^2 , \quad i =1,2,$$ $$\sum\limits_{k =1}^\infty {\left| {\psi _{ik}^{\left(3 \right)} } \right|^2 \le } C\left\| {\psi''' \left( x \right)}
\right\|_{L_2 \left( { - \pi ,\pi } \right)}^2 , \quad i =1,2.$$ Therefore, by the Weierstrass M-test (see.[@Knopp]), the series representations of $u(x,t)$ and $f(x)$ converge absolutely and uniformly in the region $\Omega .$ The convergence of the series representations of $u_{xx}(x,t)$ and $u_t(x,t)$ which are obtained by term-wise differentiation of the series representation of $u(x,t)$ can be shown is a similar way.
Uniqueness of Solution
----------------------
Suppose that there are two solution sets $\left\{ {u_1 \left( {x,t} \right),f_1 \left( x \right)} \right\}$ and $\left\{ {u_2 \left( {x,t} \right), f_2\left( x \right)} \right\}$ to the Inverse Problem IP1. Denote $$u\left( {x,t} \right) = u_1 \left( {x,t} \right) - u_2 \left( {x,t} \right),$$ and $$f\left( x \right) = f_1 \left( x \right) - f_2 \left( x
\right).$$ Then, the functions $u\left( {x,t} \right)$ and $f\left(
x \right)$ clearly satisfy equation (\[2.1\]), the boundary conditions in (\[2.3\]) and the homogeneous conditions $$\label{H_IC}
u\left({x,0} \right) = 0, \quad u\left( {x,T} \right) =
0, \quad x \in \left[ { - \pi ,\pi } \right]$$ Let us now introduce the following $$\label{u1k}
u_{1k} \left(t \right) = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {u\left( {x,t} \right)\cos \left(k+ \frac{1}{2}\right) x\,dx}, \quad k \in \mathbb{N}\cup\left\{0\right\},$$ $$\label{u2k}
u_{2k}
\left( t \right) = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {u\left( {x,t}
\right)\sin kx \, dx}, \quad k \in N,$$ $$\label{f1k}
f_{1k} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {f\left( x \right)\cos \left(k+ \frac{1}{2}\right)x\, dx}, \quad k \in \mathbb{N}\cup\left\{0\right\},$$ $$\label{f2k}
f_{2k} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi
{f\left( x \right)\sin kx \, dx}, \quad k \in N.$$ Note that the homogeneous conditions in (\[H\_IC\]) lead to $$u_{ik}(0)= u_{ik}(T)=0, \quad i=1,2,$$ and differentiating equation (\[u1k\]) gives $$u'_{1k} \left( t
\right) = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\left( {u_{xx} \left( {x,t} \right) -
\varepsilon u_{xx} \left( { - x,t} \right)} \right)\cos \left(k+ \frac{1}{2}\right)x\, dx} + f_{1k},$$ which on integrating by parts and using the conditions in (\[2.2\]) reduces to $$u'_{1k} \left( t
\right) = (\varepsilon-1) \left(k+\frac{1}{2}\right)^2 u_{1k} + f_{1k}.$$ One can then easily show that this equation together with the conditions $u_{1k}(0)=u_{1k}(T)=0$ imply that $$f_{1k} = 0, \quad u_{1k} \left( t \right) \equiv 0.$$ Similarly, for $u_{2k}$ and $f_{2k}$ as given in (\[u2k\]) and (\[f2k\]), respectively, one can show that $$f_{2k} = 0, \quad u_{2k} \left( t \right) \equiv 0.$$ Therefore, due to the completeness of the system of eigenfunctions (\[basis IP1\]) in $L_2 \left({ - \pi ,\pi } \right)$, we must have $$f\left( t \right) \equiv 0, \quad u\left(
{x,t} \right) \equiv 0, \quad \left( {x,t} \right) \in \bar{\Omega} .$$ This ends the proof of uniqueness of solution to the Inverse Problem IP1. Uniqueness of solutions to the Inverse Problems IP2, IP3 and IP4 can be proved in a similar way.
Main Results and Example Solution
=================================
Main Results
------------
The main results for the Inverse Problems IP1, IP2, IP3 and IP4 can be summarized in the following theorems:
\[th1\] Let $\varphi \left( x \right),\psi \left( x \right) \in C^2 \left[ { - \pi ,\pi } \right]$, $\varphi'''(x)$, $\psi'''(x) \in L_2(-\pi, \pi)$ and $\varphi ^{\left( i\right)} \left( { \pm \pi } \right) = \psi ^{\left( i \right)} \left( { \pm \pi } \right) = 0,i = 0,2.$ Then, a unique solution to the Inverse Problem IP1 exists and it can be written in the form $$\begin{aligned}
u\left(
{x,t} \right) = \varphi \left( x \right) &+& \sum\limits_{k = 1}^\infty {\frac{{{1-e^{ - \left( {1 + \varepsilon } \right)k^2 t}} }}{{{1 - e^{ - \left( {1 + \varepsilon } \right)k^2 T} } }} \left( \frac{ {\varphi _{1k}^{\left( 3 \right)} - \psi_{1k}^{\left( 3 \right)} }}{k^3} \right) \sin \, kx}\\
&-& \sum\limits_{k = 0}^\infty {\frac{{{1-e^{ - \left( {1 - \varepsilon }\right)\left( {k + \frac{1}{2}} \right)^2 t}} }}{{ {1 - e^{ - \left( {1 -
\varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T} }}}\left( \frac{ {\varphi_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }}{\left( {k + \frac{1}{2}} \right)^3} \right) \cos \left( {k + \frac{1}{2}} \right)x} ,\end{aligned}$$ $$\begin{aligned}
f\left( x
\right) = -\varphi'' \left( x \right) &+&\varepsilon\varphi'' \left( -x \right) +\sum\limits_{k = 1}^\infty {\frac{1 + \varepsilon}{k} \left(\frac{\varphi _{1k}^{\left( 3 \right)} - \psi
_{1k}^{\left(3 \right)} }{{{1 - e^{ - \left( {1+ \varepsilon } \right)k^2 T} }}} \right)\sin kx} \\
&-& \sum\limits_{k = 0}^\infty {\frac{\left( {1 - \varepsilon } \right)}{\left( {k + \frac{1}{2}} \right)} \left(\frac{{{\varphi
_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }
}}{{{1 - e^{ - \left( {1 - \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2 T} } }}\right)\cos \left( {k + \frac{1}{2}} \right)x},\end{aligned}$$ where $$\varphi _{1k} ^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\cos kx \, dx} , \quad \varphi
_{2k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\sin \left( {k + \frac{1}{2}} \right) x \, dx},$$ $$\psi _{1k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\cos kx \, dx}, \quad \psi _{2k}^{(3)} =
\frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\sin \left( {k + \frac{1}{2}} \right) x \, dx}.$$
\[th2\] Let $\varphi \left( x \right),\psi \left( x \right) \in C^2 \left[ { - \pi ,\pi } \right]$, $\varphi'''(x)$, $\psi'''(x) \in L_2(-\pi, \pi)$ and $\varphi' \left( { \pm \pi }\right) = \psi '\left( { \pm \pi } \right) =
0.$ Then a unique solution to the Inverse Problem IP2 exists and it can be written in the form $$\begin{aligned}
u\left( {x,t} \right) =
\varphi \left( x \right) &+&\frac{t}{T}(\psi_0-\varphi_0)+ \sum\limits_{k = 1}^\infty {\frac{{\left( {1 - e^{ - \left( {1 - \varepsilon } \right)k^2 t}
} \right)\left( {\psi _{2k}^{\left( 3 \right)}
- \varphi _{2k}^{\left( 3 \right)} } \right)}}{{\left( {1 - e^{ - \left( {1 - \varepsilon }\right)k^2 T} } \right)k^3 }}\cos kx} \\
&-&\sum\limits_{k = 0}^\infty {\frac{{\left( {1 - e^{ - \left( {1 +
\varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 t} }
\right)\left({\psi _{1k}^{\left( 3 \right)} - \varphi _{1k}^{\left( 3 \right)}} \right) }}{{\left( {1 - e^{ -\left( {1 + \varepsilon } \right)\left( {k + \frac{1}{2}}\right)^2 T} } \right)\left( {k + \frac{1}{2}} \right)^3 }}\sin \left( {k + \frac{1}{2}} \right)x} ,\end{aligned}$$ $$\begin{aligned}
f\left( x \right) = -\varphi'' \left( x \right) &+& \varepsilon \varphi'' \left( -x \right)+ \frac{\psi_0-\varphi_0}{T} -\sum\limits_{k = 1}^\infty {\frac{{\left( {1 - \varepsilon } \right)\left( {\varphi _{2k}^{\left( 3 \right)} - \psi
_{2k}^{\left( 3 \right)} } \right)}}{{k \left( {1 - e^{ - \left( {1
- \varepsilon } \right)k^2 T} } \right) }}\cos kx} \\
&+&
\sum\limits_{k = 0}^\infty {\frac{{\left( {1 + \varepsilon }
\right)\left( {\varphi _{1k}^{\left( 3 \right)} - \psi
_{1k}^{\left( 3 \right)} } \right)}}{{\left( {k + \frac{1}{2}} \right)\left( {1 - e^{ - \left( {1+ \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T} }\right) }}\sin \left( {k +\frac{1}{2}} \right)x} ,\end{aligned}$$ where $$\varphi _{0} =
\frac{1}{2\pi}\int\limits_{ - \pi }^\pi {\varphi \left( x \right)dx}, \quad\psi _{0} =
\frac{1}{2\pi}\int\limits_{ - \pi }^\pi {\psi \left( x \right)dx},$$ $$\varphi _{1k}^{(3)} = \frac{1}{\pi}\int\limits_{ -
\pi }^\pi {\varphi''' \left( x \right)\cos \left( {k +
\frac{1}{2}} \right)x \,dx}, \quad \varphi _{2k}^{(3)} =
\frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\sin kx \, dx},$$ $$\psi _{1k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi'''\left( x \right)\cos \left( {k +\frac{1}{2}} \right)x\, dx}, \quad \psi _{2k}^{(3)} =
\frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\sin kx \,dx}.$$
\[th3\] Let $\varphi \left( x \right),\psi \left( x \right) \in C^2 \left[ { - \pi ,\pi } \right]$, $\varphi'''(x)$, $\psi'''(x) \in L_2(-\pi, \pi)$ and $\varphi ^{\left( i \right)} \left( { - \pi } \right) = \varphi ^{\left( i \right)} \left( \pi \right),$ $\psi ^{\left( i \right)}
\left( { - \pi } \right) = \psi ^{\left( i \right)} \left( \pi \right),$ $i = 0,1,2.$ Then, a unique solution to the Inverse Problem IP3 exists and it can be written in the form $$\begin{aligned}
u\left({x,t} \right) = \varphi \left( x \right) &+& \frac{t}{T}(\psi_0-\varphi_0) - \sum\limits_{k =
1}^\infty {\frac{{\left( {1 - e^{ - \left( {1 - \varepsilon }
\right)k^2 t} } \right) \left( {\varphi
_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }
\right)}}{{\left( {1 - e^{ - \left( {1 -
\varepsilon } \right)k^2 T} } \right)k^3 }} \cos kx } \\
&+& \sum\limits_{k = 1}^\infty {\frac{{\left( {1 - e^{
- \left( {1 + \varepsilon } \right)k^2 t} } \right)\left( {\varphi _{1k}^{\left( 3 \right)} - \psi
_{1k}^{\left( 3 \right)} } \right)}}{{\left( {1 - e^{ - \left( {1 + \varepsilon } \right)k^2 T} }\right)k^3 }}\sin kx},\end{aligned}$$ $$\begin{aligned}
f\left( x \right) =
-\varphi'' \left( x \right) +\varepsilon \varphi'' \left( -x \right)&+& \frac{\psi_0-\varphi_0}{T}
-\sum\limits_{k = 1}^\infty
{\frac{{\left( {1 - \varepsilon } \right)\left( {\varphi
_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }
\right)}}{{\left( {1 - e^{ - \left( {1 - \varepsilon } \right)k^2
T} } \right)k }}\cos kx} \\
&+&\sum\limits_{k = 1}^\infty
{\frac{{\left( {1 + \varepsilon } \right)\left( {\varphi
_{1k}^{\left( 3 \right)} - \psi _{1k}^{\left( 3 \right)} }
\right)}}{{\left( {1 - e^{ - \left( {1 + \varepsilon } \right)k^2
T} } \right)k }}\sin kx},\end{aligned}$$ where $$\varphi _{0} =
\frac{1}{2\pi}\int\limits_{ - \pi }^\pi {\varphi \left( x \right)dx}, \quad \psi _{0} =
\frac{1}{2\pi}\int\limits_{ - \pi }^\pi {\psi \left( x \right)dx},$$ $$\varphi _{1k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\cos kx \,dx}, \quad \varphi _{2k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\varphi''' \left( x \right)\sin kx \,dx},$$ $$\psi _{1k}^{(3)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\cos kx \,dx},\quad \psi _{2k}^{(3)} = \frac{1}{\pi}\int\limits_{ -\pi }^\pi {\psi''' \left( x \right)\sin kx\,dx} .$$
\[th4\] Let $\varphi \left( x \right),\psi \left( x \right) \in C^2 \left[ { - \pi ,\pi } \right]$, $\varphi'''(x)$, $\psi'''(x) \in L_2(-\pi, \pi)$ and $\varphi
^{\left( i \right)} \left( { - \pi } \right) = - \varphi ^{\left(
i \right)} \left( \pi \right),$ $\psi ^{\left( i \right)} \left(
{ - \pi } \right) = - \psi ^{\left( i \right)} \left( \pi
\right),$ $i = 0,1,2.$ Then, a unique solution to the Inverse Problem IP4 exists and it can be written in the form $$\begin{aligned}
u\left(
{x,t} \right) = \varphi \left( x \right) &-& \sum\limits_{k =
0}^\infty {\frac{{\left( {1 - e^{ - \left( {1 - \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2 t} } \right) \left( {\varphi
_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }
\right)}}{{\left( {1 - e^{ - \left( {1 -
\varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T} }
\right)\left( {k + \frac{1}{2}} \right)^3 }}\cos \left(
{k + \frac{1}{2}} \right)x}\\
&+& \sum\limits_{k = 0}^\infty {\frac{{\left( {1
- e^{ - \left( {1 + \varepsilon } \right)\left( {k + \frac{1}{2}}
\right)^2 t} } \right) \left( {\varphi _{1k}^{\left( 3 \right)} -
\psi _{1k}^{\left( 3 \right)} } \right)}}{{\left( {1 - e^{ - \left( {1 + \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2 T} } \right)\left( {k +
\frac{1}{2}} \right)^3 }}\sin \left( {k + \frac{1}{2}}
\right)x},\end{aligned}$$
$$\begin{aligned}
f\left( x
\right) = -\varphi'' \left( x \right) + \varepsilon \varphi'' \left( -x \right) &-& \sum\limits_{k = 0}^\infty {\frac{{\left( {1 - \varepsilon } \right)\left( {\varphi
_{2k}^{\left( 3 \right)} - \psi _{2k}^{\left( 3 \right)} }
\right)}}{{\left( {k + \frac{1}{2}} \right)\left( {1 - e^{ - \left( {1 - \varepsilon }
\right)\left( {k + \frac{1}{2}} \right)^2 T} } \right)}}\cos \left( {k + \frac{1}{2}} \right)x} \\
&+& \sum\limits_{k = 0}^\infty {\frac{{\left( {1 + \varepsilon }
\right)\left( {\varphi _{1k}^{\left( 3 \right)} - \psi
_{1k}^{\left( 3 \right)} } \right)}}{{\left( {k + \frac{1}{2}} \right)\left( {1 - e^{ - \left( {1
+ \varepsilon } \right)\left( {k + \frac{1}{2}} \right)^2 T} }
\right)}}\sin \left( {k +
\frac{1}{2}} \right)x} ,\end{aligned}$$
where $$\varphi _{1k}^{(2)} = \frac{1}{\pi}\int\limits_{ -
\pi }^\pi {\varphi''' \left( x \right)\cos \left( {k + \frac{1}{2}}
\right)x \,dx}, \quad \varphi _{2k}^{(2)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi
{\varphi''' \left( x \right)\sin \left( {k + \frac{1}{2}} \right)x\,dx},$$ $$\psi _{1k}^{(2)} = \frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x\right)\cos \left( {k + \frac{1}{2}} \right)x\,dx},\quad \psi _{2k}^{(2)} =
\frac{1}{\pi}\int\limits_{ - \pi }^\pi {\psi''' \left( x \right)\sin \left( {k +\frac{1}{2}} \right)x\,dx}.$$
Example Solution
----------------
For the sake of illustration, we present here a simple example solution for the Inverse Problem IP1. For this purpose, we consider the following choice of conditions (\[2.2\]): $$u\left({x,0} \right) = 0, \quad u\left( {x,T} \right) = \sin x, \quad x \in \left[ { - \pi ,\pi } \right],$$ i.e., we have $\varphi \left( x \right)=0$ and $\psi \left( x \right)=\sin x$. Calculating the coefficients of the series solutions as given in Theorem \[th1\], we get $$u(x,t)= \frac{{{1-e^{ - \left( {1 + \varepsilon } \right) t}} }}{{{1 - e^{ - \left( {1 + \varepsilon } \right) T} } }} \, \sin x, \quad {\text{and}} \quad f(x)= \frac{{{1 + \epsilon}} }{{{1 - e^{ - \left( {1 + \varepsilon } \right) T} } }} \, \sin x.$$ These solutions are illustrated in the following figures:
\[fig2\] {height="7cm" width="7cm"} {height="7.5cm" width="7cm"}
\[fig3\] {height="7cm" width="7cm"} {height="7cm" width="7cm"}
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|
---
abstract: 'Current analysis of astronomical data are confronted with the daunting task of modeling the awkward features of astronomical data, among which heteroscedastic (point-dependent) errors, intrinsic scatter, non-ignorable data collection (selection effects), data structure, non-uniform populations (often called Malmquist bias), non-Gaussian data, and upper/lower limits. This chapter shows, by examples, how modeling all these features using Bayesian methods. In short, one just need to formalize, using maths, the logical link between the involved quantities, how the data arize and what we already known on the quantities we want to study. The posterior probability distribution summarizes what we known on the studied quantities after the data, and we should not be afraid about their actual numerical computation, because it is left to (special) Monte Carlo programs such as JAGS. As examples, we show how to predict the mass of a new object disposing of a calibrating sample, how to constraint cosmological parameters from supernovae data and how to check if the fitted data are in tension with the adopted fitting model. Examples are given with their coding. These examples can be easily used as template for completely different analysis, on totally unrelated astronomical objects, requiring to model the same awkward data features. : Astrophysics; Cosmology; Bayesian statistics; Regression; Scaling relations; Prediction; Model testing'
author:
- |
S. Andreon,$^1$[^1]\
$^1$INAF–Osservatorio Astronomico di Brera, Milano, Italy\
bibliography:
- 'springer.bib'
title: 'Understanding better (some) astronomical data using Bayesian methods '
---
Introduction
============
Astronomical data present a number of quite common awkward features (see [@AandH11review] for a review):
- [**heteroscedastic errors**]{}: error sizes vary from point to point.
- [**non-Gaussian data**]{}: the likelihood is asymmetric and thus errors are, e.g $3.4^{+2.5}_{-1.2}$. Upper/lower limits, as 2.3 at 90 % probability, are (perhaps extreme) examples of asymmetric likelihood.
- [**non uniform populations or data structure**]{}: the number of objects per unit parameter(s) is non-uniform. This is the source of the Malmquist- or Eddington- like bias that affect most astronomical quantities, as parallaxes, star and galaxy counts, mass and luminosity, galaxy cluster velocity dispersions, supernovae luminosity corrections, etc.
- [**intrinsic scatter**]{}: data often scatter more than allowed by the errors. The extra-scatter can be due to unidentified sources of errors, often called systematic errors, or indicates an intrinsic spread of the population under study, i.e. the fact that astronomical objects are not identically equal.
- [**noisy estimates of the errors**]{}: as every measurement, errors are known with a finite degree of precision. This is even more true when one is measuring complex, and somewhat model dependent, quantities like mass.
- [**non-random sampling**]{}: in simple terms, the objects in the sample are not a random sampling of those in the Universe. In some rare occasions in astronomy, sampling is planned to be non-random on purpose, but most of the times non-random sampling is due to selection effects: harder-to-observe objects are very often missed in samples.
- [**mixtures**]{}: very often, large samples include the population under interest, but also contaminating objects. Mixtures also arise when one measure the flux of a source in presence of a background (i.e. always).
- [**prior**]{}: we often known from past data or from theory that some values of the parameters are more likely than other. In order terms, we have prior knowledge about the parameter being investigated. If we known anything, not even the order of magnitude, about a parameter, it is difficult even to choose which instrument, or sample, should be used to measure the parameter.
- [**non-linear**]{}: laws of Nature can be more complicated than $y=ax+b$.
Bayesian methods allow to deal with these features (and also other ones), even all at the same time, as we illustrate in Sec 3 and 4 with two research examples, it is just matter of stating in mathematical terms our wordy statements about the nature of the measurement and of the objects being measured. The probabilistic (Bayesian) approach returns the whole (posterior) probability distribution of the parameters, very often in form of a Monte Carlo sampling of it.
In this paper we make an attempt to be clear at the cost of being non-rigorous. We defer the reader looking for rigour to general textboox, as [@gelman2004bayesian], and, to [@AandH10] for our first research example.
Parameter estimation in Bayesian Inference
==========================================
Before adressing a research example, let’s consider an idealized applied problem to explain the basics of the Bayesian approach.
Suppose one is interested in estimating the (log) mass of a galaxy cluster, $lgM$. In advance of collecting any data, we may have certain beliefs and expectations about the values of $lgM$. In fact, these thoughts are often used in deciding which instrument will be used to gather data and how this instrument may be configured. For example, if we plan to measure the mass of a poor cluster via the virial theorem, we will select a spectroscopic set up with adequate resolution, in order to avoid that velocity errors are comparable to, or larger than, the likely low velocity dispersion of poor clusters. Crystalising these thoughts in the form of a probability distribution for $lgM$ provides the prior $p(lgM)$, on which relies the feasibility section of the telescope time proposal, where instrument, configuration and exposure time are set.
For example one may believe (e.g. from the cluster being somewhat poor) that the log of the cluster mass is probably not far from $13$, plus or minus 1; this might be modeled by saying that the (prior) probability distribution of the log mass, here denoted $lgM$ is a Gaussian centred on $13$ and with $\sigma$, the standard deviation, equal to $0.5$, i.e. $lgM \sim \mathcal{N} (13,0.5^2)$.
Once the appropriate instrument and its set up have been selected, data can be collected. In our example, this means we record a measurement of log mass, say $obslgM200$, via, for example, a virial theorem analysis, i.e. measuring distances and velocities.
The likelihood describes how the noisy observation $obslgM200$ arises given a value of $lgM$. For example, we may find that the measurement technique allows us to measure masses in an unbiased way but with a standard error of 0.1 and that the error structure is Gaussian, ie. $obslgM200 \sim \mathcal{N} (lgM,0.1^2)$, where the tilde symbol reads “is drawn from" or “is distributed as". If we observe $obslgM200=13.3$ we usually summarise the above by writing $lgM=13.3\pm 0.1$.
How do we update our beliefs about the unobserved log mass $lgM$ in light of the observed measurement, $obslgM200$? Expressing this probabilistically, what is the posterior distribution of $lgM$ given $obslgM200$, i.e. $p(lgM \ | \ obslgM200)$? Bayes Theorem ([@Bayes], [@Laplace]) tells us that $$\begin{aligned}
p(lgM \ | \ obslgM200) \propto p(obslgM200 \ |\ lgM) p(lgM)
\nonumber\end{aligned}$$
i.e. the posterior (the left hand side) is equal to the product of likelihood and prior (the right hand side) times a proportionality constant of no importance in parameter estimation.
Simple algebra shows that in our example the posterior distribution of $lgM \ | \ obslgM200$ is Gaussian, with mean $\mu=\frac{13.0/0.5^2+13.3/0.1^2}{1/0.5^2+1/0.1^2}=13.29$ and $\sigma^2=\frac{1}{1/0.5^2+1/0.1^2}=0.0096$. $\mu$ is just the usual weighted average of two “input" values, the prior and the observation, with weights given by prior and observation $\sigma$’s.
From a computational point of view, only simple examples such as the one described above can generally be tackled analytically, realistic analysis should be instead tacked numerically by special (Markov Chain) Monte Carlo methods. These are included in BUGS-like programs ([@lunn2009bugs]) such as JAGS ([@JAGS]), allowing scientists to focus on formalizing in mathematical terms our wordy statements about the quantities under investigation without worrying about the numerical implementation. In the idealized example, we just need to write in an ascii file the symbolic expression of the prior, $lgM \sim \mathcal{N} (13,0.5^2)$, and of likelihood, $obslgM200 \sim \mathcal{N} (lgM,0.1^2)$ to get the posterior in form of samplings. From the Monte Carlo sampling one may directly derive mean values, standard deviations, and confidence regions. For example, for a 90 % interval it is sufficient to peak up the interval that contain 90 % of the samplings.
First example: predicting mass from a mass proxy
================================================
Mass estimates are one of the holy grails of astronomy. Since these are observationally expensive to measure, or even unmeasurable with existing facilities, astronomers use mass proxies, far less expensive to acquire: from a(n usually small) sample of objects, the researcher measures masses, $y$ and the mass proxy, $x$. Then, he regress $x$ vs $y$ and infer $y$ for those objects having only $x$. This is the way most of the times galaxy cluster masses are estimated, for example using the X-ray luminosity, X-ray temperature, $Y_X$, $Y_{SZ}$, the cluster richness or the total optical luminosity. Here we use the cluster richness, i.e. the number of member galaxies, but with minor changes this example can be adapted for other cases.
[@AandH11review] shows that predicted $y$ using the Bayesian approach illustrated here are more precise than any other method and that the Bayesian approach does not show the large systematics of other approaches. This means, in the case of masses, that more precise masses can be derived for the same input data, i.e. at the same telescope time cost.
Step 1: put in formulae what you know
-------------------------------------
### Heteroscedasticity
Clusters have widely different richnesses, and thus widely different errors. Some clusters have better determined masses than other. Heteroscedasticity means that errors have an index $i$, because they differ from point to point.
### Contamination (mixtures), non-Gaussian data and upper limits
Galaxies in the cluster direction are both cluster members and galaxies on the line of sight (fore/background). The contamination may be estimated by observing a reference line of sight (fore/background), perhaps with a $C_i$ times larger solid angle (to improve the quality of the determination).
The mathematical translation of our words, when counts are modeled as Poisson, is:
$$\begin{aligned}
obsbkg_i \sim& \mathcal{P}(nbkg_i) &\mbox{\# Poisson with intensity
} nbkg_i \\
obstot_i \sim& \mathcal{P}(nbkg_i/C_i+n200_i) &\mbox{\# Poisson with
intensity } (nbkg_i/C_i+n200_i) \end{aligned}$$
The variables $n200_i$ and $nbkg_i$ represent the true richness and the true background galaxy counts in the studied solid angles, whereas we add a prefix “obs" to indicate the observed values.
Upper limits are automatically accounted for. Suppose, for exposing simplicity, that we observed 5 galaxies, $obstot_i=5$, in the cluster direction and that in the control field direction (with $C_i=1$ for exposing simplicity) we observe four background galaxies, $obsbkg=4$. With one net galaxy and Poisson fluctuations of a few, $n200_i$ is poorly determined at best, and the conscientious researcher would probably reports an upper limits of a few. To use the information contained in the upper limit in our regression analysis we only need to list in the data file the raw measurements ($C_i=1$, $obstot_i=5$, $obsbkg_i=4$), as for the other clusters. These data will be treated independently on whether an astronomer decides to report a measurement or an upper limit, because Nature doesn’t care about astronomer decisions.
### Non-linearity and extra-scatter
The relation between mass, $M200$, and proxy, $n200$, (richness) is usually parametrized as a power-law:
$$\begin{aligned}
M200_i &\propto& n200_i^\beta \nonumber
%\label{eqn:eqn12}\end{aligned}$$
Allowing for a Gaussian intrinsic scatter $\sigma_{scat}$ (clusters of galaxies of a given richness may not all have the very same mass) and taking the log, previous equation becomes:
$$\begin{aligned}
lgM200_i \sim & \mathcal{N}(\alpha+\beta \log(n200_i),
\sigma_{scat}^2) &\mbox{\# Gaussian scatter around}
(M200_i \propto n200_i^\beta)
\label{eqn:eqn13}\end{aligned}$$
where the intercept is $\alpha$ and the slope is $\beta$.
### Noisy errors
Once logged, mass has Gaussian errors. In formulae:
$$\begin{aligned}
obslgM200_i &\sim \mathcal{N}(lgM200_i,\sigma^2_i) &\mbox{\# Gauss errors on lg mass}
\label{eqn:eq9} \end{aligned}$$
However, errors (as everything) is measured with a finite degree of precision. We assume that the measured error, $obserrlgM200_i$, is not biased (i.e. it is not systematically larger or smaller that the true error, $\sigma_i$) but somewhat noisy. If a $\chi^2$ distribution is adopted, it satisfies both our request of unbiasness and noisiness. In formulae:
$$\begin{aligned}
obserrlgM200_i^2 &\sim \sigma^2_i \chi^2_\nu / \nu &\mbox{\# Unbiased errors}
\label{eqn:eqn10}\end{aligned}$$
where the parameter $\nu$ regulates the width of the distribution, i.e. how precise measured errors are. Since we are 95% confident that quoted errors are correct up to a factor of 2,
$$\begin{aligned}
\nu &= 6 &\mbox{\# 95 \% confident within a factor 2}\end{aligned}$$
### Prior knowledge and population structure
The data used in this investigation are of quality good enough to determine all parameters, but one, to a sufficient degree of accuracy that we should not care about priors and we can safely take weak (almost uniform) priors, zeroed for un-physical values of parameters (to avoid, for example, negative richnesses). The exception is given by the prior on the errors (i.e. $\sigma_i$), for which there is only one measurement per datum. The adopted prior (eq. 11) is supported by statistical considerations (see [@AandH10] for details). The same prior is also used for the intrinsic scatter term, although any weak prior would return the same result, because this term is well determined by the data.
$$\begin{aligned}
\alpha \sim& \mathcal{N}(0.0,10^4) & \mbox{\# Almost uniform prior on intercept}\\
\beta \sim& t_1 & \mbox{\# Uniform prior on angle}\\
n200_i \sim& \mathcal{U}(0,\infty) & \mbox{\# Uniform, but positive, cluster richness}\\
nbkg_i \sim& \mathcal{U}(0,\infty) & \mbox{\# Uniform, but positive, background rate} \\
1/\sigma_i^2 \sim& \Gamma(\epsilon,\epsilon) & \mbox{\# Weak prior on error}\\
1/\sigma_{scat}^2 \sim& \Gamma(\epsilon,\epsilon)& \mbox{\# Weak prior on intrinsic scatter}
\label{eqn:eqn8}\end{aligned}$$
Richer clusters are rarer. Therefore, the prior on the cluster richness is, for sure, not uniform, contrary to our assumption (eq. 9). Modeling the population structure is un-necessary for the data used in [@AandH10] and here, but is essential if noiser richnesses were used. Indeed, [@AandH10] shows that a previous published richness-mass calibration, which uses richnesses as low as $obsn200=3$ and neglects the $n200$ structure, shows a slope biased by five times the quoted uncertainty. Therefore, the population structure cannot be overlooked in general.
Step 2: remove TeXing, perform stochastic computations and publish
------------------------------------------------------------------
At this point, we have described, using mathematical symbols, the link between the quantities that matter for our problem, and we only need to compute the posterior probability distribution of the parameters by some sort of sampling (the readers with exquisite mathematical skills may instead attempt an analytical computation).
Just Another Gibb Sampler (JAGS[^2], ([@JAGS]) can return it at the minor cost of de-TeXing equations 1 to 12 (compare them to the JAGS code below). Poisson, Normal and Uniform distributions become [`dpois, dnorm, dunif`]{}, respectively. JAGS, following BUGS ([@lunn2009bugs]), uses precisions, $prec = 1/\sigma^2$, in place of variances $\sigma^2$. Furthermore, it uses neperian logarithms, instead of decimal ones. Eq. 5 has been rewritten using the property that the $\chi^2$ is a particular form of the Gamma distribution. Eq. 3 is split in two JAGS lines for a better reading. The arrow symbol reads “take the value of". [`obsvarlgM200`]{} is the square of $obserrlgM200$. For computational advantages, $\log(n200)$ is centred at an average value of 1.5 and $\alpha$ is centred at -14.5. Finally, we replaced infinity with a large number.
The model above, when inserted in JAGS, reads:
model
{
for (i in 1:length(obstot)) {
obsbkg[i] ~ dpois(nbkg[i]) # eq 1
obstot[i] ~ dpois(nbkg[i]/C[i]+n200[i]) # eq 2
n200[i] ~ dunif(0,3000) # eq 9
nbkg[i] ~ dunif(0,3000) # eq 10
precy[i] ~ dgamma(1.0E-5,1.0E-5) # eq 12
obslgM200[i] ~ dnorm(lgM200[i],precy[i]) # eq 4
obsvarlgM200[i] ~ dgamma(0.5*nu,0.5*nu*precy[i]) # eq 5
z[i] <- alpha+14.5+beta*(log(n200[i])/2.30258-1.5) # eq 3
lgM200[i] ~ dnorm(z[i], prec.intrscat) # eq 3
}
intrscat <- 1/sqrt(prec.intrscat)
prec.intrscat ~ dgamma(1.0E-5,1.0E-5) # eq 11
alpha ~ dnorm(0.0,1.0E-4) # eq 7
beta ~ dt(0,1,1) # eq 8
nu <-6 # eq 6
}
JAGS samples the posterior distribution of all quantity of interests, such as intercept, slope and intrinsic scatter by Gibb sampling (a sort of Monte Carlo). Form these samplings, it is straightforward to compute (posterior) mean and standard deviations (by computing the average and the standard deviation!), to plot posterior marginals (by ignoring the values of the other parameters) and confidence contours, data and mean model, etc. Therefore, our effort is over, we only need to produce nice plots and summaries of our results.
![Richness-mass scaling. The solid line marks the mean fitted regression line, while the dashed line shows this mean plus or minus the intrinsic scatter $\sigma_{scat}$. The shaded region marks the 68% highest posterior credible interval for the regression. Error bars on the data points represent observed errors for both variables. The distances between the data and the regression line is due in part to the measurement error and in part to the intrinsic scatter. From [@AandH10], reproduced with permission.[]{data-label="fig:fig1"}](fig1.ps){width="8truecm"}
{width="16truecm"}
Figure 1 shows the data used in this analysis (see [@AandH10] for details), the mean scaling (solid line) and its 68% uncertainty (shaded yellow region) and the mean intrinsic scatter (dashed lines) around the mean relation. The $\pm 1$ intrinsic scatter band is not expected to contain 68% of the data points, because of the presence of measurement errors.
Figure 2 shows the posterior marginals for the intercept, slope and intrinsic scatter $\sigma_{scat}$. These marginals are reasonably well approximated by Gaussians. The intrinsic mass scatter at a given richness, $\sigma_{scat}=\sigma_{lgM200|\log n200}$, is small, $0.19\pm0.03$. (Unless otherwise stated, results of the statistical computations are quoted in the form $x\pm y$ where $x$ is the posterior mean and $y$ is the posterior standard deviation.)
The found relation is:
$$lgM200 = (0.96\pm0.15) \ (\log n200 -1.5) +14.36\pm0.04$$
Predicting masses
-----------------
As mentioned, one of the reasons why astronomers regress a quantity $x$ against another one, $y$, is to predict the latter when a direct measurement is missing (usually because observationally expensive to acquire). It is clear that the uncertainty on the predicted $y$, called $\tilde{y}$ hereafter, should account for: a) the intrinsic scatter between $y$ and $x$ (a larger scatter implies a lower quality $\tilde{y}$ estimate); b) the uncertainty of the $x$ (the larger it is, the noisier will be the $\tilde{y}$; c) the quality of the calibration between $y$ and $x$ (better determined relations should return more precise estimates of $\tilde{y}$); and d) extrapolation errors, i.e. should penalize attempts to infer $\tilde{y}$ values corresponding to $x$ values absent from the calibrator sample (e.g. outside the range sampled by it). All these requirements are satisfied using the posterior predictive distribution,
$$p( \widetilde{y} ) = \int p( \widetilde{y} | \theta ) p(\theta| y) d \theta$$
where $\theta$ are the regression parameters (intercept, slope, intrinsic scatter). This apparently ugly expression is easy to understand: one should combine (multiply) the uncertainties of the calibrating relation, $p(\theta| y)$, to the uncertainty of predicting new data if the calibrating relation were perfectly known, $p( \widetilde{y}|\theta )$. Since we are now interested in predicted values only, we get rid of non-interesting parameters ($\theta$) by marginalization (integration). Posterior predictive distributions are so basic to be introduced at page 8 of the $>700$ pages “Bayesian Data Analysis" book ([@gelman2004bayesian]) and to be offered as a standard output of JAGS. Of course, we need to list the $x$ values (richnesses), and errors of the clusters for which we want to infer $\tilde{y}$ (predicted mass) in the data file, listing for example in the data file $obsbkg=12$, $obtot=32$, $C=5$, and mass $obslgM200=NA$ (“not available”), to indicate that this quantity should be estimated using the regression computed from the points with available masses and galaxy counts. JAGS returns $p( \widetilde{y} )$ in form of sampling and therefore, as for any other parameter, a point estimate may be obtained by taking the average and a 68 % (credible) interval can be derived by taking the interval including 68 % of the samplings. Returned values behave as expected, and indeed have large errors when masses are estimated for clusters with richnesses outside the range where the calibration has been derived ([@AandM11])
Second example: Cosmological parameters from SNIa {#sec:SNIa}
=================================================
Supernovae (SNIa) are very bright objects with very similar luminosities. The luminosity spread can be made even smaller by accounting for the correlation with colour and stretch parameter (the latter is a measurement of how slowly SNIa fade), as illustrated in Figure 3 for the sample in [@Kessler09]. These features make SNIa very useful for cosmology: they can be observed far away and the relation between flux (basically the rate of photons received) and luminosity (the rate of photons emitted) is modulated by the luminosity distance (to the square), which in turns is function of the cosmological parameters. Therefore, measuring SNIa fluxes (and redshift) allows us to put constraints on cosmological parameters. The only minor complication is that SNIa luminosities are function of their colour and stretch parameter, and these parameters have an intrinsic scatter too, which in turns has to be determined from the data at the same time as the other parameters.
[@March11] shows that the Bayesian approach delivers tighter statistical constraints on the cosmological parameters over 90 % of the times, that it reduces the statistical bias typically by a factor $\sim 2-3$ and that it has better coverage properties than the usual chi-squared approach.
In this second example we can proceed a bit faster in illustrating this non-linear regression with heteroscedastic errors, non-uniform data structure and intrinsic scatter. In this example, we also briefly discuss the prior sensitivity, i.e. how much the results are affected by the chosen prior, and we also check the quality of the model fit.
Step 1: put in formulae what you know
-------------------------------------
We observe SNIa magnitudes $obsm_i$ ($=-2.5 log(flux)+c$) with Gaussian errors $\sigma_{m,i}$, i.e.
$$\begin{aligned}
obsm_i \sim& \mathcal{N}(m_i, \sigma^2_{m,i}) \end{aligned}$$
{width="12truecm"}
{width="12truecm"}
These $m_i$ are related to the distance modulus $distmod_i$, via
$$\begin{aligned}
m_i = M+distmod_i- \alpha \, x_i + \beta \, c_i \nonumber\end{aligned}$$
with a Gaussian intrinsic scatter $\sigma_{scat}$. More precisely:
$$\begin{aligned}
m_i \sim& \mathcal{N}(M+distmod_i- \alpha \, x_i + \beta \, c_i, \sigma^2_{scat}) \end{aligned}$$
where $M$ is the (unknown) mean absolute magnitude of SNIa, and $\alpha$ and $\beta$ allow to reduce the SNIa luminosity scatter by accounting for the correlation with the stretch and colour parameters.
Similarly to [@March11], the $M$, $\alpha$, $\beta$ and log $\sigma_{scat}$ priors are taken uniform in a wide range:
$$\begin{aligned}
\log_{10} \sigma_{scat} \sim& \mathcal{U}(-3, 0) \\
\alpha \sim& \mathcal{U}(-2, 2) \\
\beta \sim& \mathcal{U}(-4, 4) \\
M \sim& \mathcal{U}(-20.3, -18.3) \end{aligned}$$
$x_i$ and $c_i$ are the true value of the stretch and colour parameters, of which we observe (the noisy) $obsx_i$ and $obsc_i$ with errors $\sigma_{x,i}$ and $\sigma_{c,i}$. In formulae:
$$\begin{aligned}
obsx_i \sim& \mathcal{N}(x_i, \sigma^2_{x,i}) \\
obsc_i \sim& \mathcal{N}(c_i, \sigma^2_{c,i}) \end{aligned}$$
The key point of the modeling is that the $obsx_i$ and $obsc_i$ values scatter more than their errors, but not immensely so, see Fig 4. The presence of a non-uniform distribution induces a Malmquist-like bias if not accounted for (e.g. large $obsx_i$ values are more likely low $x_i$ values scattered to large values than vice versa, because of the larger abundance of low $x_i$ values). Therefore, we model, as [@March11] do, the individual $x_i$ and $c_i$ as drawn from independent normal distributions centered on $xm$ and $cm$ with standard deviation $R_x$ and $R_c$ respectively. In formulae:
$$\begin{aligned}
x_i \sim& \mathcal{N}(xm, R^2_{x}) \\
c_i \sim& \mathcal{N}(cm, R^2_{c}) \end{aligned}$$
We take uniform priors for $xm$ and $sc$, and uniform priors on $\log R_{x}$ and on $\log R_{c}$, between the indicated boundaries:
$$\begin{aligned}
xm \sim& \mathcal{U}(-10, +10) \\
cm \sim& \mathcal{U}(-3, +3) \\
\log_{10} R_x \sim& \mathcal{U}(-5, +2) \\
\log_{10} R_c \sim& \mathcal{U}(-5, +2) \end{aligned}$$
That’s almost all: we need to remember the definition of distance modulus:
$$\begin{aligned}
distmod_i = 25 + 5 \log_{10} dl_i \end{aligned}$$
where the luminosity distance, $dl$ is a complicate expression, involving integrals, of the redshift $z_i$ and the cosmological parameters $\Omega_\Lambda, \Omega_M, w, H_0$ (see any recent cosmology textbook for the mathematical expression).
Redshift, in the considered sample, have heteroscedastic Gaussian errors $\sigma_{z,i}$:
$$\begin{aligned}
obsz_i \sim& \mathcal{N}(z_i, \sigma^2_{z,i}) \end{aligned}$$
The redshift prior is taken uniform
$$\begin{aligned}
z_i \sim& \mathcal{U}(0,2) \end{aligned}$$
Supernovae alone do not allow to determine all cosmological parameters, so we need external prior on them, notably on $H_0$, taken from [@Freedman01] to be
$$\begin{aligned}
H_0 \sim& \mathcal{N}(72, 8^2) \end{aligned}$$
At this point, we may decide which sets of cosmological models we want to investigate using SNIa, for example a flat universe with a possible $w \ne 0$ with the following priors:
$$\begin{aligned}
\Omega_M \sim& \mathcal{U}(0, 1) \\
\Omega_k =& 0 \\
w \sim& \mathcal{U}(-4, 0) \end{aligned}$$
or a curved Universes with $w=-1$:
$$\begin{aligned}
\Omega_M \sim& \mathcal{U}(0, 1) \\
\Omega_k \sim& \mathcal{U}(-1, 0) \\
w =& -1 \end{aligned}$$
or any other set.
Both considered cosmologies have:
$$\begin{aligned}
\Omega_k =& 1- \Omega_m - \Omega_\Lambda\end{aligned}$$
Finally, one may want to use some data. As shortly mentioned, we use the compilation of 288 SNIa in [@Kessler09].
Step 2: remove TeXing, perform stochastic computations and publish
------------------------------------------------------------------
Most of the distributions above are Normal, and the posterior distribution can be almost completely analytically computed ([@March11]). However, numerical evaluation of the stochastic part of the model on an (obsolete) laptop takes about one minute, therefore there is no need for speed up. Instead, the evaluation of the luminosity distance is CPU intensive (it takes $\approx 10^3$ more times, unless approximate analytic formulae for the luminosity distance are used), because an integral has to be evaluated a number of times equal to the number of supernovae times the number of target posterior samplings, i.e. about four millions times in our numerical computation. The JAGS implementation of the luminosity distance integral is implemented as a sum over a tightly packed grid on redshift.
As the previous example, eq 15 to 32 can be de-TeXed and used in JAGS, adding one of the two set of priors, 33-35 or 36-38, depending on which problem one is interested in.
data {
# JAGS like precisions
precmag <-1/errmag/errmag
precobsc <- 1/errobsc/errobsc
precobsx <- 1/errobsx/errobsx
precz <- 1/errz/errz
# grid for distance modulus integral evaluation
for (k in 1:1500){
grid.z[k] <- (k-0.5)/1000.
}
step.grid.z <-grid.z[2]-grid.z[1]
}
model {
for (i in 1:length(obsz)) {
obsm[i] ~ dnorm(m[i],precmag[i]) # eq 15
m[i] ~ dnorm(Mm+distmod[i]- alpha* x[i] + beta*c[i], precM) # eq 16
obsc[i] ~ dnorm(c[i], precobsc[i]) # eq 22
c[i] ~ dnorm(cm,precC) # eq 24
obsx[i] ~ dnorm(x[i], precobsx[i]) # eq 21
x[i] ~ dnorm(xm, precx) # eq 23
# distmod definition & H0 term
distmod[i] <- 25 + 5/2.3026 * log(dl[i]) -5/2.3026* log(H0/300000)
z[i] ~ dunif(0,2) # eq 31
obsz[i] ~ dnorm(z[i],precz[i]) # eq 30
######### dl computation (slow and tedious)
tmp2[i] <- sum(step(z[i]-grid.z) * (1+w) / (1+grid.z)) * step.grid.z
omegade[i] <- omegal * exp(3 * tmp2[i])
xx[i] <- sum(pow((1+grid.z)^3*omegam + omegade[i] + (1+grid.z)^2*omegak,-0.5)*
*step.grid.z * step(z[i]-grid.z))
# implementing if, to avoid diving by 0 added 1e-7 to omegak
zz[1,i] <- sin(xx[i]*sqrt(abs(omegak))) * (1+z[i])/sqrt(abs(omegak+1e-7))
zz[2,i] <- xx[i] * (1+z[i])
zz[3,i] <- (exp(xx[i]*sqrt(abs(omegak)))-exp(-xx[i]*sqrt(abs(omegak))))/2 *
*(1+z[i])/sqrt(abs(omegak+1e-7))
dl[i] <- zz[b,i]
}
b <- 1 + (omegak==0) + 2*(omegak > 0)
########## end dl computation
# JAGS uses precisions
precM <- 1/ intrscatM /intrscatM
precC <- 1/ intrscatC /intrscatC
precx <- 1/ intrscatx /intrscatx
# priors
Mm~ dunif(-20.3, -18.3) # eq 20
alpha ~ dunif(-2,2.0) # eq 18
beta ~ dunif(-4,4.0) # eq 19
cm ~ dunif(-3,3) # eq 26
xm ~ dunif(-10,10) # eq 25
# uniform prior on logged quantities
intrscatM <- pow(10,lgintrscatM) # eq 17
lgintrscatM ~ dunif(-3,0) # eq 17
intrscatx <- pow(10,lgintrscatx) # eq 27
lgintrscatx ~ dunif(-5,2) # eq 27
intrscatC <- pow(10,lgintrscatC) # eq 28
lgintrscatC ~ dunif(-5,2) # eq 28
#cosmo priors
H0 ~ dnorm(72,1/8./8.) # eq 32
omegal<-1-omegam-omegak # eq 39
# cosmo priors 1st set LCDM
#omegam~dunif(0,1) # eq 36
#omegak~dunif(-1,1) # eq 37
#w <- -1 # eq 38
# cosmo priors 2nd set: wCDM
omegam~dunif(0,1) # eq 33
omegak <-0 # eq 34
w ~ dunif(-4,0) # eq 35
}
Figure 5 shows the prior (dashed blue line) and posterior (histogram) probability distribution for the three intrinsic scatter terms present in the cosmological parameter estimation: the scatter in absolute luminosity after colour and stretch corrections, ($\sigma_{scat}$), the intrinsic scatter in the distribution of the colour and stretch terms ($R_c$ and $R_x$). This plot shows that the posterior probability at intrinsic scatters near zero is approximately zero and thus that the three intrinsic scatter terms are necessary parameters for the modeling of SNIa, and not useless complications. The three posteriors are dominated by the data, being the prior quite flat in the range where the posterior is appreciably not zero (Figure 5). Therefore, any other prior choice, as long as smooth and shallow over the shown parameter range, would have returned indistinguishable results.
Not only SNIa have luminosities that depend on colour and stretch terms, but these in turns have their own probability distribution (taken Gaussian for simplicity) with a well determined width. Figure 6 depicts the Malmquist- like bias one should incur if the spread of the distribution of colour and stretch parameters is ignored: it reports the observed values (as in Fig 4), $obsx_i$ and $obsc_i$ as well as the true values $x_i$ and $c_i$ (posterior means). The effect of equations 23 and 24 is to pull values toward the mean, and more so those with large errors, to compensate the systematic shift (Malmquist-like bias) toward larger observed values.
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Figure 7 shows the probability distribution of the two the colour and stretch slopes: $\alpha=0.12\pm0.02$ and $\beta=2.70\pm0.14$ respectively. As for the intrinsic scatter terms, the posterior is dominated by the data and therefore any other prior, smooth and shallow, would have returned indistinguishable results.
Finally, Fig 8 reports perhaps the most wanted result: contours of equal probability for the cosmological parameters $\Omega_M$ and $w$.
For one dimensional marginals, we found: $\Omega_M=0.40\pm0.10$ and $w=-1.2\pm0.2$, but with non-Gaussian probability distributions.
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Model checking
--------------
The work of the careful researcher does not end by finding the parameter set that best describe the data, (s)he also checks whether the adopted model is a good description of the data, or it is misspecified, i.e. in tension with the fitted data. In the non-Bayesian paradigm this is often achieved by computing a p-value, i.e. the probability to obtain data more discrepant than those in hand once parameters are taken at the best fit value. The Bayesian version of the concept (e.g. [@gelman2004bayesian]) acknowledges that parameters are not perfectly known, and therefore one should also explore, in addition to best fit value, other values of the parameters. Therefore, discrepancy becomes a vector, instead of a scalar, of dimension $j$, that measure the distance between the data and $j$ models, one per set of parameters considered. Of course, more probable models should occur more frequently in the list to quantify that discrepancy from an unlikely model is less detrimental than discrepancy from a likely model. In practice, if parameters are explored by sampling, it is just matter of computing the discrepancy of the data in hand for each set $j$ of parameters stored in the chain, instead of relying on one single set of parameter (say, those that maximize the likelihood). Then, one repeats the computation for fake data generated from the model, and counts how many times fake data are more extreme of real data.
For example, if we want to test the modeling of the observed spread of magnitude (i.e. equation 15 and 16), let’s define:
$$\begin{aligned}
mcor_i =& M+distmod_i- \alpha \, x_i + \beta \, c_i \end{aligned}$$
We generate fake supernovae mag:
$$\begin{aligned}
m.fake_i \sim& \mathcal{N}(mcor_i, \sigma^2_{scat}) \end{aligned}$$
and fake observed values of them,
$$\begin{aligned}
obsm.fake_i \sim& \mathcal{N}(m.fake_i, \sigma^2_{m,i}) \end{aligned}$$
Then, we adopt a modified $\chi^2$ to quantify discrepancy (or its contrary, agreement). For the real data set we have:
$$\chi^2_{real, j}= \sum_i \frac{(obsm_i-mcor_{i,j})^2}{\sigma^2_{m,i}+ E^2(\sigma_{scat})}$$
where summation is over the data and $j$ refers to the index in the sampling chain.
Apart for the $j$ index, eq. 43 is just the usual $\chi^2$, the difference between observed, $obsm_i$, and true $mcor_i$, values, weighted by the expected variance, computed as quadrature sum of errors, $\sigma_{m,i}$, and supernovae mag intrinsic scatter $\sigma_{scat}$. The $\chi^2$ of the $j^{th}$ fake data set, $\chi^2_{fake,j}$ is:
$$\chi^2_{fake,j}= \sum_i \frac{(obsm.fake_{i,j}-mcor_{i,j})^2}{\sigma^2_{m,i}+ E^2(\sigma_{scat})}$$
At this point, we only need to compute for which fraction of the simulations $\chi^2_{fake,j} > \chi^2_{real, j}$ and quote the result. If the modeling is appropriate, then the computed fraction (p-value) is not extreme (far from zero or one). If not, our statistically modeling need to be revised, because the data are in disagreement with the model.
We performed 15000 simulations[^3], each one generating 288 fake measurements of SNIa measurements. In practice, we added the following three JAGS lines:
mcor[i]<-Mm+distmod[i]- alpha* x[i] + beta*c[i] # eq 40
m.fake[i] ~ dnorm(mcor, precM) # eq 41
obsm.fake[i] ~ dnorm(m.fake[i],precmag[i]) # eq 42
and we can simplify eq 16 in
m[i] ~ dnorm(mcor[i], precM) # eq 16
We found a p-value of 45 %, i.e. that the discrepancy of the data in hand is quite typical (similar to the one of the fake data). Therefore, real data are quite common and the tested part of the model shows no evidence of misspecification. The careful researcher should then move to the other parts of the model, whose detailed exploration is left as exercise.
{width="8truecm"}
In such exploration of possible model misfits, it is very useful to visually inspect several data summaries to guide the choice of which discrepancy measure one should adopt (eq. 43 or something else?), and, if the adopted model turns out to be unsatisfactory, to guide how to revise the modeling of the the tested part of the model. For example, a possible (and common) data summary is the distribution of normalized residuals, that for $obsx_i$ reads:
$$stdresobsx_i = \frac{obsx_i - E(x_i)}{\sqrt{\sigma^2_{x,i}+ E^2(R_x)}}$$
i.e. observed minus expected value of $x_i$ divided by their expected spread (the sum in quadrature of errors and intrinsic spread). A similar summary may be built for $obsc_i$ too. To first order (at least), standardized residuals should be normal distributed with standard deviation one (by construction). Fig 9 shows the distribution of normalized residuals of both $obsx_i$ and $obsc_i$, with superposed a Gaussian centered in 0 with standard deviation equal to one (in blue). Both distributions show a possible hint of asymmetry. At this point, the careful researcher may want to use a discrepancy measure sensitive to asymmetries, as the skewness index, in addition to the $\chi^2$ during model testing. While leaving the actual computation to the reader, we emphasize that if an extreme Bayesian p-value if found (on $obsx_i$ for exposing simplicity), then one may replace its modeling (eq. 23 in the case of $obsx_i$) with a distribution allowing a non-zero asymmetry and this can be easily performed in a Bayesian approach, and easily implemented in JAGS, it is just matter of replacing the adopted Gaussian with an asymmetric distribution. If instead the data exploration gives an hint of double-bumped distribution (again on $obsx_i$ for exposing simplicity), and an extreme Bayesian p-value is found when a measure of discrepancy sensitive to double-bumped distributions is adopted, then one may adopt a mixture of Gaussians, replacing (eq. 23 in the case of $obsx_i$) with
$$\begin{aligned}
obsx_i \sim& \lambda \mathcal{N}(x_i, \sigma^2_{x,i})+ (1-\lambda)\mathcal{N}(xx_i,
\sigma^2_{xx,i}) \end{aligned}$$
Even more simple is the (hypothetical) case of possible distribution (again of $obsx_i$ for exposing simplicity) with fat tails: one may adopt a Cauchy distribution. In such case, coding in JAGS is it is just matter of replacing a `dnorm` with `dt` in the JAGS implementation. And so on.
In summary, model checking consists in updating the model until it produces data similar to those in hand. One should start by carefully and attentively inspect the data and their summaries. This inspection should suggest a discrepancy measure to be used to quantify the model misfit and, if one is found, to guide the model updating. The procedure should be iterated until the model fully reproduces the data.
Summary and conclusions {#sec:finalsec}
=======================
These two analysis offer a template for modeling the common awkward features of astronomical data, namely heteroscedastic errors, non-Gaussian likelihood (inclusive of upper/lower limits), non-uniform populations or data structure, intrinsic scatter (either due to unidentified source of errors, or due to population spreads), noisy estimates of the errors, mixtures and prior knowledge.
In a Bayesian framework, learning from the data and the prior it is just matter of formalizing in mathematical terms our wordy statements about the quantities under investigation and how the data arize. The actual numerical computation of the posterior probability distribution of the parameters is left to (special) Monte Carlo programs, of which we don’t need to be afraid more than numerical methods (Monte Carlo) used to compute the integral of a function.
The great advantage of the Bayesian modeling is its high flexibility: if the data (or theory) call for a more complex modeling, or call for using distributions different from those initially taken, it is just matter of replacing them in the model, because there are no simplifications forced by the need of reaching the finishing line, as instead is the case in other modeling approaches. Furthermore, if one is interested in constraining another cosmological model, for example one with a redshift-dependent dark energy equation of state $w = w_0+w_1(1+z)/(1+z_p)$, one should just replace $w$ in the JAGS code with the above equation. This flexible, hierarchical, modeling is native with Bayesian methods.
In both our two examples, the Bayesian approach performs, unsurprisingly, better than than non-Bayesian methods obliged to discard part of the available information in order to reach the finishing line: Bayesian methods just use all the provided information.
As a final note, we remember that the careful researcher, whether using a Bayesian modeling or not, before publishing his own result should check that the numerical computation is adequate for his purpose and that the model is appropriate for the data. Therefore, if you, gentle reader, are using our examples as template, remember to include in your work a sensitivity analysis, by checking that your assumptions (both likelihood and prior) are reasonable. Some prior sensitivity analysis has been performed in both examples to emphasize the importance of showing how much conclusions (the posterior) rely on assumptions (prior). For what concerns the likelihood, i.e. misfitting, the key point consists in updating the model fomulation until it produces data similar to those in hand, as we have shown in great detail for the second example.
More in general, we hope that the template modeling shown in these two examples may be useful for any analysis confronted with modeling the awkward features of astronomical data, among which heteroscedastic (point-dependent) errors, intrinsic scatter, data structure, non-uniform population (often called Malmquist bias) and non-Gaussian data, inclusive of upper/lower limits.
The first part of this chapter is largely based on papers written in collaboration with Merrilee Hurn. It is a pleasure to thank Merrilee for the fruitful collaboration and her wise suggestions along the years, and for comments on an early draft of this chapter. Errors and inconsistencies remain my own.
[^1]: [email protected]
[^2]: http://calvin.iarc.fr/$\sim$martyn/software/jags/
[^3]: Skilled readers may note that we are dealing, by large, with Gaussian distributions, and may attempt an analytic computation.
|
---
abstract: 'In this paper we translate the necessary and sufficient conditions of Tanaka’s theorem on the finiteness of effective prolongations of a fundamental graded Lie algebras into computationally effective criteria, involving the rank of some matrices that can be explicitly constructed. Our results would apply to geometries, which are defined by assigning a structure algebra on the contact distribution.'
address:
- |
Stefano Marini: Dipartimento di Scienze Matematiche, Fisiche e Informatiche\
Unità di Matematica e Informatica\
Università degli Studi di Parma\
Parco Area delle Scienze 53/A (Campus), 43124 Parma (Italy)
- |
Costantino Medori: Dipartimento di Scienze Matematiche, Fisiche e Informatiche\
Unità di Matematica e Informatica\
Università degli Studi di Parma\
Parco Area delle Scienze 53/A (Campus), 43124 Parma (Italy)
- |
Mauro Nacinovich: Dipartimento di Matematica\
II Università degli Studi di Roma “Tor Vergata”\
Via della Ricerca Scientifica\
00133 Roma (Italy)
author:
- 'Stefano Marini, Costantino Medori, Mauro Nacinovich'
title: ' $\operatorname{\mathfrak{L}}$-prolongations of graded Lie algebras'
---
Introduction {#introduction .unnumbered}
============
The concept of $\operatorname{\mathbf{G}}$-structure was introduced to treat various interesting differential geometrical structures in a unified manner (see e.g. [@kn1964a; @kn1964; @kn1965; @kn1965a; @kn1965b; @kn1966; @Kob]). At a chosen point ${{\mathrm{p}}}_0$ of a manifold $M,$ a $\operatorname{\mathbf{G}}$-strucure can be described by the datum of a Lie algebra $\operatorname{\mathfrak{L}}\,{=}\,{Lie(\operatorname{\mathbf{G}})}$ of infinitesimal transformations, acting as linear maps on the tangent space $V{=}T_{{{\mathrm{p}}}_0}M.$ It is convenient to envision $V{\oplus}\operatorname{\mathfrak{L}}$ as an Abelian *extension* of $\operatorname{\mathfrak{L}}$ and to look for its maximal effective prolongation to read the differential invariants of the structure. When this turns out to be finite dimensional, Cartan’s method can be used to study the automorphism group and the equivalence problem for the corresponding $\operatorname{\mathbf{G}}$-structure. This *algebraic* point of view was taken up systematically in [@GS; @Sternberg], and pursued in [@Tan67; @Tan70; @AMN06; @Warhurst2007; @Ottazzi2010; @Kruglikov2011; @Ottazzi2011a; @Ottazzi2011; @alek] to study generalised contact and $CR$ structures: the datum of a smooth distribution defines on $T_{{{\mathrm{p}}}_0}M$ a structure of $\operatorname{\mathbb{Z}}$-graded nilpotent Lie algebra ${\textswab{m}}{=}{\sum}_{{\mathpzc{p}}=1}^\muup{\mathfrak{g}}_{{-}{\mathpzc{p}}}$; moreover, the *structure group* yields a Lie algebra ${\mathfrak{g}}_{0}$ of $0$-degree derivations of ${\textswab{m}},$ that encodes the geometry of $M$. The condition that the distribution be completely non integrable, or satisfies *Hörmander’s condition* at ${{\mathrm{p}}}_0,$ translates into the fact that ${\textswab{m}}$ is *fundamental*, i.e. that the part ${\mathfrak{g}}_{-1},$ tangent at ${{\mathrm{p}}}_0$ to the contact distribution, generates ${\textswab{m}}.$ In this case the action of the structure algebra on ${\textswab{m}}$ is uniquely determined by its action on ${\mathfrak{g}}_{-1}.$ Thus we prefer to consider general Lie subalgebras $\operatorname{\mathfrak{L}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}_{-1})$ and look for maximal $\operatorname{\mathfrak{L}}$-prolongations of ${\textswab{m}},$ i.e. prolongations with ${\mathfrak{g}}_{0}\,{\subseteq}\,\operatorname{\mathfrak{L}}.$
Having recently proved in [@NMSM] a finiteness result for the automorphism group of a class of homogeneous $CR$ manifolds by applying a result of N.Tanaka to a suitably filtered object, we got interested in the general preliminary problem of the finiteness of the effective $\operatorname{\mathfrak{L}}$-prolongations of general fundamental $\operatorname{\mathbb{Z}}$-graded Lie algebras, starting from rereading Tanaka’s seminal paper [@Tan70].
Our main results show that, given a subalgebra $\operatorname{\mathfrak{L}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}_{-1}),$ the finiteness of the maximal $\operatorname{\mathfrak{L}}$-prolongation is equivalent to a rank condition on some matrices which can be explicitly constructed.
Finiteness of Tanaka’s prolongations was already discussed in several different special contexts (see e.g. [@Ottazzi2011] and references therein). Here we tried to take a quite general perspective and gave effective methods toward computationally dealing with this problem.
Let us describe the contents of this work. Being essentially algebraic, all results are formulated for an arbitrary ground field ${\mathbb{K}}$ of characteristic zero.
In §\[sec-fund\] we describe fundamental graded Lie algebras as quotients of the free Lie algebra $\operatorname{\mathfrak{f}}(V)$ generated by a vector space $V$, by its graded ideals ${\mathpzc{K}}.$ After getting from [@Bou68] the maximal effective prolongation ${\mathfrak{F}}(V)$ of $\operatorname{\mathfrak{f}}(V),$ we use this result to characterise, in Theorem \[thm-tan-1-8\], the maximal effective $\operatorname{\mathfrak{L}}$-prolongation of ${\textswab{m}}$ in terms of ${\mathfrak{F}}(V)$ and ${\mathpzc{K}}.$
In §\[sect2\] we review the finiteness theorem proved by N.Tanaka in [@Tan70]. Here we make explicit the role of duality, which allows to reduce to commutative algebra. This was hinted in Serre’s appendix to [@GS] and also in [@Tan70], but, in our opinion, in a way which left part of the arguments rather obscure. We hope that our exposition would make this important theorem more understandable. Theorem \[thm-t-4-4\] reduces the finiteness of the maximal $\operatorname{\mathfrak{L}}$-prolongation to the analogous problem for an $\operatorname{\mathfrak{L}}'$[-]{}prolongation of an Abelian Lie algebra. As an application, we provide a comparison result for prolongations with different $\operatorname{\mathfrak{L}}$’s and ${\mathpzc{K}}$’s. In the following sections we provide a criterion for studying prolongations of Abelian Lie algebras and to obtain an explicit description of $\operatorname{\mathfrak{L}}'$ after knowing $\operatorname{\mathfrak{L}}$ and ${\textswab{m}}.$
In §\[sec5a\] we get an effective crierion for the finiteness of $\operatorname{\mathfrak{L}}$-prolongations of fundamental graded Lie algebras of the first kind, using duality to reduce the question to the study the co-primary decomposition of finitely generated modules over the polynomials: this boils down to computing the rank of a matrix $M_1(\operatorname{\mathfrak{L}},{\mathpzc{z}})$ of first degree homogeneous polynomials associated to $\operatorname{\mathfrak{L}}$ (see Theorem \[thm-t-4-6\]). In this way, a necessary and sufficient condition for finite prolongation can be formulated as an ellipticity condition (in the sense of Kobayashi, cf. [@Kob p.4]) for ${\mathbb{F}}\,{\otimes}\,\operatorname{\mathfrak{L}}',$ where $\operatorname{\mathfrak{L}}'$ is the *reduced* structure algebra $\operatorname{\mathfrak{L}}'.$ We illustrate this procedure by studying the classical examples of the $\operatorname{\mathbf{G}}$-structures treated, e.g. in [@Kob; @Sternberg]. In [@Ottazzi2011] the authors obtained this rank condition for contact structures, when $\operatorname{\mathfrak{L}}$ is the full linear group of ${\mathfrak{g}}_{-1}.$
When ${\textswab{m}}$ has kind $\muup{>}1,$ the effect of the terms ${\mathfrak{g}}_{-{\mathpzc{p}}}$ with ${\mathpzc{p}}{\geq}2$ is of *restricting the structure algebra $\operatorname{\mathfrak{L}}$ to a smaller algebra $\operatorname{\mathfrak{L}}'\! \! .$* The final criterion is obtained by applying Theorem \[thm-t-4-6\] to $\operatorname{\mathfrak{L}}'\! \! .$ In fact, there is a difference in the way the ${\mathfrak{g}}_{-{\mathpzc{p}}}$ summands contribute to $\operatorname{\mathfrak{L}}'$ between the cases ${\mathpzc{p}}{=}2$ and ${\mathpzc{p}}{>}2.$
In §\[sec5\] we study the maximal ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongation in the case $\muup{=}2.$ We show in Theorem \[thm-tan-4-8\] that the condition can be expressed in terms of the rank of the Lie product, considered as an alternate bilinear form on ${\mathbb{F}}{\otimes}{\mathfrak{g}}_{-1},$ for the algebraic closure ${\mathbb{F}}$ of the ground field ${\mathbb{K}}.$ We show by an example that this rank condition, that was known to be necessary over ${\mathbb{K}}$ (see e.g. [@Kob]), becomes necessary *and* sufficient when stated over ${\mathbb{F}}.$ As in §\[sec5a\], there is an equivalent formulation in terms of the rank a suitable matrix $M_2({\mathpzc{K}},{\mathpzc{z}}).$
In §\[sect7\] we show that one can take into account the non zero summands ${\mathfrak{g}}_{-{\mathpzc{p}}}$ in ${\textswab{m}},$ with ${\mathpzc{p}}\geq{3},$ by adding to $M_2({\mathpzc{K}},{\mathpzc{z}})$ a matrix $M_3({\mathpzc{K}},{\mathpzc{z}}),$ which has rank $n{=}\dim{\mathfrak{g}}_{{-}1}$ when ${\mathpzc{z}}$ does not belong to a subspace $W({\mathpzc{K}})$ of ${\mathbb{F}}{\otimes}V.$ Thus the finite dimensionality criterion of §\[sec5\] has to be checked on a smaller subspace of ${\mathpzc{z}}$’s.
In §\[sect8\], Theorem \[thm-tan-6.1\] collects the partial results of the previous sections, to state a finiteness criterion for the maximal $\operatorname{\mathfrak{L}}$-prolongation of an ${\textswab{m}}$ of any finite kind $\muup$ in terms of the rank of the matrix $(M_1(\operatorname{\mathfrak{L}},{\mathpzc{z}}),M_2({\mathpzc{K}},{\mathpzc{z}}),M_3({\mathpzc{K}},{\mathpzc{z}}))$ which is obtained by putting together the contributions coming from $\operatorname{\mathfrak{L}}$ (kind one), ${\mathfrak{g}}_{-2}$ (kind two) and ${\sum}_{{\mathpzc{p}}\leq{-3}}{\mathfrak{g}}_{{\mathpzc{p}}}$ (kind ${>}2$).
### Notation {#notation .unnumbered}
- We shall indicate by ${\mathbb{K}}$ a field of characteristic zero and by ${\mathbb{F}}$ its algebraic closure.
- The acronym FGLA stands for *fundamental graded Lie algebra*.
- The acronym EPFGLA stands for *effective prolongation of a fundamental graded Lie algebra*.
- ${\mathrm{T}}(V)={\sum}_{{\mathpzc{p}}=0}^\infty{\mathrm{T}}^{{\mathpzc{p}}}(V)$ is the tensor algebra of the vector space $V.$
- $\Lambda(V)={\sum}_{{\mathpzc{p}}=0}^n\Lambda^{{\mathpzc{p}}}(V)$ is the Grassmann algebra of an $n$-dimensional vector space $V.$
- ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ is the Lie algebra of ${\mathbb{K}}$-linear endomorphisms of the ${\mathbb{K}}$-linear space $V.$
- ${\mathfrak{gl}}_n({\mathbb{K}})$ is the Lie algebra of $n{\times}n$ matrices with entries in the field ${\mathbb{K}}.$
Fundamental graded Lie algebras and prolongations {#sec-fund}
=================================================
A $\operatorname{\mathbb{Z}}$-graded Lie ${\mathbb{K}}$-algebra $$\label{eq-ta-1-1} \tag{FGLA}
{\textswab{m}}={\sum}_{{\mathpzc{p}}\geq{1}}{\mathfrak{g}}_{-{\mathpzc{p}}},$$ is called *fundamental* if ${\mathfrak{g}}_{-1}$ is finite dimensional and generates ${\textswab{m}}$ as a Lie ${\mathbb{K}}$-algebra. We call $\muup{=}\sup\{{\mathpzc{p}}\mid{\mathfrak{g}}_{{-}{\mathpzc{p}}}{\neq}\{0\}\}$ the *kind* of ${\textswab{m}}.$
At variance with [@Tan70], we do not require here that ${\textswab{m}}$ be finite dimensional. The only FGLA with $\dim({\mathfrak{g}}_{-1}){=}1$ is the trivial one-dimensional Lie algebra. We will consider in the following FGLA’s ${\textswab{m}}$ with $\dim({\textswab{m}}){>}1.$
Let $V$ be a vector space of finite dimension $n{\geq}2$ over ${\mathbb{K}}$ and $\operatorname{\mathfrak{f}}(V)$ the free Lie ${\mathbb{K}}$-algebra generated by $V$ (see e.g. [@n1998lie; @Reu1993]). It is an FGLA of infinite kind with the natural $$\operatorname{\mathfrak{f}}(V)={\sum}_{{\mathpzc{p}}=1}^\infty \operatorname{\mathfrak{f}}_{-{\mathpzc{p}}}(V).$$ obtained by setting $\operatorname{\mathfrak{f}}_{-1}(V){=}V.$ It is characterised by the universal property:
\[prop-ta-1-1\] To every Lie ${\mathbb{K}}$-algebra $\operatorname{\mathfrak{L}}$ and to every ${\mathbb{K}}$-linear map $\phiup{:}V{\to}\operatorname{\mathfrak{L}},$ there is a unique Lie algebras homomorphism $\tilde{\phiup}:\operatorname{\mathfrak{f}}(V){\to}\operatorname{\mathfrak{L}}$ extending $\phiup.$
By Proposition \[prop-ta-1-1\] we can consider any finitely generated Lie algebra as a quotient of a free Lie algebra.
For any integer $\muup\geq{2},$ the direct sum $$\operatorname{\mathfrak{f}}_{[\muup]}(V)={\sum}_{{\mathpzc{p}}{\geq}\muup}\operatorname{\mathfrak{f}}_{-{\mathpzc{p}}}(V)$$ is a proper $\operatorname{\mathbb{Z}}$-graded ideal in $\operatorname{\mathfrak{f}}(V)$ and hence the quotient $$\operatorname{\mathfrak{f}}_{(\muup)}(V)=\operatorname{\mathfrak{f}}(V){/}\operatorname{\mathfrak{f}}_{[\muup{+}1]}(V)$$ is an FGLA of kind $\muup,$ that is called the *free* FGLA *of kind $\muup$* generated by $V$ (see e.g. [@Warhurst2007]).
Let ${\textswab{m}}$ be an FGLA over ${\mathbb{K}}$ and set $V={\mathfrak{g}}_{-1}.$ Since ${\textswab{m}}$ is generated by $V,$ there is a surjective homomorphism $$\piup:\operatorname{\mathfrak{f}}(V) \longrightarrow\!\!\!\!\rightarrow {\textswab{m}},$$ which preserves the gradations. Its kernel ${\mathpzc{K}}$ is a $\operatorname{\mathbb{Z}}$-graded ideal $$\label{t-6-10eq}
{\mathpzc{K}}={\sum}_{{\mathpzc{p}}{\geq}2}{\mathpzc{K}}_{\;-{\mathpzc{p}}}$$ and ${\textswab{m}}$ has finite kind $\muup$ iff there is a smaller integer $\muup{\geq}1$ such that $${\mathpzc{K}}\supseteq\operatorname{\mathfrak{f}}_{[\muup{+}1]}(V).$$
An ideal ${\mathpzc{K}}$ of $\operatorname{\mathfrak{f}}(V)$ is said to be *$\muup$-cofinite* if $\operatorname{\mathfrak{f}}_{[\muup{+}1]}(V)\subseteq{\mathpzc{K}}.$
The elements of a string $({\mathpzc{K}}_{\;-2},\hdots,{\mathpzc{K}}_{\;-\muup},\hdots)$ of linear subspaces ${\mathpzc{K}}_{\;-{\mathpzc{p}}}{\subseteq}\operatorname{\mathfrak{f}}_{-\operatorname{\mathfrak{p}}}(V)$ are the homogeneous summands of a $\operatorname{\mathbb{Z}}$-graded ideal of $\operatorname{\mathfrak{f}}(V)$ if and only if they satisfy the compatibility condition $$\vspace{-20pt}
[{\mathpzc{K}}_{\; -{\mathpzc{p}}},V]\subseteq{\mathpzc{K}}_{\;-({\mathpzc{p}}{+}1)},\;\;\forall {\mathpzc{p}}{\geq}2.$$
Since ${\mathbb{K}}$ has characteristic $0,$ we can canonically identify $\operatorname{\mathfrak{f}}(V)$ with a ${\mathbf{GL}}_{{\mathbb{K}}}(V)$-invariant subspace of $T(V).$
\[prop-tan-1.4\] Every FGLA ${\textswab{m}}$ is isomorphic to a quotient $\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}},$ for a $\operatorname{\mathbb{Z}}$[-]{}graded ideal ${\mathpzc{K}}$ of the form . Two FGLA’s $\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}}$ and $\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}}'$ are isomorphic if and only if ${\mathpzc{K}}$ and ${\mathpzc{K}}'$ are ${\mathbf{GL}}_{{\mathbb{K}}}(V)$-congruent.
A $\operatorname{\mathbb{Z}}$-graded Lie algebra over ${\mathbb{K}}$ $$\label{eq-ta-1-6}
{\mathfrak{g}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}}.$$ is said to be an *effective prolongation of a fundamental graded Lie algebra* if
- ${\mathfrak{g}}_{{<}0}={\sum}_{{\mathpzc{p}}{<}0}{\mathfrak{g}}_{{\mathpzc{p}}}$ is a FGLA;
- ${\mathfrak{g}}_{{\geq}0}={\sum}_{{\mathpzc{p}}{\geq}0}{\mathfrak{g}}_{{\mathpzc{p}}}$ is *effective*: this means that $$\{X\in{\mathfrak{g}}_{{\geq}0}\mid [X,{\mathfrak{g}}_{-1}]{=}\{0\}\}=\{0\}.$$
In this case, we say for short that ${\mathfrak{g}}$ is an EPFGLA of ${\mathfrak{g}}_{{<}0}.$
Let us fix a Lie subalgebra $\operatorname{\mathfrak{L}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}_{-1}),$ that we will call the *structure algebra*. An EPFGLA is said to be *of type $\operatorname{\mathfrak{L}}$*, or an $\operatorname{\mathfrak{L}}$-prolongation, if for each $A\in{\mathfrak{g}}_0$ the map ${\mathpzc{v}}\to[A,{\mathpzc{v}}],$ for ${\mathpzc{v}}\in{\mathfrak{g}}_{-1},$ is an element of $\operatorname{\mathfrak{L}}$.
\[lem-ta-1-5\] If is an EPFGLA, then $\dim({\mathfrak{g}}_{{\mathpzc{p}}}){<}\infty$ for all ${\mathpzc{p}}\in\operatorname{\mathbb{Z}}.$
Let ${\mathfrak{g}}={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}}$ be an EPFGLA and set $V={\mathfrak{g}}_{-1}.$ For each $\xiup{\in}{\mathfrak{g}},$ we define by recurrence $$\begin{cases}
\xiup({\mathpzc{v}}_0)=[\xiup,{\mathpzc{v}}_0], & \forall{\mathpzc{v}}_0\in{V},\\
\xiup({\mathpzc{v}}_0,{\mathpzc{v}}_1)=[[\xiup,{\mathpzc{v}}_0],{\mathpzc{v}}_1], &\forall{\mathpzc{v}}_0,{\mathpzc{v}}_1\in{V},\\
\xiup({\mathpzc{v}}_0,\hdots,{\mathpzc{v}}_{\mathpzc{p}})=[\xiup({\mathpzc{v}}_0,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}{-}1}),{\mathpzc{v}}_{{\mathpzc{p}}}], &\forall{\mathpzc{v}}_0,{\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}}\in{V}.
\end{cases}$$ By the effectiveness assumption, for each ${\mathpzc{p}}{\geq}0$ the map which associates to $\xiup{\in}{\mathfrak{g}}_{{\mathpzc{p}}}$ the $({\mathpzc{p}}{+}1)$ multilinear map $V^{{\mathpzc{p}}{+}1}\ni ({\mathpzc{v}}_0,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}})\to\xiup({\mathpzc{v}}_0,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}})\in{V}$ is injective from ${\mathfrak{g}}_{{\mathpzc{p}}}$ to $V{\otimes}{\mathrm{T}}^{{\mathpzc{p}}{+}1}(V^*).$
We already noted that, since ${\mathbb{K}}$ has characteristic zero, $\operatorname{\mathfrak{f}}(V)$ can be canonically identified with a Lie subalgebra of the Lie algebra of the tensor algebra ${\mathrm{T}}(V)$; ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ acts as an algebra of zero degree derivations on ${\mathrm{T}}(V),$ leaving $\operatorname{\mathfrak{f}}(V)$ invariant. Denote by $T_{\!{A}}$ the derivation of ${\mathrm{T}}(V)$ associated to $A\in{\mathfrak{gl}}_{{\mathbb{K}}}(V).$ With the Lie product defined by $$[A,X]=T_{\!{A}}(X),\;\;\forall A\in{\mathfrak{gl}}_{{\mathbb{K}}}(V),\;\forall X\in\operatorname{\mathfrak{f}}(V),$$ the direct sum $$\tilde{\operatorname{\mathfrak{f}}}(V)={\sum}_{{\mathpzc{p}}\geq{0}}\operatorname{\mathfrak{f}}_{-{\mathpzc{p}}}(V),\;\;\text{with}\;\; \operatorname{\mathfrak{f}}_0(V)={\mathfrak{gl}}_{{\mathbb{K}}}(V)$$ is an EPFGLA of $\operatorname{\mathfrak{f}}(V)$ and, for any Lie subalgebra ${\mathfrak{L}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}(V),$ the sum ${\mathfrak{L}}\oplus\operatorname{\mathfrak{f}}(V)$ is a graded Lie subalgebra of $\tilde{\operatorname{\mathfrak{f}}}(V).$
By using [@n1998lie Ch.2,§[2]{},Prop.8], we obtain
\[prop1.6\] The maximal effective prolongation ${\mathfrak{F}}(V)$ of $\operatorname{\mathfrak{f}}(V)$ is $${\mathfrak{F}}(V)={\sum}_{{\mathpzc{p}}=-\infty}^\infty \operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V),
\;\;\text{where}\;\;
\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V)={V}\otimes{\mathrm{T}}^{{\mathpzc{p}}{+}1}(V^*), \;\;\text{for}\;\; {\mathpzc{p}}{\geq}0,$$ is the space of $({\mathpzc{p}}{+}1)$-linear $V$-valued maps on $V$ and the Lie product is defined in such a way that, for $\xiup\in\operatorname{\mathfrak{f}}_{p}(V),$ with ${\mathpzc{p}}{\geq}{0},$ we have $$\label{eq1.11}
[\xiup,{\mathpzc{v}}_0]({\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}})=
\xiup({\mathpzc{v}}_0,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}}),\;\;\forall{\mathpzc{v}}_0,{\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}}\in{V}.$$
If ${\mathpzc{M}}$ is a right $\operatorname{\mathfrak{f}}(V)$-module, then we can define on ${\mathpzc{M}}{\oplus}\operatorname{\mathfrak{f}}(V)$ a Lie ${\mathbb{K}}$-algebra structure by setting $$\label{eq1star}\tag{$*$}
[(\mu,X),(\nu,Y)]=(\mu{\cdot}Y-\nu{\cdot}X,[X,Y]),\;\;\forall \mu,\nu\in{\mathpzc{M}},\;\;\forall{X},Y\in\operatorname{\mathfrak{f}}(V).$$ Let $\xiup{:}V{\to}{\mathpzc{M}}$ be any ${\mathbb{K}}$-linear map. By the universal property the ${\mathbb{K}}$-linear map $V{\ni}{\mathpzc{v}}{\to}(\xiup({\mathpzc{v}}),{\mathpzc{v}}){\in}{\mathpzc{M}}{\oplus}\operatorname{\mathfrak{f}}(V)$ extends to a Lie ${\mathbb{K}}$-algebras homomorphism $$\operatorname{\mathfrak{f}}(V)\ni X\to (D_{\xiup}(X),X)\in{\mathpzc{M}}{\oplus}\operatorname{\mathfrak{f}}(V)$$ between $\operatorname{\mathfrak{f}}(V)$ and ${\mathpzc{M}}{\oplus}\operatorname{\mathfrak{f}}(V).$ Because of , we have $D_{\xiup}{\in}{\mathpzc{Der}}(\operatorname{\mathfrak{f}}(V),{\mathpzc{M}}).$
For ${\mathpzc{p}}{\in}\operatorname{\mathbb{Z}},$ we define recursively finite dimensional ${\mathbb{K}}$-vector spaces ${\mathpzc{Der}}_{{\mathpzc{p}}}(V)$ and right $\operatorname{\mathbb{Z}}$-graded $\operatorname{\mathfrak{f}}(V)$-modules ${\mathpzc{Der}}_{({\mathpzc{p}})}(V)={\sum}_{{\mathpzc{q}}{\leq}{\mathpzc{p}}}{\mathpzc{Der}}_{{\mathpzc{q}}}(V)$ by setting $$\begin{cases}
{\mathpzc{Der}}_{{\mathpzc{p}}}(V)=\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V),\; {\mathpzc{Der}}_{({\mathpzc{p}})}(V)={\sum}_{{\mathpzc{q}}{\leq}{\mathpzc{p}}}\operatorname{\mathfrak{f}}_{{\mathpzc{q}}}(V), &\text{for ${\mathpzc{p}}{<}0,$}\\[6pt]
\left.
\begin{aligned}
&
{\mathpzc{Der}}_{{\mathpzc{p}}}(V)=\{D{\in}{\mathpzc{Der}}(\operatorname{\mathfrak{f}}(V),{\mathpzc{Der}}_{({\mathpzc{p}}{-}1)}(V)){\mid} \, D({\mathpzc{v}}){\in}{\mathpzc{Der}}_{{\mathpzc{p}}{-}1}(V),\;\forall{\mathpzc{v}}{\in}V\},
\\
&
{\mathpzc{Der}}_{({\mathpzc{p}})}(V)= {\mathpzc{Der}}_{({\mathpzc{p}}{-}1)}(V)\oplus{\mathpzc{Der}}_{{\mathpzc{p}}}(V),
\end{aligned}\right\}
&\text{for ${\mathpzc{p}}{\geq}0,$}
\end{cases}$$ By the general argument at the beginning, ${\mathpzc{Der}}_{{\mathpzc{p}}}(V){\simeq}V{\otimes}T^{{{\mathpzc{p}}}{+}1}(V^*)$ for ${\mathpzc{p}}{\geq}{0}$ and holds.
The last step is setting ${\mathpzc{Der}}_{{\mathpzc{p}}}(V){=}\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V)$ and extending the Lie ${\mathbb{K}}$-algebra structure of $\operatorname{\mathfrak{f}}(V)$ to ${\mathfrak{F}}(V).$ In doing that, we can restrain to homogeneous terms and argue by recurrence on the sums of their degrees. Let $\xiup_{i}$ denote an element of $\operatorname{\mathfrak{f}}_{{\mathpzc{p}}_i}(V)$ and consider a product $[\xiup_{1},\xiup_{2}].$ It is well defined when $\min\{{\mathpzc{p}}_1,{\mathpzc{p}}_2\}{<}0.$ If we assume it has been already defined for ${\mathpzc{p}}_1{+}{\mathpzc{p}}_2{\leq}{\mathrm{k}}$ for some ${\mathrm{k}}{\geq}0,$ we note that $[\xiup_{i},{\mathpzc{v}}]{\in}\operatorname{\mathfrak{f}}_{{\mathpzc{p}}_i{-}1}$ and hence $$\label{eq1sstar}\tag{$**$}
[ [\xiup_{1},\xiup_{2}],{\mathpzc{v}}]=[ [\xiup_{1},{\mathpzc{v}}],\xiup_{2}]
+[ \xiup_{1},[\xiup_{2},{\mathpzc{v}}]]$$ is a well defined ${\mathbb{K}}$-linear map between ${\mathpzc{v}}{\in}V$ and ${\mathpzc{Der}}_{{\mathpzc{p}}_1{+}{\mathpzc{p}}_2{-}1}(V),$ defining an element $[\xiup_{{\mathpzc{p}}_1},\xiup_{{\mathpzc{p}}_2}]$ of ${\mathpzc{Der}}_{{\mathpzc{p}}_1{+}{\mathpzc{p}}_2}(V).$ This yields a product on ${\mathfrak{F}}(V).$ To show that it satisfies the Jacobi identity $$\label{eq1ssstar} \tag{$***$}
[[\xiup_{1},\xiup_{2}],\xiup_{3}]+ [[\xiup_{2},\xiup_{3}],\xiup_{1}]
+ [[\xiup_{3},\xiup_{1}],\xiup_{2}]=0,$$ we note that follows from when $\min\{{\mathpzc{p}}_1,{\mathpzc{p}}_2,{\mathpzc{p}}_3\}{<}0$ and can be proved by recurrence on ${\mathrm{k}}{=}{\mathpzc{p}}_1{+}{\mathpzc{p}}_2{+}{\mathpzc{p}}_3$ when $\min\{{\mathpzc{p}}_1,{\mathpzc{p}}_2,{\mathpzc{p}}_3\}{\geq}0,$ because in this case an element $\xiup$ of $\operatorname{\mathfrak{f}}_{{\mathrm{k}}}(V)$ is zero iff $[\xiup,{\mathpzc{v}}]{=}0$ for all ${\mathpzc{v}}{\in}V.$
Let $\operatorname{\mathfrak{L}}$ be a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ Then $$\label{eq-ta-1-12b} \left\{\begin{aligned}
{\mathfrak{F}}(V,\operatorname{\mathfrak{L}})&=\operatorname{\mathfrak{f}}(V)\oplus\operatorname{\mathfrak{L}}\oplus{\sum}_{{\mathpzc{p}}{\geq}1}\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V,\operatorname{\mathfrak{L}}),\;\;\text{with}\;\; \\
& \operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V,\operatorname{\mathfrak{L}})=
\{\xiup\in\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V)\mid \xiup({\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}})\in\operatorname{\mathfrak{L}},\;\forall {\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}}
\in{V}\}\end{aligned}\right.$$ is the maximal EPFGLA prolongation of type $\operatorname{\mathfrak{L}}$ of $\operatorname{\mathfrak{f}}(V).$
Fix a $\operatorname{\mathbb{Z}}$-graded ideal ${\mathpzc{K}}$ of $\operatorname{\mathfrak{f}}(V),$ contained in $\operatorname{\mathfrak{f}}_{[2]}(V),$ and denote by $$\label{eq-1.13}
{\textswab{m}}({\mathpzc{K}}){=}{\sum}_{{\mathpzc{p}}{\geq}1}{\mathfrak{g}}_{-{\mathpzc{p}}}({\mathpzc{K}})$$ the FGLA defined by the quotient $\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}}.$ Let $\operatorname{\mathfrak{L}}$ be a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ Since ${\mathpzc{K}}$ is a Lie subalgebra of ${\mathfrak{F}}(V,\operatorname{\mathfrak{L}}),$ we can associate to the pair $({\mathpzc{K}},\operatorname{\mathfrak{L}})$ the normalizer $${\mathfrak{N}}({\mathpzc{K}},\operatorname{\mathfrak{L}})=\{\xiup\in{\mathfrak{F}}(V,\operatorname{\mathfrak{L}})\mid [\xiup,{\mathpzc{K}}]\subseteq{\mathpzc{K}}\}$$ of ${\mathpzc{K}}$ in ${\mathfrak{F}}(V,\operatorname{\mathfrak{L}}).$ It is the largest Lie subalgebra of ${\mathfrak{F}}(V,\operatorname{\mathfrak{L}})$ containing ${\mathpzc{K}}$ as an ideal and is $\operatorname{\mathbb{Z}}$-graded. The quotient $$\label{eq-1.14}
{\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})={\mathfrak{N}}({\mathpzc{K}},\operatorname{\mathfrak{L}})/{\mathpzc{K}}$$ has a natural $\operatorname{\mathbb{Z}}$-grading for which ${\mathfrak{g}}_{{<}0}({\mathpzc{K}},\operatorname{\mathfrak{L}}){=}{\textswab{m}}({\mathpzc{K}})$ and the natural projection ${\mathfrak{N}}({\mathpzc{K}},\operatorname{\mathfrak{L}}){\rightarrow\!\!\!\!\rightarrow}
{\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ is a $\operatorname{\mathbb{Z}}$-graded epimorphism of Lie algebras.
\[thm-tan-1-8\] The commutative diagram $$\xymatrix@1{ 0 \ar[r] &\operatorname{\mathfrak{f}}(V)\ar[r]\ar@{->>}[d] & {\mathfrak{N}}({\mathpzc{K}},\operatorname{\mathfrak{L}})\ar@{->>}[d] \\
0\ar[r] & {\textswab{m}}({\mathpzc{K}}) \ar[r] & {\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})
}$$ defines on ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ the structure of an EPFGLA of type $\operatorname{\mathfrak{L}}$ of ${\textswab{m}}({\mathpzc{K}})$ and ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ is, modulo isomorphisms, its maximal EPFGLA of type $\operatorname{\mathfrak{L}}.$
We have $$\label{eq-ta-1-12a} \left\{
\begin{aligned}
{\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})&={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}},\operatorname{\mathfrak{L}}), \;\;\text{with
${\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}},\operatorname{\mathfrak{L}})={\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})$ for ${\mathpzc{p}}{<}0,$ and}\\
{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}},\operatorname{\mathfrak{L}})&=\{\xiup\in\operatorname{\mathfrak{f}}_{{\mathpzc{p}}}(V,\operatorname{\mathfrak{L}}))
\mid [\xiup,{\mathpzc{K}}]\subseteq{\mathpzc{K}}\},\;\;\text{for ${\mathpzc{p}}{\geq}0.$} \end{aligned} \right.$$
Tanaka’s finiteness criterion {#sect2}
=============================
Left and right $V$-modules {#sec-t-1}
--------------------------
Let $V$ be an $n$-dimensional ${\mathbb{K}}$[-]{}vector space, that we consider as an abelian Lie algebra. Its universal enveloping algebra is the graded algebra $${\mathpzc{S}(V)}={\sum}_{{\mathpzc{p}}=0}^\infty{\mathpzc{S}}_{\mathpzc{p}}(V)$$ of symmetric elements of its tensor algebra. Since ${\mathbb{K}}$ has characteristic zero, ${\mathpzc{S}(V)}$ is the ring of polynomials of $V$ with coefficients in ${\mathbb{K}}.$
A right action of $V$ on a ${\mathbb{K}}$-vector space $E$ is a bilinear map $$\begin{gathered}
\label{eqt1.1}
E\times{V}\ni ({\mathpzc{e}},{\mathpzc{v}})\longrightarrow {\mathpzc{e}}{\cdot}{\mathpzc{v}}\in{E},
\\
\notag
\text{with}\qquad
({\mathpzc{e}}{\cdot}{\mathpzc{v}}_1){\cdot}{\mathpzc{v}}_2=({\mathpzc{e}}{\cdot}{\mathpzc{v}}_2){\cdot}{\mathpzc{v}}_1,\;\;
\forall{\mathpzc{e}}\in{E},\;\;\forall {\mathpzc{v}}_1,{\mathpzc{v}}_2\in{V}.\qquad\end{gathered}$$ The map extends to a right action $$E\times{\mathpzc{S}(V)}\ni ({\mathpzc{e}},{\mathpzc{a}})\longrightarrow{\mathpzc{e}}{\cdot}{\mathpzc{a}}\in{E}.$$ If $E^*$ is a ${\mathbb{K}}$-vector space which is in duality with $E$ by a pairing $$E\times{E}^*\ni ({\mathpzc{e}},\etaup)\longrightarrow\langle{\mathpzc{e}}\mid\etaup\rangle\in{\mathbb{K}},$$ a *left* action $$\label{eq-ta-2.5a}
V\times{E}^*\ni({\mathpzc{v}},\etaup)\longrightarrow{\mathpzc{v}}{\cdot}\etaup\in{E}^*$$ of $V$ on $E^*$ is *dual* of if $$\langle{\mathpzc{e}}\,|\,{\mathpzc{v}}{\cdot}\etaup\rangle=\langle{\mathpzc{e}}{\cdot}{\mathpzc{v}}\,
|\,\etaup
\rangle,\;\;\;\forall{\mathpzc{v}}{\in}V,\;\forall{\mathpzc{e}}{\in}E,\;\forall\etaup
{\in}E^*.$$ Clearly a dual left action of $V$ extends to a left action of ${\mathpzc{S}(V)}.$
A *$\operatorname{\mathbb{Z}}$-gradation* of the right $V$-module $E$ is a direct sum decomposition $$\label{eq.t.1.2}
E={\sum}_{{\mathpzc{p}}=-\infty}^\infty E_{\mathpzc{p}}\;\;\; \text{with}\;\;
E_{\mathpzc{p}}{\cdot}V\subseteq{E}_{{\mathpzc{p}}-1}\;\text{for}\; {\mathpzc{p}}\in\operatorname{\mathbb{Z}}.$$
We say that $E$ satisfies condition if $$\label{cndC} \tag{$C$}
\begin{cases}
\dim(E_p)<+\infty,\;\forall p\in\operatorname{\mathbb{Z}},\quad \\
\exists {\mathpzc{p}}_0\in\operatorname{\mathbb{Z}}\;\;{s.t.}\;\; E_{{\mathpzc{p}}}=\{0\} \;\;\text{for}\;\;
{\mathpzc{p}}{<}{\mathpzc{p}}_0,\\
{\mathpzc{e}}\in{\sum}_{h\geq{0}}E_p \;\;\text{and}\;\; {\mathpzc{e}}{\cdot}V=\{0\}
\Longrightarrow {\mathpzc{e}}=0.
\end{cases}$$
We call the $\operatorname{\mathbb{Z}}$-graded vector space $$E^*={\sum}_{h=-\infty}^\infty E^*_h, \;\;
\text{with $E^*_h=$ dual space of $E_{-h},$ for all $h\in\operatorname{\mathbb{Z}}.$}$$ the *graded dual* of $E.$ When $E$ is a graded right $V$-module, its graded dual $E^*$ is a graded left $V$-module under the action which is described on the homogeneous elements by $$\langle{\mathpzc{e}}\mid {\mathpzc{v}}{\cdot}\etaup\rangle=
\langle{\mathpzc{e}}{\cdot}{\mathpzc{v}}\mid\etaup\rangle
\;\;
\forall {\mathpzc{e}}\in{E}_{1-h},\;\forall\etaup\in{E}^*_h,\;\forall{\mathpzc{v}}\in{V}.$$
The following are equivalent
- $E$ satisfies condition ;
- $E^*$ satisfies condition $$\tag{$C^*$} \label{cndstar}
\begin{cases}
\dim(E_{\mathpzc{q}}^*)<+\infty,\;\forall {\mathpzc{q}}\in\operatorname{\mathbb{Z}},\quad \\
\exists\, {\mathpzc{q}}_{\,0}\in\operatorname{\mathbb{Z}}\;\;{s.t}\;\; E^*_{{\mathpzc{q}}}=0,\;\forall {\mathpzc{q}}>{\mathpzc{q}}_{\,0},\\
{\sum}_{{\mathpzc{q}}\geq{1}}E^*_{{\mathpzc{q}}}\;\;\text{generates}\; E^*\;
\text{as a left ${\mathpzc{S}(V)}$-module.}
\end{cases}$$
We have $E_{{\mathpzc{p}}}=\{0\}$ iff $E^*_{-{\mathpzc{p}}}=\{0\}.$
Given any basis ${\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_n$ of $V,$ the maps $$\begin{gathered}
E_{{\mathpzc{p}}}\ni{\mathpzc{e}}\to({\mathpzc{e}}{\cdot}{\mathpzc{v}}_1,\hdots,{\mathpzc{e}}{\cdot}{\mathpzc{v}}_n)\in({E}_{{\mathpzc{p}}-1})^n,
\\
({E}^*_{1-{\mathpzc{p}}})^n\ni(\etaup_1,\hdots,\etaup_n)
\to {\mathpzc{v}}_1{\cdot}\etaup_1{+}\cdots{+}{\mathpzc{v}}_n{\cdot}\etaup_n\in{E}^*_{-{\mathpzc{p}}}\end{gathered}$$ are dual of each other and all vector spaces involved are finite dimensional. Hence the first one is injective if and only if the second one is surjective. These remarks yield the statement.
Let $(V)$ denote the maximal ideal of ${\mathpzc{S}(V)},$ consisting of polynomials vanishing at $0.$ For the notions of commutative algebra that will be used below we refer to [@Bour89 Ch.IV]
\[propt.1.2\] For an $E$ satisfying condition the following are equivalent:
- $E$ is finite dimensional;
- $E^*$ is finite dimensional;
- $E^*$ is $(V)$-coprimary, i.e. all ${\mathpzc{v}}{\in}V$ are nilpotent on $E^*.$
Clearly $(i)\Leftrightarrow(ii).$ Also $(ii)\Rightarrow(iii)$ is clear. Indeed, if $E^*$ is finite dimensional, all ${\mathpzc{v}}\in{V},$ lowering the degree by one unit, are nilpotent. Vice versa, since $E^*$ is finitely generated, if the set $\mathpzc{Ass}(E^*)$ of its associated primes is $\{(V)\},$ then all $E^*_{\mathpzc{q}}$ are zero for ${\mathpzc{q}}{<}{\mathpzc{q}}_{\,0}$ for some ${\mathpzc{q}}_{\,0}\in\operatorname{\mathbb{Z}}.$
A necessary and sufficient condition for a right $V$-module $E$ satisfying condition to be infinite dimensional is that there is ${\mathpzc{v}}\in{V}$ such that $E{\cdot}{\mathpzc{v}}{=}E.$
By Proposition \[propt.1.2\] a necessary and sufficient condition for $E$ to be infinite dimensional is that $E^*$ is not $(V)$-coprimary. If ${\mathfrak{p}}_1,\hdots,{\mathfrak{p}}_m$ are the associated primes of $E^*,$ since $V{\cap}\,{\mathfrak{p}}_j=V_{\!j}$ is, for each $j,$ a proper vector subspace of $V,$ it suffices to choose a ${\mathpzc{v}}$ which does not belong to $V_1{\cup}\cdots{\cup}V_m.$
In fact, for such a ${\mathpzc{v}},$ all maps $E^*_{{\mathpzc{q}}}{\ni}\etaup{\to}
{\mathpzc{v}}{\cdot}\etaup\in{E}^*_{{\mathpzc{q}}{-}1}$ are injective. By duality, all maps $E_{{\mathpzc{p}}}\ni{{\mathpzc{e}}}{\to}{\mathpzc{e}}{\cdot}{\mathpzc{v}}\in{E}_{{\mathpzc{p}}{-}1}$ are surjective. This gives $E{\cdot}{\mathpzc{v}}{=}E.$
Vice versa, if right multiplication by ${\mathpzc{v}}$ on $E$ in surjective, by duality left multiplication by ${\mathpzc{v}}$ is injective on $E^*$ and hence $E^*$ is not $(V)$-coprimary.
It will be useful that the thesis of this corollary be verified by a ${\mathpzc{v}}$ which has a special form. To this aim we prove the following lemma.
\[lem-t.1.4\] Let $U,W$ be finite dimensional vector spaces over ${\mathbb{K}}$ and $\omegaup:U\times{W}\to{V}$ a bilinear map such that $\{\omegaup({\mathpzc{u}},{\mathpzc{w}})\mid {\mathpzc{u}}\in{U},\,{\mathpzc{w}}\in{W}\}$ spans $V.$ Then the image of $\omegaup$ is not contained in any finite union of proper linear subspaces of $V.$
It suffices to show that $\omegaup(U\times{W})$ is not contained in a finite union of hyperplanes. Let $\xiup_1,\hdots,\xiup_m\in{V}^*$ and assume that $\omegaup(U\times{W})$ is contained in $\ker(\xiup_1)\cup\cdots\cup\ker(\xiup_m).$ The condition that $\omegaup(U\times{W})$ spans $V$ implies that each quadric $Q_j=\{({\mathpzc{u}},{\mathpzc{w}})\in{U}\times{W}\mid \xiup_j(\omegaup({\mathpzc{u}},{\mathpzc{w}}))=0\}$ is properly contained in $U{\times}W.$ Hence $Q_1{\cup}\cdots{\cup}Q_m\subsetneqq{U}\times{W}$ and the $\omegaup$-image of each pair which is not contained in $Q_1{\cup}\cdots{\cup}Q_m$ does not belong to $\ker(\xiup_1){\cup}\cdots{\cup}\ker(\xiup_m).$
\[prop-t.1.4\] Let $E$ be a right $V$-module satisfying condition . If $E$ is infinite dimensional, then we can find linearly independent vectors ${\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_m$ in $V$ such that, with $$E^{(h)}=\{{\mathpzc{e}}\in{E}\mid {\mathpzc{e}}{\cdot}{\mathpzc{v}}_i=0,\;\forall i{\leq}h\},\;\; \text{for}\;\; 0{\leq}h{\leq}m,$$ the following conditions are satisfied:
- either $m{=}n$ and ${\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_n$ is a basis of $V,$ or $0{<}m{<}n$ and $E^{(m)}$ is finite dimensional;
- the maps $T_{h+1}:E^{(h)}{\ni}{\mathpzc{e}}{\to}{\mathpzc{e}}{\cdot}{\mathpzc{v}}_{h+1}{\in}{E}^{(h)}$ are surjective for $0{\leq}{h}{\leq}{m{-}1}.$
Note that $E^{(0)}=E.$ We know from Proposition \[propt.1.2\] that, if $\dim(E)=\infty,$ then $E^*$ is not $(V)$-coprimary. Hence we can find ${\mathpzc{v}}_1\in{V}$ such that the left multiplication by ${\mathpzc{v}}_1$ is injective on $E^*.$ By duality this means that the map $T_1:E\ni{\mathpzc{e}}\to{\mathpzc{e}}{\cdot}{\mathpzc{v}}_1\in{E}$ is surjective.
Let $V_1$ be a hyperplane in $V$ which does not contain ${\mathpzc{v}}_1.$ We have a natural duality pairing between $E^{(1)}$ and the quotient $E^*{/}{\mathpzc{v}}_1{\cdot}E^*,$ that we can consider as a left $V_1$-module. In case $E^{(1)}$ is infinite dimensional, $(V_1)$ is not an associated prime of $E^*{/}{\mathpzc{v}}_1{\cdot}E^*$ and hence we can find ${\mathpzc{v}}_2{\in}V_1$ which is not a zero divisor in $E^*{/}{\mathpzc{v}}_1{\cdot}E^*.$ By duality this yields a surjective $$T_{2}:E^{(1)}\ni{\mathpzc{e}}\to{\mathpzc{e}}{\cdot}{\mathpzc{v}}_2\in{E}^{(1)}.$$
The recursive argument is now clear and we get the statement after a finite number of steps.
In the hypothesis of Lemma \[lem-t.1.4\], the vectors ${\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_m$ in the statement of Proposition \[prop-t.1.4\] can be chosen of the form ${\mathpzc{v}}_i=\omegaup({\mathpzc{u}}_i,{\mathpzc{w}}_i),$ with ${\mathpzc{u}}_i\in{U}$ and ${\mathpzc{w}}_i\in{W},$ for $1{\leq}i{\leq}m.$
Right $\mathfrak{m}$-modules {#sec2}
----------------------------
Let ${\textswab{m}}={\sum}_{{\mathpzc{p}}=1}^\muup{\mathfrak{g}}_{-{\mathpzc{p}}}$ be an FGLA of finite kind $\muup{\geq}1.$
A $\operatorname{\mathbb{Z}}$-graded right ${\textswab{m}}$-module is a $\operatorname{\mathbb{Z}}$-graded ${\mathbb{K}}$-vector space on which a ${\mathbb{K}}$-bilinear map $$E\times{\textswab{m}}\ni ({\mathpzc{e}},X)\longrightarrow {\mathpzc{e}}{\cdot}X\in{E},$$ is defined, with the properties: $$\begin{cases}
( {\mathpzc{e}}{\cdot}X){\cdot}{Y}-( {\mathpzc{e}}{\cdot}Y){\cdot}{X}={\mathpzc{e}}{\cdot}[X,Y],\;\;\forall {\mathpzc{e}}\in{E},\;\forall X,Y\in{\textswab{m}},\\
E_{{\mathpzc{q}}}{\cdot}{\mathfrak{g}}_{-{\mathpzc{p}}}\subseteq{E}_{{\mathpzc{q}}-{\mathpzc{p}}},\;\;\forall {\mathpzc{q}}\in\operatorname{\mathbb{Z}},\;\forall 1{\leq}{\mathpzc{p}}{\leq}\muup.
\end{cases}$$
We say that the graded right ${\textswab{m}}$-module $E$ satisfies condition if $$\tag{$C({\textswab{m}})$}
\label{cndCi}
\begin{cases}
\dim(E_{{\mathpzc{p}}})<\infty, \;\;\forall {\mathpzc{p}}\in\operatorname{\mathbb{Z}},\\
\exists\,{\mathpzc{p}}_{0}\in\operatorname{\mathbb{Z}}\;\text{s.t.}\; E_{{\mathpzc{p}}}=0,\;\;
\forall {\mathpzc{p}}<-{\mathpzc{p}}_0,\\
{\mathpzc{e}}\in{\sum}_{{\mathpzc{p}}\geq{0}}E_{{\mathpzc{p}}}\;\;\text{and}\;\; {\mathpzc{e}}{\cdot}{\mathfrak{g}}_{-1}=\{0\}\;\Longrightarrow\; {\mathpzc{e}}=0.
\end{cases}$$
The following lemma is similar to [@Tan70 Lemma 11.4].
\[lem-t.2.3\] Assume that
- ${\textswab{m}}$ has kind $\muup{\geq}2;$
- $E$ is a graded right ${\textswab{m}}$-module satisfying condition ;
- there are $X\in{\mathfrak{g}}_{-1},$ $Y\in{\mathfrak{g}}_{1-\muup}$ and ${\mathpzc{p}}_0\in\operatorname{\mathbb{Z}}$ such that $$\psiup_{{\mathpzc{p}}}:E_{{\mathpzc{p}}}\ni {\mathpzc{e}}\longrightarrow {\mathpzc{e}}{\cdot}[X,Y]\in{E}_{p-\muup}$$ is an isomorphism for ${\mathpzc{p}}{\geq}{\mathpzc{p}}_0.$
Then $E_{{\mathpzc{p}}}=\{0\}$ for ${\mathpzc{p}}{\geq}{{\mathpzc{p}}}_0.$
We consider the right multiplication by $Z=[X,Y]$ as a linear map $R_Z$ on $E$. Note that $E_{<{{\mathpzc{q}}_{\,0}}}={\sum}_{{\mathpzc{p}}<{\mathpzc{q}}_{\,0}}E_{{\mathpzc{p}}}$ is a right ${\textswab{m}}$-submodule of $E$. Thus we can consider the *truncation* ${\sum}_{{\mathpzc{p}}\geq{\mathpzc{q}}_{\,0}}E_{{\mathpzc{p}}}$ as the quotient right ${\textswab{m}}$-module $E{/}E_{{<}{{\mathpzc{q}}_{\,0}}}.$
By substituting to $E$ its truncation, if needed, we can as well assume that $E={\sum}_{{\mathpzc{p}}\geq{\mathpzc{p}}_0-\muup}E_{{\mathpzc{p}}}.$ Then $R_Z$ has a right inverse $\Psi:E\to{E}$ such that $\Psi{\circ}R_Z$ restricts to the identity on each $E_p$ with $p{\geq}p_0,$ i.e. $\Psi$ is also a left inverse of $R_Z$ on $E_{{\mathpzc{p}}}$ for ${\mathpzc{p}}{\geq}{\mathpzc{p}}_0.$ Denote by $R_X$ and $R_Y$ the linear maps on $E$ defined by the right multiplication by $X$ and $Y,$ respectively. We claim that $$\Psi{\circ}R_X=R_X{\circ}\Psi
\quad\text{and}\quad \Psi{\circ}R_Y=R_Y{\circ}\Psi
\;\;
\text{on} \;\;
E_{{\mathpzc{p}}},\;\;
\forall {\mathpzc{p}}{\geq}{\mathpzc{p}}_0{-}1.$$ Indeed, $\Psi{\circ}R_X(E_{{\mathpzc{p}}})\cup{R}_X{\circ}\Psi(E_{{\mathpzc{p}}})\subseteq{E}_{{\mathpzc{p}}+\muup-1}$ and $\Psi{\circ}R_Y(E_{{\mathpzc{p}}})\cup{R}_Y{\circ}\Psi(E_{{\mathpzc{p}}})\subseteq{E}_{{\mathpzc{p}}+1},$ so that, since both ${\mathpzc{p}}{+}1$ and ${\mathpzc{p}}{+}\muup{-}1$ are ${\geq}{\mathpzc{p}}_0$ when ${\mathpzc{p}}\geq{\mathpzc{p}}_0{-}1,$ and $\Psi$ is a right inverse of $R_Z$ in this range, these equalities are equivalent to $$R_X=R_{Z}{\circ}R_X{\circ}\Psi\quad\text{and}\quad R_Y=R_{Z}{\circ}R_Y{\circ}\Psi,$$ and thus are verified because $R_Z$ commutes with $R_X$ and $R_Y.$ Fix ${\mathpzc{p}}{\geq}{\mathpzc{p}}_0{-1}$ and consider the finite dimensional ${\mathbb{K}}$-vector space $W={\sum}_{h=1}^{\muup}E_{{\mathpzc{p}}+h}.$ We define two endomorphisms $T_X,T_Y$ on $W$ by setting $$\begin{aligned}
T_X({\mathpzc{e}})&=
\begin{cases}
R_X{\circ}\Psi({\mathpzc{e}})\in{E}_{{\mathpzc{p}}+\muup}, &\text{if ${\mathpzc{e}}\in{E}_{{\mathpzc{p}}+1},$}\\
R_X({\mathpzc{e}})\in{E}_{h-1}, &\text{if ${\mathpzc{e}}\in{E}_{h},$ with ${\mathpzc{p}}{+}1{<}h\leq\muup,$}
\end{cases}\\
T_Y({\mathpzc{e}})&=
\begin{cases}
R_Y{\circ}\Psi({\mathpzc{e}})\in{E}_{h+1}, &\text{if ${\mathpzc{e}}\in{E}_h$ with ${\mathpzc{p}}{+}1{\leq}h<{\mathpzc{p}}{+}\muup,$}\\
R_Y({\mathpzc{e}})\in{E}_{{\mathpzc{p}}+1} & \text{if ${\mathpzc{e}}\in{E}_{{\mathpzc{p}}+{\muup}} .$}
\end{cases}
\end{aligned}$$ One easily checks that $$T_X{\circ}T_Y-T_Y{\circ}T_X=(R_X{\circ}{R}_Y-R_Y{\circ}R_X){\circ}\Psi={\mathrm{I}}_W\;\;\;\text{on $W$}$$ and hence $$\dim(W)={\mathrm{trace}}({\mathrm{I}}_W)={\mathrm{trace}}( T_X{\circ}T_Y-T_Y{\circ}T_X)=0.$$ This proves the lemma.
Let $E={\sum}_{h=-{\mathpzc{p}}_0}^\infty E_{{\mathpzc{p}}}$ be a right ${\textswab{m}}$-module, satisfying condition . Set $$\label{equ-ta-3.2}
{\mathfrak{n}}={\sum}_{h\geq{2}}{\mathfrak{g}}_{-h}$$ and $$\operatorname{\mathrm{N}}(E)={\sum}_{h=-{\mathpzc{p}}_0}^\infty\operatorname{\mathrm{N}}_{{\mathpzc{p}}}(E),\;\;\text{with}\;\; \operatorname{\mathrm{N}}_{{\mathpzc{p}}}(E)=\{{\mathpzc{e}}\in{E}_{{\mathpzc{p}}}\mid {\mathpzc{e}}{\cdot}{{\mathfrak{n}}}=\{0\}\}.$$
\[thm2.2\] Let $E$ be a $\operatorname{\mathbb{Z}}$-graded right $\operatorname{\mathbb{Z}}$-module satisfying condition . Then $E$ is finite dimensional if and only if $\operatorname{\mathrm{N}}(E)$ is finite dimensional.
We argue by recurrence on the kind $\muup$ of ${\textswab{m}}.$ In fact, when $\muup=1,$ we have ${\mathfrak{n}}=\{0\}$ and hence $\operatorname{\mathrm{N}}(E)=E$ and the statement is trivially true. Moreover, since $\operatorname{\mathrm{N}}(E)\subseteq{E},$ we only need to show that $N(E)$ is infinite dimensional when $E$ is infinite dimensional.
Assume that $\muup{>}1.$ The subspace ${\mathfrak{g}}_{-\muup}$ is an ideal of ${\textswab{m}}$ and hence ${\textswab{m}}'{=}{\textswab{m}}{/}{\mathfrak{g}}_{-\muup}$ is an FGLA of kind $\muup{-}1.$ Then $$F={\sum}_{{\mathpzc{p}}=-{\mathpzc{p}}_0}^\infty{F}_{{\mathpzc{p}}},\;\;\text{with}\;\; F_{{\mathpzc{p}}}=\{{\mathpzc{e}}\in{E}_{{\mathpzc{p}}}\mid {\mathpzc{e}}{\cdot}{\mathfrak{g}}_{{-}\muup}=\{0\}\}$$ can be viewed as a $\operatorname{\mathbb{Z}}$-graded ${\textswab{m}}'$-module which satisfies condition $C({\textswab{m}}')$.
Since $\operatorname{\mathrm{N}}(F){=}\operatorname{\mathrm{N}}(E),$ by our recursive assumption $F$ and $\operatorname{\mathrm{N}}(E)$ are either both finite, or both infinite dimensional.
Hence it will suffice to prove that, if $E$ is infinite dimensional, also $F$ is infinite dimensional. The $\operatorname{\mathbb{Z}}$-grading $${\mathrm{M}}(E)={\sum}_{{\mathpzc{p}}\in\operatorname{\mathbb{Z}}}{\mathrm{M}}_{{\mathpzc{p}}}(E),\;\;\text{where}\;\; {\mathrm{M}}_{{\mathpzc{p}}}(E)={\sum}_{j=0}^{\muup-1}E_{j+{\mathpzc{p}}{\cdot}\muup}.$$ defines on $E$ the structure of a $\operatorname{\mathbb{Z}}$-graded left $V$-module, for $V={\mathfrak{g}}_{-\muup}.$ Since, by assumption, it is infinite dimensional, by Proposition \[prop-t.1.4\] and Lemma \[lem-t.1.4\] we can find $X_1,\hdots,X_m\in{\mathfrak{g}}_{-1}$ and $Y_1,\hdots,Y_m\in{\mathfrak{g}}_{1-\muup}$ such that $$Z_1{=}[X_1,Y_1],\hdots,Z_m{=}[X_m,Y_m]$$ are linearly independent and have the properties:
- either $m{<}\dim_{{\mathbb{K}}}({\mathfrak{g}}_{-\muup})$ and $E^{(m)}=\{{\mathpzc{e}}\in{E}\mid {\mathpzc{e}}{\cdot}Z_i=0,\;\forall i=1,\hdots,m\}$ is finite dimensional, or $m{=}n$ and $Z_1,\hdots,Z_n$ is a basis of ${\mathfrak{g}}_{-\muup}$;
- with $E^{(h)}=\{{\mathpzc{e}}\in{E}\mid {\mathpzc{e}}{\cdot}Z_i=0,\;\forall i{\leq}h\},$ the maps $$T_{h+1}:E^{(h)} \ni{\mathpzc{e}}\longrightarrow {\mathpzc{e}}{\cdot}Z_{h+1}\in{E}^{(h)}$$ are surjective for $0{\leq}h{<}m.$
For all $0{\leq}h{<}m$ we obtain exact sequences $$\label{eq-t.3.4}
\begin{CD}
0 @>>> E^{(h+1)} @>>> E^{(h)} @>{{T_{h+1}}}>> E^{(h)} @>>> 0.
\end{CD}$$ Note that $E^{(0)}=E$ and that $E^{(n)}=F$ when $m{=}n.$ We want to prove that ${m}=n$ and that $F$ is infinite dimensional. We argue by contradiction. If our claim is false, then $E^{(m)}$ is finite dimensional and, from the exact sequence , we obtain that $E^{(m-1)}_{{\mathpzc{p}}}\ni{\mathpzc{e}}\to{\mathpzc{e}}{\cdot}Z_{m}\in{E}^{(m-1)}_{{\mathpzc{p}}-\muup}$ is an isomorphism for all ${\mathpzc{p}}\geq{\mathpzc{p}}_m$ for some integer ${\mathpzc{p}}_m,$ which, by Lemma \[lem-t.2.3\], implies that $E^{(m-1)}$ is finite dimensional. Using again the exact sequence for $h{=}m{-}2$ and Lemma \[lem-t.2.3\] we obtain that also $\dim_{\operatorname{{\mathbb{R}}}}(E^{(m-2)})<\infty$ and, repeating the argument, we end up getting that $E^{(0)}=E$ is finite dimensional. This yields a contradiction, proving that $F$, and thus also $\operatorname{\mathrm{N}}(E),$ must be infinite dimensional when $E$ is infinite dimensional.
Reduction to first kind
------------------------
Let us get back to the prolongations defined in §\[sec-fund\]. We will use Theorem \[thm2.2\] to show that the finiteness of the maximal effective $\operatorname{\mathfrak{L}}$-prolongation of an FGLA of finite kind $\muup{\geq}2$ is equivalent to that of the ${\mathfrak{L}}'$-prolongation of an FGLA of the first kind for a suitable ${\mathfrak{L}}'{\subseteq}{\mathfrak{L}}.$
Let ${\textswab{m}}={\sum}_{{\mathpzc{p}}=1}^\muup{\mathfrak{g}}_{-{\mathpzc{p}}}$ be an FGLA of finite kind $\muup.$ Set $V{=}{\mathfrak{g}}_{{-}1}.$ We showed in §\[sec-fund\] that ${\textswab{m}}\simeq\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}}$ for a $\operatorname{\mathbb{Z}}$-graded ideal ${\mathpzc{K}}$ of $\operatorname{\mathfrak{f}}(V),$ contained in $\operatorname{\mathfrak{f}}_{[2]}(V).$ As in , we denote by ${\mathfrak{n}}$ the ideal ${\sum}_{{\mathpzc{p}}{\leq}{-2}}{\mathfrak{g}}_{{\mathpzc{p}}}$ of ${\textswab{m}}.$ Fix a Lie subalgebra $\operatorname{\mathfrak{L}}$ of ${\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}_{-1})$ and denote by ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ the maximal EPFGLA of type $\operatorname{\mathfrak{L}}$ of ${\textswab{m}},$ that was characterised in Theorem \[thm-tan-1-8\]. We have the following finiteness criterion:
\[thm-t-4-2\] The maximal effective $\operatorname{\mathfrak{L}}$-prolongation ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ of ${\textswab{m}}({\mathpzc{K}})$ is finite dimensional if, and only if, $$\label{equ-ta-4.2}
\operatorname{\mathrm{N}}({\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}}))=\{\xiup\in{\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})\mid [\xiup,{\mathfrak{n}}]\}=\{0\}\}$$ is finite dimensional.
The statement follows by applying Theorem \[thm2.2\] to ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}}),$ considered as a right ${\textswab{m}}$-module.
Set $$\begin{gathered}
\label{eq-t-4-11a}
\operatorname{\mathfrak{a}}=({\textswab{m}}{/}{\mathfrak{n}})\oplus{\sum}_{{\mathfrak{p}}\geq{0}}\operatorname{\mathfrak{a}}_{{\mathpzc{p}}},
\;\;\text{with}\;\;
\operatorname{\mathfrak{a}}_{\mathpzc{p}}=\{\xiup\in{\mathfrak{g}}_{\mathpzc{p}}({\mathpzc{K}},\operatorname{\mathfrak{L}})
\mid [\xiup,{\mathfrak{n}}]=\{0\}\}.
\end{gathered}$$ We note that ${\textswab{m}}\oplus{\sum}_{{\mathpzc{p}}\geq{0}}\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}$ is a Lie subalgebra of ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}}),$ which contains ${\mathfrak{n}}$ as an ideal. There is a natural isomorphism of $\operatorname{\mathfrak{a}}$ with $({\textswab{m}}\oplus{\sum}_{{\mathpzc{p}}\geq{0}}\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}){/}{\mathfrak{n}},$ which defines its Lie algebra structure. Set $\operatorname{\mathfrak{a}}_{-1}={\textswab{m}}{/}{\mathfrak{n}}\simeq{V}.$
\[lem-tan-4-2\] The summand $\operatorname{\mathfrak{a}}_0$ in is given by $$\operatorname{\mathfrak{a}}_0=\operatorname{\mathfrak{a}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}})=\{A\in\operatorname{\mathfrak{L}}\mid T_A(\operatorname{\mathfrak{f}}_{[2]}(V))\subseteq{\mathpzc{K}}\}.$$
The statement follows because ${\mathfrak{n}}=\operatorname{\mathfrak{f}}_{[2]}(V){/}{\mathpzc{K}}.$
\[thm-t-4-4\] The $\operatorname{\mathbb{Z}}$-graded Lie algebra $\operatorname{\mathfrak{a}}$ of is the maximal effective prolongation of type $\operatorname{\mathfrak{a}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}})$ of ${\textswab{m}}(\operatorname{\mathfrak{f}}_{[2]}(V))$ and the following are equivalent:
- ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ is finite dimensional;
- ${\mathfrak{g}}(\operatorname{\mathfrak{f}}_{[2]}(V),\operatorname{\mathfrak{a}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}}))$ is finite dimensional.
Set $\operatorname{\mathfrak{a}}_0=\operatorname{\mathfrak{a}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}})$ and denote by ${\mathfrak{b}}{=}{\sum}_{{\mathpzc{p}}{\geq}{-}1}\!{\mathfrak{b}}_{{\mathpzc{p}}}$ the maximal EPFGLA ${\mathfrak{g}}(\operatorname{\mathfrak{f}}_{[2]}(V),\operatorname{\mathfrak{a}}_0)$ of type $\operatorname{\mathfrak{a}}_0$ of $\operatorname{\mathfrak{a}}_{-1}=\operatorname{\mathfrak{f}}(V){/}\operatorname{\mathfrak{f}}_{[2]}(V)\simeq{V}.$ By construction, ${\mathfrak{b}}_0=\operatorname{\mathfrak{a}}_0.$
Let $\piup:{\textswab{m}}\to({\textswab{m}}{/}{\mathfrak{n}})$ be the canonical projection. Then $X\in{\textswab{m}}$ acts to the right on ${\mathfrak{b}}$ by $\betaup{\cdot}X=[\betaup,\piup(X)]$ and $${\textswab{m}}\oplus{\mathfrak{g}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}})\oplus{\sum}_{{\mathpzc{p}}>0}{\mathfrak{b}}_{{\mathpzc{p}}}$$ is an effective prolongation of type $\operatorname{\mathfrak{L}}$ of ${\textswab{m}}({\mathpzc{K}}).$ Hence ${\mathfrak{b}}_{{\mathpzc{p}}}\subseteq\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}$ for ${\mathpzc{p}}{>}0.$ On the other hand, since $\operatorname{\mathfrak{a}}$ is an effective prolongation of type $\operatorname{\mathfrak{a}}_0$ of ${\textswab{m}}(\operatorname{\mathfrak{f}}_{[2]}(V)),$ we also have the opposite inclusion. This yields ${\mathfrak{b}}_{{\mathpzc{p}}}=\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}$ for all ${\mathpzc{p}}{>}0,$ proving the first part of the statement. The equivalence of $(i)$ and $(ii)$ is then a consequence of Theorem \[thm-t-4-2\].
Comparing maximal effective prolongations
-----------------------------------------
Let $V$ be a finite dimensional ${\mathbb{K}}$-vector space and ${\mathpzc{K}},{\mathpzc{K}}'$ two graded cofinite ideals in $\operatorname{\mathfrak{f}}(V),$ so that $${\textswab{m}}=\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}},\;\;\;{\textswab{m}}'=\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}}'$$ are finite dimensional FGLA’s. If ${\mathpzc{K}}\,{\subset}{\mathpzc{K}}',$ we get $${\textswab{m}}\simeq{\textswab{m}}'\oplus{\mathpzc{K}}'/{\mathpzc{K}}.$$
By the summands of positive degree of the maximal effective ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ are $$\begin{aligned}
{\mathfrak{g}}_p({\mathpzc{K}},\operatorname{\mathfrak{L}})
&=\{X\in\operatorname{\mathfrak{f}}_{p}(V,\operatorname{\mathfrak{L}})
\,\mid\,[X,{\mathpzc{K}}]\subseteq{\mathpzc{K}}\}.
\\[-8pt]\end{aligned}$$ If for the structure algebras we have the inclusion $\operatorname{\mathfrak{L}}\subseteq\operatorname{\mathfrak{L}}'\subseteq{\mathfrak{gl}}_{\mathbb{K}}(V),$ then the inclusions $\operatorname{\mathfrak{f}}_p(V,\operatorname{\mathfrak{L}})\subseteq\operatorname{\mathfrak{f}}_p(V,\operatorname{\mathfrak{L}}')$ yield $$\label{inclusion_prolongation}
{\mathfrak{g}}_{p}({\mathpzc{K}},\operatorname{\mathfrak{L}})\subseteq{\mathfrak{g}}_{p}({\mathpzc{K}},\operatorname{\mathfrak{L}}'),\;\;
\;\forall p\geq{0}.$$
We recall from Theorem \[thm-t-4-4\] that a maximal prolongation ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ is finite dimensional iff ${\mathfrak{g}}(\operatorname{\mathfrak{f}}_{[2]}(V),\operatorname{\mathfrak{a}}_{0}({\mathpzc{K}},\operatorname{\mathfrak{L}}))$ is finite dimensional.
By Lemma we have $$\operatorname{\mathfrak{a}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}})
=\{X\in\operatorname{\mathfrak{L}}\,\mid\,[X,\operatorname{\mathfrak{f}}_{[2]}(V)]\subseteq{\mathpzc{K}}\},\\$$ and hence when $\operatorname{\mathfrak{L}}\,{\subseteq}\operatorname{\mathfrak{L}}'\subseteq{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ and ${\mathpzc{K}}\,{\subseteq}\,{\mathpzc{K}}'$ we obtain $$\operatorname{\mathfrak{a}}_0({\mathpzc{K}},\operatorname{\mathfrak{L}})\subseteq\operatorname{\mathfrak{a}}_0({\mathpzc{K}}',\operatorname{\mathfrak{L}}').$$ Then the characterisation given in the proof of Theorem \[thm-t-4-4\] yields $$\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}({\mathpzc{K}},\operatorname{\mathfrak{L}}){=}{\mathfrak{g}}_{{\mathpzc{p}}}\left(\operatorname{\mathfrak{f}}_{[2]}(V),\operatorname{\mathfrak{a}}_{0}({\mathpzc{K}},\operatorname{\mathfrak{L}})\right)\subseteq{\mathfrak{g}}_{{\mathpzc{p}}}\left(\operatorname{\mathfrak{f}}_{[2]}(V),\operatorname{\mathfrak{a}}_{0}({\mathpzc{K}}',\operatorname{\mathfrak{L}}')\right){=}
\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}({\mathpzc{K}}',\operatorname{\mathfrak{L}}'),$$ for $\forall p>0$.
\[comparing\] Let $V$ be a finite dimensional ${\mathbb{K}}$-vector space, $\operatorname{\mathfrak{L}},\operatorname{\mathfrak{L}}'
{\subseteq}{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ structure algebras and ${\mathpzc{K}},{\mathpzc{K}}'$ two cofinite graded ideals of $\operatorname{\mathfrak{f}}(V).$ Assume that $$\operatorname{\mathfrak{L}}\subseteq\operatorname{\mathfrak{L}}',\;\; {\mathpzc{K}}\subseteq{\mathpzc{K}}'.$$ Then ${\mathfrak{g}}(\operatorname{\mathfrak{L}},{\mathpzc{K}})$ is finite dimensional, if ${\mathfrak{g}}(\operatorname{\mathfrak{L}}',{\mathpzc{K}}')$ is finite dimensional and ${\mathfrak{g}}(\operatorname{\mathfrak{L}}',{\mathpzc{K}}')$ is infinite dimensional if ${\mathfrak{g}}(\operatorname{\mathfrak{L}},{\mathpzc{K}})$ is infinite dimensional.
Let ${\textswab{m}}{({\mathpzc{K}})}$ be a FGLA over ${\mathbb{K}}=\operatorname{\mathbb{C}},\operatorname{{\mathbb{R}}}$ of *depth* $\mu \geq 3$ such that $\dim{\mathfrak{g}}_{-1}{=}\dim{\mathfrak{g}}_{-3}{=}2.$ Then the maximal EPFGLA ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ is finite dimensional for every $\operatorname{\mathfrak{L}}\subseteq\mathfrak{gl}
({\mathfrak{g}}_{-1})$.
In fact we use Proposition \[comparing\] over $\operatorname{\mathbb{C}}$ with a gradation of the Lie algebra $\textrm{Lie}(\operatorname{\mathbf{G}}_2)$ of the special group $\operatorname{\mathbf{G}}_2$; for ${\mathbb{K}}=\operatorname{{\mathbb{R}}}$ similar results can be found using the Lie algebra of the split real form of $\mathbf{G}_2$. Denoting by $X_{0},X_{1}$ a linear basis of ${\mathfrak{g}}_{{-}1},$ we get $$\begin{aligned}
{\mathfrak{g}}_{-1}&=\langle X_{0},X_1\rangle_{{\mathbb{K}}}\\
{\mathfrak{g}}_{-2}&=\langle [X_{0},X_1]\rangle_{{\mathbb{K}}}\\
{\mathfrak{g}}_{-3}&=\langle [[X_{0},X_1],X_0], [[X_{0},X_1],X_1]\rangle_{{\mathbb{K}}} \\
&\,\,\hdots\end{aligned}$$ After noticing that the maximal effective prolongation of $\operatorname{\mathfrak{f}}({\mathbb{K}}^{2}){/}\operatorname{\mathfrak{f}}_{{[4]}}({\mathbb{K}}^{2})$ is finite dimensional and isomorphic to the Lie algebra of $\operatorname{\mathbf{G}}_{2}$ (see e.g. [@Tan70 p.29] or [@Warhurst2007 p.62]), by using Proposition \[comparing\], we conclude that every EPFGLA ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ of any type $\operatorname{\mathfrak{L}}\,{\subseteq}\,{\mathfrak{gl}}_{2}({\mathbb{K}})$ of an ${\textswab{m}}({\mathpzc{K}})$ with ${\mathpzc{K}}{\subseteq}\operatorname{\mathfrak{f}}_{[4]}({\mathbb{K}}^{2})$ is finite dimensional.
$\operatorname{\mathfrak{L}}$-prolongations of graded Lie algebras of the first kind {#sec5a}
====================================================================================
By Theorem \[thm-t-4-4\], finiteness of the maximal $\operatorname{\mathfrak{L}}$-prolongation of an FGLA ${\textswab{m}}$ of any finite kind is equivalent to that of the $\operatorname{\mathfrak{L}}'$-prolongation of its first kind quotient $V{=}{\textswab{m}}{/}{\mathfrak{n}},$ for a Lie subalgebra $\operatorname{\mathfrak{L}}'$ of $\operatorname{\mathfrak{L}}$ that can be computed in terms of ${\textswab{m}}$ and $\operatorname{\mathfrak{L}}.$ It is therefore a key issue to establish a viable criterion for FGLA’s of the first kind.
Using duality, we will translate questions on the maximal effective prolongations of FGLA’s of the first kind to questions of commutative algebra for finitely generated modules over polynomial rings.
Let $V$ be a finite dimensional ${\mathbb{K}}$-vector space, that we will consider as a commutative Lie algebra over ${\mathbb{K}},$ and $${\mathpzc{S}}(V^*)={\sum}_{{\mathpzc{p}}=0}^\infty{\mathpzc{S}}_{{\mathpzc{p}}}(V^*)$$ the $\operatorname{\mathbb{Z}}$-graded unitary associative algebra over ${\mathbb{K}}$ of symmetric multilinear forms on $V.$ Its product is described on homogeneous forms by $$\begin{aligned}
(\xiup{\pmb{\cdot}}\etaup)({\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}+{\mathpzc{q}}}){=}
\frac{1}{({\mathpzc{p}}{+}{\mathpzc{q}})!}\!
\sum_{\sigmaup\in{\mathbf{S}}_{{\mathpzc{p}}+{\mathpzc{q}}}}\xiup({\mathpzc{v}}_{\sigmaup_1},\hdots,{\mathpzc{v}}_{\sigmaup_{\mathpzc{p}}})
\cdot\etaup({\mathpzc{v}}_{\sigmaup_{{\mathpzc{p}}+1}},\hdots,{\mathpzc{v}}_{\sigmaup_{{\mathpzc{p}}+{\mathpzc{q}}}}),\quad\\
\forall \xiup\in{\mathpzc{S}}_{{\mathpzc{p}}}(V^*),\;\forall
\etaup\in{\mathpzc{S}}_{{\mathpzc{q}}}(V^*),\; \forall
{\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}+{\mathpzc{q}}}\in{V}.
\end{aligned}$$
The duality pairing $$V\times{V}^*\ni ({\mathpzc{v}},\xiup)\longrightarrow \langle{\mathpzc{v}}\mid\xiup\rangle
\in{\mathbb{K}}$$ extends to a degree-$({-}1)$-derivation $D_{{\mathpzc{v}}}$ of ${\mathpzc{S}}(V^*),$ with $$\left\{\begin{aligned}
(D_{\mathpzc{v}}\xiup)({\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{\mathpzc{p}})=({\mathpzc{p}}{+}1)
{\cdot}\xiup({\mathpzc{v}},{\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{\mathpzc{p}}),\;
\qquad\\
\forall {\mathpzc{v}},{\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{\mathpzc{p}}\in{V},\;\forall\xiup\in{\mathpzc{S}}_{{\mathpzc{p}}+1}(V^*).
\end{aligned}\right.$$
The tensor product $$\label{equ-3-4}
{\mathpzc{X}}(V){=}{\mathpzc{S}}(V^*)\otimes{V}{=}\sum_{{\mathpzc{p}}\geq{-1}}{\mathpzc{X}}_{{\mathpzc{p}}}(V),
\;\;\text{with}\;\; {\mathpzc{X}}_{{\mathpzc{p}}}(V){=}
{\mathpzc{S}}_{{\mathpzc{p}}+1}(V^*)\otimes{V},$$ is the maximal ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongation of the commutative Lie algebra ${\textswab{m}}=V$, where each vector in $V$ is considered as a homogeneous element of degree $({-}1).$ We can identify ${\mathpzc{X}}(V)$ with the space of vector fields with polynomial coefficients on $V.$ The Lie product in ${\mathpzc{X}}(V)$ is described, on rank one elements, by $$[\xiup{\otimes}{\mathpzc{v}},\etaup{\otimes}{\mathpzc{w}}]{=}(\xiup{{\pmb{\cdot}}}(D_{{\mathpzc{v}}}\etaup))
{\otimes}{\mathpzc{w}}{-}(\etaup{{\pmb{\cdot}}}(D_{{\mathpzc{w}}}\xiup)){\otimes}{\mathpzc{v}},
\forall\xiup{\in}{\mathpzc{S}}_{{\mathpzc{p}}}(V^*),
\forall\etaup{\in}{\mathpzc{S}}_{{\mathpzc{q}}}(V^*),\forall
{\mathpzc{v}},{\mathpzc{w}}\in{V}.$$
If $\operatorname{\mathfrak{a}}_0$ is any Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V),$ then the direct sum $V\oplus\operatorname{\mathfrak{a}}_0$ is a Lie subalgebra of the Abelian extension $V\oplus{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ The maximal effective $\operatorname{\mathfrak{a}}_0$-prolongation of $V$ can be described as a Lie subalgebra of ${\mathpzc{X}}(V).$
\[prop-3-3.1\] Let $\operatorname{\mathfrak{a}}_0$ be any Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ and $$\label{eq-t-4-14}
\operatorname{\mathfrak{a}}=V\oplus\operatorname{\mathfrak{a}}_0\oplus{\sum}_{{\mathpzc{p}}\geq{1}}\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}$$ the maximal effective prolongation of type $\operatorname{\mathfrak{a}}_0$ of $V.$ Its summands of positive degree are $$\label{eq-t-4-15}
\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}=\{\xiup{\in}{\mathpzc{X}}_{{\mathpzc{p}}}(V)
\mid
\{V{\ni}{\mathpzc{v}}\to\xiup(\underset{{\mathpzc{p}}\;\text{times}}{\underbrace{{\mathpzc{w}},\hdots,{\mathpzc{w}}}},{\mathpzc{v}}){\in}{V}\}{\in}
\operatorname{\mathfrak{a}}_0,\;\forall {\mathpzc{w}}{\in}{V}\}.$$
It is known (see e.g. [@Sternberg Ch.VII §[3]{}] or [@Kob Ch.1 §[5]{}]) that the elements of $\operatorname{\mathfrak{a}}_{{\mathpzc{p}}},$ for ${\mathpzc{p}}{\geq}1,$ are the $\xiup\in{\mathpzc{S}}_{{\mathpzc{p}}{+}1}(V^*)\otimes{V}$ for which $$\{ V\ni{\mathpzc{v}}\longrightarrow \xiup({\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}},{\mathpzc{v}})\in{V}\}\in\operatorname{\mathfrak{a}}_0,\;\;\forall {\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}}\in{V}.$$ Formula follows by polarization.
Denote by $C:{\mathpzc{S}}_{{{\mathpzc{p}}}{+}1}(V^*)\otimes{V}\to{\mathpzc{S}}_{{\mathpzc{p}}}(V^*)$ the contraction map. The Casimir element ${\mathpzc{c}}$ of ${V}^*{\otimes}V$ is the sum ${\sum}_{i{=}1}^n{\epsilonup}_i{\otimes}{\mathpzc{e}}_i,$ where ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ is any basis of $V$ and ${\epsilonup}_1,\hdots,{\epsilonup}_n$ its dual basis in $V^*.$ We note that $$C(\xiup{\cdot}{\mathpzc{c}})=({\mathpzc{p}}{+}1){\cdot}\xiup,\;\;\forall\xiup\in{\mathpzc{S}}_{{\mathpzc{p}}}(V^*),$$ so that the symmetric right product by $({\mathpzc{p}}{+}1)^{{-}1}{\cdot}{\mathpzc{c}}$ is a right inverse of the contraction. In particular, $${\mathpzc{X}}_{{\mathpzc{p}}}(V)={\mathpzc{X}}'_{{\mathpzc{p}}}(V)\oplus{\mathpzc{X}}''_{{\mathpzc{p}}}(V),\;\text{with}\;
\begin{cases}
{\mathpzc{X}}'_{{\mathpzc{p}}}(V)=\ker\left(C:{\mathpzc{S}}_{{{\mathpzc{p}}}{+}1}(V^*){\otimes}{V}{\to}
{\mathpzc{S}}_{{\mathpzc{p}}}(V^*) \right),\\
{\mathpzc{X}}''_{{\mathpzc{p}}}(V)=\{\xiup{\cdot}{\mathpzc{c}}\mid \xiup{\in}{\mathpzc{S}}_{{\mathpzc{p}}}(V^*)\},
\end{cases}$$ is the decomposition of ${\mathpzc{X}}_{{\mathpzc{p}}}$ into a direct sum of irreducible ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$[-]{}modules.
Since the operators $D_{{\mathpzc{v}}},D_{{\mathpzc{w}}}$ on ${\mathpzc{S}}(V^*)$ commute, $${\mathpzc{X}}'(V)=V\oplus{\mathfrak{sl}}_{\mathbb{K}}(V)\oplus{\sum}_{{\mathpzc{p}}{\geq}1}{\mathpzc{X}}'_{{\mathpzc{p}}}(V)$$ is the maximal EPFGLA of type ${\mathfrak{sl}}_{{\mathbb{K}}}(V)$ of $V.$
\[examp-3.3\] Let $V,W$ be finite dimensional vector spaces over ${\mathbb{K}}$ and ${\mathpzc{b}}{:}V{\times}V{\to}{W}$ a non degenerate symmetric bilinear form.
*If $\operatorname{\mathfrak{a}}_0$ is the orthogonal Lie algebra ${\mathfrak{o}}_{{\mathpzc{b}}}(V),$ consisting of $X{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ such that $${\mathpzc{b}}(X({\mathpzc{v}}_1),{\mathpzc{v}}_2){+}{\mathpzc{b}}({\mathpzc{v}}_1,X({\mathpzc{v}}_2)){=}0, \;\;
\text{for all ${\mathpzc{v}}_1,{\mathpzc{v}}_2{\in}V,$}$$ then $\operatorname{\mathfrak{a}}_{1}=0.$*
An element $\xiup{\in}\operatorname{\mathfrak{a}}_1$ is a map $\xiup{:}{V}{\to}{\mathfrak{o}}_{{\mathpzc{b}}}(V)$ such that $\xiup({\mathpzc{v}}_1)({\mathpzc{v}}_2){=}
\xiup({\mathpzc{v}}_2)({\mathpzc{v}}_1)$ for all ${\mathpzc{v}}_1,{\mathpzc{v}}_2{\in}{V}.$ Then, for ${\mathpzc{v}}_1,{\mathpzc{v}}_2,{\mathpzc{v}}_3{\in}V,$ we have $$\begin{aligned}
{\mathpzc{b}}(\xiup({\mathpzc{v}}_1)({\mathpzc{v}}_2),{\mathpzc{v}}_3)=\,{-}
{\mathpzc{b}}({\mathpzc{v}}_2,\xiup({\mathpzc{v}}_1)({\mathpzc{v}}_3))=\, {-}{\mathpzc{b}}({\mathpzc{v}}_2,\xiup({\mathpzc{v}}_3)({\mathpzc{v}}_1))
= {\mathpzc{b}}(\xiup({\mathpzc{v}}_3)({\mathpzc{v}}_2),{\mathpzc{v}}_1)\\
= {\mathpzc{b}}(\xiup({\mathpzc{v}}_2)({\mathpzc{v}}_3),{\mathpzc{v}}_1) = \,{-}{\mathpzc{b}}({\mathpzc{v}}_3,\xiup({\mathpzc{v}}_2)({\mathpzc{v}}_1)) =
\, {-}{\mathpzc{b}}(\xiup({\mathpzc{v}}_1)({\mathpzc{v}}_2),{\mathpzc{v}}_3).\end{aligned}$$ Since we assumed that ${\mathpzc{b}}$ is non degenerate, this implies that $\xiup({\mathpzc{v}}_1)({\mathpzc{v}}_2)=0$ for all ${\mathpzc{v}}_2{\in}V$ and hence that $\xiup({\mathpzc{v}}_1){=}0$ for all ${\mathpzc{v}}_1{\in}V,$ i.e. that $\xiup{=}0.$
The action $$(\xiup\otimes{\mathpzc{v}}){\cdot}{\mathpzc{w}}=
(D_{\mathpzc{w}}\xiup)\otimes{\mathpzc{v}},\;\;\forall \xiup\in{\mathpzc{S}}(V^*),\;\forall
{\mathpzc{v}},{\mathpzc{w}}\in{V}$$ defines on ${\mathpzc{X}}(V){=}{\mathpzc{S}}(V^*){\otimes}V$ the structure of a $\operatorname{\mathbb{Z}}$[-]{}graded right $V$[-]{}module. Its $\operatorname{\mathbb{Z}}$-graded dual $${\mathpzc{X}}^*(V)=\sum_{{\mathpzc{p}}=-1}^\infty {\mathpzc{X}}^*_{-{\mathpzc{p}}}(V),\;\;\text{with}\;\; {\mathpzc{X}}^*_{-{\mathpzc{p}}}(V)={\mathpzc{S}}_{{\mathpzc{p}}+1}(V)\otimes{V}^*$$ has a natural dual structure of left $V$-module (see §\[sec-t-1\]). The dual action of $V$ on ${\mathpzc{X}}^*(V)$ is left multiplication, which extends to left multiplication by elements of ${\mathpzc{S}}(V).$
By using the right-$V$-module structure of ${\mathpzc{X}}(V),$ we can rewrite the summands $\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}$ in by $$\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}=\{\xiup\in{\mathpzc{X}}_{{\mathpzc{p}}}(V)
\mid \xiup{\cdot}{\mathpzc{v}}_1{\cdots}
{\mathpzc{v}}_{\mathpzc{p}}{\in}\operatorname{\mathfrak{a}}_0,\;\forall{\mathpzc{v}}_1,\hdots,{\mathpzc{v}}_{{\mathpzc{p}}}
{\in}V\}.$$ Since $\operatorname{\mathfrak{a}}$ is a right-$V$-submodule of ${\mathpzc{X}}(V),$ its graded dual $\operatorname{\mathfrak{a}}^*$ is the quotient of the left $V$-module ${\mathpzc{X}}^*(V)$ by the annihilator ${\mathpzc{M}}$ of $\operatorname{\mathfrak{a}}$ in ${\mathpzc{X}}^*(V).$
The duality pairing of $V$ and $V^*$ makes ${\mathfrak{gl}}_{{\mathbb{K}}}(V)=V{\otimes}V^*$ self-dual, its pairing being defined by the trace form $$\langle{X}\,|\,{Y}\rangle={\mathrm{trace}}(X{\cdot}Y),
\;\;\forall X,Y\in{\mathfrak{gl}}_{{\mathbb{K}}}(V).$$ Let $$\operatorname{\mathfrak{a}}_0^0=\{X\in{\mathfrak{gl}}_{{\mathbb{K}}}(V)\mid
{\mathrm{trace}}(X{\cdot}A)=0,\;\forall A\in\operatorname{\mathfrak{a}}_0\}$$ be the annihilator of $\operatorname{\mathfrak{a}}_0$ in ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ Then the annihilator ${\mathpzc{M}}$ of $\operatorname{\mathfrak{a}}$ in ${\mathpzc{X}}^*(V)$ is the graded left-$V$-module $${\mathpzc{M}}={\mathpzc{S}}(V){\cdot}\operatorname{\mathfrak{a}}_0^0\subseteq{\mathpzc{X}}^*(V).$$
\[prop-tan-3-3\] The dual of the maximal effective $\operatorname{\mathfrak{a}}_0$-prolongation $\operatorname{\mathfrak{a}}$ of $V$ is the quotient module $$\vspace{-20pt}
\operatorname{\mathfrak{a}}^*={\mathpzc{X}}^*(V){/}{\mathpzc{M}}.$$
Let $V$ be a finite dimensional ${\mathbb{K}}$-vector space and $\operatorname{\mathfrak{a}}_0$ a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ Then the following are equivalent:
1. the maximal effective $\operatorname{\mathfrak{a}}_0$-prolongation of $V$ is finite dimensional;
2. the ${\mathpzc{S}}(V)$-module $\operatorname{\mathfrak{a}}^*$ is $(V)$-coprimary;
3. ${\mathpzc{S}}_h(V){{\pmb{\cdot}}}\operatorname{\mathfrak{a}}_0^0={\mathpzc{S}}_{h{+}1}(V)\otimes{V}^*,\;\;\text{
for some $h{\geq}0.$}$
We recall that $\operatorname{\mathfrak{a}}^*$ is said to be coprimary if, for each ${\mathpzc{s}}{\in}{\mathpzc{S}}(V),$ the homothety $\{\operatorname{\mathfrak{a}}^*{\ni}\,\etaup{\to}{\mathpzc{s}}{\cdot}\etaup\,{\in}\operatorname{\mathfrak{a}}^*\}$ is either nilpotent or injective. If this is the case, the radical $\sqrt{{\mathpzc{Ann}}(\operatorname{\mathfrak{a}}^*)}$ of the ideal ${\mathpzc{Ann}}(\operatorname{\mathfrak{a}}^*){=}\{{\mathpzc{s}}{\in}{\mathpzc{S}}(V)\,{|}\, {\mathpzc{s}}{\cdot}\operatorname{\mathfrak{a}}^*{=}\{0\}\}$ is a prime ideal in ${\mathpzc{S}}(V).$ \[See e.g. [@toug1972 p.8].\]
Being $\operatorname{\mathbb{Z}}$-graded, $\operatorname{\mathfrak{a}}^*$ is finite dimensional if and only if all ${\mathpzc{v}}{\in}V$ define nilpotent homotheties on $\operatorname{\mathfrak{a}}_0^*.$ This shows that (1) and (2) are equivalent. Finally, (3) is equivalent to the fact that $\operatorname{\mathfrak{a}}_0^*,$ and hence $\operatorname{\mathfrak{a}},$ is finite dimensional.
The advantage of using duality is to reduce the question about the finite dimensionality of the maximal prolongation to an exercise on finitely generated modules over the ring of polynomials with coefficients in ${\mathbb{K}}$ and eventually to linear algebra.
Having fixed a basis $\xiup_1,\hdots,\xiup_n$ of $V^*$ we can identify ${\mathpzc{X}}^*(V)$ to ${\mathpzc{S}(V)}^n.$ Each element $X$ of $\operatorname{\mathfrak{a}}_0^0$ can be viewed as a column vector $X{\mathpzc{v}}$ of ${\mathpzc{S}(V)}^n,$ whose entries are first degree polynomials in $V.$ By taking a set $X_1,\hdots,X_m$ of generators of $\operatorname{\mathfrak{a}}_0^0$ we obtain a matrix of homogeneous first degree polynomials $$\label{eq-3.12}
M({\mathpzc{v}})=(X_1{\mathpzc{v}},\hdots,X_m{\mathpzc{v}}) \in V^{\, n{\times}m}\subset{{\mathpzc{S}(V)}}^{n{\times}m},$$ that we can use to give a finite type presentation of $\operatorname{\mathfrak{a}}^*$: $$\label{eq-3.13}
\begin{CD}
{\mathpzc{S}(V)}^m @>{M({\mathpzc{v}})}>> {\mathpzc{S}(V)}^n @>>> \operatorname{\mathfrak{a}}^* @>>>0.
\end{CD}$$
\[thm-t-5-5\] Let ${\mathpzc{J}}_0(M)$ be the ideal generated by the order $n$ minor determinants of $M({\mathpzc{v}}).$ A necessary and sufficient condition for $\operatorname{\mathfrak{a}}$ to be finite dimensional is that $\sqrt{{\mathpzc{J}}_0(M)}=(V).$
Indeed, the ideals ${\mathpzc{J}}_0(M)$ and ${\mathpzc{Ann}}(\operatorname{\mathfrak{a}}^*)=\{f{\in}{\mathpzc{S}(V)}\mid f{\cdot}\operatorname{\mathfrak{a}}^*{=}\{0\}\}$ have the same radical (see e.g. [@toug1972 ch.[I]{},§[2]{}]).
Let ${\mathbb{F}}$ be the algebraic closure of the ground field ${\mathbb{K}}$ and set $V_{({\mathbb{F}})}{=}{\mathbb{F}}{\otimes}_{{\mathbb{K}}}V.$ By taking the tensor product by ${\mathbb{F}}$ we deduce from the exact sequence $$\label{eq-3.14}
\begin{CD}
{\mathpzc{S}}(V_{({\mathbb{F}})})^m @>{M({\mathpzc{z}})}>> {\mathpzc{S}}(V_{({\mathbb{F}})}) @>>> {\mathbb{F}}\,{\otimes}_{{\mathbb{K}}}\operatorname{\mathfrak{a}}^* @>>>0.
\end{CD}$$ We observe that $\operatorname{\mathfrak{a}}^*$ is $(V)$-coprimary if and only if ${\mathbb{F}}\,{\otimes}_{{\mathbb{K}}}\!\operatorname{\mathfrak{a}}^*$ is $(V_{({\mathbb{F}})})$-coprimary. Therefore Theorem translates into
\[thm-t-4-6\] A necessary and sufficient condition for the maximal prolongation $\operatorname{\mathfrak{a}}$ to be finite dimensional is that $$\operatorname{rank}(M({\mathpzc{z}}))=n=\dim(V),\;\;\forall {\mathpzc{z}}\in{V}_{({\mathbb{F}})}{\setminus}\{0\}.$$
In fact, since ${\mathbb{F}}$ is algebraically closed, by the Nullstellensatz (see e.g. [@Hart]) the necessary and sufficient condition for an ideal ${\mathpzc{J}}$ of ${\mathpzc{S}}(V_{({\mathbb{F}})})$ to have $\sqrt{{\mathpzc{J}}}{=}(V_{({\mathbb{F}})})$ is that $\{{\mathpzc{z}}{\in}V_{({\mathbb{F}})}{\mid} \, f({\mathpzc{z}}){=}0,\;\forall f\in{\mathpzc{J}}\}{=}\{0\}.$
\[ex-tan-3-6\] Denote by $\mathfrak{co}(n,{\mathbb{K}}){=}\{A{\in}{\mathfrak{gl}}_n({\mathbb{K}})\,{\mid}\,
A^\intercal{+}A{\in}{\mathbb{K}}{\cdot}{\mathrm{I}}_n\}$ the Lie algebra of conformal transformations of ${\mathbb{K}}^n.$
#### (1)
Let $\operatorname{\mathfrak{a}}_0=\mathfrak{co}(2,{\mathbb{K}}).$ Its orthogonal in ${\mathfrak{gl}}_2({\mathbb{K}})$ consists of the traceless symmetric matrices. Take the basis consisting of $$M_1=
\begin{pmatrix}
1 & 0\\
0 & {-}1
\end{pmatrix},\;\; M_2=
\begin{pmatrix}
0 & 1\\
1 & 0
\end{pmatrix}.$$ We obtain $$M({\mathpzc{z}})=
\begin{pmatrix}
{\mathpzc{z}}_1 & {\mathpzc{z}}_2\\
{-}{\mathpzc{z}}_2 & {\mathpzc{z}}_1
\end{pmatrix},\;\; \text{with}\;\; \det(M({\mathpzc{z}}))={\mathpzc{z}}_1^2{+}{\mathpzc{z}}_2^2.$$ Since the equation ${\mathpzc{z}}_1^2{+}{\mathpzc{z}}_2^2{=}0$ has non zero solutions in ${\mathbb{F}}^2,$ by Theorem \[thm-t-4-6\], ${\mathbb{K}}^2$ has an infinite dimensional effective $\mathfrak{co}(2,{\mathbb{K}})$-prolongation.
#### (2)
Let us consider next the case $n{>}2.$ The orthogonal $\operatorname{\mathfrak{a}}_0^0$ of $\operatorname{\mathfrak{a}}_0{=}\mathfrak{co}(n,{\mathbb{K}})$ consists of the traceless $n{\times}n$ symmetric matices. As a basis of $\operatorname{\mathfrak{a}}_{0}^0$ we can take the matrices
$\Delta_h{=}(\deltaup_{1,i}\deltaup_{1,j}{-}\deltaup_{h,i}\deltaup_{h,j}),$ $(h{=}2,\hdots,n)$ and $T_{h,k}{=}(\deltaup_{h,i}\deltaup_{k,j}{+}\deltaup_{k,i}\deltaup_{h,j}),$ $(1{\leq}{h}{<}k{\leq}n).$
Accordingly, we get $$M({\mathpzc{z}})=
\begin{pmatrix}
{\mathpzc{z}}_1 & {\mathpzc{z}}_1 & \hdots & {\mathpzc{z}}_1 & {\mathpzc{z}}_2 & {\mathpzc{z}}_3 & \hdots & {\mathpzc{z}}_n & 0 & \hdots \\
{-}{\mathpzc{z}}_2 & 0 & \hdots & 0 & {\mathpzc{z}}_1 & 0 & \hdots & 0 & {\mathpzc{z}}_3 & \hdots \\
0 & {-}{\mathpzc{z}}_3 & \hdots & 0 & 0 & {\mathpzc{z}}_1 & \hdots & 0 & {\mathpzc{z}}_2 & \hdots \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots&\vdots & \hdots \\
0 & 0 & \hdots &{-}{\mathpzc{z}}_n & 0 & 0 & \hdots & {\mathpzc{z}}_1 & 0 & \hdots
\end{pmatrix}.$$ We want to show that $M({\mathpzc{z}})$ has rank $n$ when ${\mathpzc{z}}{\in}{\mathbb{F}}^n{\setminus}\{0\}.$ The minor of the $(j{-}1)$-st, $n$-th, $\hdots,$ $(n{-}2)$-nd columns is $({\mathpzc{z}}_1^2{+}{\mathpzc{z}}_j^2){\cdot}{\mathpzc{z}}_1^{n{-}2},$ for $j{=}2,\hdots,n.$ Likewise, we can show that the ideal of order $n$ minor determinants of $M({\mathpzc{z}})$ contains all polynomials $({\mathpzc{z}}_j^2{+}{\mathpzc{z}}_h^2){\cdot}{\mathpzc{z}}_j^{n{-}2}$ for $1{\leq}j{\neq}h{\leq}n.$ Denote by ${\pm}{\mathpzc{i}}$ the roots of $({-}1)$ in ${\mathbb{F}}$ and assume by contradiction that $M({\mathpzc{z}})$ has rank ${<}n$ for some ${\mathpzc{z}}{\neq}0.$ We can assume that ${\mathpzc{z}}_1{\neq}0.$ This yields ${\mathpzc{z}}_{j}\,{=}{\pm}\,{\mathpzc{i}}\,{\mathpzc{z}}_1$ for $j{=}2,\hdots,n.$ But then $$({\mathpzc{z}}_2^2{+}{\mathpzc{z}}_3^2)\cdot{\mathpzc{z}}_2^{n{-}2}=-2(\pm{\mathpzc{i}})^{n{-}2}{\mathpzc{z}}_1^n\neq{0}.$$ This contradiction proves that $M({\mathpzc{z}})$ has rank $n$ for all ${\mathpzc{z}}{\in}{\mathbb{F}}^n{\setminus}\{0\},$ showing, by Theorem \[thm-t-4-6\], that the maximal EPFGLA of type $\mathfrak{co}(n,{\mathbb{K}})$ of ${\mathbb{K}}^n$ is finite dimensional if $n{\geq}3.$
Let us fix integers $0{<}{\mathpzc{p}}{\leq}{\mathpzc{q}}$ and set ${\mathfrak{B}}=\left.\left\{
\begin{pmatrix}
0 & B^{\intercal}\\
B & 0
\end{pmatrix}\,\right| B{\in}{\mathbb{K}}^{{\mathpzc{p}}{\times}{\mathpzc{q}}}\right\}.
$ Set $n{=}{\mathpzc{p}}{+}{\mathpzc{q}}$ and let $$\operatorname{\mathfrak{a}}_0=\{X\in{\mathfrak{gl}}_{{\mathbb{K}}}(n)\mid X^\intercal B{+}B\,X\in{\mathfrak{B}},\;\forall{B}\in{\mathfrak{B}}\}$$ be the Lie algebra of ${\mathfrak{B}}$-conformal tranformations of ${\mathbb{K}}^n.$ We have $$\operatorname{\mathfrak{a}}_0{=}\left.\left\{
\begin{pmatrix}
X & 0\\
0 & Y
\end{pmatrix}\,\right| X{\in}{\mathfrak{gl}}_{{\mathbb{K}}}({\mathpzc{q}}),Y{\in}{\mathfrak{gl}}_{{\mathbb{K}}}({\mathpzc{p}})\right\}\;\text{and hence}\;
\operatorname{\mathfrak{a}}_0^0= \left.\left\{
\begin{pmatrix}
0 & E\\
F^{\intercal} & 0
\end{pmatrix}\right| E,F{\in}{\mathbb{K}}^{{\mathpzc{q}}{\times}{\mathpzc{p}}}\right\}.$$ By taking the canonical basis of ${\mathbb{K}}^{{\mathpzc{q}}{\times}{\mathpzc{p}}}$ we obtain $$M({\mathpzc{z}})=
\begin{pmatrix}
{\mathpzc{z}}_{{\mathpzc{q}}{+}1}{\mathrm{I}}_{{\mathpzc{q}}} & \hdots & {\mathpzc{z}}_n{\mathrm{I}}_{{\mathpzc{q}}} & 0 & \hdots & 0 \\
0 & \hdots& 0 & {\mathpzc{z}}_1{\mathrm{I}}_{{\mathpzc{p}}} & \hdots & {\mathpzc{z}}_{{\mathpzc{q}}}{\mathrm{I}}_{{\mathpzc{p}}}
\end{pmatrix}.$$ Clearly both $({\mathpzc{z}}_1,\hdots,{\mathpzc{z}}_{{\mathpzc{p}}})$ and $({\mathpzc{z}}_{{\mathpzc{p}}{+}1},\hdots,{\mathpzc{z}}_n)$ are associated ideals of ${\mathpzc{J}}(M)$ and hence the maximal effective $\operatorname{\mathfrak{a}}_0$-prolongation of $V$ is infinite dimensional. Note that, if we take the ${\mathfrak{B}}$-orthogonal Lie algebra $${\mathfrak{o}}_{{\mathfrak{B}}}=\{X\in{\mathfrak{gl}}_{{\mathbb{K}}}(n)\mid X^\intercal{B}+B\,X=0,\;\forall B{\in}{\mathfrak{B}}\},$$ the maximal ${\mathfrak{o}}_{{\mathfrak{B}}}$-prolongation of ${\mathbb{K}}^n$ is finite dimensional by Example \[examp-3.3\], because, if $B_1,\hdots,B_{{\mathpzc{p}}{\mathpzc{q}}}$ is a basis of the ${\mathbb{K}}$-vector space ${\mathfrak{B}},$ the symmetric bilinear form ${\mathbb{K}}^n{\times} {\mathbb{K}}^n{\ni}({\mathpzc{v}}_1,{\mathpzc{v}}_2)\to({\mathpzc{v}}_1^{\intercal}B_i{\mathpzc{v}}_2)_{1{\leq}i{\leq}{\mathpzc{p}}{\mathpzc{q}}}
\in{\mathbb{K}}^{{\mathpzc{p}}{\mathpzc{q}}}$ is nondegenerate.
\[ex-tan-3.9\] Let $V={\mathbb{K}}^{2n}$ and $$\begin{aligned}
\operatorname{\mathfrak{a}}_0&={\mathfrak{sp}}(n,{\mathbb{K}})=\{A\in{\mathfrak{gl}}_{2n}({\mathbb{K}})\mid A^\intercal\Omega+\Omega{A}=0\},\;\;\text{with}\;\;
\Omega=
\begin{pmatrix}
0 & {\mathrm{I}}_n\\
{-}{\mathrm{I}}_n& 0
\end{pmatrix},\\
&=\left.\left\{
\begin{pmatrix}
A & B \\
C & -A^\intercal
\end{pmatrix}\right| A,B,C\in{\mathfrak{gl}}_{n}({\mathbb{K}}),\; B^\intercal{=}{B},\; C^\intercal{=}C\right\}.\end{aligned}$$ Then $$\operatorname{\mathfrak{a}}_0^0=\left.\left\{
\begin{pmatrix}
A & B \\
C & A^\intercal
\end{pmatrix}\right| A,B,C\in{\mathfrak{gl}}_n({\mathbb{K}}),\; B^{\intercal}{=}{-}B,\; ,C^\intercal{=}{-}C\right\}.$$ Take any basis $A_1,\hdots,A_h$ of ${\mathfrak{gl}}_n({\mathbb{K}})$ (with $h{=}n^2$) and $B_1,\hdots,B_k$ of ${\mathfrak{o}}(n,{\mathbb{K}})$ (with $k{=}\tfrac{1}{2}n(n{-}1)$). Then we have, for ${\mathpzc{z}},{\mathpzc{w}}\in{\mathbb{K}}^n,$ $$M({\mathpzc{z}},{\mathpzc{w}})=
\begin{pmatrix}
A_1{\mathpzc{z}}& \hdots & A_h{\mathpzc{z}}& B_1{\mathpzc{w}}& \hdots & B_k{\mathpzc{w}}& 0 & \hdots & 0\\
A_1^\intercal{\mathpzc{w}}& \hdots & A_h^\intercal{\mathpzc{w}}& 0 & \hdots & 0 & B_1{\mathpzc{z}}& \hdots & B_k{\mathpzc{z}}\end{pmatrix}.$$ If we take e.g. ${\mathpzc{w}}{=}0,$ we see that, for ${\mathpzc{z}}_0{\neq}0,$ $M({\mathpzc{z}}_0,0)$ has rank $(2n{-}1),$ because $B_1{\mathpzc{z}}_0,\hdots,B_k{\mathpzc{z}}_0$ span the *orthogonal* hyperplane ${\mathpzc{z}}_0^\perp{=}\{{\mathpzc{z}}{\in}{\mathbb{K}}^n{\mid}\,{\mathpzc{z}}^\intercal{\mathpzc{z}}_0{=}0\}$ to ${\mathpzc{z}}_0$ in ${\mathbb{K}}^n$. By Theorem \[thm-t-4-6\] this implies that the maximal ${\mathfrak{sp}}(n,{\mathbb{K}})$-prolongation of ${\mathbb{K}}^{2n}$ is infinite dimensional.
The criterion of Theorem \[thm-t-4-6\] can also be expressed, in an equivalent way, as an *ellipticity condition*. In the case when, given an ${\textswab{m}}{=}{\sum}_{0{<}{\mathpzc{p}}{\leq}\muup}{{\mathfrak{g}}}_{-{\mathpzc{p}}},$ we take $\operatorname{\mathfrak{L}}{=}{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ and hence $\operatorname{\mathfrak{a}}_{0}=\{\xiup\,{\in}\,{\mathfrak{gl}}_{{\mathbb{K}}}(V)
\,{\mid}\,[\xiup,{\mathfrak{g}}_{-{\mathpzc{p}}}]\,{=}\,0,\;\text{for}\;{\mathpzc{p}}{>}1\},$ it was already considered by many authors: see e.g. [@GQS1966; @Spencer1969] and [@Ottazzi2011] for a thorough discussion.
\[thm-3.11\] Let $V$ be a finite dimensional ${\mathbb{K}}$-vector space, $\operatorname{\mathfrak{a}}_0$ a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ and ${\mathbb{F}}$ the algebraic closure of ${\mathbb{K}}.$
The maximal EPFGLA of type $\operatorname{\mathfrak{a}}_0$ of $V$ is infinite dimensional if and only if ${\mathbb{F}}{\otimes}\operatorname{\mathfrak{a}}_0$ contains an element of rank one on $V_{({\mathbb{F}})}.$
We can as well assume that ${\mathbb{K}}$ is algebraically closed.
All rank one elements of ${\mathfrak{gl}}_{\mathbb{K}}(V)$ can be written in the form ${\mathpzc{v}}{\otimes}\xiup,$ with nonzero ${\mathpzc{v}}{\in}V$ and $\xiup{\in}V^*.$ If $X{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V),$ we obtain $$\tag{$*$} \label{eq-3.*}
\langle{X}{\mathpzc{v}}\,{|}\, \xiup\rangle = {\mathrm{trace}}(X\,({\mathpzc{v}}{\otimes}\xiup)).$$ If $\operatorname{\mathfrak{a}}_0$ contains ${\mathpzc{v}}{\otimes}\xiup,$ then $X{\mathpzc{v}}{\in}\xiup^0{=}\{{\mathpzc{w}}{\in}V{\mid} \langle{\mathpzc{w}}\,{|}\,\xiup
\rangle{=}0\}$ for all $X{\in}\operatorname{\mathfrak{a}}_0^0,$ showing that $M({\mathpzc{v}})$ has rank less than $n{=}\dim(V).$ Then $\dim(\operatorname{\mathfrak{a}}){=}\infty$ by Theorem \[thm-t-4-6\].
Vice versa, if $M({\mathpzc{v}})$ has rank less than $n$ for some ${\mathpzc{v}}{\neq}0,$ we can find a nonzero $\xiup{\in}V^*$ such that $X{\mathpzc{v}}{\in}\xiup^0$ for all $X{\in}\operatorname{\mathfrak{a}}_0^0.$ Since the trace form is nondegenerate on ${\mathfrak{gl}}_{{\mathbb{K}}}(V),$ by this implies that $({\mathpzc{v}}{\otimes}\xiup)$ is a rank one element of $\operatorname{\mathfrak{a}}_0.$
Prolongation of irreducible representations {#subs-3-3}
-------------------------------------------
Assume that ${\mathbb{K}}$ is algebraically closed and let $V$ be a finite dimensional faithful irreducible representation of a reductive Lie algebra $\operatorname{\mathfrak{a}}_0$ over ${\mathbb{K}},$ having a center $\operatorname{\mathfrak{z}}_0$ of dimension ${\leq}1.$ Let $\operatorname{\mathfrak{s}}=[\operatorname{\mathfrak{a}}_0,\operatorname{\mathfrak{a}}_0]$ be the semisimple ideal of $\operatorname{\mathfrak{a}}_0,$ ${\mathfrak{h}}$ its Cartan subalgebra and ${\mathpzc{R}},$ $\Lambda_W$ its corresponding root system and weight lattice. Fix a lexicographic order on ${\mathpzc{R}},$ corresponding to the choice of a Borel subalgebra of $\operatorname{\mathfrak{s}}.$ Then $V$ is a faithful irreducible $\operatorname{\mathfrak{s}}$-module. Let $\Lambda(V){\subset}\Lambda_W$ be the set of its weights and $\phiup$ its dominant weight. If $\psiup$ is minimal in $\Lambda(V),$ then $\phiup{-}\psiup$ is dominant in $\Lambda(V{\otimes}V^*).$ Let ${\mathpzc{v}}_\phiup$ be a maximal vector in $V$ and $\xiup_{{-}\psiup}$ a maximal covector in $V^*.$ Then $v_{\phiup}\otimes\xiup_{{-}\psiup}$ is an element, of rank one on $V,$ generating an irreducible $\operatorname{\mathfrak{s}}$-sub-module $L_{\phiup{-}\psiup}$ of $V{\otimes}V^*.$ By a Theorem of Dynkin (see [@As94], [@Cahn Ch.XIV], [@Dy1952] ) we know that all nonzero elements of $L_{\phiup{-}\psiup}^0$ have rank larger than one. Hence by using Theorem \[thm-3.11\], we obtain
\[thm-3.12\] The maximal effective prolongation of type $\operatorname{\mathfrak{a}}_0$ of $V$ is infinite dimensional if and only if
$L_{\phiup{-}\psiup}{\subset}\operatorname{\mathfrak{s}}.$
The Lie algebra $\operatorname{\mathfrak{s}}$ decomposes into a direct sum $S_1{\oplus}\cdots{\oplus}S_k$ of irreducible $\operatorname{\mathfrak{s}}$-sub-modules of $V{\otimes}V^*.$ The summands $S_i$ which are distinct from $L_{\phiup{-}\psiup}$ are contained in $L_{\phiup{-}\psiup}^0.$ The statement follows from this observation.
Under the assumptions above, if $\operatorname{\mathfrak{s}}{=}[\operatorname{\mathfrak{a}}_0,\operatorname{\mathfrak{a}}_0]$ is simple, then the maximal prolongation $\operatorname{\mathfrak{a}}$ of type $\operatorname{\mathfrak{a}}_0$ of $V$ is primitive in the sense explained in [@guillemin1970]. Then Theorem \[thm-3.12\] yields easily a result about the infinite dimensionality of the maximal effective *primitive* prolongations that has been already proved by several Authors (see [@Cartan1909; @guillemin1970; @GQS1966; @Guillemin1967; @Kac1967; @kn1965b; @morimoto1970; @shnider1970; @Wil1971]).
\[prop-3.11\] Assume that ${\mathbb{K}}$ is algebraically closed, that $\operatorname{\mathfrak{a}}_0$ is reductive, with $[\operatorname{\mathfrak{a}}_0,\operatorname{\mathfrak{a}}_0]$ simple, and that $V$ is a faithful irreducible $\operatorname{\mathfrak{a}}_0$-module. Then the maximal effective prolongation of type $\operatorname{\mathfrak{a}}_0$ of $V$ is infinite dimensional if and only if one of the following is verified
- $\operatorname{\mathfrak{a}}_0$ is equal either to ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ or ${\mathfrak{sl}}_{{\mathbb{K}}}(V)$;
- $V{\simeq}{\mathbb{K}}^{2n}$ for some integer $n{\geq}2$ and $\operatorname{\mathfrak{a}}_0$ is isomorphic either to ${\mathfrak{sp}}(n,{\mathbb{K}})$ or to $\mathfrak{c}{\mathfrak{sp}}(n,{\mathbb{K}}).$
${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongations of FGLA’s of the second kind {#sec5}
==============================================================================
Let ${\textswab{m}}{=}{\mathfrak{g}}_{{-}1}\oplus{\mathfrak{g}}_{{-}2}$ be an FGLA of the second kind. Set $V{=}{\mathfrak{g}}_{{-}1}.$ By Proposition \[prop-tan-1.4\], ${\textswab{m}}$ is isomorphic to a quotient $\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}},$ for a graded ideal ${\mathpzc{K}}$ of $\operatorname{\mathfrak{f}}(V)$ with $\operatorname{\mathfrak{f}}_{[3]}(V){\subseteq}{\mathpzc{K}}{\subsetneqq}\operatorname{\mathfrak{f}}_{[2]}(V).$ We note that ${\mathpzc{K}}{=}{\mathpzc{K}}_{\;{-}2}{\oplus}\operatorname{\mathfrak{f}}_{[3]}(V)$ is a graded ideal of $\operatorname{\mathfrak{f}}(V)$ for every proper vector subspace ${\mathpzc{K}}_{\;{-}2}$ of $\operatorname{\mathfrak{f}}_{-2}(V).$
Since $\operatorname{\mathfrak{f}}_{-2}(V){=}\Lambda^2(V),$ the subspace ${\mathpzc{K}}_{\;-2}$ is the kernel of the surjective linear map $\lambdaup:\Lambda^2(V)\rightarrow\!\!\!\!\rightarrow{\mathfrak{g}}_{-2}$ associated to the bilinear map $({\mathpzc{v}}_1,{\mathpzc{v}}_2)\to [{\mathpzc{v}}_1,{\mathpzc{v}}_2]$ defined by the Lie product of elements of $V$: $$\label{eq2.5a}
\xymatrix{{V}\times{V} \ar[rr]^{({\mathpzc{v}}_1,{\mathpzc{v}}_2)
\to{\mathpzc{v}}_1{\wedge}{\mathpzc{v}}_2} \ar[dr]_{({\mathpzc{v}}_1,{\mathpzc{v}}_2)\to[{\mathpzc{v}}_1,{\mathpzc{v}}_2]\quad}
&& \Lambda^2({V}) \ar[dl]^{\lambdaup}\\
&\,{\mathfrak{g}}_{{-}2}.}$$
Let us consider $n+\binom{n}{2}$ variables, that we label by $x_i$ for $1{\leq}i{\leq}n$ and $t_{i,j}$ for $1{\leq}i{<}j{\leq}n.$ The algebra ${\textswab{m}}$ of real vector fields with polynomial coefficients generated by $${\mathfrak{g}}_{-1}\simeq\operatorname{{\mathbb{R}}}^n=\left\langle
\frac{\partial}{\partial{x}_i}+{\sum}_{h<i}x_h\,
\frac{\partial}{\partial{t}_{h,i}}
-{\sum}_{h>i}x_h\,\frac{\partial}{\partial{t}_{i,h}}
\mid 1{\leq}i{\leq}n\right\rangle$$ has ${\mathfrak{g}}_{-2}{=}[{\mathfrak{g}}_{-1},{\mathfrak{g}}_{-1}]=\left\langle
\frac{\partial}{\partial{t}_{i,j}}\mid 1{\leq}i{<}j{\leq}n
\right\rangle$ so that ${\textswab{m}}\simeq\operatorname{{\mathbb{R}}}^n\oplus\Lambda^2(\operatorname{{\mathbb{R}}}^n),$ corresponds to the case ${\mathpzc{K}}_{\;-2}=\{0\}.$
We keep the notation . For te case of FGLA’s of the second kind Proposition \[prop-tan-1.4\] reads
Two graded fundamental Lie algebras of the second kind ${\textswab{m}}({\mathpzc{K}})$ and ${\textswab{m}}({\mathpzc{K}}')$ are isomorphic if and only if the homogeneous parts of second degree ${\mathpzc{K}}_{\;{-2}}$ and ${\mathpzc{K}}'_{\;{-2}}$ of ${\mathpzc{K}}$ and ${\mathpzc{K}}'$ are ${\mathbf{GL}}_{{\mathbb{K}}}(V)$-congruent.
Fix a proper vector subspace ${\mathpzc{K}}_{\;{-}2}$ of $\operatorname{\mathfrak{f}}_{{-}2}(V){=}\Lambda^2(V)$ and the corresponding $2$-cofinite ideal ${\mathpzc{K}}{=}{\mathpzc{K}}_{\;{-}2}{\oplus}\operatorname{\mathfrak{f}}_{[3]}(V).$ By , the Lie algebra of zero-degree derivations of ${\textswab{m}}({\mathpzc{K}})$ is characterised by
\[lem-tan-6-3\] The Lie algebra ${\mathfrak{g}}_0({\mathpzc{K}})$ of the $0$-degree derivations of ${\textswab{m}}({\mathpzc{K}})$ is $$\vspace{-20pt}
\label{eq2.3}
{\mathfrak{g}}_0({\mathpzc{K}})=\{A\in{\mathfrak{gl}}_{\mathbb{K}}(V)\mid T_A({\mathpzc{K}}_{\;-2})\subseteq{\mathpzc{K}}_{\;-2}\}.$$
We recall from §\[sec-fund\] that, for ${\mathpzc{K}}{=}{\mathpzc{K}}_{\;-2}{\oplus}\operatorname{\mathfrak{f}}_{[3]}(V),$ the maximal effective canonical ${\mathfrak{gl}}_{\mathbb{K}}(V)$-prolongation of ${\textswab{m}}({\mathpzc{K}})$ is the $\operatorname{\mathbb{Z}}$[-]{}graded Lie algebra $${\mathfrak{g}}({\mathpzc{K}})={\sum}_{{\mathpzc{p}}\geq{-}2}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})$$ whose homogeneous summands are defined by $$\begin{cases}
{\mathfrak{g}}_{-2}({\mathpzc{K}}){=}\Lambda^2(V){/}{\mathpzc{K}}_{\;-2},\\
{\mathfrak{g}}_{-1}({\mathpzc{K}}){=}V,\\
{\mathfrak{g}}_h({\mathpzc{K}})={\mathpzc{Der}}_h\left({\textswab{m}},{\sum}_{{\mathpzc{p}}<h}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})\right),\;\;\text{for $h{\geq}0.$}
\end{cases}$$ The spaces ${\mathfrak{g}}_h({\mathpzc{K}})$ are defined by recurrence and consist of the degree $h$ homogeneous derivations of ${\textswab{m}}({\mathpzc{K}})$ with values in the ${\textswab{m}}({\mathpzc{K}})$-module ${\sum}_{{\mathpzc{p}}<h}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})$: an element $\alphaup$ of ${\mathfrak{g}}_h({\mathpzc{K}})$ is identified with a map $$\begin{gathered}
\alphaup:(V,\Lambda^2(V))\longrightarrow
({\mathfrak{g}}_{h-1}({\mathpzc{K}}),{\mathfrak{g}}_{h-2}({\mathpzc{K}})),\;\; \text{with}\\
\alphaup({\mathpzc{v}}{\wedge}{\mathpzc{w}})=
(\alphaup({\mathpzc{v}}))({\mathpzc{w}}){-}(\alphaup({\mathpzc{w}}))({\mathpzc{v}}),\;\;\forall {\mathpzc{v}},{\mathpzc{w}}{\in}
V,\;\;\text{and}\;\;
\alphaup(\omegaup)=0,\;\;\forall\omegaup\in{\mathpzc{K}}_{\;-2}.\end{gathered}$$ By Theorem \[thm-t-4-2\], ${\mathfrak{g}}({\mathpzc{K}})$ is finite dimensional if and only if $$\operatorname{\mathfrak{a}}({\mathpzc{K}})\,{=}\!\!\sum_{{\mathpzc{p}}=-1}^\infty \operatorname{\mathfrak{a}}_{{\mathpzc{p}}}({\mathpzc{K}}),\;\text{with}\; \operatorname{\mathfrak{a}}_{{\mathpzc{p}}}({\mathpzc{K}}){=}\{
\alphaup{\in}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}}){\mid} [\alphaup,{\mathfrak{g}}_{-2}({\mathpzc{K}})]{=}\{0\}\}$$ is finite dimensional.
We have $\operatorname{\mathfrak{a}}_0({\mathpzc{K}}_{\;-2})=\{A\in{\mathfrak{gl}}_n(\operatorname{{\mathbb{R}}})\mid T_A(\Lambda^2(\operatorname{{\mathbb{R}}}^n))\subseteq{\mathpzc{K}}_{\;-2}\}.$
Fix an identification $V{\simeq}{\mathbb{K}}^n,$ to consider the non degenerate symmetric bilinear form ${\mathpzc{b}}({\mathpzc{v}},{\mathpzc{w}})={\mathpzc{v}}^{\intercal}{\cdot}{\mathpzc{w}}$ on $V.$ It yields an isomorphism $$\rhoup:\Lambda^2(V)\longrightarrow{\mathfrak{o}}(V),\;\;\text{with}\;\;
\rhoup({\mathpzc{v}}{\wedge}{\mathpzc{w}})={\mathpzc{v}}{\cdot}{\mathpzc{w}}^{\intercal}-{\mathpzc{w}}{\cdot}{\mathpzc{v}}^\intercal,\;\forall{\mathpzc{v}},{\mathpzc{w}}\in{V},$$ between $\Lambda^2(V)$ and the orthogonal Lie algebra $${\mathfrak{o}}(V)=\{X\in{\mathfrak{gl}}_{{\mathbb{K}}}(V)\mid X^{\intercal}{+}\,X=0\}.$$ Under this identification, the action of $A{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ on $\Lambda^2(V)$ can be described by $A{\cdot}X{=}A\,X{+}X\,A^{\intercal}.$ We use $\rhoup$ to identify ${\mathpzc{K}}_{\;-2}$ with a linear subspace of ${\mathfrak{o}}(V).$ With this notation, we introduce $$\label{eq2.4}
{\mathpzc{K}}_{\;-2}^\perp=\{X\in{\mathfrak{o}}(V)\mid {\mathrm{trace}}(X{\, }K)=0,\;\forall K\in{\mathpzc{K}}_{\;-2}\}\simeq{\mathfrak{g}}_{-2}({\mathpzc{K}}).$$
For the ideal ${\mathpzc{K}}'={\mathpzc{K}}_{\;-2}^\perp\oplus\
\operatorname{\mathfrak{f}}_{[3]}(V)$ we have $${\mathfrak{g}}_0({\mathpzc{K}}')=\{A^\intercal\mid A\in{\mathfrak{g}}_0({\mathpzc{K}})\}.$$ which is a Lie algebra anti-isomorphic to ${\mathfrak{g}}_0({\mathpzc{K}}),$ but, of course, $\operatorname{\mathfrak{a}}_0({\mathpzc{K}})$ and $\operatorname{\mathfrak{a}}_0({\mathpzc{K}}')$ may turn out to be quite different.
Note that ${\mathpzc{K}}^\perp$ can be canonically identified with ${\mathfrak{g}}_{{-}2}.$
\[lem-t-6-14\] We have $ \operatorname{\mathfrak{a}}_0^0({\mathpzc{K}})=\{X{\, }Y\mid X\in{\mathfrak{o}}(V),\; Y\in{\mathpzc{K}}_{\;-2}^\perp\}.$
In fact, for $A{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V)$ and $X,Y{\in}{\mathfrak{o}}(V),$ we obtain $$\begin{aligned}
{\mathrm{trace}}((AX{+}XA^\intercal)Y)={\mathrm{trace}}(AXY)+{\mathrm{trace}}(YXA^{\intercal})=2{\mathrm{trace}}(A{\, }(XY)),\end{aligned}$$ because $(YXA^\intercal)^\intercal=A{\, }(XY).$ Thus we obtain $$\begin{aligned}
A\in\operatorname{\mathfrak{a}}_0\Leftrightarrow {\mathrm{trace}}((A{\, }X{+}X{\, }A^\intercal){\, }Y){=}
2{\mathrm{trace}}(A{\, }(Y{\, }X)) =0, \;\;
\forall X{\in}{\mathfrak{o}}(V),\forall{Y}{\in}{\mathpzc{K}}^\perp,\end{aligned}$$ proving our statement.
To apply Theorem \[thm-t-4-6\] to investigate the finite dimensionality of the maximal ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongation of ${\textswab{m}}({\mathpzc{K}}),$ we need to construct the matrix $M({\mathpzc{z}})$ in , relative to $\operatorname{\mathfrak{a}}_0^0({\mathpzc{K}}),$ that we will denote by $M_2({\mathpzc{K}},{\mathpzc{z}}).$ As usual, we denote by ${\mathbb{F}}$ be the algebraic closure of ${\mathbb{K}}$ and set $V_{({\mathbb{F}})}={\mathbb{F}}{\otimes}_{{\mathbb{K}}}V.$
We can proceed as follows. Fix generators $Y_1,\hdots,Y_m$ of ${\mathpzc{K}}^\perp$ and, for ${{{\mathpzc{z}}}}{\in}V_{({\mathbb{F}})}$ and $1{\leq}i{\leq}m,$ consider the vectors $Y_i{{{\mathpzc{z}}}}{\in}{V}_{({\mathbb{F}})}.$ By the identification $V{\simeq}{\mathbb{K}}^n,$ $$\label{eq-t-6-21}
\Phi_{{\mathpzc{K}}}({\mathpzc{z}})=(Y_1{\mathpzc{z}},\hdots,Y_m{\mathpzc{z}})
\in
({\mathbb{K}}[{\mathpzc{z}}_1,\hdots,{\mathpzc{z}}_n])^{n\times{m}},$$ is a matrix of first order homogeneous polynomials in ${\mathbb{K}}[{\mathpzc{z}}_1,\hdots,{\mathpzc{z}}_n]$ and, after choosing generators $X_1,\hdots,X_N$ of ${\mathfrak{o}}(V)$ as a ${\mathbb{K}}$-linear space, we take $$\label{equ-tan-4.16}
M_2({\mathpzc{K}},{\mathpzc{z}})=(X_1\Phi_{{\mathpzc{K}}}({\mathpzc{z}}),\hdots,X_N\Phi_{{\mathpzc{K}}}({\mathpzc{z}}))\in
({\mathbb{K}}[{\mathpzc{z}}_1,\hdots,{\mathpzc{z}}_n])^{n\times(mN)}.$$
The ${\mathfrak{o}}(V)$-orbit of a non zero vector ${\mathpzc{z}}$ of ${\mathbb{F}}^n$ spans the hyperplane $${\mathpzc{z}}^\perp=\{{\mathpzc{w}}\in{\mathbb{F}}^n\mid {\mathpzc{z}}^{\intercal}{}\,
{\mathpzc{w}}=0\}.$$ To check this fact, we can reduce to the case where ${\mathbb{F}}{=}{\mathbb{K}}.$ Take any ${\mathpzc{u}}{\in}V$ with ${\mathpzc{z}}^{\intercal}{\mathpzc{u}}{=}1.$ If ${\mathpzc{w}}{\in}{\mathpzc{z}}^\perp,$ then the matrix $X{=}{\mathpzc{w}}\,{\mathpzc{u}}^\intercal{-}{\mathpzc{u}}\,{\mathpzc{w}}^{\intercal}$ belongs to ${\mathfrak{o}}(V)$ and $X\,{\mathpzc{z}}{=}{\mathpzc{w}}.$ This shows that ${\mathpzc{z}}^{\perp}{\subset}\,{\mathfrak{o}}(V)\,{\mathpzc{z}}.$ The opposite inclusion is obvious, since ${\mathpzc{z}}^{\intercal}X{\mathpzc{z}}{=}0$ for all $X{\in}{\mathfrak{o}}(V)$ and ${\mathpzc{z}}{\in}V.$
Hence, if ${\mathpzc{z}}_1,{\mathpzc{z}}_2{\in}V_{({\mathbb{F}})}$ are linearly independent, then ${\mathfrak{o}}(n)\,{\mathpzc{z}}_1{+}{\mathfrak{o}}(n)\,{\mathpzc{z}}_2$ spans $V_{({\mathbb{F}})},$ so that $M_2({\mathpzc{K}},{\mathpzc{z}})$ has rank $n$ for all ${\mathpzc{z}}$ for which $\Phi_{{\mathpzc{K}}}({\mathpzc{z}})$ has rank ${\geq}{2}.$ We proved the following
\[prop-t-6-10\] Let ${\mathpzc{K}}_{\;-2}\subset\operatorname{\mathfrak{f}}_{-2}(V)$ and ${\mathpzc{K}}={\mathpzc{K}}_{\;-2}\oplus\operatorname{\mathfrak{f}}_{[3]}(V).$ Then the maximal ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongation ${\mathfrak{g}}({\mathpzc{K}})$ of ${\textswab{m}}({\mathpzc{K}})$ is finite dimensional if and only if $$\vspace{-20pt}
\{{\mathpzc{z}}\in{V}_{({\mathbb{F}})}\mid \operatorname{rank}(\Phi_{{\mathpzc{K}}}({\mathpzc{z}}))<{2}\}=\{0\}.$$
We can give an equivalent formulation of Proposition \[prop-t-6-10\] involving the *rank* of the ${\mathbb{F}}$-bilinear extension of the alternate bilinear form on $V$ defined by the Lie brackets.
Let ${\mathbb{F}}$ be the algebraic closure of ${\mathbb{K}}.$ We call the integer $$\ell=\inf\{\dim_{{\mathbb{F}}}([{\mathpzc{z}},V_{({\mathbb{F}})}])\mid 0\neq{\mathpzc{z}}\in{V}_{({\mathbb{F}})}\}$$ the *algebraic minimum rank* of ${\textswab{m}}({\mathpzc{K}}).$
\[thm-tan-4-8\] Let ${\mathpzc{K}}_{\;-2}\subset\operatorname{\mathfrak{f}}_{-2}(V)$ and ${\mathpzc{K}}={\mathpzc{K}}_{\;-2}\oplus\operatorname{\mathfrak{f}}_{[3]}(V).$ Then the maximal ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongation ${\mathfrak{g}}({\mathpzc{K}})$ of ${\textswab{m}}({\mathpzc{K}})$ is finite dimensional if and only if ${\textswab{m}}({\mathpzc{K}})$ has algebraic minimum rank $\ell\geq{2}.$
If we identify ${\mathpzc{z}}\,{\in}{V}_{({\mathbb{F}})}$ with the corresponding numerical vector in ${\mathbb{F}}^n,$ then ${\mathpzc{w}}^{\intercal}Y_i{\mathpzc{z}}$ is the $Y_i$-component of $[{\mathpzc{w}},{\mathpzc{z}}].$ The condition that the minimum rank of ${\textswab{m}}({\mathpzc{K}})$ is larger or equal to two is then equivalent to the fact that $\Phi_{{\mathpzc{K}}}({\mathpzc{z}})$ has rank ${\geq}2$ for all ${\mathpzc{z}}\in{V}_{({\mathbb{F}})}.$
Let ${\mathbb{K}}=\operatorname{{\mathbb{R}}},$ $n{=}4$ and ${\mathpzc{K}}=\left.\left\langle
\left(\begin{smallmatrix}
0 & A\\
{-}A & 0
\end{smallmatrix}\right)\right| A\in\operatorname{{\mathbb{R}}}^{2\times{2}}, A=A^\intercal\right\rangle.$
With ${\mathrm{J}}{=}
\left(\begin{smallmatrix}
0 & 1\\
{-}1 & 0
\end{smallmatrix}\right),$ a basis of ${\mathpzc{K}}^\perp$ is given by the matrices $$Y_1=
\begin{pmatrix}
{\mathrm{J}}& 0\\
0 & 0
\end{pmatrix},\;\; Y_2=
\begin{pmatrix}
0 & 0\\
0 & {\mathrm{J}}\end{pmatrix},\;\; Y_3=
\begin{pmatrix}
0 & {\mathrm{J}}\\
{\mathrm{J}}& 0
\end{pmatrix}.$$ This yields $$\Phi_{{\mathpzc{K}}}({\mathpzc{z}}) =
\begin{pmatrix}
{\mathpzc{z}}_2 & 0 & {\mathpzc{z}}_4\\
{-}{\mathpzc{z}}_1 & 0 & {-}{\mathpzc{z}}_3\\
0 & {\mathpzc{z}}_4 & {\mathpzc{z}}_2\\
0 & {-}{\mathpzc{z}}_3 & {-}{\mathpzc{z}}_1
\end{pmatrix}.$$ Writing $\Delta^{i,j}_{h,k}$ for the minor of the lines $i,j$ and the columns $h,k$, we obtain $$\Delta^{2,4}_{1,3}={\mathpzc{z}}_1^2,\; \; \Delta^{1,3}_{1,3}={\mathpzc{z}}_2^2,\;\;\Delta^{2,4}_{1,3}={\mathpzc{z}}_3^2,\;\;
\Delta^{1,3}_{2,3}={\mathpzc{z}}^2_4,$$ showing that ${\mathfrak{g}}({\mathpzc{K}})$ is finite dimensional.
Let ${\mathbb{K}}=\operatorname{{\mathbb{R}}},$ $n{=}4$ and ${\mathpzc{K}}^\perp$ generated by the matrices $$Y_1=
\begin{pmatrix}
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1\\
{-}1 & 0 & 0 & 0\\
0 & {-}1 & 0 & 0
\end{pmatrix},\; Y_2=
\begin{pmatrix}
0 & 0 & 0 & 1\\
0 & 0 & {-}1 & 0\\
0 & 1 & 0 & 0\\
{-}1 & 0 & 0 & 0
\end{pmatrix},\; Y_3=
\begin{pmatrix}
0 & 1 & 0 & 0 \\
{-}1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{pmatrix}.$$ Then $$\Phi_{{\mathpzc{K}}}({\mathpzc{z}})=
\begin{pmatrix}
{\mathpzc{z}}_3 & {\mathpzc{z}}_4 & {\mathpzc{z}}_2\\
{\mathpzc{z}}_4 & {-}{\mathpzc{z}}_3 & {-}{\mathpzc{z}}_1\\
{-}{\mathpzc{z}}_1 & {\mathpzc{z}}_2 & 0 \\
{-}{\mathpzc{z}}_2 & {-}{\mathpzc{z}}_1 & 0
\end{pmatrix}.$$ The matrix $\Phi_{{\mathpzc{K}}}({\mathpzc{z}})$ has rank ${\geq}2$ for all nonzero ${\mathpzc{z}}\in\operatorname{{\mathbb{R}}}^4,$ but $$\Phi_{{\mathpzc{K}}}(0,0,1,{\mathpzc{i}})=\left(
\begin{matrix}
1 & {\mathpzc{i}}& 0\\
{\mathpzc{i}}& {-}1 & 0\\
0 & 0 & 0\\
0 & 0 & 0
\end{matrix}\right)$$ has rank $1$. Thus ${\mathfrak{g}}({\mathpzc{K}})$ is infinite dimensional, although ${\textswab{m}}({\mathpzc{K}})$ has (real) minimal rank $2.$
We note that, if $\Phi_{{\mathpzc{K}}}({\mathpzc{z}})$ has at most $2$ columns, then the set $$\{{\mathpzc{z}}{\in}V_{({\mathbb{F}})}{\mid} \operatorname{rank}(\Phi_{{\mathpzc{K}}}({\mathpzc{z}})){<}2\}$$ is an algebraic affine variety of positive dimension in $V_{({\mathbb{F}})}.$ In particular, we obtain (cf. e.g. [@Kruglikov2011])
\[cor-tan-4-11\] If $\dim({\mathfrak{g}}_{-2}({\mathpzc{K}})){\leq}{2},$ then ${\mathfrak{g}}({\mathpzc{K}})$ is infinite dimensional.
${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongations of FGLA’s of higher kind {#sect7}
==========================================================================
Let $V$ be a ${\mathbb{K}}$-vector space of finite dimension $n{\geq}2$ and, for an integer $\muup\geq{3},$ fix a $\muup$-cofinite $\operatorname{\mathbb{Z}}$-graded ideal ${\mathpzc{K}}{\subset}\operatorname{\mathfrak{f}}_{[2]}(V)$ of $\operatorname{\mathfrak{f}}(V).$ Let $${\textswab{m}}({\mathpzc{K}}){=}{\sum}_{{\mathpzc{p}}=-1}^{-\muup}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})\simeq\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}}, \quad
{\mathfrak{n}}({\mathpzc{K}}){=}{\sum}_{{\mathpzc{p}}={-}{2}}^{-\muup}{\mathfrak{g}}_{{{\mathpzc{p}}}}({\mathpzc{K}})\simeq\operatorname{\mathfrak{f}}_{[2]}/{\mathpzc{K}}.$$ The canonical maximal $\operatorname{\mathbb{Z}}$-graded effective ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$-prolongation $${\mathfrak{g}}({\mathpzc{K}})={\sum}_{{\mathpzc{p}}\geq{-}\muup}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})$$ of ${\textswab{m}}({\mathpzc{K}})$ is the quotient by ${\mathpzc{K}}$ of its normaliser ${\mathfrak{N}}({\mathpzc{K}})$ in ${\mathfrak{F}}(V).$ By Tanaka’s criterion (Theorem \[thm-t-4-4\]), ${\mathfrak{g}}({\mathpzc{K}})$ is finite dimensional if, and only if, $$\begin{aligned}
\operatorname{\mathfrak{a}}({\mathpzc{K}})={\sum}_{{\mathpzc{p}}\geq{-}1}\operatorname{\mathfrak{a}}_{{\mathpzc{p}}}({\mathpzc{K}}),\quad\!\!\text{with}\; \begin{cases}\operatorname{\mathfrak{a}}_{-1} ={\textswab{m}}({\mathpzc{K}}){/}{\mathfrak{n}}({\mathpzc{K}})\simeq{V},
\\ \operatorname{\mathfrak{a}}_{{\mathpzc{p}}}({\mathpzc{K}}) {=}
\{X{\in}{\mathfrak{g}}_{{\mathpzc{p}}}({\mathpzc{K}})\,{\mid}\, [X,{\mathfrak{n}}({\mathpzc{K}})]{=}\{0\}\}\,\;\text{for}\; {\mathpzc{p}}{\geq}0,\end{cases}\end{aligned}$$ is finite dimensional. As in §\[sec5\] we get (see also Lemma \[lem-tan-4-2\]):
\[prop-t-6-3\] Let ${\mathpzc{K}}$ be a $\operatorname{\mathbb{Z}}$-graded ideal of $\operatorname{\mathfrak{f}}(V),$ contained in $\operatorname{\mathfrak{f}}_{[2]}.$ Then $$\vspace{-18pt}
\begin{cases}
{\mathfrak{g}}_0({\mathpzc{K}})=\{A\in{\mathfrak{gl}}_{\operatorname{{\mathbb{R}}}}(V)\mid T_A({\mathpzc{K}})\subset{\mathpzc{K}}\},\\
\operatorname{\mathfrak{a}}_0({\mathpzc{K}})=\{A\in{\mathfrak{gl}}_{\operatorname{{\mathbb{R}}}}(V)\mid T_A(\operatorname{\mathfrak{f}}_{-{\mathpzc{p}}}(V))\subseteq{\mathpzc{K}}_{\; -{\mathpzc{p}}},\;\forall{\mathpzc{p}}\,{\geq}{2}\}.
\end{cases}$$
Set (brackets are computed in $\operatorname{\mathfrak{f}}(V)$) $$W({\mathpzc{K}})=\{{\mathpzc{v}}{\in}V\mid [{\mathpzc{v}},\operatorname{\mathfrak{f}}_{[2]}(V)]\subset{\mathpzc{K}}\}.$$
\[prop-t-6-5\] We have the following characterisation: $$\label{eq-5.4}
\operatorname{\mathfrak{a}}_0({\mathpzc{K}})=\{A\in{\mathfrak{gl}}_{{\mathbb{K}}}(V)\mid T_A(\operatorname{\mathfrak{f}}_{{-}2}(V))\subseteq{\mathpzc{K}}_{\;-2},\;
A(V)\subseteq{W}({\mathpzc{K}})\}.$$
We use the characterisation in Proposition \[prop-t-6-3\] and the fact that $V$ generates $\operatorname{\mathfrak{f}}(V).$ Let us denote by ${\mathfrak{b}}$ the right hand side of .
We first show that ${\mathfrak{b}}{\subseteq} \operatorname{\mathfrak{a}}_0({\mathpzc{K}}).$ To this aim, we need to check that, if $A{\in}{\mathfrak{b}},$ then $T_A(\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}){\subseteq}{\mathpzc{K}}_{\;{-}{{\mathpzc{p}}}}$ for ${\mathpzc{p}}{\geq}2.$ This is true by assumption if ${\mathpzc{p}}{=}2.$ Then we argue by recurrence. We have $$\label{eq-star5} \tag{$*$}
T_A([{\mathpzc{v}},X])=[A({\mathpzc{v}}),X]+[{\mathpzc{v}},T_A(X)], \;\;\;\forall A{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V),\;\forall{\mathpzc{v}}{\in}V,\;\forall{X}{\in}\operatorname{\mathfrak{f}}(V).$$ If $T_A(\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}){\subseteq}{\mathpzc{K}}_{\;{-}{{\mathpzc{p}}}}$ for some ${\mathpzc{p}}{\geq}2,$ then, for $X{\in}\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}(V)$ and ${\mathpzc{v}}{\in}V,$ the second summand $[{\mathpzc{v}},T_A(X)]$ in the right hand side belongs to ${\mathpzc{K}}_{\;{-}p{-}1}$ because $T_A(X){\in}{\mathpzc{K}}_{\;{-}{\mathpzc{p}}}$ and ${\mathpzc{K}}$ is an ideal. If $A{\in}{\mathfrak{b}},$ then $[A({\mathpzc{v}}),X]$ belongs to ${\mathpzc{K}}_{\;{-}{\mathpzc{p}}{-}1}$ by the assumption that $A({\mathpzc{v}}){\in}W({\mathpzc{K}}).$ Since the elements of the form $[{\mathpzc{v}},X]$ with ${\mathpzc{v}}{\in}V$ and $X{\in}\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}(V)$ generate $\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}{-}1}(V),$ this shows that $T_A(\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}{-}1}){\subseteq}{\mathpzc{K}}_{\;{-}{\mathpzc{p}}{-}1},$ completing the proof of the inclusion ${\mathfrak{b}}{\subseteq}\operatorname{\mathfrak{a}}_0({\mathpzc{K}}).$
Let us prove the opposite inclusion. If $A{\in}\operatorname{\mathfrak{a}}_0({\mathpzc{K}}),$ then both the left hand side and the second summand of the right hand side of belong to ${\mathpzc{K}}_{\;{-}{\mathpzc{p}}{-}1}$ for all ${\mathpzc{v}}{\in}V$ and $X{\in}\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}(V).$ This shows that $[A({\mathpzc{v}}),\operatorname{\mathfrak{f}}_{{-}{\mathpzc{p}}}(V)]{\subseteq}{\mathpzc{K}}_{\;{-}{\mathpzc{p}}}$ for all ${\mathpzc{p}}{\geq}2$ and hence that $A(V){\subseteq}W({\mathpzc{K}}).$ Thus $\operatorname{\mathfrak{a}}_0({\mathpzc{K}}){\subseteq}{\mathfrak{b}},$ completing the proof of the proposition.
Let us fix a basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ whose first $m$ vectors ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_m$ are a basis of $W({\mathpzc{K}}).$ The matrices of the elements $A$ of ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ that map $V$ into $W({\mathpzc{K}})$ are of the form $$A=
\begin{pmatrix}
B \\
0
\end{pmatrix},\;\;\text{with}\;\; B{\in}{\mathbb{K}}^{m{\times}n}.$$ Their orthogonal in ${\mathfrak{gl}}_{{\mathbb{K}}}(V)$ consists of the matrices of the form $$Y =
\begin{pmatrix}
0 & C
\end{pmatrix},\;\;\text{with}\;\; C{\in}{\mathbb{K}}^{n\times(n{-}m)}.$$ Taking the canonical basis $Y_1,\hdots,Y_r,$ with $r{=}n{\times}(n{-}m),$ for the linear space of the matrices of this form, we obtain a matrix $$\label{eq-5.5}
M_3({\mathpzc{K}},{\mathpzc{z}})=({\mathpzc{z}}_{m{+}1}{\mathrm{I}}_n,\hdots,{\mathpzc{z}}_n{\mathrm{I}}_n).$$ Since the orthogonal of an intersection of linear subspaces is the sum of their orthogonal subspaces, the matrix of the form associated to $\operatorname{\mathfrak{a}}_0({\mathpzc{K}})$ is in this case $$\label{eq-5.6}
M({\mathpzc{z}})=
(M_2({\mathpzc{K}},{\mathpzc{z}}),M_3({\mathpzc{K}},{\mathpzc{z}})).$$ Taking into account that $M_3({\mathpzc{K}},{\mathpzc{z}})$ is described by , the matrix has rank $n$ whenever $({\mathpzc{z}}_{m{+}1},\hdots,{\mathpzc{z}}_n){\neq}(0,\hdots,0).$ Let $\Phi_{{\mathpzc{K}}}({\mathpzc{z}})$ the matrix defined by . Then Theorem \[thm-t-4-4\] yields
\[thm-ta-7.3\] Assume that ${\mathpzc{K}}$ is $\muup$-cofinite for some integer $\muup{\geq}2$ and let ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_m$ be a basis of $W({\mathpzc{K}}).$ Then ${\mathfrak{g}}({\mathpzc{K}})$ is finite dimensional if and only if $\Phi_{{\mathpzc{K}}}({\mathpzc{z}})$ has rank ${\geq}2$ for all $0{\neq}{\mathpzc{z}}{\in}W_{({\mathbb{F}})}={\mathbb{F}}{\otimes}W({\mathpzc{K}}).$
$\operatorname{\mathfrak{L}}$-prolongations of FGLA’s of general kind {#sect8}
=====================================================================
Let $\operatorname{\mathfrak{L}}$ be a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}(V).$ Fix a set $A_1,\hdots,A_k$ of generators of $$\operatorname{\mathfrak{L}}^\perp=\{A{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V)\mid {\mathrm{trace}}(AX)=0,\;\forall X{\in}\operatorname{\mathfrak{L}}\}$$ and form $$M_1(\operatorname{\mathfrak{L}},{\mathpzc{z}})=(A_1{\mathpzc{z}},\hdots,A_k{\mathpzc{z}}),\;\;{\mathpzc{z}}\in{V}_{({\mathbb{F}})}.$$ If we fix a basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ of $V,$ we may consider $M({\mathpzc{z}})$ as a matrix of homogeneous first degree polynomials of ${\mathpzc{z}}{=}({\mathpzc{z}}_1,\hdots,{\mathpzc{z}}_n)$ with coefficients in ${\mathbb{K}}.$
Let ${\textswab{m}}={\sum}_{{\mathpzc{p}}=1}^\muup{\mathfrak{g}}_{{-}{\mathpzc{p}}}$ be an FGLA of finite kind $\muup{\geq}1$ and set $V={\mathfrak{g}}_{{-}1}.$ Then ${\textswab{m}}$ is isomorphic to a quotient $\operatorname{\mathfrak{f}}(V)/{\mathpzc{K}},$ for a graded ideal ${\mathpzc{K}}$ of $\operatorname{\mathfrak{f}}(V)$ with $\operatorname{\mathfrak{f}}_{[\muup{+}1]}(V){\subseteq}{\mathpzc{K}}{\subseteq}\operatorname{\mathfrak{f}}_{[2]}(V).$ Let $\operatorname{\mathfrak{L}}$ be a Lie subalgebra of ${\mathfrak{gl}}_{{\mathbb{K}}}({\mathfrak{g}}_{{-}1}).$ We showed in §\[sec-fund\] that the maximal effective prolongation of type $\operatorname{\mathfrak{L}}$ of ${\textswab{m}}$ is the quotient by ${\mathpzc{K}}$ of its normaliser ${\mathfrak{N}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ in ${\mathfrak{F}}(V,\operatorname{\mathfrak{L}}).$
By Theorem \[thm-t-4-4\], this maximal effective $\operatorname{\mathfrak{L}}$-prolongation ${\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ of ${\textswab{m}}$ is finite dimensional if and only if the maximal effective $\operatorname{\mathfrak{a}}_0$-prolongation of $V$ is finite dimensional, where $\operatorname{\mathfrak{a}}_0$ the Lie subalgebra of $\operatorname{\mathfrak{L}}$ defined by $$\operatorname{\mathfrak{a}}_0{=}\{A{\in}\operatorname{\mathfrak{L}}\mid T_A(\operatorname{\mathfrak{f}}_{[2]}(V))\subseteq{\mathpzc{K}}\}.$$ Since the orthogonal of an intersection of linear subspaces is the sum of their orthogonal subspaces, we can use the results (and the notation) of §\[sec5a\],\[sec5\],\[sect7\] to construct the matrix $M({\mathpzc{z}})$ of Theorem \[thm-t-4-6\], obtaining $$\label{eq-6.3}
M({\mathpzc{z}})=(M_1(\operatorname{\mathfrak{L}},{\mathpzc{z}}),M_2({\mathpzc{K}},{\mathpzc{z}}),M_3({\mathpzc{K}},{\mathpzc{z}})).$$ Let $W{=}W({\mathpzc{K}})$ be the subspace of $V$ defined in §\[sect7\]. Then we obtain
\[thm-tan-6.1\] Fix the basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ of $V$ in such a way that the first $m$ vectors ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_m$ form a basis of $W({\mathpzc{K}}).$ Then a necessary and sufficient condition in order that the maximal effective $\operatorname{\mathfrak{L}}$-prolongation of ${\textswab{m}}$ be finite dimensional is that $$(M_1(\operatorname{\mathfrak{L}})({\mathpzc{z}}),M_2({\mathpzc{K}},{\mathpzc{z}}))$$ has rank $n$ for all ${\mathpzc{z}}\in{W}_{({\mathbb{F}})}{\setminus}\{0\}
=\{{\mathpzc{z}}\in{\mathbb{F}}^n{\setminus}\{0\}\mid {\mathpzc{z}}_i{=}0,\;\forall i{>}m\}.$
Let $V$ be a finite dimensional ${\mathbb{K}}$-vector space of dimension $n{\geq}3$ and ${\mathpzc{b}}:V{\times}V{\to}{\mathbb{K}}$ a nondegenerate bilinear form on $V,$ having nonzero symmetric and antisymmetric components ${\mathpzc{b}}_s$ and ${\mathpzc{b}}_a.$ Let $$\operatorname{\mathfrak{L}}=\{X{\in}{\mathfrak{gl}}_{{\mathbb{K}}}(V)\mid \exists {\mathpzc{c}}(X){\in}{\mathbb{K}}\;\text{s.t.}\;
{\mathpzc{b}}(X{\mathpzc{v}},{\mathpzc{w}}){+}{\mathpzc{b}}({\mathpzc{v}},X{\mathpzc{w}}){=}{\mathpzc{c}}(X){\mathpzc{b}}({\mathpzc{v}},{\mathpzc{w}}),\;\forall{\mathpzc{v}},{\mathpzc{w}}{\in}B\}.$$ Then it is natural to take ${\textswab{m}}{=}{\mathfrak{g}}_{{-}1}{\oplus}{\mathfrak{g}}_{{-}2},$ with ${\mathfrak{g}}_{{-}1}{=}V$ and ${\mathfrak{g}}_{{-}2}{=}{\mathbb{K}},$ defining the Lie brackets on $V$ by $$[{\mathpzc{v}},{\mathpzc{w}}]={\mathpzc{b}}_a({\mathpzc{v}},{\mathpzc{w}}),\;\;\forall{\mathpzc{v}},{\mathpzc{w}}{\in}V.$$ Then ${\textswab{m}}{\simeq}\operatorname{\mathfrak{f}}(V){/}{\mathpzc{K}},$ with ${\mathpzc{K}}{=}{\mathpzc{K}}_{{-}2}{\oplus}\operatorname{\mathfrak{f}}_{[3]}$ for $${\mathpzc{K}}_{{-}2}{=}\left.\left\{{\sum}{\mathpzc{v}}_i{\wedge}{\mathpzc{w}}_i{\in}\Lambda^2(V)\right|
{\sum}{\mathpzc{b}}_a({\mathpzc{v}}_i,{\mathpzc{w}}_i){=}0\right\}.$$ The maximal effective $\operatorname{\mathfrak{L}}$-prolongation ${\mathfrak{g}}{=}{\mathfrak{g}}({\mathpzc{K}},\operatorname{\mathfrak{L}})$ of ${\textswab{m}}$ is equal to the maximal effective $\mathfrak{co}_{{\mathpzc{b}}_s}(V)$-prolongation ${\mathfrak{g}}({\mathpzc{K}},\mathfrak{co}_{{\mathpzc{b}}_s}(V))$ of ${\textswab{m}}.$
In particular, ${\mathfrak{g}}$ is finite dimensional by Example \[ex-tan-3-6\], because $n{\geq}3,$ when ${\mathpzc{b}}_s$ is nondegenerate.
Consider now the case where ${\mathpzc{b}}_s$ has rank $0{<}{\mathpzc{p}}{<}n.$ We can find a basis ${\mathpzc{e}}_1,\hdots,{\mathpzc{e}}_n$ of $V$ such that ${\mathpzc{b}}_s$ is represented by a matrix $$B=
\begin{pmatrix}
D & 0\\
0 & 0
\end{pmatrix},\;\; \text{with}\;\; D= {\mathrm{diag}}(\lambdaup_1,\hdots,\lambdaup_{{\mathpzc{p}}})\in{\mathbb{K}}^{{\mathpzc{p}}{\times}{\mathpzc{p}}},
\;\text{with}\; \lambdaup_1{\cdots}\lambdaup_{\mathpzc{p}}{\neq}0.$$ Let $A$ be the antisymmetric $n{\times}n$ matrix representing ${\mathpzc{b}}_a$ in this basis and $A_1,\hdots,A_n$ its rows. Then the matrix $M({\mathpzc{z}})$ of has the form $$M({\mathpzc{z}})=(M_1({\mathpzc{z}}),M_2({\mathpzc{z}})),$$ with $M_1({\mathpzc{z}})=({\mathpzc{z}}_1{E}_1,\hdots,{\mathpzc{z}}_{\mathpzc{p}}{E}_{{\mathpzc{p}}})$ for invertible $n{\times}n$ matrices $E_1,\hdots,E_q$ and $$M_2({\mathpzc{z}})=
\begin{pmatrix}
A_2{\mathpzc{z}}& \hdots & A_n{\mathpzc{z}}& 0 & \hdots & 0 & \hdots \\
{-}A_1{\mathpzc{z}}& \hdots & 0 & A_{3}{\mathpzc{z}}& \hdots & A_n{\mathpzc{z}}&\hdots \\
0 & \hdots & 0 & {-}A_2({\mathpzc{z}}) & \hdots & 0 & \hdots \\
\vdots & \ddots & \vdots &\vdots & \ddots &\vdots & \hdots\\
0 & \hdots & {-}A_1{\mathpzc{z}}& 0 & \hdots & {-}A_2{\mathpzc{z}}&\hdots\\
\end{pmatrix}$$ In particular, when ${\mathpzc{z}}_i{=}0$ for $1{\leq}i{<}n,$ the first row of $M({\mathpzc{z}})$ is zero because $A$ is antisymmetric and therefore ${\mathfrak{g}}$ is infinite dimensional by Theorem \[thm-tan-6.1\] (here $W_{({\mathbb{F}})}=V_{({\mathbb{F}})}$ because ${\mathpzc{K}}_{{-}3}{=}\operatorname{\mathfrak{f}}_{[3]}(V)$).
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---
abstract: 'In this paper, we numerically study the non-Abelian statistics of the zero-energy Majorana fermions on the end of Majorana chain and show its application to quantum computing by mapping it to a spin model with special symmetry. In particular, by using transverse-field Ising model with Z2 symmetry, we verify the nontrivial non-Abelian statistics of Majorana fermions. Numerical evidence and comparison in both Majorana-representation and spin-representation are presented. The degenerate ground states of a symmetry protected spin chain therefore previde a promising platform for topological quantum computation.'
author:
- 'Xiao-Ming Zhao'
- Jing Yu
- Jing He
- 'Qiu-Bo Cheng'
- Ying Liang
- 'Su-Peng Kou'
title: 'The Simulation of Non-Abelian Statistics of Majorana Fermions in Ising Chain with Z2 Symmetry'
---
[^1]
Introduction
============
Majorana fermions have recently attracted much attention due to the potential application in topological quantum computation [@ref1]. Majorana fermions are particles that are their own antiparticles — in contrast with the case for Dirac fermions — and obey non-Abelian statistics[@ref1.1; @ref1.2; @ref1.3]. The exotic properties of Majorana fermions have attracted increasing interest from researchers[ref1.4,ref1.5,ref1.6,ref1.7,ref1.9,ref1.10]{}. Majorana fermions with zero energy (Majorana zero modes) had been predicted to be induced by vortices in two-dimensional spinless $p_{x}+ip_{y}$-wave superconductor[ref2,ref3,ref4]{}, or localize at the ends in a one-dimensional spin-polarized superconductor chain. Another creative proposal is the interface of $s$-wave superconductors and topological insulators owing to the proximity effect. On the other hand, the spin chain has been studied in depth both theoretically and experimentally. It is known that the transverse-field Ising model with Z2 symmetry is equivalent to the one-dimensional spin-polarized superconductor model[@ref5].
In this paper, we numerically study the non-Abelian statistics of the zero-energy Majorana fermions on the end of Majorana chain and show its application to quantum computing by mapping it to a spin model with special symmetry. In particular, by using transverse-field Ising model with Z2 symmetry, we verify the nontrivial quantum statistics of Majorana fermions numerically using a T-junction wire network, where the Majorana fermions can be braided by tuning local gates. We may also mimic this T-type braiding in spin representation numerically, where the two zero-energy Majorana fermion states correspond to two degenerate ground states of spin chain. In this way, we provide an easy way of detecting the fundamental non-Abelian statistics of Majorana fermions, which is useful to quantum computation.
Majorana zero modes in one-dimensional quantum spin model with Z2 symmetry
==========================================================================
It has been recognized that a one-dimensional quantum spin model with Z2 symmetry is equivalent to a one-dimensional superconductor via Jordan-Wigner transformation. Therefore, we can describe a one-dimensional spin chain using either spin representation ($\sigma $-representation) or fermion representation ($\gamma $-representation). Thus, the Majorana fermion and its statistic property can be represented in either representation.
Here, we start from the one-dimensional Ising chain with Z2 symmetry. The Hamiltonian of the Ising spin chain is given by$$\hat{H}=\sum_{n=1}^{N-1}J_{n,n+1}\sigma _{n}^{x}\sigma
_{n+1}^{x}-\sum_{n=1}^{N}\mu _{n}\vec{\sigma}_{n}$$where $J_{n,n+1}$ ( $J_{n,n+1}<0$ ) is the Ising coupling constant between two nearest-neighbour (NN) sites $n,$ $n+1$, $\mu _{n}$ ( $\mu _{n}>0$ ) is the strength of external field on site $n$, and $N$ is the total lattice number of the Ising chain. We then introduce the spin operators $\sigma
_{n}^{+},$ $\sigma _{n}^{-}$, $\sigma _{n}^{x}=\sigma _{n}^{+}+\sigma
_{n}^{-},$ and $\sigma _{n}^{y}=i(\sigma _{n}^{+}-\sigma _{n}^{-})$. The Z2 symmetry is charaterized by as spin rotation symmetry $\hat{R}=e^{i\pi
\sum_{n=1}^{N}\sigma _{n}^{z}}$, i.e., $$\hat{R}\hat{H}\hat{R}^{-1}=\hat{H}.$$ Thus, to guarantee the Z2 symmetry, the external field should be along $z$-direction, or $\vec{\sigma}_{n}\rightarrow \sigma _{n}^{z}$.
The Jordan-Wigner transformation is described by[@ref5; @ref6]$$\begin{aligned}
\sigma _{n}^{+}& =a_{n}^{+}{\displaystyle\prod\limits_{m=1}^{n-1}}a_{l}^{+}a_{l}; \notag \\
\sigma _{n}^{-}& =a_{n}{\displaystyle\prod\limits_{m=1}^{n-1}}a_{l}^{+}a_{l};
\\
\sigma _{n}^{z}& =2a_{n}^{+}a_{n}-1, \notag\end{aligned}$$where $a_{n}^{+},$ $a_{n}$ denote the creation and annihilation operators of Dirac fermions and obey the anticommutation relation $\left\{
a_{m},a_{n}^{+}\right\} =\delta _{m,n}$. By using Jordan-Wigner transformation, the Hamiltonian $\hat{H}$ can be written as $$\hat{H}=\sum_{n=1}^{N-1}J_{n,n+1}(a_{n}-a_{n}^{+})(a_{n+1}+a_{n+1}^{+})-\sum_{n=1}^{N}\mu _{n}(2a_{n}^{+}a_{n}-1).$$Then, the Majorana fermion is defined as $$\gamma _{n}^{A}=a_{n}^{+}+a_{n},\text{ }\gamma _{n}^{B}=i(a_{n}^{+}-a_{n}),$$with $\gamma _{n}^{\dag }=\gamma _{n},$ $\left\{ \gamma _{n}^{l},\gamma
_{m}^{l^{\prime }}\right\} =2\delta _{m,n}\delta _{l,l^{\prime }}$. From the definition, one can see that Majorana fermions are their own antiparticle and constitute half of an ordinary fermion. We obtain the Hamiltonian in the $\gamma $-representation [ref20]{},$$\hat{H}=-i\sum_{n=1}^{N-1}J_{n,n+1}\gamma _{n}^{B}\gamma
_{n+1}^{A}-i\sum_{n=1}^{N}\mu _{n}\gamma _{n}^{A}\gamma _{n}^{B}.$$
In the fermion representation for the Hamiltonian, we see that two Majorana fermions on one site $n$ are coupled and the coupling constant is $\mu _{n}$ (e.g., the double dark line links site A and site B inner the box $1^{\prime
}$ in Fig.1) and the two Majorana fermions on the NN sites are linked by $J_{n,n+1}$ (e.g., the single dark line between boxes 1, 2 and 3 in Fig.1). When we adiabatically turn off $\mu _{n}$ at all sites such that its value decreases from a certain value $\mu _{0}$ to zero ($\mu _{0}\rightarrow 0$), the Majorana fermions of the chain are only coupled by $J_{n,n+1}$ terms except for the two Majorana fermions at the ends (e.g., the green and blue balls in Fig.1).
To characterize the quantum states of Majorana fermions, we introduce the creation and annihilation operators of Dirac fermions, $d_{n}=(\gamma
_{n+1}^{A}+i\gamma _{n}^{B})/2$, $d_{n}^{\dag }=(\gamma _{n+1}^{A}-i\gamma
_{n}^{B})/2$. The operators of Dirac fermions are combined by two Majorana fermions at NN sites, i.e., $n$ and $n+1$. Thus, the Majorana fermions at the left (right) end of the chain $\gamma _{1}^{A}$ $(\gamma _{N}^{B})$ remain unpaired and have zero energy[@ref8; @ref9]. Here, we focus on the edge fermion and have $$d_{end}=\frac{1}{2}(\gamma _{1}^{A}+i\gamma _{N}^{B}),\text{ }d_{end}^{\dag
}=\frac{1}{2}(\gamma _{1}^{A}-i\gamma _{N}^{B}).$$It is obvious that the edge fermion has zero energy. We now define $\left\vert F\right\rangle $ to be a many-body quantum state with occupied single particle states for $E<0$ and empty single particle states $E\geq 0$. We therefore introduce a Majorana qubit that consists of two basis states $\left\vert 0\right\rangle ,$ $\left\vert 1\right\rangle $ defined as[ref6.1]{}$$\left\vert 0\right\rangle \equiv d_{end}\left\vert F\right\rangle ,\text{ }\left\vert 1\right\rangle \equiv d_{end}^{\dag }\left\vert 0\right\rangle .$$
Numerical verifying non-Abelian statistics of Majorana fermions in $\protect\gamma $-representation
===================================================================================================
In this part, we numerically study the quantum statistic of the Majorana fermions located at the end of spin chain by using one-dimensional quantum Ising model with Z2 symmetry. To explore the quantum statistic of the Majorana fermions, we take a 4-spin (i.e., 8-$\gamma $) system as an example and braid the Majorana fermions $\gamma _{1}^{A},$ $\gamma _{3}^{B}$ by seven steps using the T-type structure (see the illustration in Fig.1), which is similar to the semiconducting wire networks in Ref.[@ref11].
The parameters $J_{n,n+1}$ and $\mu _{n}$ in the original Hamiltonian are given by $J_{nB,(n+1)A}=J_{0}$ and $\mu _{n}=\mu _{0}$, respectively. We first choose the initial state with $J_{nB,(n+1)A}=0$ and $\mu _{n}=0$. Thus, there must exit two unpaired Majorana zero modes located at the left end $\left\vert \gamma _{L}(T_{0})\right\rangle =\left\vert \gamma
_{1}^{A}\right\rangle $ (green ball) and right end $\left\vert \gamma
_{R}(T_{0})\right\rangle =\left\vert \gamma _{3}^{B}\right\rangle $ (blue ball) of the Majorana chain. Here $T_{0}=0$ represents the initial time and $T_{n}$ for $n$-th step of braiding process. We denote the quantum states of Majorana modes by $\left\vert \gamma _{L}(T_{n})\right\rangle ,$ $\left\vert
\gamma _{R}(T_{n})\right\rangle ,$ $n\in (1,7)$. The Hamiltonian of the system at $T_{0}$ is given by $$H_{\gamma ,T_{0}}=-iJ_{0}\gamma _{1}^{B}\gamma _{2}^{A}-iJ_{0}\gamma
_{2}^{B}\gamma _{3}^{A}-i\mu _{0}\gamma _{1^{\prime }}^{A}\gamma _{1^{\prime
}}^{B}.$$
We then do the braiding process step by step (see Fig.1): **(a)** $\mu
_{1}|_{_{^{T_{0}}}}^{_{_{_{T_{1}}}}}(0\rightarrow \mu _{0})$, $J_{1B,2A}|_{_{^{T_{0}}}}^{_{_{_{T_{1}}}}}(J_{0}\rightarrow 0)$ (This means we adiabatically turn on $\mu _{1}$ and turn off $J_{1B,2A}$ simultaneously during the time period $t\in (T_{0},$ $T_{1})$), then $\mu _{1^{\prime
}}|_{_{^{T_{1}}}}^{_{_{_{T_{2}}}}}(\mu _{0}\rightarrow 0)$, $J_{1^{\prime
}B,2A}|_{_{^{T_{1}}}}^{_{_{_{T_{2}}}}}(0\rightarrow J_{0})$. The order of this braiding process is $1A\rightarrow 2A\rightarrow 1^{\prime }A;$ **(b)** $\mu _{3}|_{_{^{T_{2}}}}^{_{_{_{T_{3}}}}}(0\rightarrow \mu _{0})$, $J_{2B,3A}|_{_{^{T_{2}}}}^{_{_{_{T_{3}}}}}(J_{0}\rightarrow 0)$, then $J_{1^{\prime }B,2A}|_{_{^{T_{3}}}}^{_{_{_{T_{4}}}}}(J_{0}\rightarrow 0)$, $J_{1^{\prime }B,2B}|_{_{^{T_{3}}}}^{_{_{_{T_{4}}}}}(0\rightarrow J_{0})$, next, $\mu _{1}|_{_{^{T_{4}}}}^{_{_{_{T_{5}}}}}(\mu _{0}\rightarrow 0)$, $J_{1B,2A}|_{_{^{T_{4}}}}^{_{_{_{T_{5}}}}}(0\rightarrow J_{0})$. The braiding order is $3B\rightarrow 2B\rightarrow 2A\rightarrow 1A$; **(c)** $\mu
_{1^{\prime }}|_{_{^{T_{5}}}}^{_{_{_{T_{6}}}}}(0\rightarrow \mu _{0})$, $J_{1^{\prime }B,2B}|_{_{^{T_{5}}}}^{_{_{_{T_{6}}}}}(J_{0}\rightarrow 0)$, then $\mu _{3}|_{_{^{T_{6}}}}^{_{_{_{T_{7}}}}}(\mu _{0}\rightarrow 0)$, $J_{2B,3A}|_{_{^{T_{6}}}}^{_{_{_{T_{7}}}}}(0\rightarrow J_{0})$. The braiding order is $1^{\prime }A\rightarrow 2B\rightarrow 3B$.
In particular, during the time period $t\in (T_{3},$ $T_{4})$, we have $$\begin{aligned}
H_{\gamma ,T_{3}}& =-iJ_{0}\gamma _{1^{\prime }}^{B}\gamma _{2}^{A}-i\mu
_{0}\gamma _{1}^{A}\gamma _{1}^{B}-i\mu _{0}\gamma _{3}^{A}\gamma _{3}^{B},
\\
H_{\gamma ,T_{4}}& =-iJ_{0}\gamma _{1^{\prime }}^{B}\gamma _{2}^{B}-i\mu
_{0}\gamma _{1}^{A}\gamma _{1}^{B}-i\mu _{0}\gamma _{3}^{A}\gamma _{3}^{B}.\end{aligned}$$The operation during this period shifts the Majorana mode from site $2B$ to $2A$ which are on the same box $2$.
It is well known that the braiding of Majorana modes changes not only the amplitude but also the phase of the modes. We next focus on the phase difference of $\left\vert \gamma _{L}(T_{n})\right\rangle ,$ $\left\vert
\gamma _{R}(T_{n})\right\rangle $ before and after the adiabatic braiding process numerically. We diagonalize the initial Hamiltonian $H_{\gamma
,T_{0}}$ in the $\gamma $-representation and obtain two zero energy modes $\left\vert \gamma _{L}(T_{0})\right\rangle =\left\vert \gamma
_{1}^{A}\right\rangle $ and $\left\vert \gamma _{R}(T_{0})\right\rangle
=\left\vert \gamma _{3}^{B}\right\rangle $. We then define a time-evolution operator $$U(t)=\hat{T}\left\{ \exp [-i\int_{0}^{t}H(t^{\prime })dt^{\prime }]\right\} ,$$where $\hat{T}$ is the time ordering operator. Therefore, at the end of the evolution, we have $$\begin{aligned}
\left\vert \gamma _{L}(T_{7})\right\rangle =& U(T_{7})\left\vert \gamma
_{L}(T_{0})\right\rangle \notag \\
\text{ \ }\left\vert \gamma _{R}(T_{7})\right\rangle =& U(T_{7})\left\vert
\gamma _{R}(T_{0})\right\rangle .\end{aligned}$$To realize the time-evolution numerically, one may discretize the time-evolution operator employing the times slicing procedure $$U(T_{7})\approx \hat{T}\bigskip {\displaystyle\prod\limits_{i=0}^{N_{0}}}\exp [-iH(t_{i})\triangle t],\text{ \ }\triangle t=\frac{T_{7}-T_{0}}{N_{0}},$$with $\triangle t\ll \hbar /J,$ and $T_{7}-T_{0}$ being sufficiently large. We point out that it is crucial to retain the unitarity throughout the calculation $$\exp [-iH(t_{i})\triangle t]=A\exp (-i\Lambda \triangle t)A^{\dag },$$where $H(t_{i})=A\Lambda A^{\dag },$ $A$ is a unitary matrix $AA^{\dag }=I$ and $\Lambda $ is a diagonal matrix. Fig.2 shows the change in $\left\vert
\gamma _{L}(t)\right\rangle ,$ $\left\vert \gamma _{R}(t)\right\rangle $ during the braiding process. It is clearly that $\left\vert \gamma
_{L}(T_{7})\right\rangle =\left\vert \gamma _{R}(0)\right\rangle ,$ $\left\vert \gamma _{R}(T_{7})\right\rangle =-\left\vert \gamma
_{L}(0)\right\rangle $. The braiding operation therefore transforms $\gamma
_{1}^{A}$ to $\gamma _{3}^{B}$ and $\gamma _{3}^{B}$ to $-\gamma _{1}^{A}$.
Numerical verifying non-Abelian statistics of Majorana fermions in $\protect\sigma $-representation
===================================================================================================
In last section, we have verified the non-Abelian statistics numerically in $\gamma $-representation and construct a phase gate based on the qubits that is simple and easily understood[@ref11]. We then map the braiding in $\gamma $-representation to that in $\sigma $-representation by employing the Jordan-Wigner transformation [@ref13],
$$\gamma _{n}^{A}=({\displaystyle\prod\limits_{m=1}^{n-1}}\sigma
_{m}^{z})\sigma _{n}^{x},\text{ \ \ }\gamma _{n}^{B}=i({\displaystyle\prod\limits_{m=1}^{n}}\sigma _{m}^{z})\sigma _{n}^{x}.$$
It is obvious that the Majorana fermions $\gamma _{n}^{A}$ and $\gamma
_{n}^{B}$ are non-local in the $\sigma $-representation. When $J_{0}<0$ the state $\left\vert F\right\rangle $ can be written as$$\left\vert F\right\rangle =\left\vert \rightarrow \rightarrow \rightarrow
\right\rangle ,$$where $$\left\vert \rightarrow \right\rangle =\frac{\sqrt{2}}{2}\left(
\begin{array}{c}
1 \\
1\end{array}\right) ,\text{ \ }\left\vert \leftarrow \right\rangle =\frac{\sqrt{2}}{2}\left(
\begin{array}{c}
1 \\
-1\end{array}\right) .$$Then the two basis states $\left\vert 0\right\rangle $, $\left\vert
1\right\rangle $ of Majorana qubit are represented in $\sigma $-representation as $$\begin{aligned}
\left\vert 0\right\rangle & =\frac{1}{2}(\gamma _{1}^{A}+i\gamma
_{3}^{B})\left\vert F\right\rangle \notag \\
& =\frac{1}{2}(\sigma _{1}^{x}+i(i{\displaystyle\prod\limits_{m=1}^{3}}\sigma _{m}^{z})\sigma _{3}^{x})\left\vert F\right\rangle \\
& =\frac{\sqrt{2}}{2}(\left\vert \rightarrow \rightarrow \rightarrow
\right\rangle -\left\vert \leftarrow \leftarrow \leftarrow \right\rangle ),
\notag\end{aligned}$$$$\begin{aligned}
\left\vert 1\right\rangle & =\frac{1}{2}(\gamma _{1}^{A}-i\gamma
_{N}^{B})\left\vert 0\right\rangle \notag \\
& =\frac{1}{2}(\sigma _{1}^{x}-i(i{\displaystyle\prod\limits_{m=1}^{3}}\sigma _{m}^{z})\sigma _{3}^{x})\left\vert 0\right\rangle \\
& =\frac{\sqrt{2}}{2}(\left\vert \rightarrow \rightarrow \rightarrow
\right\rangle +\left\vert \leftarrow \leftarrow \leftarrow \right\rangle ).
\notag\end{aligned}$$Thus, the two quantum states of Majorana fermions correspond to two degenerate ground states of 1D transverse Ising model with Z2 symmetry.
[|m[0.5in]{}|c|c|]{} & $\gamma$-representation & $\sigma$-representation\
Fermion operator & $\gamma_{n}^{A},\gamma_{n}^{B}$ & $(\prod
\limits_{m=1}^{n-1}\sigma_{m}^{z})\sigma_{n}^{x},i(\prod
\limits_{m=1}^{n}\sigma_{m}^{z})\sigma_{n}^{x}$\
String operator & $i\prod \limits_{m=1}^{N}\gamma_{m}^{A}\gamma_{m}^{B}$ & $\prod \limits_{m=1}^{N}\sigma_{m}^{z}$\
Basis & $\frac{1}{2}(\gamma_{1}^{A}+i\gamma_{N}^{B})\left \vert F\right
\rangle $ & $\frac{\sqrt{2}}{2}(\left \vert \rightarrow \rightarrow \cdot
\rightarrow \right \rangle -\left \vert \leftarrow \leftarrow \cdot
\leftarrow \right \rangle )$\
state & $\frac{1}{2}(\gamma_{1}^{A}-i\gamma_{N}^{B})\left \vert 0\right
\rangle $ & $\frac{\sqrt{2}}{2}(\left \vert \rightarrow \rightarrow \cdot
\rightarrow \right \rangle +\left \vert \leftarrow \leftarrow \cdot
\leftarrow \right \rangle )$\
Braiding & Majorana modes & Spin rotation $\pi/2$\
process & exchange & around z-axis\
Braiding & $\gamma_{1}^{A}\rightarrow \gamma_{N}^{B}$ & $\left \vert 0\right
\rangle _{\sigma}\rightarrow e^{i\pi/2}\left \vert 0\right \rangle _{\sigma}$\
results & $\gamma_{N}^{B}\rightarrow-\gamma_{1}^{A}$ & $\left \vert 1\right
\rangle _{\sigma}\rightarrow \left \vert 1\right \rangle _{\sigma}$\
Analogy to the previous braiding process, we can obtain the Hamiltonian of the 4-spin system in different time periods $T_{n}$ as $$\begin{aligned}
H_{\sigma ,T_{0}}& =J_{0}\sigma _{1}^{x}\sigma _{2}^{x}-J_{0}\sigma
_{2}^{x}\sigma _{3}^{x}-\mu _{0}\sigma _{1^{\prime }}^{z}, \notag \\
H_{\sigma ,T_{3}}& =J_{0}\sigma _{1^{\prime }}^{x}\sigma _{2}^{x}-\mu
_{0}\sigma _{1}^{z}-\mu _{0}\sigma _{3}^{z}, \\
H_{\sigma ,T_{4}}& =J_{0}\sigma _{1^{\prime }}^{x}\sigma _{2}^{y}-\mu
_{0}\sigma _{1}^{z}-\mu _{0}\sigma _{3}^{z}. \notag\end{aligned}$$For a state $\left\vert \psi (\tau )\right\rangle $, we can define the Berry phase [@ref10] as$$\theta =i\int_{T_{0}}^{T_{7}}\left\langle \psi (\tau )\right\vert \frac{d}{d\tau }\left\vert \psi (\tau )\right\rangle d\tau .$$The changes of $\theta $ for $\left\vert 0\right\rangle $, $\left\vert
1\right\rangle $ in the process of evolution are shown in Fig.3. We find that $\theta _{0}=\frac{\pi }{4}$, $\theta _{1}=-\frac{\pi }{4}$, i.e. the phase difference of $\left\vert 0(t)\right\rangle $, $\left\vert
1(t)\right\rangle $ is $\frac{\pi }{2},$ so we have $$\binom{\left\vert 0\right\rangle }{\left\vert 1\right\rangle }\rightarrow
\left(
\begin{array}{cc}
e^{i\frac{\pi }{2}} & 0 \\
0 & 1\end{array}\right) \binom{\left\vert 0\right\rangle }{\left\vert 1\right\rangle }.$$The braiding process equals to rotating the spin at site $1,$ $2,$ $3$ in the $x$-$y$ plane from $x$-direction to $y$-direction. This process is shown in Fig.4, in which the three processes correspond to that of **(a)**, **(b)**, **(c)** in Fig1. We can also describe the results of the evolution as follow $$\begin{aligned}
& \left\vert \rightarrow \rightarrow \rightarrow \right\rangle \notag \\
\rightarrow & \frac{\sqrt{2}}{2}(\left\vert \uparrow \uparrow \uparrow
\right\rangle +e^{i\frac{\pi }{2}}\left\vert \downarrow \downarrow
\downarrow \right\rangle ) \\
=& \frac{\sqrt{2}}{2}(e^{i\frac{\pi }{4}}\left\vert \rightarrow \rightarrow
\rightarrow \right\rangle +e^{-i\frac{\pi }{4}}\left\vert \leftarrow
\leftarrow \leftarrow \right\rangle ), \notag\end{aligned}$$while the other ground state have a similar changes $$\begin{aligned}
& \left\vert \leftarrow \leftarrow \leftarrow \right\rangle \notag \\
\rightarrow & \frac{\sqrt{2}}{2}(\left\vert \uparrow \uparrow \uparrow
\right\rangle +e^{-i\frac{\pi }{2}}\left\vert \downarrow \downarrow
\downarrow \right\rangle ) \\
=& \frac{\sqrt{2}}{2}(e^{-i\frac{\pi }{4}}\left\vert \rightarrow \rightarrow
\rightarrow \right\rangle +e^{i\frac{\pi }{4}}\left\vert \leftarrow
\leftarrow \leftarrow \right\rangle ). \notag\end{aligned}$$
Finally, we show a comparison in Tab.1, in which the fermion operator, string operator, basis state, braiding process and braiding results are illustrated in both $\gamma $-representation and $\sigma $-representation. In brief, the braiding of Majorana fermion can be simulated by braiding a corresponding Ising chain with Z2 symmetry.
Conclusion
==========
In the end, we draw the conclusion. In this paper, we pointed out that the transverse-field Ising model with Z2 symmetry may simulate one-dimensional Majorana chain to braid Majorana fermions. On the one hand, in $\gamma $-representation by doing Jordon-Wigner transformation, two zero-energy Majorana fermions are localized at the left and right ends of the Majorana chain. We get numerically the transformations $\gamma _{1}^{A}\rightarrow
\gamma _{3}^{B}$ and $\gamma _{3}^{B}$ $\rightarrow $ $-\gamma _{1}^{A}$ by braiding two Majorana fermions in a T-junction. On the other hand, in $\sigma $-representation, the two degenerate ground states correspond to the degenerate quantum states of two Majorana fermions. The braiding process of the Majorana zero modes is exactly mapped to switch the spin direction from the $x$-axis to the $y$-axis in the $x$-$y$ plane. Tab.1 shows the correspondence between the two representations. Therefore, the Ising chain with Z2 symmetry can be employed to construct the phase gate in quantum computation.
This work is supported by National Basic Research Program of China (973 Program) under the grant No. 2011CB921803, 2012CB921704 and NSFC Grant No.11174035, 11474025, 11404090, 11304136, SRFDP, the Fundamental Research Funds for the Central Universities, NSF-Hebei Province under Grant No. A2015205189 and NSF-Hebei Education Department under Grant No. QN2014022.
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[^1]: Corresponding author
|
---
abstract: |
Adaptive mesh refinement (AMR) is often used when solving time-dependent partial differential equations using numerical methods. It enables time-varying regions of much higher resolution, which can be used to track discontinuities in the solution by selectively refining around those areas. The open source Clawpack software implements block-structured AMR to refine around propagating waves in the AMRClaw package. For problems where the solution must be computed over a large domain but is only of interest in a small area this approach often refines waves that will not impact the target area. We seek a method that enables the identification and refinement of only the waves that will influence the target area.
Here we show that solving the time-dependent adjoint equation and using a suitable inner product allows for a more precise refinement of the relevant waves. We present the adjoint methodology in general, and give details on how this method has been implemented in AMRClaw. Examples for linear acoustics equations are presented, and a computational performance analysis is conducted. The adjoint method is compared to AMR methods already available in the AMRClaw software, and the resulting advantages and disadvantages are discussed. The code for the examples presented is archived on Github.
author:
- Brisa N Davis
- Randall J LeVeque
title: 'Analysis and Performance Evaluation of Adjoint-Guided Adaptive Mesh Refinement for Linear Hyperbolic PDEs Using Clawpack'
---
<ccs2012> <concept> <concept\_id>10002950.10003705.10011686</concept\_id> <concept\_desc>Mathematics of computing Mathematical software performance</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10011007.10011074.10011099.10011693</concept\_id> <concept\_desc>Software and its engineering Empirical software validation</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003752.10003809</concept\_id> <concept\_desc>Theory of computation Design and analysis of algorithms</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10011007.10010940.10011003.10011002</concept\_id> <concept\_desc>Software and its engineering Software performance</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
[^1]
[^1]: This work is supported by the National Science Foundation, under an NSF Graduate Research Fellowship DGE-1256082 and grants DMS-1216732 and EAR-1331412, as well as the Graduate Presidential Fellowship, GO-MAP, University of Washington.
|
---
abstract: 'Neutralino dark matter leads to the formation of numerous earth mass dark matter haloes at redshifts $z\approx 60$ [@Diemand2005]. These abundant CDM micro-haloes have cuspy density profiles that can easily withstand the Galactic tidal field at the solar radius. Zhao, Taylor, Silk & Hooper (astro-ph/0502049) concluded that “...the majority of dark matter substructures with masses $\sim 10^{-6}M_\odot$ will be tidally disrupted due to interactions with stars in the Galactic halo”. However these authors assumed a halo density of stars that is at least an order of magnitude higher than observed. We show that the appropriate application of the impulse approximation is to the regime of multiple encounters, not single disruptive events as adopted by Zhao et al., which leads to a survival time of several Hubble times. Therefore we do not expect the tidal heating by Galactic stars to affect the abundance of micro-haloes. Disk crossing will cause some mass loss but the central cores are likely to survive and could be detected as gamma-ray sources with proper motions of several arc minutes per year.'
author:
- 'Ben Moore, Juerg Diemand, Joachim Stadel & Thomas Quinn\'
bibliography:
- 'ms.bib'
title: 'On the survival and disruption of Earth mass CDM micro-haloes'
---
Discussion {#section:introduction}
==========
The rate of energy transfer by tidal heating by a class of objects depends on several factors including their mean density ${n_*}$ and mean mass ${m_*}$. Zhao et al. assumed $n_*m_*=0.1$ for the Galactic disk and $n_*m_*=0.001$ for halo stars. Observations suggest that more appropriate numbers would be $0.05$ and $<0.0001$ for disk and halo stars respectively, [@MB1981; @Flynn1996; @Gould1998; @Ivezic2000; @Spagna2004]. Here the halo density is inferred from star counts, proper motion surveys, deep HST images and Sloan RR Lyrae counts. In their revised comment (version \#4) Zhao et al. replaced the “halo” contribution with a “bulge” component. However this does not change the observational evidence that the stellar density high above the local Galactic disk is very low. Furthermore extrapolating the Galactic bulge to the solar radius is premature since recent observations suggest that the bulge is really part of the Galactic bar [@Fr1998; @Pi2004], truncated at 4 kpc that is possibly undergoing a complex buckling instability. In which case it is unlikely that the bulge contributes to the halo density at the solar radius. These estimates of the halo density are all at least an order of magnitude lower than assumed by Zhao et al., which reduces the heating rate by non-disk stars to a minor level.
For Earth mass micro-halos with a half mass radius $r_{1/2}=0.005$ pc and $m_{halo}=10^{-6}M_\odot$ we find a critical impact parameter with a star, such that its internal energy changes by of order itself, to be $b_{min}=0.012$ pc. The disk surface density is observed to be $\approx 46 M_\odot {\rm pc}^{-2}$ [@K1989], therefore the probability of a disruptive collision per disk crossing is less than $0.02$. This neglects the clustering of stars in the disk and is a crude estimate of survival probability but demonstrates that most of the haloes are not in the single encounter disruption regime. In Diemand et al. (2005) we estimated the disruption timescale by integrating the impulsive heating $\Delta E \propto G^2r_{1/2}^2m_*^2m_{halo}/(b^4v^2)$ over all encounters and comparing to the binding energy of CDM haloes $E_{bind}\propto Gm_{halo}^2/r_{1/2}$ i.e. Binney & Tremaine (1989). With appropriate factors for the internal structure of these haloes we found a disruption timescale of $t_{dis}\approx 30$ Gyrs, assuming that the micro-haloes spend one percent of their time in the disk.
As mentioned in Diemand et al. (2005) we expect encounters with stars and molecular clouds to lead to some disruption and mass loss but that most such structures will stay intact. The impulse approximation is an approximation, as are semi-analytic models for such mass loss – tidal heating rarely leads to complete disruption and the high density central cores of orbiting systems can easily remain intact. Numerical simulations are vital to understand these complex dynamical processes. A more detailed study supported by numerical simulations in the spirit of Moore (1993) by the authors is in preparation. The most important factor in the survival statistics of micro-haloes is how many survive similar mass mergers as a Galactic mass halo is built up. However, even if just a few percent survive the hierarchical growth, many micro-haloes will lie within one parsec from the sun. Their dense cuspy cores will be sources of gamma-ray emission which could be uniquely distinguished by their high proper motions of order minutes of arc per year.
|
---
abstract: 'We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between the category of bivariant coherent sheaves and the category of sheaves of finitely generated modules.'
author:
- |
Jyh-Haur Teh\
[Department of Mathematics, National Tsing Hua University of Taiwan]{}
title: Families of algebraic varieties parametrized by topological spaces
---
Introduction
============
Algebraic varieties are defined by the zero sets of some polynomials over some fields. When the field is $\C$, with the natural analytic topology, algebraic varieties have the homotopy types of CW-complexes. Topological considerations may unveil some very interesting properties of algebraic varieties. For example, the homotopy groups of the groups of algebraic cycles of varieties which are called the Lawson homology groups, are important invariants of varieties (see [@Law; @F]). Therefore we are interested in studying varieties parametrized by topological spaces, especially by the sphere $S^n$. Since the standard unit sphere $S^n\subset
\R^{n+1}$ is a real algebraic variety, we may consider polynomials with coefficients of real rational functions on $S^n$. It is clear that the zero sets defined by such polynomials may not vary continuous. But out of a set of measure zero, they do vary continuously, namely, the Hilbert polynomial of each fibre is the same except over a set of measure zero. To study such families, we develop a foundation theory in this paper. We generalize several classical results such as Hilbert Nullstellensatz and the primary decomposition of commutative rings.
In the following, we give a brief overview of each section. In section 2, we define algebraic varieties over topological spaces and their coordinate rings. In section 3, we study Groebner ideals in polynomial rings with continuous coefficients, introduce measure topology, condensed spaces, condensed ideals, almost Groebner ideals, prove a Hilbert Nullstellensatz and an almost primary decomposition theorem. In section 4, we develop a theory of bivariant coherent sheaves over semi-topological algebraic sets, and show that there is an equivalence of the category of almost sheaves of $K[\X]$-modules and the category of bivariant quasi-coherent sheaves on $\X$.
The author thanks Li Li for his very helpful comments, and Eric Friedlander, Christian Haesemeyer, Mark Levine for their encouragement. Special thanks to my wife Yu-Wen Kao for her understanding. He also thanks the Taiwan National Center for Theoretical Sciences (Hsinchu) for providing a wonderful working environment.
Affine algebraic sets
=====================
Given a ringed space $(S, \mathcal{O}_S)$. For each open set $U\subseteq S$, we define $$(\mathcal{O}_S[x_1, ..., x_n])(U):=\mathcal{O}_S(U)[x_1, ...,
x_n]$$ where $\mathcal{O}_S(U)[x_1, ..., x_n]$ is the ring of polynomials of $n$ variables with coefficients in $\mathcal{O}_S(U)$. In other words, it is the collection of all polynomials of the form $$f(x)=\sum_{|J|\leq k}a_Jx^J$$ for some $k\geq 0$ where $J\in \Z^n_{\geq
0}$, $|J|=i_1+\cdots +i_n$ if $J=(i_1, ..., i_n)$, and all $a_J\in
\mathcal{O}_S(U)$. The degree of $f$ is defined to be the maximal $|J|$ if $a_J$ is not identically zero. The restriction maps of $\mathcal{O}_S[x_1, ..., x_n]$ are induced by restricting coefficients.
Note that $\mathcal{O}_S[x_1, ..., x_n]$ is in general not a sheaf. After we introduce the concept of a condensed space and almost sheaf, we will see that $\mathcal{O}_S[x_1, ..., x_n]$ is actually an almost sheaf.
In this paper, we denote by $S$ for a topological space and $\mathscr{C}$ the sheaf of germs of continuous complex-valued functions.
Given a topological space $(S, \mathscr{C})$. The affine $n$-space over $S$ is the set $\A^n_S:=S\times \C^n$. We denote by $\pi:S\times \C^n \rightarrow S$ the natural projection.
Given a topological space $(S, \CO)$. Let $J\subseteq
\mathscr{C}[x_1, ..., x_n](S)$ be an ideal of the global sections.
Let $$\X:=\{(s, x)|f_s(x)=0, \mbox{ for all } f\in J\}\subseteq \A^n_S$$ where $f_s$ is the restriction of $f$ to the fibre $\pi^{-1}(s)$ of $s$. $\X$ is called an affine algebraic set over $S$. It induces a presheaf of sets on $S$ defined by $$\X(U):=\X\cap \pi^{-1}(U)$$ where $U$ is an open subset of $S$.
For $U$ an open set of $S$, we write $$J|_U:=\{f|_U|f\in J\}$$
By abuse of notation, we also write $J$ for the ideal sub-presheaf of $\mathscr{C}[x_1, ..., x_n]$ induced by $J$ which is defined by $$J(U)=\mbox{ ideal of } \mathscr{C}[x_1, ..., x_n](U) \mbox{
generated by } J|_U$$
The vanishing presheaf of $\X$, denoted by $I(\X)$, is the presheaf defined by $$I(\X)(U):=\{f\in \CO[x_1, ..., x_n](U)|f(\X(U))=0\}$$
The radical presheaf of $J$ is defined by $$\sqrt{J}(U):=\{f\in \CO[x_1, ..., x_n](U)|f^n\in J(U)
\mbox{ for some } n\in \N\}$$
The fibrewise radical presheaf of $J$ is defined by $$FRad(J)(U):=\{f\in \CO[x_1, ..., x_n](U)|f_s\in \sqrt{J(U)|_s}, s\in
U\}$$ where $J(U)|_s=\{g_s|g\in J(U)\}$.
We write $$V(J)(U):=\{(s, x)|f_s(x)=0, f\in J(U)\}$$ and
$$\X_s:=\X\cap \pi^{-1}(s)$$ for $s\in S$.
\[coordinate almost sheaf\] Let $(S, \mathscr{C})$ be a topological space and $\X\subseteq
\A^n_S$ be an affine algebraic set over $S$. For an open set $U\subseteq S$, the presheaf which assigns an open set $U$ the ring $$K[\X](U):=\CO[x_1, ..., x_n](U)/I(\X)(U)$$ is called the coordinate presheaf of $\X$.
The following example shows that we can not expect an analog of Hilbert’s Nullstellensatz to hold for arbitrary ideals.
Let $S=[-1, 1]$ be the closed interval from $-1$ to $1$. Let $I=<e^{-\frac{1}{s}}>\subseteq \CO[x](S)$, $\X=V(I)=0\times \C$, $f_s(x)=s$. Then $f\in \mathscr{C}[x](S)$ and $f(\X)=0$. Since $\lim_{s\to 0}|\frac{s^n}{e^{-\frac{1}{s}}}|=\infty$ for any $n\in
\N$, therefore $f^n\notin I$ for any $n\in \N$. This shows that $f\in FRad(I)$ but $f\notin Rad(I)$.
Given a topological space $(S, \mathscr{C})$ and $I\subseteq
\CO[x_1, ..., x_n]$ be an ideal. Then $IV(I)=FRad(I)$. \[Hilbert’s Nullstellensatz\]
If $f\in IV(I)(U)$, then $f(V(I(U)))=0$. Therefore $f_s(V(I(U))|_s)=0$ for all $s\in U$. By Hilbert’s Nullstellensatz, we have $f_s\in \sqrt{I(U)|_s}$ for $s\in S$. Hence $f\in
FRad(I)(U)$. Another direction is trivial.
Let $S$ be a compact topological space. If $a\in \CO(U)$ for $U\subseteq S$, we write $|a(s)|$ for the Euclidean norm of the complex number $a(s)$. For $f, g\in \CO[x_1, ..., x_n](S)$, we define a norm by $$||f-g||:=\sup_{s\in S}\{|a_{\alpha}(s)-b_{\alpha}(s)|\ | \alpha\in
A\}$$ where $f=\sum_{\alpha\in A}a_{\alpha}(s)x^{\alpha},
g=\sum_{\alpha\in A}b_{\alpha}(s)x^{\alpha}$, and $A$ is a finite subset of $\Z^n_{\geq 0}$.
Let $S$ be a compact topological space. Let $I\subseteq
\mathscr{C}[x_1, ..., x_n](S)$ be an ideal. If $I$ is a fibrewise radical ideal, i.e., $FRad(I)=I$, then $I$ is closed in the norm defined above.
Suppose that $f_i$ approaches $f$ where $f_i\in I$ and $f\in
\CO[x_1, ..., x_n](S)$. The convergence of $f_i$ in the norm to $f$ implies that $f_i$ converges to $f$ pointwise. Since $f_i(V(I))=0$ for each $i$, we have $f(V(I))=0$. Hence $f\in IV(I)=FRad(I)=I$.
In order to prove a version of Hilbert’s Nullstellensatz to relate $IV(J)$ and $\sqrt{J}$, we need to have a bound $e$ such that if $f_{s_0}\in \sqrt{I|_{s_0}}$, then $f_{s}^e\in I|_s$ for $s$ in a neighborhood of $s_0$. For the convenience of the reader, we recall a result of Brownawell (see corollary of Theorem 1 of [@Brown87]).
(Brownawell) If $Q, P_1, ..., P_m\in \C[x_1, ..., x_n]$ have degrees $\leq D$ and if $Q$ vanishes on all the common zeros of $P_1, ..., P_m$ in $\C^n$, then there are $e\in \N$ and polynomials $B_1, ..., B_m\in
\C[x_1, ..., x_n]$ with $$e\leq e':=(\mu+1)(n+2)(D+1)^{\mu+1}$$ $$deg B_iP_i\leq eD+e'$$ where $\mu=min\{m, n\}$ such that $$Q^e=B_1P_1+\cdots +B_mP_m.$$\[Brownawell\]
With the help of topology, we can say more about the Hilbert’s Nullstellensatz for a principal ideal.
Let $S$ be a compact topological space. Let $g\in \CO[x_1, ...,
x_n](S)$ be such that $g_s$ is not is not a zero polynomial for $s\in S$. Then $FRad(<g>)=\sqrt{<g>}$ as presheaves.\[principal Hilbert Nullstellensatz\]
If $h\in FRad(<g>)(U)$, then by Theorem \[Brownawell\], there is $N$ such that $h^N_s\in <g_s>$ for all $s\in U$. Hence we can find $c_s\in \mathscr{C}[x_1, ..., x_n](U)$ such that $h^N_s=c_sg_s$. We need to show that $c$ is continuous in $s$. Fix $s_0\in S$. Then $h^N_s-h^N_{s_0}=(c_s-c_{s_0})g_s+c_{s_0}(g_s-g_{s_0})$. Since the degrees of $\{c_s|s\in U\}$ are bounded, the limit $\lim_{s \to
s_0}c_s$ taking under the norm defined above exists. So we have $\lim_{s\to s_0}(c_s-c_{s_0})g_{s_0}=0$. Since $g_{s_0}$ is not the zero polynomial which implies that $\lim_{s\to s_0}(c_s-c_{s_0})=0$ and we have $\lim_{s \to s_0}c_s=c_{s_0}$. The continuity of $c$ implies that $h\in \sqrt{<g>}(U)$.
Groebner ideals
===============
In order to generalize Hilbert Nullstellensatz, we work on nice topological spaces and ideals with nice properties. We develop the theory of Groebner ideals of the ring of continuous functions over condensed spaces. The results of this chapter play fundamental role in our theory. The main reference for Groebner ideals is [@CLS].
Fix a monomial ordering (see [@CLS Definition 2.2.1]) of $\C[x_1, ..., x_n]$. Let $S$ be a topological space. For $h\in
\CO[x_1, ..., x_n](S)$, write $$h_s(x)=a_1(s)x^{\alpha_1}+a_2(s)x^{\alpha_2}+\cdots
+a_k(s)x^{\alpha_k}$$ where $x^{\alpha_1}>x^{\alpha_2}>\cdots
>x^{\alpha_k}$ according to the monomial ordering where $\alpha_i=(a_1, ...,
a_n)$ is a multi-index. As in [@CLS], we denote by $LT(h)=a_1(s)x^{\alpha_1}$ the leading term of $h$, $LC(h)=a_1(s)$ the leading coefficient of $h$, $LM(h)=x^{\alpha_1}$ the leading monomial of $h$ and $multideg(h)=\alpha_1$ the multi degree of $h$. For $f\in \CO[x_1, ..., x_n](S)$, we say that the leading term $LT(f)$ of $f$ is nonvanishing on $S$ if $LC(f)$ is nonvanishing on $S$.
Let $S$ be a topological space. Let $f, g\in \mathscr{C}[x_1, ...,
x_n](S)$ be two nonzero polynomials. Let $\alpha=(\alpha_1, ...,
\alpha_n)=multideg(f), \beta=(\beta_1, ..., \beta_n)=multideg(g)$ and $\gamma_i=max\{\alpha_i, \beta_i\}$, $\gamma=(\gamma_1, ...,
\gamma_n)$. If $LT(f), LT(g)$ are nonvanishing on $S$, the $S$-polynomial $S(f, g)$ of $f$ and $g$ is $$S(f, g)=\frac{x^{\gamma}}{LT(f)}f-\frac{x^{\gamma}}{LT(g)}g$$
Let $S$ be a topological space. Let $G=(f_1, ..., f_r)$ where each $f_i\in \mathscr{C}[x_1, ..., x_n](S)$ and $LT(f_i)$ is nonvanishing on $S$. For any $f\in \mathscr{C}[x_1, ..., x_n](S)$, we write $\overline{f}^F$ for the remainder on division of $f$ by the ordered $r$-tuple $F$ (see [@CLS Definition 2.6.3]).
Let $S$ be a topological space. Let $f_1, ..., f_k\in
\mathscr{C}[x_1, ..., x_n](S)$ where all $LT(f_i)'s$ are nonvanishing on $S$. The set $\{f_1, ..., f_k\}$ is a Buchbergerable set if the following recursive procedure return ‘TRUE’.
REPEAT\
$G':=G$\
FOR $\{p, q\}, p\neq q \mbox{ in } G'$ DO\
$g:=\overline{S(p, q)}^{G'}$\
IF $g\neq 0$ AND $LT(g)$ nonvanishing THEN $G:=G\cup \{g\}$\
IF $g\neq 0$ AND $LT(g)$ vanishing THEN RETURN FALSE AND STOP\
UNTIL $G=G'$ RETURN TRUE AND STOP\
We say that a finite generating set $\{g_1, ..., g_k\}$ of an ideal $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$ is a Groebner basis of $I$ if
1. the ideal generated by the leading terms $<LT(g_1), ...,
LT(g_k)>$ of the generating set is same as the ideal generated by the leading terms $<LT(I)>$ of the elements of the ideals $I$ over $S$;
2. the leading terms of $g_i's$ are nonvanishing on $S$.
An ideal with a Groebner basis is called a Groebner ideal.
Throughout this paper, we fix a monomial ordering of $\C[x_1, ...,
x_n]$.
Let $(S, \mathscr{C})$ be a topological space. Let $h\in \CO[x_1,
..., x_n](S)$ and $I=<g_1, ...., g_k>$ be an ideal of $\CO[x_1, ...,
x_n](S)$ generated by $G=\{g_1, ..., g_k\}$ such that the leading terms of all $g_i's$ are nonvanishing over $S$.
1. (Continuous division algorithm) We have $h=\sum^k_{i=1}a_ig_i+r$ where all $a_i, r\in \CO[x_1, ...,
x_n](S)$, and either $r=0$ or $r$ is a linear combination, with coefficients in $\mathscr{C}(S)$, of monomials, none of which is divisible by any of $LT(g_1), ..., LT(g_k)$. (see [@CLS Theorem 2.3.3]).
2. (Buchberger’s criterion) The generating set $G$ is a Groebner basis of $I$ if and only if the remainder on division of $S(g_i,
g_j)$ by $G$ is zero for any $i\neq j$.
3. (Continuous Buchberger’s algorithm) If $G$ is Buchbergerable, a Groebner basis for $I$ can be constructed from the set $G=\{g_1, ..., g_k\}$ by Buchberger’s algorithm (see [@CLS Theorem 2.7.2]).
4. Let $\{h_1, ..., h_l\}$ be a Groebner basis of $I$. Then $h=\sum_{i=1}^la_ih_i+r$ where all $a_i, r\in \CO[x_1, ...,
x_n](S)$ and $h\in I$ if and only if $r=0$.
\[algorithms\]
\(1) Since the leading terms of all $g_i's$ are nonvanishing, they can be inverted, hence we can do the division algorithm as usual. (2) The proof is similar to the proof of [@CLS Theorem 2.6.6]. (3) The definition of a Buchberger set is to make the proof of Buchberger algorithm works. (4) The reason that $h$ has such expression follows from division algorithm and the rest follows from properties of a Groebner basis.
Let $S=[0, 1]$, $f(x, y)=x^2+sy^2$, $g(x, y)=x+y$. We use the lexicographic order such that $x^ay^b>x^cy^d$ if $a>c$ or $a=c$ and $b>d$. Let $G=\{f, g\}$. Then $$S(f, g)=-xy+sy^2, h(x, y)=\overline{S(f, g)}^{G}=(s+1)y^2$$ and the procedure ends. So $\{f, g, h\}$ is a Groebner basis for the ideal generated by $f, g$.
Let $S$ be a topological space. If $\{g_1, ..., g_k\}$ is a Groebner basis of an ideal $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$, then $<LT(g_1)_s, ..., LT(g_k)_s>=<LT(I|_s)>$ which means that $\{g_{1,
s}, ..., g_{k, s}\}$ is a Groebner basis for $I|_s$ for each $s\in
S$.
By (2) of the Lemma above, for any $i\neq j$, we have $S(g_i,
g_j)=\sum_{l=1}^kh_lg_l$ for some $h_l\in \mathscr{C}[x_1, ...,
x_n](S)$. Then for $s\in S$, $S(g_{i, s}, g_{j, s})=S(g_i,
g_j)|_s=\sum_{l=1}^kh_{l, s}g_{l, s}$. Since $I|_s=<g_{1, s}, ...,
g_{k, s}>$, by [@CLS Theorem 2.6.6], $\{g_{1, s}, ..., g_{k,
s}\}$ is a Groebner basis of $I|_s$.
Suppose that $S$ is a path-connected topological space and $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$ is a Groebner ideal. Then the Hilbert polynomial $HP_{I|_s}$ is same for $s\in S$. In particular, $\mbox{dim } V(I|_s)=\mbox{dim } V(I|_{s'})$ for any $s,
s'\in S$.
Let $f_1, ..., f_k$ be a Groebner basis for $I$. Let $\gamma:[0, 1]
\rightarrow S$ be a path such that $\gamma(0)=s$, $\gamma(1)=s'$. Then $LC(f_{i, \gamma(t)})$ is nonvanishing for any $t\in [0, 1]$. Thus $LM(f_{i, s})=LM(f_{i, s'})$ and therefore $LT(f_{i,
s'})=\frac{LC(f_{i, s'})}{LC(f_{i, s})}LT(f_{i, s})$ where $\frac{LC(f_{i, s'})}{LC(f_{i, s})}\in \mathscr{C}[x_1, ...,
x_n](S)$. So $<LT(I|_s)>=<LT(f_{1, s}), ..., LT(f_{k, s})>=<LT(f_{1,
s'}), ..., LT(f_{k, s'})>=<LT(I|_{s'})>$. The result follows from [@CLS Proposition 9.3.4] which proves that the monomial ideal $<LT(I|_s)>$ has the same Hilbert polynomial as $I|_s$.
Almost Groebner ideals
----------------------
For properties about real algebraic sets, we refer the reader to [@BCR].
(Condensed spaces) Let $S\subseteq \R^m$ be a topological subspace. A polynomial function on $S$ is a continuous function $f:S \rightarrow \C$ such that $f(x_1, ..., x_m)=\sum^n_{i=0, |\alpha|=i}c_{\alpha}x^{\alpha}$ for some $n\in \N$, $c_{\alpha}\in \C$. Let $\mathscr{P}(S)$ be the ring of all polynomials on $S$. A real algebraic subset of $S$ is the zero loci of finitely many polynomials on $S$. The space $S$ is said to be condensed if for any real algebraic subset $V\subset S$, $V\neq S$, $V$ is nowhere dense under the Euclidean topology of $S$. Let $$\mathscr{R}(S)=\{\frac{f}{g}|f, g\in \mathscr{P}(S), 0\notin
g(S)\}$$ An element of $\mathscr{R}(S)$ is called a rational function on $S$ and an element of $\mathscr{R}(S)[x_1, ..., x_n]$ is called a condensed element of $S$.
The topology of real algebraic sets are much more complicated than complex algebraic sets. Easy to see that all spheres $S^n$, connected intervals on the real line, unit disks are all condensed. See [@BCR Example 3.1.2] for real algebraic varieties that are not condensed.
(Measure topology) Suppose that $S$ is a condensed space. A closed set of $S$ in the measure topology is defined to be a set of the form $$\cap^{\infty}_{i=1}\cup^{\infty}_{j=1}V_{ij}$$ where $V_{ij}$ are some real algebraic subsets of $S$.
It is easy to check that the finite union and countably intersection of sets of this form are again a set of this form.
Let $S$ be a condensed space. Let $\Sigma$ be the collection of all closed sets of $S$ under the measure topology except the whole $S$, then $(S, \Sigma)$ is a sigma space. We will always consider a condensed space under measure topology, and as a sigma space in this way.
If $S$ is a condensed space, then any open set $U\subseteq S$ is condensed.
Let $V\subseteq U$ be a real algebraic subset of $U$ which is the zero loci of some polynomials $f_1, ..., f_k$ on $U$. We may consider $f_1, ..., f_k$ as polynomials on $S$ and let $\widetilde{V}$ be the zero loci of them in $S$. If $V$ is not nowhere dense in $U$, then there exists an open set $W\subseteq U$ such that $W\cap V$ is dense in $W$. Since $W$ is also an open subset of $S$, and $W\cap
\widetilde{V}\supset W\cap V$ is dense in $W$, $S$ is not condensed which is a contradiction.
(Condensed ideals) Suppose that $S\subseteq \R^N$ is a condensed space. An ideal $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$ is said to be condensed if $I$ is generated by some elements of $\mathscr{R}[x_1, ...,
x_n](S)$. The presheaf induced by a condensed ideal is called a condensed presheaf.
(Finite generation of condensed ideals) If $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$ is a condensed ideal where $S$ is a condensed space, then $I$ is finitely generated.\[finite generation condensed ideals\]
Let $\Sigma$ be the collection of all condensed elements of $I$. Then $\Sigma$ is an ideal of $\mathscr{P}(S)[x_1, ..., x_n]$. Since $\mathscr{P}(S)$ is Noetherian, $\mathscr{P}(S)[x_1, ..., x_n]$ is Noetherian, hence $\Sigma$ is finitely generated. Therefore $I$ is finitely generated.
(Condensed basis) Let $I\subseteq\mathscr{C}[x_1, ..., x_n](S)$ be a condensed ideal where $S$ is a condensed space. If $I$ is generated by some condensed elements $f_1, f_2, ..., f_m$, then $\{f_1, f_2, ...,
f_m\}$ is called a condensed basis of $I$.
(Sigma spaces) Let $S$ be a topological space and $\Sigma$ be a collection of closed subsets of $S$. If any countably union of elements in $\Sigma$ is again in $\Sigma$, then the pair $(S, \Sigma)$ is called a sigma space and elements of $\Sigma$ are called sets of measure zero.
(Almost sheaves) Let $S$ be a sigma space and $\mathscr{F}$ be a presheaf on $S$. An element $\sigma\in \mathscr{F}(U)$ is said to be almost zero, denoted by $a=_a 0$, if there is a set $A\subseteq S$ of measure zero such that $\sigma|_{U-A}=0$. Two elements $\sigma_1\in
\mathscr{F}(U), \sigma_2\in \mathscr{F}(V)$ are almost equal, denoted by $\sigma_1=_a \sigma_2$, if there is a set $A$ of measure zero such that $\sigma_1|_{U-A}=\sigma_2|_{V-A}$. For $\alpha\in
\mathscr{F}(V)$, we write $\alpha \in_a \mathscr{F}(U)$ if there is a set $A$ of measure zero such that $\alpha|_{V-A}\in
\mathscr{F}(U-A)$.
We say that $\mathscr{F}$ is an almost sheaf if it satisfies the following two properties:
1. (Almost Uniqueness) For $\sigma\in \mathscr{F}(U)$, if there is an open covering $U=\cup_i U_i$ such that the restriction $\sigma|_{U_i}=_a 0$ for any $i$, then $\sigma=_a 0$;
2. (Almost extension property) for any open set $U$ of $S$ and for any open covering $\{U_i\}$ of $U$, if $\sigma_i\in \mathscr{F}(U_i)$ and $\sigma_i|_{U_i\cap U_j}=_a\sigma_j|_{U_j\cap U_i}$ for any $i,
j$, then there is $\sigma \in_a \mathscr{F}(U)$ such that $\sigma|_{U_i}=_a \sigma_i$ for any $i$.
Let $S$ be a condensed space with the measure topology and $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$ be a finitely generated ideal.
1. We say that $I$ is almost Groebner if there exist a subset $A\subseteq S$ of measure zero such that $I(S-A)$ is a Groebner ideal of $\mathscr{C}[x_1, ..., x_n](S-A)$. The presheaf induced by $I$ is called an almost Groebner presheaf.
2. We say that $I$ is almost Buchbergerable if there exist a subset $A \subseteq S$ of measure zero such that there exist a generator set $\{f_1, ..., f_k\}$ of $I(S-A)$ which is a Buchbergerable set. The presheaf induced by $I$ is called an almost Buchbergerable presheaf.
Immediately from the definition and by Buchberger’s algorithm, we have the following observation.
If an ideal is almost Buchbergerable, then it is almost Groebner.
If $I\subseteq \mathscr{C}[x_1, ..., x_n](S)$ is a Groebner ideal with Groebner basis $G=\{f_1, ..., f_k\}$, then $G|_{S-A}=\{f_1|_{S-A}, ..., f_k|_{S-A}\}$ is a Groebner basis of the ideal $I(S-A)\subseteq \mathscr{C}[x_1, ..., x_n](S-A)$ for any $A\subseteq S$ of measure zero.
Let $U=S-A$. By definition, $LC(f_i)$ is nonvanishing on $S$, hence $LT(f_i)|_U=LT(f_i|_U)$. So for the $S$-polynomial, $S(f_i,
f_j)|_U=S(f_i|_U, f_j|_U)$. By Buchberger’s criterion, the remainder of $S$-polynomial $S(f_i, f_j)$ on division by $G$ is zero, hence the remainder of $S$-polynomial $S(f_i|_U, f_j|_U)$ on division by $G|_U$ is zero. Hence by Buchbergerable’s criterion, $G|_U$ is a Groebner basis for the ideal $I(U)$ in $\mathscr{C}[x_1, ...,
x_n](U)$.
The following simple example shows that we can not make any extension even for a very perfect function defined in a dense open set.
Let $I=<x+s, x-s>\subseteq \mathscr{C}[x](\R)$. Then obviously $I(\R-\{0\})=\mathscr{C}[x](\R-\{0\})$, and $V(I(\R))=\{(0, 0)\}$. So $1\in I(\R-\{0\})$ but $1\notin I(\R)$.
1. A Groebner ideal presheaf is a Groebner ideal sheaf.
2. An almost Groebner presheaf is an almost Groebner sheaf.
3. In particular, an almost Buchbergerable presheaf is an almost Buchbergerable sheaf.
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1. Suppose that $I\subseteq \mathscr{C}[x_1, ..., x_n]$ is a Groebner presheaf induced by a Groebner ideal $I\subseteq
\mathscr{C}[x_1, ..., x_n](S)$. Note that under the measure topology of $S$, $\mathscr{C}[x_1, ..., x_n]$ is a again a sheaf on $S$. Let $U$ be an open set of $S$ and $U=\cup_i U_i$ be a covering of $U$. Suppose that $f_i\in I(U_i)$ such that $f_i|_{U_i\cap U_j}=f_j|_{U_j\cap U_i}$ for any $i, j$. Then there is $f \in \mathscr{C}[x_1, ..., x_n](U)$ such that $f|_{U_i}=f_i$. Let $g_1, ..., g_k$ be a Groebner basis of $I(U)$. Then $f=\sum^k_{i=1}a_ig_i+r$ for some $a_i, r\in
\mathscr{C}[x_1, ..., x_n](U)$ where $LT(r)$ is not divisible by any of $LT(g_i)$. Then $f|_{U_i}=\sum^k_{i=1}a_ig_i|_{U_i}+r|_{U_i}\in I(U_i)$. This implies that $r|_{U_i}=0$ for all $i$. Thus $r=0$ and hence $f\in I(S)$. This proves the extension property of $I$. The uniqueness of $I$ comes directly from the uniqueness of $\mathscr{C}[x_1, ..., x_n]$.
2. Let $A\subseteq S$ be a set of measure zero such that $I(S-A)$ is a Groebner ideal of $\mathscr{C}[x_1, ..., x_n](S-A)$. Then by the result above, $I|_{S-A}$ is a Groebner sheaf and hence $I$ is an almost Groebner sheaf.
Let $S$ be a condensed space. Suppose that $I=<f_1, ...,
f_k>\subseteq \mathscr{C}[x_1, ..., x_n](S)$ is a condensed ideal where $f_1, ..., f_k\in \mathscr{P}(S)$. Then $G=\{f_1, ..., f_k\}$ is Buchbergerable on $S-A$ where $A\subsetneq S$ is a real algebraic subset. In particular, $I$ is almost Buchbergerable, and $I$ induces a Buchbergerable almost sheaf.
Let $G=\{f_1, ..., f_k\}$. The Buchberger’s algorithm does not work if the leading terms vanish somewhere. So the idea of the proof is to delete the subset of $S$ that leading terms of the $S$-polynomials vanish and this set is a real algebraic subset of $S$. Let $f_{12}=\overline{S(f_1, f_2)}^G$ and $C_{12}=$ the zero locus of the leading coefficient of $f_{12}$. On $S-C_{12}$, let $G_{12}=G\cup \{f_{12}\}$, and $f_{13}=\overline{S(f_1,
f_3)}^{G_{12}}$. Let $C_{13}$ be the zero locus of the leading coefficient of $f_{13}$. On $S-(C_{12}\cup C_{13})$, we repeat the process before. Keep doing this, we get a set $G'=\{f_1, ..., f_k,
f_{ij}, i, j=1, ..., k, i\neq j\}$ and all the leading coefficients are nonvanishing on $S-\cup_{i\neq j}C_{ij}$. Now repeat the process for $G=G'$. This process terminates after a finite number of times. Since $C_{ij}$ are all real algebraic subsets of $S$, we are done.
Suppose that $I, J\subseteq \mathscr{C}[x_1, ..., x_n](S)$ are two ideals. We say that $I$ is almost contained in $J$, denoted by $I\subseteq_a J$, if there is a measure zero subset $A\subseteq S$ such that $I(S-A)\subseteq J(S-A)$. We say that $I$ is almost equal to $J$, denote by $I=_a J$, if $I\subseteq_a J$ and $J\subseteq_a
I$. For $f\in \mathscr{C}[x_1, ..., x_n](S)$, we write $f\in_a I$ and say that $f$ is almost in $I$ if there is a set $A$ of measure zero such that $f|_{S-A}\in I(S-A)$.
Let $\sum$ be a collection of some ideals of $\mathscr{C}[x_1, ...,
x_n](S)$ where $S$ is a condensed space. If for any chain $$J_1\subseteq_a J_2 \subseteq_a J_3 \subseteq_a \cdots$$ in $\sum$, there is an $N$ such that $J_{N+k}=_aJ_N$ for all $k\in
\N$, then we say that $\sum$ has the almost ascending chain property.
Let $S$ be a condensed space. Let $R=\mathscr{C}[x_1, ..., x_n](S)$, and $AG(R)$ be the collection of all almost Groebner ideals of $R$. Then $AG(R)$ has the almost ascending chain property. \[ASC\]
Let $$J_1\subseteq_a J_2 \subseteq_a J_3 \subseteq_a \cdots$$ be an ascending chain in $AG(R)$ and $J=\cup_i J_i$. Suppose that $A_i\subset S$ is a subset of measure zero such that $J_i(S-A_i)$ is a Groebner ideal of $\mathscr{C}[x_1, ..., x_n](S-A_i)$. Let $B=\cup^{\infty}_{i=1}A_i$. Then $B$ has measure zero, and we have $J(S-B)=\cup^{\infty}_{i=1}J_i(S-B)$.
Let $M=\{LT(f)|f\in J(S-B), LC(f)=1\}$. Since each $J_i(S-B)$ is Groebner, $$LT(J(S-B))=<M>\subseteq \C[x_1, ..., x_n](S-B)$$
Consider the monomial ideal generated by $M$ in $\C[x_1, ..., x_n]$. By Dickson’s Lemma [@CLS Theorem 2.4.5], it is finitely generated, hence $LT(J(S-B))$ is finitely generated. Let $LT(J(S-B))=<a_1, ..., a_t>$ and $g_1, ..., g_t\in J(S-B)$ such that $LT(g_i)=a_i$. For $h\in J(S-B)$, by the division algorithm, we may write $$h=\sum^t_{i=1}b_ig_i+r$$ where $r=0$ or each term of $r$ is not divisible by any $LT(g_i)=a_i$. If $r\neq 0$, this implies that $LT(r)\notin
LT(J(S-B))$. This contradicts to the fact that $r\in J(S-B)$. Therefore $r=0$ and hence $J(S-B)$ is generated by $g_1, ...., g_t$. Suppose that $g_i\in J_{n_i}$ and take $N=max\{n_1, ..., n_t\}$. Then $J=_a J_N$.
Let $S$ be a condensed space and $C(R)$ be the collection of all condensed ideals of $R=\mathscr{C}[x_1, ..., x_n](S)$, $AB(R)$ be the collection of all almost Buchbergerable ideals of $R$. Then $C(R), AB(R)$ have the almost ascending chain property.
This follows from the result above and the fact that $C(R)$ and $AB(R)$ are subsets of $AG(R)$.
Suppose that $I, J \subseteq \mathscr{C}[x_1, ..., x_n](S)$ are condensed ideals. Then $I\subseteq_a J$ if and only if $f\in_a J$ for any $f\in I$.
Suppose that $I$ is generated by $f_1, ..., f_k$. Then there is a set $A_i\subseteq S$ of measure zero such that $f_i|_{S-A_i}\in
J(S-A_i)$. Let $A=\cup_{i=1}^kA_i$. Then $f_i|_{S-A}\in J(S-A)$ for all $i=1, ..., k$. This implies that $I(S-A)\subseteq J(S-A)$.
If $J$ is almost Groebner, then $\sqrt{J}=_a FRad(J)$.
It is clear that $\sqrt{J}\subseteq FRad(J)$. Let $f\in FRad(J)(U)$. Suppose that $J$ is generated by $f_1, ..., f_k\in \mathscr{C}[x_1,
..., x_n](S)$. Let $N=max\{deg \ f_i|i=1, ..., k\}$ be the maximal degree of all $f_i$. Since $f_s\in \sqrt{J(S)|_s}$, by theorem \[Brownawell\], there is a bound $e$ depends only on $n, k, N$ such that $f^e_s\in J(S)|_s$ for any $s\in S$. Since $J$ is almost Groebner, there exists real algebraic subset $A\subset S$ such that $J(S-A)$ is Groebner. Let $g_1, ..., g_l$ be a Groebner basis of $J(S-A)$. Using the division algorithm we may write $f^e|_{U-A}=\sum^l_{i=1}a_ig_i+r$ where all $a_i, r\in \CO[x_1, ...,
x_n](U-A)$. For $s\in U-A$, since $\{g_{1, s}, ..., g_{k, s}\}$ is a Groebner basis for $J(U-A)|_s$, and $f^e_s=\sum^k_{i=1}a_{i, s}g_{i,
s}+r_s\in J(U-A)|_s$, hence $r_s=0$ for $s\in U-A$ which implies that $f^e|_{U-A}=\sum^l_{i=1}a_ig_i\in J(U-A)$. So $f|_{U-A}\in
\sqrt{J}(U-A)$.
Combine this theorem and Proposition \[Hilbert’s Nullstellensatz\], we generalize the classical Hilbert’s Nullstellensatz to an equality between radical ideals and vanishing ideals.
If $J$ is an almost Groebner ideal, we have $\sqrt{J}=_a IV(J)$. \[Hilbert’s Nullstellensatz Groebner ideal\]
(Almost Hilbert Nullstellensatz) If $I$ is a condensed ideal, then $\sqrt{I} =_a IV(I) =_a FRad(I)$.
Since $I$ is a condensed ideal, $I$ is almost Buchberger and hence almost Groebner.
Let $S$ be a condensed space.
1. For a condensed ideal $J\subseteq \mathscr{C}[x_1, ..., x_n](S)$, if $g_s\in_a J(S)|_s$ for $s\in S-A$ where $A$ is a set of measure zero, then $g\in_a J$.
2. If $J$ is a radical condensed ideal, i.e, $J=\sqrt{J}$, then $J|_s=I(\X_s)$ for all $s\in S$ where $\X=V(J)$.
3. In particular, $I(\X)|_s=I(\X_s)$ for all $s\in S$ where $\X$ is the zero set of some condensed ideal of $\mathscr{C}[x_1, ...,
x_n](S)$.
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1. Let $f_1|_{S-A}, ..., f_k|_{S-A}$ be a Groebner basis for $J(S-A)$ where $A$ is a set of measure zero. Write $g|_{S-A}=\sum^k_{i=1}a_if_i|_{S-A}+r$ where $a_i, r\in
\mathscr{C}[x_1, ..., x_n](S-A)$. Since $g_s=\sum^k_{i=1}a_{i,
s}f_{i, s}+r_s\in J(S-A)|_s$ for $s\in S-A$, this implies that $r=0$ in $S-A$. Therefore $g\in_a J$.
2. Since $\X_s=V(J|_s)$, so $I(\X_s)=\sqrt{J|_s}=\sqrt{J}|_s=J|_s$.
3. By Hilbert Nullstellensatz, $I(\X)$ is a radical ideal and then the conclusion follows from the result above.
Primary decomposition
---------------------
In this section, $R=\mathscr{C}[x_1, ..., x_n](S)$ where $S$ is a condensed space. We write $C(R)$ for the collection of all condensed ideals of $R$.
(Elimination ideals) Let $J\subseteq \mathscr{C}[x_1, ..., x_n](S)$ be an ideal where $S$ is a topological space. The $l$-th elimination ideal $J_l$ is the ideal defined by $$J_l:=J\cap \mathscr{C}[x_{l+1}, ..., x_n](S)$$
The proof of the following result is exactly same as the classical case (see [@CLS Theorem 3.1.2]).
(The elimination theorem) Let $J\subseteq \mathscr{C}[x_1, ..., x_n](S)$ be a Groebner ideal and $G$ be a Groebner basis for $J$ with respect to lex order where $x_1>x_2> \cdots >x_n$. Then for any $\ell$ where $0\leq \ell \leq
n$, the set $$G_{\ell}:=G\cap \mathscr{C}[x_{\ell}, ..., x_n](S)$$ is a Groebner basis of the $\ell$-th elimination ideal $J_{\ell}$.
For the following proof, see [@CLS Theorem 4.3.11].
Let $I, J$ be ideals in $\mathscr{C}[x_1, ..., x_n](S)$. Then $$I\cap J=(tI+(1-t)J)\cap \mathscr{C}[x_1, ..., x_n](S)$$
If $I, J$ are two condensed ideals in $\mathscr{C}[x_1, ...,
x_n](S)$, then $I\cap J$ is an almost condensed ideal. \[intersecting ideals\]
Let $\{f_1, ..., f_k\}, \{g_1, ..., g_l\}$ be condensed bases for $I$ and $J$ respectively which restricted to $S-A$ are Groebner bases of $I(S-A)$ and $J(S-A)$ respectively for some real algebraic subset $A\subseteq S$. Consider the condensed ideal $$L=<tf_1, ..., tf_k, (1-t)g_1, ...,(1-t)g_l>$$ in $\mathscr{C}[x_1,
..., x_n, t](S)$. By the continuous Buchberger’s algorithm Lemma \[algorithms\], compute a Groebner basis $G$ for $L(S-A)$ with respect to lex order in which $t$ is greater than all $x_i$. By Proposition above, the elements of $G$ which do not contain the variable $t$ will form a basis of $I(S-A)\cap J(S-A)=(I\cap
J)(S-A)$.
(Almost irreducible semi-topological algebraic sets) For $J\in C(R)$ a condensed ideal, $V(J)$ is called a semi-topological algebraic set. If $V(J)\neq_a V(J_1)\cup V(J_2)$ for any $J_1, J_2\in C(R)$ with $V(J_1), V(J_2)\neq_a V(J)$, then we say $V(J)$ is almost irreducible. Otherwise, $V(J)$ is said to be almost reducible.
The following result is a simple consequence of the almost ascending chain property.
If $J\in C(R)$, then $V(J)$ can be written as a finite union of almost irreducible affine semi-topological algebraic sets.
(Almost prime ideals) Let $J\in C(R)$. The ideal $J$ is said to be almost prime if for condensed $f, g\in \mathscr{R}[x_1, ..., x_n](S)$, $fg\in_a J$, then either $f\in_a J$ or $g\in_a J$.
If we take $S$ to be a point, then $R=\mathscr{R}(S)[x_1, ...,
x_n]=\C[x_1, ..., x_n]$ and the above definition reduces to the usual definition of prime ideals.
Let $J\in C(R)$ and $V=V(J)$.
1. If $J$ is an almost radical ideal, i.e., $J=_a\sqrt{J}$, then $J$ is almost prime if and only if $V$ is almost irreducible.
2. For $J\in C(R)$, write $V(J)=_a V_1\cup \cdots \cup V_k$ where $V_i$ are almost irreducible components of $V(J)$. Then $\sqrt{J}=_aP_1\cap \cdots \cap P_k$ where $P_i=I(V_i)$ for $i=1, ..., k$.
\[irreducible components\]
1. Suppose that $J$ is almost prime. If $V$ is almost reducible, let $V=_aV(J_1)\cup V(J_2)$ where $J_1, J_2\in C(R)$ and $V(J_1)\subsetneq_a V, V(J_2)\subsetneq_a V$. Then there exists $f\in_a J_1, g\in_a J_2$, $f, g$ condensed, which are not in $J(S-A)$ for any measure zero set $A$. Because of $(fg)(V)=_a
0$, $fg\in_a IV(J)=_a \sqrt{J}=_a J$. So $f\in_a J$ or $g\in_a
J$ which is a contradiction. Hence $V$ is almost irreducible.
Now assume that $V$ is irreducible. Let $f, g\in
\mathscr{R}[x_1, ..., x_n](S)$ such that $fg\in_a J$. Since $V=_aV\cap V(fg)=_aV\cap V(f)\cup V\cap V(g)$, by Corollary \[intersecting ideals\] and the irreducibility of $V$, we must have $V\cap V(f)=_aV$ or $V\cap V(g)=_aV$. By Hilbert Nullstellensatz, this implies that $f$ or $g$ are almost in $\sqrt{J}=J$. Hence $J$ is almost prime.
2. Let $J'=_a P_1\cap \cdots \cap P_k$. Then for $f\in_a J'$, $f(V_i)=_a 0$ for all $i$ which implies that $f\in_a \sqrt{J}$. If $g\in_a \sqrt{J}$, then $g(V_i)=_a 0$ for all $i$ and hence $g\in_a I(V_i)=P_i$ for all $i$ which implies that $g\in_a J'$. Therefore $J'=_a\sqrt{J}$.
The following result is of fundamentally important in our theory.
If $J\subseteq \mathscr{C}[x_0, ...., x_n](S)$ is a condensed ideal, then $\sqrt{J}$ is almost condensed in $\mathscr{C}[x_0, ...,
x_n](S)$.
Let $V(J)=_a V_1\cup \cdots \cup V_k$ be a decomposition into almost irreducible components. By Proposition \[irreducible components\], $\sqrt{J}=_a P_1\cap \cdots \cap P_k$ where $V_i=V(P_i)$ and $P_i$ is an almost prime. Hence by Corollary \[intersecting ideals\], we know that $\sqrt{J}$ is almost condensed in $\mathscr{C}[x_0, ...,
x_n](S)$.
If $J\subseteq \mathscr{C}[x_1, ...., x_n](S)$ is an almost Groebner ideal, then $\sqrt{J}$ is almost Groebner.
Since $J$ is almost Groebner, it is almost condensed, hence $\sqrt{J}$ is almost condensed and hence almost Groebner.
Let $I, J\subseteq \mathscr{C}[x_1, ..., x_n](S)$ be two condensed ideals where $S$ is a condensed space. The ideal quotient of $I$ by $J$ is defined by $$I:J=\{f\in \mathscr{C}[x_1, ..., x_n](S)|fg\in I, \forall g\in J\}$$
We refer the reader to [@CLS Theorem 4.4.11] for the proof of the following result.
Let $J\in \mathscr{C}[x_1, ..., x_n](S)$ be an ideal where $S$ is a condensed space. If $\{f_1, ..., f_k\}$ is a condensed basis of the ideal $J\cap <g>$ where $f_1, ..., f_k, g\in \mathscr{R}(S)[x_0,
..., x_n]$, then $\{f_1/g, ..., f_k/g\}$ is a condensed basis of $I:<g>$. \[quotient single\]
Let $I, J\subseteq \mathscr{C}[x_0, ..., x_n](S)$ be condensed ideals. Then $I:J$ is also an almost condensed ideal. \[quotient ideal\]
Let $J=<g_1, ..., g_l> $ where $g_1, ..., g_l\in \mathscr{R}(S)[x_1,
..., x_n]$. By Proposition \[intersecting ideals\], $I\cap <g_i>$ is almost condensed for all $i$, then by Proposition \[quotient single\], $I:<g_i>$ is almost condensed. Since $I:<g_1,
g_2>=(I:<g_1>)\cap (I:<g_2>)$ which is almost condensed, by induction, $I:J$ is almost condensed.
(Almost primary ideals) Let $J\subseteq \mathscr{C}[x_0, ..., x_n](S)$ be a condensed ideal where $S$ is a condensed space. Suppose that for any $f,g\in
\mathscr{R}(S)[x_0, ..., x_n]$, $fg\in_a J$, we have $f\in_a J$ or $g^k\in_a J$ for some $k\in \N$. Then we say that $J$ is an almost primary ideal. \[primary\]
One of the cornerstone in commutative ring theory is that every ideal in a commutative Noetherian ring has a primary decomposition. We show that every condensed ideal has an almost primary decomposition.
(Almost primary decomposition) Every condensed ideal $J\subseteq \mathscr{C}[x_1, ..., x_n](S)$ can be written as a finite intersection of almost primary ideals where $S$ is a condensed space.
We say that a Groebner ideal $J$ is almost irreducible if $J=_a
J_1\cap J_2$ for some condensed ideals $J_1, J_2$, then either $J=_a
J_1$ or $J=_a J_2$. By using the almost ascending chain property of $C(R)$, it is not difficult to show that every condensed ideal is an intersection of finitely many almost irreducible condensed ideals. The second step is to show that an almost irreducible condensed ideal is almost primary. Suppose that $J$ is an almost irreducible condensed ideal, $f, g\in \mathscr{R}(S)[x_1, ..., x_n]$ and $fg
\in_a J$. We assume that $f\notin_a J$. There is an ascending chain: $$J:g\subseteq_a J:g^2 \subseteq_a \cdots$$ By the almost ascending chain property of $C(R)$, there is $N\in \N$ such that $J:g^N=_a J:g^{N+1}$. For $a+h_1g^N= b+h_2f$ where $a,
b\in_a J, h_1, h_2\in_a R$, we have $ga+h_1g^{N+1}=bg+h_2fg\in_a J$. Hence $h_1g^{N+1}\in_a J$ which implies that $h_1\in_a
J:g^{N+1}=J:g^N$, so we have $h_1g^N\in_a J$. This tells us that $(J+<g^N>)\cap (J+<f>)=_a J$. Since $J$ is almost irreducible and $f\notin_a J$, we have $J=_a J+<g^N>$. Hence $g^N\in_a J$.
Semi-topological Zariski topology
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In this section, we consider a condensed space $S$ with its measure topology, i.e., closed subsets of $S$ are countably intersection of countably union of real algebraic subsets of $S$. Such sets are said to have measure zero. We remind the reader that $\mathscr{C}(S)$ is the ring of all complex-valued continuous functions from $S$ to $\C$ under the Euclidean topology of $S$.
Let $\mathscr{F}$ be a presheaf on a sigma space $S$. If there is a set $A\subseteq S$ of measure zero such that $\mathscr{F}|_{S-A}$ is a sheaf, then $\mathscr{F}$ is called a strong almost sheaf.
It is clear that on a condensed space, a strong almost sheaf is an almost sheaf.
If $\X\subseteq \A^n_S$ is an affine semi-topological algebraic set over $S$, then $K[\X]$ is a strong almost sheaf on $S$.
Since $I(\X)$ is an almost Groebner sheaf, there is a set $A\subseteq S$ of measure zero such that $I(\X)|_{S-A}$ is a Groebner ideal sheaf. Let $g_1, ..., g_k$ be a Groebner basis of $I(\X)(S-A)$. To prove the extension, let $U$ be an open set, $U-A=\cup_i U_i$ and $\sigma_i\in K[\X](U_i)$ for each $i$ such that $\sigma_i|_{U_i\cap U_j}=\sigma_j|_{U_j\cap U_i}$. Let $\sigma_i=f_i+I(\X)(U_i)$. Write $f_i=\sum_ja^i_jg_j|_{U_i}+r_i$ where $a^i_j, r_i\in \mathscr{C}[x_1, ..., x_n](U_i)$ and each term of $r_i$ is not divisible by any of $LT(g_i|_{U_i})$. Since $f_i|_{U_i\cap U_j}-f_j|_{U_j\cap U_i}\in I(\X)(U_i\cap U_j)$, we have $r_i|_{U_i\cap U_j}=r_j|_{U_j\cap U_i}$. Hence there is $r\in
\mathscr{C}[x_1, ..., x_n](U-A)$ such that $r|_{U_i}=r_i$, and $(r+I(\X)(U-A))|_{U_i}=\sigma_i$ for each $i$. To prove the uniqueness, let $\sigma\in K[\X](U)$ and suppose there is an open covering $U=\cup_i U_i$ such that $\sigma|_{U_i}=0$ for all $i$. Let $\sigma|_{U_i}=f_i+I(\X)(U_i)$. Then $f_i\in I(\X)(U_i)$ and $f_i|_{U_i\cap U_j}=f_j|_{U_j\cap U_i}$ for any $i, j$. Since $I(\X)$ is an almost sheaf, there is $f\in I(\X)(U-A)$ such that $f|_{U_i-A}=f_i|_{U_i-A}$ for all $i$. Thus $\sigma|_{U-A}=f|_{U-A}+I(\X)(U-A)=I(\X)(U-A)$.
(Associated sheaves) Suppose that $S$ is a sigma space and $\mathscr{F}$ is an almost sheaf on $S$. For $\alpha\in_a \mathscr{F}(U)$, write $$[\alpha]=\{\beta|\beta \in_a \mathscr{F}(V)\mbox{ for some $V$ and }
\beta=_a \alpha\}$$
Define $$\mathscr{AF}(U):=\{[\alpha]|\alpha\in_a \mathscr{F}(U)\}$$
Suppose that $\mathscr{F}$ is an almost sheaf of rings on a sigma space $S$. For $[\alpha], [\beta]\in \mathscr{AF}(U)$, $\alpha\in
\mathscr{F}(W), \beta\in \mathscr{F}(V)$, define $$[\alpha]+[\beta]:=[\alpha|_{W\cap V}+\beta|_{W\cap V}]$$ $$[\alpha][\beta]:=[\alpha|_{W\cap V}\beta|_{W\cap V}]$$ And define restriction maps $[\alpha]|_{U'}:=[\alpha|_{U'\cap W}]$ where $U'\subseteq U$. Then $\mathscr{AF}$ is a sheaf of rings on $S$.
The sheaf $\mathscr{AF}$ is called the sheaf associated to $\mathscr{F}$.
Suppose that $\mathscr{F}, \mathscr{G}$ are two almost sheaves on a sigma space $S$. Let $\varphi:\mathscr{F} \rightarrow \mathscr{G}$ be a morphism, i.e., a morphism of presheaves.
1. We say that $\varphi$ is almost injective if $\sigma
\in \mathscr{F}(U)$ and $\varphi_U(\sigma)=_a 0$, then $\sigma
=_a 0$.
2. We say that $\varphi$ is almost surjective if for any $\beta\in \mathscr{G}(U)$, there is an open covering $U=\cup_i
U_i$ and $\alpha_i\in \mathscr{F}(U_i)$ such that $\varphi_{U_i}(\alpha_i)=_a \beta|_{U_i}$.
If $\varphi$ is almost injective and almost surjective, then we say that $\varphi$ is an almost isomorphism and say that $\mathscr{F},
\mathscr{G}$ are almost isomorphic, denoted by $\mathscr{F}\cong_a
\mathscr{G}$. If $\varphi: \mathscr{F} \rightarrow \mathscr{G}$ is a morphism between almost sheaves, then it induces a morphism $\mathscr{A}\varphi:\mathscr{AF} \rightarrow \mathscr{AG}$ between the associated sheaves defined by $$(\mathscr{A}\varphi)_U[\alpha]:=[\varphi_U(\alpha)]$$
It is not difficult to see that $\varphi$ is almost injective, surjective, isomorphic if and only if $\mathscr{A}\varphi$ is injective, surjective, isomorphic respectively. We note that if $\varphi$ is almost isomorphic, in the level of almost sheaves, there may no exist an inverse of $\varphi$. But the inverse of $\mathscr{A}\varphi$ exists.
From the definition, we see that $\mathscr{AF}(U)=\mathscr{AF}(U-A)$ where $A$ is a set of measure zero.
Let $S$ be a condensed space. If $I\subseteq \mathscr{C}[x_1, ...,
x_n](S)$ is a condensed ideal, the set $V(I)\subseteq \A^n_{S}$ is called an affine semi-topological algebraic set. The almost sheaf $K[\X]:=\mathscr{C}[x_1, ..., x_n]/I(\X)$ on $S$ is called the coordinate almost sheaf of $\X$. Call $\mathscr{A}K[\X]$ the coordinate sheaf of $\X$. An element $\sigma$ of $K[\X](U)$ is said to be condensed if there is a condensed element $g\in
\mathscr{C}[x_1, ..., x_n](U)$ such that $\sigma=g+I(\X)(U)$. And an element $[\sigma]\in \mathscr{A}K[\X](U)$ is said to be condensed if it can be represented by some condensed element in $K[\X](U)$.
(Topology for affine semi-topological algebraic sets) Let $\X\subseteq \A^n_S$ be an affine semi-topological algebraic set over $S$. For an open set $U\subseteq S$, $f \in K[\X](U)$ condensed, the principal open set defined by $f$ is the set $$D(f):=\X(U)-Z(f)$$ where $Z(f):=\{(s, x)\in \X(U)|f_s(x)=0\}$ is the zero set of $f$ over $U$. Then the collection of all sets of the form $D(f)$ forms a basis of a topology on $\X$. We call the topology generated by this basis the semi-topological Zariski topology of $\X$.
Sheaves of semi-topological modules
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(Affine localization) Given an affine semi-topological algebraic set $\X$ over $S$ and a presheaf $M$ of $K[\X]$-modules on $S$. Let $U\subseteq S$ be an open set. For $p\in \pi^{-1}(U)$, let $$m_{[p]}(U):=\{f\in K[\X](U)|f(p)\neq 0, f \mbox{ condensed}\}$$ then $m_p(U)$ is a multiplicative system. Define $$M_{[p]}(U):=(m_{[p]}(U))^{-1}M(U)$$ the localization of $M(U)$ with respect to $m_{[p]}(U)$.
Then $M_{[p]}$ is a presheaf on $S$ which is called the localization of $M$ at $p$.
We write $(M_{[p]})_{\pi(p)}$ for the stalk of $M_{[p]}$ at $\pi(p)\in S$.
(Bivariant sheaves) Let $\X$ be a semi-topological scheme over $S$. A bivariant sheaf $\mathscr{F}$ on $\X$ is an assignment such that
1. $\mathscr{F}(U)$ is a sheaf for all $U$ open in $\X$. Let $r^U_{VW}:\mathscr{F}(U)(V) \rightarrow \mathscr{F}(U)(W)$ be the restriction map where $W\subseteq V$ are open sets in $\pi(U)$.
2. For any $U'\subseteq U$ open sets of $\X$, there exists restriction map $r^{UU'}_V:\mathscr{F}(U)(V) \rightarrow
\mathscr{F}(U')(V)$ such that the following diagram commutes: $$\xymatrix{ \mathscr{F}(U)(V) \ar[r]^{r^U_{VW}} \ar[d]_{r^{UU'}_V}
& \mathscr{F}(U)(W) \ar[d]^{r^{UU'}_W}\\
\mathscr{F}(U')(V) \ar[r]^{r^{U'}_{VW}} & \mathscr{F}(U')(W)}$$ where $W\subseteq V$ are open sets in $\pi(U')$.
Suppose that $M$ is a presheaf of $K[\X]$-modules on $S$. Let $V\subseteq \X$, $U\subseteq S$ be open sets. Let $\widetilde{M}(V)(U)$ be the collection of all functions $\sigma:V\cap \pi^{-1}(U) \rightarrow \coprod_{p\in V\cap
\pi^{-1}(U)}(M_{[p]})_{\pi(p)}$ such that each of them satisfies the following conditions
1. $\sigma(p)\in (M_{[p]})_{\pi(p)}$ for each $p\in V\cap \pi^{-1}(U)$;
2. there exists open covering $V\cap \pi^{-1}(U)=\cup_i W_i$, $f_i\in K[\X](\pi(W_i)) \mbox{ condensed }, 0\notin f_i(W_i), a_i\in
M(\pi(W_i))$ such that $\sigma(q)=\frac{a_i}{f_i}\in
M_{[q]}(\pi(W_i))$ if $q\in W_i$.
We call $\widetilde{M}$ the presheaf of sheaves on $\X$ associated to $M$. If $M$ is an almost sheaf of $K[\X]$-modules, then we call $\widetilde{M}$ the almost sheaf of sheaves on $\X$ associated to $M$. If $M$ is a sheaf, then $\widetilde{M}$ is a bivariant sheaf.
(Structure sheaves) Let $\X$ be an affine semi-topological algebraic set over $S$. Define $\mathcal{O}_{\X}:=\widetilde{\mathscr{A}K[\X]}$ which is called the bivariant structure sheaf of $\X$. If $M$ is an almost sheaf of $K[\X]$-modules, then $\widetilde{\mathscr{A}M}$ is called the bivariant sheaf of $\mathcal{O}_{\X}$-modules associated to $M$.
All presheaves on a condensed space are almost sheaves. We have the following two functors: $$\mathscr{A}:Cat(Ash K[\X]-mod) \rightarrow
Cat(\mathscr{A}K[\X]-mod)$$ $$\widetilde{ }:Cat(\mathscr{A}K[\X]-mod) \rightarrow Cat(BSh
\mathcal{O}_{\X}-mod)$$ where $Cat(Ash K[\X]-mod)$ is the category of almost sheaves of sheaves of $K[\X]$-modules, $Cat(\mathscr{A}K[\X]-mod)$ is the category of sheaves of $\mathscr{A}K[\X]$-modules, and $Cat(BSh \mathcal{O}_{\X}-mod)$ is the category of bivariant sheaves of $\mathcal{O}_{\X}$-modules.
(Almost Noetherian space) Let $S$ be a condensed space and $\pi:\X \rightarrow S$ a space over $S$. The space $\X$ is said to be an almost Noetherian space if for any chain of open sets $U_1\subseteq U_2 \subseteq \cdots $, there is a set $A\subseteq S$ of measure zero and a number $N$ such that $U_N-\pi^{-1}(A)=U_{N+k}-\pi^{-1}(A)$ for all $k>0$.
Suppose that $\X\subseteq \A^n_S$ is an affine semi-topological algebraic set over $S$, then $\X$ is almost Noetherian.
Let $P:\mathscr{C}[x_1, ..., x_n] \rightarrow K[\X]$ be the natural quotient map. First we note that $K[\X]$ has the almost ascending chain property, i.e., for any open set $U\subseteq S$, if $$I_1\subseteq I_2 \subseteq \cdots$$ is a chain of condensed ideals of $K[\X](U)$, since $I(\X)(U)$ is almost Groebner, $$P^{-1}(I_1) \subseteq P^{-1}(I_2) \subseteq P^{-1}(I_3) \subseteq
\cdots$$ is a chain of almost Groebner ideals in $\mathscr{C}[x_1,
..., x_n](U)$, hence by the almost ascending chain property, the chain almost stabilizes at some $N$ and so is the chain in $K[\X](U)$.
If $$D(f_1) \subseteq \cup^2_{i=1} D(f_i) \subseteq \cup^3_{i=1}D(f_i)
\subseteq \cdots$$ is a chain of open sets in $\X$ where $f_i\in K[\X](S-A_i)$ where $A_i$ is of measure zero, we get a chain $$<f_1|_{S-A}>\subseteq <\{f_i|_{S-A}\}^2_{i=1}>\subseteq
<\{f_i|_{S-A}\}^3_{i=1}>\subseteq \cdots$$ in $K[\X](S-A)$ where $A=\cup_i A_i$. Then by the almost ascending chain property of $K[\X]$, the sequence almost stabilizes at some $N$.
Since the semi-topological Zariski topology of $\X$ is generated by sets of the form $D(f_i)$, an open set $U\subseteq \X$ can be written as $$U=\cup^{\infty}_{i=1}D(f_i)$$ for countably many $f_i$. Then $$D(f_1) \subseteq \cup^2_{i=1}D(f_i) \subseteq \cup^3_{i=1}D(f_i)
\subseteq \cdots$$ and hence $U=_a \cup^N_{i=1}D(f_i)$ for some $N$.
Hence without loss of generality, it is enough to consider chains of open sets of $\X$ of the form $$D(f_1) \subseteq \cup^2_{i=1}D(f_i) \subseteq \cup^3_{i=1}D(f_i)
\subseteq \cdots$$ but then by previous argument, we know that this sequence almost stabilizes at some $N$. Hence $\X$ is almost Noetherian.
Let $\X$ be an affine semi-topological algebraic set over a condensed space $S$ and $M$ an almost sheaf of $K[\X]$-modules. Then
1. For $p\in \X$, the stalk $(\widetilde{M})_{p}$ is isomorphic to the stalk $(M_{[p]})_{\pi(p)}$ as a $(K[\X]_{[p]})_{\pi(p)}$-module.
2. If $f\in K[\X]$ is condensed, then $M_f\cong_a \widetilde{M}(D(f))$ as almost sheaves of $K[\X]_f$-modules.
3. In particular, $M\cong_a \widetilde{M}(\X)$ as almost sheaves of $K[\X]$-modules.
\[module principal open set\]
1. For $\sigma\in (\widetilde{M})_{p}$, there exists a neighborhood $U\subseteq \X$ of $p$ such that $\sigma\in \widetilde{M}(U)$. Define $\varphi(\sigma)=\sigma(p)$, then we get a well defined $(K[\X]_{[p]})_{\pi(p)}$-homomorphism $\varphi:(\widetilde{M})_{p} \rightarrow (M_{[p]})_{\pi(p)}$. To prove that $\varphi$ is surjective, consider $\alpha\in
(M_{[p]})_{\pi(p)}$. Then $\alpha=\frac{a}{f}$ in a neighborhood $W\subseteq S$ of $\pi(p)$ where $a\in M(W), f\in K[\X](W),
f(p)\neq 0$, $f$ condensed on $W$. Hence $D(f)\subseteq \X$ is a neighborhood of $p$ and $\frac{a}{f}\in \widetilde{M}(D(f))$ which implies $\frac{a}{f}\in (\widetilde{M})_{p}$ and hence $\varphi(\frac{a}{f})=\alpha$. To show that $\varphi$ is injective, we assume that $\sigma, \eta\in (\widetilde{M})_{p}$ such that $\sigma(p)=\eta(p)$. Then we may take a neighborhood $U\subseteq \X$ of $p$ such that $\sigma=\frac{a}{f},
\eta=\frac{b}{g}$ in $\widetilde{M}(U)$ where $a, b\in M(U), f,
g\in m_{[p]}(U)$. Then there is $h\in m_{[p]}(U)$ such that $h(ag-bf)=0$ which implies that $\frac{a}{f}=\frac{b}{g}$ in the neighborhood $W'=D(f)\cap D(g)\cap D(h)$ of $p$. Then $\sigma=\eta$ in $\widetilde{M}(W)$ and therefore $\sigma=\eta$ in $(\widetilde{M})_{p}$.
2. Let $U$ be an open subset of $S$ and $f\in K[\X](U)$ condensed in $U$. If $\eta=\frac{a}{f^k}\in M(U)_f$ for some $a\in M(U)$, $\eta$ induces an element $\widetilde{\eta}\in
\widetilde{M}(D(f))(U)$ that assigns each $p\in D(f)\cap
\pi^{-1}(U)$, the element $\frac{a}{f^k}\in (M_{[p]})_{\pi(p)}$. Define $\psi: M(U)_f \rightarrow \widetilde{M}(D(f))(U)$ by sending $\eta$ to $\widetilde{\eta}$.
Suppose that $\psi(\frac{a}{f^k})=\psi(\frac{b}{f^l})$. Then $\frac{a}{f^k}=\frac{b}{f^l}\in (M_{[p]})_{\pi(p)}$ for $p\in
D(f)\cap \pi^{-1}(U)$. So there is an open neighborhood $V\subseteq U$ of $\pi(p)$ such that $\frac{a}{f^k}=\frac{b}{f^l}$ in $M_{[p]}(V)$. And hence there is $h\in m_{[p]}(V)$ such that $h(f^la-f^kb)=0$ in $M(V)$. Let $$\mathscr{A}=\{g\in K[\X](V)| g \mbox{ condensed }, g(f^la-f^kb)=0\}$$ and $I$ be the ideal of $K[\X](V)$ generated by $\mathscr{A}$. Since for any $q\in D(f|_V)$, there is $g\in \mathscr{A}$ such that $g(q)\neq 0$. Therefore $V(I)\cap D(f|_V)=\emptyset$. We have $f(V(I))=0$. Since $I$ is a condensed ideal, by Hilbert Nullstellensatz, $f|_V^N\in_a \mathscr{A}$. Hence $f^N(f^la-f^kb)=_a0$ in $M(V)$ which implies that $\frac{a}{f^k}=_a\frac{b}{f^l}$ in $M(U)_f$. This proves that $\psi$ is almost injective.
Let $\sigma \in \widetilde{M}(D(f))(U)$. Then there exists open cover $\{V_i\}$ of $D(f)$ such that on $V_i$, $\sigma|_{V_i}=\frac{a_i}{g_i}$ where $g_i\in K[\X](\pi(V_i))$ is not vanishing in $V_i$ and condensed in $\pi(V_i)$. Hence $V_i\subseteq D(g_i)$. Since the semi-topological Zariski topology of $\X$ is generated by principal open sets, we may assume $V_i=D(h_i)$ for some $h_i\in K[\X](U)$ where $h_i$ is condensed in $\pi(U)$. Then $D(h_i)\subseteq D(g_i)$, we have $h_i\in \sqrt{<g_i>}$ for each $i$. So $h^m_i=cg_i$ for some $c\in \mathscr{C}[x_1, ..., x_n](U)$ and $\frac{a_i}{g_i}=\frac{ca_i}{h^m_i}$. Replacing $h_i$ by $h^m_i$ and $a_i$ by $ca_i$, we may assume that $D(f)$ is covered by the open subsets $D(h_i)$, and that $\sigma$ is represented by $\frac{a_i}{h_i}$ in $D(h_i)$.
Since $\X$ is almost Noetherian, we may assume that $D(f)$ is almost covered by $D(h_1), ..., D(h_t)$ where each $h_i\in
K[\X](W)$.
On $D(h_i)\cap D(h_j)=D(h_ih_j)$, $\frac{a_i}{h_i}$ and $\frac{a_j}{h_j}$ both represent $\sigma$. Applying the injectivity proved above, we have $\frac{a_i}{h_i}=\frac{a_j}{h_j}$ in $M(\pi(D(h_ih_j)))_{h_ih_j}$. Therefore, there is $m\in \N$ such that $(h_ih_j)^m(h_ja_i-h_ia_j)=0$ which implies $h^{m+1}_j(h^m_ia_i)-h^{m+1}_i(h^m_ja_j)=0$. If we replace $a_i$ by $h^m_ia_i$ and $h_i$ by $h^{m+1}_i$, then on $D(h_i)$, $\sigma$ is again represented by $\frac{a_i}{h_i}$ and we have $h_ja_i=h_ia_j$ on $D(h_i)\cap D(h_j)$.
Let $J=<h_1, ..., h_t>\subseteq K[\X](W)$ be the condensed ideal generated by $h_1, ..., h_t$. Since $D(f|_W)\subseteq
\cup^t_{i=1}D(h_i)$, so $f|_W\in \sqrt{J}$ this implies that $f^m|_W=\sum^t_{i=1}b_ih_i$ for some $b_i\in K[\X](W)$. Let $a=\sum^t_{i=1}b_ia_i\in K[\X](W)$. Then on $D(h_j)$, $h_ja=\sum^t_{i=1}b_ih_ja_i=\sum^t_{i=1}b_ih_ia_j=f^m|_Wa_j$. So $$\frac{a}{f^m|_W}=\frac{a_j}{h_j}$$ on $D(h_j)$. Since $\frac{a}{f^m|_W}\in M(W)_f$, this shows that $\psi$ is almost surjective.
3. For any $U\subseteq S$ open, by result above, we have $\widetilde{M}(\X)(U)=\widetilde{M}(\pi^{-1}(U))(U)\cong M(U)$ and hence $M$ is almost isomorphic to $\widetilde{M}(\X)$ as $K[\X]$-modules.
If $M$ is an almost sheaf, and $\alpha\in M(U)$, we write $[\alpha]$ for the class of $\alpha$ in $\mathscr{A}M(U)$.
Suppose that $M$ is an almost sheaf of $K[\X]$-modules over a condensed space $S$ where $\X$ is a semi-topological algebraic set. Then $\mathscr{A}(\widetilde{M})\cong \widetilde{\mathscr{A}M}$.
For any $[\sigma] \in \mathscr{A}(\widetilde{M})(U)$, there is an open cover $(U)=\cup_i V_i$, $U-A=\cup_i(V_i-A)$, and $a_i\in
M(\pi(V_i)), f_i\in K[\X](\pi(V_i))$, $0\notin f_i(V_i)$ such that $\sigma(p)=\frac{a_i}{f_i}\in (M_{[p]})_{\pi(p)}$ for $p\in V_i$. We define $\sigma'(p)=\frac{[a_i]}{f_i}\in
((\mathscr{A}M)_{[p]})_{\pi(p)}$. Then $\sigma' \in
\widetilde{\mathscr{A}M}(U)$. Not difficult to check the assignment $\sigma\mapsto \sigma'$ defines an isomorphism $$\mathscr{A}(\widetilde{M})\rightarrow \widetilde{\mathscr{A}M}$$
Suppose $\X$ is a semi-topological algebraic set over $S$. Then $\mathscr{A}M \cong \widetilde{\mathscr{A}M}(\X)$.
Since $M\cong_a \widetilde{M}(\X)$, $\mathscr{A}{M}\cong
\mathscr{A}\widetilde{M}(\X)\cong \widetilde{\mathscr{A}M}(\X)$.
Bivariant coherent sheaves
--------------------------
Let $\X$ be an affine semi-topological algebraic set over $S$, and $\mathscr{F}$ a sheaf of $\mathcal{O}_{\X}$-modules. We say that $\mathscr{F}$ is quasi-coherent if $\X$ can be covered by affine open subsets $U_i$ such that for each $i$, $\widetilde{\mathscr{A}M}_i\cong \mathscr{F}|_{U_i}$ where $M_i$ is an almost sheaf of $K[U_i]$-module. We say that $\mathscr{F}$ is coherent if furthermore each $M_i$ can be taken to be an almost sheaf of finitely generated $K[U_i]$-module.
We denote by $\mathscr{AS}(\X)$ the category of associated sheaves of almost sheaves of finitely generated $K[\X]$-modules and $\mathscr{CS}(\X)$ the category of bivariant coherent sheaves of $\mathcal{O}_{\X}$-modules.
Let $\X, \Y$ be affine semi-topological algebraic sets over a condensed space $S$.
1. The map $\mathscr{A}M \rightarrow \widetilde{\mathscr{A}M}$ is an exact fully faithful functor from $\mathscr{AS}(\X)$ to $\mathscr{CS}(\X)$.
2. If $M, N$ are $K[\X]$-modules, then $(M\otimes_{K[\X]} N)\widetilde \ \cong
\widetilde{M}\otimes_{\mathcal{O}_{\X}}\widetilde{N}$.
3. If $\{M_i\}$ is any family of $\mathcal{O}_{\X}$-modules, then $(\bigoplus M_i)\widetilde \
\cong \bigoplus\widetilde{M_i}$.
A morphism of $K[\X]$-modules $\varphi:\mathscr{A}M \rightarrow
\mathscr{A}N$ induces maps $\varphi_{[p]}:\mathscr{A}M_{[p]}
\rightarrow \mathscr{A}N_{[p]}$ for each $p\in \X$ where on each open set $U\subseteq S$, $\varphi_{[p],
U}(\frac{[a]}{f})=\frac{\varphi_U([a])}{f}$ for $\frac{[a]}{f}\in
\mathscr{A}M_{[p]}(U)$. It therefore induces a map in the stalk $\varphi_{[p], \pi(p)}:(\mathscr{A}M_{[p]})_{\pi(p)} \rightarrow
(\mathscr{A}N_{[p]})_{\pi(p)}$ and hence a map $$\widetilde{\varphi}:\widetilde{\mathscr{A}M} \rightarrow \widetilde{\mathscr{A}N}$$ Hence the map $$\widetilde{}:\mathscr{AS}(\X) \rightarrow \mathscr{CS}(\X)$$ defined by $$\mathscr{A}M\mapsto \widetilde{\mathscr{A}M}$$ is a functor.
It has an inverse functor $\Gamma:\mathscr{CS}(\X) \rightarrow
\mathscr{AS}(\X)$ defined by $$\mathscr{F}\mapsto \mathscr{AF}(\X)$$ and $$\Gamma:Hom_{\mathcal{O}_{\X}}(\widetilde{\mathscr{A}M}, \widetilde{\mathscr{A}N}) \rightarrow Hom_{\mathscr{A}K[\X]}(\mathscr{A}M,
\mathscr{A}N)$$ defined by $$\Gamma(\psi)=\mathscr{A}\psi_{\X}$$ where we use the identification $\widetilde{\mathscr{A}M}(\X)\cong \mathscr{A}M,
\widetilde{\mathscr{A}N}(\X)\cong \mathscr{A}N$ and $\psi_{\X}:\widetilde{\mathscr{A}M}(\X) \rightarrow
\widetilde{\mathscr{A}N}(\X)$ is the homomorphism of $\psi$ on $\X$. Therefore the functor $\widetilde{}$ is fully faithful. The exactness follows from the exactness of localization.
Let $\X, \Y$ be two semi-topological algebraic sets over a condensed space $S$. A semi-topological $K$-morphism is a pair $$(\psi, \phi):(\X, K[\X]) \rightarrow (\Y, K[\Y])$$ which satisfy the following properties:
1. $\psi:\X \rightarrow \Y$ is continuous and the following diagram commutes $$\xymatrix{\X \ar[rr]^{\psi} \ar[rd] && \Y \ar[ld] \\
& S &}$$
2. $\phi:K[\Y] \rightarrow K[\X]$ is a morphism of almost sheaves such that $\phi(m^{\Y}_{[\psi(p)]}(U))\subseteq m^{\X}_{[p]}(U)$ for any open sets $U\subseteq S$
The following proof was suggested by Li Li.
Suppose that $\X, \Y$ are affine semi-topological algebraic sets over $S$ and $(\psi, \phi):(\X, K[\X]) \rightarrow (\Y, K[\Y])$ is a semi-topological $K$-morphism.
1. If $N$ is a presheaf of $K[\X]$-modules and $p\in \X$, then $N\otimes_{K[\X]}K[\X]_{[p]}\cong N_{[p]}$ as presheaves of $K[\X]$-modules.
2. If $M$ is a presheaf of $K[\Y]$-module, then $(\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X])_{[p]}\cong
\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X]_{[p]}$.
3. Let $N$ be a presheaf of $K[\X]$-modules. Consider $N_{[p]}$ as a presheaf of $K[\Y]$-modules through $\phi$, then $((N_{[p]})_{[q]})_{\pi_2(q)}\cong (N_{[p]})_{\pi_1(p)}$ as presheaves of $K[\Y]_{[p]}$-modules.
4. If $M$ is an almost sheaf of $K[\Y]$-modules, then $\varphi^*(\widetilde{\mathscr{A}M})\cong
\widetilde{(\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}
\mathscr{A}K[\X])}\cong
\widetilde{\mathscr{A}(M\otimes_{K[\Y]}K[\X])}$.
5. If $N$ is a presheaf of $K[\X]$-modules, let $_{\Y}N$ denote $N$ as a presheaf of $K[\Y]$-modules through $\varphi$, then $\psi_{*}\widetilde{\mathscr{A}N}\cong
\widetilde{(_{\Y}\mathscr{A}N)}$.
\[pullback\]
1. Define a presheaf morphism $\varphi:N\times K[\X]_{[p]}
\rightarrow N_{[p]}$ by assigning every open set $U\subseteq S$, a homomorphism $\varphi_U:(N\times K[\X]_{[p]})(U) \rightarrow
N_{[p]}(U)$ which is defined by $$(\frac{f}{g}, b)\mapsto \frac{fb}{g}$$ where $\frac{f}{g}\in
K[\X]_{[p]}(U), b\in M(U)$. Since $\varphi_U$ is $K[\X](U)$-linear, by the universal property of tensor product, we get a map $$\varphi'_U:(N\otimes_{K[\X]} K[\X]_{[p]})(U) \rightarrow N_{[p]}(U)$$ and hence a morphism $$\varphi':N\otimes_{K[\X]} K[\X]_{[p]}
\rightarrow N_{[p]}$$ of $K[\X]_{[p]}$-modules. By [@AM Proposition 3.5], $\varphi'_U$ is an isomorphism of $K[\X]_{[p]}(U)$-modules for all $U$ and hence $\varphi'$ is a presheaf isomorphism of $K[\X]_{[p]}$-modules.
2. By (1), $(\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X])_{[p]}\cong
(\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X])\otimes_{\mathscr{A}K[\X]}\mathscr{A}K[\X]_{[p]}\cong
\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}(\mathscr{A}K[\X]\otimes_{\mathscr{A}K[\X]}\mathscr{A}K[\X]_{[p]})\cong
\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X]_{[p]}$
3. Define $\psi:((N_{[p]})_{[q]})_{\pi_2(q)}\rightarrow
(N_{[p]})_{\pi_1(p)}$ by $$\frac{\frac{a}{f}}{g} \mapsto \frac{a}{\phi(g)f}$$ for $\frac{a}{f}\in N_{[p]}(U)$ and $g\in m^{\Y}_{[q]}(U)$ for some $U\subseteq S$ a neighborhood of $q$. This map is clearly an isomorphism.
4. $\varphi^*(\widetilde{\mathscr{A}M})_p=(\varphi^{-1}\widetilde{\mathscr{A}M}\otimes_{\varphi^{-1}\mathcal{O}_{\Y}}\mathcal{O}_{\X})_p
\cong
(\mathscr{A}M_{[q]})_{\pi(q)}\otimes_{(\mathscr{A}K[\Y]_{[q]})_{\pi(q)}}(\mathscr{A}K[\X]_{[p]})_{\pi(p)}\cong
(\mathscr{A}M_{[q]})_{\pi(q)}\otimes_{(\mathscr{A}K[\Y]_{[q]})_{\pi(q)}}((\mathscr{A}K[\X]_{[p]})_{[q]})_{\pi(q)}
\cong
((\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X]_{[p]})_{[q]})_{\pi(q)}\cong
(((\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X])_{[p]})_{[q]})_{\pi(q)}\cong
((\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X])_{[p]})_{\pi(p)}\cong
(\widetilde{\mathscr{A}M\otimes_{\mathscr{A}K[\Y]}\mathscr{A}K[\X]})_p$ for each $p\in \X$.
5. $(\psi_*\widetilde{\mathscr{A}N})(\Y)=\widetilde{\mathscr{A}N}(\psi^{-1}(\Y))=\widetilde{\mathscr{A}N}(\X)\cong
_{\Y}\mathscr{A}N=\widetilde{(_{\Y}\mathscr{A}N)}(\Y)$, hence $\psi_*\widetilde{\mathscr{A}N}\cong _{\Y}\mathscr{A}N$.
Let $\X$ be a semi-topological over a condensed space $S$, and $\mathscr{F}$ a sheaf on $\X$. We say that $\alpha\in
\mathscr{F}(U)$ is almost zero, denoted by $\alpha=_a0$, if there is a set $A\subseteq S$ of measure zero such that $\alpha|_{U-\pi^{-1}(A)}=0$. And we write $\beta\in_a
\mathscr{F}(U)$ if there is a set $B\subseteq S$ of measure zero such that $\beta\in \mathscr{F}(U-\pi^{-1}(B))$.
Let $\mathscr{F}$ be a quasi-coherent sheaf on an affine semi-topological algebraic set $\X$ over $S$. Then the restriction map from $\mathscr{F}(U)$ to $\mathscr{F}(U-A)$ where $A$ is a set of measure zero is an isomorphism.
Suppose that $\X$ is covered by $U_i$ and $\mathscr{F}|_{U_i}\cong
\widetilde{\mathscr{AN}_i}$. Let $A$ be a set of measure zero. Since $U-A=\cup_i(U\cap U_i-A)$, $\mathscr{F}(U\cap
U_i-A)=\widetilde{\mathscr{AN}}_i(U\cap U_i-A)$. From the following commutative diagram $$\xymatrix{ \mathscr{F}(U\cap U_i) \ar[r]^{\cong} \ar[d] &
\widetilde{\mathscr{AN}}_i(U\cap U_i) \ar[d]^{\cong}\\
\mathscr{F}(U\cap U_i-A) \ar[r]^{\cong} &
\widetilde{\mathscr{AN}}_i(U\cap U_i-A)}$$ we see that the restriction from $\mathscr{F}(U\cap U_i)$ to $\mathscr{F}(U\cap
U_i-A)$ is an isomorphism. And these isomorphisms patch an isomorphism between $\mathscr{F}(U)$ and $\mathscr{F}(U-A)$.
1. If $M$ is a presheaf of $K[\X]$-modules generated by global sections, then $\mathscr{A}M$ is a finitely generated $\mathscr{A}K[\X]$-module.
2. $\mathscr{AA}M\cong \mathscr{A}M$.
We have develop enough machinery such that the proof of the following result is similar to [@Hart Lemma 5.3].
\[extending sections\] Let $\X$ be an affine condensed scheme over $S$, $f\in K[\X](S)$ condensed, and let $\mathscr{F}$ be a quasi-coherent sheaf on $\X$.
1. If $\alpha\in \Gamma(\X, \mathscr{F})$ whose restriction to $D(f)$ is 0, then for some $n>0$, $f^n\alpha=_a 0$.
2. Suppose that $\alpha\in \mathscr{F}(D(f))$, then for some $n>0$, there is an almost global section $\beta\in_a \mathscr{F}(\X)$ such that $\beta|_{D(f)}=f^n\alpha$.
Suppose that $\X$ is an affine semi-topological algebraic set over $S$. Then a bivariant sheaf $\mathscr{F}$ of $\mathcal{O}_{\X}$-modules is quasi-coherent if and only if for every affine open subset $U\subseteq \X$, there is an almost sheaf of $K[U]$-modules $M$ such that $\mathscr{F}|_U\cong
\widetilde{\mathscr{A}M}$. Furthermore, $\mathscr{F}$ is bivariant coherent if and only the same is true, with the extra condition that $M$ is an almost sheaf of finitely generated $K[U]$-modules.
As in the proof above, we may cover $\X$ by finitely many $D(g_i)$ where $g_i\in \mathscr{K}[\X](S)$ such that $\mathscr{F}|_{D(g_i)}\cong \widetilde{\mathscr{AN}}_i$. By Lemma above, $\mathscr{F}|_{D(g_i)}\cong \mathscr{AM}_{g_i}$, hence we get an isomorphism $\varphi_i:\mathscr{AM}_{g_i} \rightarrow
\mathscr{AN}_i$. Since $\widetilde{\mathscr{AM}_{g_i}}\cong
\widetilde{\mathscr{AM}}|_{D(g_i)}$, we get an isomorphism $$\widetilde{\varphi}_i:\widetilde{\mathscr{AM}} \rightarrow
\mathscr{F}|_{D(g_i)}$$ These isomorphisms patch together to give us an isomorphism $$\widetilde{\varphi}:\widetilde{\mathscr{AM}} \rightarrow
\mathscr{F}$$
Suppose that $\X$ is an affine condensed scheme over $S$. The functor $M\mapsto \widetilde{M}$ is an equivalence of categories between the category of sheaves of $K[\X]$-modules and the category of bivariant quasi-coherent sheaves. Its inverse is given by $\mathscr{F}\mapsto \mathscr{F}(\X)$. The same functors gives an equivalence of categories between the category of sheaves of finitely generated $K[\X]$-modules and the category of bivariant coherent sheaves.
[1]{} Bochnak, J., Coste, M., Roy, M-F., Real algebraic geometry, A series of Modern Surveys in Mathematics, 36, 1998, Springer, New York.
Brownawell, D., Bounds for the degrees in the Nullstellensatz, Ann. Math., 126, no. 3, 1987, 577–591.
Cox, D., Little, J., O’Shea, D., Ideals, varieties, and algorithms, 1996, 2nd, Springer-Verlag, New York.
E. Friedlander, [*Algebraic cycles, Chow varieties, and Lawson homology*]{}, Compositio Math. [**77**]{} (1991), 55-93.
H.B. Lawson, [*Algebraic cycles and homotopy theory*]{}, Annals of Math. [**129**]{} (1989), 253-291.
Friedlander, E., Lawson, H.B., A theory of algebraic cocycles, Annals of Math., 136, 1992, 361–428.
Hartshorne, R., Algebraic geometry, 1977, Springer-Verlag, New York.
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abstract: 'For a graph $G=(V,E)$, a Roman $\{2\}$-dominating function (R2DF)$f:V\rightarrow \{0,1,2\}$ has the property that for every vertex $v\in V$ with $f(v)=0$, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors $x,y\in N(v)$ having $f(x)=f(y)=1$. The weight of a R2DF is the sum $f(V)=\sum_{v\in V}{f(v)}$, and the minimum weight of a R2DF is the Roman $\{2\}$-domination number $\gamma_{\{R2\}}(G)$. A R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman $\{2\}$-domination number $i_{\{R2\}}(G)$ is the minimum weight of an independent Roman $\{2\}$-dominating function on $G$. In this paper, we show that the decision problem associated with $\gamma_{\{R2\}}(G)$ is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of $i_{\{R2\}}(T)$ for any tree $T$. This answers an open problem raised by Rahmouni and Chellali \[Independent Roman $\{2\}$-domination in graphs, Discrete Applied Mathematics 236 (2018), 408-414\]. Chellali, Haynes, Hedetniemi and McRae [@chellali2016roman] have showed that Roman $\{2\}$-domination number can be computed for the class of trees in linear time. As a generalization, we present a linear time algorithm for solving the Roman $\{2\}$-domination problem in block graphs.'
author:
- |
Hangdi Chen and Changhong Lu\
School of Mathematical Sciences,\
Shanghai Key Laboratory of PMMP,\
East China Normal University,\
Shanghai 200241, P. R. China\
\
Email: [email protected]\
Email: [email protected]
title: 'A Note on Roman {2}-domination problem in graphs[^1]'
---
Introduction
============
Let $G=(V,E)$ be a simple graph. The *open neighborhood* $N(v)$ of a vertex $v$ consists of the vertices adjacent to $v$ and its *closed neighborhood* is $N[v]=N(v)\cup\{v\}$. We denote $N^2[v]=\bigcup_{u\in N[v]} N[u]$. For an edge $e=uv$, it is said that $u$ (resp. $v$) is incident to $e$, denoted by $u\in e$ (resp. $v\in e$). A *vertex cover* of $G$ is a subset $V'\subseteq V$ such that for each edge $uv\in E$, at least one of $u$ and $v$ belongs to $V'$. A *Roman dominating function (RDF)* on graph $G$ is a function $f:V\rightarrow \{0,1,2\}$ satisfying the condition that every vertex $u$ for which $f(u)=0$ is adjacent to at least one vertex $v$ for which $f(v)=2$. The *weight of a Roman dominating function* $f$ is the value $f(V)=\sum_{v\in V}{f(v)}$. The minimum weight of a Roman dominating function on a graph $G$ is called the *Roman domination number* $\gamma_R(G)$ of $G$. Roman domination and its variations have been studied in a number of recent papers (see, for example, [@Chambers2009Extremal; @Cockayne2004Roman; @Liu2012Upper]).
Chellali, Haynes, Hedetniemi and McRae [@chellali2016roman] introduced a variant of Roman dominating functions. For a graph $G=(V,E)$, a *Roman $\{2\}$-dominating function* $f:V\rightarrow \{0,1,2\}$ has the slightly different property that only for every vertex $v\in V$ with $f(v)=0$, $f(N(v))\ge2$, that is, either there exists a neighbor $u\in N(v)$, with $f(u)=2$, or at least two neighbors $x,y\in N(u)$ have $f(x)=f(y)=1$. The *weight of a Roman $\{2\}$-dominating function* is the sum $f(V)=\sum_{v\in V}{f(v)}$, and the minimum weight of a Roman $\{2\}$-dominating function $f$ is the *Roman $\{2\}$-domination number*, denoted $\gamma_{\{R2\}}(G)$. Roman $\{2\}$-domination is also called Italian domination by some scholars ([@henning2017italian]). Suppose that $f:V\rightarrow \{0,1,2\}$ is a $R2DF$ on a graph $G=(V,E)$. Let $V_i=\{v|f(v)=i\}$, for $i\in\{0,1,2\}$. If $V_1\cup V_2$ is an independent set, then $f$ is called an *independent Roman $\{2\}$-dominating function (IR$2$DF)*, which was introduced by Rahmouni and Chellali [@Rahmouni2018Independent] in a recent paper. The minimum weight of an independent Roman $\{2\}$-dominating function $f$ is the *independent Roman $\{2\}$-domination number*, denoted $i_{\{R2\}}(G)$. The authors in [@chellali2016roman; @Rahmouni2018Independent] have showed that the associated decision problems for Roman $\{2\}$-domination and independent Roman $\{2\}$-domination are NP-complete for bipartite graphs. In [@Rahmouni2018Independent], the authors raised some interesting open problems, one of which is whether there is a linear algorithm for computing $i_{\{R2\}}(T)$ for any tree $T$.
A graph $G=(V,E)$ is a *split graph* if $V$ can be partitioned into $C$ and $I$, where $C$ is a clique and $I$ is an independent set of $G$. Split graph is an important subclass of chordal graphs (see [@chang2013algorithmic]). Chordal graph is one of the classical classes in the perfect graph theory, see the book by Golumbic [@golumbic1980algorithmic]. It turns out to be very important in the domination theory.
A *cut-vertex* is any vertex whose removal increases the number of connected components. A maximal connected induced subgraph without a cut-vertex is called a *block* of $G$. A graph $G$ is a *block graph* if every block in $G$ is a complete graph. There are widely research on variations of domination in block graphs (see, for example, [@chang1989total; @chen2010labelling; @pradhan2018computing; @xu2006power]).
In this paper, we first show that the decision problem associated with $\gamma_{\{R2\}}(G)$ is NP-complete for split graphs. Then, we give a linear time algorithm for computing $i_{\{R2\}}(T)$ in any tree $T$. Moreover, we present a linear time algorithm for solving the Roman $\{2\}$-domination problem in block graphs.
Complexity result
=================
In this section, we consider the decision problem associated with Roman {2}-dominating functions.
ROMAN {2}-DOMINATING FUNCTION(R2D)
[**INSTANCE:**]{} A graph $G=(V,E)$ and a positive integer $k\le|V|$.
[**QUESTION:**]{} Does $G$ have a Roman {2}-dominating function of weight at most $k$?
We show that this problem is NP-complete by reducing the well-known NP-complete problem Vertex Cover(VC) to R2D.
VERTEX COVER(VC)
[**INSTANCE:**]{} A graph $G=(V,E)$ and a positive integer $k\le|V|$.
[**QUESTION:**]{} Is there a vertex cover of size $k$ or less for $G$?
\[thm1\] R2D is NP-complete for split graphs.
R2D is a member of NP, since we can check in polynomial time that a function $f:V\rightarrow \{0,1,2\}$ has weight at most $k$ and is a Roman $\{2\}$-dominating function. The proof is given by reducing the VC problem in general graphs to the R2D problem in split graphs.
Let $G=(V,E)$ be a graph with $V=\{v_1,v_2,\cdots,v_n\}$ and $E=\{e_1,e_2,\cdots,e_m\}$. Let $V^1=\{v_1',v_2',\cdots,v_n'\}$. We construct the graph $G'=(V',E')$ with: $$V'=V^1\cup V\cup E,$$ $$E'=\{v_iv_j | v_i\neq v_j , v_i\in V, v_j\in V \}\cup \{v_iv_i' | i=1,...,n\} \cup \{ve|v\in e,e\in E\} .$$ Notice that $G'$ is a split graph whose vertex set $V'$ is the disjoint union of the clique $V$ and the independent set $V^1\cup E$. It is clear that $G'$ can be constructed in polynomial time from $G$.
If $G$ has a vertex cover $C$ of size at most $k$, let $f: V'\rightarrow \{0,1,2\}$ be a function defined as follows. $$f(v)=\begin{cases}
2, \text{ if } v\in C \\
1, \text{ if } v\in V^1, v'\in V-C \text{ and } vv'\in E'\\
0, \text{ otherwise}
\end{cases}$$ It is clear that $f$ is a Roman $\{2\}$-dominating function of $G'$ with weight at most $2k+(n-k)$.
On the other hand, suppose that $G'$ has a Roman $\{2\}$-dominating function of weight at most $2k+(n-k)$. Among all such functions, let $g=(V_0,V_1,V_2)$ be one chosen so that
1. $|V^1\cap V_2|$ is minimized.
2. Subject to Conditions (C1): $|E\cap V_0|$ is maximized.
3. Subject to Conditions (C1) and (C2): $|V\cap V_1|$ is minimized.
4. Subject to Conditions (C1), (C2) and (C3): the weight of $g$ is minimized.
We make the following remarks.
(i) No vertex in $V^1$ belongs to $V_2$. Indeed, suppose to the contrary that $g(v_i')=2$ for some $i$. We reassign $0$ to $v_i'$ instead of $2$ and reassign $2$ to $v_i$. Then it provides a R2DF on $G'$ of weight at most $2k+(n-k)$ but with less vertices of $V^1$ assigned $2$, contradicting the condition (C1) in the choice of $g$.
(ii) No vertex in $E$ belongs to $V_2$. Indeed, suppose that $g(e)=2$ for some $e\in E$ and $v_j, v_k\in e$. By reassigning $0$ to $e$ instead of $2$ and reassigning $2$ to $v_j$ instead of $g(v_j)$, we obtain a R2DF on $G'$ of weight at most $2k+(n-k)$ but with more vertices of $E$ assigned $0$, contradicting the condition (C2) in the choice of $g$.
(iii) No vertex in $E$ belongs to $V_1$. Suppose that $g(e)=1$ for some $e\in E$ and $v_j, v_k\in e$. If $g(v_j')=0$, then $g(v_j)=2$(by the definition of R2DF). By reassigning $0$ to $e$ instead of $1$, we obtain a R2DF on $G'$ of weight at most $2k+(n-k)$ but with more vertices of $E$ assigned $0$, contradicting the condition (C2) in the choice of $g$. Hence we may assume that $g(v_j')=1$(by item(i)). Clearly we can reassign $2$ to $v_j$ instead of $0$, $0$ to $v_j'$ instead of $1$ and $0$ to $e$ instead of $1$. We also obtain a R2DF on $G'$ of weight at most $2k+(n-k)$ but with more vertices of $E$ assigned $0$, contradicting the condition (C2) in the choice of $g$.
(iv) No vertex in $V$ belongs to $V_1$. Suppose to the contrary that $g(v_i)=1$ for some $i$, then $g(v_i')=1$(by item(i) and the definition of R2DF). We reassign $0$ to $v_i'$ instead of $1$ and $2$ to $v_i$ instead of $1$. It provides a R2DF on $G'$ of weight at most $2k+(n-k)$ but with less vertices of $V$ assigned $1$, contradicting the condition (C3) in the choice of $g$.
(v) If a vertex in $V$ is assigned $2$, then its neighbor in $V^1$ is assigned $0$ by the condition (C4) in the choice of $g$.
(vi) If a vertex in $V$ is assigned $0$, then its neighbor in $V^1$ is assigned $1$ by the definition of R2DF and item(i).
Therefore, according to the previous items, we conclude that $V^1\cap V_2=\emptyset$, $E\subseteq V_0$, and $V\cap V_1=\emptyset$. Hence $V_2\subseteq V$. Let $C=\{v|g(v)=2\}$. Since each vertex in $E\cup (V-C)$ belongs to $V_0$ in $G'$, it is clear that $C$ is a vertex cover of $G$ by the definition of R2DF. Then $g(V^1)+g(V)+g(E)=2|C|+(n-|C|)\le 2k+(n-k)$, implying that $|C|\le k$. Consequently, $C$ is a vertex cover for $G$ of size at most $k$.
Since the vertex cover problem is NP-complete, the Roman $\{2\}$-domination problem is NP-complete for split graphs.
Independent Roman $\{2\}$-domination in trees
=============================================
In this section, a linear time dynamic programming style algorithm is given to compute the exact value of independent Roman $\{2\}$-dominating number in any tree. This algorithm is constructed using the methodology of Wimer [@wimer1987linear].
A *rooted tree* is a pair $(T,r)$ with $T$ is a tree and $r$ is a vertex of $T$. A rooted tree $(T,r)$ is trivial if $V(T)={r}$. Given two rooted trees $(T_1,r_1)$ and $(T_2,r_2)$ with $V(T_1)\cap V(T_2)=\emptyset$, the composition of them is $(T_1,r_1)\circ (T_2,r_2)=(T,r_1)$ with $V(T)=V(T_1)\cup V(T_2)$ and $E(T)=E(T_1)\cup E(T_2)\cup \{r_1r_2\}$. It is clear that any rooted tree can be constructed recursively from trivial rooted trees using the defined composition.
Let $f:V(T)\rightarrow \{0,1,2\}$ be a function on $T$. Then $f$ splits two functions $f_1$ and $f_2$ according to this decomposition. We express this as follows: $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, where $r=r_1$, $f_1=f|_{T_1}$ and $f_2=f|_{T_2}$. For each $1\le i\le 2$, $f|_{T_i}$ is a function that $f$ restricted to the vertices of $T_i$. On the other hand, let $f_1:V(T)\rightarrow \{0,1,2\}$ (resp. $f_2$) be a function on $T_1$ (resp. $T_2$). We can define the composition as follows: $(T_1,f_1,r_1)\circ (T_2,f_2,r_2)=(T,f,r)$, where $V(T)=V(T_1)\cup V(T_2)$, $E(T)=E(T_1)\cup E(T_2)\cup \{r_1r_2\}$, $r=r_1$ and $f=f_1\circ f_2:V(T)\rightarrow \{0,1,2\}$ with $f(v)=f_i(v)$ if $v\in V(T_i)$, $i=1,2$. Suppose that $M$ and $N$ are the sets of possible tree turples. If $(T_1,f_1,r_1)\in M$ and $(T_2,f_2,r_2)\in N$, we use $M\circ N$ to denote the set of $(T,f,r)$. In our paper, we sometimes use $T-r$ to mean $T-\{r\}$. Before presenting the algorithm, let us give the following observation.
Let $f$ be an IR2DF of $T$ and $f_1=f|_{T_1}$ (resp. $f_2=f|_{T_2}$). If $f_1(r_1)\ne 0$ (resp. $f_2(r_2)\ne 0$), then $f_1$ (resp. $f_2$) is an IR2DF of $T_1$ (resp. $T_2$). If $f_1(r_1)=0$ (resp. $f_2(r_2)=0$), then $f_1$ (resp. $f_2$) may not be an IR2DF of $T_1$ (resp. $T_2$), but $f_1$ (resp. $f_2$) restricted to the vertices of $T_1-r_1$ (resp. $T_2-r_2$) is an IR2DF of $T_1-r_1$ (resp. $T_2-r_2$).
In order to construct an algorithm for computing independent Roman $\{2\}$-domination number, we must characterize the possible tree-subset tuples $(T,f,r)$. For this purpose, we introduce some additional notations as follows:
$IR2DF(T)=\{ f~|~ f$ is an IR2DF of $T\}$;\
$IR2DF_r(T)=\{ f~|~ f\notin IR2DF(T) $ and $f|_{T-r}\in IR2DF(T-r)\}$.
Then we consider the following five classes:
$A=\{(T,f,r)~|~ f\in IR2DF(T)$ and $f(r)=2\}$;\
$B=\{(T,f,r)~|~ f\in IR2DF(T)$ and $f(r)=1\}$;\
$C=\{(T,f,r)~|~ f\in IR2DF(T)$ and $f(r)=0\}$;\
$D=\{(T,f,r)~|~ f\in IR2DF_r(T)$ and $f(N[r])=1\}$;\
$E=\{(T,f,r)~|~ f\in IR2DF_r(T)$ and $f(N[r])=0\}$. Next, we provide some Lemmas.
\[lem2\] $A=(A\circ C)\cup (A\circ D)\cup (A\circ E)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(A\circ C)\cup (A\circ D)\cup (A\circ E)\subseteq A$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is clear that the following items are true.
(i) If $(T_1,f_1,r_1)\in A$ and $(T_2,f_2,r_2)\in C$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A$.
(ii) If $(T_1,f_1,r_1)\in A$ and $(T_2,f_2,r_2)\in D$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A$.
(iii) If $(T_1,f_1,r_1)\in A$ and $(T_2,f_2,r_2)\in E$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A$.
Now we prove that $A\subseteq(A\circ C)\cup (A\circ D)\cup (A\circ E)$. Let $(T,f,r)\in A$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then $f_1(r_1)=f(r)=2$. Since $f\in IR2DF(T)$ and $f_1=f|_{T_1}$, then $f_1\in IR2DF(T_1)$. So $(T_1,f_1,r_1)\in A$. From the independence of $V_1\cup V_2$, we have $f_2(r_2)=f(r_2)=0$. If $f_2\in IR2DF(T_2),$ then we obtain $(T_2,f_2,r_2)\in C$. If $f_2\notin IR2DF(T_2),$ then $(T_2,f_2,r_2)\in D$ or $ E$. Hence, we conclude that $A\subseteq(A\circ C)\cup (A\circ D)\cup (A\circ E)$.
\[lem3\] $B=(B\circ C)\cup (B\circ D)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(B\circ C)\cup (B\circ D)\subseteq B$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$.
(i) If $(T_1,f_1,r_1)\in B$ and $(T_2,f_2,r_2)\in C$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B$.
(ii) If $(T_1,f_1,r_1)\in B$ and $(T_2,f_2,r_2)\in D$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B$.
It is easy to check the previous items. Then we conclude that $(B\circ C)\cup (B\circ D)\subseteq B$.
Next we need to show $B\subseteq(B\circ C)\cup (B\circ D)$ . Let $(T,f,r)\in B$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then $f_1(r_1)=f(r)=1$. It is clear that $f_1\in IR2DF(T_1).$ So we conclude that $(T_1,f_1,r_1)\in B$. From the definition of IR2DF, we must have $f_2(r_2)=f(r_2)=0$. If $f_2\in IR2DF(T_2),$ then we obtain $(T_2,f_2,r_2)\in C$. If $f_2\notin IR2DF(T_2),$ then $f_2(N_{T_2}[r_2])=1$ and $f_2|_{T_2-r_2}\in IR2DF(T_2-r_2)$ using the fact that $(T,f,r)\in B$. Therefore, we have $f_2\in IR2DF_{r_2}(T_2)$, implying that $(T_2,f_2,r_2)\in D$. Hence, we deduce that $B\subseteq(B\circ C)\cup (B\circ D).$
\[lem4\] $C=(C\circ A)\cup (C\circ B)\cup (C\circ C)\cup (D\circ A)\cup (D\circ B)\cup (E\circ A)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(C\circ A)\cup (C\circ B)\cup (C\circ C)\cup (D\circ A)\cup (D\circ B)\cup (E\circ A)\subseteq C$ . Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is easy to check the following remarks by definitions.
(i) If $(T_1,f_1,r_1)\in C$ and $(T_2,f_2,r_2)\in A$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in C$.
(ii) If $(T_1,f_1,r_1)\in C$ and $(T_2,f_2,r_2)\in B$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in C$.
(iii) If $(T_1,f_1,r_1)\in C$ and $(T_2,f_2,r_2)\in C$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in C$.
(iv) If $(T_1,f_1,r_1)\in D$ and $(T_2,f_2,r_2)\in A$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in C$.
(v) If $(T_1,f_1,r_1)\in D$ and $(T_2,f_2,r_2)\in B$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in C$.
(vi) If $(T_1,f_1,r_1)\in E$ and $(T_2,f_2,r_2)\in A$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in C$.
Therefore, we need to prove $C\subseteq(C\circ A)\cup (C\circ B)\cup (C\circ C)\cup (D\circ A)\cup (D\circ B)\cup (E\circ A).$ Let $(T,f,r)\in C$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then $f\in IR2DF(T)$ and $f_1(r_1)=f(r)=0$. Consider the following cases.
[**Case 1.**]{} $f(r_2)=2.$ Using the fact that $f\in IR2DF(T)$ and $f_2=f|_{T_2}$, then $f_2\in IR2DF(T_2)$. It means that $(T_2,f_2,r_2)\in A$. If $f_1\in IR2DF(T_1)$, then we obtain that $(T_1,f_1,r_1)\in C$. If $f_1\notin IR2DF(T_1)$, we have $(T_1,f_1,r_1)\in D$ or $ E$.
[**Case 2.**]{} $f(r_2)=1.$ Since $f\in IR2DF(T)$ and $f_2=f|_{T_2}$, then $f_2\in IR2DF(T_2)$. So $(T_2,f_2,r_2)\in B$. If $f_1\in IR2DF(T_1)$, then we deduce $(T_1,f_1,r_1)\in C$. If $f_1\notin IR2DF(T_1)$, therefore, it implies that $(T_1,f_1,r_1)\in D$.
[**Case 3.**]{} $f(r_2)=0.$ It is clear that $f_1$ and $f_2$ are both IR2DF. Then we obtain that $(T_1,f_1,r_1)\in C$ and $(T_2,f_2,r_2)\in C$.
Therefore, we obtain $C\subseteq(C\circ A)\cup (C\circ B)\cup (C\circ C)\cup (D\circ A)\cup (D\circ B)\cup (E\circ A).$
\[lem5\] $D=(D\circ C)\cup (E\circ B)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(D\circ C)\cup (E\circ B)\subseteq D$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is easy to check the following remarks by definitions.
(i) If $(T_1,f_1,r_1)\in D$ and $(T_2,f_2,r_2)\in C$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in D$.
(ii) If $(T_1,f_1,r_1)\in E$ and $(T_2,f_2,r_2)\in B$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in D$.
On the other hand, we show $D\subseteq(D\circ C)\cup (E\circ B).$ Let $(T,f,r)\in D$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$. Then $f_1(r_1)=f(r)=0$. By the definition of $D$ and $f_2=f|_{T_2}$, $f_2\in IR2DF(T_2)$. Using the fact that $f(N_T[r_1])=1$, we deduce that $f(r_2)<2$. Consider the following cases.
[**Case 1.**]{} $f(r_2)=1.$ It is clear that $(T_2,f_2,r_2)\in B$ because $f_2$ is an IR2DF of $T_2$. Since $f_1=f|_{T_1}$ and $f_1(N_{T_1}[r_1])=0$, we obtain $f_1|_{T_1-r_1}\in IR2DF(T_1-r_1)$. Hence, we have $f_1\in IR2DF_{r_1}(T_1)$, implying that $(T_1,f_1,r_1)\in E$.
[**Case 2.**]{} $f(r_2)=0.$ Then $f_2$ is an IR2DF of $T_2$, implying that $(T_2,f_2,r_2)\in C$. Using the fact that $f(N_T[r_1])=1$ and $f(r_2)=0$, we know that $f_1(N_{T_1}[r_1])=1$. It is clear that $f_1\in IR2DF_{r_1}(T_1)$. It implies that $(T_1,f_1,r_1)\in D$.
Consequently, we deduce that $D\subseteq(D\circ C)\cup (E\circ B).$
\[lem5\] $E=E\circ C$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. If $(T_1,f_1,r_1)\in E$ and $(T_2,f_2,r_2)\in C$, then it is clear that $(T,f,r)\in E$. Hence, $(E\circ C)\subseteq E$.
On the other hand, let $(T,f,r)\in E$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$. Then $f_1(r_1)=f(r)=0$. By the definition of $E$, we deduce that $f(r_2)=0$. Using the fact that $(T,f,r)\in E$, we have that $f_2\in IR2DF(T_2).$ So $(T_2,f_2,r_2)\in C$. Notice that $(T,f,r)\in E$, we have $f_1(N_{T_1}[r_1])=0$, implying that $(T_1,f_1,r_1)\notin D$. We can easily check that $f_1\in IR2DF_{r_1}(T_1)$. Hence, we have $(T_1,f_1,r_1)\in E$, implying that $E\subseteq(E\circ C).$
Let $T=(V,E)$ be a tree with $n$ vertices. It is well known that the vertices of $T$ have an ordering $v_1,v_2,\cdots,v_n$ such that for each $1\le i\le n-1$, $v_i$ is adjacent to exactly one vertex $v_j$ with $j>i$ (see [@west2001introduction]). The ordering is called a tree ordering, where the only neighbor $v_j$ with $j>i$ is called the father of $v_i$ and $v_i$ is a child of $v_j$. For each $1\le i\le n-1$, the father of $v_i$ is denoted by $F(v_i)=v_j$.
The final step is to define the initial vector. In this case, for a tree, the only basis graph is a single vertex. It is easy to obtain that the initial vector is $(2,1,\infty,\infty,0),$ where $'\infty'$ means undefined. Now, we are ready to present the algorithm.
$i_{\{R2\}}(T)=\min\{l[n,1], l[n,2], l[n,3]\}$
From the above argument, we can obtain the following theorem.
\[thm3\] Algorithm INDEPENDENT-ROMAN $\{2\}$-DOM-IN-TREE can output the independent Roman $\{2\}$-domination number of any tree $T=(V,E)$ in linear time $O(n)$, where $n=|V|$.
Roman $\{2\}$-domination in block graphs
========================================
Let $G$ be a connected block graph and $G$ isn’t a complete graph. The *block-cutpoint graph* of $G$ is a bipartite graph $T_G$ in which one partite set consists of the cut-vertices of $G$, and the other has a vertex $h_i$ for each block $H_i$ of $G$. We include $vh_i$ as an edge of $T_G$ if and only if $v\in H_i$, where $h_i$ is called *block-vertex*. Since two blocks in a graph share at most one vertex, then $T_G$ is a tree. We can easily get $T_G$ in linear time (see [@west2001introduction]).
Let $H$ be a block and $I$ be the set of cut-vertices of $H$. We say $H$ is a block of *type 0* if $|H|=|I|$ and $H$ is a block of *type 1* if $|H|=|I|+1$. If $|H|\ge |I|+2$, we say $H$ is a block of *type 2*. Next, we give the definition of *induced Roman {2}-domination function*.
A function $f_*$ is called an induced Roman {2}-domination function $(R2DF_*)$ of $T_G$, if there is a R2DF $f$ of $G$, such that
$$f_*(v)=\begin{cases}
f(v), \text{ if } v \text{ is a cut-vertex of G}\\
f(H)-f(I), \text{ if } v(=h) \text{ is a block-vertex of } T_G \\
\end{cases}$$
Therefore, we can transform Roman {2}-domination problem on $G$ to induced Roman {2}-domination problem on $T_G$. Then, we show how to verify whether a given funtion of $T_G$ is a $R2DF_*$ or not.
There exists a R2DF $f$ of weight $\gamma_{R2}(G)$, which satisfies the following conditions.
1\. If $H$ is a block of type 1, $v$ isn’t a cut-vertex and $v\in H$, then $f(v)\in \{0,1\}$.
2\. If $H$ is a block of type 2, $v$ isn’t a cut-vertex and $v\in H$, then $f(v)=0$.
Let $f$ be a R2DF of weight $\gamma_{R2}(G)$ and $u$ be a cut-vertex of $H$, where $H$ isn’t a block of type 0 and $f(u)=max_{v_0\in I} f(v_0)$. If $f(v)=2$, we can reassign $0$ to $v$ and $2$ to $u$. Hence, $f(v)\in \{0,1\}$. Futhermore, if $H$ is a block of type 2, we suppose that there exists a vertex $v\in H$ such that $f(v)=1$. If $f(u)\ge 1$, then we can reassign $2$ to $u$ and $0$ to $v$, a contradiction. Suppose that $f(u)=0$, then there exists a vertex $w\in H$, such that $w$ isn’t a cut-vertex and $f(w)\ge 1$. We reassign $2$ to $u$ and $0$ to $v,w,$ a contradiction.
The function $f_*$ is a $R2DF_*$ of $T_G$ with its corresponding function $f$ satisfying $Lemma$ $8$ if and only if $f_*$ satisfies the following conditions.
1\. If $H$ is a block of type 1, then $f_*(h)=0$ or $1$.
2\. If $H$ is a block of type 0 or 2, then $f_*(h)=0$.
3\. If $v$ is a cut-vertex with $f_*(v)=0$, then $\exists u\in N^2_{T_G}(v)$ such that $f_*(u)=2$ or $\exists u_1,u_2\in N^2_{T_G}(v)$ such that $f_*(u_1)=f_*(u_2)=1$.
4\. If $H$ is a block of type 1 or 2 with $f_*(h)=0$, then $\exists u\in N_{T_G}(h)$ such that $f_*(u)=2$ or $\exists u_1,u_2\in N_{T_G}(h)$ such that $f_*(u_1)=f_*(u_2)=1$.
5\. The weight of $f_*$ is $\gamma_{R2}(G)$.
Let $f_*$ be a $R2DF_*$ of $T_G$ with its corresponding function $f$ satisfying $Lemma$ $8$. We first show $f_*$ satisfies the above conditions. It is clear that the above item 1 and item 2 are true. Suppose that $v$ is a cut-vertex of $G$ with $f_*(v)=0$, then $f(v)=0.$ If there exists a neighbor $u\in N(v)$ with $f(u)=2$, then $u$ is a cut-vertex and $u\in N_{T_G}^2[v]$. It means that $\exists u\in N_{T_G}^2[v]$ such that $f_*(u)=2$. Otherwise, there exists at least two neighbors $x,y\in N(v)$ having $f(x)=f(y)=1$. If $x$ and $y$ are cut-vertices, then we obtain $x,y\in N_{T_G}^2[v]$ having $f_*(x)=f_*(y)=1$. If at least one of $x$ and $y$ isn’t a cut-vertex, without loss of generality we can assume $x$ isn’t a cut-vertex and $H$ is a block containing $x$. We deduce that $H$ is a block of type 1, implying that $f_*(h)=1$. So item 3 holds. Suppose that $H$ is a block of type 1 or 2 with $f_*(h)=0$ and $I$ is the set of cut-vertices of $H$. By $Lemma$ 8, we deduce that $f(v)=0$ for each $v\in H-I$. Since $f$ is a $R2DF$, then $\exists u\in N(v)$ such that $f(u)=2$ or $\exists u_1,u_2\in N(v)$ such that $f(u_1)=f(u_2)=1$. It is clear that $u$ is a cut-vertex. So do $u_1$, $u_2$. It means that $f_*(u)=2$ and $f_*(u_1)=f_*(u_2)=1$. We obtain item 4. It is easy to check that $f_*$ satisfies item 5.
On the other hand, let $f_*$ be a function satisfying the above conditions. Define $f$ as follows. $$f(v)=\begin{cases}
f_*(v), \text{ if } v \text{ is a cut-vertex}\\
f_*(h), \text{ if } v \text{ isn't a cut-vertex with } v\in H\\
\end{cases}$$ Then, we show $f$ is a R2DF satisfying $Lemma$ 8. Suppose that $H$ is a block and $v$ isn’t a cut-vertex with $v\in H$. If $H$ is a block of type 1, by the above item 1, we have $f(v)=f_*(h)\in \{0,1\}.$ If $H$ is a block of type 2, by the above item 2, we obtain $f(v)=f_*(h)=0$. It is clear that $f(V_G)=f_*(V_{T_G})=\gamma_{R2}(G)$.
Suppose that $v$ is a cut-vertex with $f(v)=f_*(v)=0$. If $\exists u\in N_{T_G}^2[v]$ such that $f_*(u)=2$, by items 1-3, we deduce that $u$ is a cut-vertex and $u\in N_G(v)$. Otherwise, $\exists h_1,h_2\in N_{T_G}^2[v]$ such that $f_*(h_1)=f_*(h_2)=1$. If $h_1$ and $h_2$ are both cut-vertices, then we have $h_1,h_2\in N_G(v)$ and $f(h_1)=f(h_2)=1$. If at least one of $h_1$ and $h_2$ isn’t a cut-vertex, without loss of generality, we can assume $h_1$ isn’t a cut-vertex and $h_1$ represent block $H_1$ in $T_G$. We deduce that $H_1$ is a block of type 1. Hence, $\exists v_1\in H_1$ and $v_1$ isn’t a cut-vertex such that $f(v_1)=f_*(h_1)=1$. Therefore, we obtain $f(N(v))\ge 2$.
Suppose that $H$ is a block containing $v$ and $v$ isn’t a cut-vertex with $f(v)=f_*(h)=0$. Hence, we deduce $H$ is a block of type 1 or 2. Since item 4, we have that $\exists u\in N_{T_G}(h)$ such that $f_*(u)=2$ or $\exists u_1,u_2\in N_{T_G}(h)$ such that $f_*(u_1)=f_*(u_2)=1$. It is clear that $u$ is a cut-vertex and $u\in N_G(v)$. So do $u_1,u_2$. We also obtain $f(u)=f_*(u)=2$ and $f(u_1)=f(u_2)=1$. Therefore, we deduce $f(N(v))\ge 2$.
By $Theorem$ $9$, we can easily verify whether a given function of $T_G$ is a $R2DF_*$. Then, we continue to use the method of tree composition and decomposition in Section 3. For convenience, $T_G$ is denoted by $T$ if there is no ambiguity. Suppose that $T$ is a tree rooted at $r$ and $f:V(T)\rightarrow \{0,1,2\}$ is a function on $T$. $T'$ is defined as a new tree rooted at $r'$ and $f':V(T')\rightarrow \{0,1,2\}$ is a function on $T'$, where $V(T')=V(T)\cup \{r'\}$ and $E(T')=E(T)\cup \{rr'\}$, $f'|_T=f$.
In order to construct an algorithm for computing Roman $\{2\}$-domination number, we must characterize the possible tree-subset tuples $(T,f,r)$. For this purpose, we introduce some additional notations as follows:
$CVX(T)=\{ r~|~ r$ is a cut-vertex of $G\}$;\
$BVX(T)=\{ r~|~ r$ is a block-vertex of $T\}$;\
$R2DF_*(T)=\{ f~|~ f$ is a $R2DF_*$ of $T\}$;\
$F_1(T)=\{ f~|~ f\in R2DF_*(T)$ with $f(r)=1\}$;\
$F_2(T)=\{ f~|~ f\in R2DF_*(T)$ with $f(r)=2\}$;\
$R2DF_*(T^{+1})=\{ f~|~ f\notin R2DF_*(T) $, $f'\in F_1(T')$ and $f'|_T=f\}$;\
$R2DF_*(T^{+2})=\{ f~|~ f\notin R2DF_*(T) $, $f'\in F_2(T')$ and $f'|_T=f\}-R2DF_*(T^{+1})$.
Then we consider the following eleven classes:
$A_1=\{(T,f,r)~|~ f\in R2DF_*(T),$ $r\in CVX(T)$ and $f(r)=2\}$;\
$A_2=\{(T,f,r)~|~ f\in R2DF_*(T),$ $r\in CVX(T)$ and $f(r)=1\}$;\
$A_3=\{(T,f,r)~|~ f\in R2DF_*(T),$ $r\in CVX(T)$ and $f(r)=0\}$;\
$A_4=\{(T,f,r)~|~ f\in R2DF_*(T^{+1}),$ $r\in CVX(T)\}$;\
$A_5=\{(T,f,r)~|~ f\in R2DF_*(T^{+2}),$ $r\in CVX(T)\}$;\
$B_1=\{(T,f,r)~|~ f\in R2DF_*(T),$ $r\in BVX(T)$ and $f(N[r])\ge2\}$;\
$B_2=\{(T,f,r)~|~ f\in R2DF_*(T),$ $r\in BVX(T)$ and $f(N[r])=1\}$;\
$B_3=\{(T,f,r)~|~ f\in R2DF_*(T),$ $r\in BVX(T)$ and $f(N[r])=0\}$;\
$B_4=\{(T,f,r)~|~ f\in R2DF_*(T^{+1}),$ $r\in BVX(T)$ and $f(N[r])=1\}$;\
$B_5=\{(T,f,r)~|~ f\in R2DF_*(T^{+1}),$ $r\in BVX(T)$ and $f(N[r])=0\}$;\
$B_6=\{(T,f,r)~|~ f\in R2DF_*(T^{+2}),$ $r\in BVX(T)\}$. In order to give the algorithm, we present the following Lemmas.
\[lem10\] $A_1=(A_1\circ B_1)\cup (A_1\circ B_2)\cup (A_1\circ B_3)\cup (A_1\circ B_4)\cup (A_1\circ B_5)\cup (A_1\circ B_6)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(A_1\circ B_1)\cup (A_1\circ B_2)\cup (A_1\circ B_3)\cup (A_1\circ B_4)\cup (A_1\circ B_5)\cup (A_1\circ B_6)\subseteq A_1$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. For each $1\le i\le 6$, if $(T_1,f_1,r_1)\in A_1$ and $(T_2,f_2,r_2)\in B_i$, it is clear that $f$ is a $R2DF_*$ of $T$, $r\in CVX(T)$ and $f(r)=f(r_1)=2$. We deduce that $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_1$. It means that $(A_1\circ B_1)\cup (A_1\circ B_2)\cup (A_1\circ B_3)\cup (A_1\circ B_4)\cup (A_1\circ B_5)\cup (A_1\circ B_6)\subseteq A_1$.
Now we prove that $A_1\subseteq(A_1\circ B_1)\cup (A_1\circ B_2)\cup (A_1\circ B_3)\cup (A_1\circ B_4)\cup (A_1\circ B_5)\cup (A_1\circ B_6)$. Let $(T,f,r)\in A_1$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then $f_1(r_1)=f(r)=2$. Since $f\in R2DF_*(T)$ and $f_1=f|_{T_1}$, $f_1\in R2DF_*(T_1)$ and $r_1\in CVX(T_1)$. So $(T_1,f_1,r_1)\in A_1$ and $r_2\in BVX(T_2)$. If $f_2\in R2DF_*(T_2)$, then we obtain $(T_2,f_2,r_2)\in B_1,$ $B_2$ or $B_3$. If $f_2\notin R2DF_*(T_2)$, then $(T_2,f_2,r_2)\in B_4,$ $B_5$ or $B_6$. Hence, we conclude that $A_1\subseteq(A_1\circ B_1)\cup (A_1\circ B_2)\cup (A_1\circ B_3)\cup (A_1\circ B_4)\cup (A_1\circ B_5)\cup (A_1\circ B_6)$.
\[lem11\] $A_2=(A_2\circ B_1)\cup (A_2\circ B_2)\cup(A_2\circ B_3)\cup(A_2\circ B_4)\cup(A_2\circ B_5)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(A_2\circ B_1)\cup (A_2\circ B_2)\cup(A_2\circ B_3)\cup(A_2\circ B_4)\cup(A_2\circ B_5)\subseteq A_2$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. For each $1\le i\le 5$, if $(T_1,f_1,r_1)\in A_2$ and $(T_2,f_2,r_2)\in B_i$, it is clear that $f$ is a $R2DF_*$ of $T$, $r\in CVX(T)$ and $f(r)=f(r_1)=1$. We conclude that $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_2$, implying that $(A_2\circ B_1)\cup (A_2\circ B_2)\cup(A_2\circ B_3)\cup(A_2\circ B_4)\cup(A_2\circ B_5)\subseteq A_2$.
Then we need to show $A_2\subseteq(A_2\circ B_1)\cup (A_2\circ B_2)\cup(A_2\circ B_3)\cup(A_2\circ B_4)\cup(A_2\circ B_5)$. Let $(T,f,r)\in A_2$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then $f_1(r_1)=f(r)=1$. It is clear that $f_1$ is a $R2DF_*$ of $T_1$ and $r_1\in CVX(T_1)$. So we conclude that $(T_1,f_1,r_1)\in A_2$ and $r_2\in BVX(T_2)$. If $f_2$ is a $R2DF_*$ of $T_2$, then we obtain $(T_2,f_2,r_2)\in B_1,B_2$ or $B_3$. If $f_2$ is not a $R2DF_*$ of $T_2$, then $f_2(N_{T_2}[r_2])\le 1$ and $f_2\in R2DF_*(T_2^{+1})$ by using the fact that $(T,f,r)\in A_2$. Therefore, we have $(T_2,f_2,r_2)\in B_4$ or $B_5$. Hence, we deduce that $A_2\subseteq(A_2\circ B_1)\cup (A_2\circ B_2)\cup(A_2\circ B_3)\cup(A_2\circ B_4)\cup(A_2\circ B_5)$.
\[lem12\] $A_3=(A_3\circ B_1)\cup (A_3\circ B_2)\cup (A_3\circ B_3)\cup (A_4\circ B_1)\cup (A_4\circ B_2)\cup (A_5\circ B_1)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(A_3\circ B_1)\cup (A_3\circ B_2)\cup (A_3\circ B_3)\cup (A_4\circ B_1)\cup (A_4\circ B_2)\cup (A_5\circ B_1)\subseteq A_3$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. We make some remarks.
(i) For each $1\le i\le3$, if $(T_1,f_1,r_1)\in A_3$ and $(T_2,f_2,r_2)\in B_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_3$. Indeed, if $(T_1,f_1,r_1)\in A_3$ and $(T_2,f_2,r_2)\in B_i$, then $f_1$ is a $R2DF_*$ of $T_1$ and $f_2$ is a $R2DF_*$ of $T_2$. Hence, $f$ is a $R2DF_*$ of $T$, $r\in CVX(T)$ and $f(r)=0$. Then We obtain $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_3$.
(ii) For each $1\le i\le2$, if $(T_1,f_1,r_1)\in A_4$ and $(T_2,f_2,r_2)\in B_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_3$. Indeed, if $(T_1,f_1,r_1)\in A_4$, then we have that $f_1\in R2DF_*(T_1^{+1})$, $r\in CVX(T)$, $f(r)=0$ and $f(N^2_{T_1}[r])=1$. By the definition of $B_i$, we obtain $f(N^2_{T}[r])\ge 2$ and $f\in iR2DF(T)$. It means that $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_3$.
(iii) If $(T_1,f_1,r_1)\in A_5$ and $(T_2,f_2,r_2)\in B_1$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_3$. Indeed, if $(T_1,f_1,r_1)\in A_5$, then we have that $f_1\in R2DF_*(T_1^{+2})$, $r\in CVX(T)$, $f(r)=0$ and $f(N^2_{T_1}[r])=0$. By the definition of $B_1$, we obtain $f(N^2_{T}[r])\ge 2$ and $f\in R2DF_*(T)$. It means that $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_3$.
Therefore, we need to prove $A_3\subseteq(A_3\circ B_1)\cup (A_3\circ B_2)\cup (A_3\circ B_3)\cup (A_4\circ B_1)\cup (A_4\circ B_2)\cup (A_5\circ B_1).$ Let $(T,f,r)\in A_3$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then we have that $f_1(r_1)=f(r)=0$, $r_1\in CVX(T_1)$ and $f_2\in R2DF_*(T_2)$. So $r_2\in BVX(T_2)$. If $f_1\in R2DF_*(T_1)$, then we obtain $(T_1,f_1,r_1)\in A_3$, implying that $(T_2,f_2,r_2)\in B_1,B_2$ or $B_3$. Suppose that $f_1\notin R2DF_*(T_1)$. Consider the following cases.
[**Case 1.**]{} $f_1(N^2_{T_1}[r_1])=1.$ Then we obtain $f_1\in R2DF_*(T_1^{+1})$, implying that $(T_1,f_1,r_1)\in A_4$. Since $(T,f,r)\in A_3$, we have $f_2(N_{T_2}[r_2])\ge1$. So $(T_2,f_2,r_2)\in B_1$ or $B_2$.
[**Case 2.**]{} $f_1(N^2_{T_1}[r_1])=0.$ So we have $f_1\in R2DF_*(T_1^{+2})$. Then $(T_1,f_1,r_1)\in A_5$. Since $(T,f,r)\in A_3$, we obtain $f_2(N_{T_2}[r_2])\ge2$. Hence, $(T_2,f_2,r_2)\in B_1$.
So $A_3\subseteq(A_3\circ B_1)\cup (A_3\circ B_2)\cup (A_3\circ B_3)\cup (A_4\circ B_1)\cup (A_4\circ B_2)\cup (A_5\circ B_1).$
\[lem13\] $A_4=(A_4\circ B_3)\cup (A_5\circ B_2)$.
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(A_4\circ B_3)\cup (A_5\circ B_2)\subseteq A_4$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is easy to check the following remarks by definitions.
(i) If $(T_1,f_1,r_1)\in A_4$ and $(T_2,f_2,r_2)\in B_3$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_4$.
(ii) If $(T_1,f_1,r_1)\in A_5$ and $(T_2,f_2,r_2)\in B_2$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in A_4$.
Therefore, we obtain that $(A_4\circ B_3)\cup (A_5\circ B_2)\subseteq A_4$.
On the other hand, we show $A_4\subseteq(A_4\circ B_3)\cup (A_5\circ B_2)$. Let $(T,f,r)\in A_4$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$. Then we have that $f\in R2DF_*(T^{+1})$ and $r_1\in CVX(T_1)$, implying that $f(N_T^2[r_1])=1$. It means that $r_2\in BVX(T_2)$. By the definition of $A_4$ and $f_2=f|_{T_2}$, $f_2\in R2DF_*(T_2)$. Using the fact that $f(N_T^2[r_1])=1$, we deduce that $f_2(N[r_2])<2$. Consider the following cases.
[**Case 1.**]{} $f_2(N[r_2])=1.$ It is clear that $(T_2,f_2,r_2)\in B_2$. Since $f_1(N^2_{T_1}[r_1])=f(N^2_T[r_1])-f_2(N[r_2])=0,$ we obtain $(T_1,f_1,r_1)\in A_5$.
[**Case 2.**]{} $f_2(N[r_2])=0.$ Then $(T_2,f_2,r_2)\in B_3$. Using the fact that $f_1(N^2_{T_1}[r_1])=f(N^2_T[r_1])-f_2(N[r_2])=1,$ we know $(T_1,f_1,r_1)\in A_4$.
Consequently, we deduce that $A_4\subseteq(A_4\circ B_3)\cup (A_5\circ B_2)$.
\[lem14\] $A_5=A_5\circ B_3$.
It is easy to check that $(A_5\circ B_3)\subseteq A_5$ by the definitions. On the other hand, let $(T,f,r)\in A_5$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$. Then we obtain $f\in R2DF_*(T^{+2})$, $r_1\in CVX(T_1)$ and $f_1(N^2[r_1])=f(N^2[r])=0$. It implies that $(T_1,f_1,r_1)\in A_5$ and $r_2\in BVX(T_2)$. Using the fact that $(T,f,r)\in A_5$, we deduce that $f_2(N[r_2])=0$ and $f_2\in R2DF_*(T_2)$. Therefore, $(T_2,f_2,r_2)\in B_3$. Then $A_5\subseteq(A_5\circ B_3).$
\[lem15\] $B_1=(B_1\circ A_1)\cup (B_1\circ A_2)\cup (B_1\circ A_3)\cup (B_1\circ A_4)\cup (B_1\circ A_5)\cup (B_2\circ A_1)\cup (B_2\circ A_2)\cup (B_3\circ A_1)\cup (B_4\circ A_1)\cup (B_4\circ A_2)\cup (B_5\circ A_1)\cup (B_6\circ A_1).$
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(B_1\circ A_1)\cup (B_1\circ A_2)\cup (B_1\circ A_3)\cup (B_1\circ A_4)\cup (B_1\circ A_5)\cup (B_2\circ A_1)\cup (B_2\circ A_2)\cup (B_3\circ A_1)\cup (B_4\circ A_1)\cup (B_4\circ A_2)\cup (B_5\circ A_1)\cup (B_6\circ A_1)\subseteq B_1$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. We make some remarks.
(i) For each $1\le i\le5$, if $(T_1,f_1,r_1)\in B_1$ and $(T_2,f_2,r_2)\in A_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_1$. It is easy to check it by the definitions of $B_1$ and $A_i$.
(ii) For each $2\le i\le6$, if $(T_1,f_1,r_1)\in B_i$ and $(T_2,f_2,r_2)\in A_1$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_1$. We can easily check it by definitions too.
(iii) For each $i\in \{2,4\}$, if $(T_1,f_1,r_1)\in B_i$ and $(T_2,f_2,r_2)\in A_2$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_1$. Indeed, it is clear that $f\in R2DF_*(T)$, $r\in BVX(T)$ and $f(N[r])=f_1(N[r_1])+f_2(r_2)=2.$ Hence, $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_1.$
Therefore, we need to prove $B_1\subseteq(B_1\circ A_1)\cup (B_1\circ A_2)\cup (B_1\circ A_3)\cup (B_1\circ A_4)\cup (B_1\circ A_5)\cup (B_2\circ A_1)\cup (B_2\circ A_2)\cup (B_3\circ A_1)\cup (B_4\circ A_1)\cup (B_4\circ A_2)\cup (B_5\circ A_1)\cup (B_6\circ A_1).$ Let $(T,f,r)\in B_1$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then we have that $f\in R2DF_*(T)$, $r_1\in BVX(T_1)$ and $f(N[r])\ge 2$. It means that $r_2\in CVX(T_2)$. Consider the following cases.
[**Case 1.**]{} $f(r_2)=2.$ Then we have $f_2\in R2DF_*(T_2)$, impling that $(T_2,f_2,r_2)\in A_1$. If $f_1\in R2DF_*(T_1)$, we obtain $(T_1,f_1,r_1)\in B_1,$ $B_2$ or $B_3$. Suppose that $f_1\notin R2DF_*(T_1)$, then $f_1\in R2DF_*(T_1^{+1})$ or $f_1\in R2DF_*(T_1^{+2})$. Hence, $(T_1,f_1,r_1)\in B_4$, $B_5$ or $B_6$.
[**Case 2.**]{} $f(r_2)=1.$ It is clear that $(T_2,f_2,r_2)\in A_2$. We also have that $f_1(N[r_1])=f(N[r])-f_2(r_2)\ge 2-1\ge 1$. If $f_1\in R2DF_*(T_1)$, we obtain $(T_1,f_1,r_1)\in B_1$ or $B_2$. Suppose that $f_1\notin R2DF_*(T_1)$, then $f_1\in R2DF_*(T_1^{+1})$. Therefore, $(T_1,f_1,r_1)\in B_4$.
[**Case 3.**]{} $f(r_2)=0.$ Then we obtain $f_1(N[r_1])=f(N[r])-f_2(r_2)\ge 2$ and $f_1\in R2DF_*(T_1)$, implying that $(T_1,f_1,r_1)\in B_1$. If $f_2\in R2DF_*(T_2)$, we deduce that $(T_1,f_1,r_1)\in A_3$. Suppose that $f_2\notin R2DF_*(T_2)$, then $f_2\in R2DF_*(T_2^{+1})$ or $f_2\in R2DF_*(T_2^{+2})$. Therefore, $(T_2,f_2,r_2)\in A_4$ or $A_5$.
Hence, $B_1\subseteq(B_1\circ A_1)\cup (B_1\circ A_2)\cup (B_1\circ A_3)\cup (B_1\circ A_4)\cup (B_1\circ A_5)\cup (B_2\circ A_1)\cup (B_2\circ A_2)\cup (B_3\circ A_1)\cup (B_4\circ A_1)\cup (B_4\circ A_2)\cup (B_5\circ A_1)\cup (B_6\circ A_1).$
\[lem16\] $B_2=(B_2\circ A_3)\cup (B_2\circ A_4)\cup (B_3\circ A_2)\cup (B_5\circ A_2).$
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(B_2\circ A_3)\cup (B_2\circ A_4)\cup (B_3\circ A_2)\cup (B_5\circ A_2)\subseteq B_2$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. We make some remarks.
(i) For each $3\le i\le4$, if $(T_1,f_1,r_1)\in B_2$ and $(T_2,f_2,r_2)\in A_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_2$. It is easy to check it by the definitions.
(ii) For each $i\in \{3,5\}$, if $(T_1,f_1,r_1)\in B_i$ and $(T_2,f_2,r_2)\in A_2$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_2$. Indeed, if $(T_1,f_1,r_1)\in B_i$ and $(T_2,f_2,r_2)\in A_2$, we obtain that $f\in R2DF_*(T)$, $r\in BVX(T)$ and $f(N[r])=f_1(N[r_1])+f_2(r_2)=1.$ Hence, we deduce $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_2$.
According to the previous items, we deduce that $(B_2\circ A_3)\cup (B_2\circ A_4)\cup (B_3\circ A_2)\cup (B_5\circ A_2)\subseteq B_2$.
Therefore, we need to prove $B_2\subseteq(B_2\circ A_3)\cup (B_2\circ A_4)\cup (B_3\circ A_2)\cup (B_5\circ A_2).$ Let $(T,f,r)\in B_2$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then we have that $f\in R2DF_*(T)$, $r_1\in BVX(T_1)$ and $f(N[r])=1$. It implies $r_2\in CVX(T_2)$. Consider the following cases.
[**Case 1.**]{} $f(r_2)=1.$ Then we have $f_1(N[r_1])=f(N[r])-f(r_2)=0$ and $f_2(r_2)=1$, implying that $f_2\in R2DF_*(T_2)$. So $(T_2,f_2,r_2)\in A_2$. If $f_1\in R2DF_*(T_1)$, we obtain $(T_1,f_1,r_1)\in B_3$. Suppose that $f_1\notin R2DF_*(T_1)$, then $f_1(r_1)=0$ because $f\in R2DF_*(T)$. Since $f_1(N[r_1])=0$, we have that $(T_1,f_1,r_1)\in B_5$.
[**Case 2.**]{} $f(r_2)=0.$ It is clear that $f_1(N[r_1])=f(N[r])-f(r_2)=1$. Since $f_1=f|_{T_1}$ and $f\in R2DF_*(T)$, we have $f_1\in R2DF_*(T_1)$. Hence, $(T_1,f_1,r_1)\in B_2$. If $f_2\in R2DF_*(T_2)$, we deduce that $(T_2,f_2,r_2)\in A_3$. Suppose that $f_2\notin R2DF_*(T_2)$, then $f_2(N^2[r_2])=1$. It implies $f_2\in R2DF_*(T_2^{+1})$. Therefore, $(T_2,f_2,r_2)\in A_4$.
Hence, $B_2\subseteq(B_2\circ A_3)\cup (B_2\circ A_4)\cup (B_3\circ A_2)\cup (B_5\circ A_2).$
\[lem17\] $B_3=B_3\circ A_3$.
It is easy to check that $(B_3\circ A_3)\subseteq B_3$ by the definitions. On the other hand, let $(T,f,r)\in B_3$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$. Then we obtain $f_1(N[r_1])=f(N[r])=0$, $r_1\in BVX(T_1)$ and $f(r_2)=0$. It means that $r_2\in CVX(T_2)$. Since $f\in R2DF_*(T)$ and $f(r_2)=0$, we obtain that $f_1\in R2DF_*(T_1)$, implying that $(T_1,f_1,r_1)\in B_3$. Using the fact that $f_1(N[r_1])=0$ and $f(r_2)=0$, we deduce that $f_2\in R2DF_*(T_2)$. Therefore, $(T_2,f_2,r_2)\in A_3$. Then $B_3\subseteq(B_3\circ A_3).$
\[lem18\] $B_4=(B_2\circ A_5)\cup (B_4\circ A_3)\cup (B_4\circ A_4)\cup (B_4\circ A_5)\cup (B_6\circ A_2).$
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(B_2\circ A_5)\cup (B_4\circ A_3)\cup (B_4\circ A_4)\cup (B_4\circ A_5)\cup (B_6\circ A_2)\subseteq B_4$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is easy to check the following remarks by definitions.
(i) If $(T_1,f_1,r_1)\in B_2$ and $(T_2,f_2,r_2)\in A_5$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_4$.
(ii) For each $3\le i\le 5$, if $(T_1,f_1,r_1)\in B_4$ and $(T_2,f_2,r_2)\in A_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_4$.
(iii) If $(T_1,f_1,r_1)\in B_6$ and $(T_2,f_2,r_2)\in A_2$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_4$.
Therefore, we need to prove $B_4\subseteq(B_2\circ A_5)\cup (B_4\circ A_3)\cup (B_4\circ A_4)\cup (B_4\circ A_5)\cup (B_6\circ A_2).$ Let $(T,f,r)\in B_4$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then we have that $f\in R2DF_*(T^{+1})$, $r_1\in BVX(T_1)$ and $f(N[r])=1$. It implies $r_2\in CVX(T_2)$. Consider the following cases.
[**Case 1.**]{} $f(r_2)=1.$ Then we have $f_1(N[r_1])=f(N[r])-f(r_2)=0$ and $f_2(r_2)=1$, implying that $f_2\in R2DF_*(T_2)$. So $(T_2,f_2,r_2)\in A_2$ and $f_1\notin R2DF_*(T_1)$. Since $f_1(N[r_1])=0$ and $(T,f,r)\in B_4$, we obtain $(T_1,f_1,r_1)\in B_6$.
[**Case 2.**]{} $f(r_2)=0.$ It is clear that $f_1(N[r_1])=f(N[r])-f(r_2)=1$. If $f_2\in R2DF_*(T_2)$, we deduce that $(T_2,f_2,r_2)\in A_3$, implying $(T_1,f_1,r_1)\in B_4$. Suppose that $f_2\notin R2DF_*(T_2)$, then $f_2(N^2[r_2])=0$ or 1. If $f_2(N^2[r_2])=0$, we obtain $(T_2,f_2,r_2)\in A_5$. Then, we have $(T_1,f_1,r_1)\in B_2$ or $B_4$. If $f_2(N^2[r_2])=1$, we obtain $(T_2,f_2,r_2)\in A_4$. Then, we have $(T_1,f_1,r_1)\in B_4$. Hence, $B_4\subseteq(B_2\circ A_5)\cup (B_4\circ A_3)\cup (B_4\circ A_4)\cup (B_4\circ A_5)\cup (B_6\circ A_2).$
\[lem19\] $B_5=(B_3\circ A_4)\cup (B_5\circ A_3)\cup (B_5\circ A_4).$
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(B_3\circ A_4)\cup (B_5\circ A_3)\cup (B_5\circ A_4)\subseteq B_5$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is easy to check the following remarks by definitions.
(i) If $(T_1,f_1,r_1)\in B_3$ and $(T_2,f_2,r_2)\in A_4$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_5$.
(ii) For each $3\le i\le 4$, if $(T_1,f_1,r_1)\in B_5$ and $(T_2,f_2,r_2)\in A_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_5$.
Therefore, we need to prove $B_5\subseteq(B_3\circ A_4)\cup (B_5\circ A_3)\cup (B_5\circ A_4).$ Let $(T,f,r)\in B_5$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then we have that $f\in R2DF_*(T^{+1})$, $r_1\in BVX(T_1)$ and $f(N[r])=0$. It implies $r_2\in CVX(T_2)$ and $f_2(r_2)=f(r_2)=0$. Consider the following cases.
[**Case 1.**]{} If $f_2\in R2DF_*(T_2)$, then we have $(T_2,f_2,r_2)\in A_3$ and $f_1\notin R2DF_*(T_1)$. Since $f_1(N[r_1])=0$ and $(T,f,r)\in B_5$, we obtain $(T_1,f_1,r_1)\in B_5$.
[**Case 2.**]{} If $f_2\notin R2DF_*(T_2)$, we deduce that $(T_2,f_2,r_2)\in A_4$. It is clear that $(T_1,f_1,r_1)\in B_3$ or $B_5$.
Hence, $B_5\subseteq(B_3\circ A_4)\cup (B_5\circ A_3)\cup (B_5\circ A_4).$
\[lem20\] $B_6=(B_3\circ A_5)\cup (B_5\circ A_5)\cup (B_6\circ A_3)\cup (B_6\circ A_4)\cup (B_6\circ A_5).$
Let $(T,r)=(T_1,r_1)\circ (T_2,r_2)$ and $r=r_1$. We first show that $(B_3\circ A_5)\cup (B_5\circ A_5)\cup (B_6\circ A_3)\cup (B_6\circ A_4)\cup (B_6\circ A_5)\subseteq B_6$. Suppose that $f_1$ (resp. $f_2$) is a function on $T_1$ (resp. $T_2$). Define $f$ as the function on $T$ with $f|_{T{}_1}=f_1$ and $f|_{T{}_2}=f_2$. It is easy to check the following remarks by definitions.
(i) For each $i\in\{3,5\}$, if $(T_1,f_1,r_1)\in B_i$ and $(T_2,f_2,r_2)\in A_5$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_6$.
(ii) For each $3\le i\le5$, if $(T_1,f_1,r_1)\in B_6$ and $(T_2,f_2,r_2)\in A_i$, then $(T_1,f_1,r_1)\circ(T_2,f_2,r_2)\in B_6$.
Therefore, we need to prove $B_6\subseteq(B_3\circ A_5)\cup (B_5\circ A_5)\cup (B_6\circ A_3)\cup (B_6\circ A_4)\cup (B_6\circ A_5).$ Let $(T,f,r)\in B_6$ and $(T,f,r)=(T_1,f_1,r_1)\circ (T_2,f_2,r_2)$, then we have that $f\in R2DF_*(T^{+2})$, $r_1\in BVX(T_1)$ and $f(N[r])=0$. It implies $r_2\in CVX(T_2)$. Consider the following cases.
[**Case 1.**]{} $f_1\in R2DF_*(T_1).$ Since $f_1(N[r_1])=f(N[r])=0$, we have $(T_1,f_1,r_1)\in B_3$. It implies $(T_2,f_2,r_2)\in A_5$.
[**Case 2.**]{} $f_1\notin R2DF_*(T_1).$ Since $f_1(N[r_1])=f(N[r])=0$, then we obtain $(T_1,f_1,r_1)\in B_5$ or $B_6$. If $(T_1,f_1,r_1)\in B_5$, we have $f_1\in R2DF_*(T_1^{+1})$. Since $f\in R2DF_*(T^{+2})$, it means that $f_2\in R2DF_*(T_2^{+2})$. Then we deduce $(T_2,f_2,r_2)\in A_5$. If $(T_1,f_1,r_1)\in B_6$, we have $f_1\in R2DF_*(T_1^{+2})$. Since $(T,f,r)\in B_6$, we deduce that $f_2(r_2)=0$. So we obtain $(T_2,f_2,r_2)\in A_3$, $A_4$ or $A_5$.
Hence, $B_6\subseteq(B_3\circ A_5)\cup (B_5\circ A_5)\cup (B_6\circ A_3)\cup (B_6\circ A_4)\cup (B_6\circ A_5).$
The final step is to define the initial vector. In this case, for a block-cutpoint graphs, the only basis graph is a single vertex. It is clear that if $v$ is a cut-vertex, then the initial vector is $(2,1,\infty,\infty,0,\infty)$; if $v$ is a block-vertex and its corresponding block is a block of type 0, then the initial vector is $(\infty,\infty,0,\infty,\infty,\infty)$; if $v$ is a block-vertex and its corresponding block is a block of type 1, then the initial vector is $(\infty,1,\infty,\infty,\infty,0)$; if $v$ is a block-vertex and its corresponding block is a block of type 2, then the initial vector is $(\infty,\infty,\infty,\infty,\infty,0)$. Among them, $'\infty'$ means undefined. Now, we are ready to present the algorithm.\
$\gamma_{\{R2\}}(G)=\min\{h[n,1], h[n,2], h[n,3]\}$
From the above argument, we can obtain the following theorem.
\[thm3\] Algorithm ROMAN $\{2\}$-DOM-IN-BLOCK can output the Roman $\{2\}$-domination number of any block graphs $G=(V,E)$ in linear time $O(n)$, where $n=|V|$.
[10]{}
G. J. Chang. Total domination in block graphs. , 8(1):53–57, 1989.
L. Chen, C. Lu, and Z. Zeng. Labelling algorithms for paired-domination problems in block and interval graphs. , 19(4):457–470, 2010.
D. Pradhan and A. Jha. On computing a minimum secure dominating set in block graphs. , 35(2):613–631, 2018.
G. Xu, L. Kang, E. Shan, and M. Zhao. Power domination in block graphs. , 359(1-3):299–305, 2006.
E. W. Chambers, B. Kinnersley, N. Prince, and D. B. West, Extremal problems for roman domination, SIAM Journal on Discrete Mathematics. $\mathbf{23}$(2009) 1575-1586.
G. J. Chang, Algorithmic aspects of domination in graphs, Handbook of Combinatorial Optimization. pages 221–282, 2013.
M. Chellali, T. W. Haynes, S. T. Hedetniemi, and A. A. McRae, Roman $\{$2$\}$-domination, Discrete Applied Mathematics. $\mathbf{204}$(2016) 22-28.
E. J. Cockayne, P. A. D. Jr, S. M. Hedetniemi, and S. T. Hedetniemi, Roman domination in graphs, Discrete Mathematics. $\mathbf{278}$(2004) 11-22.
M. Golumbic, Algorithmic graph theory and perfect graphs, acad. Press, New York, 1980.
M. A. Henning and W. F. Klostermeyer, Italian domination in trees, Discrete Applied Mathematics. $\mathbf{217}$(2017) 557-564.
C. H. Liu and G. J. Chang, Upper bounds on Roman domination numbers of graphs, Discrete Applied Mathematics. $\mathbf{312}$(2012) 1386-1391.
A. Rahmouni and M. Chellali, Independent Roman {2}-domination in graphs, Discrete Applied Mathematics. $\mathbf{236}$(2018) 408-414.
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T. V. Wimer, Linear algorithms on k-terminal graphs, Ph.D.Thesis, Clemson University, 1987.
[^1]: Supported in part by National Natural Science Foundation of China (No. 11371008) and Science and Technology Commission of Shanghai Municipality(No. 18dz2271000)
|
---
abstract: 'We study the spin dependence of accretion onto rotating Kerr black holes using analytic techniques. In its linear regime, angular momentum transport in MHD turbulent accretion flow involves the generation of radial magnetic field connecting plasma in a differentially rotating flow. We take a first principles approach, highlighting the constraint that limits the generation and amplification of radial magnetic fields, stemming from the transfer of energy from mechanical to magnetic form. Because the energy transferred in magnetic form is ultimately constrained by gravitational potential energy or Killing energy, the spin-dependence of the latter allows us to derive spin-dependent constraints on the success of the accreting plasma to expel its angular momentum and accrete. We find an inverse relationship between this ability and black hole spin. If this radial magnetic field generation forms the basis for angular momentum transfer in accretion flows, accretion rates involving Kerr black holes are expected to be lower as the black hole spin increases in the prograde sense.'
author:
- David Garofalo
title: Spacetime constraints on accreting black holes
---
Introduction
============
Magnetized accretion onto rotating black holes forms the basis for models of X-ray binaries (XRBs), active galactic nuclei (AGN) and their smaller counterparts, microquasars. Balbus & Hawley (1991) showed how the magnetorotational instability (MRI) - discovered independently by Velikhov and Chandrasekhar - could lead to angular momentum transport in differentially rotating MHD flow (Velikhov, 1959; Chandrasekhar, 1961), such as in magnetized accretion onto rotating black holes. For a sufficiently small ratio of magnetic pressure to gas pressure, the highly conducting flow advects the magnetic field with it in a process described as flux-freezing. This means that the magnetic field is subject to being deformed and stretched as determined by the gas motion, producing magnetic connections between spatially distinct regions. If gas at an inner accretion disk radius, $r_{in}$, is coupled magnetically to gas at an outer radius, $r_{out}$, the magnetic field transports angular momentum from $r_{in}$ to $r_{out}$, decreasing the angular momentum of gas at $r_{in}$ and increasing the angular momentum of gas at $r_{out}$. As a consequence of this, gas at $r_{in}$, which loses angular momentum, spirals inward to smaller radial values and larger circular velocities. Gas at $r_{out}$, instead, migrates to larger radial values compared to $r_{out}$, compatible with the angular momentum acquired, and settles in circular orbits with lower circular velocities. As long as inward spiraling gas continues to be magnetically coupled to outward migrating gas, angular momentum is transferred at a greater rate as the coupling distance increases. Due to the energetically less favorable configurations, larger coupling distances are limited, and eventually suffer magnetic reconnection. This study highlights the constraints from energy conservation on coupling distance, ultimately suggesting that general relativity imposes a coupling distance constraint that depends monotonically on the spin of the black hole. The monotonic nature of this dependence, suggests that jets and outflows produced in magnetized black hole accretion flows, may bear the signature of black hole spin. Section \[Work\] describes the study and presents the results. Section \[Discussion\] concludes.
Spin-dependence of accretion flows {#Work}
==================================
Under the assumption that the MRI is the basic mechanism behind angular momentum transport, we measure the ability of a thin accretion disk to expel its angular momentum and accrete, by determining the amount of work done to extract the angular momentum that exists over a fixed proper distance between $r_{in}$ and $r_{out}$. For a fixed mass density flow between $r_{out}$ and $r_{in}$, the angular momentum difference between $r_{out}$ and $r_{in}$ must be deposited outward of $r_{out}$. We assume that the ability to extract that angular momentum difference is the same for all black hole spins and show how this assumption becomes increasingly problematic at higher prograde spin. From conservation laws in the Kerr metric, the angular momentum difference that must be extracted to produce a given accretion rate, increases for fixed proper distance as the black hole spin increases in the prograde sense. A work-kinetic energy argument applied to angular momentum extraction follows, motivating a spin-dependent constraint. We begin by introducing two conserved quantities in the Kerr spacetime that originate from time-translation and azimuthal angle-invariance that are identified as energy and angular momentum per unit mass for circular geodesic orbits in the equatorial plane of a rotating black hole. From the Kerr metric we have
$$E = -\epsilon_{\nu}p^{\nu}= (1-\frac{2M}{r})\dot{t} +
\frac{2Ma}{r}\dot{\phi}$$
$$L = \psi_{\nu}p^{\nu} = -\frac{2Ma}{r}\dot{t} +
\frac{(r^{2}+a^{2})^{2}-\Delta a^{2}}{r^{2}}\dot{\phi}$$
where $\epsilon$ and $\psi$ are the two spacetime Killing vectors for Boyer-Lindquist coordinates, $p^{\nu}$ is the 4-momentum, dot implies differentiation with respect to proper time, $M$ and $a$ are the mass and spin parameters, $r$ is the radial coordinate, $t$ is the time coordinate, $\phi$ is the azimuthal angle coordinate, and $\Delta =
r^2 - 2Mr + a^2$. Replacing $\dot{t}$ and $\dot{\phi}$ in terms of metric components, one obtains the following explicit forms for E and L (Bardeen et al, 1972).
$$E = \frac{r^{3/2}-2Mr^{1/2}\pm aM^{1/2}} {r^{3/4}(r^{3/2}-3Mr^{1/2}\pm
2aM^{1/2})^{1/2}}$$
$$L = \pm M^{1/2}\frac{r^{2}\mp 2aM^{1/2}r^{1/2}+a^{2}}
{r^{3/4}(r^{3/2}-3Mr^{1/2}\pm2aM^{1/2})^{1/2}}$$
with upper sign for prograde orbits and lower sign for retrograde orbits.
As mentioned, we choose a fixed proper distance, R, from the marginally stable circular orbit and determine the energy difference between the marginally stable orbit and the radial location for which the proper distance from the marginally stable circular orbit is R. We show the spin dependence of this energy difference in figure \[Work-spin\] and refer to it as work resulting from the difference in kinetic energy of the gas parcel between $r_{in}$ and $r_{out}$. We conclude that if the accretion rate is to remain spin-independent, the gas must lose greater energy per mass density, per unit coordinate time, as the spin becomes more prograde.
Given that it is the MRI behind the transfer of angular momentum and energy between $r_{in}$ and $r_{out}$, we motivate the existence of a spin-dependence in the energy transferred from mechanical form in the gas, to magnetic form in the following way. We imagine that two gas parcels, initially adjacent and connected via magnetic field (at some radial position that is intermediate between $r_{out}$ and $r_{in}$), begin to slowly migrate apart. If the two initially adjacent gas parcels become separated by a proper distance R, as assumed, and their radial velocities are small, the energy transferred from kinetic to magnetic form will roughly be equal to that of figure \[Work-spin\] for the specific spin. Therefore, in order for accretion rates to be spin-independent, the energy in the poloidal component of the magnetic field, must increase with increase in prograde spin. But, magnetic reconnection is progressively more likely to occur before the proper coupling distance becomes R, as the spin increases in the prograde direction. The overall conclusion, thus, is that lower accretion rates are energetically preferred as the spin increases in the prograde direction. One might be tempted to consider the spin dependence of the ability to transport angular momentum by focusing on the work required to extract a fixed amount of angular momentum. In other words, instead of approaching the question by determining the work done over a fixed proper distance, consider the work required to extract a fixed angular momentum difference. Although we claim this not to be the appropriate approach, it is perhaps noteworthy that if one were to calculate work vs. spin this way, the trend would be the same (i.e. that greater work is required to extract that fixed angular momentum difference as the spin increases in the prograde direction). Either way, the direct connection assumed here between the energy difference over a fixed proper distance and the likelihood of magnetic reconnection, is speculative. It could be that magnetic reconnection is related to aspects of the Kerr geometry in other non-trivial ways that are not addressed here. Also, since the constraints discussed here depend on the spin parameter of the black hole, they are purely relativistic, so they produce no Newtonian counterpart to this study.
Conclusion {#Discussion}
==========
This work highlights features of spacetime that appear to be intrinsically suited to influence the character of MHD accretion flows. Because MHD accretion flows in AGN are linked to outflows that influence the galactic and intergalactic medium (Kormendy & Richstone, 1995; Magorrian et al. 1998; Marconi & Hunt, 2003; Gebhardt et al. 2000; Ferrarese & Merritt, 2000; Tremaine et al, 2002), black hole spin may produce signatures or even influence the evolution of galaxies (Garofalo, 2009). If the MRI forms the foundation for angular momentum transport in magnetized accretion flows, and, given that its modus operandi is based on magnetically connected spatially separated regions, we have derived constraints that influence the spatial connectivity of such regions. Our suggestion is that the spin dependence of the conservation laws may be reflected in the flow in terms of dynamical constraints. Our analysis in section \[Work\] points to the need for greater conversion of energy to magnetic form in order for the accretion rate to be insensitive to black hole spin. From an energetic viewpoint, then, this suggests that the magnetized flow will be subjected to a spin dependence involving lower accretion rates at higher prograde spin. If MHD turbulence is non-local (e.g. Guan et al., 2009), our analysis applies. We note that accretion rates in GRMHD simulations of adiabatic black hole accretion flows, do in fact decrease as the spin increases in the prograde direction (De Villiers, 2003; McKinney, 2004).
Acknowledgments
===============
The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. D.G. is supported by the NASA Postdoctoral Program at NASA JPL administered by Oak Ridge Associated Universities through contract with NASA.
References
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|
---
abstract: 'The 1994 data published by the H1 collaboration are compared with models based on Regge phenomenology. The $x_{{I\!\!P}}$ dependence of the data can be described in a model based on the exchange of a dominant diffractive (pomeron) trajectory with additional sub-leading reggeon contributions. The dynamics of the Pomeron structure is studied within the framework of perturbative QCD and new parton distributions are obtained. These parton distributions will allow a direct test of factorisation breaking at Tevatron.'
address: |
DAPNIA/SPP, Commissariat à l’Energie Atomique, Saclay,\
F-91191 Gif-sur-Yvette Cedex
author:
- 'C. Royon'
title: QCD analysis of the diffractive structure functions measured at HERA and factorisation breaking at Tevatron
---
Regge parametrization
=====================
The 1994 data are first investigated in the framework of a Regge phenomenological model [@F2d94]. The 1994 data are subjected to a fit in which a single factorisable trajectory (${I\!\!P}$) is exchanged such that: $$\begin{aligned}
F_2^{D(3)}(Q^2,\beta,x_{{I\!\!P}})=
f_{{I\!\!P}/ p} (x_{{I\!\!P}}) F_2^{{I\!\!P}} (Q^2,\beta) \ .
\label{ff1}\end{aligned}$$ In this parameterization, $F_2^{{I\!\!P}}$ can be interpreted as the structure function of the pomeron [@IS]. The value of $F_2^{{I\!\!P}}$ is treated as a free parameter at each point in $\beta$ and $Q^2$. The pomeron flux takes a Regge form with a linear trajectory $\alpha_{{I\!\!P}}(t)=\alpha_{{I\!\!P}}(0)+\alpha^{'}_{{I\!\!P}} t$, such that $$\begin{aligned}
f_{{I\!\!P}/ p} (x_{{I\!\!P}})= \int^{t_{min}}_{t_{cut}} \frac{e^{B_{{I\!\!P}}t}}
{x_{{I\!\!P}}^{2 \alpha_{{I\!\!P}}(t) -1}} {\rm d} t \ ,
\label{flux}\end{aligned}$$ where $|t_{min}|$ is the minimum kinematically allowed value of $|t|$ and $t_{cut}=-1$ GeV$^2$ is the limit of the measurement. The value of $\alpha_{{I\!\!P}}(0)$ is a free parameter and $B_{{I\!\!P}}$ and $\alpha^{'}_{{I\!\!P}}$ are taken from hadron-hadron data [@F2d94]. The fit with a single trajectory does not give a good description of the data in the same way as it is observed at $Q^2 = 0$ [@gammap] that secondary trajectories in addition to the pomeron are required to describe diffractive $ep$ data.
A much better fit is obtained when both a leading (${I\!\!P}$) and a sub-leading (${I\!\!R}$) trajectory are considered in the same way as in formula (\[ff1\]), where the values of $F_2^{{I\!\!P}}$ and $ F_2^{{I\!\!R}}$ are treated as free parameters at each point in $\beta$ and $Q^2$, $\alpha_{{I\!\!P}}(0)$ and $\alpha_{{I\!\!R}}(0)$ being two free parameters. The flux factor for the secondary trajectory takes the same form as equation (\[flux\]), with $B_{{I\!\!R}}$, and $\alpha^{'}_{{I\!\!R}}$ again taken from hadron-hadron data [@F2d94]. This fit yields to the following value of ${\alpha_{I\!\!P}}(0) = 1.203 \pm 0.020 \ ({\rm stat.})
\pm 0.013 \ ({\rm syst.}) ^{+0.030}_{-0.035} \ ({\rm model})$ [@F2d94] and is significantly larger than values extracted from soft hadronic data ($\alpha_{{I\!\!P}} \sim 1.08$). The quality of the fit is similar if interference between the two trajectories is introduced.
QCD fits and the structure of the Pomeron
=========================================
It has been suggested that the $Q^2$ evolution of the Pomeron structure function may be understood in terms of parton dynamics from perturbative QCD where parton densities are evolved according to DGLAP [@dglap] equations [@IS; @F2d94], using the GRV parametrization for $F_2^{{I\!\!R}}$ [@GRVpion].
For the pomeron, a quark flavour singlet distribution ($z{ {S}}_{q}(z,Q^2)=u+\bar{u}+d+\bar{d}+s+\bar{s}$) and a gluon distribution ($z{\it {G}}(z,Q^2)$) are parameterized in terms of coefficients $C_j^{(S)}$ and $C_j^{(G)}$ at $Q^2_0=3$ GeV$^2$ such that : $$\begin{aligned}
z{\it {S}}(z,Q^2=Q_0^2) \left[
\sum_{j=1}^n C_j^{(S)} \cdot P_j(2z-1) \right]^2
\cdot e^{\frac{a}{z-1}} \\
z{\it {G}}(z,Q^2=Q_0^2) \left[
\sum_{j=1}^n C_j^{(G)} \cdot P_j(2z-1) \right]^2
\cdot e^{\frac{a}{z-1}}\end{aligned}$$ where $z=x_{i/I\!\!P}$ is the fractional momentum of the pomeron carried by the struck parton, $P_j(\zeta)$ is the $j^{th}$ member in a set of Chebyshev polynomials, which are chosen such that $P_1=1$, $P_2=\zeta$ and $P_{j+1}(\zeta)=2\zeta P_{j}(\zeta)-P_{j-1}
(\zeta)$. Some details about the fits can be found in Reference [@Laurent].
A sum of $n=3$ orthonormal polynomials is used so that the input distributions are free to adopt a large range of forms for a given number of parameters. The exponential factor is needed to ensure a correct convergence close to $z$=1.
The trajectory intercepts are fixed to $\alpha_{{I\!\!P}} = 1.20$ and $\alpha_{{I\!\!R}} = 0.62$. Only data points of H1 with $\beta \le 0.65$, $M_X > 2$ GeV and $y \le 0.45$ are included in the fit in order to avoid large higher twist effects and the region that may be most strongly affected by a non zero value of $R$, the longitudinal to transverse cross-section ratio.
Results of the QCD fits
=======================
The resulting parton densities of the Pomeron are presented in figure \[f1\]. As it was noticed in the 1994 $F_2^D$ paper [@F2d94], we find two possible fits quoted here as fit 1 and fit 2. Each fit shows a large gluonic content. The quark contribution is quite similar for both fits, but the gluon distribution tends to be quite different at high values of $z$. This can be easily explained as no data above $z=0.65$ are included in the fits. Thus there is no constraint from the data at high $z$. The quark densities is on the contrary more constrained in this region with the DGLAP evolution. Both fits show similar $\chi^2$ (the $\chi^2$ per degree of freedom is about 1.2) [^1]. Adding the 1995 data points into the fits also allows to get a better constraint on initial parton densities at $Q_0^2 = 3$ GeV$^2$ compared to the fits performed with 1994 data points alone. For the gluon density presented in figure \[f1\], we have determined that $ \frac{\delta G}{G} \simeq 25 \%$ for $z$ below 0.6.
The result of the fit is presented in figure \[f2\] together with the experimental values for 1994 data points ; we see on this figure the good agreement of the QCD prediction and the data points, which supports the validity of description of the Pomeron in terms of partons following a QCD dynamics.
We have also tried to extend the QCD fits to lower $Q^2$ (below $3$ GeV$^2$) using the 1995 $F_2^D$ measurement. The $\chi^2$ of the fit turns out to increase ($\chi^2/ndf = 1.6$, adding 35 low $Q^2$ points to the 171 points) [@chr]. This can be illustrated in figure 2 of Reference [@chr] where changes of slopes of scaling violations for $Q^2$ below and above $3$ GeV$^2$ can be seen. It may indicate that breaking of perturbative QCD has already occured in this region.
The idea would then to use these parton distributions and to compare with the measurements at Tevatron in order to study factorisation breaking. The roman pots which will be available in the D0 experiment at Run II will allow a direct comparison with the results obtained from the HERA parton distributions. It will be possible to know where factorisation breaking takes place at Tevatron, e.g. is it at low or high $\beta$?
Acknowledgments
===============
The results described in the present contribution come from a fruitful collaboration with J.Bartels, H.Jung R.Peschanski and L.Schoeffel.
[9]{} H1 Collab., C.Adloff et al., Z. Phys. C76 (1997) 613. H1 Collab., C. Adloff et al., Z. Phys. C74 (1997) 221. G.Altarelli, G.Parisi, Nucl. Phys. B126 (1977) 298.\
V.N.Gribov, L.N. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 438 and 675. G. Ingelman, P. Schlein, Phys. Lett. B152 (1985) 256. M. Glück, E. Reya, A. Vogt, Z. Phys. C53 (1992) 651. V.S.Fadin, E.A.Kuraev, L.N.Lipatov Phys. Lett. B60 (1975) 50.\
I.I.Balitsky, L.N.Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. L.Schoeffel, N.I.M.A423 (1999) 439. C.Royon for the H1 collaboration, talk given at the DIS99 conference, Zeuthen (Allemagne), 19-23/04/99, preprint hep-ph/9908216
[^1]: Fit 2 is a bit disfavoured compared to fit 1 (its $\chi^2$ by degree of freedom is 1.3 compared to 1.2 for fit 1) and is quite instable: changing a little the parameters modifies the gluon distribution at high $z$.
|
---
abstract: 'We consider ensembles of planar maps with two marked vertices at distance $k$ from each other and look at the closed line separating these vertices and lying at distance $d$ from the first one ($d<k$). This line divides the map into two components, the hull at distance $d$ which corresponds to the part of the map lying on the same side as the first vertex and its complementary. The number of faces within the hull is called the hull volume and the length of the separating line the hull perimeter. We study the statistics of the hull volume and perimeter for arbitrary $d$ and $k$ in the limit of infinitely large planar quadrangulations, triangulations and Eulerian triangulations. We consider more precisely situations where both $d$ and $k$ become large with the ratio $d/k$ remaining finite. For infinitely large maps, two regimes may be encountered: either the hull has a finite volume and its complementary is infinitely large, or the hull itself has an infinite volume and its complementary is of finite size. We compute the probability for the map to be in either regime as a function of $d/k$ as well as a number of universal statistical laws for the hull perimeter and volume when maps are conditioned to be in one regime or the other.'
address: 'Institut de physique théorique, Université Paris Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette'
author:
- Emmanuel Guitter
bibliography:
- 'hullrefined.bib'
title: Refined universal laws for hull volumes and perimeters in large planar maps
---
Introduction {#sec:introduction}
============
The study of random planar maps, which are connected graphs embedded on the sphere, has been for more than fifty years the subject of some intense activity among combinatorialists and probabilists, as well as among physicists in various domains. Very recently, some special attention was paid to statistical properties of the *hull* in random planar maps, a problem which may be stated as follows: consider an ensemble of planar maps having two marked vertices at graph distance $k$ from each other. For any non-negative $d$ strictly less than $k$, we may find a closed line “at distance $d$" (i.e. made of edges connecting vertices at distance $d$ or so) from the first vertex and separating the two marked vertices from each other. Several prescriptions may be adopted for a univocal definition of this separating line but they all eventually give rise to similar statistical properties. The separating line divides de facto the map into two connected components, each containing one of the marked vertices. The *hull at distance $d$* corresponds to the part of the map lying on the same side as the first vertex (i.e. that from which distances are measured). The geometrical characteristics of this hull for arbitrary $d$ and $k$ provide random variables whose statistics may be studied by various techniques. In particular, the statistics of the *volume of the hull*. which is its number of faces, and of the *hull perimeter*, which is the length of the separating line, have been the subject of several investigations [@Krikun03; @Krikun05; @CLG14a; @CLG14b; @G16a; @Men16; @G16b].
In a recent paper [@G16a], we presented a number of results on the statistics of the hull perimeter at distance $d$ for planar triangulations (maps with faces of degree three) and quadrangulations (faces of degree four) in a universal regime of infinitely large maps where both $d$ and $k$ are large and *remain of the same order* (i.e. the ratio $d/k$ is kept fixed). As we shall see, for such a regime, although the hull perimeter remains finite (but large, of order $d^2$), the volume of the hull at distance $d$ may very well be itself strictly infinite. We will compute below the probability for this to happen, a probability which remains non-zero for large $d$ and $k$ (unless $d/k\to 0$). In particular, if we wish a non-trivial description of the hull volume statistics, we have to *condition the map configurations so that their hull volume remains finite*. More generally, we may reconsider the statistics of the hull perimeter by separating the contribution coming from the set of map configurations with a finite hull volume from that coming from the set of map configurations with an infinite hull volume. More simply, we may consider the *hull perimeter conditional statistics* obtained by limiting the configurations to either set of configurations. It is the subject of the present paper to give a precise description of this *refined hull statistics* where we control the finite or infinite nature of the hull volume. Most of the obtained laws crucially depend on the value of $d/k$ but are the same for planar triangulations and planar quadrangulations, as well as for Eulerian triangulations (maps with alternating black and white triangular faces).
The paper is organized as follows: we first present in Section \[sec:summary\] a summary of our results and give explicit expressions for the probability to have a finite or an infinite hull volume as a function of $d/k$ (Sect. \[sec:inandout\]), as well as for the conditional probability density for the hull perimeter in both situations (Sect. \[sec:pd\]). We then give (Sect. \[sec:jointlaw\]) the joint law for the hull perimeter and hull volume, assuming that the latter is finite. Section \[sec:strategy\] presents the strategy that we use for our calculations which is based on already known generating functions whose expressions are recalled in the case of quadrangulations (Sect. \[sec:genfunc\]). We explain in details (Sect. \[sec:extract\] ) how to extract from these generating functions the desired statistical results. This strategy is implemented for quadrangulations in Section \[sec:explicit\] where we compute the probability to have a finite or an infinite hull volume (Sect.\[sec:explprobs\]), the probability density for the hull perimeter in both regimes (Sect. \[sec:explperimlaw\]) and the joint law for the hull perimeter and volume when the latter is finite (Sect. \[sec:expljointlaw\]). Section \[sec:Other\] briefly discusses triangulations and Eulerian triangulations for which the same universal laws as those found in the previous sections for quadrangulations are recovered. We gather a few concluding remarks in Section \[sec:conclusion\] and present additional non-universal expressions at finite $d$ and $k$ in appendix A.
Summary of the results {#sec:summary}
======================
The results presented in this paper have been obtained for three families of planar maps: (i) planar quadrangulations, i.e. planar maps whose all faces have degree four, (ii) planar triangulations, i.e. planar maps whose all faces have degree three and (iii) planar Eulerian triangulations, which are planar triangulations whose faces are colored in black and white with adjacent faces being of different color. For all these families, we obtain in the limit of large maps *the same laws* for hull volumes and perimeters, up to two non-universal normalization factors, one for the volume and one for the perimeter (called $f$ and $c$ respectively). The hull volumes and perimeters are defined as follows: for the three families of maps and for some integer $k\geq 1$, we consider more precisely $k$-pointed-rooted maps, i.e. maps with a marked vertex $x_0$ (called the origin) and a marked oriented edge pointing from a vertex $x_1$ at graph distance $k$ from the origin $x_0$ to a neighbor of $x_1$ at distance $k\!-\!1$ (such neighbor always exists)[^1]. Given $k \geq 3$ and some integer $d$ in the range $2\leq d\leq k\!-\!1$, there exists a simple closed line along edges of the map, “at distance $d$" from the origin[^2] which separates the origin $x_0$ from $x_1$. Several prescription are possible for a univocal definition of this *separating line* and we will adopt here that proposed in [@G16a] in cases (i) and (ii) and in [@G16b] for case (iii). We expect that other choices should not modify our results, except possibly for the value of the perimeter normalization factor $c$. The *hull at distance $d$ in our $k$-pointed-rooted map* is defined as the domain of the map lying *on the same side as the origin* of the separating line at distance $d$. Its *volume ${\mathcal V}(d)$* is its number of faces and its *perimeter ${\mathcal L}(d)$* the length (i.e. number of edges) of its boundary, namely the length of the separating line at distance $d$ itself.
![An illustration of map configurations in the out- and in-regimes. The map is represented schematically with vertices placed at a height equal to their distance from the origin $x_0$. The vertices at distance $d$ from $x_0$ form a number of closed curves at height $d$, one of which (in red) separates $x_0$ from $x_1$ and defines the separating line at distance $d$. The part of the map lying on the same side of this separating line as $x_0$ constitutes the hull at distance $d$ (here in light blue). For maps with an infinite volume, the configuration is in the out-regime if the hull volume remains finite (configuration on the left) or it is in the in-regime if the hull volume itself becomes infinite (configuration on the right).[]{data-label="fig:hullboundary"}](hullboundary.pdf){width="11cm"}
The out- and in-regimes {#sec:inandout}
-----------------------
Our results deal with the statistics of uniformly drawn $k$-pointed-rooted maps in the families (i), (ii) or (iii) having a *fixed number of faces* $N$, and for a *fixed value of the parameter $k$* and, more precisely, with *the limit $N\to \infty$ of this ensemble*, keeping $k$ finite. This corresponds to the so called *local limit* of infinitely large maps and, as in [@G16a], we shall denote by $P_k(\{\cdot\})$ the probability of some event $\{\cdot\}$ and $E_k(\{\cdot\})$ the expectation value of some quantity $\{\cdot\}$ in this limit.
In the limit $N\to \infty$, two situations may occur: either the volume $\mathcal V(d)$ remains finite and the number of faces $N\!-\!{\mathcal V}(d)$ of the complementary of the hull (namely the part of the map lying on the same side of the separating line as $x_1$) is infinite. This situation will be referred to as the *“out-regime"* in the following. Or the volume ${\mathcal V}(d)$ is itself infinite while the number $N\!-\!{\mathcal V}(d)$ of faces of the complementary of the hull remains finite. This situation will be referred to as the *“in-regime"* in the following. The case where both ${\mathcal V}(d)$ and $N\!-\!{\mathcal V}(d)$ would be infinite is expected to be suppressed when $N\to \infty$ (i.e. the number of configurations in this regime does not grow with $N$ as fast as that in the out- and in-regimes). Situations in the out- and in- regime are illustrated in figure \[fig:hullboundary\].
The main novelty in this paper is that our laws will discriminate between situations where the map configurations are in the out- or in the in-regime. We shall use accordingly the notations $$\begin{split}
&P^{\rm{out}}_{k,d}(\{\cdot\})=P_k\big(\{\cdot\}\ \hbox{and}\ {\mathcal V}(d)\ \hbox{finite}\big)\ ,
\quad P^{\rm{in}}_{k,d}(\{\cdot\})=P_k\big(\{\cdot\}\ \hbox{and}\ {\mathcal V}(d)\ \hbox{infinite}\big)\ ,\\
& E^{\rm{out}}_{k,d}(\{\cdot\})=E_k\big(\{\cdot\}\!\times\!\theta_{\rm finite}({\mathcal V}(d)\big)\ ,
\quad E^{\rm{in}}_{k,d}(\{\cdot\})=E_k\big(\{\cdot\}\!\times\!(1-\theta_{\rm finite}({\mathcal V}(d))\big)\,\\
\end{split}$$ with $\theta_{\rm finite}({\mathcal V})=1$ if ${\mathcal V}$ is finite and $0$ otherwise. Alternatively, we will consider conditional probabilities and conditioned expectation values, defined respectively as $$\begin{split}
&P_k(\{\cdot\}|{\mathcal V}(d)\ \hbox{finite})=\frac{P^{\rm{out}}_{k,d}(\{\cdot\})}{P_k({\mathcal V}(d)\ \hbox{finite})}\ ,
\quad P_k(\{\cdot\}|{\mathcal V}(d)\ \hbox{infinite})=\frac{P^{\rm{in}}_{k,d}(\{\cdot\})}{P_k({\mathcal V}(d)\ \hbox{infinite})}\ .\\
&E_k(\{\cdot\}|{\mathcal V}(d)\ \hbox{finite})=\frac{E^{\rm{out}}_{k,d}(\{\cdot\})}{P_k({\mathcal V}(d)\ \hbox{finite})}\ ,
\quad E_k(\{\cdot\}|{\mathcal V}(d)\ \hbox{infinite})=\frac{E^{\rm{in}}_{k,d}(\{\cdot\})}{P_k({\mathcal V}(d)\ \hbox{infinite})}\ ,\\
\end{split}$$ where $P_k({\mathcal V}(d)\ \hbox{finite})=E_k\big(\theta_{\rm finite}({\mathcal V}(d)\big)=1-P_k({\mathcal V}(d)\ \hbox{infinite})$. .3cm Universal laws may are obtained when $k$ and $d$ themselves become large simultameously, i.e. upon taking the limit $k\to \infty$, $d\to \infty$ with $d/k$ fixed (necessarily between $0$ and $1$). We set accordingly $$u\equiv \frac{d}{k}\ , \qquad 0\leq u\leq 1\ ,$$ and our universal results will deal with configurations having *a fixed value of $u$*.
As in [@G16a], we insist on that we first let $N\to \infty$, and only then take the limit of large $k$ and $d$. In particular, this is to be contrasted with the so called scaling limit where $N$, $k$ and $d$ would tend simultaneously to infinity with $k\sim d\sim N^{1/4}$. Note also that our universal laws describe a broader regime than that explored in most papers so far on the hull statistics [@Krikun03; @Krikun05; @CLG14a; @CLG14b; @Men16], where the hull boundary is defined as a closed line *separating some origin vertex $x_0$ from infinity* in pointed maps of infinite size. This latter, more restricted, regime may be recovered in our framework by sending first $k\to \infty$ with $d$ kept finite, and only then letting eventually $d\to \infty$. As we shall discuss, the results obtained for this latter order of limits match precisely those obtained by taking the limit $u\to 0$ of our results and they may thus be considered as particular instances of our more general laws for arbitrary $u$. To be precise, we *observe* that, for all the observables $\{\cdot\}_{d}$ depending on $d$ that we consider, we have the equivalence $$\begin{split}
&\lim_{u\to 0} \Big( \lim_{k\to \infty} P_{k}(\{\cdot\}_{k\, u})\Big)=\lim_{d\to \infty}\Big( \lim_{k\to \infty} P_{k}(\{\cdot\}_{d})\Big)\ ,\\
&\lim_{u\to 0} \Big( \lim_{k\to \infty} E_{k}(\{\cdot\}_{k\, u})\Big)=\lim_{d\to \infty}\Big( \lim_{k\to \infty} E_{k}(\{\cdot\}_{d})\Big)\ .\
\end{split}
\label{eq:utozero}$$ This equivalence is not a surprise since the limit $u\to 0$ describes precisely situations where the distance $d$ does not scale with $k$. We have however no rigorous argument to state that the above identity (based on an inversion of limits) should hold in all generality for any observable $\{\cdot\}_{d}$[^3] .
![A plot of the probability $p^{\rm out}(u)$ (in red) and the complementary probability $p^{\rm in}(u)$ (in blue) as a function of $u=d/k$, as given by .[]{data-label="fig:probasoutin"}](probasoutin.pdf){width="6.5cm"}
.3cm Our first result is an expression, for a given $u$, of the probability that a randomly picked $k$-pointed-rooted map be in the out- or in the in-regime. We find: $$\begin{split}
&p^{\rm out}(u)\equiv \lim_{k\to \infty} P_k({\mathcal V}(k\, u)\ \hbox{finite})=\frac{1}{4} \left(4-7 u^6+3 u^7\right)\ ,\\
&p^{\rm in}(u)\equiv \lim_{k\to \infty} P_k({\mathcal V}(k\, u)\ \hbox{infinite})=\frac{1}{4} (7-3 u)\, u^6\ ,\\
\end{split}
\label{eq:poutin}$$ with of course $p^{\rm out}(u)+p^{\rm in}(u)=1$. Note that these probabilities involve no normalization factor and are the same for the three families (i) (ii) and (iii) that we considered. They are represented in figure \[fig:probasoutin\]. For $u\to 0$, we have $p^{\rm out}(u)\to 1$ and $p^{\rm in}(u)\to 0$ so that the map configuration is in the out-regime with probability $1$. This is a first manifestation of the equivalence above. Indeed, sending $k\to \infty$ first before letting $d$ be large ensures that the connected component containing $x_1$ (i.e. the complementary of the hull at distance $d$) has some infinite volume, hence the configuration necessarily lies in the out-regime. On the other hand, when $u\to 1$, we see that $p^{\rm out}(u)\to 0$ and $p^{\rm in}(u)\to 1$. This corresponds to situations where the vertex $x_1$ remains at a finite distance from the separating line at distance $d$, with $d$ becoming infinitely large. In such a situation, the connected part containing $x_1$ has a finite volume with probability $1$. This result may be explained heuristically as follows: a rough estimate of the probability $p^{\rm out}(1)$ is given by the ratio of the length of the line at distance $d$ separating $x_0$ from infinity by the length of the boundary of the ball of radius $d$ with origin $x_0$. Indeed, the first length measures the number of ways to place $x_1$ “just above"[^4] the line separating $x_0$ from infinity while the second length is an equivalent measure of the number of ways of placing $x_1$ anywhere just above a line at distance $d$. Since the first length typically grows like $d^2$ [@Krikun03; @Krikun05; @CLG14a; @CLG14b; @G16a; @Men16; @G16b] while the second length grows like $d^3$ (recall that random maps have fractal dimension $4$ [@AmWa95; @ChSc04]), the ratio vanishes as $1/d$ when $d\to \infty$ hence $p^{\rm out}(1)$ vanishes.
Probability density for the rescaled perimeter in the out- and in-regimes {#sec:pd}
-------------------------------------------------------------------------
Our second result concerns the probability density for the hull perimeter at distance $d=k\, u$ in the out- and in-regimes. For large $d$, ${\mathcal L}(d)$ scales as $d^2$ so a finite probability density is obtained for the *rescaled perimeter* $$L(d)\equiv \frac{{\mathcal L}(d)}{d^2}\ .$$ We define more precisely the probability densities $$\begin{split}
&D^{\rm out}(L,u)\equiv \lim_{k\to \infty} \frac{1}{dL} P^{\rm out}_{k,k\, u}\left(L\leq L(k\, u) < L+dL\right)\ ,\\
&D^{\rm in}(L,u)\equiv \lim_{k\to \infty} \frac{1}{dL} P^{\rm in}_{k,k\, u}\left(L\leq L(k\, u) < L+dL\right)\ ,
\end{split}$$ for which we find the following explicit expressions: $$\begin{split}
&\hskip -0.8cm D^{\rm out}(L,u)=
\frac{(1-u)^4}{2 c \sqrt{\pi } u} \\ &\hskip 0.2cm \times e^{-B X} \left(-2 \sqrt{X} ((X-10) X-2)+e^X \sqrt{\pi } X (X (2 X-5)+6) \left(1-\text{erf}\left(\sqrt{X}\right)\right)\right)\ ,\\
&\hskip -0.8cm D^{\rm in}(L,u)=
\frac{u^5}{2 c \sqrt{\pi } (1-u)^2}\\&\hskip 0.2cm \times
e^{-B X} (B X+2) \left(2 \sqrt{X} (X+1)-e^X \sqrt{\pi } X (2 X+3) \left(1-\text{erf}\left(\sqrt{X}\right)\right)\right)\ ,\\
&\hskip -0.8cm \hbox{where}\ X\equiv X(L,u)=\frac{u^2}{(1-u)^2}\frac{L}{c}\ , \qquad B\equiv B(u)=\frac{(1-u)^2}{u^2}\ .\\
\label{eq:probainoutL}
\end{split}$$ Here $c$ is a normalization factor given in cases (i), (ii) and (iii) respectively by: $$\hbox{(i):}\ c=\frac{1}{3}\ , \qquad \hbox{(ii):}\ c=\frac{1}{2}\ , \qquad \hbox{(iii):}\ c=\frac{1}{4}\ .
\label{eq:cval}$$ Note that by definition, we have the normalizations $$\int_0^\infty D^{\rm out}(L,u)\, dL= p^{\rm out}(u)\ , \qquad \int_0^\infty D^{\rm in}(L,u)\, dL= p^{\rm in}(u)\ ,$$ a result which may be checked directly from the explicit expressions and . Note also that the ratio $D^{\rm out}(L,u)/p^{\rm out}(u)$ (resp. $D^{\rm in}(L,u)/p^{\rm in}(u)$) denotes, at fixed $u=k/d$, the probability density for the rescaled perimeter ${\mathcal L}(d)/d^2$ for map configurations *conditioned to be in the out-regime* (resp. in the in-regime), with an integral over $L$ now normalized to $1$.
![The conditional probability density $D^{\rm out}(L,u)/p^{\rm out}(u)$ as a function of $L$ for increasing values of $u$ (following the arrow) and its $u\to 0$ and $u\to 1$ limits, as given by . []{data-label="fig:probaoutL"}](probaoutL.pdf){width="9cm"}
.3cm Let us now analyze these latter conditional probability densities in more details. Let us first assume that the map configuration lies in the out-regime: the conditional probability density $D^{\rm out}(L,u)/p^{\rm out}(u)$, as displayed in figure \[fig:probaoutL\], varies for increasing $u$ between its $u\to 0$ and $u\to 1$ limits given, from the general expression , by $$\begin{split}
&\lim_{u\to 0} \frac{D^{\rm out}(L,u)}{p^{\rm out}(u)}=2\, \sqrt{L}\, \frac{e^{-\frac{L}{c}}}{c^{3/2} \sqrt{\pi }}\ ,\\
&\lim_{u\to 1} \frac{D^{\rm out}(L,u)}{p^{\rm out}(u)}=\frac{4}{3}(\sqrt{L})^3\frac{e^{-\frac{L}{c}}}{c^{5/2} \sqrt{\pi }}\ .\\
\end{split}
\label{eq:uzeroone}$$ It is easy to verify that the $u\to 0$ expression above reproduces precisely the result obtained by first sending $k\to \infty$, and then $d\to\infty$, in agreement with the announced equivalence . As expected, this expression therefore matches that of Krikun [@Krikun03; @Krikun05] and of Curien and Le Gall [@CLG14a; @CLG14b] concerning the probability density for the length of the line at distance $d$ separating some origin $x_0$ from infinity in large pointed maps of the family at hand (with possibly different values of $c$ due to inequivalent prescriptions for the definition of the separating line). Note that the requirement that the configuration be in the out-regime is actually not constraining for $u\to 0$ since $p^{\rm out}(0)=1$.
For $u\to 1$, the requirement to be in the out-regime restricts the set of configurations to those where we have chosen $x_1$ in the vicinity (i.e. just above) the line separating $x_0$ from infinity (so that the domain in which $x_1$ lies is infinite). As just discussed, this line has a length $L\, d^2$ with density probability $2\, \sqrt{L}\, e^{-\frac{L}{c}}/(c^{3/2} \sqrt{\pi })$ while the number of choices for $x_1$ is (for fixed $d$) proportional to $L$. The conditional probability density for $L(d)$ in the in-regime is thus expected to be $$\frac{L\times 2\, \sqrt{L}\, \frac{e^{-\frac{L}{c}}}{c^{3/2} \sqrt{\pi }}}{\int_0^\infty L\times 2\, \sqrt{L}\, \frac{e^{-\frac{L}{c}}}{c^{3/2} \sqrt{\pi }}\, dL}
= \frac{4}{3}(\sqrt{L})^3\frac{e^{-\frac{L}{c}}}{c^{5/2} \sqrt{\pi }}\ ,$$ and this is precisely the result obtained above.
![The conditional probability density $D^{\rm in}(L,u)/p^{\rm in}(u)$ as a function of $L$ for increasing values of $u$ (following the arrow) and its $u\to 0$ limit, as given by []{data-label="fig:probainL"}](probainL.pdf){width="10cm"}
Let us now assume that the map configuration lies in the in-regime and discuss the corresponding conditional probability density for $L(d)$. As displayed in figure \[fig:probainL\], $D^{\rm in}(L,u)/p^{\rm in}(u)$ varies for increasing $u$ between its $u\to 0$ limit given, from the general expression , by $$\lim_{u\to 0} \frac{D^{\rm in}(L,u)}{p^{\rm in}(u)}=\frac{4}{7}\, \sqrt{L}\, (2 c+L)\, \frac{e^{-\frac{L}{c}}}{ c^{5/2} \sqrt{\pi }}
\label{eq:uzeroin}$$ and a degenerate $u\to 1$ limit where only the rescaled length $L=0$ is selected. Recall that $p^{\rm in}(u)\to 0$ for $u\to 0$ and the limiting law just above for $u\to 0$ therefore describes a very restricted set of configurations where the connected domain containing $x_1$, although $k$ becomes arbitrary larger than $d$, remains of finite volume. As for the $u\to 1$ limit, the fact that the probability density concentrates around $L=0$ means that ${\mathcal L}(d)$ scales less rapidly than $d^2$ in this limit and that some new appropriate rescaling is required. As already discussed in [@G16a], a non-trivial law is in fact obtained by switching to the variable $X$ in , i.e. considering the probability density for the rescaled length $$X(k,d)\equiv \frac{{\mathcal L}(d)}{c\, (k^2-d^2)}=\frac{u^2}{c\, (1-u)^2}\ L(d) \ ,$$ where the coefficient $c$ is arbitrarily included in the definition of $X(k,d)$ so as to have the same limiting law for the three map families (i), (ii) and (iii). Setting $B=(1-u)^2/u^2$ as in , so that $L(d)=c\ B\, X(k,d)$, the probability density for $X(k,k\, u)$ is given for $u\to 1$ by $$\lim_{u\to 1} c\, B\, \frac{D^{\rm in}(c\, B\, X,u)}{p^{\rm in}(u)}=\frac{2 \sqrt{X} (X+1)-e^X \sqrt{\pi } X (2 X+3) \left(1-\text{erf}\left(\sqrt{X}\right)\right)}{\sqrt{\pi }}\ .$$ This result matches that of [@G16a] found for a statistics where the out- and in-regimes are not discriminated, as it should since, for $u\to1$, the requirement to be in the in-regime is not constraining ($p^{\rm in}(1)=1$). The probability density for $X(k,k\, u)$ for increasing values of $u$ and its universal limit above when $u\to 1$ are displayed in figure \[fig:probainLrescaled\].
![The conditional probability density in the in-regime for the variable $X= L/(c B)$, at $u=1/8,2/8,3/8,\cdots$ and in the limit $u\to 1$.[]{data-label="fig:probainLrescaled"}](probainLrescaled.pdf){width="10cm"}
.3cm It is interesting to measure the relative contribution of the out- and in-regimes to the “total" probability density for the rescaled length $L(d)$, i.e. the probability density obtained irrespectively of whether ${\mathcal V}(d)$ is finite or not, namely $$D(L,u)\equiv \lim_{k\to \infty} \frac{1}{dL} P_{k}\left(L\leq L(k\, u) < L+dL\right)=
D^{\rm out}(L,u)+D^{\rm in}(L,u)\ .$$
![ The relative contribution of the probability densities $D^{\rm out}(L,u)$ (in red) and $D^{\rm in}(L,u)$ (in blue) to the total probability density $D(L,u)=D^{\rm out}(L,u)+D^{\rm in}(L,u)$ (in black) as a function of $L$ for the indicated four values of $u$. []{data-label="fig:pLtot"}](pLtot.pdf){width="13cm"}
The reader will easily check that the expression for $D(L,u)$ resulting from the explicit forms matches precisely the expression for $D(L,u)$ found in [@G16a], as it should. We have represented in figure \[fig:pLtot\] the probability density $D(L,u)$ for various values of $u$ as well as its two components $D^{\rm out}(L,u)$ and $D^{\rm in}(L,u)$. As expected, $D(L,u)$ is dominated by the contribution of the out-regime at small enough $u$ (in practice up to $u\sim 1/2$) and by starts feeling the in-regime contribution when $u$ approaches $1$. This latter contribution moreover dominates the $u\to 1$ limit for small $L$. In particular, the appearance in $D(L,u)$ of a peak around $L=0$ when $u$ is large enough, which was observed in [@G16a] but remained quite mysterious is simply explained by the domination of the in-regime for $u\to 1$. No such peak ever appears in the contribution $D^{\rm out}(L,u)$ of the out-regime.
From the laws , we may also compare the expectation value of $L(d)$ in the out- and in-regime to that obtained whithout conditioning: we have respectively $$\begin{split}
&\hskip -.5cm \lim_{k\to \infty} E_{k}\left(L(k\, u)\Big|{\mathcal V}(k\, u)\ \hbox{finite}\right)
=\frac{\lim
\limits_{k\to \infty}E_{k,k\, u}^{\rm out}\big(L(k\, u)\big)}{p^{\rm out}(u)}=\frac{3 c \left(4\!+\!4u\!-\!21 u^6\!+\!17 u^7\!-\!4 u^8\right)}{2 \left(4\!-\!7 u^6\!+\!3 u^7\right)}\ ,\\
&\hskip -.5cm\lim_{k\to \infty} E_{k}\left(L(k\, u)\Big|{\mathcal V}(k\, u)\ \hbox{infinite}\right)
=\frac{\lim\limits_{k\to \infty}E_{k,k\, u}^{\rm in}\big(L(k\, u)\big)}{p^{\rm in}(u)}=\frac{3 c (9\!-\!4 u) (1\!-\!u)}{2 (7\!-\!3 u)}\ , \\
&\hskip -.5cm\lim_{k\to \infty} E_{k}\left(L(k\, u)\right)
=p^{\rm out}(u)\ \frac{3 c \left(4+4u-21 u^6+17 u^7-4 u^8\right)}{2 \left(4-7 u^6+3 u^7\right)}+p^{\rm in}(u)\ \frac{3 c (9-4 u) (1-u)}{2 (7-3 u)}\\
& \hskip 1.9cm =\frac{3}{2} c \left(1+u-3u^6+u^7\right)\ ,\\
\end{split}
\label{eq:expectperim}$$ where the last expression matches the result of [@G16a].
To end this section, let us discuss the probability $\pi^{\rm out}(L,u)$ (resp. $\pi^{\rm in}(L,u)$ to be in the out-regime (resp. in the in-regime), knowing that the rescaled length $L(d)$ is equal to $L$ (with as before $u=d/k fixed$), namely $$\pi^{\rm out}(L,u)=\lim_{k\to \infty} P_{k}\left({\mathcal V}(k\, u)\ \hbox{finite}\Big| L(k\, u)=L\right)
= \frac{D^{\rm out}(L,u)}{D(L,u)}=1-\pi^{\rm in}(L,u)\ .$$ We have plotted in figure \[fig:outorin\] the quantities $\pi^{\rm out}(L,u)$ and $\pi^{\rm in}(L,u)$ as a function of $L$ for various values of $u$. For $u\to 0$, we have $\pi^{\rm out}(L,0)=1$ and $\pi^{\rm in}(L,0)=0$ irrespectively of $L$. For $u\to 1$, we have the limiting expression: $$\pi^{\rm out}(L,1)=\frac{28 L^3}{6 c^3+3 L c^2+28 L^3}=1-\pi^{\rm in}(L,1)\ .$$
![The probabilities $\pi^{\rm out}(L,u)$ (in red) and $\pi^{\rm in}(L,u)$ (in blue) to be in the out- or in the in-regime, knowing the value $L$ of the rescaled perimeter $L(k\, u)$ for fixed $u$ and in the limit $k\to \infty$. This probabilities are represented as a function of $L$ for the indicated values of $u$.[]{data-label="fig:outorin"}](outorin.pdf){width="12cm"}
Figure \[fig:Uoutorin\] displays the same probabilities $\pi^{\rm out}(L,u)$ and $\pi^{\rm in}(L,u)$, now as a function of $u$ for various values of $L$. For $L\to 0$, we have the limiting expression $$\pi^{\rm out}(0,u)=\frac{(1-u)^6}{\left(1-2u+2 u^2\right) \left(1-4u+5 u^2-2u^3+u^4\right)}=1-\pi^{\rm in}(0,u)$$ (note the remarkable symmetry $\pi^{\rm out}(0,u)=\pi^{\rm in}(0,1-u)$). For $L\to \infty$, $\pi^{\rm out}(L,u)$ tends to $1$ and $\pi^{\rm in}(L,u)$ to $0$, irrespectively of $u$.
![The probabilities $\pi^{\rm out}(L,u)$ (in red) and $\pi^{\rm in}(L,u)$ (in blue) to be in the out- or in the in-regime, knowing the value $L$ of the rescaled perimeter $L(k\, u)$ for fixed $u$ and in the limit $k\to \infty$. This probabilities are represented as a function of $u$ for the indicated values of $L$.[]{data-label="fig:Uoutorin"}](Uoutorin.pdf){width="12cm"}
Joint law for the rescaled perimeter and volume in the out-regime {#sec:jointlaw}
-----------------------------------------------------------------
Our third result concerns the joint law for the hull perimeter ${\mathcal L}(d)$ and the hull volume ${\mathcal V}(d)$. Of course such law is non-trivial only if the hull volume is finite, i.e. if we condition the configurations to be in the out-regime. From now on, all our results will thus be *conditioned to be in the out-regime*. For Large $d$, ${\mathcal V}(d)$ scale as $d^4$ and we therefore introduce the rescaled volume $$V(d)\equiv \frac{{\mathcal V}(d)}{d^4}\ .$$ Our main result is the following expectation value $$\begin{split}
&
\lim_{k\to \infty} E_{k}\left(e^{-\sigma\, V(k\, u)-\tau\, L(k\, u)}\Big|{\mathcal V}(k\, u)\ \hbox{finite}\right)
=\frac{(1-u)^6}{u^3\, p^{\rm out}(u)} \times \frac{(f \sigma )^{3/4} \cosh \left(\frac{1}{2}(f \sigma)^{1/4}\right)}{8 \sinh^3 \left(\frac{1}{2}(f \sigma)^{1/4}\right)}
\\ & \hskip 10.cm \times M\big(\mu(\sigma,\tau,u)\big) \ ,\\
&\\
& \hbox{where}\ M(\mu)= \frac{1}{\mu ^4}\left(3 \mu ^2-5 \mu +6+\frac{4 \mu ^5+16 \mu ^4-7 \mu ^2-40 \mu -24}{4 (1+\mu )^{5/2}}\right)\\
&\hbox{and}\ \mu(\sigma,\tau,u)=\frac{(1-u)^2}{u^2}\, \left(
c\, \tau\, +\frac{\sqrt{f \sigma }}{4}\, \left(\coth^2 \left(\frac{1}{2}(f \sigma)^{1/4}\right)-\frac{2}{3}\right)\right)-1\ ,\\
\end{split}
\label{eq:expsigmatau}$$ with $p^{\rm out}(u)$ as in and where $f$ is a normalization factor given in cases (i), (ii) and (iii) respectively by $$\hbox{(i):}\ f=36\ , \qquad \hbox{(ii):}\ f=192\ , \qquad \hbox{(iii):}\ f=16\ .
\label{eq:fval}$$ Setting $\tau=0$ and expanding at first order in $\sigma$, we immediately deduce that, in particular $$\hskip -.8cm \lim_{k\to \infty} E_{k}\left(V(k\, u)\Big|{\mathcal V}(k\, u)\ \hbox{finite}\right)=\frac{f }{480}\frac{\left(20+12 u-77 u^6+57 u^7-12 u^8\right)}{ \left(4-7 u^6+3 u^7\right)}\ ,$$ a quantity which increases from $f/96$ at $u=0$ to $7f/480$ at $u=1$. .3cm The expectation value above has a simple limit when $u\to 0$, namely $$\begin{split}
& \hskip -1.2cm \lim_{u\to 0}\Big( \lim_{k\to \infty} E_{k}\left(e^{-\sigma\, V(k\, u)-\tau\, L(k\, u)}\right)\Big)\\
& \hskip 3.cm =\frac{(f \sigma)^{3 /4}\cosh\left(\frac{1}{2}(f \sigma)^{1/4}\right)}{8 \sinh^3\left(\frac{1}{2}(f \sigma)^{1/4}\right)\left(c\, \tau\, +\frac{\sqrt{f \sigma }}{4}\, \left(\coth^2 \left(\frac{1}{2}(f \sigma)^{1/4}\right)-\frac{2}{3}\right)\right)^{3/2}}\\
\end{split}$$ (note that the condition that ${\mathcal V}(d)$ is finite is automatically satisfied in the limit $u\to 0$ since $p^{\rm out}(0)=1$). For $\tau=0$, this expression simplifies into $$\lim_{u\to 0}\Big( \lim_{k\to \infty} E_{k}\left(e^{-\sigma\, V(k\, u)}\right)\Big)=\frac{\cosh\left(\frac{1}{2}(f \sigma)^{1/4}\right)}{ \sinh^3\left(\frac{1}{2}(f \sigma)^{1/4}\right)\left(\coth^2 \left(\frac{1}{2}(f \sigma)^{1/4}\right)-\frac{2}{3}\right)^{3/2}}$$ and we recover here a result by Curien and Le Gall [@CLG14b][^5], in agreement with the equivalence principle . When $u\to 1$, we get another interesting limit $$\begin{split}
& \hskip -1.2cm \lim_{k\to \infty} E_{k}\left(e^{-\sigma\, V(k)-\tau\, L(k)}\Big|{\mathcal V}(k)\ \hbox{finite}\right)\\
& \hskip 3.cm
=\frac{(f \sigma)^{3 /4}\cosh\left(\frac{1}{2}(f \sigma)^{1/4}\right)}{8 \sinh^3\left(\frac{1}{2}(f \sigma)^{1/4}\right)\left(c\, \tau\, +\frac{\sqrt{f \sigma }}{4}\, \left(\coth^2 \left(\frac{1}{2}(f \sigma)^{1/4}\right)-\frac{2}{3}\right)\right)^{5/2}}\ .\\
\end{split}$$ .3cm Performing an inverse Laplace transform on the variable $\tau$, we may extract from the expectation value of $e^{-\sigma\, V(d)}$ knowing the value $L$ of $L(d)$ in the out-regime. We find (see Section \[sec:expljointlaw\] for details) that $$\begin{split}
&\hskip -1.2cm \lim_{k\to \infty} E_{k}\left(e^{-\sigma\, V(k\, u)}\Big|{\mathcal V}(k\, u)\ \hbox{finite and}\ L(k\, u)
=L\right) \\
& =\frac{1}{8} e^{-\frac{L}{c} \left(\frac{\sqrt{f \sigma }}{4} \left(\coth ^2\left(\frac{1}{2}(f \sigma)^{1/4}\right)-\frac{2}{3}\right)-1\right)} (f \sigma )^{3/4}
\frac{ \cosh \left(\frac{1}{2}(f \sigma )^{1/4}\right)}{ \sinh^3\left(\frac{1}{2}(f \sigma )^{1/4}\right)}\ .\\
\end{split}
\label{eq:VcondL}$$ Note that this quantity turns out to be *independent of $u$* and is thus equal to its limit for $u\to 0$. In agreement with the equivalence , our result thus reproduces, *now for any $u$*, the expression found by Ménard in Ref. [@Men16] in a limit where $k\to \infty$ before $d$ becomes large.
We have in particular $$\lim_{k\to \infty} E_{k}\left(V(k\, u)\Big|{\mathcal V}(k\, u)\ \hbox{finite and}\ L(k\, u)=L\right)
=\frac{f (c+L)}{240 c}$$ independently of $u$. The fact that the law for the rescaled volume $V(d)$, knowing the rescaled perimeter $L(d)$, is independent of $u$ is not so surprising. Indeed, $u$ measures the distance $k=d/u$ from the origin at which the marked vertex $x_1$ lies. Once the perimeter ${\mathcal L}(d)$ is fixed, the hull, whenever finite, is insensitive to the position of the second marked vertex. The law for its volume ${\mathcal V}(d)$ depends only on $d$ and ${\mathcal L}(d)$, and, by simple scaling, it translates into a law for the rescaled volume $V(d)$ depending on the rescaled perimeter $L(d)$ only. Note that, on the other hand, fixing the hull perimeter ${\mathcal L}(d)$ has some influence on the possible choices for the position of $x_1$ as a function of its distance $k=d/u$ from the origin $x_0$. This in return explains why the law for $L(d)$ and consequently that for $V(d)$ in the out-regime both depend on $u$ for fixed $k$, as displayed in .
Derivation of the results: the strategy {#sec:strategy}
=======================================
Let us now come to the derivation of our results and explain the strategy behind our calculations. To simplify the discussion, we will focus here on the family (i) of quadrangulations. The cases (ii) of triangulations and that (iii) of Eulerian triangulations are amenable to exactly the same type of treatment and we will briefly discuss them in Section \[sec:Other\] below.
Generating functions {#sec:genfunc}
--------------------
The main ingredient is the generating function $G(k,d,g,h,\alpha)$ of planar $k$-pointed-rooted quadrangulations, enumerated with a weight $$g^{N-{\mathcal V}(d)}\ h^{{\mathcal V}(d)}\ \alpha^{{\mathcal L}(d)}\ ,$$ where $N$ is the total number of faces, and ${\mathcal L}(d)$ and ${\mathcal V}(d)$ are respectively the perimeter and volume of the hull at distance $d$ (note that ${\mathcal V}(d)\leq N$ by definition and we assume $k\geq 3$ and $2\leq d\leq k-1$). To define precisely the hull at distance $d$, we use the construction discussed in [@G16a]. Then $G(k,d,g,h,\alpha)$ may be given an explicit expression as follows: we use for the weights $g$ and $h$ the parametrization $$g=\frac{x(1+x+x^2)}{(1+4x+x^2)^2}\ , \qquad h=\frac{y(1+y+y^2)}{(1+4y+y^2)^2}\ ,
\label{eq:gh}$$ with $x$ and $y$ real between $0$ and $1$ (so that the generating function is well-defined for real $g$ and $h$ in the range $0\leq g,h \leq 1/12$). We also introduce the quantity $$T_\infty(z)= \frac{z(1+4z+z^2)}{(1+z+z^2)^2}\ ,$$ where $z$ will be taken equal to $x$ or $y$ depending on the formula at hand. We have, from [@G16a], $$G(k,d,g,h,\alpha)= \underbrace{ \mathcal{K}\big( \mathcal{K}\big(\cdots \big( \mathcal{K}\big(}_{k-d\ \hbox{\scriptsize times}}\alpha^2\, T_{d}(y)\big)\big)\big)\big)- \underbrace{ \mathcal{K}\big( \mathcal{K}\big(\cdots \big( \mathcal{K}\big(}_{k-d\ \hbox{\scriptsize times}}\alpha^2\, T_{d-1}(y)\big)\big)\big)\big)\ ,
\label{eq:Gform}$$ where $T_d(y)$ is defined by $$T_d(y)= T_\infty(y)\, \frac{(1-y^{d-1})(1-y^{d+4})}{(1-y^{d+1})(1-y^{d+2})}\ ,$$ and ${\mathcal K}\equiv \mathcal{K}(x)$ is an operator (depending on $x$ only), which satisfies the relation (which fully determines it): $$\mathcal{K}\left(
T_\infty(x)\, \frac{(1-\lambda\, x^{-1})(1-\lambda\, x^{4})}{(1-\lambda\, x)(1-\lambda\, x^{2})}\right)=T_\infty(x)\, \frac{(1-\lambda)(1-\lambda\, x^{5})}{(1-\lambda\, x^{2})(1-\lambda\, x^{3})}
\label{eq:propK}$$ for any arbitrary[^6] $\lambda$ .
![A schematic picture of the bijection between a $k$-pointed-rooted planar quadrangulation (left) and a $k$-slice (right), as obtained by cutting the quadrangulation along the leftmost shortest path from $x_1$ to $x_0$ (taking the root-edge of the map as first step). The light blue and light gray domains are supposedly filled with faces of degree four. Left: the separating line at distance $d$ (i.e. visiting alternately vertices at distance $d$ and $d-1$ from $x_0$) delimits the hull at distance $d$ (top part in light blue). Right: the image of this line connects the right- and left-boundaries of the $k$-slice and delimits an upper part containing $x_0$ (in light blue), which is the image of the hull, from a lower part containing $x_1$.[]{data-label="fig:dividing"}](dividing.pdf){width="12cm"}
![A schematic picture of the successive decompositions of a $k$-slice obtained by cutting along $(k-d)$ successive separating lines at respective distance $d$, $d+1$, $\cdots$, $k-1$ from $x_0$ and, for the $m$-th such line ($1\le m\leq k-d$), by cutting along the leftmost shortest paths to $x_0$ from the $\mathcal{L}(d+m-1)/2$ vertices on this line lying at distance $d+m-1$ from $x_0$ (see text). Here $k=d+2$ and for the level $m=1$, we represented only the leftmost shortest paths (in brown) lying within one particular sub-slice delimited by the leftmost shortest paths (in blue) at level $2$.[]{data-label="fig:constrhull"}](constrhull.pdf){width="12cm"}
The origin of the above formula can be found in Refs. [@G15b] and [@G16a]. We invite the reader to consult these references for details. Let us still briefly discuss the underlying decomposition of $k$-pointed-rooted quadrangulations on which the formula is based. As displayed in figure \[fig:dividing\], a $k$-pointed-rooted quadrangulation may be unwrapped into what is called a *$k$-slice* by cutting it along some particular path of length $k$, namely the leftmost among shortest paths (along edges of the map) from $x_1$ to $x_0$ having the root edge (i.e. the edge joining $x_1$ to its chosen neighbor at distance $k\!-\!1$ from $x_0$) as first step. The resulting $k$-slice has a a left- and a right-boundary of respective lengths $k$ and $k\!-\!1$ linking the image of the root-edge in the $k$-slice (the so-called slice *base*) to the image of $x_0$ (the so-called slice *apex*). The passage from the $k$-pointed-rooted quadrangulation to the $k$-slice is a bijection so $G(k,d,g,h,\alpha)$ may also be viewed as the generating function for $k$-slices with appropriate weights. The hull boundary at distance $d$ on the quadrangulation becomes a simple dividing line which links the right-and left-boundaries of the $k$-slice and separates it into an upper part, corresponding to the hull at distance $d$ in the original map and a complementary lower part. The dividing line visits alternately vertices at distance $d\!-\!1$ and $d$ from the apex, starting at the unique vertex along the right-boundary at distance $d\!-\!1$ from the apex and ending at the unique vertex along the left-boundary at distance $d\!-\!1$ from the apex. The upper part can be decomposed into a number of $d'$-slices with $d'\leq d$ by cutting it along the leftmost shortest paths to $x_0$ starting from all the vertices at distance $d$ from $x_0$ along the dividing line. Since these vertices represent half of the vertices along the dividing line, the number of $d'$-slices is precisely $\mathcal{L}(d)/2$ (note that $\mathcal{L}(d)$ is necessarily even for quadrangulations). Each of these slices is enumerated by a quantity $T_d(y)$ equal to the *generating function of $d'$-slices with $2\leq d'\leq d$ and a weight $h$ per face* (see [@G15b] for a precise definition), where $h$ and $y$ are related via . The expression for $T_d(y)$ is that given just above, as computed in [@G15b]. The juxtaposition of the $d'$-slices results in a total weight $(T_d(y))^{\mathcal{L}(d)/2}$ but, in order to impose that the maximum value of $d'$ for all the $d'$-slices is actually *exactly equal to* $d$, we must eventually subtract the weight of those configurations where all $d'$ would be less than $d\!-\!1$, namely $(T_{d-1}(y))^{\mathcal{L}(d)/2}$. Incorporating the desired weight $\alpha^{\mathcal{L}(d)}$, the generating function of the upper part eventually reads $$\left(\alpha^2\, T_d(y)\right)^{\mathcal{L}(d)/2}-\left(\alpha^2 T_{d-1}(y)\right)^{\mathcal{L}(d)/2}
\label{eq:upperpart}$$ for the contribution of those configurations having a fixed value $\mathcal{L}(d)$ of the hull perimeter at distance $d$. This explains why the expression of $G(k,d,g,h,\alpha)$ is a difference of two terms, corresponding to the action of the same operator $\mathcal{K}^{\circ^{(k-d)}}$ on $\alpha^2\, T_d(y)$ and $\alpha^2\, T_{d-1}(y)$ respectively.
To understand the origin of this operator, which incorporates the contribution of the lower part, we proceed by recursion upon drawing the images of the successive hull boundaries at distance $d\!+\!1$, $d\!+\!2, \cdots$ until we reach the hull boundary at distance $k$ which reduces to the line of length $\mathcal{L}(k)=2$ formed by the concatenation of root-edge of the $k$-slice and the first edge (starting from $x_1$) of the left-boundary (see figure \[fig:constrhull\] where $k=d+2$). Looking at the hull boundary at distance $d\!+\!1$, we perform the same decomposition of the part above this boundary as we did before, by splitting it into $\mathcal{L}(d+1)/2$ slices upon cutting along the leftmost shortest paths to the apex starting from all the boundary vertices at distance $d+1$ (blue lines in figure \[fig:constrhull\]). This creates $d''$-slices ${\mathcal S}_i$, $i=1,\cdots \mathcal{L}(d+1)/2$, each of them satisfying $d''\leq d+1$ and encompassing a number $\mathcal {L}_i$ of the previous $d'$-slices. These $\mathcal {L}_i$ $d'$-slices contribute a weight $\left(\alpha^2\, T_d(y)\right)^{\mathcal{L}_i/2}$ to the first term in (with $\mathcal{L}(d)=\sum_{i=1}^{\mathcal{L}(d+1)/2} \mathcal{L}_i$) while the generating function for the part of the slice ${\mathcal S}_i$ lying below the hull boundary at distance $d$, which *depends only on the (half-)length $\mathcal{L}_i/2$* may be written has $[T^{\mathcal{L}_i/2}] \mathcal {K}(T)$ for some operator $\mathcal {K}(T)$ depending on $g$ only (or equivalently on $x$ via ). This operator was computed in [@G15b] and, as explained in [@G16a], satisfies the property above. Summing over all values of $\mathcal{L}(d)$, hence on all values of $\mathcal{L}_i$, each of the $d''$-slices contributes a weight $$\mathcal{K}\left(\alpha^2\, T_d(y)\right)$$ to the sum over $\mathcal{L}(d)$ of the first term in . Taking into account the $\mathcal{L}(d+1)/2$ $d''$-slices, we end up with a contribution $$\left(\mathcal{K}\left(\alpha^2\, T_d(y)\right)\right)^{\mathcal{L}(d+1)/2}-\left(\mathcal{K}\left(\alpha^2 T_{d-1}(y)\right)\right)^{\mathcal{L}(d+1)/2}$$ for the part above the hull-boundary at distance $d+1$ of those configurations with a fixed value $\mathcal{L}(d+1)$ of the hull perimeter at distance $d+1$. Repeating the argument $k-d$ times immediately yields the desired expression since $\mathcal{L}(k)=2$ by construction. .3cm In order to have a more tractable expression, we may now perform explicitly the $k\!-\!d$ iterations of the operator ${\mathcal K}$ in . This leads immediately to the more explicit formula $$\begin{split}
& \hskip -1.cm G(k,d,g,h,\alpha)= H\big(k-d,x,\alpha^2T_d(y)\big)-H\big(k-d,x,\alpha^2T_{d-1}(y)\big)\\
& \hbox{where}\ H(k,x,T)=T_\infty(x)\, \frac{\big(1-\lambda(x,T)\, x^{k-1}\big)\big(1-\lambda(x,T)\, x^{k+4}\big)}{\big(1-\lambda(x,T)\, x^{k+1}\big)\big(1-\lambda(x,T)\, x^{k+2}\big)} \\
\end{split}
\label{eq:Gexpr}$$ provided $\lambda(x,T)$ is defined through $$T_\infty(x)\, \frac{\big(1-\lambda(x,T)\, x^{-1}\big)\big(1-\lambda(x,T)\, x^{4}\big)}{\big(1-\lambda(x,T) \, x\big)\big(1-\lambda(x,T) \, x^{2}\big)}=T \ ,$$ namely -1.cm $$\begin{split}
&\\ &\\
&\hskip -1.cm \lambda(x,T)=\frac{T_\infty(x)(1\!+\!x^5)\!-x^2T (1+x)\!-\!\sqrt{\left(T_\infty(x)(1\!+\!x^5)\!-\!x^2T (1\!+\!x)\right)^2\!-\!4\, x^5(T_\infty(x)\!-\!T)^2}}
{2\, x^4\, (T_\infty(x)\!-\!T)}\ , \\
\end{split}
\label{eq:lambdaexpr}$$ These latest expressions and will be our starting point for explicit calculations. .3cm A last quantity of interest is the generating function of $F(k,g)$ of planar $k$-pointed-rooted quadrangulations with a weight $g$ per face. We have clearly $F(k,g)=G(k,d,g,g,1)$ for any $d\leq k-1$ and we easily obtain from the above formulas that $\lambda\big(x,T_d(x)\big)=x^d$ so that $$F(k,g)=T_\infty(x)\left( \frac{(1-x^{k-1})(1-x^{k+4})}{(1-x^{k+1})(1-x^{k+2})}-\frac{(1-x^{k-2})(1-x^{k+3})}{(1-x^{k})(1-x^{k+1})}\right)\ .$$
Sending ${\boldsymbol N\to \infty}$: the out- and in-regimes {#sec:extract}
------------------------------------------------------------
Let us now explain how we can extract from the above generating functions results on the $N\to \infty$ limit, imposing that the configurations are either in the out- or the in-regime. To simplify the notations, let us omit for a while the dependence of $G(k,d,g,h,\alpha)$ in $\alpha$, $k$ and $d$ and write $G(k,d,g,h,\alpha)=G(g,h)$, as well as $N-{\mathcal V}(d)=n_1$ and ${\mathcal V(d)}=n_2$. We also denote by $G_{n_1,n_2}$ the coefficient $[g^{n_1}h^{n_2}]G(g,h)$. We are then interested in the large $N$ limit of the quantity $$\sum_{n_1,n_2\atop n_1+n_2=N} G_{n_1,n_2}\ ,
\label{eq:sumn1n2}$$ which we wish to extract from the knowledge of the generating function $G(g,h)$. As mentioned earlier, we also assume that when $N\to \infty$, the sum in is dominated by two contributions, that with $n_1\to \infty$, $n_2$ staying finite, which corresponds to what we called the out-regime, and that with $n_2\to \infty$, $n_1$ staying finite, which we called the in-regime, while the contribution where both $n_1$ and $n_2$ become infinite simultaneously is algebraically suppressed for large $N$. To describe the out-regime, we must consider the $n_1\to \infty$ behavior of $G_{n_1,n_2}$ which is encoded in the singular behavior of $G(g,h)$ when $g$ reaches some critical value $g^*$ (the radius of convergence of the series in $g$, possibly depending on $h$ and the other parameters). Similarly, properties of the in-regime are encoded in the singular behavior of $G(g,h)$ when $h$ reaches some critical value $h^*$ (possibly depending on $g$ and the other parameters). For the generating function $G(k,d,g,h,\alpha)$ of interest, the singularities appear when either $x\to 1$ or $y\to 1$, irrespectively of $k$, $d$ and $\alpha$, i.e., from , for $g\to 1/12$ or $h\to 1/12$. We therefore have $g^*=1/12$ (independently of $h$ and the other parameters) and $h^*=1/12$ (independently of $g$ and the other parameters). More precisely, we have expansions of the form $$\begin{split}
&G(g,h)=\mathfrak{g}_0(h)+\mathfrak{g}_2(h)(1-12 g)+\mathfrak{g}_3(h)(1-12 g)^{3/2}+O((1-12g)^2)\ ,\\
&G(g,h)=\tilde{\mathfrak{g}}_0(g)+\tilde{\mathfrak{g}}_2(g)(1-12 h)+\tilde{\mathfrak{g}}_3(g)(1-12 h)^{3/2}+O((1-12h)^2)\ .\\
\end{split}$$ (where all the functions implicitly depend on $k$, $d$ and $\alpha$). Note in particular that $G(1/12,h)=\mathfrak{g}_0(h)$ and $G(g,1/12)=\tilde{\mathfrak{g}}_0(g)$ are finite and that there are no square-root singularities.
Taking the term of order $h^{n_2}$ in the first expansion above, we deduce the singular part $$\left(\sum_{n_1} G_{n_1,n_2} g^{n_1}\right) \Big|_{\rm sing.}=[h^{n_2}] \mathfrak{g}_3(h) \times (1-12 g)^{3/2}$$ from which we deduce the large $n_1$ behavior $$G_{n_1,n_2} \underset{n_1\to \infty}{\sim}[h^{n_2}] \mathfrak{g}_3(h)\times \frac{3}{4} \frac{12^{n_1}}{\sqrt{\pi} n_1^{5/2}}$$ so that $$\sum_{n_2} G_{N-n_2,n_2} \underset{N \to \infty}{\sim}\frac{3}{4} \frac{12^{N}}{\sqrt{\pi} N^{5/2}} \sum_{n_2} [h^{n_2}] \mathfrak{g}_3(h) \times 12^{-n_2}
= \frac{3}{4} \frac{12^{N}}{\sqrt{\pi} N^{5/2}} \mathfrak{g}_3\left(\frac{1}{12}\right) \ .$$ This represents precisely *the contribution of the out-regime* to the large $N$ limit of the sum . If, more generally, we wish to control the volume $\mathcal{V}(d)$ in the out-regime, we may consider $$\sum_{n_2} G_{N-n_2,n_2}\, \rho^{n_2} \underset{N \to \infty}{\sim}\frac{3}{4} \frac{12^{N}}{\sqrt{\pi} N^{5/2}} \sum_{n_2} [h^{n_2}] \mathfrak{g}_3(h) \times \left(\frac{\rho}{12}\right)^{n_2}
= \frac{3}{4} \frac{12^{N}}{\sqrt{\pi} N^{5/2}} \mathfrak{g}_3\left(\frac{\rho}{12}\right) \ .$$ By a similar argument, the in-regime contribution to the sum behaves as $$\sum_{n_1} G_{n_1,N-n_1} \underset{N \to \infty}{\sim} \frac{3}{4} \frac{12^{N}}{\sqrt{\pi} N^{5/2}} \tilde{\mathfrak{g}}_3\left(\frac{1}{12}\right)\ .$$ For $F(g)\equiv F(k,g)$, we have an expansion of the form $$F(g)=\mathfrak{f}_0+\mathfrak{f}_2(1-12 g)+\mathfrak{f}_3(1-12 g)^{3/2}+O((1-12g)^2)$$ which yields the large $N$ estimate $$[g^N]F(g) \underset{N \to \infty}{\sim} \frac{3}{4} \frac{12^{N}}{\sqrt{\pi} N^{5/2}} \mathfrak{f}_3 \ .$$ By taking the appropriate ratios, we eventually deduce the large $N$ limit of the desired expectation values, namely $$E_{k,d}^{\rm out}\left(\rho^{\mathcal{V}(d)}\, \alpha^{\mathcal{L}(d)}\right)
= \frac{\mathfrak{g}_3\left(\frac{\rho}{12},k,d,\alpha\right)}{\mathfrak{f}_3(k)}\ ,
\label{eq:asympout}$$ where we re-introduced explicitly the dependence in $k$, $d$ and $\alpha$ of $\mathfrak{g}_3(h)\equiv \mathfrak{g}_3(h,k,d,\alpha)$ and $\mathfrak{f}_3\equiv \mathfrak{f}_3(k)$, and $$E_{k,d}^{\rm in}\left(\alpha^{\mathcal{L}(d)}\right)
= \frac{\tilde{\mathfrak{g}}_3\left(\frac{1}{12},k,d,\alpha\right)}{\mathfrak{f}_3(k)}
\label{eq:asympin}$$ (with the more explicit dependence $\tilde{\mathfrak{g}}_3(g)\equiv \tilde{\mathfrak{g}}_3(g,k,d,\alpha)$). This reduces our problem to estimating the quantities $\mathfrak{g}_3(h)$, $\tilde{\mathfrak{g}}_3(g)$ and $\mathfrak{f}_3$ from our explicit expressions for $G(k,d,g,h,\alpha)$ and $F(k,g)$.
Derivation of the results: explicit calculations {#sec:explicit}
================================================
Since the expression for $G(k,d,g,h,\alpha)$ is quite involved, explicit calculations may be difficult to perform in all generalities for finite $k$ and $d$ and some of our results will hold only in the limit of large $k$ and $d$. Still, the simplest questions may be solved exactly for finite $k$ and $d$: this is the case for the probability to be in the out- or the in-regime, as we discuss now.
Results at finite ${\boldsymbol k}$ and ${\boldsymbol d}$: the probability to be in the out- or in-regime {#sec:explprobs}
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If we wish to compute the probability to be in the out- or in-regime, we may set $\alpha=1$ and $\rho=1$ in and since we do not measure the hull perimeter nor the hull volume. More precisely, we have $$\begin{split}
&P_k(\mathcal{V}(d)\ \hbox{finite})=E_{k,d}^{\rm out}(1)= \frac{\mathfrak{g}_3\left(\frac{1}{12},k,d,1\right)}{\mathfrak{f}_3(k)}\ ,\\
&P_k(\mathcal{V}(d)\ \hbox{infinite})=E_{k,d}^{\rm in}(1)= \frac{\tilde{\mathfrak{g}}_3\left(\frac{1}{12},k,d,1\right)}{\mathfrak{f}_3(k)}\ .\\
\end{split}$$ To compute $\mathfrak{f}_3(k)$, we set $$g=\frac{1}{12}(1-\epsilon^4)\leftrightarrow \epsilon=(1-12 g)^{1/4}\ ,$$ and, from the corresponding small $\epsilon$ expansion of $x$, $$x=1-\sqrt{6}\, \epsilon +3\, \epsilon ^2-\frac{5}{2} \sqrt{\frac{3}{2}}\, \epsilon ^3+3\, \epsilon ^4-\frac{39}{16} \sqrt{\frac{3}{2}}\, \epsilon ^5+3\, \epsilon ^6-\frac{157}{64}
\sqrt{\frac{3}{2}} \, \epsilon ^7+3 \epsilon ^8-\cdots\ ,$$ we immediately get the small $\epsilon$ expansion of $F(k,g)$. As expected, we find terms of order $\epsilon^0$, $\epsilon^4=(1-12 g)$ and $\epsilon^6=(1-12 g)^{3/2}$ but no term of odd order in $\epsilon$ (as a consequence of the $x\to 1/x$ symmetry of all the formulas) and, more importantly, no term of order $\epsilon^2=(1-12 g)^{1/2}$. The coefficient of $\epsilon^6$ in the expansion yields: $$\mathfrak{f}_3(k)= \frac{4 \left(k^2+2 k-1\right) \left(5 k^4+20 k^3+27 k^2+14 k+4\right)}{35 k (k+1) (k+2)}\ .
\label{eq:f3}$$ To compute $\mathfrak{g}_3\left(1/12,k,d,1\right)$, we have to consider the expansion of $G(k,d,g,1/12,1)$ when $g\to 1/12$. Note that setting $h=1/12$ amounts to setting $y=1$, in which case $T_d(y)$ simplifies into $$T_d(1)=\frac{2}{3}\ \frac{(d-1)(d+4)}{(d+1)(d+2)}\ .$$ We may easily compute the singularity of the function $H(k,x,T)$ (as defined in and ) for $\epsilon\to 0$. Again, the leading singularity corresponds to the $\epsilon^6$ term and we find explicitly $$\begin{split}
&
\hskip -1.cm H(k,x,T)\Big|_{\rm sing.}=\mathfrak{h}_3\big(k,Y(T)\big) (1-12 g)^{3/2}\\
&\hskip -1.cm \hbox{with}\ \mathfrak{h}_3(k,Y)= \frac{k}{840\, Y \left((2 k+Y)^2-1\right)^2}\Big(
105 (k+Y)^8+420 \left(k^2-3\right) (k+Y)^6\\& -210 \left(k^4+6 k^2+49\right) (k+Y)^4-4 \left(75 k^6-567 k^4-1715 k^2-2273\right) (k+Y)^2\\
& -(k-5) (k-1) (k+1)
(k+5) \left(15 k^4+138 k^2-217\right)\Big)\\
& \hskip -1.cm \hbox{and}\ Y(T)=\sqrt{\frac{3 T-50}{3 T-2}}\ .
\\
\end{split}
\label{eq:Hsing}$$ Note that we have the particularly simple expression $$Y\big(T_d(1)\big)=2d+3$$ so that, from , $$\mathfrak{g}_3\left(\frac{1}{12},k,d,1\right)=\mathfrak{h}_3(k-d,2d+3)-\mathfrak{h}_3(k-d,2d+1)$$ and $$\begin{split}
&\hskip -1.cm P_k({\mathcal V}(d)\ \hbox{finite})=\frac{1}{\mathfrak{f}_3(k)}\left(\mathfrak{h}_3(k-d,2d+3)-\mathfrak{h}_3(k-d,2d+1)\right)\\
& \hskip -1.cm = \frac{1}{\mathfrak{f}_3(k)}\Bigg(
\frac{1}{105 (2d+3) (k+1)^2 (k+2)^2}\times \\ & \hskip .3cm\Big((2 d+3) (k-1) (k+1) (k+2) (k+4) \left(15 k^4+90 k^3+237 k^2+306 k+140\right)\\ &\hskip .3cm - (2 k+3) (d-1) (d+1) (d+2) (d+4) \left(15 d^4+90 d^3+237 d^2+306 d+140\right)\Big)\\
& -\frac{1}{105 (2 d+1) k^2 (k+1)^2}\times \\ &
\hskip .3cm\big((2 d+1) (k-2) k (k+1) (k+3) \left(15 k^4+30 k^3+57 k^2+42 k-4\right)\\ &\hskip .3cm -(2k+1)(d-2) d (d+1) (d+3) \left(15 d^4+30 d^3+57 d^2+42 d-4\right)\big)
\Bigg)\\
\end{split}$$ with $\mathfrak{f}_3(k)$ as in above.
Let us now compute the probability to be in the in-regime which requires the knowledge of $\tilde{\mathfrak{g}}_3\left(1/12,k,d,1\right)$. We thus have to consider the expansion of $G(k,d,1/12,h,1)$ when $h\to 1/12$. Note that setting $g=1/12$ now amounts to setting $x=1$[^7], in which case we find the simple expression $$H(k,1,T)=\frac{2}{3}\frac{\left(2 k+Y\left(T\right)\right)^2-25}{\left(2 k+Y\left(T\right)\right)^2-1}
\label{eq:HkoneT}$$ with $Y(T)$ as in . To compute the desired singularity, we now set $$h=\frac{1}{12}(1-\eta^4)\leftrightarrow \eta=(1-12 h)^{1/4}\ ,$$ so that $$y=1-\sqrt{6}\, \eta +3\, \eta ^2-\frac{5}{2} \sqrt{\frac{3}{2}}\, \eta ^3+3\, \eta ^4-\frac{39}{16} \sqrt{\frac{3}{2}}\, \eta ^5+3\,\eta ^6-\frac{157}{64}
\sqrt{\frac{3}{2}} \, \eta ^7+3 \eta ^8-\cdots\ .$$ We have the expansion $$\begin{split}
&\hskip -1.cm Y\big(T_d(y)\big)=(2d+3)-\frac{(d-1) (d+1) (d+2) (d+4) \left(9 d^2+27 d+10\right)}{30 (2 d+3)}\eta^4\\
& +
\frac{(d-1) (d+1) (d+2) (d+4) \left(15 d^4+90 d^3+237 d^2+306 d+140\right)}{210 (2 d+3)}\eta^6+\cdots \\
\end{split}
\label{eq:expTd}$$ which yields eventually $$\begin{split}
&
\hskip -1.cm H\big(k,1,T_d(y)\big)\Big|_{\rm sing.}=\tilde{\mathfrak{h}}_3(k,d) (1-12 h)^{3/2}\\
&\hskip -1.cm \hbox{with}\ \tilde{\mathfrak{h}}_3(k,d)= \frac{(d-1) (d+1) (d+2) (d+4) \left(15 d^4+90 d^3+237 d^2+306 d+140\right) (2 d+2 k+3)}{105 (2 d+3) (d+k+1)^2 (d+k+2)^2}\ .
\\
\end{split}$$ We end up with $$\begin{split}
&\hskip -1.cm P_k({\mathcal V}(d)\ \hbox{infinite})=\frac{1}{\mathfrak{f}_3(k)}\left(\tilde{\mathfrak{h}}_3(k-d,d)-\tilde{\mathfrak{h}}_3(k-d,d-1)\right)\\
& \hskip -1.cm = \frac{1}{\mathfrak{f}_3(k)}\Bigg(
\frac{(2 k+3)(d-1) (d+1) (d+2) (d+4) \left(15 d^4+90 d^3+237 d^2+306 d+140\right)}{105 (2 d+3) (k+1)^2 (k+2)^2}\Big)\\
& \hskip 1.cm-\frac{(2 k+1)(d-2) d (d+1) (d+3) \left(15 d^4+30 d^3+57 d^2+42 d-4\right) }{105 (2 d+1) k^2 (k+1)^2}
\Bigg)\ .\\
\end{split}$$ It is easily verified from their explicit expressions that $$P_k({\mathcal V}(d)\ \hbox{finite})+P_k({\mathcal V}(d)\ \hbox{infinite})=1$$ for any fixed $k$ and $d$, as expected. This corroborates the absence of some regime other than the out- and in-regimes and justifies a posteriori our statement that the contribution of configurations where both the hull and its complementary would have infinite volumes is negligible at large $N$. For $k\to\infty$ and $d\to \infty$ with $u=d/k$ fixed, we immediately obtain $$\begin{split}
& \lim_{k\to \infty} P_k({\mathcal V}(k\, u)\ \hbox{finite})=\frac{1}{4} \left(4-7 u^6+3 u^7\right)\ ,\\
&\lim_{k\to \infty} P_k({\mathcal V}(k\, u)\ \hbox{infinite})=\frac{1}{4} (7-3 u)\, u^6\ ,\\
\end{split}$$ which is precisely the announced result . Figure \[fig:probasoutinfinite\] shows a comparison between the limiting expressions $p^{\rm out}(u)$ and $p^{\rm in}(u)$ vs $u$ (as given by ) and the corresponding finite $k$ and $d$ expressions (as given above) $P_k({\mathcal V}(d)\ \hbox{finite})$ and $P_k({\mathcal V}(d)\ \hbox{infinite})$ vs $d/k$ for $k=50$ and $2\leq d\leq 49$.
![A comparison between the probability $P_k({\mathcal V}(d)\ \hbox{finite})$ (respectively $P_k({\mathcal V}(d)\ \hbox{infinite})$) vs $d/k$ for $k=50$ and $2\leq d\leq 49$ and its limiting expression $p^{\rm out}(u)$ (respectively $p^{\rm in}(u)$) vs $u$, as given by .[]{data-label="fig:probasoutinfinite"}](probasoutinfinite.pdf){width="7.5cm"}
.3cm Another quantity which may be easily computed for finite $k$ and $d$ is the expectation value of the perimeter in the out-regime, $E_{k}\left({\mathcal L}(d)\Big|{\mathcal V}(d)\ \hbox{finite}\right)$, as well as that in the in-regime, $E_{k}\left({\mathcal L}(d)\Big|{\mathcal V}(d)\ \hbox{infinite}\right)$. Details of the computation are discussed in Appendix A.
Law for the perimeter at large ${\boldsymbol k}$ and ${\boldsymbol d}$ in the out- and in-regimes {#sec:explperimlaw}
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To describe the statistics of the perimeter in the out-regime, we have to look at $G(k,d,g,h,\alpha)$ for arbitrary $\alpha$ and for $h=1/12$ (i.e. $y=1$). We consider here the large $k$ and $d$ limit by setting $d=k\, u$ (with $0\leq u\leq 1$) and letting $k\to \infty$. In this limit, ${\mathcal L}(d)$ growths like $d^2=(k\, u)^2$ and the large $k$ statistics of the perimeter is captured by setting $$\alpha=e^{-\frac{\tau}{(k\, u)^2}}$$ with $\tau$ finite. From and , we need the large $k$ behavior of $\mathfrak{h}_3(k,Y(T))$ for $T=\alpha^2 T_{d}(1)$ and $T=\alpha^2 T_{d-1}(1)$ which involves the associated expansions of $Y(T)$, namely $$\begin{split}
&Y\left(e^{-2 \frac{\tau}{(k\, u)^2}} T_{k\, u}(1)\right)= 2 \sqrt{\frac{3}{3+\tau}}\, k\, u+ 3 \left(\frac{3}{3+\tau}\right)^{3/2}+O\left(\frac{1}{k}\right)\ ,\\
&Y\left(e^{-2 \frac{\tau}{(k\, u)^2}} T_{k\, u-1}(1)\right)= 2 \sqrt{\frac{3}{3+\tau}}\, k\, u+ \left(\frac{3}{3+\tau}\right)^{3/2}+O\left(\frac{1}{k}\right)\ .\\
\end{split}$$ Using the expansion $$\mathfrak{f}_3=\frac{4}{7} k^3+O\left(k^2\right)\ ,$$ we obtain[^8] $$\begin{split}
& \hskip -1.cm \lim_{k\to \infty} E_{k,k\, u}^{\rm out}\left(e^{-\tau\, L(k\, u)}\right)\\
&=\lim_{k\to \infty}
\frac{\mathfrak{h}_3\left(k\!-\!k\, u,2 \sqrt{\frac{3}{3+\tau}}\, k\, u\!+\! 3 \left(\frac{3}{3+\tau}\right)^{3/2} \right)\!-\!\mathfrak{h}_3\left(k\!-\!k\, u,2 \sqrt{\frac{3}{3+\tau}}\, k\, u\!+\! \left(\frac{3}{3+\tau}\right)^{3/2}\right)}{\frac{4}{7} k^3}\\
&=\frac{(1-u)^6 }{u^3} M\big(\mu(\tau,u)\big) \\
&\hskip -1.cm \hbox{with}\ M(\mu)= \frac{1}{\mu ^4}\left(3 \mu ^2-5 \mu +6+\frac{4 \mu ^5+16 \mu ^4-7 \mu ^2-40 \mu -24}{4 (1+\mu )^{5/2}}\right)\\
& \hskip -1.cm \hbox{and}\ \mu(\tau,u)=\frac{(1-u)^2 }{u^2}\left(1+\frac{\tau}{3}\right)-1\ .
\end{split}
\label{eq:Laplaceperim}$$ Introducing the inverse Laplace transform of $M(\mu)$, i.e. the quantity $\check{M}(X)$ such that $\displaystyle{\int_0^\infty e^{-\mu\, X}\check{M}(X)= M(\mu)}$, this quantity is easily computed and reads $$\hskip -1.cm \check{M}(X)= \frac{e^{-X}}{2\sqrt{\pi}} \left(-2 \sqrt{X} ((X-10) X-2)+e^X \sqrt{\pi } X (X (2 X-5)+6) \left(1-\text{erf}\left(\sqrt{X}\right)\right)\right)\ .$$ From the linear relation $$\mu(\tau,u)=\frac{B(u)}{3}\, \tau+\big(B(u)-1\big)\ , \qquad B(u)=\frac{(1-u)^2 }{u^2}\ ,
\label{eq:muBu}$$ we immediately deduce, taking the inverse Laplace transform of , that $$\hskip -1.cm D^{\rm out}(L,u)= \frac{(1-u)^6 }{u^3} \frac{3}{B(u)}\, e^{(1-B(u)) X(u)} \check{M}(X(u))\ \hbox{with}\ X(u)=\frac{3 L}{B(u)}\ .$$ This is precisely the expression for $c=1/3$. .3cm To compute its counterpart $D^{\rm in}(L,u)$ in the in-regime, we use the explicit expression of $H(k,1,T)$ to derive the identity $$\begin{split}
&\hskip -1.cm H(k-k\, u,1,T(y))\Big|_{\rm sing.}=\frac{32 \, (\alpha_0+2 k (1-u)) \, \alpha_3}{\left(4 k^2 (1-u)^2+4 \alpha_0 \, k (1-u)+\alpha_0^2-1\right)^2} (1-12 h)^{3/2}\, \\
&\hskip -1.cm\hbox{whenever}\ Y\big(T(y)\big)= \alpha_0+\alpha_2 \eta^4+\alpha_3 \eta^6+\cdots \ ,\\
\end{split}$$ where, as before, $\eta=(1-12h)^{1/4}$. It implies that $$\begin{split}
& \hskip -1.cm \lim_{k\to \infty} E_{k,k\, u}^{\rm in}\left(e^{-\tau\, L(k\, u)}\right)\\
&=\lim_{k\to \infty}
\frac{1}{{\frac{4}{7} k^3}}\Bigg(\frac{32 (\alpha_0(\tau,k\, u)+2 k (1-u)) \alpha_3(\tau,k\, u)}{\left(4 k^2 (1-u)^2+4 \alpha_0(\tau,k\, u)\, k (1-u)+(\alpha_0(\tau,k\, u))^2-1\right)^2} \\
&\hskip 2cm - \frac{32 (\tilde{\alpha}_0(\tau,k\, u)+2 k (1-u)) \tilde{\alpha}_3(\tau,k\, u)}{\left(4 k^2 (1-u)^2+4 \tilde{\alpha}_0(\tau,k\, u)\, k (1-u)+(\tilde{\alpha}_0(\tau,k\, u))^2-1\right)^2} \Bigg)\\
&\hskip -1.cm\hbox{whenever}\ Y\left(e^{-2 \frac{\tau}{(k\, u)^2}} T_{k\, u}(y)\right)= \alpha_0(\tau,k\, u)+\alpha_2(\tau,k\, u) \eta^4+\alpha_3(\tau,k\, u) \eta^6+\cdots \ ,\\
&\hskip -1.cm\hbox{and}\ Y\left(e^{-2 \frac{\tau}{(k\, u)^2}} T_{k\, u-1}(y)\right)= \tilde{\alpha}_0(\tau,k\, u)+\tilde{\alpha}_2(\tau,k\, u) \eta^4+\tilde{\alpha}_3(\tau,k\, u) \eta^6+\cdots \ .\\
\end{split}
\label{eq:Laplaceperimbis}$$ Using the easily computed large $k$ expansions $$\begin{split}
&\hskip -1.cm \alpha_0(\tau,k\, u)= 2 \sqrt{\frac{3}{3+\tau}}\, (k\, u)+ 3 \left(\frac{3}{3+\tau}\right)^{3/2}+O\left(\frac{1}{k\, u}\right)\ ,\\
&\hskip -1.cm \tilde{\alpha}_0(\tau,k\, u)= 2 \sqrt{\frac{3}{3+\tau}}\, (k\, u)+ \left(\frac{3}{3+\tau}\right)^{3/2}+O\left(\frac{1}{k}\right)\ ,\\
&\hskip -1.cm \alpha_3(\tau,k\, u)= \left(\frac{3}{3+\tau}\right)^{3/2} \, \frac{(k\, u)^7}{28}+ 3 \, (21+4\tau) \left(\frac{3}{3+\tau}\right)^{5/2} \, \frac{(k\, u)^6}{168}+O\left((k\, u)^5\right)\ ,\\
&\hskip -1.cm \tilde{\alpha}_3(\tau,k\, u)= \left(\frac{3}{3+\tau}\right)^{3/2} \, \frac{(k\, u)^7}{28}+ (21+4\tau) \left(\frac{3}{3+\tau}\right)^{5/2} \, \frac{(k\, u)^6}{168}+O\left((k\, u)^5\right)\ ,\\
\end{split}$$ we deduce[^9] $$\begin{split}
& \hskip -1.cm \lim_{k\to \infty} E_{k,k\, u}^{\rm in}\left(e^{-\tau\, L(k\, u)}\right)= u^3 Q(\mu(\tau,u),B(u))\\
& \hskip -1.cm \hbox{with}\ Q(\mu,B)=\frac{1}{\mu^4}
\left(-\!3 \mu ^2\!-\!3 B \mu\!-\!4 \mu\!-\!6 B\!+\!\frac{4 \mu ^3\!+\!3 B \mu ^2\!+\!20 \mu ^2\!+\!24 B \mu\!+\!16 \mu\!+\!24 B}{4 \sqrt{1+\mu}}\right)
\\
\end{split}$$ and $\mu(\tau,u)$ and $B(u)$ as in . Introducing the inverse Laplace transform $\check{Q}(X,B)$ of $Q(\mu,B)$, easily computed to be $$\hskip -1.cm \check{Q}(X,B)=\frac{e^{-X}}{2\sqrt{\pi}} (B X+2) \left(2 \sqrt{X} (X+1)-e^X \sqrt{\pi } X (2 X+3) \left(1-\text{erf}\left(\sqrt{X}\right)\right)\right)\ ,$$ we immediately deduce, that $$\hskip -1.cm D^{\rm in}(L,u)= u^3 \frac{3}{B(u)}\, e^{(1-B(u)) X(u)} \check{Q}(X(u),B(u))\ \hbox{with}\ X(u)=\frac{3 L}{B(u)}\ .$$ This is precisely the expression for $c=1/3$.
Joint law for the volume and perimeter at large ${\boldsymbol k}$ and ${\boldsymbol d}$ in the out-regime {#sec:expljointlaw}
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We now wish to control, in addition to the perimeter, the volume of the hull. Of course, this is non-trivial only if we condition the map configurations to be in the out-regime where this volume is finite. We are now interested in $G(k,d,g,h,\alpha)$ for arbitrary $\alpha$ and arbitrary $h=\rho/12$ (with $0\leq \rho\leq 1$). We consider again only the large $k$ and $d$ limit with fixed $u=d/k$, a limit where ${\mathcal L}(d)$ growths like $d^2=(k\, u)^2$ while ${\mathcal V}(d)$ growths like $d^4=(k\, u)^4$. We therefore set $$\alpha=e^{-\frac{\tau}{(k\, u)^2}}\ , \qquad \rho=e^{-\frac{\sigma}{(k\, u)^4}}$$ with $\rho$ and $\sigma$ remaining finite. From the relation between $h$ and $y$, taking the form of $\rho$ above amounts to setting $$y=e^{-\sqrt{6}\frac{\sigma^{1/4}}{k\, u}+O\left(\frac{1}{(k\, u)^3}\right)}\ .$$ We then have the expansions $$\begin{split}
&\hskip -.8cmY\left(\alpha^2 T_{k\,u}(y)\right)= 2\frac{1-u}{\sqrt{1+\mu}}\, k +9\sqrt{6}\, \frac{(1-u)^3}{u^3\, (1+\mu)^{3/2}} \frac{(1+W)(2+W)\sigma^{3/4}}{W^3}+O\left(\frac{1}{k}\right) \\
&\hskip -.8cmY\left(\alpha^2 T_{k\,u-1}(y)\right)= 2\frac{1-u}{\sqrt{1+\mu}}\, k +3\sqrt{6}\, \frac{(1-u)^3}{u^3\, (1+\mu)^{3/2}} \frac{(1+W)(2+W)\sigma^{3/4}}{W^3}+O\left(\frac{1}{k}\right) \\
&\hskip -.8cm \hbox{with}\ \mu\equiv \mu(\sigma,\tau,u)=\frac{(1-u)^2}{u^2}\, \left(
\frac{\tau}{3}+\frac{3\sqrt{\sigma }}{2}\left(\coth^2\left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)-\frac{2}{3}\right)\right)-1\\
&\hskip -.8cm\hbox{and}\ W\equiv W(\sigma)=e^{\sqrt{6} \, \sigma^{1/4}}-1\\
\end{split}$$ so that, eventually, $$\begin{split}
& \hskip -1.cm \lim_{k\to \infty} E_{k,k\, u}^{\rm out}\left(e^{-\tau\, L(k\, u)-\sigma\, V(k\, u)}\right)\\
&=\lim_{k\to \infty}\frac{1}{\frac{4}{7} k^3}\Bigg(
\mathfrak{h}_3\left(k\!-\!k\, u,
2\frac{1-u}{\sqrt{1+\mu}}\, k +9\sqrt{6}\, \frac{(1-u)^3}{u^3\, (1+\mu)^{3/2}} \frac{(1+W)(2+W)\sigma^{3/4}}{W^3} \right)\\
&\hskip 1.5cm \!-\!\mathfrak{h}_3\left(k\!-\!k\, u,
2\frac{1-u}{\sqrt{1+\mu}}\, k +3\sqrt{6}\, \frac{(1-u)^3}{u^3\, (1+\mu)^{3/2}} \frac{(1+W)(2+W)\sigma^{3/4}}{W^3}\right)\Bigg) \\
&=3 \sqrt{6}\ \frac{(u-1)^6}{u^3} \frac{(W+1) (W+2)}{W^3} \sigma ^{3/4}\, M(\mu)\\
&= \left(\frac{3}{2}\right)^{3/2}\ \frac{(u-1)^6}{u^3}\frac{ \cosh \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}{\sinh^3 \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}\, \sigma ^{3/4}\, M(\mu)
\end{split}
\label{eq:tausigmader}$$ with $\mu=\mu(\sigma,\tau,u)$ and $W=W(\sigma)$ as above, and where the function $M(\mu)$ has the same expression as in . Normalizing by $p^{\rm out}(u)$, this yields precisely the announced expression with $c=1/3$ and $f=36$.
From the linear relation between $\mu$ and $\tau$ $$\begin{split}
&\hskip -1.cm \mu(\sigma,\tau,u)= B(u)\, \frac{\tau}{3}+\big(A(\sigma) B(u)-1\big)\ , \\
&\hskip -1.cm \hbox{with}\ B(u)=\frac{(1-u)^2 }{u^2}\ \ \hbox{and}\ \ A(\sigma)=\frac{3\sqrt{\sigma }}{2}\left(\coth^2\left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)-\frac{2}{3}\right)\ , \\
\end{split}$$ we immediately deduce, taking the inverse Laplace transform of , that $$\begin{split}
&\hskip -1.2cm \lim_{k\to \infty} E_{k}\left(e^{-\sigma\, V(k\, u)}\Big|{\mathcal V}(k\, u)\ \hbox{finite and}\ L(k\, u)
=L\right) \\
&= \frac{1}{D^{\rm out}(L,u)}\left(\frac{3}{2}\right)^{3/2}\ \frac{(u-1)^6}{u^3}\frac{ \cosh \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}{\sinh^3 \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}\, \sigma ^{3/4}\\
& \hskip 2.cm \times \frac{3}{B(u)}\, e^{(1-A(\sigma)\, B(u)) X(u)} \check{M}(X(u))\quad \hbox{with}\ X(u)=\frac{3 L}{B(u)}\\
& =\left(\frac{3}{2}\right)^{3/2} \frac{ \cosh \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}{\sinh^3 \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}\, \sigma ^{3/4}\, e^{\left(1-A(\sigma)\right)\, B(u) X(u)}\\
& =\left(\frac{3}{2}\right)^{3/2} \frac{ \cosh \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}{\sinh^3 \left(\sqrt{\frac{3}{2}}\, \sigma^{1/4}\right)}\, \sigma ^{3/4}\, e^{-3\, L\, \left(A(\sigma)-1\right)}\ .\\
\end{split}$$ This is precisely the announced result for $c=1/3$ and $f=36$. Remarkably, all the $u$ dependences dropped out upon normalizing by $D^{\rm out}(L,u)$ and the above conditional probability is thus *independent of $u$*.
Other families of maps {#sec:Other}
======================
The strategy presented in this paper may be applied to other families of maps provided that a coding by slices exists and that the decomposition of the corresponding slices along dividing lines at a fixed distance from their apex is fully understood. This is the case for planar triangulations, as explained in [@G15a], and for planar Eulerian triangulations, as explained in [@G16b]. We have reproduced and adapted the computations above to deal with these two other families of maps. We do not display here our calculations since they are quite tedious and give no really new information. Indeed, we find that, in the limit of large $k$ and $d$ with fixed $u=d/k$, all the laws that we obtained for quadrangulations have exactly the same expressions for triangulations and Eulerian triangulations, up to a global normalization for the rescaled length $L(d)=\mathcal{L}(d)/d^2$ and a global normalization for the rescaled volume $V(d)=\mathcal{V}(d)/d^4$. If we adopt the definitions of [@G15a] and [@G16b] for the hull at distance $d$ in triangulation and Eulerian triangulations respectively, these normalizations amount to change in our various laws of Section \[sec:summary\] the values $c=1/3$ and $f=36$ found in Section \[sec:explicit\] for quadrangulations to the values displayed in and .
The origin of the scaling factor $f$ is easily found in the relation between the weight $h$ per face in the hull and the variable $y$ which is “conjugated" to the distance $d$ in the slice generating function $T_d(y)$ (by this, we mean that $d$ appears in $T_d(y)$ via the combination $y^d$ only). We have for the three families (i), (ii) and (iii) of maps (see for instance [@G15a; @G15b; @G16b]) $$\hskip -1.cm \hbox{(i):}\ h(y)=\frac{y \left(1+y+y^2\right)}{\left(1+4y+y^2\right)^2}\ , \ \
\hbox{(ii):}\ h(y)= \frac{\sqrt{y (1+y)}}{\left(1+10 y+ y^2\right)^{3/4}}\ , \ \
\hbox{(iii):}\ h(y)= \frac{y \left(1+y^2\right)}{(1+y)^4}\ .$$ The desired singularities are obtained for $h(y)=h(1)\rho$ where $\rho=e^{-\sigma/d^4}$ if we wish to capture the large $d$ behavior of the rescaled volume. Setting $h(y)=h(1)e^{-\sigma/d^4}$ amounts, at large $d$ to setting $y=e^{-(f\, \sigma)^{1/4}/d}$ with, from the above relations $f=36$ in case (i), $f=192$ in case (ii) and $f=16$ in case (iii). All the universal laws involving $y^d\simeq e^{-(f\sigma)^{1/4}}$, the quantity $\sigma$ always appears via the combination $(f \, \sigma)^{1/4}$ in our various laws. This explains the origin of $f$. .3cm To understand the origin of the normalization factor $c$, the simplest quantity to compute is probably the limiting expectation value $$\lim_{k\to\infty}E_{k,d}^{\rm out}\left(\alpha^{\mathcal{L}(d)}\right)\ .$$ In the case of quadrangulations, we have from the large $k$ expansion $$\mathfrak{h}_3(k,T)=\frac{1}{7}\, k^4 + \frac{2\, Y}{7}\, k^3 + O(k^2)$$ so that (after normalization by $\mathfrak{f}_3(k)\sim (4/7)k^3$) $$\begin{split}
&\lim_{k\to\infty}E_{k,d}^{\rm out}\left(\alpha^{\mathcal{L}(d)}\right)=\frac{1}{2}\left(Y\left(\alpha^2\, T_d(1)\right)-Y\left(\alpha^2\, T_{d-1}(1)\right)\right)\\
&
= \frac{1}{2} \left(\sqrt{\frac{6 \alpha ^2+(d+1) (d+2) \left(25-\alpha ^2\right)}{6 \alpha ^2+(d+1) (d+2) \left(1-\alpha
^2\right)}}-\sqrt{\frac{6 \alpha ^2+d (d+1) \left(25-\alpha ^2\right)}{6 \alpha ^2+d (d+1) \left(1-\alpha ^2\right)}}\right)\ .\\
\end{split}$$ Using (for $C>0$) $$\hskip -1.2cm \sqrt{\frac{C^2-\beta}{1-\beta}}=C+2\sum_{p\geq 1}\beta^p\, A_p(C)\ \ \hbox{with}\ A_p(C)=\frac{1}{C^{2p-1}}\, \sum_{q=0}^{p-1}{p-1\choose q}{2q+1\choose q}\left(\frac{C^2-1}{4}\right)^{q+1}$$ here with $C^2=25$, we deduce that $$\hbox{(i):}\ \lim_{k\to\infty}P_{k,d}^{\rm out}\left({\mathcal{L}(d)=2p}\right)=A_p(5)\, \left(\left(\frac{(d-1)(d+4)}{d+1)(d+2)}\right)^p-\left(\frac{(d-2)(d+3)}{d(d+1)}\right)^p\right)\ ,$$ where the subscript “out" is irrelevant since for finite $d$ and infinite $k$, map configurations are necessarily in the out-regime. A similar calculation for the families (ii) and (iii) yields $$\begin{split}
& \hbox{(ii):}\ \lim_{k\to\infty}P_{k,d}^{\rm out}\left({\mathcal{L}(d)=p}\right)=A_p(3)\, \left(\left(\frac{d(d+3)}{d+1)(d+2)}\right)^p-\left(\frac{(d-1)(d+2)}{d(d+1)}\right)^p\right)\ ,\\
& \hbox{(iii):}\ \lim_{k\to\infty}P_{k,d}^{\rm out}\left({\mathcal{L}(d)=2p}\right)=2 A_p(3)\, \left(\left(\frac{(d-1)(d+5)}{d+1)(d+3)}\right)^p-\left(\frac{(d-2)(d+4)}{d(d+2)}\right)^p\right)\\
\end{split}$$ (note that $\mathcal{L}(d)$ is necessarily even in case (iii) but has arbitrary parity in case (ii)). From the large $p$ behavior $A_p(C)\sim \sqrt{C^2-1}/(2\sqrt{\pi\, p})$, we immediately deduce, taking $d$ and $p$ large with $p/d^2=L/2$ (case (i) and (iii)) or $p/d^2=L$ (case (ii)) the following probability densities for the three families of maps: $$\begin{split}
& \hbox{(i):}\ \lim_{d\to \infty} \left(\lim_{k\to\infty} \frac{1}{dL} P^{\rm out}_{k,d}\left(L\leq L(d) < L+dL\right)\right)= 6\, \sqrt{3}\, \sqrt{\frac{L}{\pi}}\, e^{-3\, L}\ , \\
& \hbox{(ii):}\ \lim_{d\to \infty} \left(\lim_{k\to\infty} \frac{1}{dL} P^{\rm out}_{k,d}\left(L\leq L(d) < L+dL\right)\right)= 4\, \sqrt{2}\, \sqrt{\frac{L}{\pi}}\, e^{-2\, L}\ , \\
& \hbox{(iiI):}\ \lim_{d\to \infty} \left(\lim_{k\to\infty} \frac{1}{dL} P^{\rm out}_{k,d}\left(L\leq L(d) < L+dL\right)\right)= 16\, \sqrt{\frac{L}{\pi}}\, e^{-4\, L}\ . \\
\end{split}$$ In agreement with the equivalence principle , these law reproduce the general form for the limit $u\to 0$ of $D^{\rm out}(L,u)/p^{\rm out}(u)$ (recall that $p^{\rm out}(u)
\to 1$ for $u\to 0$) via the identification $c=1/3$ in case (i), $c=1/2$ in case (ii) and $c=1/4$ in case (iii).
We end this section by giving for bookkeeping the (non-universal) expression for the probability $P_k({\mathcal V}(d)\ \hbox{infinite})$ at finite $k$ and $d$ for the families (ii) and (iii). We find $$\begin{split}
&\hskip -1.cm \hbox{(ii):}\ P_k({\mathcal V}(d)\ \hbox{infinite})=\frac{k^2 (k+1)^2}{2 (2 k+1) \left(5 k^6+15 k^5+14 k^4+3 k^3-k^2-1\right)}\\
& \hskip 1.cm \times \Bigg(\frac{d (d+1) (d+2) (d+3) \left(10 d^4+60 d^3+146 d^2+168 d+71\right)}{(2 d+3) (k+1)^3}\\
&\hskip 2.cm - \frac{(d-1) d (d+1) (d+2) \left(10 d^4+20 d^3+26 d^2+16 d-1\right)}{(2 d+1) k^3}\Bigg) \ ,\\
&\hskip -1.cm \hbox{(iii):}\ P_k({\mathcal V}(d)\ \hbox{infinite})= \frac{k (k+1) (k+2) (k+3)}{2 (2 k+3) \left(10 k^6+90 k^5+283 k^4+348 k^3+103 k^2-42 k-36\right)}\\
& \hskip 1.cm\times \Bigg( \frac{(d-1) (d+1) (d+3) (d+5) \left(10 d^4+80 d^3+256 d^2+384 d+189\right) (k+2)}{(d+2) (k+1)^2 (k+3)^2} \\
& \hskip 2.cm - \frac{(d-2) d (d+2) (d+4) \left(10 d^4+40 d^3+76 d^2+72 d-9\right) (k+1)}{ (d+1) k^2 (k+2)^2}\Bigg)\ .\\
\end{split}$$ When $k,d\to \infty$ and $d/k=u$, both expressions tend to $p^{\rm in}(u)=(7-3u)u^6/4$.
Conclusion {#sec:conclusion}
==========
![A plot of the expectation value of the (properly normalized) volume $\mathcal{W}(d)\equiv N-\mathcal{V}(d)$ in the in-regime as a function of $u=k/d$ for large $k$ and $d$ (here with $f=36$).[]{data-label="fig:complementvol"}](complementvol.pdf){width="8cm"}
In this paper, we explored the statistics of hull perimeters for three families of infinitely large planar maps: quadrangulations, triangulations and Eulerian triangulations, with a particular emphasis on the influence on this statistics of the constraint that the map configurations either yield a finite hull volume or not. In the case where the hull volume is finite, we also discussed the statistics of this volume itself, as well as its coupling to the hull perimeter statistics. Our study, based on an accurate coding of $k$-pointed-rooted planar maps by $k$-slices, makes a crucial use of a particular *recursive decomposition of these slices* obtained by cutting them along lines which precisely follow hull boundaries for increasing distances $d$ (see figure \[fig:constrhull\] for an illustration) for $d<k$. This decomposition, initiated in [@G15a] for triangulations, and then extended in [@G15b; @G16b] for the two other families of maps, may be used to address many other questions of the type discussed here, either for the same geometry, i.e. within pointed-rooted maps, or for other more involved geometries.
Among other quantities which may be computed within the above geometry of $k$-pointed-rooted maps are the statistics of the volume $\mathcal{W}(d)\equiv N-\mathcal{V}(d)$ of the complementary of the hull at distance $d$, i.e. the component containing the marked vertex $x_1$. In the in-regime, $\mathcal{W}(d)$ is finite and we may compute its limiting universal expectation value for large $k$ and $d$. We find $$\hskip -1.cm \lim_{k\to \infty}E_{k}\left(\frac{\mathcal{W}(k\,
u)}{k^4}\Big|\mathcal{V}(k\, u)\ \hbox{infinite}\right)=f\, \frac{(1-u) \left(14+16\, u+16\, u^2+16\, u^3-39\, u^4+12\, u^5\right)}{480 (7-3 u)}\ .$$ This quantity is plotted in figure \[fig:complementvol\] for $f=36$ (quadrangulations).
Concerning other tractable geometries, we recall that pointed *maps with a boundary* (i.e. maps with a distinguished external face of arbitrary degree) may be decomposed into sequences of slices and our recursive decomposition of slices gives a direct access to the statistics of a generalized hull at distance $d$ whose boundary would separate the pointed vertex from the external face (assuming that all vertices of the boundary are at a distance strictly larger that $d$ from the pointed vertex).
To conclude, many other families of maps (for instance maps with prescribed face degrees) may be coded by slices and, even if a recursion relation of the type of Ref. [@G15a; @G15b; @G16b] is not known in general for these slices[^10], the actual form of the associated slice generating functions is known in many cases [@BG12]. This might be enough to address the hull statistics for these maps since, as the reader noticed, the actual expression for the operator $\mathcal{K}$ describing the action of one step of the recursion is not really needed. What is needed is an equation of the form which displays the result of this operator on properly parametrized generating functions. This equation itself is moreover directly read off the explicit expression of the slice generating functions themselves (here for quadrangulations). Slices associated with maps with arbitrary face degrees have generating functions whose expressions are of the same general form (although more involved in general) as that for quadrangulations (see [@BG12]). The actual knowledge of these expressions might thus be sufficient to infer the hull statistics for the corresponding maps.
Expectation value of the perimeter at finite $k$ and $d$ in the out- and in-regimes
===================================================================================
Computing the expectation value of the perimeter simply involves computing the quantity $\partial_\alpha G(k,d,g,h,\alpha)\Big|_{\alpha=1}$, which itself, from , simply requires an expression for the quantity $$2 T\, \partial_T H(k,x,T)\ .$$ In the out-regime, we need to estimate the singularity of this latter quantity when $g\to 1/12$ ($x\to 1$). We find $$\begin{split}
&
\hskip -1.cm 2 T\, \partial_T H(k,x,T)\Big|_{\rm sing.}=2 T\, \partial_T \mathfrak{h}_3\big(k,Y(T)\big) (1-12 g)^{3/2}\\
&\hskip 2.cm = \mathfrak{dh}_3\big(k,Y(T)\big) (1-12 g)^{3/2}\\
& \hskip -1.cm \hbox{with}\ \mathfrak{dh}_3(k,Y)=\frac{(25-Y^2)(1-Y^2)}{24Y}\partial_Y\mathfrak{h}_3(k,Y)
\\ &= \frac{k (25-Y^2)(1-Y^2)}{20160 Y^3 (2 k+Y-1)^3 (2 k+Y+1)^3} \big(315 Y^{10}+3780 k Y^9+19740 k^2 Y^8-1995 Y^8\\ & +60480 k^3 Y^7-20160 k Y^7+120960 k^4 Y^6-82320 k^2 Y^6+16590 Y^6+161280 k^5
Y^5\\ & -174720 k^3 Y^5+71400 k Y^5+138240 k^6 Y^4-209664 k^4 Y^4+101640 k^2 Y^4+3594 Y^4\\ & +69120 k^7 Y^3-139776 k^5 Y^3+60480 k^3 Y^3-26784 k Y^3+15360 k^8 Y^2-39936 k^6
Y^2\\ & +24192 k^4 Y^2-54224 k^2 Y^2-36217 Y^2-65100 k Y-21700 k^2+5425\big)\ .\\
\end{split}$$ Setting $h=1/12$ ($y=1$) and $\alpha=1$ so that the values of interest are $Y(T_d(1))=2d+3$ and $Y(T_{d-1}(1))=2d+1$, we immediately deduce, upon normalization, that $$E_{k}\left({\mathcal L}(d)\Big|{\mathcal V}(d)\ \hbox{finite}\right)=\frac{\mathfrak{dh}_3(k-d,2d+3)-\mathfrak{dh}_3(k-d,2d+1)}{\mathfrak{h}_3(k-d,2d+3)-\mathfrak{h}_3(k-d,2d+1)}\ .$$ This immediately yields an explicit expression (which we do not reproduce here) for the expectation value of the perimeter at finite $k$ and $d$ in the out-regime. It is then easily verified that at large $k$ and $d$, ${\mathcal L}(d)$ scales as $d^2$ and that, for $k,d\to \infty$ and $d/k=u$ fixed, the expression for the expectation value of $L(d)={\mathcal L}(d)/d^2$ simplifies into the formula given in the first line of , with here $c=1/3$.
In the in-regime, we now set $g=1/12$ ($x=1$). Using, from , $$2T\, \partial_T H(k,1,T)=\frac{\big(25-Y(T)^2\big)\big(1-Y(T)^2\big)}{24Y(T)} \frac{32 (2 k+Y(T))}{(2 k+Y(T)-1)^2 (2 k+Y(T)+1)^2}$$ and plugging the expansion for $Y(T_d(y))$ when $y\to 1$ ($\eta\to 0$), we deduce $$\begin{split}
&
\hskip -1.cm 2T\, \partial_T H\big(k,1,T_d(y)\big)\Big|_{\rm sing.}=\widetilde{\delta\mathfrak{h}}_3(k,d) (1-12 h)^{3/2}\\
&\hskip -1.cm \hbox{with}\ \widetilde{\delta\mathfrak{h}}_3(k,d)=\frac{2 (d-1) (d+1) (d+2) (d+4)}{315 (2 d+3)^3 (d+k+1)^3
(d+k+2)^3} \left(15 d^4+90 d^3+237 d^2+306 d+140\right)\\ & \times \big(6 k d^6+12 k^2 d^5+54 k d^5+24 d^5+6 k^3 d^4+90 k^2 d^4+240 k d^4+180 d^4+36 k^3
d^3\\ & +270 k^2 d^3+630 k d^3+534 d^3+68 k^3 d^2+405 k^2 d^2+937 k d^2+783 d^2+42 k^3 d\\ &+285 k^2 d+705 k d+567 d-2 k^3+63 k^2+203 k+162\big)\ .
\\
\end{split}$$ We immediately deduce, upon normalization, that $$E_{k}\left({\mathcal L}(d)\Big|{\mathcal V}(d)\ \hbox{infinite}\right)=\frac{\widetilde{\mathfrak{dh}}_3(k-d,d)-\widetilde{\mathfrak{dh}}_3(k-d,d-1)}{\tilde{\mathfrak{h}}_3(k-d,d)-\tilde{\mathfrak{h}}_3(k-d,d-1)}$$ which again yields an explicit expression (not reproduced here) for the expectation value of the perimeter at finite $k$ and $d$ in the in-regime. It is again easily verified that, for $k,d\to \infty$ and $d/k=u$ fixed, the expression for the expectation value of $L(d)={\mathcal L}(d)/d^2$ simplifies into the formula given in the second line of , with here $c=1/3$.
Figure \[fig:perimeteroutfinite\] (respectively figure \[fig:perimeterinfinite\]) shows a comparison between the limiting expression given in the first (respectively the second) line of with $c=1/3$ vs $u$ and the finite $k$ and $d$ expression for $E_{k}\left(L(d)\Big|{\mathcal V}(d)\ \hbox{finite}\right)$ (respectively $E_{k}\left(L(d)\Big|{\mathcal V}(d)\ \hbox{infinite}\right)$) vs $d/k$ for $k=50$, $100$, $500$, and $2000$ and $2\leq d\leq k-1$.
![Plots of the expectation value of the rescaled hull perimeter $L(d)$ in the out-regime as a function of $d/k$ for $k=50$, $100$, $500$, and $2000$. In red: the corresponding limiting law for large $k$ and $d$, as given by the first line of .[]{data-label="fig:perimeteroutfinite"}](perimeteroutfinite.pdf){width="8cm"}
![Plots of the expectation value of the rescaled hull perimeter $L(d)$ in the in-regime as a function of $d/k$ for $k=50$, $100$, $500$, and $2000$. In blue: the corresponding limiting law for large $k$ and $d$, as given by the second line of .[]{data-label="fig:perimeterinfinite"}](perimeterinfinite.pdf){width="8cm"}
Acknowledgements {#acknowledgements .unnumbered}
================
The author acknowledges the support of the grant ANR-14-CE25-0014 (ANR GRAAL).
[^1]: In case (iii), we use more precisely some natural “oriented graph distance" using oriented paths keeping black faces on their left, see [@G16b].
[^2]: In practice, the line may be chosen in case (ii) so as to visit only vertices at distance $d$, and in case (i) and (iii) so as to visit alternately vertices at distance $d$ and $d-1$.
[^3]: If the observable depends on both $d$ and $k$, the equivalence clearly cannot be true in general as seen by taking for instance the expectation of $d^2/(k+d^2)$ equal to $1$ or $0$ according to the order of the limits.
[^4]: By “just above", we mean at a distance from the origin larger than that of the line by a quantity remaining bounded when $d$ becomes large.
[^5]: The expression ${3}^{3/2} \cosh \left( (2\sigma)^{1/4}\, s/\sqrt{8/3}\right) \left(\cosh ^2\left((2\sigma)^{1/4}\, s/\sqrt{8/3}\right)+2\right)^{-3/2}$ of [@CLG14b] is indeed fully equivalent under the correspondence $s=(2f)^{1/4}/\sqrt{3}$.
[^6]: In practice, as explained in [@G16a], $\lambda$ must be small enough and this condition precisely dictates the branch of solution chosen in below.
[^7]: More precisely, we must take the limit $x\to 1^-$.
[^8]: Note that, for $\tau=0$, $\frac{(1-u)^6 }{u^3} M\big(\mu(0,u)\big)=\frac{1}{4} \left(4-7 u^6+3 u^7\right)=p^{\rm out}(u)$ as it should.
[^9]: Note that, for $\tau=0$, $ u^3 Q(\mu(0,u),B(u))=\frac{1}{4} (7-3 u)\, u^6=p^{\rm in}(u)$ as it should.
[^10]: Other recursions are known however.
|
---
abstract: 'We present three ordinal notation systems representing ordinals below $\epsn$ in type theory, using recent type-theoretical innovations such as mutual inductive-inductive definitions and higher inductive types. We show how ordinal arithmetic can be developed for these systems, and how they admit a transfinite induction principle. We prove that all three notation systems are equivalent, so that we can transport results between them using the univalence principle. All our constructions have been implemented in cubical Agda.'
author:
- Fredrik Nordvall Forsberg
- Chuangjie Xu
- Neil Ghani
bibliography:
- 'ordbib.bib'
title: |
Three Equivalent Ordinal Notation Systems\
in Cubical Agda
---
<ccs2012> <concept> <concept\_id>10003752.10003790.10003792</concept\_id> <concept\_desc>Theory of computation Proof theory</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003752.10003790.10011740</concept\_id> <concept\_desc>Theory of computation Type theory</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Introduction {#sec:intro}
============
Ordinals and ordinal notation systems play an important role in program verification, since they can be used to prove termination of programs — using ordinals to verify that programs terminate was suggested already by Turing [@turing:ordinals]. The idea is to assign an ordinal to each input, and then prove that the assigned ordinal decreases for each recursive call. Hence the program must terminate by the well-foundedness of the order on ordinals. At first, such proofs were carried out using pen and paper [@Floyd:1967; @dershowitz:termination], but with advances in proof assistants, also machine-checked proofs can be produced [@MV:ord:acl2; @schmitt:ord:key]. As a first step, one must then represent ordinals inside a theorem prover. This is usually done via some kind of ordinal notation system (however see Blanchette *et al*. [@BPT:card:isa] for well-orders encoded directly in Isabelle/HOL, and Schmitt [@schmitt:ord:key] for an axiomatic method, which is implemented in the KeY program verification system). Typically, ordinals are represented by trees [@dershowitz:ord:tree; @DR:ord:list]; for instance, binary trees can represent the ordinals below $\epsn$ as follows: the leaf represents 0, and a tree with subtrees representing ordinals $\alpha$ and $\beta$ represents the sum $\omega^\alpha + \beta$. However, an ordinal may have multiple such representations. As a result, traditional approaches to ordinal notation systems [@buchholz:notation; @schuette:book; @takeuti:book] usually have to single out a subset of ordinal terms in order to provide unique representations. In this paper, we show how modern type-theoretic features can be used to directly give faithful representations of ordinals below $\epsn$.
The first feature we use is mutual inductive-inductive definitions [@nordvallforsberg2013thesis], which are well supported in the proof assistant Agda. This allows us to define an ordinal notation system for ordinals below $\epsn$, simultaneously with an order relation on it (Section \[sec:mutual\]). This means that we can recover uniqueness of representation, by insisting that subtrees representing ordinals are given in a decreasing order. This is similar to the traditional approach which first freely generate ordinal terms, and then later restrict attention to a subset of well-behaved terms (Section \[sec:subset\]). The advantage of the mutual approach is that there are no intermediate “junk” terms, and that the more precise types often suggests necessary lemmas to prove. However this is mostly an ergonomic advantage, since the two approaches are equivalent (Section \[sec:equiv\]).
We also use the feature of higher inductive types [@lumsdaine:shulman:hit] that has recently been added to Agda under the `–cubical` flag [@VMA:cubical:agda]. We define a different ordinal notation system for ordinals below $\epsn$ as a quotient inductive type [@QIITs], where we represent ordinals by finite hereditary multisets (Section \[sec:hit\]). Path constructors are used to identify multiple representations of the same ordinal, so that we again recover uniqueness. Also this approach is equivalent to the other two approaches (Section \[sec:equiv\]).
Different representations are convenient for different purposes. For instance, the higher inductive type approach to define the ordinal notation system is convenient for defining the commutative Hessenberg sum of ordinals (Section \[sec:arith\]), while the mutual representation is convenient for proving transfinite induction (Section \[sec:ti\]). Using the univalence principle [@hottbook], we can transport such constructions and properties between the different ordinal notation systems as needed.
#### Contributions
We make the following contributions:
- We give two to our knowledge new ordinal notation systems in type theory, representing ordinals below $\epsn$. These can be used to verify termination of programs inside type-theory-based proof assistants such as Agda.
- We prove that our ordinal notation systems are equivalent, and also equivalent to a third, well-known ordinal notation system based on a predicate of being in Cantor normal form. This allows us to transport constructions and properties between them using the univalence principle.
- We prove that our ordinal notation systems allow the principle of transfinite induction. This, and the rest of the development, is completely computational and axiom-free, in particular we do not need to assume excluded middle or countable choice.
- In general, we show how recent features of Agda such as simultaneous definitions and higher inductive types can be used to obtain user-friendly constructions, and how to work around common pitfalls.
#### Agda Formalization
Our full Agda development can be found at <https://doi.org/10.5281/zenodo.3588624>.
Cubical Agda {#sec:cubical:agda}
============
We start by giving a brief introduction to cubical Agda, an implementation of Cubical Type Theory [@CCHM] in the Agda proof assistant [@norell:thesis]. We refer to the Agda Wiki[^1] and the Agda User Manual[^2] for more resources on Agda, and to Vezzosi, M[örtberg]{} and Abel [@VMA:cubical:agda] for the technical details of the cubical extension of type theory.
Agda has a hierarchy of *universes* called s. The Cubical Agda library[^3] renames them to s to avoid the confusion with the notion of set in Homotopy Type Theory [@hottbook]. The lowest universe is now called , and it lives in . More generally, there is a universe for each , where is the successor function of universe levels.
We make use of Agda features such as *mixfix operators*, *implicit arguments* and *generalizable variables* to improve the readability of our Agda code. In turn, they work as follows: A mixfix operator may contain one or more name parts and one or more underscores . When applied, its arguments go in place of the underscore. For instance, when using the maximum function , we can write which is the same as . The symbol also has other usages: when an argument is not (explicitly) needed in a definition, or a term can be inferred by Agda’s unifier, we can replace it by . We can even omit using implicit arguments, which are declared using curly braces . For instance, if we define then type-checks, because the type checker knows and and hence can infer that the implicit argument is (the lowest ), and that is . To explicitly give an implicit argument, we just enclose it in curly braces. For example, we can also write . We often want our types and functions to be universe polymorphic by adding arguments in the declaration as in the above example. We can further omit by using generalizable variables: throughout our Agda development, we declare and then bindings for them are inserted automatically in declarations where they are not bound explicitly. For instance, now the identity function can be declared as where is implicitly universally quantified. We also use generalizable arguments for the different notions of ordinal terms considered in this paper.
Agda supports simultaneous definition of several mutually dependent data types such as in the schemes of *inductive-recursive* [@dybjer2000IR] and *inductive-inductive* [@nordvallforsberg2013thesis] definitions. Both schemes permit the simultaneous definition of an inductive type , together with a type family over ; the difference between them lies in whether is defined recursively over the inductive structure of , or if is itself inductively defined. The type is allowed to refer to and vice versa, so that one may for instance define simultaneously with a predicate or relation on . In this paper, we will use this to define a type of ordinal notations simultaneously with their order relation (Section \[sec:mutual\]). The Agda syntax for mutual definitions is to place the type signature of all the mutually defined data types and/or functions before their definitions.
The cubical mode extends Agda with various features from Cubical Type Theory [@CCHM]. To use Agda’s cubical mode, we have to place at the top of the file. First of all, cubical Agda has a primitive *interval* type with two distinguished endpoints and . Paths in a type , representing equality between elements of , are functions ; hence they can be introduced using $\lambda$-abstraction and eliminated using function application. There is a special primitive which can be considered as the type of *dependent paths* whose endpoints are in different types. The type of non-dependent paths is defined by where tells Agda to bind the implicit argument declared in the type of to a variable also named , which is used in the definition of . In this paper, we will need the following path-related proofs from the cubical Agda library: A type is called a *proposition* if all its elements are identical, and is called a *set* if all its path spaces are propositions. In Agda, this is formulated as follows: These univalent concepts play an important role in the development of mathematics in Homotopy Type Theory. Cubical Agda also supports a general schema of *higher inductive types* [@cubicalHITs], a generalization of inductive types allowing constructors to produce paths. In this paper, we will construct an ordinal notation system as a higher inductive type (Section \[sec:hit\]).
Another important concept from Homotopy Type Theory is the notion of type equivalence. We say that two types and are *equivalent*, and write , if there is a function $f : \AB{A} \to \AB{B}$ with an two-sided inverse $g : \AB{B} \to \AB{A}$, and if the proofs that $f$ and $g$ are inverses are coherent in a suitable sense. Importantly, every isomorphism ( a function with a two-sided inverse, but without coherence conditions on the inverse proofs) gives rise to an equivalence, we have where we have written for the type of isomorphisms between and . The *univalence principle* is provable in cubical Agda. In particular, there is a function generating a path between two types from a proof that they are equivalent. We will use univalence to construct paths between equivalent systems of ordinal notations (Section \[sec:equiv\]) and then transport various constructions and proofs between them along these paths (Sections \[sec:arith\] and \[sec:ti\]).
We will also use the following standard Agda data types:
- The empty type (with no constructors)
- Coproduct types (disjoint unions)
- $\Sigma$-types (dependent pairs)
- Cartesian products (non-dependent pairs)
- The natural numbers, and the standard order relation on them
When the type of a variable $x$ can be inferred, we will adopt the notational convention for , and similarly for .
When reasoning using chains of equations, we may write for readability, where and . This desugars to uses of transitivity , but has the advantage of keeping , and explicit.
Notation Systems for Ordinals Below epsilon-zero
================================================
The classical set-theoretic theory of ordinals defines an ordinal to be a set $\alpha$ which is transitive ( $x \in \alpha \to x \subseteq \alpha$) and connected ( ${x \not= y} \to {x \in y} \vee {y \in x}$ for any $x,y\in\alpha$). For program verification, the perhaps most important consequence of this definition is that $\in$ is a well ordering on ordinals — we hence often write $\alpha < \beta$ for $\alpha \in \beta$ — since this implies that properties of ordinals can be proven by transfinite induction, which in turn implies that there can be no infinitely descending chains of ordinals $$\alpha_0 > \alpha_1 > \alpha_2 > \ldots$$ — in other words, any process that can be assigned a decreasing sequence of ordinals must terminate.
Obviously the empty set $\emptyset$ is an ordinal (commonly denoted 0), and if $\alpha$ is an ordinal, it is not hard to see that its *successor* $\alpha + 1 \eqdef \alpha \cup \{\alpha\}$ is also an ordinal. This way, we can construct all finite ordinals 1 = 0 + 1, 2 = 1 + 1, 3 = 2 + 1, …, and then take their limit $\omega = \{0, 1, 2, 3, \ldots\}$. We can then continue constructing $\omega + 1$, $\omega + 2$, …, eventually reaching $\omega + \omega = \omega \cdot 2$, then $\omega \cdot 3$, …and thus eventually $\omega\cdot \omega = \omega^2$. Iterating this process, we can construct $\omega^\omega$, and then take the limit of the sequence $$\omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}, \ldots$$ The resulting ordinal is denoted $\epsn$, and is the minimal ordinal $\alpha$ such that $\omega^\alpha = \alpha$. It is well known that every ordinal $\alpha$ can be written uniquely in so-called Cantor normal form $$\alpha = \omega^{\beta_1} + \omega^{\beta_2} + \cdots + \omega^{\beta_{n}}
\qquad \text{with } \beta_1 \geq \beta_2 \geq \cdots \geq \beta_{n}$$ for some natural number $n$ and ordinals $\beta_i$ (the special case $\alpha = 0$ is written as the empty sum with $n = 0$). Our primary interest in $\epsn$ is that if $\alpha < \epsn$, then every exponent $\beta_i$ in the Cantor normal form of $\alpha$ satisfies $\beta_i < \alpha$. Hence if we in turn write $\beta_i = \omega^{\gamma_1} + \cdots + \omega^{\gamma_{m}}$ in Cantor normal form, we discover a decreasing sequence $$\alpha > \beta_i > \gamma_j > \ldots$$ of ordinals, which hence must terminate in finitely many steps. As a result, we have a finitary *notation system* which we can hope to implement inside a theorem prover in order to represent the ideal concept of ordinals below $\epsn$ in it. In the rest of this section, we explore three different approaches for achieving this in Agda.
The Subset Approach SigmaOrd {#sec:subset}
----------------------------
Traditional approaches to ordinal notation systems such as Buchholz [@buchholz:notation], Schütte [@schuette:book] and Takeuti [@takeuti:book] usually start by generating ordinal terms inductively, and then single out a subset in order to provide a unique representation for ordinals. Along this direction, we construct a notation system of ordinals below $\epsn$ as a sigma type in an Agda module .
The first step is to define ordinal terms, which are simply *binary trees*, albeit with highly suggestive constructor names: The idea is that represents the ordinal 0, and represents $\omega^\alpha + \beta$ if and represent $\alpha$ and $\beta$ respectively. However $\omega^\alpha + \beta$ might not be in Cantor normal form, and might have multiple such representations, because no order constraint has been imposed in the exponents occurring in . To remedy this flaw, we define an *ordering* on trees as follows (where ): The first constructor states that is smaller than any other tree, and the constructors and say that non- trees are compared lexicographically. However, this is not a well-founded order on ! To recover well-foundedness, we must restrict to trees that are in Cantor normal form. Towards this, we define the non-strict order in terms of the strict order : Then we can define the predicate of being in *Cantor normal form*: is in normal form, and is in normal form if also and are, and in addition is greater than or equal to the first exponent in , formally expressed using the following function: We construct the predicate formally using the following indexed inductive definition: For instance, if are in Cantor normal form and , then is inhabited.
Finally, we can form the subset of trees in Cantor normal form by the following dependent pair type: We are justified in using the “subset” terminology, because we can prove that is proof-irrelevant, the proof of which in turn relies on the following facts: Therefore, equality on is determined only by the component, we can prove For the formal proofs, we refer to our Agda development. This approach gives a faithful representation of ordinals below $\epsn$, but it is sometimes inconvenient to work with, one has to explicitly prove that all operations preserve being in Cantor normal form. Agda’s termination checker is often happier with curried functions, which further discourages use of as a programming abstraction.
The Mutual Approach MutualOrd {#sec:mutual}
-----------------------------
Instead of considering an imprecise type of trees, including trees not in Cantor normal form that do not represent ordinals, we can use Agda’s support for mutual definitions to directly generate trees in Cantor normal form only, by simultaneously defining ordinal terms and an ordering on them. The idea is to additionally require the term representing an ordinal $a$ to be greater than or equal to the first exponent of the term representing an ordinal $b$ when forming the term representing $\omega^a+b$. Hence we also need to define the operation which computes the first exponent of an ordinal term simultaneously. All in all, in a module we define simultaneously by where we write . Obviously this is very similar to the definitions in Section \[sec:subset\], but this time, *every* term of type satisfies the order constraint because of the third argument of the constructor . This means that every term that we can form is already in Cantor normal form, and there is no need for a separate predicate.
Because of the coproduct hidden in the constructor argument , and the function occurring in it, is a nested [@bird1998nested] inductive-inductive-recursive [@nordvallforsberg2013thesis] definition. However, by replacing the constructor with a coproduct argument by two constructors (one for each summand), and by defining the *graph* of inductively instead of the function itself recursively, it is possible to define an equivalent non-nested, non-inductive-recursive type. This justifies the soundness of our current definition.
Just like in the subset approach, we can prove that the order relation is proof-irrelevant, there is a proof of for every and . However, because of the mutual nature of the definitions, the following facts has to be proved simultaneously: One advantage when working with a tighter type such as compared to the looser is that the right lemma is often naturally suggested in the course of a construction: for example, when proving , the lemma falls more or less immediately out as required by one of the subgoals.
For later use in Section \[sec:arith\], we note that we can prove (constructively) that the ordering is trichotomous, The proof is the same as a simpler proof for from Section \[sec:subset\], except that we have to make essential use of .
The Higher Inductive Approach HITOrd {#sec:hit}
------------------------------------
In our third approach, an ordinal may have multiple representations, but we ensure that all of them are identical in the sense of Cubical Type Theory. We do this by defining a *higher inductive type*, which is given by freely generated terms and paths between them. Instead of representing an ordinal by a list of ordinal representations (the exponents in its Cantor normal form), where the order matters, we instead consider finite multisets of ordinal representations, where the order of elements does not matter. Such finite multisets can be defined in a first-order way as a higher inductive type, as in Licata [@licata:hott:talk]. Because the elements of the multiset again are ordinal representations, what we need is a higher inductive type of so-called finite hereditary multisets. We make the following definition in the module : This is a higher inductive type, since it is given by listing its generating term constructors and , as well as its generating path constructors and . Cubical Agda supports higher inductive types natively, and their soundness is guaranteed by the cubical sets model [@cubicalHITs]. As hinted at by the name of the constructor , our intention for a term is no longer to represent the non-commutative sum of ordinals $\omega^\alpha + \beta$ where $\alpha$ and $\beta$ are represented by and respectively, but rather the commutative *Hessenberg sum* $\omega^{\alpha} \oplus \beta$ (see Section \[sec:hessenberg-arith\]). This is justified by the inclusion of the path constructor , which states that terms with permuted exponents are identical, as illustrated by the following example (using equational reasoning combinators from the end of Section \[sec:cubical:agda\]): Adding just the constructor would result in a lack of higher-dimensional coherence ( we would expect to be the reflexivity path), and so we also include the constructor which forces to be a set. This means that we can prove the following recursion principle for : This recursion principle states that there is a function from to any other type which is closed under the same “constructors” as . In other words, to define a function using the recursion principle, needs to be a set, and one needs not only a point of and an operator , but also a proof of a “swap” rule for . This stops us from defining a function with by , since this would require $$a\; \AgdaSymbol{=}\; a
\AgdaOperator{⋆} \AgdaSymbol{(}b \AgdaOperator{⋆} c \AgdaSymbol{)}\AgdaSpace{}\; \AgdaFunction{≡}\; \AgdaSpace{} \AgdaSymbol{(}b
\AgdaOperator{⋆} a\AgdaSymbol{)} \AgdaOperator{⋆} c \;\AgdaSymbol{=}\; b$$ for any $a$, $b$ , which is clearly not true. In general, the recursion principle can be used to define *non-dependent* functions out of that respect the additional path constructors (we will make use of this in Section \[sec:equiv\]). Similarly, to prove properties of , we will make use of the following induction principle for propositions: Since the motive $\AgdaBound{P}\;\AgdaBound{x}$ is a proposition for every $\AgdaBound{x}$ by assumption, we do not need to ask for any methods involving path constructors — there are no non-trivial paths in $\AgdaBound{P}\;\AgdaBound{x}$. Both the recursion principle and the induction principle for propositions are instances of the full induction principle, which can be proven by pattern matching in cubical Agda.
Equivalences Between the Three Approaches {#sec:equiv}
-----------------------------------------
We now wish to show that all three approaches are in fact equivalent, in the strong sense of Homotopy Type Theory. To show , it suffices to construct an isomorphism between and . Hence we construct isomorphisms between and , and between and . In a new module , we import the previous modules: Since many names are shared between the imported modules ( both and define and ), we use the short module names , and to qualify ambiguous names, we write and to refer to the concepts from , and and for the ones from .
### is Equivalent to
We first construct a function from to . To help Agda’s termination checker, we define in curried form — in fact the first component of the sigma type can even be kept implicit. Because is defined simultaneously with its ordering, when defining we have to simultaneously prove that it is monotone: We omit the easy proofs of and here, but give the definition of since it is computationally relevant:
When implementing , we also need the curried equivalent of specialised to the image of , which can be defined using the path induction principle. Unfortunately, this detour trips up Agda’s termination checker. We work around this by converting a given path to an inductively defined propositional equality using the following construction: Here is the builtin module defining propositional equality as inductively generated by the constructor . With this in hand, we can pattern match directly on the produced propositional equality instead of using path induction when implementing , which placates the termination checker: Hopefully the termination checker of cubical Agda will be fixed to accept a direct proof in future versions.
For the reverse direction, we convert to , and then show that the resulting trees are in Cantor normal form: We have omitted the easy proofs that is monotone: Putting all the pieces together, we can now define maps from to and vice versa: The proofs that the two compositions of and are identities rely on the fact that the orderings are proof-irrelevant; more precisely, they use the lemmas and : Since every isomorphism can be extended to an equivalence (using ), and we have just constructed an isomorphism between and , we have proven:
and are equivalent, there is a proof .
Using , one direction of the univalence principle, we get a path from to .
and are identical, there is a proof .
### is Equivalent to
A translation from to is easy: we simply forget about the order witnesses. The other direction is more interesting. We need a binary operation satisfying the “swap” rule in order to use the recursion principle of . For this purpose, we notice that both and admit a list structure: is the empty list; and in and respectively, is the head and is the tail. All lists in are in descending order, while those in are quotiented by permutations so that it is impossible to access the order of elements in lists. Coming back to the binary operation with this list-structure intuition in mind, we see that needs to add its first argument (regarding it as an element) into its second (regarding it as a list) such that different orders of doing this result in the same list. One operation satisfying these requirements is list insertion. Again, we simultaneously need to prove that preserves the ordering, since is simultaneously defined with it. The function implements the standard algorithm for list insertion (slightly obfuscated by our choice of constructor names). Similarly the proof follows the same call structure to show that is order-preserving. Here is a proof that for every . Using that is trichotomous, using to compare any two elements, we can prove that satisfies the swap rule: Hence we can use and the recursion principle for to define and then show that and form an isomorphism (the step case is using equational reasoning combinators, as explained in Section \[sec:cubical:agda\]): We omit the easy proof of the lemma used in the final step. For the other direction, we use the induction principle for propositions: This is using the following lemma: Putting everything together, we have proven:
and are equivalent, there is a proof .
and are identical, there is a proof .
Ordinal Arithmetic {#sec:arith}
==================
In this section, we demonstrate the usability of our definitions by showing how well-known arithmetic operations can be defined on them. We have two quite different data structures representing ordinals below $\epsn$: hereditary descending lists and finite hereditary multisets . It is more convenient and efficient to construct the ordinary arithmetic operations on , because comparing the “heads” suffices for the constructions rather than iterating through the whole ordinal terms. On the other hand, constructing the commutative arithmetic operations such as Hessenberg sums and products is easier and more natural on , because orders do not play a role in the constructions. Hence we implement ordinary ordinal addition and multiplication on , and Hessenberg addition and multiplication on . We prove some properties of the operations, and then transport the constructions and proofs between them using the path .
Ordinary Addition and Multiplication {#sec:ordinary-arith}
------------------------------------
Ordinal arithmetic extends addition and multiplication from the natural numbers to all ordinals, including transfinite ones. It is famously non-commutative: $1 + \omega = \omega$, but $\omega + 1 > \omega$. On , we have to define addition whilst simultaneously proving the property that it preserves the ordering. The interesting case of this well-known algorithm, when both summands are non-zero, is guided by the fact that ordinals of the form $\omega^\beta$ are so-called additive principal ordinals, if $\gamma<\omega^\beta$ then $\gamma + \omega^\beta = \omega^\beta$ (after defining addition, this is not hard to prove for ). In particular if $\alpha < \beta$, then $\omega^\alpha < \omega^\beta$ and hence $\omega^\alpha + \omega^\beta = \omega^\beta$. The proof that addition preserves the ordering again follows the same structure as addition itself. The construction of an element of contains also a proof that it is in Cantor normal form. When implementing above, the construction (more precisely, the last case when ) explicitly tells us what property of is required to show that the sum is in Cantor normal form, and we are led to prove this property simultaneously. In the traditional subset approach, one usually constructs addition on all ordinal terms, and then proves that it preserves Cantor normal form. However one has to figure out what property of addition is needed for the proof oneself. The above example of a “construction-guided” proof demonstrates one advantage of the mutual approach.
Moving from programs to proofs, consider the following type stating that a given binary operation is associative: We can construct an easy but lengthy proof that on is associative — the lengthiness is due to the use of a case distinction on in the definition of . Now, using the path , we can transport both the operation of addition and the proof that it is associative to an associative operation on : where is a dependent path from to .
Similarly, we can implement the standard multiplication algorithm for ordinals in Cantor normal form where . Since every case is implemented in terms of previously defined functions, there is no need to prove any simultaneous lemma about preservation of the order this time. Again, we can transport this definition to get multiplication on for free: Let us look at some examples. We define the representation of the ordinal 1 by = and the one of $\omega$ by = . The following examples illustrate that ordinal addition and multiplication are not commutative: for addition, we have $1+\omega = \omega \not= \omega +1$, where the equality is definitional, , it computes: Similarly, for multiplication, we have $2 \cdot \omega = \omega \not= \omega + \omega = \omega \cdot 2$: For the examples of , we define , = and = . The operations of addition and multiplication on are obtained by transporting those on along . We get this path using (one direction of) the univalence axiom which is constructively provable in cubical Agda. Therefore, closed terms of constructed using these operations can be evaluated into normal form, for instance Again, note that both equalities are definitional.
Hessenberg Addition and Multiplication {#sec:hessenberg-arith}
--------------------------------------
Hessenberg arithmetic [@hessenberg] is a variant of ordinal arithmetic which is commutative and associative, but not continuous in its second argument. On , Hessenberg addition is simply implemented as the concatenation operation on finite multisets. Here we define it by pattern matching on the first argument, which is equivalent to using the recursion principle. Note that we also have to produce clauses for and , corresponding to proving that the defined function preserves the generating paths. For instance, for , we have to prove that our definition gives identical results for swapped exponents, , a path $\AgdaOperator{\AgdaInductiveConstructor{ω\textasciicircum{}}}~\AgdaBound{a}~\AgdaOperator{\AgdaInductiveConstructor{⊕}}~\AgdaOperator{\AgdaInductiveConstructor{ω\textasciicircum{}}}~\AgdaBound{b}~\AgdaOperator{\AgdaInductiveConstructor{⊕}}~\AgdaSymbol{(}\AgdaBound{c}~\AgdaOperator{\AgdaFunction{⊕}}~\AgdaBound{y}\AgdaSymbol{)}~\AgdaOperator{\AgdaFunction{≡}}~\AgdaOperator{\AgdaInductiveConstructor{ω\textasciicircum{}}}~\AgdaBound{b}~\AgdaOperator{\AgdaInductiveConstructor{⊕}}~\AgdaOperator{\AgdaInductiveConstructor{ω\textasciicircum{}}}~\AgdaBound{a}~\AgdaOperator{\AgdaInductiveConstructor{⊕}}~\AgdaSymbol{(}\AgdaBound{c}~\AgdaOperator{\AgdaFunction{⊕}}~\AgdaBound{y}\AgdaSymbol{)}$, which is again an instance of : Our goal is now to justify the notation in the constructor name for by showing that is commutative. First we define the property of being commutative: Next we can use the induction principle for propositions to prove that indeed is commutative. The base case is given by a simple lemma and the heavy work of the step case is done by the lemmas which are also proved using the induction principle . Using these lemmas, the proof is as follows: By transporting along the reversed path we get a commutative operation on : where is a dependent path from to .
We also implement Hessenberg multiplication on , which is essentially pairwise concatenation of elements in finite multisets. We first define which concatenates every element of with . Again, we are asked to prove that this respects swapping exponents and set-truncation. Then we define Hessenberg multiplication by using this operation to concatenate to every exponent of : where is easily proved using and . Finally we can again transport to get Hessenberg multiplication on : Let us look at some examples. Hessenberg addition on can be viewed as a concatenation operation, as illustrated below: Again, because univalence is computational in cubical Agda, the transported Hessenberg operations on compute. For instance, we have the following definitional equalities — note that these equations are not true for ordinary addition and multiplication.
Transfinite Induction {#sec:ti}
=====================
In this section, we prove transfinite induction for , and then transport it to transfinite induction for . Already defining an ordering on by hand is non-trivial, and usually requires several auxiliary concepts such as a subset relation for multisets and multiset operations such as union and subtraction [@dershowitz:termination; @BFT:nested:mset]. Now we can simply transport the ordering on to . Similarly, it seems easier to prove transfinite induction for and then transport the proof to if needed, rather than proving it directly.
The Transported Ordering on HITOrd
----------------------------------
We firstly transport the ordering on to as follows: We can further transport the properties of to . For instance, let us define the property of decidability We can easily prove and then transport it to get where is a dependent path from to .
Now we demonstrate that the transported property computes, like the transported constructions in Section \[sec:arith\]. To simplify the examples, we turn into a boolean-valued function by where assigns : to the left summand and : to the right. Here are some examples: Again, note that all equalities displayed are definitional.
Transfinite Induction {#transfinite-induction}
---------------------
Transfinite induction for a type $A$ with respect to a relation on $A$ says that if for every $x$ in $A$ a property $P(x)$ is provable assuming that $P(y)$ holds for all $y < x$, then $P(x)$ holds for every $x$. It is well-known that transfinite induction is logically equivalent to every element of $A$ being accessible, in the following sense: The proof of transfinite induction uses . We now show that every element of is accessible: The base case is trivial. We show the non-zero case using the following two lemmas which are simultaneously proved. The idea is that, to prove the accessibility of , we have to show that is accessible for any . There are three cases: (1) If is , then we are done. (2) If is with , then we use . (3) If is with and , then we use .
Combining and , we can now prove:
Transfinite induction holds for , there is a proof .
Transporting along our path , we also have:
Transfinite induction holds for , there is a proof .
All Strictly Descending Sequences are Finite
--------------------------------------------
Now we consider a simple application of transfinite induction: to prove that all strictly descending sequences of ordinals below $\epsn$ are finite. Formulating this faithfully in Agda is not easy when representing sequences as functions from the natural numbers, and one often ends up with the negative formulation “there is no strictly descending sequence” instead. One may replace finiteness by eventual zeroness, but this would contradict the strictly descending condition. As a stronger and *computational* formulation, we introduce the following notion: Note that it is not enough to require only in the second summand, as that would allow to “restart” at stage . This notion is obviously weaker than the notion of being strictly descending: The following facts of pseudo-descendingness are trivial but play an important role in the proof. where inequality of natural numbers is inductively defined in the standard way. Moreover, we say that a sequence $f$ is *eventually zero* if we can find an $n$ such that $f(i)$ takes the value zero for every $i$ after $n$: One can easily prove the following fact of eventual-zeroness: Now we can formulate our result positively as follows:
Every pseudo-descending sequence is eventually zero, there is a proof\
.
We prove the statement using transfinite induction on , we use the following motive: We have to prove the following induction step: It consists of two cases: (1) If , then by the fact . Hence is eventually zero by the hypothesis , and so is by the fact . (2) If , then is constantly zero by the fact . Hence we can take .
The algorithm encoded in the above proof checks the values of , , …in turn, until it finds a zero point. By construction, it will thus find the least such that for all . The transfinite induction principle proves that this procedure is terminating, using the assumption of pseudo-descendingness.
Because strict descendingness implies the pseudo notion, the negative formulation is a simple corollary.
There is no strictly descending sequence, there is a proof .
Comparison with Related Work
============================
In this section, we compare existing work with our development.
#### Trees as Ordinals
The relationship between ordinals — especially ordinals below $\epsn$ — and various tree structures is of course well known, and more or less folklore. Dershowitz [@dershowitz:ord:tree] gives an overview of different ordinal representations using finite trees, and Dershowitz and Reingold [@DR:ord:list] construct binary trees using Lisp-like list structures. This is similar to our definition , but our systems provide *unique* representations of ordinals. Jervell [@jervell:finiteTrees] gives a clever total ordering on finite trees with $\epsn$ the supremum of all binary trees. It is not straightforward to encode and work with this ordering in a proof assistant.
#### Ordinals in Type Theory
Surprisingly large ordinals can be constructed in basic Martin-Löf Type Theory with primitive type of (countable) ordinals, but no recursion principle for it. Coquand, Hancock and Setzer [@coquand:ord] show that already in this setting, one can reach $\phi_{\epsn}(0)$, where $\phi_{\alpha}$ is the Veblen hierarchy. Hancock [@hancock:thesis] uses a class of predicate transformers called lenses to give a clean proof of (half of) Hancock’s conjecture: Martin-Löf Type Theory with $n$ universes can reach $\phi_{\phi_{\phi_{\ldots}(0)}(0)}(0)$ with $n$ nestings of $\phi_{\epsn}(0)$. In contrast, in our work we are not restricting ourselves to a spartan type theory, but try to take full advantage of all of Agda, with the goal of producing an easy-to-use representation. It is clear that we can draw much inspiration from this line of work when going beyond $\epsn$. See also Setzer [@Setzer:prooftheory] for a general overview of the ordinals that can be constructed in different variations of type theory.
#### Formalisations
Several formalisations of ordinals and ordinal notation systems exist in the literature. Manolios and Vroon[@MV:ord:acl2] represents ordinals below $\epsn$ in the ACL2 theorem prover, based on a variation of Cantor normal form with $$\omega^{\beta_1}c_1 + \ldots + \omega^{\beta_n}c_n \quad \text{with $\beta_1 > \ldots > \beta_n$ and all $c_i$ finite}$$ This is similar to our representation, except that there are no mechanical guarantees that given inputs actually are in Cantor normal form. They also provide algorithms for ordinal arithmetic and comparisons of ordinals, but their correctness proofs have to assume that the given inputs are in Cantor normal form. In contrast, it is not possible to construct ordinal terms not in Cantor normal form in our systems. Similarly, Castéran and Contejean [@CC:ord:coq] and Grimm [@grimm:ord:coq] develop significant theories of ordinals below $\epsn$ in Coq, including arithmetic operations and transfinite induction. This is again similar to our approach (a choice perhaps made because Coq to date does not support simultaneous definitions or higher inductive types, which are needed for the and approaches respectively).
#### Finite Multisets
In Isabelle/HOL, Blanchette, Fleury and Traytel [@BFT:nested:mset] define an inductive datatype of hereditary multisets to represent ordinals below $\epsn$, similar to our $\HO$ . The representation relies on the notion of multisets in Isabelle’s standard library, which are defined as natural number-valued functions with a finite support. This can be constructively problematic, for instance when defining ordinal exponentiation. In contrast, our use of higher inductive types to define multisets means that our datatypes are reassuringly first-order. Because hereditary multisets are viewed as a subtype of nested multisets, the nested multiset ordering and its well-foundedness proof are “lifted” to the hereditary multisets using the sophisticated machinery in Isabelle. However, defining the nested multiset ordering [@dershowitz:termination] is non-trivial and proving its well-foundedness is challenging as admitted in [@BFT:nested:mset]. In comparison, our ordinal notation system is convenient to work with for instance to prove its well-foundedness. By showing that it is equivalent to hereditary multisets , we obtain also a well-foundedness proof for the latter.
Concluding Remarks
==================
We have used modern features of cubical Agda such as simultaneous definitions and higher inductive types to faithfully represent ordinals below $\epsn$, and shown that our definitions are easy to work with by defining common operations on, and proofs about, our ordinal notation systems. Our development is fully constructive.
Of course, in the world of ordinals, $\epsn$ is tiny; already Martin-Löf Type Theory with W-types and only one universe has proof-theoretic strength well beyond $\epsn$ [@Setzer:prooftheory], and simultaneous inductive-recursive definitions are known to increase the proof-theoretic strength even further (a consequence of Hancock’s Conjecture [@hancock:thesis]). Similarly Lumsdaine and Shulman [@lumsdaine:shulman:hit] show that adding recursive higher inductive types increases the power of type theory by considering in particular a higher inductive type encoding of a variation of Brouwer tree ordinals. To verify termination of programs exhausting the strength of such systems, one would have to define even stronger ordinal notation systems. We conjecture that powerful definitional principles such as simultaneous inductive-recursive definitions and higher inductive types — perhaps combined — can be used to faithfully represent also larger ordinals, and hence be useful for such program verification problems.
We thank Nicolai Kraus, Helmut Schwichtenberg, Ryota Akiyoshi, Nils Köpp, Masahiko Sato and Anders Mörtberg for many interesting and illuminating discussions, and the anonymous reviewers for their helpful suggestions and comments. This work was supported by funding from the \[grant number \], the , and the .
[^1]: Agda Wiki: <https://wiki.portal.chalmers.se/agda/pmwiki.php>
[^2]: Agda User Manual: <https://agda.readthedocs.io/>
[^3]: Cubical Agda library: <https://github.com/agda/cubical>
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---
abstract: 'Given a sequence of complete Riemannian manifolds $(M_n)$ of the same dimension, we construct a complete Riemannian manifold $M$ such that for all $p \in (1,\infty)$ the $L^p$-norm of the Riesz transform on $M$ dominates the $L^p$-norm of the Riesz transform on $M_n$ for all $n$. Thus we establish the following dichotomy: given $p$ and $d$, either there is a uniform $L^p$ bound on the Riesz transform over all complete $d$-dimensional Riemannian manifolds, or there exists a complete Riemannian manifold with Riesz transform unbounded on $L^p$.'
address:
- |
Delft Institute of Applied Mathematics\
Delft University of Technology\
P.O. Box 5031\
2600 GA Delft\
The Netherlands
- |
School of Mathematics\
The University of Edinburgh and Maxwell Institute for Mathematical Sciences\
James Clerk Maxwell Building\
Rm 5210 The King’s Buildings\
Peter Guthrie Tait Road\
Edinburgh\
EH9 3FD\
United Kingdom
author:
- Alex Amenta
- Leonardo Tolomeo
bibliography:
- 'riesz\_failure.bib'
title: A dichotomy concerning uniform boundedness of Riesz transforms on Riemannian Manifolds
---
Introduction
============
Given a Riemannian manifold $M$, one can consider the Riesz transform $R := \nabla (-\Delta)^{\frac12}$, where $\nabla$ is the Riemannian gradient and $\Delta$ is the (negative) Laplace–Beltrami operator. In the Euclidean case $M = {\mathbb R}^n$, this can be identified with the vector of classical Riesz transforms $(R_1,\ldots,R_n)$, as can be seen by writing $R$ as a Fourier multiplier (see [@lG14 5.1.4]).
It is easy to show that $R$ is bounded from $L^2(M)$ to $L^2(M;TM)$, and substantially harder to determine whether $R$ extends to a bounded map from $L^p(M)$ to $L^p(M;TM)$ for $p \neq 2$. We let $$R_p(M) := \sup_{{\left\|{f}\right\|}_{L^p} \le 1} {\left\|{R(f)}\right\|}_{L^p}$$ denote the (possibly infinite) $L^p$-norm of the Riesz transform on $M$. Various conditions, often involving the heat kernel on $M$ and its gradient, are known to imply finiteness of $R_p(M)$; see for example [@AC05; @ACDH04; @BF16; @gC17; @CCH06; @CCFR17; @CD99; @CD03; @hL99; @LZ17]. These results usually entail finiteness of $R_p(M)$ for all $p \in (1,2)$, or for some range of $p > 2$. On the other hand, there exist manifolds $M$ for which $R_p(M)$ is known to be infinite for some (or all) $p > 2$: see [@aA17; @gC17; @CCH06; @CCFR17; @CD99; @hL99].
When $M$ has finite volume we abuse notation and write $L^p(M)$ to denote the space of $p$-integrable functions *with mean zero*. This modification ensures that $(-\Delta)^{-1/2}$ is densely defined. When $M$ has infinite volume, $L^p(M)$ denotes the usual Lebesgue space.
The Euclidean case is now classical: for all $p \in (1,\infty)$ there is a constant $C_p < \infty$ such that $R_p({\mathbb R}^n) \leq C_p < \infty$ for all $n \in {\mathbb N}$ ([@eS83]). This behaviour is expected to persist for all complete Riemannian manifolds, at least for $p < 2$. More precisely, in [@CD03] it is conjectured that for all $p \in (1,2)$ there exists a constant $C_p < \infty$ such that $R_p(M) \leq C_p$ for all complete Riemannian manifolds $M$. Such uniform bounds have been proven for all $p \in (1,\infty)$ under curvature assumptions; rather than provide an overview of the vast literature on this topic we simply point to the recent paper [@DDS18] and references therein.
One could weaken the conjecture slightly and guess that $R_p(M)$ is finite for all $M$, given $p \in (1,2)$. In this article we show that this can only hold if the bound is uniform among all manifolds of a fixed dimension. This observation follows from the following dichotomy.
\[thm:main\] Fix $d \in {\mathbb N}$ and $p \in (1,\infty)$. Then the following dichotomy holds: either
- there exists a constant $C_{p,d} < \infty$ such that $R_p(M) \leq C_{p,d}$ for all complete $d$-dimensional Riemannian manifolds $M$, or
- there exists a complete $(d+1)$-dimensional Riemannian manifold $M$ such that $R_p(M) = \infty$.
This follows from the following proposition, which we prove by an explicit construction.
\[prop:construction\] Fix $d \geq 1$, and let $(M_n)_{n \in {\mathbb N}}$ be a sequence of complete $d$-dimensional Riemannian manifolds. Then there exists a complete Riemannian manifold $M$ of dimension $d+1$ such that for all $p \in (1,\infty)$, $$R_p(M) \geq \sup_{n \in {\mathbb N}}R_p(M_n).$$
The main implication of Theorem \[thm:main\] is as follows: to construct a manifold $M$ for which $R_p(M) = \infty$ for some $p \in (1,2)$, it suffices to construct a sequence $(M_n)_{n \in {\mathbb N}}$ of manifolds of equal dimension such that $R_p(M_n) \to \infty$ as $n \to \infty$. Thus one is led to consider lower bounds for $L^p$-norms of Riesz transforms. These seem not to have been considered in the literature, excluding of course the well-known computation of the $L^p$-norm of the Hilbert transform (the Riesz transform on ${\mathbb R}$) [@sP72]. We hope that our contribution will provoke further interest in such lower bounds.
Preliminary lemmas
==================
We begin with some basic lemmas. The first says that the range of the Laplace-Beltrami operator is dense in $L^p$, and the second relates the Riesz transform on a manifold $M$ with that on the $M$-cylinder $M \times {\mathbb R}$. These cylinders play a key role in the proof of our main theorem.
\[lem:range-lap-dens\] Let $M$ be a complete Riemannian manifold. Then the set $S := \Delta(C_c^\infty(M))$ is dense in $L^p(M)$ for all $p \in (1,\infty)$ (recalling that we write $L^p(M)$ for the space of $p$-integrable mean zero functions when $M$ has finite volume).
Let $H \in L^{p'}(M)$ be such that ${\left\langleH,F\right\rangle} = 0$ for every $F \in S$. Then ${\left\langleH,\Delta G\right\rangle} = 0$ for every test function $G$, so $H$ is harmonic. By [@sY76 Theorem 3], it follows that $H$ is constant, and the result follows.
\[lem:cyl-R\] Let $M$ be a complete Riemannian manifold. Then $$R_p(M\times {\mathbb R}) \ge R_p(M).$$
Consider the following modification of the Riesz transform on $M \times {\mathbb R}$: $$\tilde R := \nabla_M (-\Delta_{M\times{\mathbb R}})^{-\frac12} = \nabla_M(-\Delta_M - \partial_t^2)^{-\frac12}.$$ This is just the projection of $R$ onto the first summand of the tangent bundle $T(M \times {\mathbb R}) = TM \oplus T{\mathbb R}$, so we have that $$\label{eqn:tilde-R-est}
\| \tilde RF \|_{L^p} \le \| RF \|_{L^p}.$$ Let $F \in C^\infty_c(M \times {\mathbb R})$, and for all $\lambda > 0$ consider the function $$F_\lambda(x,t) := \lambda^{\frac1p} F(x,\lambda t),$$ which satisfies ${\left\|{F_\lambda}\right\|}_{L^p(M\times{\mathbb R})} = {\left\|{F}\right\|}_{L^p(M\times {\mathbb R})}$. Rescaling the operator $\tilde R$ in the variable $t$, we define $$\tilde R_\lambda := \nabla_M(-\Delta_M - \lambda^2 \partial_t^2)^{-\frac12},$$ so that $$\label{eqn:tilde-R}
\| \tilde R F_\lambda \|_{L^p} = \| \tilde R_\lambda F \|_{L^p}.$$ Now take $f \in C^\infty_c(M) \cap D((-\Delta_M)^{-\frac12})$ and $\rho \in C^\infty_c({\mathbb R})$ such that ${\left\|{\rho}\right\|}_{L^p({\mathbb R})} = 1$, and consider the function $F(x,t) = f(x)\rho(t)$. Since $\Delta_M$ and $\partial_t^2$ commute, and the function $$G_\lambda(x,y) = \left(\frac{x}{x+\lambda^2 y}\right)^\frac12$$ is bounded by $1$ for $(x,y) > 0$, and $G_\lambda \to 1$ pointwise as $\lambda\to0$, we have $$\lim_{\lambda\to0}(-\Delta_M - \lambda^2 \partial_t^2)^{-\frac12}F = \lim_{\lambda\to0}G_\lambda(-\Delta_M,-\partial_t^2) (-\Delta_M)^{-\frac12} f \otimes \rho = (-\Delta_M)^{-\frac12} f \otimes \rho$$ in $L^2$, and thus also as distributions. Therefore $\tilde R_\lambda F \to R f \otimes \rho$ as distributions, and so $$\liminf_{\lambda \to 0} \| \tilde R_\lambda F \|_{L^p(M\times {\mathbb R})} \ge {\left\|{R f \otimes \rho}\right\|}_{L^p(M\times {\mathbb R})} = {\left\|{Rf}\right\|}_{L^p(M)}.$$ Combining this with and , and the fact that $C^\infty_c(M) \cap D((-\Delta_M)^{-\frac12})$ is dense in $L^p(M)$,[^1] yields $R_p(M\times {\mathbb R}) \ge R_p(M)$.
Proof of the main theorem
=========================
In this section we carry out the construction that proves Proposition \[prop:construction\], which implies Theorem \[thm:main\].
Consider a sequence $(M_n)_{n \in {\mathbb N}}$ of complete $d$-dimensional Riemannian manifolds. We will connect the $M_n$-cylinders $(M_n \times {\mathbb R})_{n \in {\mathbb N}}$ along a ${\mathbb T}^d$-cylinder ${\mathbb T}^d \times {\mathbb R}$ as follows.[^2] For each $n \in {\mathbb N}$ fix a coordinate chart $U_n \subset M_n \times (-1/2,1/2)$ and a small ball $B_n \subset U_n$. Similarly, for each $n \in {\mathbb N}$ choose a small coordinate chart $U^\prime_n \subset {\mathbb T}^n \times {\mathbb R}$ such that the charts $(U^\prime_n)_{n \in {\mathbb N}}$ are pairwise disjoint, and a small ball $B^\prime_n \subset U^\prime_n$. For each $n \in {\mathbb N}$, glue the manifold $(M_n \times {\mathbb R}) \setminus B_n$ to $({\mathbb T}^n \times {\mathbb R}) \setminus B_n^\prime$ along the boundaries of $B_n$ and $B_n^\prime$; this is possible since both these balls are ‘Euclidean’ balls sitting inside coordinate charts. This results in a $C^0$-Riemannian manifold $(M,g^\prime)$, which is $C^\infty$ away from the set $\Sigma = \cup_n \partial B_n$ on which we glued the manifolds together. Mollify the metric to get a $C^\infty$-Riemannian manifold $(M,g)$ such that $g = g^\prime$ away from the $\varepsilon$-neighbourhood of $\Sigma$ for some very small $\varepsilon$. An artist’s impression of this construction, with $M_n = S^1$ for each $n$, is shown in Figure \[fig:construction\].
![Construction of $M$ from $(M_n)_{n \in {\mathbb N}}$.[]{data-label="fig:construction"}](diagram.eps){width="\textwidth"}
For each $n \in {\mathbb N}$ we have an inclusion map $$i_n \colon M_n \times (1,\infty) \to M$$ which is an isometry. From here on we fix $n$ and just write $i = i_n$. Functions on $M$ can be pulled back to $M_n \times (1,\infty)$; the pullback map is denoted $i^*$, so that for $f \colon M \to {\mathbb R}$ the function $i^* f \colon M_n \times (1,\infty) \to {\mathbb R}$ is defined by $$i^*f(x,t) = f(i(x,t)).$$ On the other hand, for $g \colon M_n \times (1,\infty) \to {\mathbb R}$ we can define a pushforward $i_* g \colon M \to {\mathbb R}$ by setting $i_* g(i(x,t)) := g(x,t)$ on $i(M_n \times (1,\infty))$ and extending by zero to the rest of $M$. For a function $g \colon M_n \times {\mathbb R}\to {\mathbb R}$ and for $s \in {\mathbb R}$ we let $\tau_s g \colon M_n \times {\mathbb R}\to {\mathbb R}$ be the translated function $\tau_s g(x,t) := g(x,t-s)$. Similarly if $g \colon M_n \times (1,\infty) \to {\mathbb R}$ we can define $\tau_s g \colon M_n \times (1 + s, \infty) \to {\mathbb R}$. These concepts apply equally well to vector fields in place of functions.
We will need the following lemma, which relates the heat flow on $M_n \times {\mathbb R}$ to the one on $M$.
\[heat\_flow\_convergence\] Let $F \colon M_n\times{\mathbb R}\to {\mathbb R}$ be smooth and compactly supported, and fix $\sigma>0$. Then for every $(x,t) \in M_n\times {\mathbb R}$, $$\lim_{s\to +\infty} (e^{\sigma\Delta_M}i_*\tau_s F) (i(x,t+s)) = (e^{\sigma\Delta_{M_n\times{\mathbb R}}}F)(x,t).$$
Let $W_{x,t}(\sigma)$ be a Brownian motion on $M_n\times{\mathbb R}$ at time $\sigma$ starting from the point $(x,t)$. Since the generator $\frac12\Delta_{M_n\times{\mathbb R}}$ satisfies $\frac12i_*\circ\Delta_{M\times {\mathbb R}}|_{i(M_n\times (1,+\infty))} = \frac12\Delta_{M}|_{i(M_n\times (1,+\infty))}$, defining the stopping time $$T(x,t):= \inf\left\{s : W_{x,t}(s) \in M_n \times (-\infty,1) \right\},$$ we have that $i(W_{x,t}(\sigma))$ is a Brownian motion on $M$ for $\sigma < T(x,t)$. Therefore there exists a Brownian motion $\tilde W_{i(x,t)}(\sigma)$ on $M$ such that $\tilde W(\sigma) = i(W(\sigma))$ for $\sigma<T$; if ${\overline}W$ is a Brownian motion on $M$, we can take for example $$\tilde W_{i(x,t)}(\sigma) = \left\{
\begin{aligned}
&i(W_{x,t}(\sigma)) &\text{if } \sigma < T, \\
&{\overline}W_{i(W_{x,t}(T))}(\sigma - T) &\text{if } \sigma\ge T.
\end{aligned}\right.$$ We have that $$\begin{aligned}
&(e^{\sigma\Delta_M}i_*\tau_s F) (i(x,t+s)) \\
&= {\mathbb E}[(i_*\tau_s F)(\tilde W_{i(x,t+s)}(2\sigma))]\\
&= {\mathbb E}[(i_*\tau_s F)(\tilde W_{i(x,t+s)}(2\sigma)) {\mathbb 1}_{2\sigma<T}] + {\mathbb E}[(i_*\tau_s F)(\tilde W_{i(x,t+s)}(2\sigma)){\mathbb 1}_{2\sigma\ge T}]\\
&= {\mathbb E}[(\tau_s F)(W_{x,t+s}(2\sigma)){\mathbb 1}_{2\sigma<T}] + {\mathbb E}[(i_*\tau_s F)(\tilde W_{i(x,t+s)}(2\sigma)){\mathbb 1}_{2\sigma\ge T}]\\
& \begin{multlined}
={\mathbb E}[(\tau_s F)(W_{x,t+s}(2\sigma))] \\
- {\mathbb E}[(\tau_s F)(W_{x,t+s}(2\sigma)){\mathbb 1}_{2\sigma\ge T}] + {\mathbb E}[(i_*\tau_s F)(\tilde W_{i(x,t+s)}(2\sigma)){\mathbb 1}_{2\sigma\ge T}]
\end{multlined}\\
&\begin{multlined}
= (e^{\sigma\Delta_{M_n\times{\mathbb R}}}\tau_sF)(x,t+s)\\
- {\mathbb E}[(\tau_s F)(W_{x,t+s}(2\sigma)){\mathbb 1}_{2\sigma\ge T}] + {\mathbb E}[(i_*\tau_s F)(\tilde W_{i(x,t+s)}(2\sigma)){\mathbb 1}_{2\sigma\ge T}].
\end{multlined}\end{aligned}$$ Therefore $$\left|(e^{\sigma\Delta_M}i_*\tau_s F) (i(x,t+s)) - (e^{\sigma\Delta_{M_n\times{\mathbb R}}}\tau_sF)(x,t+s)\right|
\le 2 {\left\|{F}\right\|}_{L^\infty}{\mathbb P}(T(x,t+s)\le2\sigma).$$ Since $\Delta_{M_n\times {\mathbb R}}$ is translation invariant in the ${\mathbb R}$ coordinate, we have that $$\begin{aligned}
{\mathbb P}(T(x,t+s)\le 2\sigma) &\le {\mathbb P}\big(\{W_{x,t+s}(\sigma') \in M_n\times (-\infty,1) \text{ for some }\sigma'\le2\sigma+1\}\big) \\
&={\mathbb P}\big(\{W_{x,t}(\sigma') \in M_n\times (-\infty,1-s) \text{ for some }\sigma'\le2\sigma+1\}\big)\end{aligned}$$ and by continuity of $W_{x,t}(\cdot)$, this tends to $0$ as $s \to \infty$. Thus we find that $$\lim_{s\to +\infty} \bigg( (e^{\sigma\Delta_M}i_*\tau_s F) (i(x,t+s)) - (e^{\sigma\Delta_{M_n\times{\mathbb R}}}\tau_sF)(x,t+s) \bigg) = 0.$$ The conclusion follows from translation invariance of $\Delta_{M_n\times{\mathbb R}}$ in ${\mathbb R}$.
We return to the proof of Proposition \[prop:construction\]. Fix ${\varepsilon}> 0$, and choose $F = \Delta_{M_n \times {\mathbb R}} H$ for some $H \in C^\infty_c(M_n \times {\mathbb R})$ with ${\left\|{F}\right\|}_{L^p} = 1$ such that $${\left\|{R_{M_n\times{\mathbb R}} F}\right\|}_{L^p} \ge (R_p(M_n)-{\varepsilon}) \wedge {\varepsilon}^{-1}.$$ Such a function exists by Lemmas \[lem:range-lap-dens\] and \[lem:cyl-R\]. We claim that $$\label{eqn:dist-limit}
\lim_{s \to +\infty} \tau_{-s} i^*R_{M}(i_* \tau_s F) = R_{M_n\times {\mathbb R}} F$$ as distributions. Assuming for the moment, we have $$\begin{aligned}
\limsup_{s \to \infty} {\left\|{R_{M}(i_* \tau_s F)}\right\|}_{L^p(M)} &\ge \limsup_{s \to \infty} {\left\|{i^*R_{M}(i_* \tau_s F)}\right\|}_{L^p(M_n\times{\mathbb R})}\\
&= \limsup_{s \to \infty} {\left\|{\tau_{-s}i^*R_{M}(i_* \tau_s F)}\right\|}_{L^p(M_n\times{\mathbb R})}\\
&\ge {\left\|{R_{M_n\times {\mathbb R}} F}\right\|}_{L^p(M_n\times{\mathbb R})} \ge R_p(M_n)-{\varepsilon},\end{aligned}$$ while for all $s \in {\mathbb R}$ $${\left\|{i_* \tau_s F}\right\|}_{L^p(M)} \leq {\left\|{\tau_s F}\right\|}_{L^p(M_n\times{\mathbb R})} = {\left\|{F}\right\|}_{L^p(M_n\times{\mathbb R})}\le 1.$$ The result follows, so it remains to prove .
For $s$ sufficiently large, we have that $$i_*\tau_sF = i_*\tau_s(\Delta_{M_n \times {\mathbb R}} H) = i_*(\Delta_{M_n \times {\mathbb R}}\tau_s H) = \Delta_M i_* \tau_s H,$$ therefore $i_*\tau_sF \in D(\Delta_M^{-1}) \subseteq D((-\Delta_M)^{-\frac12})$, and hence $$R(i_*\tau_sF) = \nabla \left((-\Delta)^{-\frac12}_Mi_*\tau_sF\right)$$ as a distribution. To test the distributional convergence, let $X$ be a smooth compactly supported vector field in $M_n \times {\mathbb R}$. For large $s$ we have that $$\begin{aligned}
{\left\langle\tau_{-s} i^*R_{M}(i_* \tau_s F),X\right\rangle} &= {\left\langleR_{M}(i_* \tau_s F),i_*\tau_sX\right\rangle} \\
&= {\left\langle(-\Delta)^{-\frac12}_Mi_*\tau_sF,\operatorname{div}(i_*\tau_sX)\right\rangle} \\
&= {\left\langle(-\Delta)^{-\frac12}_Mi_*\tau_sF,i_*\tau_s\operatorname{div}(X)\right\rangle}.\end{aligned}$$ Therefore it is enough to show that for every $G \in C^\infty_c(M_n\times {\mathbb R})$, $$\label{limit}
\lim_{s\to \infty}{\left\langle(-\Delta)^{-\frac12}_Mi_*\tau_sF,i_*\tau_sG\right\rangle} = {\left\langle(-\Delta)^{-\frac12}_{M_n\times{\mathbb R}} F,G\right\rangle}.$$ By the well-known formula $$(-\Delta)^{-\frac12} = \pi^{-\frac12} \int_0^{+\infty} \sigma^{-\frac12}e^{\sigma\Delta} \, d\sigma,$$ is equivalent to showing that $$\label{eqn:limit-integrals}
\lim_{s\to \infty}\int_0^{+\infty} \sigma^{-\frac12} {\left\langlee^{\sigma\Delta_M}i_*\tau_sF,i_*\tau_sG\right\rangle} \, d\sigma
= \int_0^{+\infty} \sigma^{-\frac12} {\left\langlee^{\sigma\Delta_{M_n\times{\mathbb R}}} F,G\right\rangle} \, d\sigma.$$ Note that $$\left|\sigma^{-\frac12} {\left\langlee^{\sigma\Delta_M}i_*\tau_sF,i_*\tau_sG\right\rangle}\right| \le \sigma^{-\frac12}
{\left\|{i_*\tau_sF}\right\|}_{L^2}{\left\|{i_*\tau_sG}\right\|}_{L^2}\le \sigma^{-\frac12} {\left\|{F}\right\|}_{L^2}{\left\|{G}\right\|}_{L^2}$$ and $$\resizebox{\textwidth}{!}{$\displaystyle
\left|\sigma^{-\frac12} {\left\langlee^{\sigma\Delta_M}i_*\tau_sF,i_*\tau_sG\right\rangle}\right| = \left|\sigma^{-\frac32} {\left\langlee^{\sigma\Delta_M}\sigma\Delta_Mi_*\tau_sH,i_*\tau_sG\right\rangle}\right| \lesssim \sigma^{-\frac32} {\left\|{H}\right\|}_{L^2}{\left\|{G}\right\|}_{L^2}.$}$$ Since the function $\min(\sigma^{-\frac12},\sigma^{-\frac32})$ is integrable, by dominated convergence will be proved if we show $$\label{eqn:limit-integrand}
\lim_{s\to \infty} {\left\langlee^{\sigma\Delta_M}i_*\tau_sF,i_*\tau_sG\right\rangle} = {\left\langlee^{\sigma\Delta_{M_n\times{\mathbb R}}} F,G\right\rangle}$$ for every $\sigma > 0$. We show by writing $$\begin{aligned}
\lim_{s\to\infty}{\left\langlee^{\sigma\Delta_M}i_*\tau_sF,i_*\tau_sG\right\rangle} & = \lim_{s\to\infty}{\left\langle\tau_{-s}i^*e^{\sigma\Delta_M}i_*\tau_sF,G\right\rangle}\\
& = \lim_{s\to\infty} \int_{1-s}^{+\infty} \int_{M_n} (e^{\sigma\Delta_M}i_*\tau_s F) (i(x,t+s))G(x,t) \, dx \, dt \\
& = \int_{\mathbb R}\int_{M_n} (e^{\sigma\Delta_{M_n\times{\mathbb R}}} F)(x,t) G(x,t) \, dx \, dt\\
& = {\left\langlee^{\sigma\Delta_{M_n\times{\mathbb R}}} F,G\right\rangle},\end{aligned}$$ using Lemma \[heat\_flow\_convergence\] and dominated convergence (by ${\left\|{F}\right\|}_{L^\infty} |G(x,t)|$). This completes the proof of Proposition \[prop:construction\], and hence establishes Theorem \[thm:main\].
[^1]: This follows from the inclusion $D((-\Delta_M)^{-\frac12}) \supseteq D((-\Delta_M)^{-1}) \supseteq \Delta_M(C_c^\infty(M))$, which is dense by Lemma \[lem:range-lap-dens\]. See also [@MR3445205 Lemma 2.2]. Again, recall that $L^p(M)$ denotes the corresponding space of mean zero functions when $M$ has finite volume.
[^2]: Of course, one could connect the $M_n$-cylinders to each other directly, without needing the ${\mathbb T}^d$-cylinder. This would work just as well.
|
---
abstract: 'We consider the stationary incompressible Navier Stokes equation in the exterior of a disk $B\subset \mathbb{R}^{2}$ with non-zero Dirichlet boundary conditions on the disk and zero boundary conditions at infinity. We prove the existence of solutions for an open set of boundary conditions without symmetry.'
author:
- |
Matthieu Hillairet\
[Université Paris Dauphine ]{}\
[Place du Maréchal De Lattre De Tassigny]{}\
[75775 Paris Cedex 16 - France]{}\
[[email protected]]{}
- |
Peter Wittwer[^1]\
[University of Geneva]{}\
[24, Quai Ernest Ansermet]{}\
[1205 Geneva - Switzerland]{}\
[ [email protected] ]{}
title: |
On the existence of solutions to the\
planar exterior Navier Stokes system
---
Introduction
============
In this paper we consider the incompressible Navier Stokes equations in an exterior domain: $$\left \{
\begin{array}
[c]{r}\Delta \mathbf{u}-\nabla p=\mathbf{u}\cdot \nabla \mathbf{u}\text{ },\\
\operatorname{div}\mathbf{u}=0\text{ },
\end{array}
\right. \text{ in $\mathbb{R}^{2}\setminus \overline{B}$ }, \label{NS}$$ with $B$ a smooth bounded domain, with non-zero Dirichlet boundary conditions on $\partial B$, and zero boundary conditions at infinity:$$\mathbf{u}_{|_{\partial B}}=\mathbf{u}^{\ast}\text{ },\quad \lim_{|\mathbf{x}|\rightarrow \infty}\mathbf{u}(\mathbf{x})=0\text{ }. \label{BC}$$ Of particular interest is the case of boundary data $\mathbf{u}^{\ast}$ with zero flux: $$\int_{\partial B}\mathbf{u}^{\ast}\cdot \mathbf{n}\text{ }\mathrm{d}\sigma=0\text{ }. \label{fluxnul}$$ We note that, since the size of $B$ is arbitrary, we have set without restriction of generality all the physical constants in (\[NS\]) equal to one.
The above system is a special case of the exterior Navier Stokes problem: $$\left \{
\begin{array}
[c]{r}-\left( \mathbf{u}\cdot \nabla \right) \mathbf{u}-\lambda \partial
_{1}\mathbf{u}+\Delta \mathbf{u}-\nabla p=0\text{ },\\
\nabla \cdot \mathbf{u}=0\text{ },
\end{array}
\right. \text{ in }\mathbb{R}^{n}\setminus \overline{B}\text{ },
\label{eq_NSE3}$$ with $n=2$ or $3$, with $B$ a smooth bounded domain in $\mathbb{R}^{n}$, with boundary conditions (\[BC\]), and with $\lambda \in \{0,1\}$ distinguishing between the case of a flow around $B$ ($\lambda=0$) and a flow past $B$ ($\lambda=1$), respectively. The system (\[NS\])-(\[fluxnul\]) corresponds to $n=2$ and $\lambda=0$. The case $\lambda=0$ is in many respects more complicated than the case $\lambda=1$, and, whereas the picture is rather complete for $n=3$, the case $n=2$, $\lambda=0$, presents particular difficulties. The difficulty with the classical method for solving the Navier Stokes equations consists in the fact that the linearization around $\mathbf{u=0}$ is given by the Stokes system, which, for $n=2$, does not admit a solution satisfying (\[BC\]), unless the domain $B$ and the boundary data $\mathbf{u}^{\ast}$ satisfy certain symmetry conditions. This fact is known as the Stokes paradox. [For completeness we note that if one relaxes the no flux condition (\[fluxnul\]), there exists a two parameter family of solutions to (\[NS\])-(\[fluxnul\]), the so called Hamel solutions, see [@Galdi94II]. [These examples emphasize that the decay of solutions can be arbitrary slow and that uniqueness might be lost for some boundary data. However, these solutions have flux larger than one, and are far from the regime which we will consider here.]{}]{}
In what follows we construct a new class of solutions to (\[NS\])-(\[fluxnul\]), by linearizing not around $\mathbf{u=0}$, but around $\mathbf{u}=\mu \mathbf{x}^{\perp}/\left \vert \mathbf{x}\right \vert ^{2}$, with $\left \vert \mu \right \vert > \sqrt{48}$. This improves the decay of the solutions to the vorticity equation, yielding vorticities decaying at infinity generically faster than $\left \vert \mathbf{x}\right \vert ^{-2}$, instead of like $\left \vert \mathbf{x}\right \vert ^{-1}$ as would be the case for the Stokes equation, thus avoiding the Stokes paradox when reconstructing $\mathbf{u}$ *via* the Bio-Savart law.
To put our problem into a wider context, we briefly recall the concept of weak solutions for (\[eq\_NSE3\]), (\[BC\]) [ (also known as generalized solutions or $D$–solutions),]{} and the method of J. Leray [@Leray33] for proving the existence of such weak solutions.
Given $\mathbf{u}^{\ast}\in H^{1/2}(\partial{B})$ satisfying (\[fluxnul\]), a function $\mathbf{u}$ which satisfies the following conditions is called a weak solution to (\[eq\_NSE3\]), (\[BC\]):
1. $\mathbf{u}\in D^{1,2}(\mathbb{R}^{n}\setminus \overline{B})$, where $D^{1,2}(\mathbb{R}^{n}\setminus \overline{B})$ is the subset of $L_{loc}^{1}(\mathbb{R}^{2}\setminus \overline{B})$ containing functions with gradient in $L^{2}(\mathbb{R}^{n}\setminus \overline{B}),$
2. $\mathbf{u}$ is divergence-free and $\mathbf{u}=\mathbf{u}^{\ast}$ on $\partial{B}$,
3. for all divergence-free vector fields $\mathbf{w}\in
C_{c}^{\infty}(\mathbb{R}^{n}\setminus B)$, [there holds]{}:$$\int_{\mathbb{R}^{n}\setminus \overline{B}}\nabla \mathbf{u}\colon
\nabla \mathbf{w} + \int_{\mathbb{R}^{n}\setminus \overline{B}}(\left(
\mathbf{u}\cdot \nabla \right) \mathbf{u}+\lambda \partial_{1}\mathbf{u})\cdot \mathbf{w}=0\text{ }.$$
The method of J. Leray to prove the existence of solutions according to this definition, and a posteriori to (\[eq\_NSE3\]), (\[BC\]), in the sense of distributions, consists in the following steps:
- First, one introduces a sequence of approximate problems by restricting (\[eq\_NSE3\]) to bounded subsets $\Omega \subset \mathbb{R}^{n} $ containing $B$, with zero Dirichlet boundary conditions on $\partial \Omega \setminus
\partial{B}$.
- Second, one proves the existence of (weak) solutions to all these approximate problems.
- Third, one shows that for any sequence of bounded subsets exhausting $\mathbb{R}^{n}\setminus \overline{B}$, there exists a subsequence, such that the corresponding approximate solutions converge to a weak solution of (\[eq\_NSE3\]), (\[BC\]).
- Finally, given a weak solution $\mathbf{u}$, a pressure $p$ an be constructed *via* De Rham’s theory, such that the equations (\[eq\_NSE3\]) are satisfied in $\mathcal{D}^{\prime}(\mathbb{R}^{n}\setminus \overline{B})$.
See also [@Hillairet07d; @HillairetSerre03; @Serre87; @Weinberger72; @Weinberger73], where this method has been adapted [to a similar system with more general boundary conditions]{}. Note that if $B$ has a smooth boundary, the ellipticity of the Stokes operator (see [@Galdi94I Section IX.1]) and the smoothness of $\mathbf{u}^{\ast}$ imply that weak solutions are smooth. Therefore, for smooth data, the only possible shortcoming of weak solutions is that they may not satisfy the boundary condition at infinity in a point-wise sense. [Much]{} work has been devoted to clarify the situation in various cases (see [@Galdi94II] for more details):
For $n=3$, the condition $\mathbf{u}\in D^{1,2}(\mathbb{R}^{3}\setminus
\overline{B})$ implies that weak solutions tend to zero at infinity. The exact decay can be obtained by various methods yielding the following results:
- for $\lambda=1$, there exists a solution that decays like the fundamental solution of the Oseen equation (the linear system obtained from (\[eq\_NSE3\]) by deleting the nonlinear convective terms) [@Babenko73; @Farwig92; @Finn59]. This result can be obtain by a detailed analysis of the Oseen equation with a source term in the usual Sobolev spaces [@Babenko73; @Finn59], and also in weighted Sobolev spaces [@Farwig92].
- for $\lambda=0$ and sufficiently small boundary data, there exists a unique weak solution, and this solution decays like a Landau solution [@KorolevSverak10], a special solution of the nonlinear system which decays like $1/|\mathbf{x}|.$ This result can been obtained by constructing first a strong solution to (\[BC\]), (\[eq\_NSE3\]), which is asymptotic to the Landau solution, by perturbative techniques. Using the known decay of this particular solution as an input [@Galdi94II Section IX.9], one then proves a weak-strong uniqueness result for small data.
For $n=2$, the situation is more delicate since the condition $\mathbf{u}\in
D^{1,2}(\mathbb{R}^{2}\setminus \overline{B})$ does not guarantee that the boundary condition at infinity is satisfied:
- For $\lambda=1$, the relevant linear system is again the Oseen equation, but the results concerning the decay are limited to small data, since, as for the case $n=3$, $\lambda=0$, perturbative techniques are used to prove the existence of a strong solution decaying at infinity like the fundamental solution of the Oseen equation. This solution is then again used as an input to a weak strong uniqueness argument in order to show the decay of weak solutions. These results can be found in [@Galdi93].
- The case $\lambda=0$ remains largely open. [As we already pointed out, the]{} problem is that the solution to the Stokes equation with boundary data $\mathbf{u}^{\ast}\neq0$ diverges at infinity, unless one makes additional assumptions on the domain $B$ and the data $\mathbf{u}^{\ast}$. Partial results for the Navier Stokes system [with symmetric data]{} can be found in [@Galdi02; @Russo10; @Russopp; @PileckasRusso12] .
From now on we limit the discussion to the case where $B$ is a disk of radius one. We choose $\mathbf{x}=(x,y)$ Cartesian coordinates with the origin at the center of $B$, $(r,\theta)\in \Omega:=(0,\infty)\times(-\pi,\pi)$ the associated polar coordinates, and $(\mathbf{e}_{r},\mathbf{e}_{\theta})$ the corresponding local orthonormal basis. For the function $\mathbf{u}$ we have in polar coordinates: $$\mathbf{u}(r,\theta)=u_{r}(r,\theta)\mathbf{e}_{r}+u_{\theta}(r,\theta
)\mathbf{e}_{\theta},\quad \forall \text{ }(r,\theta)\in \Omega \,. \label{pc}$$
The following theorem is our main result:
\[thm\_main\] Let $\mu_{0}>{\mu_{crit}}\equiv \sqrt{48}$ and $\mathbf{u}^{\ast}\in C^{\infty}(\partial B)$ satisfying be sufficiently close to ${\mathbf{u}_{\mu_{0}}^{\ast}:=}\mu_{0}\mathbf{e}_{\theta}$. Then, the equations (\[NS\]), (\[BC\]), with boundary condition $\mathbf{u}^{\ast}$, have at least one solution $(\mathbf{u},p)\in C^{\infty}(\mathbb{R}^{2}\setminus \overline{B})^{2}\times C^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$. Moreover, there exist $\mu$ close to $\mu_{0}$ such that: $$\lim_{r\rightarrow \infty}\text{ }r\left \Vert \mathbf{u}(r,\theta)-\frac
{\mu \mathbf{e}_{\theta}}{r};L^{\infty}(-\pi,\pi)\right \Vert =0\text{ }.
\label{bound}$$
If the pair $(u(x,y),v(x,y))$ is a solution for the boundary condition $(u^{\ast}(x,y),v^{\ast}(x,y))$, then the pair $(u(x,-y),-v(x,-y))$ is a solution for the boundary condition $(u^{\ast}(x,-y),-v^{\ast}(x,-y))$. Thus, our result extends to $\mu_{0}<-{\mu_{crit}}$.
If $\mathbf{u}(r,\theta)$ is a solution for the boundary condition** **$\mathbf{u}^{\ast}$on the complement of the unit disk, then for all $\lambda>0$, $\lambda \mathbf{u}(\lambda r,\theta)$ is a solution for the boundary condition $\lambda \mathbf{u}^{\ast}$ on the complement of the disk of radius $\lambda^{-1}$.
The restriction to the case where $B$ is a disk is for the sake of simplicity only. This permits to rewrite the system in polar coordinates, yielding explicit expressions for the solutions. We expect that with more work the results can be generalized to arbitrary smooth $B$.
To prove Theorem \[thm\_main\] we proceed as follows: We fix $\mu>\mu_{crit}$ and consider the pair $(\mathbf{u}_{\mu},p_{\mu})$: $$\mathbf{u}_{\mu}(r,\theta)=\dfrac{\mu \mathbf{e}_{\theta}}{r}\text{ },\qquad
p_{\mu}(r,\theta)=-\frac{1}{2}\dfrac{\mu^{2}}{r^{2}}\text{ },\qquad
\forall \,(r,\theta)\in{\Omega}\text{ }, \label{explicit}$$ which is an exact solution to (\[NS\]), (\[BC\]). Next we set, $(\mathbf{u},p)=(\mathbf{u}_{\mu}+\mathbf{v},p=p_{\mu}+$ ${q})$ and prove, that for all sufficiently small boundary conditions $\mathbf{v}^{\ast}$ satisfying$$\int_{\partial B}\mathbf{v}^{\ast}\cdot \mathbf{n}\text{ }\mathrm{d}\sigma=0\text{ },$$ there existence of a solution $(\mathbf{v},$ ${q})\in C^{\infty
}(\mathbb{R}^{2}\setminus \overline{B})^{2}\times C^{\infty
}(\mathbb{R}^{2}\setminus \overline{B})$ such that $\left. \mathbf{v}\right \vert _{\partial B}=\mathbf{v}^{\ast}+\mu_{\ast}$, for some $\mu_{\ast
}>{\mu_{crit}}$ depending on $\mu$ and $\mathbf{v}^{\ast}$. In a final step, we show that this function can be inverted, giving $\mu$ as a function of $\mu_{\ast}$ and $\mathbf{v}^{\ast}$, thus yielding Theorem \[thm\_main\].
The feasibility of our approach relies on the fact [that ]{}the system obtained by linearizing (\[NS\]), (\[BC\]) around the explicit solution $(\mathbf{u}_{\mu},p_{\mu})$ can be analyzed explicitly. As mentioned above, when compared with the case $\mu=0$,* i.e.*, the Stokes equation, the vorticity decays for $\mu>{\mu_{crit}}$ faster than ${1/r^{2}}$, instead of like ${1/r}$, such that $\mathbf{u}$ can be shown [to]{} decay faster than $1/r$ at infinity, making the nonlinearity subcritical. Introducing suitable function spaces, we are then able to solve the full non-linear system by a classical fixed-point argument.
Dynamical system formulation
============================
Let $(\mathbf{u},p)\in C^{\infty}(\mathbb{R}^{2}\setminus B)$ be a solution to (\[NS\]), (\[BC\]), satisfying (\[fluxnul\]). We first make the construction of the stream-function $\psi$ associated with $\mathbf{u}$ precise. Since $\mathbf{u}\in C^{\infty
}(\mathbb{R}^{2}\setminus B)$, we have in particular that $\mathbf{u}^{\ast
}\in C^{\infty}(\partial B)$. Let $\mathbf{u}_{int} \in C^{\infty}(\overline{B})$ satisfy $\mathbf{u}_{int}=\mathbf{u}^{\ast}$ on $\partial{B}$. Such a function exists since $\mathbf{u}^{\ast}$ satisfies (\[fluxnul\]). For instance, $\mathbf{u}_{int}$ can be the solution to the Stokes equations on $B$, with boundary condition $\mathbf{u}^{\ast}$ on $\partial B$. Then, setting: $${\mathbf{\bar{u}}}(x,y)=\left \{
\begin{array}
[c]{ll}\mathbf{u} & \text{in $\mathbb{R}^{2}\setminus \overline{B\,}$},\\
\mathbf{u}_{int} & \text{in $\overline{B}$}\,,
\end{array}
\right.$$ we obtain a continuous divergence-free vector-field on the whole of $\mathbb{R}^{2}$. Furthermore, this function is smooth on both sides of $\partial B$ so that there exists $\psi \in C^{1}(\mathbb{R}^{2})\cap C^{\infty}(\overline{B})\cap C^{\infty}(\mathbb{R}^{2}\setminus{B})$ satisfying $\mathbf{u}=\nabla^{\bot}\psi$.
Instead of (\[NS\]), (\[BC\]), we consider now the equation for the stream function $\psi$ and the vorticity $\omega=\nabla \times \mathbf{u}$, $$\left \{
\begin{array}
[c]{rcl}\Delta \psi & = & -\omega \\
\Delta \omega & = & \mathbf{u}\cdot \nabla \omega
\end{array}
\right. \quad \text{ in $\mathbb{R}^{2}\setminus \overline{B}\,.$}\nonumber$$ For the function $\mathbf{u}$ we have in polar coordinates (\[pc\]), and the vorticity becomes:$$\omega=\dfrac{1}{r}\partial_{r}(ru_{\theta})-\dfrac{1}{r}\partial_{\theta
}u_{r},\quad \forall \text{ }(r,\theta)\in \Omega \,.$$ For the boundary data we have:$$\mathbf{u}^{\ast}(\theta)=u_{r}^{\ast}(\theta)\mathbf{e}_{r}+u_{\theta}^{\ast
}(\theta)\mathbf{e}_{\theta}\text{ },\quad \forall \text{ }\theta \in(-\pi
,\pi)\,.$$ In polar coordinates we get the following equations for the stream function $\psi$ and the vorticity $\omega$: $$\left \{
\begin{array}
[c]{rcl}\partial_{rr}\psi+\dfrac{1}{r}\partial_{r}\psi+\dfrac{1}{r^{2}}\partial
_{\theta \theta}\psi & = & -\omega \,,\\[6pt]\partial_{rr}\omega+\dfrac{1}{r}\partial_{r}\omega+\dfrac{1}{r^{2}}\partial_{\theta \theta}\omega & = & u_{r}\partial_{r}\omega+\dfrac{u_{\theta}}{r}\partial_{\theta}\omega \,,
\end{array}
\right. \quad \forall \text{ }(r,\theta)\in \Omega \text{ }, \label{NS_polar}$$ and$$\left \{
\begin{array}
[c]{rcl}\smallskip u_{r} & = & \dfrac{\partial_{\theta}\psi}{r}\,,\\
u_{\theta} & = & -\partial_{r}\psi \,,
\end{array}
\right. \text{ }\forall \text{ }(r,\theta)\in \Omega \text{ },
\label{comp_polar}$$ together with the boundary conditions:$$\left \{
\begin{array}
[c]{rclcrcl}u_{r}(1,\theta) & = & u_{r}^{\ast}(\theta)\,, & & \lim_{r\rightarrow \infty}{u}_{r}(r,\theta) & = & 0\,,\\[6pt]u_{\theta}(1,\theta) & = & u_{\theta}^{\ast}(\theta)\,, & & \lim
_{r\rightarrow \infty}{u}_{\theta}(r,\theta) & = & 0\,,
\end{array}
\right.
\quad \forall \text{ }\theta \in(-\pi,\pi)\,. \label{bc_polar}$$ For the exact solution $(\mathbf{u}_{\mu},p_{\mu})$ given by (\[explicit\]) we have in polar coordinates for the corresponding stream-function-vorticity pair $(\psi_{\mu},\omega_{\mu})$, for all $\mu \in \mathbb{R}$: $$\left \{
\begin{array}
[c]{ccc}\smallskip \psi_{\mu}(r,\theta) & = & -{\mu \ln(r)}\,,\\
\omega_{\mu}(r,\theta) & = & 0\text{ },
\end{array}
\right. \qquad \forall \text{ }(r,\theta)\in \Omega \, \text{$.$}$$ In order to prove Theorem \[thm\_main\] we construct, as explained above, a solution which is a perturbation of the explicit solutions $(\mathbf{u}_{\mu
},p_{\mu})$. We therefore set $\psi=\psi_{\mu}+\gamma$ and $\omega=\omega
_{\mu}+w$. Substituting this Ansatz into (\[NS\_polar\]), (\[comp\_polar\]), we obtain the following equivalent system for the unknowns $(\gamma,w)$: $$\left \{
\begin{array}
[c]{lcl}\partial_{rr}\gamma+\frac{1}{r}\partial_{r}\gamma+\dfrac{1}{r^{2}}\partial_{\theta \theta}\gamma & = & -w\,,\\[6pt]\partial_{rr}w+\frac{1}{r}\partial_{r}w+\dfrac{1}{r^{2}}\partial_{\theta
\theta}w-\dfrac{\mu}{r^{2}}\partial_{\theta}w & = & {\dfrac{\partial_{\theta
}\gamma}{r}\partial_{r}w-\dfrac{\partial_{r}\gamma}{r}\partial_{\theta}w}\text{ },
\end{array}
\right. \quad \text{ }\forall \text{ }(r,\theta)\in \Omega \text{ },
\label{NS_pert}$$ with the boundary conditions:$$\left \{
\begin{array}
[c]{lcl}\smallskip \partial_{\theta}\gamma(1,\theta) & = & v_{r}^{\ast}(\theta)\text{
},\\
\smallskip \partial_{r}\gamma(1,\theta) & = & -v_{\theta}^{\ast}(\theta)\text{
},\\
{\lim_{r\rightarrow \infty}}\, \left( |\gamma(r,\theta)|+\left \vert
\partial_{r}\gamma(r,\theta)\right \vert \right) & = & 0\text{ },
\end{array}
\right. \quad \text{ }\forall \text{ }\theta \in(-\pi,\pi)\text{ },
\label{BC_pert}$$ for certain $(v_{r}^{\ast}(\theta),v_{\theta}^{\ast}(\theta))$ to be defined later on, satisfying:$$\int_{\partial B}v_{r}^{\ast}\text{ }\mathrm{d}\sigma=0\text{ }, \label{abc}$$ and which are small in a sense to be made precise.
Following the method developed in [@HillairetWittwer09], we solve (\[NS\_pert\]), (\[BC\_pert\]), for data ${(v_{r}^{\ast},v_{\theta}^{\ast})}$, by interpreting the radial coordinate $r$ as a time and by expanding in a Fourier series: $$\gamma(r,\theta)=\sum_{n\in \mathbb{Z}}\gamma_{n}(r)e^{in\theta}\text{ },\qquad
w(r,\theta)=\sum_{n\in \mathbb{Z}}w_{n}(r)e^{in\theta}\text{ }.$$
#### Notation.
To unburden the notation we write for the Fourier series of $\gamma$ and $w$:$${\hat{\gamma}=(\gamma_{n})_{n\in \mathbb{Z}}\,,\quad \hat{w}=(w_{n})_{n\in \mathbb{Z}}\,,} \label{notation}$$ and analogously for all other functions.
From (\[NS\_pert\]), (\[BC\_pert\]) we obtain, for $n\in \mathbb{Z}$, the following system of ordinary differential [equations:]{}$$\left \{
\begin{array}
[c]{lll}\smallskip \partial_{rr}\gamma_{n}+\dfrac{1}{r}\partial_{r}\gamma_{n}-\dfrac{n^{2}}{r^{2}}\gamma_{n} & = & -w_{n}\,,\\
\partial_{rr}w_{n}+\dfrac{1}{r}\partial_{r}w_{n}-\dfrac{i\mu n+n^{2}}{r^{2}}w_{n} & = & F_{n}\,,
\end{array}
\right. \quad \text{ on }(1,\infty)\,, \label{Fourier}$$ with the source term $F_{n}$ given by: $$F_{n}= - {\dfrac{i}{r}}{\sum_{k+l=n}}\left( k\,w_{k}\, \partial_{r}\gamma
_{l}-l\, \gamma_{l}\, \partial_{r}w_{k}\right) \,, \label{eq_Fn}$$ and with the boundary conditions:$$\left \{
\begin{array}
[c]{lcl}\smallskip in\gamma_{n}(1) & = & v_{r,n}^{\ast}\text{ },\\
\smallskip - \partial_{r}\gamma_{n}(1) & = & v_{\theta,n}^{\ast}\text{ },\\
{\lim_{r\rightarrow \infty}}\, \left( |\gamma_{n}(r)|+\left \vert \partial
_{r}\gamma_{n}(r)\right \vert \right) & = & 0\text{ },
\end{array}
\right. \quad \forall \,n\in \mathbb{Z}\setminus \{0\} \,. \label{eq_bcFn}$$ Note that $v_{r,0}^{\ast}=0$ by assumption (\[abc\]) [and that]{} the value of $\gamma_{0}(1)$ is irrelevant, *i.e.*, the stream function is only unique up to an additive constant. As we [show later in this section, the value $v_{\theta,0}^{\ast}$]{} cannot be chosen freely if one wants the solution $\gamma_{0}$ to satisfy the boundary condition at infinity.
For convenience, we [first solve with boundary conditions:$$\left \{
\begin{array}
[c]{lcl}\smallskip \gamma_{n}(1) & = & \gamma_{n}^{\ast}\text{ },\\
\smallskip w_{n}(1) & = & \omega_{n}^{\ast}\text{ },\\
{\lim_{r\rightarrow \infty}}\, \left( |\gamma_{n}(r)|+\left \vert
w_{n}(r)\right \vert \right) & = & 0\text{ },
\end{array}
\right. \quad \forall \,n\in \mathbb{Z}\setminus \{0\} \,, \label{BC_Fourier}$$ instead of (\[BC\_Fourier\]).]{} Once the solution is constructed we then re-express the solution in terms of the original boundary conditions.
Assuming that the functions $F_{n}$ are continuous and decay sufficiently rapidly at infinity, there exits exactly one solution to (\[Fourier\]) satisfying . Since the Green’s function of equations (\[Fourier\]) are $r\mapsto r^{\pm \left \vert n\right \vert }$ and $r\mapsto
r^{\pm \zeta_{n}}$, respectively, where $\zeta_{n}=\sqrt{n^{2}+i\mu n}$, with $\mathcal{R}e(\sqrt{z}) >0$ for $z \in \mathbb C \setminus (-\infty,0]$, the solutions are given by the following explicit expressions:$$\left \{
\begin{array}
[c]{lll}\bigskip \gamma_{n}(r) & = & \dfrac{\overline{\gamma}_{n}}{r^{|n|}}+{\displaystyle \int_{r}^{\infty}}
\dfrac{sw_{n}(s)}{2|n|}\left( \dfrac{r}{s}\right) ^{|n|}\text{$\,
\mathrm{d}s$}+{\displaystyle \int_{1}^{r}}
\dfrac{sw_{n}(s)}{2|n|}\left( \dfrac{s}{r}\right) ^{|n|}\text{$\,
\mathrm{d}s$}\,,\\
w_{n}(r) & = & \dfrac{\overline{w}_{n}}{r^{\zeta_{n}}}-{\displaystyle \int_{r}^{\infty}}
\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{r}{s}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s$}-{\displaystyle \int_{1}^{r}}
\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{s}{r}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s$}\,,
\end{array}
\right. \quad \text{ for }n\in \mathbb{Z}\setminus \{0\} \,. \label{eq_wngamman}$$ with:$$\left \{
\begin{array}
[c]{lll}\medskip \overline{\gamma}_{n} & = & \gamma_{n}^{\ast}-{\displaystyle \int_{1}^{\infty}}
\dfrac{sw_{n}(s)}{2|n|}\left( \dfrac{1}{s}\right) ^{|n|}\text{$\,
\mathrm{d}s$}\,,\\
\overline{w}_{n} & = & w_{n}^{\ast}+{\displaystyle \int_{1}^{\infty}}
\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{1}{s}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s\,,$}\end{array}
\right. \quad \text{ for }n\in \mathbb{Z}\setminus \{0\} \,.
\label{eq_wngammanb}$$ For $n=0$, there still exist solutions to (\[Fourier\]) decaying at infinity, but these solutions exist only for exactly one boundary condition. The reason is that for $n=0$ the Green’s functions for the equations in (\[Fourier\]) are $r\mapsto1$ and $r\mapsto \ln r$, which do not decay at infinity. The solutions decaying at infinity are:$${\left \{
\begin{array}
[c]{lll}\medskip \gamma_{0}(r) & = &
{\displaystyle \int_{r}^{\infty}}
\dfrac{1}{s}{\displaystyle \int_{s}^{\infty}}
t\,w_{0}(t)\text{$\, \mathrm{d}t\, \mathrm{d}s$}\,,\\[10pt]w_{0}(r) & = & -{\displaystyle \int_{r}^{\infty}}
\dfrac{1}{s}{\displaystyle \int_{s}^{\infty}}
t\,F_{0}(t)\text{$\, \mathrm{d}t\, \mathrm{d}s$}\,.
\end{array}
\right. } \label{eq_w0gamm0}$$ [We recall that the value of $\gamma_{0}(1)$ is irrelevant]{}. The value of $w_{0}(1)$ fixes the value of [$\partial_{r}\gamma_{0}(1)= - v_{\theta,0}^{\ast}$ as a function of $\mu$. Once the solution is constructed, we will show that $\mu \mapsto \mu - \partial_{r}\gamma_{0}(1)=:\mu^{\ast}$ can be inverted, which then shows the existence of a solution for an open set of boundary conditions.]{}
Functional framework and main result
====================================
We now introduce the function spaces which we use to solve the system [(\[eq\_bcFn\])–(\[eq\_w0gamm0\])]{}. We use the notation introduced in (\[notation\]):
\[spaces\]Given $\kappa>0$, $\alpha>0$ and $m\in \mathbb{N}$, such that $m<\kappa$, we set: $$\begin{array}
[c]{rcl}\mathcal{B}_{\kappa} & := & \{ \hat{\varphi}^{\ast}\in \mathbb{C}^{\mathbb{Z}}\text{ such that }{\sup_{n\in \mathbb{Z}}}(1+|n|)^{\kappa}|\varphi_{n}^{\ast
}|<\infty \} \text{ },\\[10pt]\mathcal{B}_{\kappa}^{0} & := & \{ \hat{\varphi}^{\ast}\in \mathcal{B}_{\kappa
}\text{ such that }\varphi_{0}^{\ast}=0\} \text{ },
\end{array}$$ and $$\begin{array}
[c]{rcl}\mathcal{B}_{\alpha,\kappa} & := & \{ \hat{\varphi}\in({C}([1,\infty
);\mathbb{C}))^{\mathbb{Z}},\text{ such that }\ {\sup_{n\in \mathbb{Z}}\,
\sup_{r\in \lbrack1,\infty)}}r^{\alpha}(1+|n|)^{\kappa}|\varphi_{n}(r)|<\infty \} \text{ },\\[10pt]\mathcal{U}_{\alpha,\kappa}^{m} & := & \{ \hat{\varphi}\in({C}^{m}([1,\infty);\mathbb{C}))^{\mathbb{Z}},\text{ such that }(\partial_{r}^{l}\varphi_{n})_{n\in \mathbb{Z}}\in \mathcal{B}_{\alpha+l,\kappa
-l},\ \text{for all }0\leq \text{$l\leq m$}\} \text{ }.
\end{array}$$
These function spaces are reminiscent of weighted Sobolev spaces, and permit to obtain sharp estimates on the decay of solutions to [(\[eq\_wngamman\])–(\[eq\_w0gamm0\])]{}. The spaces with one lower index (mainly $\mathcal{B}_{\kappa}^{0}$) are used for the boundary data, whereas the spaces with two lower indices (mainly $\mathcal{U}_{\alpha,\kappa}^{m}$) will be used for [solving (\[eq\_wngamman\])–(\[eq\_w0gamm0\])]{}.
The spaces introduced in Definition \[spaces\] satisfy the following straightforward properties. Given $\alpha>0$, $\kappa>0$, and $m\in \mathbb{N}$, such that $m<\kappa$, we have:
1. The spaces $\mathcal{B}_{\kappa}$, $\mathcal{B}_{\alpha,\kappa}$, $\mathcal{U}_{\alpha,\kappa}^{m}$ are Banach spaces when equipped with their respective norms: $$\Vert \hat{w}\ ;\ \mathcal{B}_{\kappa}\Vert=\sup_{n\in \mathbb{N}}(1+|n|)^{\kappa}|w_{n}|,\qquad \Vert \hat{\varphi}\ ;\ \mathcal{B}_{\alpha,\kappa}\Vert=\sup_{n\in \mathbb{Z}}\sup_{r\in \lbrack1,\infty
)}r^{\alpha}(1+|n|)^{\kappa}|\varphi_{n}(r)|\,,$$$$\Vert \hat{\varphi}\ ;\ \mathcal{U}_{\alpha,\kappa}^{m}\Vert=\sum_{l=0}^{m}\Vert(\partial_{r}^{l}\varphi_{n})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\alpha+l,\kappa-l}\Vert \,.$$
2. Given $\alpha \geq \alpha^{\prime}$ and $\kappa \geq \kappa^{\prime}$ we have the embedding $\mathcal{B}_{\alpha,\kappa}\subset \mathcal{B}_{\alpha^{\prime},\kappa^{\prime}}$ together with the bound: $$\Vert \hat{\varphi}\ ;\mathcal{B}_{\alpha^{\prime},\kappa^{\prime}}\Vert
\leq \Vert \hat{\varphi}\ ;\mathcal{B}_{\alpha,\kappa}\Vert \,,\qquad \forall \,
\hat{\varphi}\in \mathcal{B}_{\alpha,\kappa}\,. \label{eq_emb}$$
3. The space $\mathcal{B}_{\kappa}^{0}$ is a closed subspace of $\mathcal{B}_{\kappa}$, and thus also a Banach space.
We now formulate the problem of finding a solution to [(\[eq\_bcFn\]]{})–(\[eq\_w0gamm0\]) in such a way that we can apply the inverse map theorem on our function spaces:
\[lem\_S\] Given ${\mu>\mu_{crit}}$, let $\alpha>0$ be sufficiently small and $\kappa >0$. Then, the map $\mathcal{S}_{\mu}\colon \mathcal{B}_{4+2\alpha,\kappa}\times \mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0}\rightarrow \mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha
+2,\kappa+2}^{2}$, which associates to the triple $(\hat{F},\hat{\gamma}^{\ast},\hat{w}^{\ast})$ the pair $\left( \hat{\gamma},\hat{w}\right) $ by virtue of equations (\[eq\_wngamman\]), (\[eq\_w0gamm0\]), together with (\[BC\_Fourier\]), (\[eq\_wngammanb\]), is linear and continuous.
The notion of $\alpha$ small enough will be made precise in the [last section.]{}
\[lem\_NL\] Let $\alpha>0$ and $\kappa>0$. Then, the map $\mathcal{NL}\colon \left( \mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2}\right) ^{2}\rightarrow \mathcal{B}_{4+2\alpha
,\kappa+1}$, defined by$$\mathcal{NL}[(\hat{\gamma}_{a},\hat{w}_{a}),(\hat{\gamma}_{b},\hat{w}_{b})]= \left( r \mapsto -{\frac{i}{r}}{\sum_{k+l=n}}\left( k\,w_{k}^{a}(r)\, \partial_{r}\gamma_{l}^{b}(r)-l\, \gamma_{l}^{a}(r)\, \partial_{r}w_{k}^{b}(r)\right) \right)_{n\in \mathbb Z} \,,$$ is bilinear and continuous.
The proofs of these lemmas are postponed to Section \[sec\_technical\]. [We also introduce the trace operator $\Gamma_{1}$: $$\begin{array}
[c]{rrcl}{\Gamma}_{1}\colon & \mathcal{U}_{\alpha_{1},\kappa_{1}}^{2}\times
\mathcal{U}_{\alpha_{2},\kappa_{2}}^{2} & \rightarrow & \mathcal{B}_{\kappa_{1}-1}^{0}\times \mathcal{B}_{\kappa_{1}-1}^{0}\,,\\
& (\hat{\gamma},\hat{w}) & \longmapsto & ((in\gamma_{n}(1))_{n\in \mathbb{Z}},((\delta_{n,0}-1)\partial_{r}\gamma_{n}(1))_{n\in \mathbb{Z}})\,,
\end{array}$$ where $\delta_{n,m}$ is the Kronecker symbol. This map is linear and continuous for arbitrary $(\alpha_{i},\kappa_{i})\in(0,\infty)^{2}$, $(i=1,2)$. ]{}
To compute [solutions to (\[eq\_bcFn\])]{}–(\[eq\_w0gamm0\]), we introduce a map $\Phi_{\mu}$, which allows to solve the differential equations and constrain the trace on $r=1$ in one step. Namely, given [$\mu>\mu_{crit}$,]{} $\alpha$ sufficiently small and $\kappa>0$, we set: $$\begin{array}
[c]{rrcl}\Phi_{\mu}\colon & (\mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2})\times(\mathcal{B}_{\kappa+4}^{0}\times
\mathcal{B}_{\kappa+2}^{0}) & \longrightarrow & (\mathcal{U}_{\alpha,\kappa
+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2})\times(\mathcal{B}_{\kappa
+3}^{0}\times \mathcal{B}_{\kappa+3}^{0})\, \\[10pt]
&
\begin{pmatrix}
\hat{x}=(\hat{\gamma},\hat{w})\\[4pt]\hat{x}^{\ast}=(\hat{\gamma}^{\ast},\hat{w}^{\ast})
\end{pmatrix}
& \longmapsto &
\begin{pmatrix}
\mathcal{S}_{\mu}(\mathcal{NL}(\hat{x},\hat{x}),\hat{x}^{\ast})-\hat{x}\\[4pt]\Gamma_{1}[\mathcal{S}_{\mu}(\mathcal{NL}(\hat{x},\hat{x}),\hat{x}^{\ast})]
\end{pmatrix}
\end{array}
\label{PhiMu}$$ By definition, if $(\hat{x}=(\hat{\gamma},\hat{w}),\hat{x}^{\ast}=(\hat
{\gamma}^{\ast},\hat{w}^{\ast}))$ is a solution to $\Phi_{\mu}(\hat{x},\hat
{x}^{\ast})=(0,(\hat{v}_{r}^{\ast},\hat{v}_{\theta}^{\ast}))$, then $(\hat{\gamma},\hat{w})$ satisfies [(\[Fourier\])-]{}. This motivates the following notion of $\kappa$–solutions:
Given an exponent $\kappa>0$, an angular velocity [$\mu>\mu_{crit}$]{}, and a boundary condition $\hat{\mathbf{v}}^{\ast}:=(\hat{v}_{r}^{\ast},\hat
{v}_{\theta}^{\ast})\in \mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa
+3}^{0}$, we call [*$\kappa$–solution for the boundary condition $\hat{\mathbf{v}}^{\ast}$ and the asymptotic angular velocity $\mu$*]{} a pair $\hat{x}=(\hat{\gamma},\hat{w})$, such that, for sufficiently small $\alpha>0$ and some $\hat{x}^{\ast}=(\hat{\gamma}^{\ast},\hat{w}^{\ast})\in
\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0}$:
- $\hat{x}\in \mathcal{U}_{\alpha+4,\kappa}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2},$
- $\Phi_{\mu}(\hat{x},\hat{x}^{*}) = (0,\hat{\mathbf{v}}^{*}).$
The remaining sections are devoted to the proof of the following result:
\[thm\_main\_tech\] Given $\kappa>0$ and [$\mu_{0}>\mu_{crit}$]{} there exists $\varepsilon_{\kappa,\mu_{0}}>0$ and an open interval $I_{\kappa,\mu_{0}}\ni \mu_{0}$ such that, given $\hat{\mathbf{v}}^{\ast}\in B(\mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}\ ;\ \varepsilon_{\kappa
,\mu_{0}})$ and $\mu^{\ast}\in$ ${I_{\kappa,\mu_{0}}}$, there exists [$\mu
>\mu_{crit}$]{} and a $\kappa$–solution $(\hat{\gamma},\hat{w})$ for the boundary condition $\hat{\mathbf{v}}^{\ast}$ and the asymptotic angular velocity $\mu$, satisfying the condition $\mu - \partial_{r}\gamma_{0}(1)
=\mu^{\ast}$.
As mentioned above, the notion of [$\alpha$]{} small enough will be made precise in the last section. Before entering into the details of the proof of Theorem \[thm\_main\_tech\], we explain why it implies Theorem \[thm\_main\].
\[Proof of [Theorem \[thm\_main\]]{}\]Let [$\mu_{0}>\mu_{crit}$]{} and [$\kappa>1$]{}. Applying Theorem \[thm\_main\_tech\] yields a ball of initial conditions with positive radius $\varepsilon_{\kappa,\mu_{0}}>0$ and an open neighborhood $I_{\kappa,\mu_{0}}$ of $\mu_{0}$.
For $\mathbf{u}^{\ast}\in C^{\infty}(\partial B),$ we define : $$\mu^{\ast} = \dfrac{1}{2\pi}\int_{0}^{2\pi}u_{\theta}^{\ast}(s)\,
\mathrm{d}\text{$s$}\,,$$ and the sequences $\hat{u}^{\ast}$ and $\hat{v}^{\ast}$ by $u_0=v_0 =0$ and: $$u_{n}^{\ast} = \dfrac{1}{2\pi}\int_{0}^{2\pi}u_{r}^{\ast}(\theta)e^{-in\theta
}\text{ \textrm{d}$\theta$}\,,\qquad v_{n}^{\ast}= \dfrac{1}{2\pi}\int
_{0}^{2\pi}u_{\theta}(\theta)e^{-in\theta}\text{ \textrm{d}$\theta$}\,,\qquad \forall \,n\in \mathbb{Z}\setminus \{0\} \,.$$ The regularity of $\mathbf{u}^{\ast}$ yields that $(\hat{u}^{\ast},\hat{v}^{\ast
})\in \mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}$ and $$u_{r}^{\ast}(\theta)= \sum_{n\in \mathbb{Z}}u_{n}^{\ast}e^{in\theta}\,,\qquad
u_{\theta}^{\ast}(\theta)=\mu^{\ast} + \sum_{n\in \mathbb{Z}}v_{n}^{\ast
}e^{in\theta}\,.$$ We now assume that: $$\Vert(\hat{u}^{\ast},\hat{v}^{\ast})\ ;\ \mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}\Vert<\varepsilon_{\kappa,\mu_{0}}\,,\quad
\mu^{\ast}\in I_{\kappa,\mu_{0}}\,, \label{eq_2}$$ which makes the meaning of $\mathbf{u}^{\ast}$ sufficiently close to $\mu
_{0}\mathbf{e}_{\theta}$ in the statement of Theorem \[thm\_main\] precise.
Consequently, the assumptions of Theorem \[thm\_main\_tech\] are satisfied, which yields that there exists [$\mu>\mu_{crit}$]{} and a $\kappa$–solution $(\hat{\gamma},\hat{w})$ for the boundary data [$(\hat{u}^{\ast},\hat{v}^{\ast})$]{} satisfying $\mu - \partial_{r}\gamma_{0}(1)=\mu^{\ast}$. Let $$w(r,\theta)=\sum_{n\in \mathbb{Z}}w_{n}(r)e^{in\theta}\,,\qquad \gamma
(r,\theta)=\sum_{n\in \mathbb{Z}}\gamma_{n}(r)e^{in\theta}\,,\quad
\forall{\,(r,\theta)\in \Omega \,.}$$ Because $\kappa>1$, classical results from the theory of Fourier series yield that:
- $w\in C^{2}(\mathbb{R}^{2}\setminus \overline{B})$ and $\gamma \in C^{4}(\mathbb{R}^{2}\setminus \overline{B})$ so that $\mathbf{v}=\nabla^{\bot}\gamma \in C^{3}(\mathbb{R}^{2}\setminus \overline{B})$,
- $\mathbf{v}\cdot \nabla w\in C^{1}(\mathbb{R}^{2}\setminus
\overline{B})$ with:$$\mathbf{v}\cdot \nabla w(r,\theta)=\sum_{n\in \mathbb{Z}}\left[ - \frac{i}{r}{\sum_{k+l=n}}\left( l\,w_{l}(r)\, \partial_{r}\gamma_{k}(r)-k\, \gamma_{k}(r)\, \partial_{r}w_{l}(r)\right) \right] e^{in\theta}\,, \quad \forall \, (r,\theta) \in \Omega\,.$$
Let $$\Delta{w}-{\dfrac{\mu}{r}}\ \mathbf{e}_{\theta}\cdot \nabla w-\mathbf{v}\cdot \nabla w=\colon \varphi \in{C}([1,\infty);{C}([-\pi,\pi]))\,.$$ Because $(\hat{\gamma},\hat{w})$ satisfies (\[Fourier\]), all the Fourier coefficients of $\varphi$ vanish identically on $[1,\infty)$. Hence, $(\gamma,w)$ is a solution of $$\left \{
\begin{array}
[c]{rcl}\Delta \gamma & = & - w\,,\\
\Delta{w}-{\dfrac{\mu}{r}}\ \mathbf{e}_{\theta}\cdot \nabla w & = &
\mathbf{v}\cdot \nabla w\,,
\end{array}
\right. \quad \text{in $\mathbb{R}^{2}\setminus \overline{B}\,.$} \label{eq_3}$$ Straightforward manipulations of the Fourier series of $\gamma$ yield that: $$|\mathbf{v}(r,\theta)|\leq \dfrac{\Vert \gamma \ ;\ \mathcal{U}_{\alpha,\kappa
+4}^{2}\Vert}{r^{\alpha+1}}\,,\quad \forall \,{(r,\theta)\in \Omega}\,.
\label{eq_decayv}$$ Since $\mathbf{u}^{\ast}$ has zero flux, *i.e.*, since $v^*_{r}$ has zero average, we have that $\mathbf{v}(1,\theta)=v_{r}^{\ast
}(\theta)e_{r}+v_{\theta}^{\ast}(\theta)\mathbf{e}_{\theta},$ for all $\theta \in (-\pi,\pi)$, where: $$\begin{aligned}
v_{r}^{\ast}(\theta) & = \sum_{n\in \mathbb{Z}}in\gamma_{n}(1)e^{in\theta
}= \sum_{n\in \mathbb{Z}}u_{n}^{\ast}e^{in\theta}=u_{r}^{\ast}(\theta)\,,\\
v_{\theta}^{\ast}(1,\theta) & = - \sum_{n\in \mathbb{Z}}\partial_{r}\gamma
_{n}(1)e^{in\theta}=u_{\theta}^{\ast}(\theta) - \partial_{r}\gamma_{0}(1)-\mu^{\ast}\,.\end{aligned}$$ Therefore, if we set $\psi:=\psi_{\mu}+\gamma$, $\omega:=w$, $\mathbf{u}:=\nabla^{\bot}\psi=\mathbf{u}_{\mu}+\mathbf{v}$, then the pair $(\psi
,\omega)$ is a solution to $$\left \{
\begin{array}
[c]{rcl}\Delta \psi & = & -\omega \,,\\
\Delta \omega & = & \mathbf{u}\cdot \nabla \omega \,,
\end{array}
\right. \quad \text{in $\mathbb{R}^{2}\setminus \overline{B}$}\,, \label{eq_4}$$ and the following boundary conditions are satisfied (recall that $\mu - \partial_r \gamma_0(1)=\mu^{\ast}$ by construction of $\hat{\gamma}$): $$\mathbf{u}=\mathbf{u}^{\ast}\,,\quad \text{on $\partial B\,,$}\qquad
\lim_{r\rightarrow \infty}|\mathbf{u}(r,\theta)|=0\,.$$ The inequality implies that the boundary condition at infinity is satisfied in the following more precise sense: $$\lim_{r\rightarrow \infty}\left \Vert r\left( \mathbf{u}(r,\theta)-\dfrac
{\mu \mathbf{e}_{\theta}}{r}\right) ;{L^{\infty}((-\pi,\pi))}\right \Vert
=\lim_{r\rightarrow \infty}r\Vert v(r,\theta);{L^{\infty}((-\pi,\pi))}\Vert=0\,.$$
To complete the proof, we need to show how to obtain the Navier Stokes equations from the relations between $\mathbf{u}$, $\psi$ and $\omega$, together with . First, multiplying (\[eq\_4\]) by $\varphi \in C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$ yields: $${-\int_{\mathbb{R}^{2}\setminus \overline{B}}\nabla w\colon \nabla \varphi}=\int_{\mathbb{R}^{2}\setminus \overline{B}}\left[ \mathbf{u}\cdot \nabla
w\right] \varphi \,.$$ We have the basic identities: $$w=\nabla \times\mathbf{u}\,,\qquad \mathbf{u}\cdot \nabla
w=\nabla \times\left[ \mathbf{u}\cdot \nabla \mathbf{u}\right] \,,
\label{eq_identities}$$ from which we obtain, after integration by parts, that for any given $\varphi \in C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$: $${-\int_{\mathbb{R}^{2}\setminus \overline{B}}\nabla \mathbf{u}\colon \nabla
\nabla^{\bot}\varphi}=\int_{\mathbb{R}^{2}\setminus \overline{B}}\left[
\mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot \nabla^{\bot}\varphi \,.
\label{eq_0}$$ This identity yields the pressure $p$ *via* De Rham’s theory, modulo the difficulty, that not all the divergence-free velocity-fields $\mathbf{w}\in C_{c}^{\infty}(\mathbb{R}^{2}\setminus
\overline{B})$ of compact support can be written in the form $\nabla^{\bot}\varphi$ with $\varphi \in C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$. More precisely, if a smooth velocity-field $\mathbf{v}$ satisfies $\nabla \times\mathbf{v}=0$ in $\mathbb{R}^{2}\setminus \overline{B}$, then $\mathbf{v}$ is the gradient of a function up to a contribution of the form $C\mathbf{x}^{\bot
}/|\mathbf{x}|^{2}$. We now show that this contribution vanishes in our case.
Let $\Phi_{0}\in C^{\infty}(\mathbb{R})$ be such that $\mathrm{supp}(\Phi_{0}^{\prime})\subset \subset(1,2)$ and $\Phi_0(0) = 0,$ $\Phi_0(2)=1,$ and let $\mathbf{w}_{0}=\nabla^{\bot}\Phi_{0}$. In polar coordinates we have $\mathbf{w}_{0}(r,\theta)=-\Phi_{0}^{\prime}(r)\mathbf{e}_{\theta}$, for all $(r,\theta
)\in \Omega$. Given a divergence-free $\mathbf{w}\in
C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$ we define $\mathbf{\widetilde{w}}$ and $\varphi$ by: $$\mathbf{\widetilde{w}}=\mathbf{w}-M_{\mathbf{w}}\mathbf{w}_{0}\quad \text{with}\quad \left[ \int_{1}^{\infty}w_{\theta}(r,0)\, \mathrm{d}r\right] =\colon
M_{\mathbf{w}},\qquad \varphi(r,\theta)=\int_{0}^{r}{w}_{\theta}(s,\theta
)\text{$\, \mathrm{d}s$}-M_{\mathbf{w}}\Phi_{0}(r)\,.$$ By definition of $M_{\mathbf{w}}$, we have that $\varphi \in C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$ and $\nabla^{\bot}\varphi=\mathbf{\widetilde{w}}$, so that implies: $${-\int_{\mathbb{R}^{2}\setminus \overline{B}}\nabla \mathbf{u}\colon \nabla
\widetilde{\mathbf{w}}}=\int_{\mathbb{R}^{2}\setminus \overline{B}}\left[
\mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot \widetilde{\mathbf{w}}\,.$$ Replacing $\widetilde{\mathbf{w}}$ by its definition yields that, for any divergence-free $\mathbf{w}\in C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$, we have: $${-\int_{\mathbb{R}^{2}\setminus \overline{B}}\nabla \mathbf{u}\colon
\nabla{\mathbf{w}}}=\int_{\mathbb{R}^{2}\setminus \overline{B}}\left[
\mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot{\mathbf{w}-}M_{\mathbf{w}}\int_{\mathbb{R}^{2}\setminus \overline{B}}\left( \left[ \mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot{\mathbf{w}_{0}+\nabla \mathbf{u}\colon \nabla{\mathbf{w}_{0}}}\right) \,.$$ Let $I_{0}$ be the last integral in the previous equality. We then have: $$I_{0}=\lim_{N\rightarrow \infty}\int_{B(\mathbb{R}^{2},N)\setminus \overline{B}}\left( \left[ \mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot{\mathbf{w}_{0}+\nabla \mathbf{u}\colon \nabla{\mathbf{w}_{0}}}\right) \,.$$ Integrating by parts, we obtain, for all $N>1$: $$\begin{aligned}
\int_{B(\mathbb{R}^{2},N)\setminus \overline{B}}\left( \left[ \mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot{\mathbf{w}_{0}+\nabla \mathbf{u}\colon \nabla{\mathbf{w}_{0}}}\right) & =\int_{\partial B(\mathbb{R}^{2},N)}\left( \left[ \Phi_{0}\mathbf{u}\cdot \nabla \mathbf{u}\right]
\cdot \mathbf{n}^{\bot}{+{\Phi_{0}^{\prime}}\partial_{r}\mathbf{u}\cdot \mathbf{n}^{\bot}}\right) \text{ \textrm{d}$\sigma$}\\
& {-\int_{B(\mathbb{R}^{2},N)}\left( \Phi_{0}\mathbf{u}\cdot \nabla
\omega+\nabla \omega \cdot \nabla \Phi_{0}\right) }\\
& =\int_{\partial B(\mathbb{R}^{2},N)}\left( \Phi_{0}\left[ {(\mathbf{u}\cdot \nabla \mathbf{u})\cdot \mathbf{n}^{\bot}-\partial_{\mathbf n}\omega}\right]
{+{\Phi_{0}^{\prime}}}\partial_{r}\mathbf{u}\cdot \mathbf{n}^{\bot}\right)
\text{ \textrm{d}$\sigma$}\,,\end{aligned}$$ where, in order to get the last identity, we have again used that $\mathbf{u}\cdot \nabla \omega=\Delta \omega$, in $\mathbb{R}^{2}\setminus
\overline{B}$. Since $\mathbf{u}$ decays like $1/r$, $\nabla \mathbf{u}$ decays like $1/r^{2}$, and $\nabla \omega$ like $1/r^{3}$. This yields that $I_{0}=0$ in the limit $N\rightarrow \infty$. Finally, we have: $${-\int_{\mathbb{R}^{2}\setminus \overline{B}}\nabla \mathbf{u}\colon
\nabla{\mathbf{w}}}=\int_{\mathbb{R}^{2}\setminus \overline{B}}\left[
\mathbf{u}\cdot \nabla \mathbf{u}\right] \cdot{\mathbf{w}}\,,$$ for any divergence-free vector-field $\mathbf{w}\in C_{c}^{\infty}(\mathbb{R}^{2}\setminus \overline{B})$, and De Rham’s theory (see [@TemamB Remark 1.5]) implies the existence of a pressure $p$ such that [ ]{}is satisfied.
Proof of **Theorem \[thm\_main\_tech\]**
========================================
In this section $\kappa>0$ and [$\mu_{0}>\mu_{crit}$]{} are fixed. First, we set $\mu_{-}=\left( \mu_{0}+{\mu_{crit}}\right) /2$ and $\mu_{+}=(2\mu_{0}+$ ${\mu_{crit}})/2$ so that $I:=[\mu_{-},\mu_{+}]$ satisfies$$\mu_{0}\in \lbrack \mu_{-},\mu_{+}]\subset \left( {\mu_{crit}},\infty \right)
\,.$$ We also set: $$\label{eq_alpha}
\alpha:=\dfrac{1}{4}\min \left( \frac{1}{\sqrt{2}}\left[ \sqrt{ 1+{|\mu_{-}|}^{2} }+1\right] ^{{1}/{2}}-2,1\right) \,.$$ We emphasize that, because ${\mu_{0}>\mu_{crit}}$, we have $\alpha\in
(0,1/4]$. Let: $$\begin{array}
[c]{rrcl}\Phi_{\mu}\colon & (\mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2})\times(\mathcal{B}_{\kappa+4}^{0}\times
\mathcal{B}_{\kappa+2}^{0}) & \longrightarrow & (\mathcal{U}_{\alpha
,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2})\times
(\mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0})\, \\[10pt]
&
\begin{pmatrix}
\hat{x}=(\hat{\gamma},\hat{w})\\[4pt]\hat{x}^{\ast}=(\hat{\gamma}^{\ast},\hat{w}^{\ast})
\end{pmatrix}
& \longmapsto &
\begin{pmatrix}
\mathcal{S}_{\mu}(\mathcal{NL}(\hat{x},\hat{x}),\hat{x}^{\ast})-\hat{x}\\[4pt]\Gamma_{1}[\mathcal{S}_{\mu}(\mathcal{NL}(\hat{x},\hat{x}),\hat{x}^{\ast})]
\end{pmatrix}
\end{array}
\label{mapPHI}$$ We will show in the next section that for $\mu \in I$ the map $\Phi_{\mu}$ is well defined. We split the proof of Theorem \[thm\_main\_tech\] into two steps. First, we show that we can construct a $\kappa$–solution for any sufficiently small boundary condition $(\hat{u}^{\ast},\hat{v}^{\ast})\in \mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}$ and an interval of asymptotic angular velocities $\mu$. In a second step, we analyze the dependence of this solution as a function of $\mu$.
We have the following abstract result:
\[prop\_abstractresult\] Let $I$ be a compact interval and $X$, $Y$ two Banach spaces. Assume that $\Phi \colon \mu \in I\mapsto \Phi_{\mu}$ satisfies:
- $\Phi \in{C}( I ; {C}^{1}(X ; Y))$,
- $\Phi_{\mu}(x) = 0$ for all $\mu \in I$,
- $D\Phi_{\mu}(0)$ is one-to-one and onto, and has a continuous inverse for all $\mu \in I.$
Then, there exist positive constants $\eta_{x}$ and $\eta_{y}$, such that $\Phi_{\mu}$ is a $C^{1}$-diffeomorphism from $B_{X}(0,\eta_{x})$ onto $\Phi_{\mu}\left( B_{X}(0,\eta_{x})\right) \supset B_{Y}(0,\eta_{y})$. Furthermore, the family of inverse maps $\Phi^{-1}\colon \mu \in I\mapsto \left(
\Phi_{\mu}\right) ^{-1}$ satisfies $$\Phi^{-1}\in C(I;C^{1}(B_{Y}(0,\eta_{y});B_{X}(0,\eta_{x})))\,.$$
The proof is standard, but for the sake of completeness we recall the main ingredients. Given the hypothesis of Proposition \[prop\_abstractresult\], the map $\Phi_{\mu}$ satisfies the assumptions of the inverse function theorem for arbitrary $\mu \in I$, so that there exits $\eta_{x,\mu}>0$ and $\eta_{y,\mu}>0$ such that $\Phi_{\mu}$ is a $C^{1}$-diffeomorphism from $B_{X}(0,\eta_{x,\mu})$ onto $\Phi_{\mu}\left( B_{X}(0,\eta_{x,\mu})\right)
\supset B_{Y}(0,\eta_{\mu,y})$. Since $\Phi$ is continuous, it is clear that these constants can be chosen independently of $\mu$, locally in $\mu$. By a compactness argument, we can therefore find constants $\eta_{x}>0$ and $\eta_{y}>0$ such that $\Phi_{\mu}\colon B_{X}(0,\eta_{x})\rightarrow
\Phi_{\mu}\left( B_{X}(0,\eta_{x})\right) \supset B_{Y}(0,\eta_{y})$ is a $C^{1}$ diffeomorphism for arbitrary $\mu \in I$. We now show that $\Phi
^{-1}\in C(I\times B_{Y}(0,\eta_{y}))$. The proof that $\Phi^{-1}\in
C(I ; C^{1}(B_{Y}(0,\eta_{y});B_{X}(0,\eta_{x})))$ is then obtained by differentiating (with respect to $x$) the identity $\Phi^{-1}_{\mu} \circ \Phi_{\mu}(x) = x$ which holds true on $B_{X}(0,\eta_{x}).$
Given $(\mu,\tilde{\mu})\in I^{2}$ and $(y,\tilde{y})\in \lbrack B_{Y}(0,\eta_{y})]^{2}$, we denote: $$x=\Phi_{\mu}^{-1}({y})\,,\qquad \tilde{x}=\Phi_{\tilde{\mu}}^{-1}({\tilde{y}})\,.$$ By construction we have: $$\begin{aligned}
y-\tilde{y} & =\Phi_{\mu}(x)-\Phi_{\tilde{\mu}}(\tilde{x}),\\
& =D\Phi_{\mu}(0)[x-\tilde{x}]+\Phi_{\mu}(\tilde{x})-\Phi_{\tilde{\mu}}(\tilde{x})+o(|x-\tilde{x}|)\\
& =D\Phi_{\mu}(0)[x-\tilde{x}]+o(|\mu-\tilde{\mu}|)+o(\Vert x-\tilde
{x};X\Vert)\,.\end{aligned}$$ Consequently, reducing the size of $\eta_{x}$ and $\eta_{y}$ if necessary, we get that: $$\Vert x-\tilde{x} \, ; \, X \Vert \leq2\Vert \lbrack D\Phi_{\mu}(0)]^{-1};\mathcal{L}_{c}(Y;X)\Vert \left[ \Vert y-\tilde{y};Y\Vert+o(|\mu-\tilde{\mu}|)\right]
\,,$$ where $\mathcal{L}_c(Y;X)$ denotes the set of continuous linear map $Y \to X.$ This completes the proof.
We now show that we can apply Proposition \[prop\_abstractresult\] to the map $\Phi$ as defined in (\[mapPHI\]) To this end, we remark that $\Phi_{\mu}$ depends on $\mu$ only through $\mu \mapsto \mathcal{S}_{\mu}$, and that for all $\mu \in I$ the differential $D\Phi_{\mu}(0)$ is: $$D\Phi_{\mu}(0)[\hat{x},\hat{x}^{\ast}]=(\Gamma_{1}\mathcal{S}_{\mu}(0,\hat
{x}^{\ast}),-\hat{x})\,,\quad \forall \,(\hat{x},\hat{x}^*)\in(\mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2})\times(\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0})\,.$$ Let $\mathcal{T}_{\mu}(\hat{x}^*):=\Gamma_{1}\mathcal{S}_{\mu}(0,\hat{x}^*)$. Since $\Phi$ is a combination of linear and bilinear maps, it suffices to apply Lemma \[lem\_NL\] and the following lemma in order to check that the assumptions of Proposition \[prop\_abstractresult\] are satisfied:
\[lem\_Smu\] Let $\alpha$ be given by , then the restriction of the map $\mathcal{S}\colon \mu
\mapsto \mathcal{S}_{\mu}$ to $I:=[\mu_{-},\mu_{+}]$ satisfies:
- $\mathcal{S}\in C(I;\mathcal{L}_{c}(\mathcal{B}_{\kappa,2\alpha+4}\times(\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0})\ ;\ \mathcal{U}_{\alpha,\kappa+4}^{2}\times
\mathcal{U}_{\alpha+2,\kappa+2}^{2}))$,
- $\mathcal{T}_{\mu}$ is a one to one and onto map $\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0}\rightarrow \mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}$ with continuous inverse.
We postpone the proof of this technical lemma to the next section. We now apply Proposition \[prop\_abstractresult\] to $\Phi$, but restrict the image to the component $\hat{x}$. This yields the following result:
There exists a map $\Psi \colon \mu \mapsto \Psi_{\mu}$ satisfying $$\Psi \in C(I;C^{1}(B_{\mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa
+3}^{0}}(0,\varepsilon_{\kappa})\ ;\ \mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2}))\, \,,\newline$$ such that, for all $\mathbf{v}^{\ast}\in B_{\mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}}(0,\varepsilon_{\kappa})$ and $\mu \in I$, $\Psi_{\mu}(\mathbf{v}^{\ast})$ is a $\kappa$–solution for the boundary condition $\mathbf{v}^{\ast}$ with respect to the asymptotic angular velocity $\mu$.
In a final step we [show how]{} to prescribe the zero mode of the solution $\Psi_{\mu}(\mathbf{v}^{\ast})$ by using the dependence of the solution on $\mu$. Let $\eta=(\mu_{0}-$ ${\mu_{crit}})/4,$ and consider a boundary condition $$\mathbf{v}^{\ast}\in B_{\mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa
+3}^{0}}(0,\varepsilon_{\kappa})\,.$$ [Let $(\hat{\psi}_{\mu},\hat{w}_{\mu}):=\Psi_{\mu}(\mathbf{v}^{\ast}),$ for all $\mu \in I$]{}. Using that $\Psi_{\mu}(0)=0$, and restricting the size of $\varepsilon_{\kappa}$ if necessary, we can assume that $$|\partial_{r}\psi_{\mu,0}(1)|\leq \Vert \hat{\psi}_{\mu}\,;\, \mathcal{U}_{\alpha,\kappa}^{2}\Vert \leq \eta \,,\qquad \forall \, \mu \in I\,.$$ [Consequently]{}, the map $\mu \mapsto \mu-\partial_{r}\psi_{\mu,0}(0)$, [which is continuous from $I$ to $\mathbb{R}$, because $\mu \mapsto \Psi_{\mu
}(0,\mathbf{v}^{\ast})$ is continuous,]{} satisfies: $$\mu_{-}-\partial_{r}\psi_{\mu_{-},0}(1)\leq \mu_{-}+\eta<\mu_{0}\,,\qquad
\mu_{+}-\partial_{r}\psi_{\mu_{+},0}(0)\geq \mu_{+}-\eta>\mu_{0}\,.$$ Hence the image of this map contains an open interval $I_{\kappa,\mu_{0}}$ containing $\mu_{0}$. This completes the proof of Theorem \[thm\_main\_tech\].
Proof of main lemmas {#sec_technical}
====================
This section contains the proof of the technical lemmas which have been used without proof in the previous sections. First we prove Lemma \[lem\_NL\], which is standard. We then give proofs of Lemmas \[lem\_S\] and \[lem\_Smu\] which are more delicate.
\[Proof of [Lemma \[lem\_NL\]]{}\]Let $\hat{F} = \mathcal{NL}((\hat{\gamma}^{a},\hat{w}^{a}),(\hat{\gamma}^{b},\hat{w}^{b}))$, for $(\hat{\gamma}^{i},\hat{w}^{i})\in \mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2}$, $i=\{a,b\}$. First, we note that for $\kappa>0$ and $n\in \mathbb{N}$, the following series converge: $$\sum_{l\in \mathbb{Z}}\dfrac{1}{(1+|l|)^{\kappa+1}(1+|n-l|)^{\kappa+3}}\,,\qquad \sum_{k\in \mathbb{Z}}\dfrac{1}{(1+|k|)^{\kappa+3}(1+|n-k|)^{\kappa
+1}}\,. \label{series}$$ Consequently, we have for all $(n,k,l)\in \mathbb{Z}^{3}$:$$|lw_{l}^{a}(r)\partial_{r}\gamma_{n-l}^{b}(r)|\leq \dfrac{1}{r^{3+2\alpha}}\dfrac{\Vert(w_{n}^{a})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\alpha+2,\kappa
+2}\Vert \, \Vert(\partial_{r}\gamma_{n}^{b})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\alpha+1,\kappa+3}\Vert}{(1+|l|)^{\kappa+1}(1+|n-l|)^{\kappa+3}}\,,$$ and $$|k\gamma_{k}^{a}(r)\partial_{r}w_{n-k}^{b}(r)|\leq \dfrac{1}{r^{3+2\alpha}}\dfrac{\Vert(\gamma_{n}^{a})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\alpha
,\kappa+4}\Vert \, \Vert(\partial_{r}w_{n}^{b})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\alpha+3,\kappa+1}\Vert \ }{(1+|k|)^{\kappa+3}(1+|n-k|)^{\kappa+1}}\,,$$ and therefore the series defining $F_{n}$ is converging. We now bound the series (\[series\]). By symmetry, it is sufficient to consider only the first series and $n\geq0$. We split the sum into two parts:$$\begin{aligned}
S_{n/2}^{-} & ={\sum_{l=-\infty}^{\left[ n/2\right] }}\dfrac
{1}{(1+|l|)^{\kappa+1}(1+|n-l|)^{\kappa+3}}\\
& \leq C_{\kappa}\left( \dfrac{1}{1+|n|}\right) ^{\kappa+3}{\sum
_{l\in \mathbb{Z}}}\dfrac{1}{(1+|l|)^{\kappa+1}} \leq C_{\kappa}\left( \dfrac{1}{1+|n|}\right) ^{\kappa+3}\,,\end{aligned}$$ and$$\begin{aligned}
S_{n/2}^{+} & ={\sum_{\left[ n/2\right] +1}^{\infty}}\dfrac{1}{(1+|l|)^{\kappa+1}(1+|n-l|)^{{\kappa+3}}}\\
& \leq C_{\kappa}\left( \dfrac{1}{1+|n|}\right) ^{\kappa+1}\sum_{l\in\mathbb Z}\dfrac{1}{(1+|l|)^{\kappa+3}} \leq C_{\kappa}\left( \dfrac{1}{1+|n|}\right) ^{\kappa+1}\,.\end{aligned}$$ This shows that $ \hat{F}\in \mathcal{B}_{4+2\alpha,\kappa+1}$.
Proof of Lemma \[lem\_S\]
-------------------------
From now on, we assume $\kappa >0.$ We recall that: $$\zeta_{n}:=\left[ n^{2}+i\mu n\right] ^{\frac{1}{2}}\,,\quad \forall
\,n\in \mathbb{Z}\,.$$ In what follows, we use without mention the following properties of $\zeta
_{n}$: $$|\zeta_{n}|=|n|\left( 1+\left( \frac{\mu}{n}\right) ^{2}\right) ^{\frac
{1}{4}}\,,\quad \forall \,n\in \mathbb{Z}\setminus \{0\} \,,$$ and$$\begin{array}
[c]{rcl}\xi_{n}:=\mathcal{R}e(\zeta_{n}) & = & \dfrac{|n|}{\sqrt{2}}\left[ \left(
1+\left( \dfrac{\mu}{n}\right) ^{2}\right) ^{\frac{1}{2}}+1\right]
^{\frac{1}{2}}\,,\\
\mathcal{I}m(\zeta_{n}) & = & \dfrac{n}{\sqrt{2}}\left[ \left( 1+\left(
\dfrac{\mu}{n}\right) ^{2}\right) ^{\frac{1}{2}}-1\right] ^{\frac{1}{2}}\,,
\end{array}
\quad \forall \,n\in \mathbb{Z}\setminus \{0\} \,. \label{def_win}$$ We note that $\xi_{n}$ is an increasing function of $|n|$ so that its minimal value (over $n\in \mathbb{Z}\setminus \{0\}$) is reached for $n=\pm1$ and is equal to: $$\rho_{\mu}:=\dfrac{1}{\sqrt{2}}\left[ \left( 1+\mu^{2}\right) ^{\frac{1}{2}}+1\right] ^{\frac{1}{2}}\,.$$ Let$$\alpha_{\mu}:= \dfrac{1}{2}\min(\rho_{\mu}-2,1)\,.$$ For $\mu>{\mu_{crit}}$ we have $\rho_{\mu}>2$, so that $\alpha_{\mu}>0$. We choose $\alpha \in(0,\alpha_{\mu})$ from now on. This is the smallness condition that is mentioned in Lemma \[lem\_S\].
With the above conventions, we first analyze the equations which determine $\hat{w}$:
\[prop\_S\_w\] Given $\hat{F}\in \mathcal{B}_{2\alpha+4,\kappa}$ and $\hat{w}^{\ast}\in \mathcal{B}_{\kappa+2}^{0}$, the equations: $$\begin{aligned}
w_{n}(r) & =\dfrac{\overline{w}_{n}}{r^{\zeta_{n}}}+\int_{r}^{\infty}\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{r}{s}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s$}+\int_{1}^{r}\dfrac{sF_{n}(s)}{2\zeta_{n}}\left(
\dfrac{s}{r}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s$}\,, \label{prf_gamman}\\[6pt]
w_{0}(r) & =-\int_{r}^{\infty}\dfrac{1}{s}\int_{s}^{\infty}t\,F_{0}(t)\text{$\, \mathrm{d}t\, \mathrm{d}s$}\,,\end{aligned}$$ with: $$\overline{w}_{n}=w_{n}^{\ast}-\int_{1}^{\infty}\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{1}{s}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s\,,$}$$ define $C^{2}$ functions. Moreover, we have $\hat{w}\in \mathcal{U}_{\alpha+2,\kappa+2}^{2}$, and there exists a constant $C_{\alpha,\mu
}<\infty$, depending only on $\alpha$ and $\mu$, such that $$\Vert(w_{n})_{n\in \mathbb{Z}}\ ;\ \mathcal{U}_{\alpha+2,\kappa+2}^{2}\Vert \leq
C_{\alpha,\mu}\, \left[ \Vert(F_{n})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{4+2\alpha,\kappa
+2}\Vert+\Vert(w_{n}^{\ast})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\kappa+2}^{0}\Vert \right] \,. \label{regw}$$
We only prove ; existence and continuity follow in a straightforward way. We first treat the case $n\neq0$ and then the case $n=0$. Throughout the proof, we use the shorthand $M_{F}$ for $\Vert(F_{n})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{4+2\alpha,\kappa}\Vert$.
**Case $n\neq0$.** We split the expression defining $w_{n}$ according to (\[prf\_gamman\]): $$w_{n}(r)=\dfrac{\overline{w}_{n}}{r^{\zeta_{n}}}+I_{1}^{n}(r)+I_{\infty}^{n}(r)\,.$$ By definition of the norm on the space $\mathcal{B}_{4+2\alpha,\kappa}$, we have: $$|F_{n}(s)|\leq \dfrac{M_{F}}{(1+|n|)^{\kappa}\,s^{4+2\alpha}}\,,\qquad
\forall \,s\geq1\,.$$ Using that $\xi_{n}\geq \rho_{\mu}>\alpha+2$, and that $c |n| \leq |\zeta_{n}| \leq C(1+\sqrt{|\mu|})|n|$ and $|n| \leq \xi_{n} \leq (1+\sqrt{|\mu|})|n|$ for large values of $|n|$, we get that:$$\begin{aligned}
|I_{1}^{n}(r)| & =\left \vert {\int_{1}^{r}}\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{s}{r}\right) ^{\zeta_{n}}\text{$\, \mathrm{d}s$}\right \vert
\nonumber \\[10pt]
\ & \leq \dfrac{M_{F}}{(1+|n|)^{\kappa}|\zeta_{n}|}\ \dfrac{1}{r^{\xi_{n}}}\ {{\int_{1}^{r}}}s^{1+\xi_{n}-(4+2\alpha)}\text{$\, \mathrm{d}s$}\nonumber \\[10pt]
\ & \leq C_{\alpha,{\mu}}\, \dfrac{M_{F}}{(1+|n|)^{\kappa+2}}\dfrac{\left(
1+r^{\xi_{n}-(2\alpha+2)}\right) }{r^{\xi_{n}}}\,, \label{In1}$$ for all $r\geq1$. Here we used that, by our smallness condition on $\alpha,$ we have $\xi_n - (2\alpha+2) >-1$. We also have: $$\begin{aligned}
|I_{\infty}^{n}(r)| & =\left \vert {\int_{r}^{\infty}}\dfrac{sF_{n}(s)}{2\zeta_{n}}\left( \dfrac{r}{s}\right) ^{\zeta_{n}}\text{$\,
\mathrm{d}s$}\right \vert \nonumber \\[10pt]
& \leq \dfrac{M_{F}}{(1+|n|)^{\kappa}|\zeta_{n}|}\ r^{\xi_{n}}\ {{\int
_{r}^{\infty}}}s^{1-\xi_{n}-(2\alpha+4)}\text{$\, \mathrm{d}s$}\nonumber \\[10pt]
& \leq C_{\alpha,{\mu}}\, \dfrac{M_{F}}{(1+|n|)^{\kappa+2}}\dfrac{1}{r^{2\alpha+2}}\,, \label{Ininf}$$ for all $r\geq1.$ Using these bounds for $r=1$, we obtain:$$|\overline{w}_{n}|\leq|w_{n}^{\ast}|+C_{\alpha,{\mu}}\, \dfrac{M_{F}}{(1+|n|)^{\kappa+2}}\,. \label{wnbar}$$ Plugging (\[In1\]), (\[Ininf\]) and (\[wnbar\]) into (\[prf\_gamman\]) and recalling that $\xi_{n}>\alpha+2$ yields: $$|w_{n}(r)|\leq C_{\alpha,\mu}\, \dfrac{M_{F}+\Vert \hat{w}^{\ast}\,;\,
\mathcal{B}_{\kappa+2}^{0}\Vert}{(1+|n|)^{\kappa+2}}\dfrac{1}{r^{\alpha+2}}\,,\qquad \forall \,r\geq1\,.$$ Differentiating (\[prf\_gamman\]) with respect to $r$, we obtain: $$\partial_{r}w_{n}(r)=-\dfrac{\zeta_{n}\overline{w}_{n}}{r^{\zeta_{n}+1}}+\frac{\zeta_{n}}{r}I_{1}^{n}(r)-\frac{\zeta_{n}}{r}I_{\infty}^{n}(r)\,,\quad \forall \,r\geq1\,.$$ To summarize, when we differentiate $w_n$ with respect to $r$, the decay in $r$ increases by one power, and the decay in $n$ decreases by one power. This observation allows us to bound $\partial_{r}w_{n}$ in the indicated function spaces. Finally, since the expression defining $\hat
{w}$ define a solution of , we plug the bounds on $w_{n}$ and $\partial_{r}w_{n}$ into this equation and get a bound for $\partial
_{rr}w_{n}(r)$. We obtain that there exists a constant $C_{\alpha,\mu}$, depending only on $\alpha$ and $\mu$, such that$$\dfrac{r^{2}|\partial_{rr}w_{n}(r)|}{(1+|n|)^{2}}+\dfrac{r|\partial_{r}w_{n}(r)|}{(1+|n|)}+|w_{n}(r)|\leq C_{\alpha,{\mu}}\, \dfrac{M_{F}+\Vert
\hat{w}^{\ast}\,;\, \mathcal{B}_{\kappa+2}^{0}\Vert}{(1+|n|)^{\kappa+2}}\dfrac{1}{r^{\alpha+2}}\,,\qquad \forall \,r\geq1\,.$$ We emphasize here that the constant $C_{\alpha,\mu}$ depends on $\alpha$ and $\mu.$ Nevertheless, it is clear from the computations above that, when $\alpha$ is fixed and $\mu$ varies in a a compact interval $I \subset \mathbb R,$ this constant remains uniformly bounded.
**Case $n=0$.** Proceeding as in the case $n\neq0$, we get the bound:$$|w_{0}(r)|\leq M_{F}\int_{r}^{\infty}\dfrac{1}{s}\int_{s}^{\infty}\dfrac
{1}{t^{4+2\alpha}}\, \text{$\, \mathrm{d}t\, \mathrm{d}s$}\leq C_{{\alpha}}\dfrac{M_{F}}{r^{2+\alpha}}\,,\quad \forall \,r\geq1\,.$$ Similarly, one shows $$|\partial_{r}w_{0}(r)|\leq \dfrac{M_{F}}{r}\int_{r}^{\infty}\dfrac
{1}{s^{4+2\alpha}}\text{$\, \mathrm{d}s\,$}\leq C_{{\alpha}}\dfrac{M_{F}}{r^{3+\alpha}}\,,\quad \forall \,r\geq1\,,$$ and we again conclude, by recalling the differential equation satisfied by $w_{0}$ (see for $n=0$), that: $${r^{2}|\partial_{rr}w_{0}(r)|}+{r|\partial_{r}w_{0}(r)|}+|w_{0}(r)|\leq
C_{\alpha}\, \dfrac{M_{F}+\Vert \hat{w}^{\ast}\,;\, \mathcal{B}_{\kappa+2}^{0}\Vert}{r^{2+\alpha}}\, \,,\qquad \forall \,r\geq1\,.$$ This completes the proof.
We next consider the equation satisfied by $\hat{\gamma}$:
\[lem\_S\_gamma\] Given $\hat{\phi}\in \mathcal{B}_{\alpha+2,\kappa+2}$ and $\hat{\gamma}^{\ast}\in \mathcal{B}_{\kappa+4}^{0}$, the equations: $$\begin{aligned}
\gamma_{n}(r) & =\dfrac{\overline{\gamma}_{n}}{r^{|n|}}-\int_{r}^{\infty
}\dfrac{s\phi_{n}(s)}{2|n|}\left( \dfrac{r}{s}\right) ^{|n|}\text{$\,
\mathrm{d}s$}-\int_{1}^{r}\dfrac{s\phi_{n}(s)}{2|n|}\left( \dfrac{s}{r}\right) ^{|n|}\text{$\, \mathrm{d}s$}\,,\\[10pt]
\gamma_{0}(r) & =\int_{r}^{\infty}\dfrac{1}{s}\int_{s}^{\infty}t\, \phi
_{0}(t)\text{$\, \mathrm{d}t\, \mathrm{d}s$}\,,\end{aligned}$$ with $$\overline{\gamma}_{n}=\gamma_{n}^{\ast} + \int_{1}^{\infty}\dfrac{s\phi_{n}(s)}{2|n|}\left( \dfrac{1}{s}\right) ^{|n|}\text{$\, \mathrm{d}s$}\,,$$ define $C^{2}$ functions. Moreover, $\hat{\gamma}\in \mathcal{U}_{\alpha
,\kappa+4}^{2}$ and there exists a constant $C_{\alpha}<\infty$, depending only on $\alpha$, such that $$\Vert(\gamma_{n})_{n\in \mathbb{Z}}\ ;\ \mathcal{U}_{\alpha,\kappa+4}^{2}\Vert \leq C_{\alpha}\left[ \Vert(\phi_{n})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\alpha+2,\kappa+2}\Vert+\Vert(\gamma_{n}^{\ast})_{n\in \mathbb{Z}}\ ;\ \mathcal{B}_{\kappa+4}^{0}\Vert \right] \,.
\label{gammareg}$$
The proof is identical to the proof of Proposition \[prop\_S\_w\] and is left to the reader. Lemma \[lem\_S\] is a straightforward consequence of Proposition \[lem\_S\_gamma\] and Proposition \[prop\_S\_w\].
Proof of Lemma \[lem\_Smu\], first item
---------------------------------------
Let $I=[\mu_{-},\mu_{+}] \subset (\mu_{crit},\infty)$ and $\alpha$ be given by . In particular, we have $\alpha < \min \{ \alpha_{\mu},\mu \in
I\}$ so that, applying the results of the preceding section, it follows that $\mathcal{S}_{\mu}\colon \mathcal{B}_{4+2\alpha,\kappa}\times(\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0})\longrightarrow \mathcal{U}_{\alpha,\kappa+4}^{2}\times \mathcal{U}_{\alpha+2,\kappa+2}^{2}$ is a well-defined continuous linear map, for all values of $\mu \in I$. We now show that the map $\mathcal{S}\colon \mu \mapsto
\mathcal{S}_{\mu}$ is also continuous. This amounts to show that, for arbitrary $\mu_{0}\in I$ and $\mu \in I,$ there exists a constant $C_{\mu}$ which converges to zero as $\mu$ converges to $\mu_{0}$, such that, for arbitrary $(\hat{F},\hat{x}^{\ast})\in \mathcal{B}_{4+2\alpha,\kappa}\times
(\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0})$: $$\Vert \mathcal{S}_{\mu_{0}}(\hat{F},\hat{x}^{\ast})-\mathcal{S}_{{\mu}}(\hat
{F},\hat{x}^{\ast})\,;\, \mathcal{U}_{\alpha,\kappa+4}^{2}\times
\mathcal{U}_{\alpha+2,\kappa+2}^{2}\Vert \leq C_{\mu}\Vert(\hat{F},\hat
{x}^{\ast})\,;\, \mathcal{B}_{4+2\alpha,\kappa}\times(\mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0})\Vert \,.$$ Given $\mu \in I$, we let $(\hat{\gamma}[\mu],\hat{w}[\mu]):=\mathcal{S}_{{\mu}}(\hat{F},\hat{x}^{\ast})$. Since $\hat{\gamma}[\mu]$ is obtained from $\hat{w}[\mu]$ *via* an equation which does not depend on $\mu$, we can apply directly Proposition \[prop\_S\_w\], yielding that, for arbitrary $(\mu,\tilde{\mu})\in I^{2}$: $$\Vert \hat{\gamma}[\mu]-\hat{\gamma}[\tilde{\mu}]\,;\, \mathcal{U}_{\alpha
,\kappa+4}^{2}\Vert \leq C_{\alpha}\Vert \hat{w}[\mu]-\hat
{w}[\tilde{\mu}]\,;\, \mathcal{U}_{\alpha+2,\kappa+2}^{2}\Vert \,.$$ Hence, it suffices to prove that $\hat{w}[\mu]$ is continuous with respect to $\mu$, in order to obtain the continuity of $\mathcal{S}$.
To show the continuity of $\hat{w}[\mu]$, we first remark that $w_{0}[\mu]$ does not depend $\mu$, so that we only detail the case $n\neq0$. Let $n\neq 0,$ we split ${w}_{n}[\mu]$ into three terms:$$w_{n}[\mu](r)=W_{b}^{n}[\mu](r)+I_{1}^{n}[\mu](r)+I_{\infty}^{n}[\mu](r)\,,$$ where: $$W_{b}^{n}[\mu](r)=\dfrac{\bar{w}_{n}[\mu]}{r^{\zeta_{n}[\mu]}}\,,\quad I_{1}^{n}[\mu](r)={\int_{1}^{r}}\dfrac{sF_{n}(s)}{2\zeta_{n}[\mu]}\left( \dfrac{s}{r}\right) ^{\zeta_{n}[\mu]}\text{$\, \mathrm{d}s$}\,,\quad I_{\infty}^{n}[\mu](r)=\int_{r}^{\infty}\dfrac{sF_{n}(s)}{2\zeta_{n}[\mu]}\left( \dfrac{r}{s}\right) ^{\zeta_{n}[\mu]}\text{$\, \mathrm{d}s$}\,.$$ We recall that $$\begin{aligned}
\partial_{r}w_{n}[\mu](r) & =-\dfrac{\zeta_{n}[\mu]}{r}W_{b}^{n}[\mu
](r)-\dfrac{\zeta_{n}[\mu]}{r}I_{1}^{n}[\mu](r)+\dfrac{\zeta_{n}[\mu]}{r}I_{\infty}^{n}[\mu](r)\,,\label{eq_derw1}\\[10pt]
\partial_{rr}w_{n}[\mu](r) & =F_{n}(r)-\dfrac{\partial_{r}w_{n}[\mu](r)}{r} + \dfrac{(n^{2}+i\mu n)}{r^2}w_{n}[\mu](r)\,. \label{eq_derw2}$$ Note also that $\zeta_{n}[\mu]=(n^{2}+i\mu n)^{1/2}$ is a continuous function of $\mu$ uniformly in $n$. Indeed, since the square root is analytic in a neighborhood of $1$, we have for sufficiently large $n$ (uniformly in $\mu \in I$): $$|\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]|=|n|\left \vert \left( 1+\dfrac{i\mu
}{n}\right) ^{\frac{1}{2}}-\left( 1+\dfrac{i\tilde{\mu}}{n}\right)
^{\frac{1}{2}}\right \vert \leq C|\mu-\tilde{\mu}|\,.$$ We also have the bound $|\zeta_{n}[\mu]|\leq c|n|,$ with $c$ independent of $\mu \in I$. Introducing these bounds into - shows that the continuity of $\hat{w}[\mu]$ follows from the continuity of $(\hat{W}_{b}[\mu],\hat{I}_{1}[\mu],\hat
{I}_{\infty}[\mu])$ in $\mathcal{B}_{\alpha+2,\kappa+2}$. For consistency, the three sequences, which are only defined for $n\neq0,$ are completed by $0$ for $n=0$.
To begin with, we consider the continuity of $\mu \mapsto \hat{I}_{1}[\mu]$. Let $(\mu,\tilde{\mu})\in I^{2}$, and assume that $|\zeta_{n}[\mu]-\zeta
_{n}[\tilde{\mu}]|<\alpha/2$, uniformly in $n$. We have: $$I_{1}^{n}[\mu]-I_{1}^{n}[\tilde{\mu}]=J_{1}(r)+J_{2}(r)\,,$$ where: $$\begin{aligned}
J_{1}(r) & =\int_{1}^{r}\dfrac{sF_{n}(s)}{2\zeta_{n}[\mu]}\left[
\dfrac{\zeta_{n}[\tilde{\mu}]-\zeta_{n}[\mu]}{\zeta_{n}[\tilde{\mu}]}\right]
\left( \dfrac{s}{r}\right) ^{\zeta_{n}[\mu]}\, \mathrm{d}\text{$s$}\,,\\[6pt]
J_{2}(r) & =\int_{1}^{r}\dfrac{sF_{n}(s)}{2\zeta_{n}[\tilde{\mu}]}\left[
1-\left( \dfrac{s}{r}\right) ^{\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]}\right] \left( \dfrac{s}{r}\right) ^{\zeta_{n}[\tilde{\mu}]}\,
\mathrm{d}\text{$s$}\,.\end{aligned}$$ We have, uniformly in $n$: $$\begin{aligned}
\left \vert \dfrac{\zeta_{n}[\tilde{\mu}]-\zeta_{n}[\mu]}{\zeta_{n}[\tilde{\mu
}]}\right \vert & \leq c|\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]|=o(1)\,,\\[10pt]
\left \vert 1-\left( \dfrac{s}{r}\right) ^{\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]}\right \vert & \leq c\left \vert \zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]\right \vert \ln(r)r^{|\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]|}=o(1)\ln(r)r^{\alpha/2}\,.\end{aligned}$$ We introduce these uniform bounds in $J_1$ and $J_2$ and redo the computations in the proof of [Proposition \[prop\_S\_w\]]{} (see ). We get: $$|J_{1}(r)|\leq \dfrac{\Vert \hat{F};\mathcal{B}_{4+2\alpha}\Vert
}{(1+|n|^{\kappa+2})}\ \dfrac{o(1)}{r^{2+2\alpha}}\,,\quad|J_{2}(r)|\leq \dfrac{\Vert \hat{F};\mathcal{B}_{4+2\alpha}\Vert}{(1+|n|^{\kappa
+2})}\ \dfrac{o(1)\ln(r)}{r^{2+3\alpha/2}}\,,$$ where the term $o(1)$ denotes a constant converging to $0$ when $\mu-\tilde{\mu} \rightarrow0$, uniformly in $n\,$. Finally, we have that, for all $n\in \mathbb{Z}\setminus \{0\}$: $$|I_{1}^{n}[\mu]-I_{1}^{n}[\tilde{\mu}]|\leq \dfrac{\Vert \hat{F};\mathcal{B}_{4+2\alpha}\Vert}{(1+|n|^{\kappa+2})}\ \dfrac{o(1)}{r^{2+\alpha}}\,, \label{eq_diffI1}$$
We now prove the continuity of $\mu \mapsto \hat{I}_{\infty}[\mu]$. For any $(\mu,\tilde{\mu})\in I$ and $n\neq0$, we perform a similar splitting: $$I_{\infty}^{n}[\mu]-I_{\infty}^{n}[\tilde{\mu}]=J_{1}(r)+J_{2}(r)\,,$$ where: $$\begin{aligned}
J_{1}(r) & =\int_{r}^{\infty}\dfrac{sF_{n}(s)}{2\zeta_{n}[\mu]}\left[
\dfrac{\zeta_{n}[\tilde{\mu}]-\zeta_{n}[\mu]}{\zeta_{n}[\tilde{\mu}]}\right]
\left( \dfrac{r}{s}\right) ^{\zeta_{n}[\mu]}\mathrm{d}\text{$s$}\,,\\[6pt]
J_{2}(r) & =\int_{r}^{\infty}\dfrac{sF_{n}(s)}{2\zeta_{n}[\tilde{\mu}]}\left[ 1-\left( \dfrac{r}{s}\right) ^{\zeta_{n}[\mu]-\zeta_{n}[\tilde
{\mu}]}\right] \left( \dfrac{r}{s}\right) ^{\zeta_{n}[\tilde{\mu}]}\mathrm{d}\text{$s$}\,.\end{aligned}$$ As in the preceding bound we have, uniformly in $n$: $$\begin{aligned}
\left \vert \dfrac{\zeta_{n}[\tilde{\mu}]-\zeta_{n}[\mu]}{\zeta_{n}[\tilde{\mu
}]}\right \vert & \leq c|\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]|=o(1)\,,\\[10pt]
\left \vert 1-\left( \dfrac{s}{r}\right) ^{\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]}\right \vert & \leq o(1)\ln \left( \dfrac{s}{r}\right) \left(
\dfrac{s}{r}\right) ^{\alpha/2}\,,\end{aligned}$$ where we have used that $|\zeta_{n}[\mu]-\zeta_{n}[\tilde{\mu}]|\leq \alpha/2$. We can therefore redo the computations in the proof of [Proposition \[prop\_S\_w\]]{} (see ). This yields, for all $n\in\mathbb Z \setminus\{0\}$: $$|J_{1}(r)| + |J_2(r)| \leq \dfrac{\Vert \hat{F};\mathcal{B}_{4+2\alpha,\kappa}\Vert
}{(1+|n|^{\kappa+2})}\ \dfrac{o(1)}{r^{2+\alpha}}\,.$$ As in the preceding estimate we conclude that: $$\Vert \hat{I}_{\infty}[\mu]-\hat{I}_{\infty}[\tilde{\mu}];\mathcal{B}_{\alpha
_{-}+2,\kappa+2}\Vert=o(1)\Vert \hat{F};\mathcal{B}_{4+2\alpha,\kappa}\Vert \,.$$ Finally, we prove the continuity of $\mu \mapsto \hat{W}_b[\mu]$: $$\begin{aligned}
\left| W^n_b[\mu](r) - W^n_b[\tilde{\mu}](r) \right| &=& \left| \dfrac{\bar{w}_n[\mu] - \bar{w}_n[\tilde{\mu}]}{r^{\zeta_n[\mu]}} + \dfrac{\bar{w}_n[\tilde{\mu}]}{r^{\zeta_n[\mu_-]}} \left( \dfrac{1}{r^{\zeta_n[\mu]- \zeta_n[\mu_-]}} - \dfrac{1}{r^{\zeta_n[\tilde{\mu}]- \zeta_n[\mu_-]}}\right) \right| \,, \\[10pt]
&\leq &
\left| \dfrac{\bar{w}_n[\mu] - \bar{w}_n[\tilde{\mu}]}{r^{\zeta_n[\mu]}} \right| + \left| \dfrac{\bar{w}_n[\tilde{\mu}]}{r^{\zeta_n[\mu_-]}} \left( \dfrac{1}{r^{\zeta_n[\mu]- \zeta_n[\mu_-]}} - \dfrac{1}{r^{\zeta_n[\tilde{\mu}]- \zeta_n[\mu_-]}}\right) \right| \,, \\[10pt]
&\leq &
\dfrac{|\bar{w}_n[\mu] - \bar{w}_n[\tilde{\mu}]|}{r^{2+2\alpha}} + \dfrac{|\bar{w}_n[\tilde{\mu}]|}{r^{2+\alpha}} \dfrac{1}{r^{\alpha}} \left|\left( \dfrac{1}{r^{\zeta_n[\mu]- \zeta_n[\mu_-]}} - \dfrac{1}{r^{\zeta_n[\tilde{\mu}]- \zeta_n[\mu_-]}}\right) \right|\,,\end{aligned}$$ where we have used that $\mathcal{R}e(\zeta_n[\mu]) \geq \mathcal{R}e(\zeta_n[\mu_-]) > 2+ 2\alpha.$ At this point we note that the bound which we obtained above for $I_{\infty}^n$ in $r=1$ yields: $$|\bar{w}_n[\mu] - \bar{w}_n[\tilde{\mu}]| = \dfrac{o(1)}{(1+|n|)^{\kappa+2}}\Vert \hat{F};\mathcal{B}_{4+2\alpha,\kappa}\Vert\,.$$ As $\mu \mapsto \zeta_n[\mu]$ is continuous in $\mu$ (uniformly in $n$) and satisfies $\mathcal{R}e(\zeta_n[\mu]) \geq \mathcal{R}e(\zeta_n[\mu_-]),$ for all $\mu \in I,$ we also have: $$\left\| \dfrac{1}{r^{\alpha}} \left( \dfrac{1}{r^{\zeta_n[\mu]- \zeta_n[\mu_-]}} - \dfrac{1}{r^{\zeta_n[\tilde{\mu}]- \zeta_n[\mu_-]}}\right) \, ; \, L^{\infty}(1,\infty) \right\| = o(1)\,,$$ where $o(1)$ is uniform in $n.$ By combination, this yields, for all $n \in \mathbb Z \setminus \{0\}$: $$\left|
W^n_b[\mu](r) - W^n_b[\tilde{\mu}](r)
\right| \leq \dfrac{o(1)}{(1+|n|^{\kappa+2}) r^{2+\alpha}} \left[ \Vert \hat{F};\mathcal{B}_{4+2\alpha,\kappa}\Vert + \Vert \hat{w}^*;\mathcal{B}^0_{\kappa+2}\Vert \right]\,.$$ This completes the proof of the first item in Lemma \[lem\_Smu\].
Proof of Lemma \[lem\_Smu\], second item
----------------------------------------
In this paragraph, we prove that the map $\mathcal{T}_{1}$ is one-to-one and onto with a continuous inverse. Given $\hat{x}^{\ast}=(\hat{\gamma}^{\ast},\hat{w}^{\ast
})\in \mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0}$, we set $(\hat{\gamma},\hat{w})=\mathcal{S}_{\mu}(0,\hat{x}^*)$. A straightforward computation shows: $$w_{n}(r)=\dfrac{w_{n}^{\ast}}{r^{\zeta_{n}}}\qquad \gamma_{n}(r)=\dfrac
{\bar{\gamma}_{n}}{r^{|n|}} + \int_{r}^{\infty}\dfrac{sw_{n}(s)}{2|n|}\left(
\dfrac{r}{s}\right) ^{|n|}\text{$\, \mathrm{d}s$} + \int_{1}^{r}\dfrac
{sw_{n}(s)}{2|n|}\left( \dfrac{s}{r}\right) ^{|n|}\text{$\, \mathrm{d}s$}\,,\quad \forall \,r\geq1\,,$$ where: $$\bar{\gamma}_{n}=\gamma_{n}^{\ast}-\int_{1}^{\infty}\dfrac{sw_{n}(s)}{2|n|}\left( \dfrac{1}{s}\right) ^{|n|}\text{$\, \mathrm{d}s$}\,.$$ Therefore, we have, for all $n\in \mathbb{Z}\setminus \{0\}$: $$\gamma_{n}(1)=\gamma_{n}^{\ast}\,,$$ together with:$$\partial_{r}\gamma_{n}(1)=-|n|\gamma_{n}^{\ast} + {\int_{1}^{\infty}}sw_{n}(s)\left( \dfrac{1}{s}\right) ^{|n|}\text{$\, \mathrm{d}s$}=-|n|\gamma_{n}^{\ast} - \dfrac{w_{n}^{\ast}}{2-|n|-\zeta_{n}}\,,$$ so that $(\hat{v}_r^{\ast},\hat{v}_{\theta}^{\ast})=\mathcal{T}_{1}(\hat{x}^{\ast})$ satisfies:
- $v_{r,0}^{\ast}=v_{\theta,0}^{\ast}=0$,
- $v_{r,n}^{\ast}=in\gamma_{n}^{\ast}$, and $v_{\theta,n}^{\ast}= -\partial_r \gamma_n(1) = |n|\gamma
_{n}^{\ast} + \dfrac{w_{n}^{\ast}}{2-|n|-\zeta_{n}}$, for all $n\in
\mathbb{Z}\setminus \{0\}$.
This shows that the map $\mathcal{T}_{1}$ is one-to-one and onto. Indeed, the inverse map is given by: $$\mathcal{T}_{1}^{-1}[\hat{v}_{r}^{\prime},\hat{v}_{\theta}^{\prime})]=(\hat{\gamma}^{\prime},\hat{w}^{\prime})\,,$$ where:
- $\gamma_{0}^{\prime}=w_{0}^{\prime}=0,$
- $\gamma_{n}^{\prime}=\dfrac{v_{r,n}^{\prime}}{in}$, and $w_{n}^{\prime
}= (2-|n|-\zeta_{n})\left( v_{\theta,n}^{\prime} - \dfrac{|n|v_{r,n}^{\prime}}{in}\right) $ for all $n\in \mathbb{Z}\setminus \{0\}$.
It is therefore clear that $\mathcal{T}_{1}^{-1}\in \mathcal{L}_{c}(\mathcal{B}_{\kappa+3}^{0}\times \mathcal{B}_{\kappa+3}^{0}\ ;\ \mathcal{B}_{\kappa+4}^{0}\times \mathcal{B}_{\kappa+2}^{0})$. This completes the proof.
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[^1]: Work supported in part by the Swiss National Science Foundation.
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abstract: 'We present a new derivation of relativistic dissipative hydrodynamic equations, which invokes the second law of thermodynamics for the entropy four-current expressed in terms of the single-particle phase-space distribution function obtained from Grad’s 14-moment approximation. This derivation is complete in the sense that all the second-order transport coefficients are uniquely determined within a single theoretical framework. In particular, this removes the long-standing ambiguity in the relaxation time for bulk viscosity thereby eliminating one of the uncertainties in the extraction of the shear viscosity to entropy density ratio from confrontation with the anisotropic flow data in relativistic heavy-ion collisions. We find that in the one-dimensional scaling expansion, these transport coefficients prevent the occurrence of cavitation even for rather large values of the bulk viscosity estimated in lattice QCD.'
author:
- 'Amaresh Jaiswal, Rajeev S. Bhalerao and Subrata Pal'
title: 'Complete relativistic second-order dissipative hydrodynamics from the entropy principle'
---
Relativistic fluid dynamics has been quite successful in explaining the various collective phenomena observed in cosmology, astrophysics and the physics of high-energy heavy-ion collisions. The earliest theories of relativistic dissipative hydrodynamics by Eckart [@Eckart:1940zz] and Landau-Lifshitz [@Landau] were based on the assumption that the entropy four-current is first order in dissipative quantities, which led to parabolic differential equations that suffered from acausality. The second-order Israel-Stewart (IS) theory [@Israel:1979wp] with the entropy current quadratic in dissipative quantities led to hyperbolic equations and thus restored causality.
Application of the second-order viscous hydrodynamics to high-energy heavy-ion collisions has evoked widespread interest ever since a surprisingly small value for the shear viscosity to entropy density ratio $\eta/s$ was estimated from the analysis of the elliptic flow data [@Romatschke:2007mq]. Indeed the estimated $\eta/s$ was close to the conjectured lower bound $\eta/s|_{\rm KSS} = 1/4\pi$ [@Policastro:2001yc; @Kovtun:2004de]. This led to the claim that the quark-gluon plasma (QGP) formed at the Relativistic Heavy-Ion Collider (RHIC) was the most perfect fluid ever observed. A precise estimate of $\eta/s$ is vital to the understanding of the properties of the QCD matter.
In this Communication, we provide a solution to one of the major uncertainties that hinders an accurate extraction of the viscous corrections to the ideal fluid behavior, namely the inadequate knowledge of the second-order transport coefficients. In the standard derivation of second-order evolution equations for dissipative quantities from the requirement of positive divergence of the entropy four-current, the most general algebraic form of the entropy current is parameterized in terms of unknown thermodynamic coefficients [@Israel:1979wp]. These coefficients which are related to relaxation times and coupling lengths of the shear and bulk pressures and heat current, however, remain undetermined within the framework of thermodynamics alone [@Muronga:2003ta]. While kinetic theory for massless particles [@Baier:2006um] and strongly coupled ${\cal N}=4$ supersymmetric Yang-Mills theory [@Baier:2007ix] predict different shear relaxation times $\tau_\pi = 3/2\pi T$ and $(2-\ln 2)/2\pi T$, respectively, for $\eta/s = 1/4\pi$, the bulk relaxation time $\tau_\Pi$ remains completely ambiguous. Hence ad hoc choices have been made for the value of $\tau_\Pi$ in hydrodynamic studies [@Fries:2008ts; @Denicol:2009am; @Song:2009rh; @Rajagopal:2009yw].
Lattice QCD studies for gluonic plasma in fact predict large values of bulk viscosity to entropy density ratio, $\zeta/s$, of about (6-25) $\eta/s|_{\rm KSS}$ near the QCD phase-transition temperature $T_c$ [@Meyer:2007dy]. This would translate into large values of the bulk pressure and bulk relaxation time, and may affect the evolution of the system significantly [@Denicol:2009am; @Song:2009rh]. Further, the large bulk pressure could result in a negative longitudinal pressure leading to mechanical instabilities (cavitation) whereby the fluid breaks up into droplets [@Torrieri:2008ip; @Rajagopal:2009yw; @Bhatt:2010cy]. Thus the theoretical uncertainties arising from the absence of reliable estimates for the second-order transport coefficients should be eliminated for a proper understanding of the system evolution.
We present here a formal derivation of the dissipative hydrodynamic equations where all the second-order transport coefficients get determined uniquely within a single theoretical framework. This is achieved by invoking the second law of thermodynamics for the generalized entropy four-current expressed in terms of the phase-space distribution function given by Grad’s 14-moment approximation. Significance of these coefficients is demonstrated in one-dimensional scaling expansion of the viscous medium.
Hydrodynamic evolution of a medium is governed by the conservation equations for the energy-momentum tensor and particle current [@deGroot] $$\begin{aligned}
\label{NTD}
T^{\mu\nu} &= \int dp \ p^\mu p^\nu (f+\bar f) = \epsilon u^\mu u^\nu-(P+\Pi)\Delta ^{\mu \nu}
+ \pi^{\mu\nu}, \nonumber\\
N^\mu &= \int dp \ p^\mu (f-\bar f) = nu^\mu + n^\mu,\end{aligned}$$ where $dp = g d{\bf p}/[(2 \pi)^3\sqrt{{\bf p}^2+m^2}]$, $g$ and $m$ being the degeneracy factor and particle rest mass, $p^{\mu}$ is the particle four-momentum, $f\equiv f(x,p)$ is the phase-space distribution function for particles and $\bar f$ for antiparticles. The above integral expressions assume the system to be dilute so that the effects of interaction are small [@deGroot]. In the above tensor decompositions, $\epsilon,
P, n$ are respectively energy density, pressure, net number density, and the dissipative quantities are the bulk viscous pressure $(\Pi)$, shear stress tensor $(\pi^{\mu\nu})$ and particle diffusion current $(n^\mu)$. Here $\Delta^{\mu\nu}=g^{\mu\nu}-u^\mu u^\nu$ is the projection operator on the three-space orthogonal to the hydrodynamic four-velocity $u^\mu$ defined in the Landau frame: $T^{\mu\nu}
u_\nu=\epsilon u^\mu$.
Energy-momentum conservation, $\partial_\mu T^{\mu\nu} =0$ and current conservation, $\partial_\mu N^{\mu}=0$ yield the fundamental evolution equations for $\epsilon$, $u^\mu$ and $n$. $$\begin{aligned}
\label{evol}
D\epsilon + (\epsilon+P+\Pi)\partial_\mu u^\mu - \pi^{\mu\nu}\nabla_{(\mu} u_{\nu)} &= 0, \nonumber\\
(\epsilon+P+\Pi)D u^\alpha - \nabla^\alpha (P+\Pi) + \Delta^\alpha_\nu \partial_\mu \pi^{\mu\nu} &= 0, \nonumber\\
Dn + n\partial_\mu u^\mu + \partial_\mu n^{\mu} &=0.\end{aligned}$$ We use the standard notation $A^{(\alpha}B^{\beta )} = (A^\alpha
B^\beta + A^\beta B^\alpha)/2$, $D=u^\mu\partial_\mu$, and $\nabla^\alpha = \Delta^{\mu\alpha}\partial_\mu$. Even if the equation of state is given, the system of Eqs. (\[evol\]) is not closed unless the evolution equations for the dissipative quantities $\Pi$, $\pi^{\mu\nu}$, $n^\mu$ are specified.
Traditionally the dissipative equations have been obtained by invoking the second law of thermodynamics, viz., $\partial_\mu S^\mu \geq 0$, where the entropy four-current $S^\mu$ is given by [@Israel:1979wp; @Baier:2006um; @Muronga:2003ta] $$\begin{aligned}
\label{AEFC}
S^\mu =& \ P\beta u^\mu - \alpha N^\mu + \beta u_\nu T^{\mu \nu}-Q^\mu(\delta N^\mu, \delta T^{\mu \nu}) \nonumber \\
=& \ s u^\mu -
\frac{\mu n^\mu}{T}-
\left(\beta_0\Pi^2 - \beta_1 n_\nu n^\nu
+ \beta_2\pi_{\rho\sigma} \pi^{\rho\sigma}\right) \frac{u^\mu}{2T} \nonumber\\
&- \left(\alpha_0\Pi\Delta^{\mu\nu} + \alpha_1\pi^{\mu\nu}\right)\frac{n_\nu}{T}.\end{aligned}$$ Here $\beta=1/T$ is the inverse temperature, $\mu$ is the chemical potential, $\alpha=\beta \mu$, and $Q^\mu$ is a function of deviations from local equilibrium. The second equality is obtained by using the definition of the equilibrium entropy density $s=\beta (\epsilon+P-\mu n)$ and Taylor-expanding $Q^\mu$ to second order in dissipative fluxes. In this expansion, $\beta_i(\epsilon,n) \geq 0$ and $\alpha_i(\epsilon,n) \geq 0$ are the thermodynamic coefficients corresponding to pure and mixed terms. These coefficients can be obtained within the kinetic theory approach such as the IS theory [@Israel:1979wp]. However, it is important to note that they cannot be determined solely from thermodynamics using Eq. (\[AEFC\]) and as a consequence the evolution equations remain incomplete.
In contrast to the above approach, our starting point for the derivation of the dissipative evolution equations is the entropy four-current expression generalized from Boltzmann’s H-function: $$\begin{aligned}
\label{EFC}
S^\mu_{r=0} &=& -\int dp ~p^\mu \left[ f \left(\ln f - 1\right)+(f \to \bar f) \right] , \nonumber \\
S^\mu_{r=\pm 1} &=& -\int dp ~p^\mu \left[ \left(f \ln f + r\tilde f \ln \tilde f\right)+(f \to \bar f) \right],\quad\end{aligned}$$ where $\tilde f \equiv 1 - rf$ and $r = 1,-1,0$ for Fermi, Bose, and Boltzmann gas, respectively. The divergence of $S^\mu_{r=0,\pm 1}$ leads to $$\label{EFCD}
\partial_\mu S^\mu = -\int dp ~p^\mu \left[ \left(\partial_\mu f\right)
\ln (f/\tilde f) +(f \to \bar f) \right] .$$
For small departures from equilibrium, $f$ and $\bar f$ can be written as $f =
f_0 + \delta f$ and $\bar f = \bar f_0 + \delta \bar f$. The equilibrium distribution functions are defined as $f_0 = [\exp(\beta u\cdot p -\alpha) + r]^{-1}$ and $\bar f_0 = [\exp(\beta u\cdot p +\alpha) + r]^{-1}$, where $\beta=1/T$ and $\alpha=\mu/T$ are obtained from the equilibrium matching conditions $n\equiv n_0$ and $\epsilon \equiv \epsilon_0$.
To proceed further, we take recourse to Grad’s 14-moment approximation [@Grad] for the single particle distribution in orthogonal basis [@Denicol:2010xn; @Jaiswal:2012qm] $$\label{G14}
f = f_0 + f_0 \tilde f_0 \phi, ~~~ \phi = \lambda_\Pi \Pi + \lambda_n n_\alpha p^\alpha
+ \lambda_\pi \pi_{\alpha\beta} p^\alpha p^\beta, \quad$$ and similarly for $\bar f$. The coefficients ($\lambda_\Pi, \lambda_n, \lambda_\pi$) are assumed to be independent of four-momentum $p^{\mu}$ and are functions of $(\epsilon, \alpha, \beta)$. From Eqs. (\[EFCD\]) and (\[G14\]), we get $$\begin{aligned}
\label{EFCD1}
\partial_\mu S^\mu = -\!\!\int\! dp ~ p^\mu \Big[
& \left(\partial_\mu f\right)
\left\{ \ln \! \left( \frac{f_0}{\tilde f_0} \right)\! + \ln \!
\left(\! 1\! + \frac{\phi}{1-rf_0\phi}\right) \! \right\} \nonumber \\
& + (f \to \bar f, ~ f_0 \to \bar f_0) \Big].\end{aligned}$$ The $\phi$-independent terms on the right vanish due to energy-momentum and current conservation equations. To obtain second-order evolution equations for dissipative quantities, one should consider $S^\mu$ up to the same order. Hence $\partial_\mu S^\mu$ necessarily becomes third-order. Expanding the $\phi$-dependent terms in Eq. (\[EFCD1\]) and retaining all terms up to third order in gradients (where $\phi$ is linear in dissipative quantities), we get $$\begin{aligned}
\label{EFCD2}
\partial_\mu S^\mu = & -\int dp ~p^\mu \Big[ \Big \{
\phi\left(\partial_\mu f_0\right) -
\phi^2 (\tilde f_0 -1/2)(\partial_{\mu}f_0) \nonumber \\
&+\phi^2 \partial_\mu (f_0 \tilde f_0) + \phi f_0 \tilde f_0 (\partial_\mu \phi) \Big \}
+ (f_0 \to \bar f_0) \Big].\end{aligned}$$ The various integrals in the above equation can be decomposed into hydrodynamic tensor degrees of freedom via the definitions: $$\begin{aligned}
\label{TDI}
I^{\mu_1\mu_2\cdots\mu_n}_\pm \equiv& \int dp \ p^{\mu_1} \cdots p^{\mu_n} (f_0 \pm \bar f_0)
= I_{n0}^\pm u^{\mu_1} \cdots u^{\mu_n} \nonumber \\
& + I_{n1}^\pm (\Delta^{\mu_1\mu_2} u^{\mu_3} \cdots u^{\mu_n} + \mathrm{perms}) + \cdots,\end{aligned}$$ where ‘perms’ denotes all non-trivial permutations of the Lorentz indices. We similarly define $J^{\mu_1\mu_2\cdots\mu_n}_\pm $ and $K^{\mu_1\mu_2\cdots\mu_n}_\pm $ where the momentum integrals are weighted with $f_0 \tilde f_0 \pm (f_0 \to \bar f_0)$ and $f_0 \tilde f_0^2 \pm (f_0 \to \bar f_0)$, and are tensor decomposed with coefficients $J_{nq}^\pm$ and $K_{nq}^\pm$, respectively. All these coefficients can be obtained by suitable contractions of the integrals and are related to each other by $$\begin{aligned}
\label{ICR}
2K_{nq}^\pm &= J_{nq}^\pm + \frac{1}{\beta}\big[- J_{n-1,q-1}^\pm + (n-2q)J_{n-1,q}^\pm\big], \nonumber \\
J_{nq}^\pm &= \frac{1}{\beta}\left[-I_{n-1,q-1}^\pm + (n-2q)I_{n-1,q}^\pm \right],\end{aligned}$$ and also satisfy the differential relations $$\begin{aligned}
\label{ICDR}
2K_{nq}^\pm &= J_{nq}^\pm - \frac{d}{d\beta}J_{n-1,q}^\pm = J_{nq}^\pm + \frac{d}{d\alpha}J_{nq}^\pm, \nonumber \\
J_{nq}^\pm &= -\frac{d}{d\beta}I_{n-1,q}^\pm = \frac{d}{d\alpha}I_{nq}^\pm. \end{aligned}$$ With the help of these relations and Grad’s 14-moment approximation, Eq. (\[EFCD2\]) reduces to $$\begin{aligned}
\label{EFCD3}
\partial_\mu S^\mu =
& -\! \beta\Pi\Big[ \theta +\! \beta_0\dot\Pi
+\! \beta_{\Pi\Pi} \Pi\theta
+ \alpha_0 \nabla_\mu n^{\mu}
+ \psi\alpha_{n\Pi } n_\mu \dot u^\mu \nonumber \\
&+ \psi\alpha_{\Pi n} n_\mu \nabla^{\mu}\alpha \Big]
\!-\! \beta n^\mu \Big[ T\nabla_\mu \alpha - \beta_1\dot n_\mu
- \beta_{nn} n_\mu \theta \nonumber \\
&+ \alpha_0\nabla_{\mu}\Pi
+ \alpha_1\nabla_\nu \pi^\nu_\mu
+ \tilde \psi\alpha_{n\Pi } \Pi\dot u_\mu
+ \tilde \psi\alpha_{\Pi n}\Pi\nabla_{\mu}\alpha \nonumber \\
&+ \tilde \chi\alpha_{\pi n} \pi^\nu_\mu \nabla_\nu \alpha
+ \tilde \chi\alpha_{n\pi } \pi^\nu_\mu \dot u_\nu \Big]
\!\!+\! \beta\pi^{\mu\nu}\Big[ \sigma_{\mu\nu}
\!-\! \beta_2\dot\pi_{\mu\nu} \nonumber \\
&- \beta_{\pi\pi}\theta\pi_{\mu\nu}
- \alpha_1 \nabla_{\langle\mu}n_{\nu\rangle}
- \chi\alpha_{\pi n}n_{\langle\mu}\nabla_{\nu\rangle}\alpha \nonumber \\
&- \chi\alpha_{n\pi }n_{\langle\mu}\dot u_{\nu\rangle} \Big],\end{aligned}$$ where $\alpha_i,~\beta_i,~\alpha_{XY},~\beta_{XX}$ are known functions of $\beta,~\alpha$ and the integral coefficients $I_{nq}^\pm,~J_{nq}^\pm$ and $K_{nq}^\pm$. Two new parameters $\psi$ and $\chi$ with $\tilde \psi =
1-\psi$ and $\tilde \chi = 1-\chi$ are introduced to ‘share’ the contributions stemming from the cross terms of $\Pi$ and $\pi^{\mu\nu}$ with $n^{\mu}$.
The second law of thermodynamics, $\partial_{\mu}S^{\mu}\ge 0$, is guaranteed to be satisfied if we impose linear relationships between thermodynamical fluxes and extended thermodynamic forces, leading to the following evolution equations for bulk, charge current and shear $$\begin{aligned}
\Pi &= -\zeta\Big[ \theta
+ \beta_0 \dot \Pi
+ \beta_{\Pi\Pi} \Pi \theta
+ \alpha_0 \nabla_\mu n^\mu \nonumber \\
&\quad\quad\quad + \psi\alpha_{n\Pi} n_\mu \dot u^\mu
+ \psi\alpha_{\Pi n} n_\mu \nabla^\mu \alpha \Big], \label{bulk} \\
n^{\mu} &= \lambda \Big[ T \nabla^\mu \alpha
- \beta_1\dot n^{\langle\mu\rangle}
- \beta_{nn} n^\mu \theta
+ \alpha_0 \nabla^\mu \Pi \nonumber \\
&\quad\quad\; + \alpha_1 \Delta^\mu_\rho \nabla_\nu \pi^{\rho\nu}
+ \tilde \psi\alpha_{n\Pi} \Pi \dot u^{\langle\mu\rangle}
+ \tilde \psi\alpha_{\Pi n} \Pi \nabla^\mu \alpha \nonumber \\
&\quad\quad\; + \tilde \chi\alpha_{\pi n} \pi_\nu^\mu \nabla^\nu \alpha
+ \tilde \chi\alpha_{n\pi} \pi_\nu^\mu \dot u^\nu \Big], \label{current} \\
\pi^{\mu\nu} &= 2\eta\Big[ \sigma^{\mu\nu}
- \beta_2\dot\pi^{\langle\mu\nu\rangle}
- \beta_{\pi\pi}\theta\pi^{\mu\nu}
- \alpha_1 \nabla^{\langle\mu}n^{\nu\rangle} \nonumber \\
&\quad\quad\quad - \chi\alpha_{\pi n} n^{\langle\mu} \nabla^{\nu\rangle} \alpha
- \chi\alpha_{n\pi } n^{\langle\mu} \dot u^{\nu\rangle} \Big] , \label{shear}\end{aligned}$$ with the coefficients of charge conductivity, bulk and shear viscosity, viz. $\lambda, \zeta,\eta \ge 0$. The notations, $A^{\langle\mu\rangle} = \Delta^{\mu}_{\nu}A^{\nu}$ and $B^{\langle\mu\nu\rangle} =
\Delta^{\mu\nu}_{\alpha\beta}B^{\alpha\beta}$ represent space-like and traceless symmetric projections respectively, both orthogonal to $u^{\mu}$, where $\Delta^{\mu\nu}_{\alpha\beta} =
[\Delta^{\mu}_{\alpha}\Delta^{\nu}_{\beta} +
\Delta^{\mu}_{\beta}\Delta^{\nu}_{\alpha} -
(2/3)\Delta^{\mu\nu}\Delta_{\alpha\beta}]/2$. It may be noted that although the forms of the Eqs. (\[bulk\])-(\[shear\]) are the same as in the standard Israel-Stewart theory [@Israel:1979wp; @Muronga:2003ta], all the transport coefficients are explicitly determined in the present derivation: $$\begin{aligned}
\beta_0 &= \lambda^2_\Pi J_{10}^+ /\beta,\quad
\beta_1 = -\lambda^2_n J_{31}^+ /\beta,\quad
\beta_2 = 2\lambda^2_\pi J_{52}^+ /\beta, \nonumber \\
\alpha_0 &= \lambda_\Pi \lambda_n J_{21}^+ /\beta,\quad
\alpha_1 = -2\lambda_\pi \lambda_n J_{42}^+ /\beta. \label{alphas}\end{aligned}$$ As a consequence, the relaxation times defined as, $$\label{RT}
\tau_{\Pi} = \zeta\,\beta_0, \quad
\tau_{n} = \lambda\,\beta_1, \quad
\tau_{\pi} = 2\,\eta\,\beta_2 ,$$ can be obtained directly. With $\lambda_\Pi = -1/J_{21}^+$, $\lambda_n=1/J_{21}^-$, $\lambda_{\pi}=1/(2J_{42}^+)$, $n=I_{10}^-$, $\epsilon=I_{20}^+$, and $P=-I_{21}^+$, the expressions for $\beta_1,\alpha_0,\alpha_1$ simplify to $$\label{B1A0A1}
\beta_1 = (\epsilon+P)/n^2, \quad \alpha_0 = \alpha_1 = 1/n.$$ For a classical Boltzmann gas ($\tilde f_0=1$), the coefficients $\beta_0$ and $\beta_2$ take the simple forms $$\label{B0B2}
\beta_0 = 1/P,\quad
\beta_2 = 3/(\epsilon+P) + m^2\beta^2P/[2(\epsilon+P)^2].$$ Equations (\[bulk\])-(\[shear\]) in conjunction with the second-order transport coefficients (\[B1A0A1\]) and (\[B0B2\]) constitute one of the main results in the present work. These coefficients are obtained consistently within the same theoretical framework. In contrast, in the standard derivation from entropy principles [@Israel:1979wp], the transport coefficients have to be estimated from an alternate theory. For instance, in the IS derivation based on kinetic theory, these involve complicated expressions which in the photon limit ($m \beta \to 0$) reduce to [@Israel:1976tn] $$\label{IST}
\beta_0^{IS} = 216/(m^4 \beta^4 P), \quad \beta_2^{IS} = 3/4P.$$ An alternate derivation from kinetic theory (KT) using directly the definition of dissipative currents yields [@Denicol:2010xn] $$\begin{aligned}
\label{DKR}
\beta_0^{KT} =& \Big[ \left(\frac{1}{3}-c_s^2 \right)(\epsilon+P)-\frac{2}{9}(\epsilon-3P) \nonumber \\
&~ - \frac{m^4}{9}{\left\langle (u.p)^{-2} \right\rangle} \Big]^{-1}, \nonumber \\
\beta_2^{KT} =& \, \frac{1}{2} \left[ \frac{4P}{5}
+\frac{1}{15}(\epsilon-3P)
- \frac{m^4}{15}{\left\langle (u.p)^{-2} \right\rangle}\right]^{-1},\end{aligned}$$ where $c_s$ is the speed of sound and ${\left\langle \cdots \right\rangle}\equiv \int dp(\cdots)f_0$. A field-theoretical (FT) approach gives [@Huang:2011ez] $$\begin{aligned}
\label{HK}
\beta_0^{FT} =& \left[ \left(\frac{1}{3}-c_s^2 \right)(\epsilon+P)-\frac{a}{9}(\epsilon-3P)\right]^{-1},
\nonumber \\
\beta_2^{FT} =& \, 1/[2(3-a)P],\end{aligned}$$ where $a=2$ for charged scalar bosons and $a=3$ for fermions. We find that our expression for $\beta_2$ (Eq. (\[B0B2\])) in the massless limit, agrees with the IS result (Eq. (\[IST\])) and also with those obtained in Refs. [@Baier:2006um; @El:2009vj]. Thus the shear relaxation times $\tau_\pi$ (Eq. (\[RT\])) obtained here and in these studies are also identical. As $\beta_0$ in Eqs. (\[IST\])-(\[HK\]) diverge in the massless limit, so does the bulk relaxation time $\tau_\Pi$ (Eq. (\[RT\])), thereby stopping the evolution of the bulk pressure. It is important to note that $\beta_0$ in Eq. (\[B0B2\]) and hence $\tau_\Pi$ in the present calculation remain finite in this limit. A detailed comparison of IS, KT and FT results can be found in [@Denicol:2010br]. The two parameters $\psi$ and $\chi$ occurring in Eq. (\[EFCD3\]) remain undetermined as in [@Israel:1979wp]; however, these do not contribute to the scaling expansion.
To demonstrate the numerical significance of the new coefficients derived here, we consider the evolution equations in the boost-invariant Bjorken hydrodynamics at vanishing net baryon number density [@Bjorken:1982qr]. In terms of the coordinates ($\tau,x,y,\eta$) where $\tau = \sqrt{t^2-z^2}$ and $\eta=\tanh^{-1}(z/t)$, the initial four-velocity becomes $u^\mu=(1,0,0,0)$. For this scenario $n^\mu=0$ and the evolution equations for $\epsilon$, $\pi \equiv -\tau^2 \pi^{\eta \eta}$ and $\Pi$ reduce to $$\begin{aligned}
\frac{d\epsilon}{d\tau} &= -\frac{1}{\tau}\left(\epsilon + P + \Pi -\pi\right), \label{BED} \\
\tau_{\pi}\frac{d\pi}{d\tau} &= \frac{4\eta}{3\tau} - \pi - \frac{4\tau_{\pi}}{3\tau}\pi, \label{Bshear} \\
\tau_{\Pi}\frac{d\Pi}{d\tau} &= -\frac{\zeta}{\tau} - \Pi - \frac{4\tau_{\Pi}}{3\tau}\Pi \label{Bbulk}.\end{aligned}$$ Noting that $\beta_0=1/P$, $\beta_2=3/(\epsilon+P)$ and $s=(\epsilon+P)/T$, the relaxation times defined in Eq. (\[RT\]) reduce to $$\label{Mtaus}
\tau_{\Pi} = \frac{\epsilon+P}{PT}\left(\frac{\zeta}{s}\right), \quad
\tau_{\pi} = \frac{6}{T}\left(\frac{\eta}{s}\right).$$
We have used the state-of-the-art equation of state [@Huovinen:2009yb], which is based on a recent lattice QCD result [@Bazavov:2009zn]. For $\zeta/s$ at $T \geq T_c \approx 184$ MeV, we use the parametrized form [@Rajagopal:2009yw] of the lattice QCD results of Meyer [@Meyer:2007dy] which suggest a peak near $T_c$. At $T<T_c$, the sharp drop in $\zeta/s$ reflects its extremely small value found in the hadron resonance gas model [@Prakash:1993bt]; see inset of Fig. \[tauT\]. For the $\eta/s$ ratio, we use the minimal KSS bound [@Kovtun:2004de] value of $1/4\pi$.
In the absence of any reliable prediction for the bulk relaxation time $\tau_\Pi$, it has been customary to keep it fixed or set it equal to the shear relaxation time $\tau_\pi$ or parametrize it in such a way that it captures critical slowing-down of the medium near $T_c$ due to growing correlation lengths [@Fries:2008ts; @Denicol:2009am; @Song:2009rh; @Rajagopal:2009yw]. Since $\zeta/s$ has a peak near the phase transition, the $\tau_\Pi$ obtained here (Eq. (\[Mtaus\])) and shown in Fig. \[tauT\], [*naturally*]{} captures the phenomenon of critical slowing-down.
The evolution equations (\[BED\])-(\[Bbulk\]) are solved simultaneously with an initial temperature $T_0 = 310$ MeV [@Rajagopal:2009yw] and initial time $\tau_0 = 0.5$ fm/c typical for the RHIC energy scan. We take initial values for bulk stress and shear stress, $\Pi= \pi = 0$ GeV/fm$^3$ which corresponds to an isotropic initial pressure configuration.
Figure \[Piphi\](a) shows time evolution of the shear pressure $\pi$ and the magnitude of the bulk pressure $\Pi$. At early times $\tau \lesssim 2$ fm/c or equivalently at $T \gtrsim 1.2 T_c$, shear dominates bulk. This implies that eccentricity-driven elliptic flow which develops early in the system would be controlled more by the shear pressure [@Song:2009rh]. At later times (when $T \sim T_c$), the large value of $\zeta/s$ makes the bulk pressure dominant. This leads to sizeable entropy generation (Eq. (\[EFCD3\])) and consequently enhanced particle production.
Figure \[Piphi\](a) also compares the $\Pi$ evolution for bulk relaxation time, $\tau_{\Pi}$, calculated from Eq. (\[Mtaus\]) (solid line) and $\tau_{\Pi} = \tau_{\pi}$ (dashed line). At early times, the larger value of $\tau_{\Pi}$ in the latter case (see Fig. \[tauT\]) results in a relatively smaller growth of $|\Pi|$ as evident from Eq. (\[Bbulk\]). Near $T_c$, the rapid increase in $\zeta/s$ causes $|\Pi|$ to increase. Subsequently the longitudinal pressure $P_L= (P+\Pi-\pi)$ vanishes leading to cavitation [@Fries:2008ts; @Rajagopal:2009yw; @Torrieri:2008ip; @Bhatt:2010cy]. In contrast, with our $\tau_{\Pi}$, this rise in $\zeta/s$ is overcompensated by a faster increase in $\tau_{\Pi}$ thereby slowing down the evolution of $\Pi$. This behavior prevents the onset of cavitation and guarantees the applicability of hydrodynamics with bulk and shear up to temperatures well below $T_c$ into the hadronic phase. Furthermore, this slowing down of the medium followed by its rapid expansion, has the right trend to explain the identical-pion correlation measurements (Hanbury Brown-Twiss puzzle) [@Paech:2006st; @Pratt:2008qv].
The absence of cavitation in our calculation is clearly evident in Fig. \[Piphi\](b) which shows the variation of pressure anisotropy, $P_L/P_T = (P+\Pi-\pi)/(P+\Pi+\pi/2)$, with temperature. Near $T_c$, the longitudinal pressure $P_L$ vanishes if one assumes $\tau_{\Pi} =
\tau_{\pi}$ (dashed line) leading to cavitation, whereas it is found to be positive for all temperatures with $\tau_\Pi$ derived here (solid line). In fact, we have found that in the latter case, cavitation is completely avoided for the entire range of $\zeta/s$ values ($0.5 < \zeta/s < 2.0$ near $T_c$) estimated in lattice QCD [@Meyer:2007dy]. The sizeable difference between the $\Pi
= 0$ case (dot-dashed line) and the $\tau_\Pi=\zeta/P$ case (solid line) clearly underscores the importance of bulk pressure near $T_c$, which can have significant implications for the elliptic flow $v_2$ [@Denicol:2009am] thus affecting the extraction of $\eta/s$. Further, the large bulk pressure when incorporated in the freezeout prescription could also affect the final particle abundances and spectra.
We have also found that the evolution of $\Pi$ is insensitive to the choice of initial conditions such as $\Pi(\tau_0)=0$ and the Navier-Stokes value $-\zeta(T_0)/\tau_0$. This is due to very small $\tau_\Pi$ at early times (or higher temperatures) which causes $\Pi$ to quickly lose the memory of its initial condition and to relax to the same value at $\tau \gtrsim 1$ fm/c.
To summarize, we have presented a new derivation of the relativistic dissipative hydrodynamic equations from entropy considerations. We arrive at the same form of dissipative evolution equations as in the standard derivation but with all second-order transport coefficients such as the relaxation times and the entropy flux coefficients determined consistently within the same framework. We find that in the Bjorken scenario, although the bulk pressure can be large, the relaxation time derived here prevents the onset of cavitation due to the critical slowing down of bulk evolution near $T_c$.
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abstract: 'Modern physics is founded on two mainstays: mathematical modelling and empirical verification. These two assumptions are prerequisite for the objectivity of scientific discourse. Here we show, however, that they are contradictory, leading to the ‘experiment paradox’. We reveal that any experiment performed on a physical system is — by necessity — invasive and thus establishes inevitable limits to the accuracy of any mathematical model. We track its manifestations in both classical and quantum physics and show how it is overcome ‘in practice’ via the concept of environment. We argue that the scientific pragmatism ordains two methodological principles of compressibility and stability.'
author:
- 'Michał Eckstein${}^{1,2}$, Paweł Horodecki${}^{3,4}$'
bibliography:
- 'causality\_bib.bib'
title: The experiment paradox in physics
---
Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre, Faculty of Mathematics, Physics and Informatics, University of Gdańsk, Wita Stwosza 57, 80-308 Gdańsk
Copernicus Center for Interdisciplinary Studies, ul. Szczepańska 1/5, 31-011 Kraków, Poland
International Centre for Theory of Quantum Technologies, University of Gdańsk, Wita Stwosza 63, 80-308 Gdańsk, Poland
Faculty of Applied Physics and Mathematics, National Quantum Information Centre, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland
The methodology of physics, pioneered by Archimedes, Galileo and Newton, has been crystallising with the development of formal languages and statistical analysis. Its “unreasonable effectiveness”[@Wigner] is founded on two basic assumptions: Firstly, physical systems are modelled via mathematical structures, which guarantee the universality and objectivity of the description. Secondly, the models are verifiable via a comparison of their predictions against empirical data.
The question whether there exists an ‘ultimate’ mathematical model of physical reality (or an overarching “law of physics”), and whether it is intelligible, is controversial and has long been the subject of philosophical debate[@Wheeler83; @Deutsch86; @Weinberg94; @HawkingGodel; @Heller2009]. This problem, however, seems irrelevant for the practice of doing physics. Indeed, because of the finiteness of the available resources — any theory and any set of empirical data must eventually be expressed as a finite combination of some intersubjective symbols — no model can become arbitrarily accurate. Yet, we have to assume that our ignorance is the only source of the uncertainty, for otherwise we would decree “\[…\] that there exist aspects of the natural world that are fundamentally inaccessible to science.”[@Deutsch86]
The second inexorable[@EllisSilk] pillar of modern physics is the falsifiability of mathematical models against empirical data. Again, in practice the hypotheses can only be confirmed at some confidence level, yet we have to assume that this is solely caused by our incapabilities. In order for the experiments to be conclusive and repeatable, we have to assure that they are free — that is the input cannot be correlated with the studied physical system, until the experiment is actually performed. The pertinence of this assumption has been recognised only recently on the occasion of the Bell tests[@AspectBell2015; @HallBell; @Bell_quasars] demonstrating the predictive supremacy of quantum mechanics over “hidden variables” explanations. Heuristically, it has been voiced by Stephen Hawking and George F.R. Ellis: “\[…\] the whole of our philosophy of science is based on the assumption that one is free to perform any experiment.”[@HawkingEllis p. 189]
Local physics
=============
The prodigious success of physics relies on the fact that one can probe physical systems *locally*, that is abstracting from the ‘rest of the world’. The possibility of making a cut between a system and its ‘compliment’ has been implicitly assumed in the scientific discourse from its dawn, but has acquired a concrete shape only at the beginning of the 20th century. Voiced by Michael Faraday and sharpened by Albert Einstein the principle of locality brought about the concept of spacetime consisting of events[@Haag]. The formalisation of the notion of an event gave it an operational sense and thus established a rigorous link between the mathematical theory and ‘real world’ experiments.
The Einsteinian notion of spacetime (that is a 4-dimensional smooth Lorentzian manifold[@Wald]) could be and has been questioned from different standpoints[@Earman89; @QGreview]. Nevertheless, any physical theory must eventually make reference to the *effective spacetime*, in which the experimental data is being gathered and shared. This comes about as follows:
Let us firstly note that intersubjective data is but a (finite) set of values of some registers. For sake of simplicity we shall assume that the registers are bits, as any more complicated universal description of the data sets can eventually be rewritten in the binary form. Now observe that any data always comes in a sequence, that is as a totally ordered set of bits. Hence, it has an inbuilt time-ordering informing that the bit $b_1$ has been input, acquired or communicated before the bit $b_2$. Furthermore, given two data sequences one needs two different labels, say $a$ and $b$, to distinguish them. The notation $\{b_1,b_2,b_3,\ldots\}$ purports that the bit $b_2$ has been obtained at a place “$b$” in space and a time “2”. Thus, any bit of the data uniquely specifies an *event* – a point in space and time.
The events are intersubjective – because the data is so. This is not at variance with the fact that one can relabel the bits by ascribing them some explicit spacetime coordinates, such as $\{(b_1,t_1,x_1),(b_2,t_2,x_2),\ldots\}$. However, when doing so, one has to specify a ‘covariance law’ allowing one to unambiguously translate the local data from one coordinate chart to another. Any such covariance law must hence eventually refer to an invariant object — call it a spacetime —, which is common to all observers. The demand of covariance is not a falsifiable physical principle, but a methodological assumption[@Barbour2001] — a prerequisite for the intersubjectivity of physics.
To accommodate the local data any physical model must include a description of the effective spacetime. But the actual purpose of a theory is to provide explanations, which go beyond the directly observed phenomena. Therefore, the effective spacetime ought to contain also the *potential events*. These are not directly associated with any empirical data, but they are necessary for a consistent explanation of the data. For example, the detection of a photon is an actual event, whereas the photon emission from a star is a potential event indicated by the quantum theory. Viewed from a different angle, the potential events are associated with empty registers, denoted by $\emptyset$, for the data. This gives justice to the slogan: “Unperformed experiments have no results.”[@NoResults]
However, a conceivable explanation of the actual events in terms of the past potential ones is not a sufficient criterion for a credible theory of physical reality. Indeed, the gist of physics (and, actually, of the entire science) is the ability to *predict* future events. In particular, in order to conceive technology of any sort we must be able to precisely forecast the future behaviour of the designed devices. To enable predictions a theory must firstly make a distinction between the past potential events and the future ones. More precisely, it has to specify a *causal structure* telling us which of the past potential events could have influenced the actual ones and which of the future events might be affected by the present ones. For instance, in Newton’s theory the future abuts the past, so one cannot exclude in principle an immediate influence of a phenomenon in some remote part of the universe on our empirical data. On the other hand, in Einstein’s theory the regions of potential influence are constrained by the light cone structure.
The inconsonance of local experiments {#sec:incons}
=====================================
A *prediction* of a theoretical model ${\mathcal{M}}$ is a claim about some future potential events. A sharp prediction takes the form of a conditional claim: “If ${d_{\text{in}}}^p$ was input and ${\mathcal{M}}$ is valid, then ${d_{\text{out}}}^p$ ought to be registered.” In such a case a single experiment with ${d_{\text{in}}}= {d_{\text{in}}}^p$, but ${d_{\text{out}}}\neq {d_{\text{out}}}^p$ would be sufficient to falsify the model ${\mathcal{M}}$. In general, one formulates — more modest — *statistical predictions*. These are expressed as conditional probabilities $P({d_{\text{out}}}^p \, \vert \, {\mathcal{M}}, {d_{\text{in}}}^p)$ and they require multiple independent experiments with ${d_{\text{in}}}= {d_{\text{in}}}^p$ to validate ${\mathcal{M}}$ at a prescribed confidence level, which we decree as satisfactory. Any two competing models ${\mathcal{M}}$ and ${\mathcal{M}}'$ of a given phenomenon ought to be discernible, ${\mathcal{M}}\not\equiv {\mathcal{M}}'$, that is there must exist at least one experimental setting for which $P({d_{\text{out}}}^p \, \vert \, {\mathcal{M}}, {d_{\text{in}}}^p) \neq P({d_{\text{out}}}^p \, \vert \, {\mathcal{M}}', {d_{\text{in}}}^p)$. Let us stress that at the level of falsification it is irrelevant whether the model ${\mathcal{M}}$ is fundamentally probabilistic — as quantum models are —, or effectively statistical — as a result of ignorance of some of the system’s aspects, for instance its microscopic structure.
For an experiment to be trustworthy, one has to warrant that the input data is *free*, that is independent of the history of the physical system at hand. Concretely, we have to ensure the statistical independence of ${d_{\text{in}}}$ and past states ${\Omega_{\text{past}}}$ pertaining to the system. The set ${\Omega_{\text{past}}}= {\Omega_{\text{past}}}({\mathcal{M}})$ involves both the actual events associated with the existing data, as well as the past potential events entailed by the model ${\mathcal{M}}$. In other words, *a model ${\mathcal{M}}$ must not imply correlations between* ${\Omega_{\text{past}}}$ *and the potential future events related with* ${d_{\text{in}}}$. For if it would do so, it would induce a statistical bias in what could be tested and how, hence *a priori* excluding a part of physical reality from our empirical cognition. If multiple experiments with different ${d_{\text{in}}}$’s are performed, so that $P({d_{\text{in}}})$ can be defined a posteriori, we can express the demand of freedom as $P({d_{\text{in}}}\, \vert \, {\Omega_{\text{past}}}) = P({d_{\text{in}}})$.
Any experiment has to involve at least one free bit of input data, as the experiment might or might not be actually performed. Note that “experiment not done” is indeed an objective information, which corresponds to a definite event ${d_{\text{in}}}= 0$. In this case, the event associated with ${d_{\text{out}}}$ remains potential, ${d_{\text{out}}}= \emptyset$, as “Unperformed experiments have no results.”[@NoResults] On the other hand, if the experiment was done, ${d_{\text{out}}}$ must take a definite value. For even if the experimental apparatus did not register any signal, this mere fact corresponds to an objective information (cf. [@QIandGR]).
Let us fix a model ${\mathcal{M}}$ of a chosen physical system $F$. Let us also suppose that $F$ is embedded in an environment $E$, which is *not* modelled within ${\mathcal{M}}$, but does affect the experimental outcomes. This simply signifies that ${\mathcal{M}}= {\mathcal{M}}(F;E)$ includes some noise and/or free parameters. On the physical side, it means that $F$ did and does interact with $E$, which is *natural*, i.e. in principle modellable.
Suppose now that an experiment with some ${d_{\text{in}}}$ and ${d_{\text{out}}}$ has been performed, to check the validity of ${\mathcal{M}}$. Since ${d_{\text{in}}}$ is an (intersubjective, that is ‘classical’) information it must be physical, hence it had to be supported (or ‘written’) in a form of ‘matter’ $G$. Whatever model of $G$ we would consider, it must be allowed to interact with the studied physical system $F$, because it actually did so in the experiment just performed.
Let us suppose that the model ${\mathcal{M}}$ did encompass the interaction of $F$ and $G$. If $G$ is a part of $F$, that is ${\mathcal{M}}(F,G;E) \equiv {\mathcal{M}}(F;E)$, then, clearly, ${d_{\text{in}}}$ could not have been free. Suppose then that $G$ was included in the environment part: ${\mathcal{M}}(F;E,G) \equiv {\mathcal{M}}(F;E)$. But then we admit that the entire ‘experiment’ was actually a natural, that is modellable, phenomenon. Consequently, even if no correlations between ${d_{\text{in}}}$ and ${\Omega_{\text{past}}}$ were assumed within ${\mathcal{M}}$, there exists an extended model $\overline{{\mathcal{M}}} = \overline{{\mathcal{M}}}(F,E)$ providing a natural explanation of the entire ‘experiment’ hence, a statistical dependence between ${d_{\text{in}}}$ and $\overline{\Omega}_{\text{past}} = \overline{\Omega}_{\text{past}}(\overline{{\mathcal{M}}})$.
Therefore, in order to guarantee the freedom of ${d_{\text{in}}}$, we have to assume its independence in *any* conceivable model $$\begin{aligned}
P({d_{\text{in}}}\, \vert \, {\Omega_{\text{past}}}({\mathcal{M}})) = P({d_{\text{in}}}), \quad \text{ for all } {\mathcal{M}}. \end{aligned}$$ In other words, the experimental input ${d_{\text{in}}}$ has to be *random* (cf. [@RandomnessAmp2012]). But then, once the experiment was performed, we must update the model to take into account the interaction of $F$ with $G$, symbolically: ${\mathcal{M}}'(F,G;E) \not\equiv {\mathcal{M}}(F;E)$. Note that such a change has *global* consequences — it affects, in general, not only the future potential events, but also the past ones, because ${\Omega_{\text{past}}}= {\Omega_{\text{past}}}({\mathcal{M}})$.
This is what we might call the *experiment paradox*: We must assume that the experimental input is free in order to perform credible tests of theoretical models, but then we allow for ‘non-physical’ interventions, which are — by assumption — not modellable.
Faces of the experiment paradox
===============================
We now unravel the manifestations of the experiment paradox in well-established physical theories.
A convenient universal framework for both classical and quantum mechanics uses the language of *states* – encoding the properties of a given physical system $F$ and *observables* – measurable physical quantities[@Strocchi]. Any observable $A$ has a spectrum ${\mathrm{sp}}(A) \subset {\mathbb{R}}$, that is a set of possible measurement outcomes and any state $\rho$ defines a probability distribution $\mu_{\rho,A}$ over the set ${\mathrm{sp}}(A)$.
Models of local physical phenomena are formulated in terms of dynamical equations $$\begin{aligned}
\label{evo}
f(\rho(t),t) = 0, \quad \text{for} \quad t \in [0,T],\end{aligned}$$ where $t$ is a time parameter and $f$ is functional (typically, a linear differential operator) acting on the space of states. Such a model specifies the time-evolution of the system’s state $\rho(t)$ from an initial condition $$\begin{aligned}
\label{const}
g(\rho(t),t) \vert_{t=0} = 0,\end{aligned}$$ determined by a (collection of) functionals $g$. It involves, in particular, the initial state $\rho(0)$.
The predictions of the model are then formulated as follows: If the system was initially described by and an observable $A$ was measured at a time $t > 0$, then an outcome $a\in{\mathrm{sp}}(A)$ will be obtained with probability $\mu_{\rho(t),A}(a)$[^1]. Hence, to test a model determined by equation one has to *prepare* the system in an initial condition and then measure an observable $A$ at some time $t \in (0,T]$. Multiple experiments with different inputs would tell us whether the predicted probabilities match the observed ones.
Note that the experimental input listed above is indeed free within model , because the latter specifies neither the initial conditions (The model does specify the admissible forms of the initial conditions, but not the numerical values.) nor the observable $A$ and the measurement time $t$. However, we have to admit that the studied system had been in *some* state (for instance, the vacuum state) before it was prepared by the experimentalist.
Quantum theory, as opposed to classical mechanics, provides a formal operation corresponding to the reset of system’s state – the von Neumann projective measurement. But, the admission of projective measurements leads to the notorious measurement paradox (see Box 1).
To circumvent the ‘resetting problem’ we could construct an extended model $$\begin{aligned}
\label{evo2}
f(\rho(t),t) = 0, \quad \text{for} \quad t \in [t_0,T],\end{aligned}$$ describing the evolution of the system from some earlier time $t_0 < 0$ until $T$. Then, we assume that its dynamics has been *perturbed* $$\begin{aligned}
\label{evoj}
f(\rho(t),t) = j(t), \quad \text{for} \quad t \in [t_0,0),\end{aligned}$$ with a suitable *source* $j$, so that the desired condition at $t=0$ is met, regardless of the primordial system’s initial conditions at $t=t_0$.
But, clearly, models and are different and the introduction of a source term is invasive. Had the experiment not been performed, the object would evolve according to equation rather than . If, on the other hand, we attempt to model the source itself we lose (or rather, shift to another level of complexity) its tunability — hence the ‘preparation paradox’ (see Box 1).
The same line of reasoning could be followed in the Heisenberg picture, in which the system’s state remains steady, but the observables evolve in time. Let us also note that the source term $j$ need not be a function — it could be, for instance, a time-dependent Hamiltonian appended to the Schrödinger equation of a given system.
In conclusion, regardless of whether the theory entails that the measurement — i.e. information acquisition — disturbs the system or not, the preparation procedure is always invasive.
Fortunately, the experimental outcomes typically depend very weakly on how the system has been prepared. The ‘triggering effects’, that is the details of the source $j$, can usually be alleviated below the noise level shaped by the uncontrolled interaction of the system with its environment.
The experiment paradox is, however, more salient in the cosmological context, which does not leave room for any environment. Modern cosmological models are formulated in the framework of field theory. Let us emphasise that the fields do not evolve per se — a solution to field equations specifies the field content in the entire (effective) spacetime. Therefore, any disturbance coming ‘from outside’ would effectuate a global change. In other words, a local terrestrial experiment affects both future and past states of the Universe (see Figure \[fig:cosmos\]). Note also that cosmological observations are indeed genuine experiments for, firstly, they might but need not be effectuated and, secondly, they involve a number of tunable free parameters, such as the telescope’s location and direction or electromagnetic spectrum sensitivity range.
As an illustration, let us consider a cosmological model based on Einstein’s equations $$\begin{aligned}
\label{Einstein}
G_{\mu\nu} = \tfrac{8 \pi G}{c^4} T_{\mu\nu},\end{aligned}$$ with a matter energy–momentum tensor $T_{\mu\nu}$ (possibly including the “dark energy”, i.e. the cosmological constant term $\Lambda g_{\mu\nu}$). The geometrical Bianchi identity $\nabla^{\mu} G_{\mu\nu} = 0$ implies the local covariant conservation of energy and momentum $\nabla^{\mu} T_{\mu\nu} = 0$[@Wald]. But, if an ‘external’ source term $j_{\nu}$ is introduced into , the conservation law is violated, $\nabla^{\mu} T_{\mu\nu} = -j_{\nu}$, explicitly breaking general covariance. In other words, if one introduces into the universe some information which was not there, one creates *ex nihilo* a local source of energy–momentum.
In quantum field theory, whereas the energy and momentum need not be conserved locally, the suitable expectation values ought to be conserved. Concretely, if $\hat{T}_{\mu\nu}$ is the energy–momentum operator constructed from quantum matter fields, then $$\begin{aligned}
\label{cons}
\nabla_{\mu} {\langle}\psi \vert \hat{T}^{\mu\nu} \vert \psi {\rangle}= 0\end{aligned}$$ should hold[@Bertlmann] for any state vector $\vert \psi {\rangle}$. The introduction of a, possibly quantum, source $\hat{j}$ violates the constraint leading to the Einstein anomaly and, eventually, to the breakdown of general covariance[@Bertlmann] (see also [@Bednorz]).
In order to perceive the experiment paradox from the perspective of ‘cosmic evolution’ we firstly need to choose a time function — that is an observer —, which fixes an effective splitting of the global spacetime into space and time[@Wald]. Secondly, one has to assure that equations allow for a well-defined Cauchy problem[@HR09]. The latter consists in imposing initial data on a time-slice, say at observer’s time $t=0$, and studying its (maximal) hyperbolic development (see Figure \[fig:cosmos\]). This guarantees that both past and future field configurations are uniquely derived from the imposed initial data. The objectivity of the evolution is guaranteed by general covariance, which enables unequivocal transcription of the time-slice field configurations for different observers.
Now, a free perturbation, or an abrupt change of initial data on a time-slice in a region $K$ of space inflicts a change in both causal future $J^+(K)$ and causal past $J^-(K)$ of $K$. The problem persists in the context of quantum field theory, because of the “time-slice axiom”[@Haag]. This is independent from the fact that projective measurements are as harmful to quantum field theory as they are to the non-relativistic quantum theory.
![\[fig:cosmos\]The conformal diagram for the Minkowski spacetime. The field content of the entire spacetime is uniquely determined by initial data imposed on a Cauchy hypersurface $S$. Consequently, a free intervention effectuated in the region $K$ induces a change both in the causal future $J^+(K)$ and the causal past $J^-(K)$ of $K$. More generally, the outer diamond could serve as an illustration for the maximal Cauchy development of the hypersurface $S$.](Pandora_spacetime.png)
Consequences for foundations of physics
=======================================
If we define the ‘fundamental level of physics’ as being *in principle* subject to both arbitrarily accurate modelling and experimental probing, then it does not exist — because of the unveiled paradox. Furthermore, the assumption about the existence of ‘ontic’ random events is a methodological necessity (see Box 2). Yet, there is no decisive procedure to check *post factum* whether an event was random or not.
The experiment paradox has profound consequences for the philosophy of science, which shall be discussed elsewhere[@EHH_phil]. Nevertheless, from the practical point of view, one may adopt the perspective that “all models are wrong, but some of them are useful.”[@Box]
The ‘usefulness’ of theoretical models is quantified by their explanatory and predictive power. These rely on two key properties: *compressibility* and *stability*. The former means that we can describe large sets of empirical data within a tight theoretical scheme based on several overarching rules. Notwithstanding, the theory itself might need to be expressed in terms of sophisticated mathematical structures[@Dirac63]. By “stability” we understand the independence of the laws of nature from the testing procedures. It guarantees the repeatability of experiments and, eventually, enables the construction of trustworthy devices.
As expressed in a compressed quote from Albert Einstein: “Everything should be made as simple as possible, but no simpler.”[@EinsteinQuote]
We are grateful to Michael Heller, Ryszard Horodecki and Tomasz Miller for inspiring discussions and enlightening comments on the manuscript.
The work of ME was supported by the National Science Centre in Poland under the research grant Sonatina (2017/24/C/ST2/00322). PH acknowledges support by the Foundation for Polish Science through IRAP project co-financed by EU within Smart Growth Operational Programme (contract no. 2018/MAB/5).
**BOX 1: Chains of causal reasoning**
1\) The measurement paradox
Let $F$ be a quantum system described by the quantum state $\rho$ and suppose that we choose to measure an observable $A$, according to our free input ${d_{\text{in}}}$. Then, the standard von Neumann postulate of quantum mechanics implies that after the measurement the system’s state jumps abruptly to $\rho'$ – one of the (pure) eigenstates of the observable $A$. Such a ‘non-physical’ intervention can be given a natural explanation by embedding $F$ in a (quantum) environment $E$ and invoking the formal equivalence of projective measurements on $F$ with a unitary evolution on $F \otimes E$[@vonNeumann]. But then we face the notorious Wigner’s friend paradox[@WignerFriend] and are eventually forced to conclude[@RennerMW] that no definite ${d_{\text{out}}}$ can ever be consistently produced.
2\) The preparation paradox
The preparation paradox is the ‘mirror’ version of the measurement paradox. Suppose that we have obtained an output ${d_{\text{out}}}$ from an experiment performed on the system $F$. By seeking a ‘purely natural’ explanation of ${d_{\text{out}}}$ we have to embed $F$ in an environment $E$, the interaction with which caused ${d_{\text{out}}}$. But then no ${d_{\text{in}}}$ has ever occurred and we have never actually prepared the system in any way.
3\) Superdeterminism
One could maintain that we never actually prepare physical systems or perturb them – we just observe them evolving without effectuating any disturbance. But such a superdeterministic viewpoint excludes *a priori* the possibility of any interaction with the ‘physical world’, in particular it disallows any experiments. This firstly annihilates the explanatory power of science and, secondly, it is highly unpractical for it excludes *a priori* the existence of devices functioning according to our inputs.
4\) Scientific Pragmatism
In scientific practice we assume that our interactions with the studied system $F$ encoded in ${d_{\text{in}}}$ do not have natural, i.e. modellable, causes and that the obtained information ${d_{\text{out}}}$ is always definite and objective. In order to save the model’s consistency we have to warrant that our interventions do not affect the past states of the system $F$. To this end we need to embed it in a suitable environment $E$, which absorbs the ‘retrocausal’ effects ($E \to \widetilde{E}$) and enables a consistent description of system’s history by multiple observers. Whether we wish our interventions to affect the system’s future states ($F \neq F'$, $E = E'$) or not ($F = F'$, $E \neq E'$) depends on whether we work in the observational paradigm (as, for instance, in cosmology) or in the engineering one.
**BOX 2: Random events in nature**
Any experiment requires at least one free, i.e. *random* bit (see Section \[sec:incons\]). Quantum theory implies that some of the events related to projective measurements are indeed random, hence quantum phenomena could be used as ‘sources of randomness’. However, in order to test the quantum theory itself we need to perform an experiment, the free input of which does not rely on quantum theory. In general, we are always bound to *assume* the freedom of some events in order to trust the experiments, as there exist no decisive procedure to check whether a finite binary sequence comes from a ‘source of randomness’ or a complex deterministic algorithm[@Claude].
This insight uncovers a captivating similarity between theories and experiments. Any formal theory is based on a collection of axioms, from which theorems are deduced. The axioms cannot be ultimately proven true or false — this is a general fact stemming from the limiting theorems in formal systems. In practice, we assume the axioms to be true and check whether such a premise is useful for deriving new results. Similarly, any experiment is based on a set of free bits (the ‘input’), which facilitate an explanation — within a theoretical scheme — for the ‘output’. We do not ask whether these bits were ‘truly random’ or not, but rather if the assumption about their freedom is useful or not.
Let us illustrate this observation with two concrete examples:
Consider an experiment consisting of multiple tosses of a coin. The assumption that its binary outcomes are random is fairly useful[@Oversimplifying]. Yet, a coin is admittedly a macroscopic object following a definite trajectory determined by the gravitational attraction and air drag. Hence, we *could* in principle establish a model of the coin toss, which would ‘explain’ the outcomes. Clearly, such a model would be very complex and would rely on a number of unknown parameters. Furthermore, it would likely be unstable – new ‘experiments’ (or, rather, ‘observations’) would require ad hoc adjustments. In consequence, although we could provide a deterministic model of a coin toss, it would be fairly useless.
Let us now turn to the Bell test[@BellThm; @AspectBell2015] – a foundational experiment aimed at demonstrating the intrinsic randomness of quantum measurements. In a typical scenario two parties independently perform measurements on a shared pair of entangled particles. Assuming that the “locality” and “fair sampling” loopholes have been closed[@AspectBell2015], the Bell–CHSH theorem[@BellThm; @CHSH] says that if the measurement outcomes are determined by a “hidden variable” $\lambda$ *and* the settings of the parties’ devices are free, that is uncorrelated with $\lambda$, then a certain measure of correlations $S$ between the outcomes is bounded, $S \leq 2$. Yet, numerous experiments have shown with a high statistical significance that the value of $S$ exceeds 2, reaching the quantum bound[@Cirelson] $S = 2\sqrt{2}$. Consequently, one could conclude that there is no “hidden variable” explanation and the outcomes are ‘truly random’, as implied by quantum mechanics. Alternatively, one could give up the “measurement independence” assumption and provide a fully deterministic explanation[@HallBell]. However, the adequacy of such a ‘hidden variable’ model is highly questionable, as, for instance, it would require the triggering of a common cause mechanism at the early stages of the Universe’s evolution[@Bell_quasars].
[^1]: If the observable $A$ has a continuous spectrum, then the ‘outcome’ is specified within some interval $[a-\delta,a+\delta]$ and the probability is given by $\mu_{\rho(t),A}([a-\delta,a+\delta])$.
|
---
abstract: 'In this article we explore the critical end point in the $T-\mu$ phase diagram of a thermomagnetic nonlocal Nambu–Jona-Lasinio model in the weak field limit. We work with the Gaussian regulator, and find that a crossover takes place at $\mu, B=0$. The crossover turns to a first order phase transition as the chemical potential or the magnetic field increase. The critical end point of the phase diagram occurs at a higher temperature and lower chemical potential as the magnetic field increases. This result is in accordance to similar findings in other effective models. We also find there is a critical magnetic field, for which a first order phase transition takes place even at $\mu=0$.'
address:
- 'Santiago College, Avenida Camino Los Trapenses 4007, Lo Barnechea, Santiago, Chile. '
- 'Instituto de Ciencias Básicas, Universidad Diego Portales, Casilla 298-V, Santiago, Chile.'
- 'Centro de Investigación y Desarrollo en Ciencias Aeroespaciales (CIDCA), Fuerza Aérea de Chile, Santiago, Chile.'
author:
- 'F. Márquez'
- 'R. Zamora'
bibliography:
- 'BNJL.bib'
title: 'Critical end point in a thermo-magnetic nonlocal NJL model'
---
Introduction
============
The main concern of this article is to study the critical end point (CEP) of the QCD phase diagram. Hadronic matter exists in either a chirally symmetric phase or a phase were chiral symmetry is spontaneously broken. The transition between these phases can be a first order or second order phase transition, or even a crossover. The type of transition occurring depends on the chemical potential and temperature at which it occurs. At low $\mu$ (and high $T$) the transition is either a crossover or a second order phase transition. At high $\mu$ (and low $T$) the transition is a first order one. The $(\mu, T)$ point separating both transitions in the phase diagram is called a CEP. The existence of the CEP in QCD was suggested a few decades ago [@Asakawa:1989bq; @Barducci:1989wi; @Barducci:1989eu; @Barducci:1993bh]. To determine the position of the CEP in the phase diagramm one must work within the realm of nonperturbative QCD. Therefore, a number of different approaches are used to investigate this, namely lattice QCD [@Fodor:2004nz; @Fodor:2001pe], the linear sigma model [@renato2], the Nambu–Jona-Lasinio (NJL) model [@Costa:2008gr; @Avancini:2012ee; @Costa:2010zw; @Costa:2007ie] and its nonlocal variant (nNJL) [@Scoccolannjl; @Contrera:2010kz].\
More recently, the study of the QCD phase diagram in the presence of a magnetic field has been addressed in many articles [@Boomsma01; @Loewe1; @Agasian; @Fraga; @Fraga2; @Andersen; @Nosotros01; @renato1; @renato2; @renato3; @renato4; @renato5; @Gamayun; @Marquez:2016fvb]. The magnetic field has been shown to have an effect on both the order of the phase transition and the critical temperature and chemical potential at which it occurs [@renato2]. Therefore, the magnetic field will have an effect on CEP position. Such a scenario may be found in heavy ion collisions, where a magnetic field is produced in presence of hadronic matter [@heavy01].\
The NJL model was originally proposed as model of interacting nucleons [@Nambu1; @Nambu2] and later reinterpreted as a model of interacting quarks [@Volkov; @Hatsuda]. The nNJL model was then introduced as way of including confinement in the model [@Birse01; @Birse02; @Fede02]. The nNJL has also shown good agreement with lattice data [@Scoccolannjl; @Contrera:2010kz]. Therefore, the nNJL model is not only used to study confinement, but also to study nonpertubative properties of QCD. We will use this model in presence of a homogeneous magnetic field, in order to study the behavior of the CEP under such conditions.\
The article is organized as follows. In Sec. 2, the model is presented and the uniform magnetic field is introduced. In Sec. 3 our results are presented and in Sec. 4 we discuss our conclusions and final remarks.\
Thermo-magnetic NJL Model.
==========================
The nNJL model is described through the Euclidean Lagrangian $$\mathcal{L}_E=\left[\bar{\psi}(x)(-i\slashed{\partial}+m)\psi(x)-\frac{G}{2}j_a(x)j_a(x)\right],\label{lagrangiannjl}$$ with $\psi(x)$ being the quark field. The nonlocal aspects of the model are incorporated through the nonlocal currents $j_a(x)$ $$j_a(x)=\int d^4y\,d^4z\,r(y-x)r(z-x)\bar{\psi}(x)\Gamma_a\psi(z),$$ where $\Gamma_a=(1,i\gamma^5\vec{\tau})$ and $r(x)$ is the so-called regulator of the model in the configuration space. If $r(x)=\delta(x)$ then we would recover the original NJL model. It is usual to bosonize the model through the incorporation of a scalar ($\sigma$) and a pseudoscalar ($\vec{\pi}$) field. Then, in the mean field approximation, $$\begin{aligned}
\sigma&=&\bar{\sigma}+\delta\sigma\\
\vec{\pi}&=&\delta\vec{\pi},\end{aligned}$$ where $\bar{\sigma}$ is the vacuum expectation value of the scalar field, serving as an order parameter for the chiral phase transition. The vacuum expectation value of the pseudoscalar field is taken to be null because of isospin symmetry. Quark fields can then be integrated out of the model [@Scoccola02; @Scoccola04] and the mean field effective action can be obtained. $$\Gamma^{MF}=V_4\left[\frac{\bar{\sigma}^2}{2G}-2N_c\int\frac{d^4q_E}{(2\pi)^4}\operatorname{tr}\ln S_E^{-1}(q_E)\right],$$ with $N_f=2$, the number of light-quark flavors and $N_c=3$, the number of colors in the model. Here, $S_E(q_E)$ is the Euclidean effective propagator $$S_E=\frac{-\slashed{q}_E+\Sigma(q_E^2)}{q_E^2+\Sigma^2(q_E^2)}.\label{Euprop}$$ Here, $\Sigma(q_E^2)$ is the constituent quark mass $$\Sigma(q_E^2)=m+\bar{\sigma}r^2(q_E^2).$$ Finite temperature ($T$) and chemical potential ($\mu$) effects can be incorporated through the imaginary time formalism (ITF) or Matsubara formalism. To do so, one can make the following substitutions $$\begin{aligned}
V_4&\rightarrow&V/T\\
q_4&\rightarrow&-q_n\\
\int\frac{dq_4}{2\pi}&\rightarrow&T\sum_n,\end{aligned}$$ where $q_n$ includes the Matsubara frequencies $$q_n\equiv(2n+1)\pi T+i\mu.$$ With this, the propagator in Eq. (\[Euprop\]) will now look like $$S_E(q_n,\boldsymbol{q},T)=\frac{\gamma^4q_n-\boldsymbol{\gamma}\cdot\boldsymbol{q}+\Sigma(q_n,\boldsymbol{q})}{q_n^2+\boldsymbol{q}^2+\Sigma^2(q_n,\boldsymbol{q})}.\label{Euprop2}$$ It is worth noting that the propagator in Eq. (\[Euprop2\]) has no singularities. Since there are no poles at some $p^2$, the definition of an effective mass for the particle with such propagator is not clear and therefore the quasiparticle interpretation cannot be made.\
The $\sigma$ field will evolve with temperature. This evolution can be computed through the grand canonical thermodynamical potential in the mean field approximation $\Omega_{MF}(\bar{\sigma},T,\mu)=(T/V)\Gamma_{MF}(\bar{\sigma},T,\mu)$ [@Kapusta01]. Then the value of $\bar{\sigma}$ must be at the minimum of the potential where $\partial\Omega_{MF}/\partial\bar{\sigma}=0$, which means $$\left.\frac{\bar{\sigma}}{G}=2N_cT\sum_n\int\frac{d^3q}{(2\pi)^3}
r^2(q_E^2)\operatorname{tr}S_E(q_E)
\right|_{q_4=-q_n}.
\label{gap}$$ From this equation one can get the temperature evolution of $\bar{\sigma}$. All of the computations have been made in ITF. Similar derivations are readily available in the literature (see for example [@Scoccola02; @Scoccola06]).\
We are interested in studying the model coupled to a homogeneous magnetic field. The derivative in the Lagrangian (\[lagrangiannjl\]) is replaced by a covariant derivative $$D_\mu=\partial_\mu + ie_fA_\mu,$$ where $A^{\mu}$ is the vector potential corresponding to a homogeneous external magnetic field $\boldsymbol{B}=|\boldsymbol{B}|\hat{z}$ and $e_f$ is the electric charge of the quark fields (i.e. $e_u = 2e/3$ and $e_d = -e/3$). In the symmetric gauge, $$A^{\mu}= \frac{B}{2}(0,-y,x,0).$$ The Schwinger proper time representation for the propagator in the Euclidean space is given by [@Schwinger]
$$\begin{gathered}
S_E(q_E)=\int_0^\infty ds \frac{e^{-s(q_{\|}^2+q_\perp^2
\frac{\tanh (eBs)}{eBs} + M^2)}}{\cosh (eBs)}
\\\times\biggl[\left(\cosh (eBs) -i \gamma_1 \gamma_2 \sinh (eBs)\right)
(M-{{\not \! q_{\|}}}) - \frac{{{\not \! q_\bot}}}{\cosh(eBs)} \biggr],\end{gathered}$$
with $q_\|^2=q_0^2+q_3^2$, $q_\bot^2=q_1^2+q_2^2$ and where $e$ is the charge of the particle and $B$ is the magnetic field.\
For simplicity, we will consider the weak magnetic field case. By weak we mean that the magnetic field is weak with respect with the dominant energy scale in the problem, i.e. $eB<T^2$ or $eB < \mu^2$ [@mexicanos]. The Euclidean fermionic propagator in this region can be written as [@Taiwan] $$\begin{aligned}
&&S_E(q_E) = \frac{(\Sigma(q_E^2)-{{\not \! q_E}})}{q_E^2+\Sigma^2(q_E^2)} - i\frac{\gamma_1 \gamma_2(eB)( \Sigma(q_E^2)-{{\not \! q}}_{E \|})}{(q_E^2+\Sigma^2(q_E^2))^2} \nonumber \\
&+&\frac{2 (eB)^2 q_{E \bot}^2}{(q_E^2+\Sigma^2(q_E^2))^4} \nonumber \\
&\times&\biggl[ (\Sigma(q_E^2)-{{\not \! q}}_{E \|})
+ \frac{{{\not \! q}}_{E \bot}(\Sigma^2(q_E^2)+{q}_{E \|}^2)}{q_{E \bot}^2} \biggr]. \label{debil}\end{aligned}$$
Results
=======
Throughout this work, we will use the Gaussian regulator for the nNJL model in the Euclidean momentum space, i.e. $$r^2(q_E)={\rm e}^{-q_E^2/\Lambda^2}.$$ For the parameters of the model, we take [@LoeweMorales] $m=10.5$ MeV, $\Lambda=627$ MeV and $G=5\times10^{-5}$ MeV$^{-2}$. With this set of parameters we have $\bar{\sigma}_0=339$ MeV.\
The gap equation is solved for different values of the magnetic field and the chemical potential, obtaining the behavior of $\bar{\sigma}(T)$ for each pair of $(eB,\mu)$ values. This solution is found by numerical computation. This allows to determine the critical temperature for chiral phase transition, as well as the nature of the phase transition, namely if it is a first or second order phase transition, or rather a crossover.\
![$T-\mu$ phase diagram of the model, for different values of the magnetic field. Values of $eB$ are indicated in the figure. Red points signal a crossover while blue ones signal a first order phase transition. The stars show the position of the critical end point.[]{data-label="F1"}](F1.pdf)
Figure \[F1\] shows the $T-\mu$ phase diagram for different values of the magnetic field in terms of the pion mass $m_\pi=134.97$ MeV [@valor]. In all cases, we initially have a crossover that, at high enough chemical potential, turns to a first order phase transition. We can also see that the CEP moves to the left of the phase diagram as the magnetic field increases. It is worth noting that an opposite behavior is found in [@inagaki], indicating that the CEP behavior is a model-dependent phenomenon. However, similar results have been found in [@renato2]. This means that for higher magnetic fields, the critical temperature increases while the critical chemical potential decreases.\
![Behavior of the chemical potential of the critical end point as a function of the magnetic field. The number below data points indicate the value of the temperature of the CEP for each case.[]{data-label="F3"}](F3.pdf)
Figure \[F3\] shows the behavior of the chemical potential for the CEP as a function of the magnetic field. As can be seen from the figure, the chemical potential decreases as the magnetic field increases. Furthermore, at $eB=1.1 m_\pi^2$ MeV$^2$, the chemical potential for the critical endpoint is null, meaning that there is no longer a crossover in the phase diagram, but rather a first order phase transition at every critical temperature, therefore we can no longer define a CEP.\
![Behavior of the temperature of the critical end point as a function of the magnetic field. The number below data points indicate the value of the chemical potential of the CEP for each case.[]{data-label="F2"}](F2.pdf)
Figure \[F2\] shows the behavior of the temperature of the CEP as a function of the magnetic field. As can be seen from the figure, the temperature of the CEP increases as a function of the magnetic field. However, at $T=128$ MeV, the CEP can no longer be defined, as there is no crossover in the model.\
![Behavior of the temperature of the CEP as a function of the chemical potential of the CEP. The number below data points indicate the value of the magnetic field for each case. The values of the magnetic field are normalized by the pion mass.[]{data-label="F4"}](F4.pdf)
Figure \[F4\] shows the behavior of the temperature of the CEP as a function of the chemical potential of the CEP. As the chemical potential increases, the temperature decreases.
Conclusions
===========
In this article, we obtained the $T-\mu$ phase diagram for a thermo-magnetic nNJL model. We find that there is a chiral phase transition that can either be a crossover or a first order phase transition at $B=0$, depending on the value of the chemical potential. One can define a CEP as the set of $(T,\mu, eB)$ values that separate the crossover from the first order phase transition. We find that as the magnetic field increases the temperature of the CEP also increases, while the chemical potential of the CEP decreases. This is in agreement with results obtained in different effective models when working in the mean-field approximation. Furthermore, it has been shown [@renato4] that inverse magnetic catalysis will be found when going beyond mean field. We studied the beahvior of the CEP in the phase diagram. If the magnetic field is high enough ($eB>1.1 m_{\pi}^2$), the CEP will vanish, meaning that we will no longer have a crossover in our model, but rather a phase transition for any temperature. At this point is no longer possible to define a CEP. It is worth noting that, while this will occur at higher $B$, the magnetic field is still weaker than the dominant energy scale. Therefore, the weak field approximation is still valid in this region.
Acknowledgements
================
R. Zamora would like to thank support from CONICYT FONDECYT Iniciación under grant No. 11160234.
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---
abstract: 'We study the quantitative stability of Serrin’s symmetry problem and it’s connection with a dynamic model for contact angle motion of quasi-static capillary drops. We prove a new stability result which is both linear and depends only on a weak norm $$\big\||Du|^2- 1\big\|_{L^2(\partial \Omega)}.$$ This improvement is particularly important to us since the $L^2(\partial \Omega)$ norm squared of $|Du|^2-1$ is exactly the energy dissipation rate of the associated dynamic model. Combining the energy estimate for the dynamic model with the new stability result for the equilibrium problem yields an exponential rate of convergence to the steady state for regular solutions of the contact angle motion problem. As far as we are aware this is one of the first applications of a stability estimate for a geometric minimization problem to show dynamic stability of an associated gradient flow.'
address: 'Department of Mathematics, The University of Chicago, Chicago, IL 60637, USA'
author:
- 'William M. Feldman'
bibliography:
- 'DropletRatesArticles.bib'
title: 'Stability of Serrin’s Problem and Dynamic Stability of a Model for Contact Angle Motion'
---
[^1]
Introduction {#sec: Droplet Problem}
============
We consider the solutions of the following free boundary problem, for $(x,t) \in {\mathbb{R}}^N \times [0,\infty)$, $$\label{e.dropletprob}\tag{\textup{P}}
\left\{
\begin{array}{lll}
-\Delta u(x,t) = \lambda(t) & \hbox{ in } & \Omega_t(u) = \{u(\cdot,t)>0\} \vspace{1.5mm}\\
\mathcal{V}_n = \tfrac{\partial_t u}{|Du|} = F(|Du|) & \hbox{ on } & \Gamma_t(u) = \partial \Omega_t(u),
\end{array}\right.$$ where $\mathcal{V}_n$ means the normal velocity of $\Gamma_t(u)$ and $\lambda(t)$ is a Lagrange multiplier enforcing the volume constraint, $$\int u(\cdot,t) \ dx = \text{Vol} \ \hbox{ for all } \ t>0.$$ The above problem is formulated under the assumption that $\Omega_t(u)$ remains connected, in general each connected component should have its own Lagrange multiplier $\lambda(t)$ and the multipliers could change discontinuously at times when disjoint components merge.
Problem [(\[e.dropletprob\])]{} can be posed entirely in terms of the domain $\Omega_t$. Fixing the domain $\Omega_t = \Omega$ the solution $u_\Omega$ and the Lagrange multiplier $\lambda(\Omega)$ are determined by, $$\label{e.ulambdadef}
\begin{array}{c}
u_\Omega := \operatorname*{\arg\!\min}\{ \int_\Omega |Dv|^2dx: v \in H^1_0(\Omega), \int_\Omega v \ dx = \text{Vol} \} \\ \\
\lambda(\Omega) := \min \{ \int_\Omega |Dv|^2dx: v \in H^1_0(\Omega), \int_\Omega v \ dx = \text{Vol} \}
\end{array}$$ The energy $\lambda(\Omega)$, when $\textup{Vol} = 1$, is sometimes called the torsional rigidity of the domain $\Omega$.
This problem is a quasi-static model for the contact angle driven motion of a liquid droplet on a solid surface. The model is derived under the assumption of small volume or small magnitude $Du$ so that the surface area of the graph can be replaced by the Dirichlet energy. The quasi-static assumption is that the time scale for the trend to equilibrium of the droplet profile $u$ is much smaller than the time scale for the motion of the contact line $\Gamma_t(u)$. For more information about the derivation of the model see [@Greenspan:1976aa; @greenspan]. The problem can be exactly solved in $N=1$, we will take $N \geq 2$ from now on. The physically relevant dimensions are $N=1,2$ and the problem can be exactly solved in $N=1$. We will take $N \geq 2$ from now on.
The function $F: [0,\infty) \to {\mathbb{R}}$ determines the normal velocity of the free boundary based on the contact angle of the graph $(x,u(x,t))$ with the surface $(x,0)$. $F$ is always assumed to satisfy $$\label{e.F hyp}
\hbox{$F$ is smooth, $F'>0$, and $F(1)=0$.}$$ The monotonicity implies that the problem [(\[e.dropletprob\])]{} has a formal comparison principle when $\lambda(t)$ is a given function of time, although we emphasize that there is not a comparison principle, at least in the standard sense, for the volume constrained problem. With the condition $F(1)=0$, the free boundary problem [(\[e.dropletprob\])]{} can be thought of, formally, as a gradient flow for the energy, $$\label{e.Jdef}
\begin{array}{c}
\mathcal{J}(\Omega) := \int_\Omega |Du_{\Omega}|^2 dx + |\Omega| = \lambda(\Omega) \textup{Vol} + |\Omega|
\end{array}$$ in an appropriate metric on bounded subsets of ${\mathbb{R}}^N$.
To motivate the gradient flow structure one can compute the following energy decay estimate for smooth solutions of [(\[e.dropletprob\])]{}, $$\frac{d}{dt}\mathcal{J}(\Omega_t) = \int_{\Gamma_t} (1-|Du|^2)F(|Du|) \leq 0,$$ in the case $F(|Du|) = |Du|^2-1$ this has a particularly appealing form, $$\label{e.dissipation}
\frac{d}{dt}\mathcal{J}(\Omega_t) = -\int_{\Gamma_t} (|Du|^2-1)^2.$$ See below in Section \[sec: exponential rate\] for the full computation. Thus, at least at a formal level, solution of [(\[e.dropletprob\])]{} are driven by the energy dissipation to stationary solutions, $$\label{e.serrin}\tag{S}
\left\{
\begin{array}{lll}
-\Delta u = \lambda(\Omega) & \hbox{ in } & \Omega(u), \vspace{1.5mm}\\
|Du| = 1 & \hbox{ on } & \Gamma(u), \vspace{1.5mm}\\
\int_{\Omega} u \ dx = \textup{Vol}
\end{array}\right.$$ This overdetermined boundary value problem was originally studied by Serrin [@Serrin71], and slightly afterwards with a different approach by Weinberger [@Weinberger:1971aa]. They proved that when $\partial \Omega$ is $C^2$ then, modulo a translation, $$\Omega = B_{r_*}$$ with $$r_*(\textup{Vol})^{N+1} = \omega_N^{-1}(N+2)\textup{Vol} \ \hbox{ and } \ \lambda(B_{r_*},\textup{Vol}) = N\left(\tfrac{\omega_N}{N+2}\right)^{\frac{1}{N+1}}\textup{Vol}^{-\frac{1}{N+1}}.$$ The goal of this paper is to prove a quantitative stability estimate for Serrin’s symmetry result. More precisely we would like to establish an estimate which controls the excess energy, $$\mathcal{J}(\Omega) - \mathcal{J}(B_{r_*})$$ by the energy dissipation, $$\int_{\partial \Omega} (|Du|^2 - 1)^2 d \sigma .$$ The ideal scenario, from the perspective of the energy dissipation estimate, would be to establish the following inequality, $$\mathcal{J}(\Omega) - \mathcal{J}(B_{r_*}) \leq C \int_{\partial \Omega} (|Du|^2 - 1)^2 d \sigma$$ with the constant $C$ depending only on quantities which can be controlled by the available a-priori estimates for the flow [(\[e.dropletprob\])]{}. Then the energy decay would yield, $$\frac{d}{dt}[\mathcal{J}(\Omega_t) - \mathcal{J}(B_{r_*})]= -\int_{\Gamma_t} (|Du|^2-1)^2 \leq - C[\mathcal{J}(\Omega_t) - \mathcal{J}(B_{r_*})]$$ establishing the expected exponential rate of convergence of the energy, $$\mathcal{J}(\Omega_t) - \mathcal{J}(B_{r_*}) \leq [\mathcal{J}(\Omega_0) - \mathcal{J}(B_{r_*})] e^{-Ct}.$$ This is the result we establish, conditional on $\Omega_t$ being a global in time regular solution. Such regularity is expected to hold for solutions with initial data $\Omega_0$ close to $B_{r_*}$ in a Lipschitz distance. The regularity hypothesis will be discussed further below in Section \[sec: hypothesis\]
Stability results for the Serrin Problem
----------------------------------------
First we will discuss the stability estimate of Serrin’s symmetry problem [(\[e.serrin\])]{}. Before describing the results of this paper we will discuss the existing literature on this problem.
As mentioned before the radial symmetry of solutions to [(\[e.serrin\])]{} was first proved by Serrin [@Serrin71]. His proof used the method of moving planes, used before by Alexandrov [@Alexandrov] to show that constant mean curvature hypersurfaces in ${\mathbb{R}}^N$ are spheres. Slightly afterwards Weinberger [@Weinberger:1971aa] discovered a very short proof also based on a maximum principle type argument. More recently there has been interest in the stability estimates of Serrin’s problem. Aftalion, Busca and Reichel [@Aftalion:1999aa] made the moving planes method quantitative and proved a stability estimate in Hausdorff distance for $C^{2,\alpha}$ domains, basically they obtain a logarithmic stability estimate, $$R - r \leq C|\log\||Du| - 1\|_{C^1(\partial \Omega)}|^{-1/N},$$ where $r,R$ are respectively the in-radius and the out-radius of $\Omega$ and $u=u_\Omega$ is the torsion solution defined in [(\[e.ulambdadef\])]{}. Since then the moving planes based method has been studied further by Ciraolo, Magnanini [@Ciraolo:2014aa] and Ciraolo, Magnanini and Vespri [@Ciraolo:2016aa] obtaining linear stability in terms of the Lipschitz semi-norm, $$R - r \leq C [|Du|]_{\partial \Omega} \ \hbox{ with } \ [f]_{\partial \Omega} = \sup_{\substack{x \neq y \\x,y \in \partial \Omega}}\frac{|f(x)-f(y)|}{|x-y|}$$ for $C^{2,\alpha}$ domains. A different approach to the symmetry problem, and to the stability estimates, was developed by Brandolini, Nitsch, Salani and Trombetti in several papers [@BNST08; @BNST-alt; @BNST09] where they also found applications of their method to symmetry problems for $k$-Hessian type equations. The advantage of their method is that it relies more on integration by parts identities than the maximum principle and thus attains estimates in a weaker norm. The result of [@BNST08] gives the following Hölder stability estimate which, at least in one direction, depends only on the $L^1(\partial \Omega)$ norm, suppose that, $$\|Du\|_{C^0(\partial \Omega)} \leq 1+ \delta \ \hbox{ and } \ \||Du| - 1\|_{L^1(\partial \Omega)} \leq \delta |\partial \Omega|,$$ then there is a finite collection of balls $\{B_i\}_{i=1}^k$ such that, $$||B_i| - |B_{r_{*,k}}|| \leq C \delta^{\beta} \ \hbox{ and } \ |\Omega \Delta \cup_i B_i | \leq C \delta^{\beta} \ \hbox{ with } \ \beta = \tfrac{1}{4N+13}.$$ Note that with only the measure theoretic bound it is possible in $N\geq 3$ for $\Omega$ to be close to a union of finitely many balls of radius close to $r_{*,k} = r_*(\textup{Vol}/k)$ joined perhaps by long thin tentacles. In $N=2$ this should also be possible, but only with adjacent balls connected by very short necks. We will avoid both of these issues with our regularity assumption, anyway such configurations do not seem to be relevant to the study of the dynamic problem [(\[e.dropletprob\])]{}.
Our contribution, which takes the approach of [@BNST08] as a starting point, is a stability estimate which depends only on a weak norm, the $L^2(\partial \Omega)$ norm of $|Du|^2-1$, and has linear order. Our result is still perhaps not optimal in terms of the regularity assumption on $\partial \Omega$, but it is optimal in the sense that it allows to prove the exponential convergence rate for [(\[e.dropletprob\])]{}.
Our result is also connected with a similar kind of stability estimate for hypersurfaces with almost constant mean curvature, i.e. a stability estimate associated with Alexandrov’s [@Alexandrov] symmetry result for constant mean curvature hypersurfaces in ${\mathbb{R}}^N$. In fact the connection between these two symmetry results is more than just an analogy, following the ideas of [@MR0474149], Ros [@MR996826] was able to use Serrin’s symmetry result to prove Alexandrov’s theorem. The stability problem for Alexandrov’s theorem has been recently studied by Ciraolo and Maggi [@Ciraolo:2017aa], and also by Krummel and Maggi [@MR3627438] where one of their results is an $L^2$ type bound similar to ours, under the assumption that $\partial \Omega$ can be written as a $C^{1,1}$ graph over the sphere. In fact our result, since it gives control of the asymmetry in terms of the $L^2(\partial \Omega)$ norm of $|Du|^2-1$, could be used to prove a similar symmetry result for almost constant mean curvature hypersurfaces, this is already explained in quite a bit of detail in [@Ciraolo:2017aa] (see the introduction and Lemma $2.3$).
We were made aware, after we completed this work, of a paper by Magnanini and Poggesi [@Magnanini:2016aa] studying the Alexandrov problem which used several similar ideas to our paper. Basically, our Proposition \[prop: fund est\] is close to their Theorem $2.1$, but the methods need to diverge after that point since the estimate for Serrin’s problem is in a weaker norm.
We mention one last connection, with the Faber-Krahn inequality which is typically stated for the first Dirichlet eigenvalue but also has a version for the torsional rigidity $\lambda(\Omega)$ (defined in [(\[e.ulambdadef\])]{}). In this case the Faber-Krahn inequality says, $$\label{e.FK}
\lambda(\Omega) \geq \lambda(B) \ \hbox{ for the ball $B$ with } \ |B| = |\Omega|.$$ This follows from the fact that the Dirichlet energy is non-increasing with respect to Schwarz symmetrization. We can derive immediately from the Faber-Krahn inequality that any minimizer of $\mathcal{J}$ over the class of open sets of ${\mathbb{R}}^N$ must be radially symmetric, then direct computation yields that $B_{r_*}$ is the only energy minimizer. Thus the Faber-Krahn inequality, and its related stability results, have an important connection with the energy $\mathcal{J}$ and its gradient flows. In a recent paper Brasco, De Philippis and Velichkov [@Brasco:2015aa] have proven an optimal stability result for the Faber-Krahn (as well as a range of related inequalities). Basically, in our current understanding, the relationship between our result and the stability of the Faber-Krahn inequality is analogous to the relationship between the stability of almost constant mean curvature hypersurfaces [@Ciraolo:2017aa] and the stability of the isoperimetric inequality (Figalli, Maggi and Pratelli [@Figalli:2010aa]).
The Faber-Krahn and Isoperimetric stability results are proving a lower bound $$E(\Omega) - E(B) \gtrsim d(\Omega,B)^2,$$ where $E$ is the associated energy functional, Dirichlet or perimeter, $B$ is the energy minimizer and $d$ is an appropriate distance. Results of the type considered in our paper or in [@Ciraolo:2017aa] are proving a lower bound of $$g_\Omega({\nabla}E(\Omega),{\nabla}E(\Omega)) \gtrsim d(\Omega,B)^2$$ where, formally, $g$ is a metric on a manifold of subsets of ${\mathbb{R}}^N$ associated to the distance $d$. For our problem, $$g_\Omega(f,g) = \int_{\partial \Omega} f g \ d \sigma.$$ This interpretation suggests why this type of gradient stability estimate may require more regularity than energy stability estimates, and also explains the fundamental connection with gradient flows. For smooth functions on finite dimensional spaces both of these estimates follow from a lower bound of ${\nabla}^2 E$, but as of yet we do not see how to manifest such a connection in our setting.
Now we make explicit the assumptions on the domain $\Omega \subset {\mathbb{R}}^N$ which we use for our result.
1. $\Omega$ is connected and has $C^2$ boundary. \[a\]
2. $\Omega$ has an interior ball of radius $\rho_0>0$ at every boundary point. \[b\]
3. $\Omega$ is a $L_0$-John domain, i.e. there is a base point $x_0 \in \Omega$ such that each point $x \in \Omega$ can be joined to $x_0$ by a curve $\gamma : [0,1] \to \Omega$ such that, \[c\] $$d(\gamma(t),\partial \Omega) \geq L_0^{-1} |\gamma(t)-x|.$$
Assumption is qualitative, assumption implies quantitative non-degeneracy of $u$ near $\partial \Omega$ and assumption is a quantification of connectivity and it is exactly what is needed to prove Poincaré-type inequalities in $\Omega$. Instead of we could assume that $\Omega$ is a Lipschitz domain. One thing we would like to emphasize is that, although we do require a certain amount of regularity, these assumptions do not place us in a perturbative regime.
To simplify notation we will just take $r_0 = \rho_0$.
\[thm: main stability\] Suppose that $\Omega$ satisfies Assumptions , , and . Then for some ball $B_{r_*}$ of radius $r_*$, $$\label{e.opt}
\frac{|\Omega \Delta B_{r_*}|}{|B_{r_*}|} \leq C\left(N,L_0,\frac{\textup{diam}(\Omega)}{\rho_0},\frac{\textup{diam}(\Omega)}{r_*}\right)\left(\frac{1}{r_*^{N-1}} \int_{\partial \Omega} (|Du|^2 - 1)^2 d\sigma\right)^{1/2}.$$
See the beginning of Section \[sec: stability\] for an outline of the proof. Further discussion about possible modifications of the assumptions can be found throughout Section \[sec: stability\] in the course of the proof.
Linear stability of equilibria for the droplet model {#sec: hypothesis}
----------------------------------------------------
Next we explain the application of Theorem \[thm: main stability\] to the dynamic stability of the droplet problem [(\[e.dropletprob\])]{}.
Before we describe our result we discuss the literature on [(\[e.dropletprob\])]{}. For more details and literature see [@FKdrops]. The quasi-static limit leading to problems of the form [(\[e.dropletprob\])]{} was first studied by Greenspan [@Greenspan:1976aa; @greenspan]. Grunewald and Kim [@GrunewaldKim11] construct global in time weak (energy) solutions from general initial data by a discrete gradient flow scheme. The author and Kim [@FKdrops] constructed global in time time viscosity solutions which converge to the equilibrium (although without a rate) under a certain approximate reflection symmetry condition on the initial data (see Section \[sec: exponential rate\] where we recall the precise condition). See also Glasner [@MR2144627; @MR2221703; @glasner], Glasner, Kim [@GlasnerKim09] and Mellet [@MR3268920]. A different approach was taken by Escher and Guidotti [@Escher:2015aa] who show local existence of smooth solutions by writing the equation in a coordinate system adapted to the initial data. Most relevant to our result, although by a completely different method, is the work of Guidotti [@Guidotti:2015aa] showing the stability of the equilibrium state for [(\[e.dropletprob\])]{} by a perturbative approach.
We also show the dynamic stability of the equilibrium state for [(\[e.dropletprob\])]{} with exponential convergence, but our method, which was outlined above, is completely different from [@Guidotti:2015aa]. We use the energy dissipation estimate [(\[e.dissipation\])]{} in combination with our new stability result Theorem \[thm: main stability\] to obtain the exponential rate. As far as we are aware this is one of the first applications of a stability estimate for a geometric minimization problem to show dynamic stability of an associated gradient flow. The main benefits of our approach are $(1)$ it does not require to be in the perturbative regime, $(2)$ it has an appealing connection with the gradient flow structure of the problem, $(3)$ given the stability estimate Theorem \[thm: main stability\] and regularity theory for the flow the energy decay is just an elementary Grönwall argument. We state our result below.
\[thm: dynamic stability\] Suppose that the solution $\Omega_t$ of the droplet problem [(\[e.dropletprob\])]{} satisfies assumptions , , and uniformly for all $t>0$ and $\textup{diam}(\Omega)$ remains bounded. Then there are constants $C,c>0$ depending on $N$ and the suprema in $t$ of $L_0,\frac{\textup{diam}(\Omega_t)}{\rho_0}$, and $\frac{\textup{diam}(\Omega_t)}{r_*}$ such that, $$\inf_{x \in {\mathbb{R}}^N} |\Omega_t \Delta B_{r_*}(x)| \leq C(\mathcal{J}(\Omega_0) - \mathcal{J}(B_{r_*}))e^{-cr_*^{-1}t}.$$
Since the convergence result is conditional on the propagation of assumptions , , and we must explain when this is expected to be true. This will be explained in greater detail in Section \[sec: conditions\] but we give a brief summary here. The result of the author and Kim [@FKdrops] implies that assumption and the diameter bound will be propagated for initial data satisfying a certain strong star-shapedness type condition. The regularity assumptions and have not been studied yet for this problem, but initial data with small local Lipschitz constant is expected to regularize immediately, see Choi, Jerison and Kim [@Choi:2009aa; @CJK] for a regularity result for a similar quasi-static problem.
Notation
--------
We will use $C,c>0$ to denote dimension dependent constants which may change from line to line. If the constant has additional dependencies on some parameters $A_1,A_2,\dots$ we will write $C(A_1,A_2,\dots)$. The only exception to this rule should be in heuristic remarks (outside of proofs) explaining ideas where we will not include all the dependencies of the constants.
Acknowledgments
---------------
I would like to thank Inwon Kim, Francesco Maggi, and John Garnett for inspiring discussions and helpful comments.
Stability result for regular boundary {#sec: stability}
=====================================
We explain the strategy to obtain the stability estimate of Theorem \[thm: main stability\]. The starting point is the ideas of [@BNST08; @BNST-alt; @BNST09; @BNST0] where it was discovered that the symmetry property of Serrin’s problem is closely related to a certain arithmetic-geometric mean inequality. We are able to improve on their calculation to obtain the following fundamental estimate, $$\int_\Omega u \left( \left(\tfrac{\Delta u}{N}\right)^2- \tfrac{1}{{ N \choose 2 }}S_2(D^2u)\right) dx \leq C_N\int_{\partial\Omega}\left| \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du\right| \left||Du|^2-1\right| \ d \sigma.$$ where $S_2$ is the second symmetric function of the eigenvalues and is defined below in Section \[sec: symmetric\]. Via the AM-GM inequality this yields, $$\label{e.fundest0}
\int_\Omega u(x) |D^2u(x)+\tfrac{\lambda(\Omega)}{N}{\textup{Id}}|^2 dx \leq C_N\int_{\partial\Omega}\left| \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du\right| \left||Du|^2-1\right| \ d \sigma.$$ We will need to exploit this weighted Sobolev norm estimate to obtain symmetry. We explain the idea of how to use [(\[e.fundest0\])]{}. Let $x_0 \in \Omega(u)$ be a point where $\max u$ is obtained. Define, $$\label{e.wdef1}
w(x) = u(x) - (u(x_0) - \tfrac{\lambda(\Omega)}{2N}|x-x_0|^2) \ \hbox{ with } \ w(x_0) = 0, \ Dw(x_0) = 0,$$ and $w$ is harmonic. Then we derive from [(\[e.fundest0\])]{}, $$\int_\Omega u(x) |D^2w|^2 dx \leq \|Dw\|_{L^2(\partial \Omega)}\||Du|^2-1\|_{L^2(\partial \Omega)}.$$ Now one is tempted to use a trace theorem for the $L^2$ norm, weighted by $u(x)$, to bound, $$\label{e.trace thm 0}
\|Dw\|_{L^2(\partial \Omega)} \leq C \left( \int_\Omega u(x) |D^2w|^2 dx \right)^{1/2}.$$ When $u$ is comparable to the distance function to ${\mathbb{R}}^N \setminus \Omega$ this estimate has the correct scaling, so perhaps it seems reasonable to expect. Now in reality the estimate [(\[e.trace thm 0\])]{} is false in general, it is an endpoint case which fails for some logarithmically singular functions, however a very similar estimate *does* hold for harmonic functions (like $w,Dw$) via a simple integration by parts identity, see Section \[sec: trace thm\]. Anyway, pretending we can use [(\[e.trace thm 0\])]{}, we would obtain, $$\int_\Omega u(x) |D^2w|^2 dx \leq C \int_{\partial \Omega} (|Du|^2-1)^2 \ d \sigma \ \hbox{ and } \ \int_{\partial \Omega} |Dw|^2 dx \leq C \int_{\partial \Omega} (|Du|^2-1)^2 \ d \sigma.$$ From the second estimate we can easily derive, $$\int_{\partial \Omega} (\tfrac{\lambda(\Omega)}{N}|x-x_0| - 1)^2 dx \leq C \int_{\partial \Omega} (|Du|^2-1)^2 \ d \sigma,$$ which is now obviously a type of distance estimate to the ball $B_{\lambda(\Omega)/N}(x_0)$. With some more work we can obtain the estimate in measure. This is the basic outline of the proof.
Convexity inequalities and $k$-Hessian equations {#sec: symmetric}
------------------------------------------------
Just as the AM-GM inequality underlies the isoperimetric inequality and it’s corresponding stability estimates, a closely related convexity inequality underlies the stability of Serrin’s problem. Let $M$ be an $N \times N$ symmetric matrix with real entries, and call the $N$ real eigenvalues of $M$, $\mu_1,\dots, \mu_N$. The $k$-th symmetric function of the eigenvalues of $M$ is, $$S_k(M) = \tfrac{1}{{N \choose k }}\sum_{ i_1 < \cdots < i_k} \mu_{i_1}\cdots\mu_{i_k}.$$ When $k=1$ this is the trace, and when $k=N$ it is the determinant. There is a classical refinement of the arithmetic geometric mean inequality which gives, $$\label{eqn: general amgm}
\det(M)^{1/N} = S_N(M)^{1/N} \leq \cdots \leq S_{2}(M)^{1/2} \leq S_1(M) = \tfrac{1}{N}{\textup{Tr}}(M),$$ with strict inequality holding unless $\mu_1 = \cdots =\mu_N$.
As was discovered by [@BNST08; @BNST09; @BNST-alt], the symmetry of solutions of [(\[e.serrin\])]{} is fundamentally related to the AM-GM inequality between ${\textup{Tr}}$ and $S_2$. We note that this inequality can be further quantified as:
\[lem: quadratic growth\] There is a dimensional constant $c_N>0$ so that, $$S_1(M)^2-S_2(M)\geq c_N|M-S_1(M){\textup{Id}}|^2.$$
This can be checked by direct computation.
We state several more useful facts about $S_2(M)$, proofs of these identities can be found in Reilly [@reilly1974], see also Wang [@Wang:2009aa] where these $k$-Hessian equations are studied. If $M$ has entries $m_{ij}$ then we call, $$S_2^{ij}(M) = \frac{\partial S_2(M)}{\partial m_{ij}},$$ and since $S_2(M)$ is homogeneous of degree $2$ on ${\mathbb{R}}^{2N}$ we have the identity, $$S_2(M) = \tfrac{1}{2}S_2^{ij}(M)m_{ij}.$$ Now $S_2^{ij}$ is divergence free, $$D_{i}S_2^{ij}(D^2u) = 0,$$ and so $S_2(D^2u)$ can be written in divergence form, $$S_2(D^2u) = \tfrac{1}{2}D_i(S^{ij}_2(D^2u)D_ju).$$ Finally we have the following identity relating $S_2(D^2u)$ with the curvature of the level sets of $u$, $$\frac{S_2^{ij}(D^2u)D_iuD_ju}{|Du|^2} = \kappa |Du| \ \hbox{ where } \ \kappa = |Du|^{-1} {\textup{Tr}}((Id-\widehat{Du}\otimes\widehat{Du})D^2u).$$
Integration by parts identities
-------------------------------
Before proceeding with the proofs we establish several integration by parts identities which will be useful later. This first identity was also used in an important way in [@BNST08], $$\begin{aligned}
\int_{\Omega} |Du|^2 dx &= \int_\Omega |Du|^2\frac{-\Delta u}{\lambda(\Omega)} \ dx \\
&= \tfrac{2}{\lambda(\Omega)}\int_\Omega \langle D^2u Du,Du\rangle dx + \tfrac{1}{\lambda(\Omega)}\int_{\partial\Omega} |Du|^3 d\sigma(x) \\
& = \tfrac{2}{\lambda(\Omega)}\int_\Omega |Du|^2\Delta u - {\textup{Tr}}((Id-\widehat{Du}\otimes\widehat{Du})D^2u)|Du|^2 \ dx + \tfrac{1}{\lambda(\Omega)}\int_{\partial\Omega} |Du|^3 d\sigma(x) .\end{aligned}$$ Recalling from above that $\kappa = |Du|^{-1} {\textup{Tr}}((Id-\widehat{Du}\otimes\widehat{Du})D^2u)$ is $N-1$ times the mean curvature of the level sets of $u$ and rearranging above, $$\label{e.identity1}
\int_\Omega \kappa |Du|^3 dx= -\tfrac{3\lambda(\Omega)^2}{2}\textup{Vol}+\tfrac{1}{2} \int_{\partial \Omega} |Du|^3 d\sigma(x).$$ It turns out that the term $\int_{\partial \Omega} |Du|^3 d\sigma(x)$ in the above identity is not so convenient to work with by itself, we establish the following new identity,
\[lem: cube\] For any $x_0 \in {\mathbb{R}}^N$, $$\label{e.cube}
\int_{\partial\Omega} |Du|^3 d \sigma = \tfrac{\lambda(\Omega)^2(N+2)}{N}\textup{Vol} - \int_{\partial\Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du,\nu \rangle (|Du|^2-1) \ d \sigma.$$
Using this identity in [(\[e.identity1\])]{} gives, $$\label{e.identity2}
\int_\Omega \kappa |Du|^3 dx= -\tfrac{N-1}{N}\lambda(\Omega)^2\textup{Vol}-\tfrac{1}{2} \int_{\partial\Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du,\nu \rangle (|Du|^2-1) \ d \sigma$$
We will need a Pohozaev type identity, for any $x_0 \in {\mathbb{R}}^N$, $$\begin{aligned}
-\int_\Omega \langle x-x_0, D u \rangle \Delta u \ dx&= \int_\Omega |D u|^2+\langle x-x_0,D^2u Du\rangle \ dx + \int_{\partial \Omega} \langle x-x_0, D u \rangle |Du| d \sigma \\
& = \lambda(\Omega)\textup{Vol}+\int_\Omega \langle x-x_0,\tfrac{1}{2}D(|Du|^2)\rangle \ dx-\int_{\partial \Omega} \langle x-x_0, \nu \rangle |Du|^2 d \sigma \\
&= -\frac{N-2}{2}\lambda(\Omega)\textup{Vol} -\tfrac{1}{2}\int_{\partial \Omega} \langle x-x_0, \nu \rangle |Du|^2 d \sigma
\end{aligned}$$ and on the other hand by using the equation $-\Delta u = \lambda(\Omega)$ and then integrating by parts, $$-\int_\Omega \langle x-x_0, D u \rangle \Delta u \ dx = -N\lambda(\Omega)\textup{Vol}.$$ Combining these identities we find, $$\int_{\partial \Omega} \langle \tfrac{\lambda(\Omega)}{N}(x-x_0), \nu \rangle |Du|^2 d \sigma= \frac{N+2}{2N}\lambda(\Omega)^2\textup{Vol}.$$ Now we can compute the desired identity, $$\begin{aligned}
\int_{\partial\Omega} |Du|^3 d \sigma &= -\int_{\partial\Omega} |Du|^2\langle Du,\nu \rangle \ d \sigma \\
&= -\int_{\partial\Omega} (|Du|^2-1)\langle Du,\nu \rangle \ d \sigma - \int_{\partial\Omega} \langle Du,\nu \rangle \ d \sigma \\
&=\int_{\partial\Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0),\nu \rangle |Du|^2 \ d\sigma-\int_{\partial\Omega} (|Du|^2-1)\langle Du+\tfrac{\lambda(\Omega)}{N}(x-x_0),\nu \rangle \ d \sigma \\
&\cdots+ \int_{\Omega} (-\Delta u) \ d \sigma -\int_{\partial \Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0),\nu \rangle \ d \sigma\\
&=\tfrac{\lambda(\Omega)^2(N+2)}{N}\textup{Vol}-\int_{\partial\Omega} (|Du|^2-1)\langle Du+\tfrac{\lambda(\Omega)}{N}(x-x_0),\nu \rangle \ d \sigma
\end{aligned}$$ where the last two terms were equal using $-\Delta u = \lambda(\Omega)$ in $\Omega$ and divergence theorem to evaluate $\int_{\partial \Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0),\nu \rangle \ d \sigma = \int_{\Omega} \lambda(\Omega) \ dx$.
A weighted $L^2$ estimate on the Hessian {#sec: hessian est}
----------------------------------------
In this section we establish the fundamental estimate which, with sufficient boundary regularity, we will be able to exploit to obtain the stability result. We state the result as a Proposition,
\[prop: fund est\] For any $x_0 \in {\mathbb{R}}^N$, $$\int_\Omega u \left( \left(\tfrac{\Delta u}{N}\right)^2- \tfrac{1}{{ N \choose 2 }}S_2(D^2u)\right) dx \leq C_N\int_{\partial\Omega}\left| \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du\right| \left||Du|^2-1\right| \ d \sigma.$$
Using Lemma \[lem: quadratic growth\] we also have the estimate, $$\label{e.fund est}
\int_\Omega u(x) |D^2u(x)+\tfrac{\lambda(\Omega)}{N}{\textup{Id}}|^2 dx \leq C_N\int_{\partial\Omega}\left| \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du\right| \left||Du|^2-1\right| \ d \sigma$$
The main ideas of the computation have already been established in [@BNST08; @BNST-alt], however we have made an important improvement. In the result of [@BNST08] (see [@BNST08] Lemma $3.1$) the analogous estimates had mismatched homogeneity, quadratic in $D^2u(x)+\tfrac{\lambda(\Omega)}{N}{\textup{Id}}$ and linear in $|Du|-1$. Our estimate [(\[e.fund est\])]{} is quadratic on both the left hand side and the right hand side.
The beginning of the computation follows the idea of [@BNST08], $$\begin{aligned}
\textup{Vol} = \int_\Omega \left( \lambda(\Omega)^{-1}\Delta u\right)^2 u \ dx
& \geq \tfrac{N^2}{\lambda(\Omega)^2}\int_\Omega \frac{1}{{ N \choose 2 }}S_2(D^2u) u \ dx \\
& = \tfrac{N}{(N-1)\lambda(\Omega)^2}\int_\Omega uS_2^{ij}(D^2u)D_{ij}^2u \ dx \\
& = \tfrac{N}{(N-1)\lambda(\Omega)^2}\int_\Omega uD_i(S_2^{ij}(D^2u)D_{j}u) dx \\
& = -\tfrac{N}{(N-1)\lambda(\Omega)^2}\int_\Omega S_2^{ij}(D^2u)D_{i}uD_ju \ dx \\
& = -\tfrac{N}{(N-1)\lambda(\Omega)^2}\int_\Omega \kappa |Du|^3 \ dx \\
& = \tfrac{3N}{2(N-1)}\textup{Vol} - \tfrac{N}{2(N-1)\lambda(\Omega)^2} \int_{\partial\Omega} |Du|^3 d \sigma(x).\end{aligned}$$ Here is where our computation diverges from that of [@BNST08]. We insist on achieving an error estimate which is quadratic in $|Du|-1$ to match the quadratic homogeneity of $[{ N \choose 2 }^{-1}S_2(D^2u) - N^{-1}\Delta u]$. Instead of adding and subtracting $1$ in last boundary integral above, which leads to a linear order error term $|Du|^3 - 1$, we use the identity [(\[e.cube\])]{} and find a quadratic term. We finish the computation using [(\[e.cube\])]{}, $$\begin{aligned}
\textup{Vol} = \int_\Omega \left( \lambda(\Omega)^{-1}\Delta u\right)^2 u \ dx
& \geq \tfrac{N^2}{\lambda(\Omega)^2}\int_\Omega \frac{1}{{ N \choose 2 }}S_2(D^2u) u \ dx \\
& = \tfrac{3N}{2(N-1)}\textup{Vol} - \tfrac{N}{2(N-1)\lambda(\Omega)^2} \int_{\partial\Omega} |Du|^3 d \sigma(x) \\
&= \textup{Vol} + \tfrac{N}{2(N-1)\lambda(\Omega)^2} \int_{\partial\Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du,\nu \rangle (|Du|^2-1) \ d \sigma,\end{aligned}$$ where $x_0$ is an arbitrary point in ${\mathbb{R}}^N$. Rearranging the above computation, what we have finally established is, $$\int_\Omega u \left( \left(\tfrac{\Delta u}{N}\right)^2- \tfrac{1}{{ N \choose 2 }}S_2(D^2u)\right) dx \leq \tfrac{-1}{2N(N-1)}\int_{\partial\Omega}\langle \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du,\nu \rangle (|Du|^2-1) \ d \sigma.$$ Finally this was the desired estimate of Proposition \[prop: fund est\].
A weighted norm trace theorem for harmonic functions {#sec: trace thm}
----------------------------------------------------
In this section we make rigorous the trace theorem claimed in [(\[e.trace thm 0\])]{}. For a given measurable weight function $\omega\geq 0$ we will write $L^2_\omega$ for the weighted $L^2$-space with norm, $$\|f\|_{L^2_w} = \int |f|^2 \omega(x) \ dx.$$ We are interested to prove the following type of estimate, $$\int_{\partial \Omega} |f-(f)_\Omega|^2 \ d\sigma \leq C \int_{\Omega} |Df|^2 u(x) \ dx.$$ To simplify matters we start by replacing the weight $u(x)$ by the distance function to ${\mathbb{R}}^N \setminus \Omega$ which we call $d_\Omega(x)$. Now we consider the possible validity of the following trace estimate $$\int_{\partial \Omega} |f - (f)_{\partial \Omega}|^2 \ dx \leq C \int_\Omega |Df|^2 d_\Omega(x) \ dx.$$ This estimate is an endpoint case and it is *false* for general smooth $f$. Take as an example $|\log d_\Omega(x)|^{\alpha}$ for any $0 < \alpha <1/2$, the gradient has finite $L^2_{d_\Omega}(\Omega)$ norm, but obviously there cannot be a trace on $\partial \Omega$.
In order to obtain a valid estimate we will need to use that the particular $f$ we will be working with satisfies a stronger property, it is harmonic in $\Omega$. We make use of the following identity. Let $f$ be a harmonic function in $\Omega$ then, $$\begin{aligned}
\int_{\Omega} |Df|^2u(x) \ dx &= -\int_{\Omega} f D f \cdot D u \ dx \\
&= \int_{\Omega} f^2 \Delta u + f D f \cdot D u \ dx - \int_{\partial \Omega} f^2 Du \cdot \nu \ d\sigma.
\end{aligned}$$ Using the middle equality we find, $$-\int_{\Omega} f D f \cdot D u \ dx = \frac{1}{2} \int_{\Omega} f^2 \Delta u \ dx+\frac{1}{2}\int_{\partial \Omega} f^2 |Du|\ d\sigma.$$ This can be rearranged to, $$\label{e.traceid}
\int_{\partial \Omega} f^2 |Du| \ d\sigma = 2\int_{\Omega} |Df|^2u(x) \ dx +\lambda(\Omega)\int_{\Omega} f^2 \ dx,$$ holding for every $f$ harmonic in $\Omega$ and continuous up to $\partial \Omega$.
Now we can state the main result of this section which, in particular, will apply to (a slight modification of) the function $Dw$ defined in [(\[e.wdef1\])]{} in Section \[sec: hessian est\].
\[prop: trace\] Suppose that $\Omega$ is an $L_0$-John domain with base point $x_0$ and $f$ is harmonic in $\Omega$, continuous up to $\partial \Omega$, with $f(x_0) = 0$. Then we have the following estimates, $$\int_{\partial \Omega} f^2 |Du| \ d\sigma \leq 2 \int_{\Omega} |Df|^2u(x) \ dx + CL_0^N\lambda(\Omega)|\Omega|^{1/N}\int_{\Omega} |Df|^2d_\Omega(x) \ dx$$
Now all that we need to go from the trace identity [(\[e.traceid\])]{} to Proposition \[prop: trace\] is to control $\|f\|_{L^2(\Omega)}$ by the weighted $L^2_{u}$ norm of $Df$. We will use a Poincaré inequality with weighted norm proved by Hurri-Syrjänen [@Hurri-Syrjanen:1994aa]. This is the only place where we use the assumption that $\Omega$ is a John domain, as it is exactly suited to this type of weighted Poicaré inequality. We have a small wrinkle which is that $f$ does not have mean $0$ on $\Omega$, instead we need to use that $f(x_0) = 0$ and that $f$ is harmonic. This will require us to state the result of [@Hurri-Syrjanen:1994aa] carefully.
The following result is not exactly what is stated in [@Hurri-Syrjanen:1994aa] but an inspection of the proof will see that this result is indeed proven there.
\[lem: dist weighted poincare\] Let $\Omega$ a $L_0$-John domain in ${\mathbb{R}}^n$ with base point $x_0$ and $d_\Omega(x)$ be the distance function to ${\mathbb{R}}^N \setminus \Omega$. There is a dimensional constant $C>0$ such that for any smooth $f: \Omega \to {\mathbb{R}}$, $$\| f - (f)_{Q_0}\|_{L^{\frac{2N}{N-1}}(\Omega)} \leq CL_0^N\|Df\|_{L^2_{d_\Omega}(\Omega)},$$ where $Q_0$ is any cube centered at $x_0$ with $\textup{diam}(Q_0) \leq d(Q_0,\partial \Omega) \leq 4 \ \textup{diam}(Q_0)$.
We remark that since we only need a bound for the $L^2(\Omega)$ norm in [(\[e.traceid\])]{} the assumptions, either on $\Omega$ or on the weight, can be weakened somewhat and we could still derive a version of Proposition \[prop: trace\].
Now since we have carefully stated Theorem \[lem: dist weighted poincare\] we can use the property that $f$ is harmonic with $f(x_0) = 0$ in place of having $(f)_\Omega = 0$.
\[lem: poincare\] Let $\Omega$ a $L_0$-John domain in ${\mathbb{R}}^n$ with base point $x_0$ and $d_\Omega(x)$ be the distance function to ${\mathbb{R}}^N \setminus \Omega$. There is a dimensional constant $C>0$ such that if $f$ is a harmonic function in $\Omega$ with $f(x_0) = 0$ then, $$\| f \|_{L^{\frac{2N}{N-1}}(\Omega)} \leq CL_0^N\|Df\|_{L^2_{d_\Omega}(\Omega)}.$$
From the Lemma and identity [(\[e.traceid\])]{}, Proposition \[prop: trace\] follows immediately.
Call $B_0 = B_{\textup{diam}(Q_0)/2}(x_0)$. Then by the assumption on $Q_0$, $$Q_0 \subset B_0 \subset \Omega.$$ Since $f$ is harmonic and $f(x_0) = 0$ we have by the mean value property, $$(f)_{B_0}= 0.$$ The usual Poincaré inequality in $B_0$ implies then that, $$\begin{aligned}
\| f \|_{L^{\frac{2N}{N-1}}(Q_0)} \leq \| f \|_{L^{\frac{2N}{N-1}}(B_0)} &\leq C|Q_0|^{\frac{1}{N}+\frac{N-1}{2N}-\frac{1}{2}}\|Df\|_{L^2(B_0)} \\
&\leq C|Q_0|^{\frac{1}{N}+\frac{N-1}{2N}-\frac{1}{2}-\frac{1}{2N}}\|Df\|_{L^2_{d_\Omega}(B_0)} \\
&\leq C\|Df\|_{L^2_{d_\Omega}(\Omega)}.
\end{aligned}$$ Now we apply the weighted Poincaré inequality Theorem \[lem: dist weighted poincare\] and the above estimate, $$\| f\|_{L^{\frac{2N}{N-1}}(\Omega)} \leq \| f - (f)_{Q_0}\|_{L^{\frac{2N}{N-1}}(\Omega)} + \left(|\Omega|/|Q_0|\right)^{\frac{N-1}{2N}}\| (f)_{Q_0} \|_{L^{\frac{2N}{N-1}}(Q_0)} \leq CL_0^N\|Df\|_{L^2_{d_\Omega}(\Omega)}$$ which is the desired result. Note we have used the John condition with base point $x_0$ and a point $x \in \Omega$ with $|x-x_0| \geq \textup{diam}(\Omega)/2$ to find that $|Q_0| \geq \textup{diam}(\Omega)^N/L_0^N$.
An $L^2$ distance estimate on $\partial \Omega$
-----------------------------------------------
Now we are able apply the trace inequality Proposition \[prop: trace\] in combination with Proposition \[prop: fund est\] to obtain an $L^2$-type estimate on the distance of $\partial \Omega$ to $\partial B_{N/\lambda(\Omega)}$.
\[prop: l2 dist est\] Suppose that $\Omega$ satisfies assumptions , and . Then there exists $C>0$ depending on $N,L_0,\frac{\textup{diam}(\Omega)}{\rho_0}$ such that, $$\inf_{x_0 \in {\mathbb{R}}^N}\left( \int_{\partial \Omega} (\tfrac{\lambda(\Omega)}{N}|x-x_0| - 1)^2 \ d\sigma(x)\right)^{1/2} \leq C\left(\int_{\partial \Omega} (|Du|^2-1)^2 \ d \sigma(x)\right)^{1/2}.$$
The proof is mainly carrying out the formal argument found in Section \[sec: hessian est\], we use the interior ball condition, assumption , to apply the Hopf Lemma and obtain a lower bound $$u(x) \geq c_N\frac{\rho_0}{\textup{diam}(\Omega)} d_\Omega(x)$$ and then apply the trace inequality Proposition \[prop: trace\]. At this stage we can also see some indication that the interior ball assumption (or a slight weakening) on $\partial \Omega$ is necessary. When $\partial \Omega$ has only Lipschitz regularity $|Du|^{-1}$ will no longer be in $L^{\infty}(\partial \Omega)$ and $u(x)$ will no longer have a lower bound by the distance function.
The first thing we prove is the lower bound, $$\label{e.ulb}
u(x) \geq c_N\textup{diam}(\Omega)^{-1} \rho_0 d_\Omega(x).$$ Let $x_0 \in \Omega$ and call $y \in \partial \Omega$ to be the point achieving $d_\Omega(x_0) = |x_0-y|$. There is a ball $B_{\rho_0}(z)$ touching $\partial \Omega$ from the inside at $y$. Since $\partial \Omega$ is $C^1$ we must have, $$\frac{y-z}{|y-z|} = \frac{y-x_0}{|y-x_0|} = \nu(y) \ \hbox{ the inward unit normal vector to $\partial \Omega$ at $y$.}$$ We construct a barrier to get a lower bound on $u(z)$, $$\varphi(x) = \frac{\lambda(\Omega)}{2N}(\rho_0^2 - |x-z|^2).$$ By the maximum principle $u(x) \geq \varphi(x)$ and, $$u(x_0) \geq \varphi(x_0) \geq c_N \lambda(\Omega)\rho_0 (\rho_0 -|x_0-z|) = c_N \lambda(\Omega)\rho_0d_\Omega(x_0).$$ Finally by the monotonicity of $\lambda(\Omega)$, $$\label{e.llb}
\lambda(\Omega) \geq \lambda(B_{\textup{diam}(\Omega)/2}) = c_N \textup{diam}(\Omega)^{-1},$$ and now we have [(\[e.ulb\])]{}.
Now we make rigorous the heuristic argument described at the beginning of the section. Let $x_0$ be the base point from the John domain property, assumption . Now define, $$v(x) = a - \frac{\lambda(\Omega)}{2N}|x-x_1|^2 \ \hbox{ and } \ w = u - v$$ where $$\label{e.pchoice}
x_1 = x_0 - \tfrac{N}{\lambda(\Omega)}Du(x_0) \ \hbox{ and } \ a = u(x_0) + \tfrac{N}{2\lambda(\Omega)}|Du(x_0)|^2$$ are chosen so that $w(x_0) = |Dw(x_0)| = 0$. Without loss we can assume $x_1 = 0$. Now $w$ as defined is harmonic in $\Omega$ and satisfies $$w(x_0) = |Dw(x_0)| = 0.$$ Now we can apply the weighted trace theorem Proposition \[prop: trace\] and [(\[e.ulb\])]{} to obtain, $$\begin{aligned}
\int_{\partial \Omega} |Dw|^2 \ d\sigma &\leq C\frac{\textup{diam}(\Omega)}{\rho_0}\int_{\partial \Omega} |Dw|^2|Du| \ d\sigma \\
&\leq C\frac{\textup{diam}(\Omega)}{\rho_0}\int_{\Omega} |D^2w|^2u(x) \ dx +CL_0^N\frac{\textup{diam}(\Omega)}{\rho_0}\int_{\Omega} |D^2w|^2d_\Omega(x) \ dx \\
&\leq C \left(L_0,\frac{\textup{diam}(\Omega)}{\rho_0}\right)\int_{\Omega} |D^2w|^2u(x) \ dx.
\end{aligned}$$ Next we apply Proposition \[prop: fund est\] to find, dropping the dependencies of $C$, $$\int_{\partial \Omega} |Dw|^2 \ d\sigma \leq C \int_{\partial \Omega} \left| \tfrac{\lambda(\Omega)}{N}(x-x_0) +Du\right| \left||Du|^2-1\right| \ d\sigma$$ Noting that $\frac{\lambda(\Omega)}{N}(x-x_0) +Du = Dw$ and applying Cauchy-Schwarz we get, $$\left(\int_{\partial \Omega} |Dw|^2 \ d\sigma\right)^{1/2} \leq C \left(\int_{\partial \Omega} (|Du|^2-1)^2 \ d\sigma\right)^{1/2}.$$ Now we are almost finished, we just need a triangle inequality, $$\begin{aligned}
\|Dw\|_{L^2(\partial \Omega)} &\geq \left\|\tfrac{\lambda(\Omega)}{N}|x-x_0| - |Du|\right\|_{L^2(\partial \Omega)} \\
&\geq \left\|\tfrac{\lambda(\Omega)}{N}|x-x_0| - 1\right\|_{L^2(\partial \Omega)} - \big\||Du|-1\big\|_{L^2(\partial \Omega)},
\end{aligned}$$ then we finish using $||Du| - 1| \leq ||Du|^2-1|$ since that inequality is true for all positive reals.
The distance estimate in measure
--------------------------------
Now we are able to prove the stability estimate in measure, the main result of Theorem \[thm: main stability\], which we restate here:
\[thm: main stability 1\] If $\Omega$ satisfies assumptions , and then, $$\inf_{x_0 \in {\mathbb{R}}^n}\frac{|\Omega \Delta B_{r_*}(x_0)| }{|B_{r_*}|} \leq C\left(L_0,\frac{\textup{diam}(\Omega)}{\rho_0},\frac{\textup{diam}(\Omega)}{r_*}\right)\left(\frac{1}{r_*^{N-1}}\int_{\partial \Omega}(|Du|^2-1)^2 \ d \sigma \right)^{1/2}.$$
In $N=2$ the result can be improved to get an estimate in Hausdorff distance, control of $\|D^2w\|_{L^2_{u}(\Omega)}$ will give control of $\|Dw\|_{L^4(\Omega)}$ by the Poincaré inequality Lemma \[lem: poincare\] which then gives control of $\|w\|_{L^{\infty}(\Omega)}$.
The main additional estimate to go from Proposition \[prop: l2 dist est\] to Theorem \[thm: main stability\] is the following bound between the $L^2$ pseudo-distance on $\partial \Omega$ and the distance in measure.
\[lem: distance est\] Suppose that $E$ is a set with Lipschitz boundary and $r>0$ such that $|E| \leq K |B_r|$ and the in-radius has $r_{in}(E) \geq r/K$. Then it holds, $$\frac{|E \Delta B_r|}{|B_r|} \leq C(N,K) \left(\frac{1}{r^{N-1}}\int_{\partial E} \left(\frac{|x|}{r}-1\right)^2 d\sigma(x)\right)^{1/2}.$$
We remark that the assumption that $E$ has Lipschitz boundary is not really necessary, the same result, appropriately stated, would hold for sets of finite perimeter.
We will return to the proof of Lemma \[lem: distance est\] below. First we complete the proof of Theorem \[thm: main stability\].
We call as before $x_0$ to be the base point from the John domain property and, $$v(x) = a - \frac{\lambda(\Omega)}{2N}|x-x_1|^2 \ \hbox{ and } \ w = u - v$$ where, as in [(\[e.pchoice\])]{}, the parameters $a,x_1$ are chosen so that $w(x_0) = Dw(x_0) = 0$. Without loss we can assume $x_1 = 0$. Recall, as [(\[e.llb\])]{} and using $B_{\textup{diam}(\Omega)/2L_0}(x_0) \subset \Omega$ which follows from , $$c_N \textup{diam}(\Omega)^{-1}\leq \lambda(\Omega) \leq c_NL_0 \textup{diam}(\Omega)^{-1}.$$ Then we get the following bounds, $$B_{c_N N/\lambda(\Omega)}(x_0) \subset \Omega \ \hbox{ and } \ |\Omega| \leq C(N,L_0)|B_{N/\lambda(\Omega)}|.$$ To shorten the notation we give the name $\mu_0$ to the non-dimensional ratio $\textup{diam}(\Omega)/\rho_0$.
Now we can apply Lemma \[lem: distance est\] and Proposition \[prop: l2 dist est\] to find $$\begin{aligned}
\frac{|\Omega \Delta B_{N/\lambda(\Omega)}|}{|B_{N/\lambda(\Omega)}|} &\leq C(L_0) \left(\lambda(\Omega)^{N-1}\int_{\partial \Omega} \left(\tfrac{\lambda(\Omega)}{N}|x|-1\right)^2 d\sigma(x)\right)^{1/2} \notag\\
& \leq C(L_0,\mu_0)\left(\lambda(\Omega)^{N-1}\int_{\partial \Omega} (|Du|^2-1)^2 \ d \sigma(x)\right)^{1/2} \label{e.measureest1}.\end{aligned}$$ Next we aim for an estimate of, $$\frac{N}{\lambda(\Omega)} - r_*.$$ We still have a piece of information we have not used, which is the volume constraint, we use it in the following way, $$\begin{aligned}
\lambda(\Omega)\textup{Vol} &= \int_{\Omega} |Du|^2 \ dx \notag \\
&= \int_{B_{N/\lambda(\Omega)}} |Dv|^2 \ dx + \int_{\Omega}|Du|^2- |Dv|^2 \ dx \label{e.dirichletsplit} \\
& \quad \quad \quad \cdots+ \int({\bf 1}_{\Omega \setminus B_{N/\lambda(\Omega)}} - {\bf 1}_{ B_{N/\lambda(\Omega)} \setminus \Omega})|Dv|^2 \ dx \notag\end{aligned}$$ We look at each term individually, the first term is, $$\int_{B_{N/\lambda(\Omega)}} |Dv|^2 \ dx = \frac{N}{N+2}|B_{N/\lambda(\Omega)}|.$$ The remaining terms need to be estimated, for the middle term we use the Poincaré-type inequality Lemma \[lem: poincare\], $$\begin{aligned}
\left|\int_{\Omega}|Du|^2- |Dv|^2 \ dx\right| &\leq \|Du+Dv\|_{L^{\frac{2N}{N+1}}(\Omega)}\|Dw\|_{L^{\frac{2N}{N-1}}(\Omega)}\\
&\leq C\left(L_0,\mu_0\right)(\|Dv\|_{L^{\frac{2N}{N+1}}(\Omega)}+|\Omega|^{\frac{1}{2N}}\|Du\|_{L^{2}(\Omega)})\|D^2w\|_{L^{2}_{d_\Omega}(\Omega)} \\
&\leq C\left(L_0,\mu_0\right)(|\Omega|^{\frac{N+1}{2N}}\lambda(\Omega)\textup{diam}(\Omega)+|\Omega|^{\frac{1}{2N}}\lambda(\Omega)^{1/2}\textup{Vol}^{1/2})\|D^2w\|_{L^{2}_{d_\Omega} }\\
&\leq C\left(L_0,\mu_0\right)(|\Omega|^{\frac{N+1}{2N}}+r_*^{\frac{N+1}{2}})\||Du|^2-1\|_{L^{2}(\partial \Omega)}
\end{aligned}$$ In the last inequality we have bounded $\lambda(\Omega) \textup{diam}(\Omega)$ by $c_N\textup{diam}(\Omega)/\rho_0 = c_N \mu_0$, similarly for $|\Omega|^{\frac{1}{2N}}\lambda(\Omega)^{1/2}$, and we have used Proposition \[prop: fund est\]. For the final term of [(\[e.dirichletsplit\])]{} we estimate, $$\begin{aligned}
\left|\int({\bf 1}_{\Omega \setminus B_{N/\lambda(\Omega)}} - {\bf 1}_{ B_{N/\lambda(\Omega)} \setminus \Omega})|Dv|^2 \ dx\right| &\leq C(1+\lambda(\Omega)^2\textup{diam}(\Omega)^2)|\Omega \Delta B_{N/\lambda(\Omega}| \\
&\leq C(\mu_0)|\Omega \Delta B_{N/\lambda(\Omega}|
\end{aligned}$$ where we have just estimated $|Dv|$ by it’s supremum on the region of integration, which depends on the maximum between $\textup{diam}(\Omega)$ and $N/\lambda(\Omega)$.
Combining the estimates on each of the terms in [(\[e.dirichletsplit\])]{} we end up with the estimate, $$\begin{aligned}
\lambda(\Omega)\left| \left(\frac{N}{\lambda(\Omega)}\right)^N - r_*^N\right| &\leq \left| \left(\frac{N}{\lambda(\Omega)}\right)^{N+1} - \omega_N^{-1}(N+2)\textup{Vol}\right| \\
&\leq C(L_0,\mu_0)(|\Omega|^{\frac{N+1}{2N}}+r_*^{\frac{N+1}{2}})\||Du|^2-1\|_{L^{2}(\partial \Omega)} \\
& \quad \cdots +C(\mu_0)|\Omega \Delta B_{N/\lambda(\Omega}|
\end{aligned}$$ where for the first inequality we have used the elementary inequalities $|x^{n} - 1| \geq |x-1|$ for all $x>0$ and $n \geq1$ and, $$|x^n - a^n| = a^n|(x/a)^n-1| \geq a^n|(x/a)-1| = a^{n-1}|x-a|.$$ Now we can combine the estimate for $|\Omega \Delta B_{N/\lambda(\Omega)}|$ from [(\[e.measureest1\])]{} with the above estimates for $\frac{N}{\lambda(\Omega)} - r_*$, $$\begin{aligned}
|\Omega \Delta B_{r_*}| &\leq |\Omega \Delta B_{N/\lambda(\Omega)}| + |B_{N/\lambda(\Omega)} \Delta B_{r_*}| \\
&\leq C(\mu_0)|\Omega \Delta B_{N/\lambda(\Omega}| + C(L_0,\mu_0)(|\Omega|^{\frac{N+1}{2N}}+r_*^{\frac{N+1}{2}})\||Du|^2-1\|_{L^2(\partial \Omega)} \\
& \leq C(L_0,\mu_0)(\textup{diam}(\Omega)^{\frac{N+1}{2}}+r_*^{\frac{N+1}{2}})\||Du|^2-1\|_{L^2(\partial \Omega)}.
\end{aligned}$$ Dividing on both sides by $r_*^{N}$ yields the desired estimate, $$\frac{|\Omega \Delta B_{r_*}| }{|B_{r_*}|} \leq C\left(L_0,\mu_0,\frac{\textup{diam}(\Omega)}{r_*}\right)\left(\frac{1}{r_*^{N-1}}\int_{\partial \Omega}(|Du|^2-1)^2 \ d \sigma \right)^{1/2}.$$
Now we return to the proof of Lemma \[lem: distance est\].
Let ${\varepsilon}>0$ to be chosen later and call $A_{\varepsilon}$ to be the annulus $B_{(1+{\varepsilon})r}\setminus B_{(1-{\varepsilon})r}$. We may rewrite, $$|{E} \Delta B_r| = |{E} \cap A_{\varepsilon}|+|{E} \setminus B_{(1+{\varepsilon})r}|+|B_{(1-{\varepsilon})r}\setminus {E}|.$$ For the first term we can estimate easily, $$|{E} \cap A_{\varepsilon}|\leq |A_{\varepsilon}| = ((1+{\varepsilon})^N-(1-{\varepsilon})^N)|B_r| \leq C_N{\varepsilon}|B_r| ,$$ as long as we choose ${\varepsilon}\leq 1$. For the second term we use the co-area formula to rewrite, $$|{E} \setminus B_{(1+{\varepsilon})r}| = \int_{(1+{\varepsilon})r}^\infty \mathcal{H}^{N-1}({E} \cap \partial B_s) ds$$ Then using the divergence theorem, $$0 \leq \int_{{E} \setminus B_s} {\nabla}\cdot (\frac{x}{|x|}) \ dx = \int_{\partial {E} \setminus B_s} \frac{x}{|x|} \cdot \nu \ d \sigma(x) - \int_{{E} \cap \partial B_s} d \sigma(x),$$ and so we have, $$\mathcal{H}^{N-1}({E} \cap \partial B_s) \leq \mathcal{H}^{N-1}(\partial {E} \setminus B_s).$$ Now on $\partial {E} \setminus B_s$ we have $1 \leq (r^{-1}s-1)^{-2}(r^{-1}|x|-1)^2$ and therefore, $$\begin{aligned}
|{E} \setminus B_{(1+{\varepsilon})r}| &\leq \int_{(1+{\varepsilon})r}^\infty \mathcal{H}^{N-1}(\partial {E} \setminus B_s) ds \\
&\leq \left(\int_{(1+{\varepsilon})r}^\infty (r^{-1}s-1)^{-2} \ ds\right) \left( \int_{\partial {E}} (r^{-1}|x|-1)^2 \ d\sigma(x)\right).
\end{aligned}$$ Calculating the integral above yields $$\notag
|{E} \setminus B_{(1+{\varepsilon})r}| \leq \frac{r}{{\varepsilon}} \int_{\partial {E}} (r^{-1}|x|-1)_+^2 d\sigma(x).$$ Now we choose ${\varepsilon}$ so that the two terms in the estimate are of the same size, we can choose, $${\varepsilon}^2 = \frac{1}{r^{N-1}}\int_{\partial E} (r^{-1}|x|-1)_+^2 d\sigma(x).$$ If ${\varepsilon}\leq 1$ as chosen then combining the estimates we obtain, $$\label{e.outerbd}
\frac{|E \setminus B_r|}{|B_r|} \leq C(N) \left(\frac{1}{r^{N-1}}\int_{\partial E} \left(\frac{|x|}{r}-1\right)^2 d\sigma(x)\right)^{1/2}$$ otherwise, $$\frac{|E \setminus B_r|}{|B_r|} \leq (1+K) \leq (1+K) {\varepsilon}^{1/2} = (1+K)\left(\frac{1}{r^{N-1}}\int_{\partial E} \left(\frac{|x|}{r}-1\right)^2 d\sigma(x)\right)^{1/2}.$$ Either way the desired result holds.
Next we will obtain, by a similar argument, $$\label{e.innerbd}
\frac{| B_r \setminus E|}{|B_r|} \leq C(N,K)\left(\frac{1}{r^{N-1}}\int_{\partial E} \left(\frac{|x|}{r}-1\right)^2 d\sigma(x)\right)^{1/2}.$$ By the assumption there exists $x_0$ with $B_{r/K}(x_0) \subset \Omega$. If $|x_0| \geq (1-\frac{1}{2K})r$ then, $$\frac{|E \setminus B_r|}{|B_r|} \geq c(N,K) \geq c(N,K)\frac{| B_r \setminus E|}{|B_r|},$$ and [(\[e.innerbd\])]{} follows from [(\[e.outerbd\])]{}. Otherwise we can take $|x_0| \leq (1-\frac{1}{2K})r$. Now let us take $h$ to be the harmonic function, $$\left\{
\begin{array}{l}
-\Delta h = 0 \ \hbox{ in } \ B_r \setminus B_{r/4K}(x_0) \vspace{1.5mm}\\
h = r \ \hbox{ on } \ \partial B_r \ \hbox{ and } \ h = 0 \ \hbox{ on } \ \partial B_{r/4K}(x_0)
\end{array}\right.$$ It follows from Hopf Lemma and the star-shapedness of $B_r$ with respect to $B_{r/4K}(x_0)$ that there is a constant $c(N,K)$ such that, $$D h \cdot (x-x_0) \geq c(N,K) \ \hbox{ on } \ \partial B_r \cup \partial B_{r/4K}(x_0).$$ It is easy to check that $Dh \cdot (x-x_0)$ is harmonic so actually we have, $$|Dh| \geq c(N,K) \ \hbox{ in } \ B_{r} \setminus B_{r/4K}(x_0).$$ A standard barrier argument and the sub-harmonicity of $|Dh|$ shows $|Dh| \leq C(N,K)$. We use the divergence theorem, using again that $\frac{1}{4K}B_r (x_0)\subset B_r \subset E$, for all $ 0 < s < r$, $$-\int_{\partial {E}\cap \{ h < s\} } Dh \cdot \nu_E \ d \sigma(x) + \int_{{E}^C \cap \partial \{ h <s\}} |Dh|d \sigma(x) = \int_{ \{ h <s \} \setminus {E}} \Delta h \ dx = 0 .$$ Thus we obtain, using the bounds on $|Dh|$, $$\mathcal{H}^{N-1}({E}^C \cap \{ h =s\}) \leq C(N,K)\mathcal{H}^{N-1}(\partial {E}\cap \{ h < s\}).$$ Let ${\varepsilon}>0$, to be chosen, we use the co-area formula with the level set function $h$, $$\begin{aligned}
|B_{(1-{\varepsilon})r} \setminus E| & \leq C(N,K)\int_{B_{(1-{\varepsilon})r}} {{\bf 1}}_{E^C}|Dh|dx\\
&\leq C\int_{0}^{(1-c{\varepsilon})r} \mathcal{H}^{N-1}({E}^C \cap \{ h = s\}) ds \\
&\leq C\int_{0}^{(1-c{\varepsilon})r} \mathcal{H}^{N-1}(\partial {E}\cap \{ h < s\}) ds \\
&\leq C \left(\int_{0}^{(1-c{\varepsilon})r} (r^{-1}s-1)^{-2} \ ds\right) \left( \int_{\partial {E}} (r^{-1}|x|-1)^2 \ d\sigma(x)\right) \\
& \leq C\frac{r}{{\varepsilon}} \int_{\partial {E}} (r^{-1}|x|-1)_+^2 d\sigma(x).
\end{aligned}$$ The rest of the proof is the same as the argument for [(\[e.outerbd\])]{} above, choosing as before ${\varepsilon}^2 =\frac{1}{r^{N-1}}\int_{\partial E} (r^{-1}|x|-1)_+^2 d\sigma(x)$.
Exponential convergence to equilibrium conditional on regularity {#sec: exponential rate}
================================================================
In this final section we discuss the application of our quantitative stability result to the long time behavior of the contact angle motion problem [(\[e.dropletprob\])]{}. We recall the problem, $$\label{e.dropletprob2}
\left\{
\begin{array}{lll}
-\Delta u(x,t) = \lambda(t) & \hbox{ in } & \Omega_t(u) = \{u(\cdot,t)>0\} \vspace{1.5mm}\\
\tfrac{\partial_t u}{|Du|} = |Du|^2-1 & \hbox{ on } & \Gamma_t(u) = \partial \Omega_t(u),
\end{array}\right.$$ where $\lambda(t)$ is a Lagrange multiplier enforcing the volume constraint, $$\int u(\cdot,t) \ dx = \text{Vol} \ \hbox{ for all } \ t>0.$$ We compute the energy decay estimate which was stated in the introduction, $$\begin{aligned}
\frac{d}{dt}\mathcal{J}(\Omega_t) &= \int_{\Omega_t} 2Du\cdot Du_t+ \int_{\Gamma_t} (|Du|^2+1)(|Du|^2-1) \\
&= -2\lambda(t)\int_{\Omega_t} u_t +\int_{\Gamma_t} 2u_t Du\cdot n+(|Du|^2+1)(|Du|^2-1) \\
&= -2\lambda(t) \frac{d}{dt}\left(\int_{\Omega_t} u\right)+\int_{\Gamma_t} -2 (|Du|^2-1)|Du|^2+(|Du|^2+1)(|Du|^2-1) \\
&= -\int_{\Gamma_t} (|Du|^2-1)^2 \leq 0.\end{aligned}$$ Thus we have obtained, $$\label{e.energyest}
\frac{d}{dt}(\mathcal{J}(\Omega_t) - \mathcal{J}(B_{r_*}))= -\int_{\Gamma_t} (|Du|^2-1)^2,$$ or in integrated form, $$\label{e.intform}
\mathcal{J}(\Omega_t) - \mathcal{J}(B_{r_*})= \mathcal{J}(\Omega_0) - \mathcal{J}(B_{r_*}) - \int_0^t\int_{\Gamma_s} (|Du|^2-1)^2 \ d\sigma ds.$$ The problem with using this estimate directly, at least in $N \geq 3$, is that the stability result Theorem \[thm: main stability\] controls the measure difference squared $|\Omega_t \Delta B_{r_*}|^2$ by the energy dissipation and it is not clear whether $|\Omega_t \Delta B_{r_*}|^2$ controls the energy gap. Due to this issue we take a different approach, applying the Grönwall argument directly to $|\Omega_t \Delta B_{r_*}|^2$.
For this we need the optimal quantitative Faber-Krahn inequality proven recently by Brasco, De Philippis and Velichkov [@Brasco:2015aa]. Basically this inequality gives the sharp lower bound quadratic growth of the energy $\mathcal{J}(\Omega)$ near it’s minimum in terms of the $L^1$ distance.
\[thm: faber-krahn\] There exists a positive constant $c_N$ depending only on dimension such that for every open set $\Omega \subset {\mathbb{R}}^N$ with finite measure and any ball $B$, $$|\Omega|^{\frac{2}{N}+1}\lambda(\Omega) - |B|^{\frac{2}{N}+1}\lambda(B) \geq c_N \textup{Vol}^2 \mathcal{A}(\Omega)^2,$$ where $\mathcal{A}(\Omega)$ is the asymmetry, the infimum over all balls $B \subset {\mathbb{R}}^N$ of $|\Omega \Delta B|/|B|$.
From this Theorem we can easily derive a stability estimate of the capillary energy.
\[cor: fk cor\] For every open set $\Omega \subset {\mathbb{R}}^N$ with finite measure and $B_r*$ the ball with minimal energy for $\mathcal{J}$, $$\label{e.fkcor}
(\mathcal{J}(\Omega) - \mathcal{J}(B_{r_*}))^{1/2} \geq c\left(N,\frac{r_*}{|\Omega|^{\frac{1}{N}}}\right)\mathcal{J}(B_{r_*})^{1/2}\frac{|\Omega \Delta B_{r_*}|}{|B_{r_*}|}.$$
We postpone the proof of the Corollary till the end of the section. The argument to show that $B_{r_*}$ minimizes the energy $\mathcal{J}$ over all open sets $\Omega$ with finite measure goes as follows. First let $B_r$ be the ball with volume $|B_r| = |\Omega|$. Then the Polya-Szegö principle implies that the Schwarz symmetrization of $u_\Omega$ has the same volume but lower Dirichlet energy than $u_\Omega$ and so $$\mathcal{J}(B_r) \leq \mathcal{J}(\Omega).$$ Explicit computation of the radially symmetric solutions, see Appendix \[sec: computations\], shows that $B_{r_*}$ has the minimal energy among all balls. For these two steps we have separate stability estimates, respectively, the Theorem of [@Brasco:2015aa] copied above, and the calculus computation in Appendix \[sec: computations\].
We make several smaller comments about Theorem \[thm: dynamic stability\] before we go to the proof.
We expect that the convergence modulo translation can be upgraded to convergence to a unique ball $B_{r_*}(x_*)$ using the ideas in [@FKdrops] (Proposition $5.2$) with some extra work.
In $N=2$ it should be possible to get the stability estimate Theorem \[thm: main stability\] in Hausdorff distance. Then one can show a quadratic upper bound on the energy growth near the minimum (in Hausdorff distance) and apply a Grönwall argument directly to the energy. To go from the exponential convergence of the energy to convergence in measure or in Hausdorff distance one would still need a stability estimate for the Faber-Krahn inequality, although the optimal scaling is not necessary in that case.
Now we prove Theorem \[thm: dynamic stability\], it is very simple given the set up.
We define, $${\varepsilon}(t)^{1/2} =\inf_{x \in {\mathbb{R}}^N} \frac{|\Omega \Delta B_{r_*}(x)|}{|B_{r_*}|}$$ Instead of trying to use the energy dissipation estimate [(\[e.intform\])]{} directly, we apply the quantitive Faber-Krahn inequality [(\[e.fkcor\])]{} in combination with our stability result Theorem \[thm: main stability\] to obtain, $$\label{e.intform2}
r_*^N{\varepsilon}(t) \leq C(\mathcal{J}(\Omega_0) - \mathcal{J}(B_{r_*})) - cr_*^{N-1}\int_0^t {\varepsilon}(t) ds,$$ and therefore, $${\varepsilon}(t) \leq Cr_*^{-N}(\mathcal{J}(\Omega_0) - \mathcal{J}(B_{r_*}))e^{-cr_*^{-1}t}.$$
We carry out the argument described above using the stability estimates, $$\mathcal{J}(\Omega) - \mathcal{J}(B_r) = \lambda(\Omega) - \lambda(B_r) \geq c_N\textup{Vol}^2r^{-3N-2}|\Omega \Delta B_r|$$ by Theorem \[thm: faber-krahn\]. By the explicit computation in Appendix \[sec: computations\] we have, $$\mathcal{J}(B_r) - \mathcal{J}(B_{r_*}) \geq c_N r^{N-2} |r-r_*|^2 \geq c_N r^{-N} |r^N-r_*^N|^2.$$ Thus, $$\begin{aligned}
\mathcal{J}(\Omega) - \mathcal{J}(B_{r_*}) &\geq c_N(\textup{Vol}^2r^{-(3N+2)}+r^{-N})|\Omega \Delta B_{r_*}|^2 \\
&= c_N(r_*^{2N}\textup{Vol}^2r^{-(3N+2)}+r_*^{2N}r^{-N})\left(\frac{|\Omega \Delta B_{r_*}|}{|B_{r_*}|}\right)^2 \\
& \geq c_Nf\left(\frac{r_*}{|\Omega|^{1/N}}\right)\mathcal{J}(B_{r_*})\left(\frac{|\Omega \Delta B_{r_*}|}{|B_{r_*}|}\right)^2
\end{aligned}$$ where the function $f(s) = s^{3N+2}+s^N$.
Conditions for regularity {#sec: conditions}
-------------------------
Now we make more precise a set of conditions on the initial data under which the regularity assumed in Theorem \[thm: dynamic stability\] is expected to be true. First we recall the geometric condition introduced in [@FKdrops].
A domain $\Omega$ is said to have the *$\rho$-reflection* property if $B_\rho(0) \subset \Omega$ and for every half space $H \subset {\mathbb{R}}^N$ which does not intersect $B_\rho(0)$ and the corresponding reflection operator $R$, $$\Omega \cap H \subset R(\Omega) \cap H.$$
The property of $\rho$-reflection is preserved by the flow due to the comparison principle, note that $u(Rx)$ has the same associated Lagrange multiplier as $u$. This actually requires a bit of work to prove since the comparison required does not have strict ordering at the initial time.
The $\rho$-reflection property is in a sense a quantified version of the moving planes method. If $\Omega$ has $\rho$-reflection with $\rho=0$ then $\Omega$ is a ball around $0$. Sets with $\rho$-reflection satisfy also a strong star-shapedness property as long as $\partial \Omega$ stays away from $B_\rho$. Precisely, $$\sup_{x \in \partial \Omega} |x| - \inf_{x \in \partial \Omega} |x| \leq 4\rho$$ and $$\hbox{ $\Omega$ is star-shaped with respect to $B_r(0)$ with } \ r = (\inf_{x \in \partial \Omega} |x|^2 - \rho^2)^{1/2},$$ see [@FKdrops] Lemmas $3.23$ and $3.24$. This is a quantified Lipschitz condition on $\partial \Omega$, and the local Lipschitz constant can be made arbitrarily small if $\rho$ is small.
The $\rho$-reflection property was used to establish the long time existence of viscosity solutions to [(\[e.dropletprob\])]{} in [@FKdrops].
Suppose that $\Omega_0$ has the $\rho$-reflection property for some $0 \leq \rho < \frac{1}{10}\textup{Vol}^{\frac{1}{N+1}}$, then there exists a global in time viscosity solution of [(\[e.dropletprob\])]{} which has the $\rho$-reflection property for all $t>0$.
Any initial data satisfying the assumptions of the above result will preserve the John domain property assumption globally in time, it is an easy consequence of strong star-shapedness. We remark that with this regularity property the exponential rate for convergence in measure from Theorem \[thm: dynamic stability\] can be upgraded to convergence in Hausdorff distance and convergence of the energy.
The next component is the local regularity from assumptions and . In analogy to the results of Choi, Jerison and Kim [@Choi:2009aa; @CJK] on the Hele-Shaw flow we expect that initial data with small Lipschitz constant will be smooth at positive times. Thus for initial data $\Omega_0$ with $\rho$-reflection $\rho>0$ sufficiently small depending on the dimension, we would expect the free boundary to be globally $C^{1,\alpha}$ in $x,t$.
Computations {#sec: computations}
============
We record here several useful computations related to the minimal energy shape. For a ball of radius $r$ the droplet height profile is given by, $$u(x) = \tfrac{\lambda(B_r)}{2N}(r^2 - |x|^2).$$ To enforce the volume constraint we require $$\begin{aligned}
\textup{Vol} &= \int_{B_r}\tfrac{\lambda(B_r)}{2N}(r^2 - |x|^2) dx \\
& = \tfrac{\lambda(B_r)}{2N}\omega_N(1-\tfrac{N}{N+2})r^{N+2} \\
& = \tfrac{\lambda(B_r)}{N(N+2)}\omega_Nr^{N+2}.\end{aligned}$$ This determines the Lagrange multiplier, $$\lambda(B_r) = \frac{N(N+2)}{\omega_Nr^{N+2}}\textup{Vol}.$$ From this we can compute, $$\frac{d^2}{dr^2} \mathcal{J}(B_r) = \frac{d^2}{dr^2}(\lambda(B_r)\textup{Vol} + |B_r|) = \frac{N(N+2)^2(N+3)}{\omega_N}\frac{\textup{Vol}}{r^{N+4}} + N(N-1)\omega_N r^{N-2},$$ From which we have, $$\frac{d^2}{dr^2} \mathcal{J}(B_r) \geq N(N-1)\omega_N r^{N-2}$$ which we use for the stability estimate.
We continue with computing $r_*$. In order that $|Du| = 1$ on $\partial B_{r_*}$ we must have, $$r_* = N/\lambda(B_{r_*}) = \frac{\omega_Nr_*^{N+2}}{(N+2)}\textup{Vol}^{-1},$$ which determines the optimal radius, $$r_*^{N+1} = \omega_N^{-1}(N+2)\textup{Vol}$$ Also we see, $$\lambda(B_{r_*}) = N\left(\tfrac{\omega_N}{N+2}\right)^{\frac{1}{N+1}}\textup{Vol}^{-\frac{1}{N+1}}.$$ From here we can calculate the minimal energy, $$\begin{aligned}
\mathcal{J}(B_{r_*}) &= \lambda(B_{r_*})\textup{Vol}+\omega_Nr_*^N \\
&= N\left(\tfrac{\omega_N}{N+2}\right)^{\frac{1}{N+1}}\textup{Vol}^{\frac{N}{N+1}}+ \omega_N\left(\tfrac{N+2}{\omega_N}\right)^{\frac{N}{N+1}}\textup{Vol}^{\frac{N}{N+1}} \\
& = \omega_N^{\frac{1}{N+1}}(N+2)^{-\frac{1}{N+1}}\left(2N+1\right)\textup{Vol}^{\frac{N}{N+1}}
\end{aligned}$$
[^1]: W. M. Feldman partially supported by NSF-RTG grant DMS-1246999.
|
---
author:
- 'Martin Ratajczak, Sebastian Tschiatschek, and Franz Pernkopf, [^1]'
bibliography:
- 'IEEEabrv.bib'
- 'icml2014.bib'
title: |
Sum-Product Networks\
for Sequence Labeling
---
[^1]: This work was supported by the Austrian Science Fund (FWF) under the project number P25244-N15 and P27803-N15. Furthermore, we acknowledge NVIDIA for providing GPU computing resources.
|
**Lie symmetries of generalized Burgers equations:\
application to boundary-value problems**
O.O. Vaneeva$^{\dag 1}$, C. Sophocleous$^{\ddag 2}$ and P.G.L. Leach$^{\ddag\S 3}$
*${}^\dag$ Institute of Mathematics of National Academy of Sciences of Ukraine,\
$\phantom{{}^\dag}$ 3 Tereshchenkivska Str., Kyiv-4, 01601 Ukraine\
${}^\ddag$ Department of Mathematics and Statistics, University of Cyprus, Nicosia CY 1678, Cyprus\
${}^\S$ School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X54001,\
$\phantom{{}^\S}$ Durban 4000, South Africa*
[**Keywords:**]{} Group classification; Burgers Equation; Lie symmetries; Boundary Value Problems\
[**MSC 2010:**]{} 35Q35, 34A34, 22E70
Introduction
============
=-1 The Burgers equation, $
u_t + uu_x + u_{xx} = 0,
$ is one of the simplest nonlinear $(1 + 1)$ evolution equations. Nevertheless it has a long history as it was already known to Forsyth [@Forsyth06] and discussed by Bateman not many years later [@Bateman15]. However, it was a serious contribution made by Burgers which led to its present name [@Burgers48]. Burgers equation has been used to describe many processes in fluid mechanics and a variety of other fields which do seem to be rather disparate. It’s remarkable feature is that it can be transformed to the standard heat equation by means of the Hopf–Cole transformation [@Hopf50; @Cole51].
In 1989 Hammerton and Crighton [@HammertonCrighton1989] derived the generalized Burgers equation describing the propagation of weakly nonlinear acoustic waves under the influence of geometrical spreading and thermoviscous diffusion, which in non-dimensional variables can be reduced to the form $$u_t+uu_x=g(t)u_{xx}.$$
In this paper we investigate Lie symmetries of a generalized Burgers equation of the form $$\label{Eq_GenBurgers}
u_t+a(u^n)_x=g(t)u_{xx},$$ where $a$ is a nonzero constant, $g$ is an arbitrary smooth nonvanishing function and $n\neq0,1$. The enhanced group classification for this class is achieved and comes to complete the results that exist in the literature . This generalization of Burgers equation was recently treated by Abd-el-Malek and Helal who used what can only be described as a very ingenious method to obtain the solution. Here we present an alternate approach to solving the equation with associated boundary conditions and we are of the opinion that this approach is easier to implement and somewhat more transparent.
The paper is organized as follows. In Section 2 we determine the equivalence group. In Section 3 we calculate the Lie point symmetries of (\[Eq\_GenBurgers\]) and in Section 4 we demonstrate the solution of the boundary-value problem.
Equivalence transformations
===========================
The first step of solving a group classification problem is to find the equivalence group admitted by a given class [@Ovsiannikov1982]. Notions of different kinds of equivalence group can be found, e.g., in [@vane2012b]. We were able to study all admissible (form-preserving ) transformations in class , in other words we described all point transformations that link equations from the class. The results of the study are given in the following statements.
The usual equivalence group $G^{\sim}$ of class comprises the transformations $$\begin{array}{l}
\tilde t=\delta_1t+\delta_2,\quad \tilde x=\delta_3x+\delta_4,\quad
\tilde u=\delta_5u, \quad
\tilde a=\dfrac{\delta_3}{\delta_1}\delta_5^{1-n}a, \quad
\tilde g=\dfrac{{\delta_3}^2}{\delta_1} g,\quad \tilde n=n,
\end{array}$$ where $\delta_j,$ $j=1,\dots,5$, are arbitrary constants with $\delta_1\delta_3\delta_5\not=0$.
It appears that for $n=2$ class admits a nontrivial conditional equivalence group which is wider than $G^{\sim}$.
The generalized equivalence group $\hat
G^{\sim}_{2}$ of the class, $$\label{Eq_GenBurgers_n2}
u_t+a(u^2)_x=g(t)u_{xx},$$ comprises the transformations $$\begin{array}{l}
\tilde t=\dfrac{\alpha t+\beta}{\gamma t+\delta},\quad \tilde x=\dfrac{\kappa x +\mu_1t+\mu_0}{\gamma t+\delta},\quad
\tilde u=\dfrac{\sigma}{2a(\alpha\delta-\beta\gamma)}\left(2a\kappa(\gamma t+\delta)u-\kappa\gamma x+\mu_1\delta-\mu_0\gamma\right), \\[2ex]
\tilde a=\dfrac{a}{\sigma} \quad \mbox{\rm and} \quad
\tilde g=\dfrac{\kappa^2}{\alpha\delta-\beta\gamma} g,
\end{array}$$ where $\alpha, \beta, \gamma, \delta, \kappa, \mu_1, \mu_0, \sigma$ are constants defined up to a nonzero multiplier, $\alpha\delta-\beta\gamma\neq0$ and $\kappa\sigma\not=0$.
Let the equations $u_t+a(u^n)_x =g(t)u_{xx}$ and $\tilde u_{\tilde t}+\tilde a(\tilde u^{\tilde n})_{\tilde x} =\tilde g(\tilde t)\tilde u_{\tilde x\tilde x}$ be connected by a point transformation $\mathcal{T}$ in the variables $t$, $x$ and $u$. Then $\tilde n=n$, and the transformation $\mathcal{T}$ is the projection of a transformation from $G^{\sim}$ or $\hat G^{\sim}_{2}$ on the space $(t,x,u)$, if $n\not=2$ or $n=2$, respectively.
Suppose that an equation from class is connected with an equation $$\label{Eq_GenBurgers_tilde}
{\tilde u}_{\tilde t}+\tilde a(\tilde u^{\tilde n})_{\tilde x}=\tilde g(\tilde t){\tilde u}_{\tilde x\tilde x}$$ from the same class by a point transformation $
\tilde t=T(t,x,u),$ $\tilde x=X(t,x,u),$ $\tilde u=U(t,x,u),
$ where $|\partial(T,X,U)/\partial(t,x,u)|\ne0$. It is known that for evolution equations we have the restrictions, $T_x=T_u=0$, on the general form of admissible transformations and moreover for equations of the form $u_t=F(t,x,u)u_{xx}+G(t,x,u,u_x)$ we necessarily have the condition $X_u=0$ . Therefore it is enough to consider a transformation, $\mathcal T$, of the form $$\tilde t=T(t), \quad \tilde x=X(t,x), \quad \tilde u=U(t,x,u),$$ where $T_tX_xU_u\ne0$. After we change the variables in , we obtain an equation in the variables without tildes. It should be an identity on the manifold $\mathcal L$ determined by in the second-order jet space $J^2$ with the independent variables $(t,x)$ and the dependent variable $u$. To involve the constraint between variables of $J^2$ on the manifold $\mathcal L$, we substitute the expression of $u_t$ implied by equation . The splitting of this identity with respect to the derivatives $u_{xx}$ and $u_x$ results in the determining equations for the functions $T$, $X$ and $U$ $$\begin{gathered}
\label{1}
U_{uu}=0,\qquad
\tilde g T_t -g X_x^2=0,\\\label{3}
X_t U_x-X_x U_t+\tilde g T_t\left(\frac{U_x}{X_x}\right)_x-\tilde a\tilde n T_t U^{\tilde n-1}U_x=0 \quad \mbox{\rm and} \\\label{4}
anu^{n-1}-\tilde a\tilde n\frac{T_t}{X_x}U^{\tilde n-1}+2\tilde g\frac{T_t}{X_x^2}\frac{U_{xu}}{U_u}-\tilde g T_t\frac{X_{xx}}{X_x^3}+\frac{X_t}{X_x}=0.\end{gathered}$$ Equations imply that $$U=\eta^1(t,x)u+\eta^0(t,x),\quad X=\varphi(t)x+\psi(t) \quad\mbox{and}\quad\tilde g=\frac{\varphi^2}{T_t}g.$$ Here the functions $\eta^i(t,x)$, $i=1,2$, $\varphi(t)$, and $\psi(t)$ are arbitrary smooth functions of their arguments and $\eta^1\varphi\neq0.$
When we use the differential consequences of the fourth equation with respect to $u$, we get that the arbitrary element $n$ is invariant under the action of a point transformation, i.e., $\tilde n =n.$ Also we obtain that $\eta^0=0$ $\forall$ $ n\neq2.$ After we substitute the expressions for $U$, $X$ and $\tilde g$ into the third and fourth determining equations, we can split them with respect to $u$. Further consideration varies depending upon whether $n\neq2$ or $n=2$.
[**I**]{}. If $n\neq2$, then splitting of and results in the equations $$\begin{gathered}
\eta^1_x=0, \quad \eta^1(\varphi_tx+\psi_t)+2\varphi g\eta^1_x=0,\\
\varphi\eta^1_t=\eta^1_x(\varphi_tx+\psi_t)+\varphi g\eta^1_{xx}, \quad \mbox{\rm and} \\
\varphi a \eta^1=\tilde a (\eta^1)^nT_t.\end{gathered}$$ The general solution of this system is given by $$T=\delta_1t+\delta_2,\quad\varphi=\delta_3,\quad\psi=\delta_4,\quad \eta^1=\delta_5,$$ where $\delta_j,$ $j=1,\dots,5,$ are arbitrary constants with $\delta_1\delta_3\delta_5\neq0.$ Then $\tilde a=\dfrac{\delta_3}{\delta_1}\delta_5^{1-n}a $ and $\tilde g=\dfrac{{\delta_3}^2}{\delta_1} g.$
The statement of Theorem 1 is proved.
[**II**]{}. If $n=2$, then splitting of equations and leads to the system $$\begin{gathered}
\eta^1_x=0, \quad \eta^1(\varphi_tx+\psi_t)+2\varphi g\eta^1_x-2\tilde a T_t\eta^0\eta^1=0,\\
\varphi\eta^1_t=\eta^1_x(\varphi_tx+\psi_t)+\varphi g\eta^1_{xx}-2\tilde a T_t(\eta^0\eta^1)_x, \\
\varphi\eta^0_t=\eta^0_x(\varphi_tx+\psi_t)+\varphi g\eta^0_{xx}-2\tilde a T_t\eta^0\eta^0_x, \quad \mbox{\rm and} \\
\varphi a =\tilde a \eta^1T_t.\end{gathered}$$From this system we initially obtain forms of $\eta^1 $ and $\eta^0$ as $$\eta^1=\sigma\frac{\varphi}{T_t},\quad\eta^0=\frac{\sigma}{2aT_t}(\phi_tx+\psi_t),\quad \tilde a=\frac{a}{\sigma},$$ where $\sigma$ is a nonzero constant. The remaining equations for the functions $T,$ $\varphi$ and $\psi$ are $$\left(\frac{\varphi^2}{T_t}\right)_t=0,\quad\left(\frac{\varphi_t}{T_t}\right)_t=0,\quad\left(\frac{\psi_t}{T_t}\right)_t=0,$$ the general solution of which can be written as $$T=\frac{\alpha t+\beta}{\gamma t+\delta},\quad \varphi=\frac{\kappa}{\gamma t+\delta},\quad \psi=\frac{\mu_1 t+\mu_0}{\gamma t+\delta},$$ where $\alpha$, $ \beta$, $\gamma$, $\delta$, $\kappa$, $\mu_1$ and $ \mu_0$ are constants defined up to a nonzero multiplier, $\alpha\delta-\beta\gamma\neq0$ and $\kappa\not=0$. When we substitute the functions $T,$ $\varphi$ and $\psi$ into the formulas for $\eta^1 $ and $\eta^0$, we get exactly the statement of Theorem 2. The group $\hat G^{\sim}_2$ is called generalized since transformation component for $u$ depends on arbitrary element $a$ of the class.
We have found that all admissible transformations in class are exhausted by those presented in Theorems 1 and 2. Therefore, Theorem 3 is proved.
Note that if $a=1/n$ the group $\hat G^\sim_2$ was found previously in (see also ) in the course of study of form-preserving (admissible) transformations for the class of generalized Burgers equations, $u_t+uu_x+f(t,x)u_{xx}=0.$
Lie symmetries
==============
We perform the group classification of class within the framework of the classical Lie approach [@Olver1986; @Ovsiannikov1982]. Naturally one performs the group classification for class up to $G^\sim$-equivalence and for its subclass up to $\hat G^\sim_2$-equivalence.
We search for operators of the form $\Gamma=\tau(t,x,u)\partial_t+\xi(t,x,u)\partial_x+\theta(t,x,u)\partial_u$ which generate one-parameter groups of point-symmetry transformations of an equation from class . Any such vector field, $\Gamma$, satisfies the infinitesimal invariance criterion, i.e., the action of the second prolongation, $\Gamma^{(2)}$, of the operator $\Gamma$ on equation results in the conditions being an identity for all solutions of this equation. Namely, we require that $$\label{c1a}
\Gamma^{(2)}\{u_t+anu^{n-1}u_x-g(t)u_{xx}\}=0$$ identically, modulo equation .
The criterion of infinitesimal invariance implies that $$\tau=\tau(t),\quad
\xi=\xi(t,x), \quad
\theta=\theta^1(t,x)u+\theta^0(t,x),$$ where $\tau$, $\xi$, $\theta^1$ and $\theta^0$ are arbitrary smooth functions of their variables. The remaining determining equations have the form $$\begin{gathered}
\label{deteq1}
2g\xi_x=(g\tau)_t,\\\label{deteq2}
an\theta^1_xu^{n+1}+an\theta^0_xu^{n}+(\theta^1_t-g\theta^1_{xx})u^2+(\theta^0_t-g\theta^0_{xx})u=0,\\\label{deteq3}
an(\tau_t-\xi_x+(n-1)\theta^1)u^{n+1}+an(n-1)\theta^0u^n+(g\xi_{xx}-2g\theta^1_x-\xi_t)u^2=0.\end{gathered}$$ It is easy to see from that $\xi_{xx}=0.$ The second and the third equations can be split with respect to different powers of $u$. Special cases of splitting arise if $n=0,1,2$. If $n=0$ or $n=1$, equations are linear and are excluded from consideration (Lie symmetries of second-order linear differential equations in two dimensions were studied over the century ago by S. Lie [@Lie1881].) Therefore we investigate two cases, $n\neq2$ and $n=2$, separately.
[**I.**]{} If $n\neq2$, then we immediately get that $\theta^1=c_0,$ where $c_0$ is an arbitrary constant, $\theta^0=0$. $$\tau=c_1t+c_2,\quad \xi=(c_1+(n-1)c_0)x+c_3,\quad \theta=c_0u.$$ The classifying equation on $g$ has the form $$\label{c1}
(c_1t+c_2)g_t=(c_1+2(n-1)c_0)g.$$ Further consideration is performed using [*the method of furcate split*]{} suggested in . For any operator $\Gamma$ from maximal Lie invariance algebra $A^{\max}$ equation gives some equations on $g$ of the general form $$(p\,t+q)g_t=sg,$$ where $p,q,s={\rm const}$. In general for all operators from $A^{\max}$ the number $k$ of such independent equations is no greater than 2 otherwise they form an incompatible system on $g$. There exist three inequivalent cases for the value of $k$ given by $k=0,$ $k=1$ and $k=2$.
If $k=0,$ then is identically zero and $c_1=c_2=c_0=0$. So, if $g$ is arbitrary, we obtain that the kernel of maximal Lie invariance algebras of equations from is the one-dimensional algebra $\langle\partial_x\rangle$.
If $k=1,$ then $g\in\{\varepsilon e^t,\varepsilon\, t^{\rho}\}\!\!\mod G^{\sim}$, where $\varepsilon=\pm1$ and $\rho\ne0$. In the exponential case, $g=\varepsilon e^t$, we have $$\Gamma=2(n-1)c_0\partial_t+((n-1)c_0x+c_3)\partial_x+c_0u\partial_u.$$ If $g=\varepsilon\, t^\rho$ and $\rho\neq0,$ then $$\Gamma=c_1t\partial_t+\left(\tfrac12(\rho+1)c_1x+c_3\right)\partial_x+ \tfrac12\displaystyle{\tfrac{\rho-1}{n-1}}c_1u\partial_u.$$ In both these cases the maximal Lie-invariance algebras are two-dimensional with basis operators presented in Cases 2 and 3 of Table 1.
If $k=2$, $g=1\bmod G^\sim.$ The infinitesimal operator takes the form $$\Gamma=(c_1t+c_2)\partial_t+\left(\tfrac12c_1x+c_3\right)\partial_x- \displaystyle{\tfrac1{2(n-1)}}c_1u\partial_u.$$ Therefore we have proven that, if $g$ is a constant, the maximal Lie invariance algebra of with $n\neq2$ is three-dimensional spanned by operators presented in Case 4 of Table 1.
[**II.**]{} If $n=2$, then splitting of and results in the system $$\begin{gathered}
\theta^1_x=0,\quad 2a\theta^0_x+\theta^1_t=0,\quad \theta^0_t-g\theta^0_{xx}=0,\\
\tau_t-\xi_x+\theta^1=0,\quad 2a\theta^0-\xi_t=0.\end{gathered}$$ The general solution of this system is $$\begin{gathered}
\tau=c_2t^2+c_1t+c_0,\quad \xi=\left(c_2t+\frac12c_1+c_5\right)x+2ac_3t+c_4,\\
\theta^1=-c_2t-\frac12c_1+c_5,\quad \theta^0=\frac1{2a}c_2x+c_3,\end{gathered}$$ where $c_i$, $i=0,\dots,5,$ are arbitrary constants. The classifying equation takes the form $$\label{c2}
(c_2t^2+c_1t+c_0)g_t=2c_5\,g.$$ For any operator $\Gamma$ from the maximal Lie invariance algebra $A^{\max}$ equation gives some equations for $g$ of the general form $$\label{Eqn2clas}
(p\,t^2+q\,t+r)g_t=s\,g,$$ where $p,q,r,s={\rm const}$. As in the previous case the number, $k$, of such independent equations is no greater than 2 otherwise they form an incompatible system on $g(t)$. So three inequivalent cases for the value of $k$ should be considered, namely $k=0,$ $k=1$ and $k=2.$
If $k=0,$ then is identically zero and $c_0=c_1=c_2=c_5=0$. So, if $g(t)$ is arbitrary, we obtain that the kernel of the maximal Lie invariance algebras of equations from is the two-dimensional algebra $\langle\partial_x,\,2at\partial_x+\partial_u\rangle$.
If $k=1$, the following statement is true.
\[LemmaOntransOfCoeffsOfClassifyingSystem2\] Up to $\hat G^{\sim}_{2}$-equivalence the parameter quadruple $(p,q,r,s)$ can be assumed to belong to the set $$\{(0,1,0,\bar s),\ (0,0,1,1),\ (1,0,1,s')\},$$ where $\bar s$, $s'$ are nonzero constants, $\bar s>0$.
**Table .** Group classification of the class $u_t+a(u^n)_x=g(t)u_{xx}$, $n\neq0,1$.\
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
no. $n$ $g$ Basis operators of $A^{\rm max}$
----- --------- ---------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------
1 $\neq2$ $\forall$ $\partial_x$
2 $\neq2$ $\varepsilon t^\rho$ $\partial_x,\quad
2t\partial_t+(\rho+1)x\partial_x+ \displaystyle{\frac{\rho-1}{n-1}}u\partial_u$
3 $\neq2$ $\varepsilon e^t$ $\partial_x,\quad 2\partial_t+x\partial_x+\frac1{n-1}u\partial_u$
4 $\neq2$ $1$ $\partial_x,\quad\partial_t,\quad
2t\partial_t+x\partial_x-\frac1{n-1}u\partial_u$
5 $2$ $\forall$ $\partial_x,\quad t\partial_x+\partial_u$
6 $2$ $\varepsilon t^\rho$ $\partial_x,\quad t\partial_x+\partial_u,\quad
2t\partial_t+(\rho+1)x\partial_x+(\rho-1)u\partial_u$
7 $2$ $\varepsilon e^t$ $\partial_x,\quad t\partial_x+\partial_u,\quad 2\partial_t+x\partial_x+u\partial_u$
8 $2$ $\varepsilon e^{2\rho\arctan t}$ $\partial_x,\quad t\partial_x+\partial_u,\quad(t^2+1)\partial_t+(t+\rho)x\partial_x+(x+(\rho-t)u)\partial_u$
9 $2$ $1$ $\partial_x,\quad t\partial_x+\partial_u,\quad\partial_t,\quad 2t\partial_t+x\partial_x-u\partial_u, \quad t^2\partial_t+tx\partial_x+(x-tu)\partial_u$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Here $\varepsilon=\pm1\bmod G^\sim$ and $\rho$ is a nonzero constant. In all cases $a=1/n\bmod G^\sim$. In Case 6 we can set, $\bmod\hat G^\sim_{2}$, either $\rho>0$ or $\rho<0$.
Combined with multiplication by a nonzero constant, each transformation from the equivalence group $\hat G^{\sim}_{2}$ can be extended to the coefficient quadruple of equation as $$\begin{gathered}
\begin{array}{l}
\tilde p=\nu(p\delta^2-q\gamma\delta+r\gamma^2),\quad
\tilde q=\nu(-2p\beta\delta+q(\alpha\delta+\beta\gamma)-2r\alpha\gamma),
\\[1ex]
\tilde r=\nu(p\beta^2-q\alpha\beta+r\alpha^2),
\quad\tilde s=\nu s\Delta.
\end{array}\end{gathered}$$ Here $\Delta=\beta\gamma-\alpha\delta$ and $\nu$ is an arbitrary nonzero constant.
There are only three $\hat G^\sim_2$-inequivalent values of the triple $(p,q,r)$ depending upon the sign of $D=q^2-4pr$, $$\begin{gathered}
(0,1,0)\quad\mbox{if}\quad D>0, \quad
(0,0,1)\quad\mbox{if}\quad D=0 \quad \mbox{\rm and} \quad
(1,0,1)\quad\mbox{if}\quad D<0.\end{gathered}$$ Indeed, if $D>0$, then there exist two linearly independent pairs $(\delta,\gamma)$ and $(\alpha,\beta)$ such that $p\delta^2-q\gamma\delta+r\gamma^2=0$ and $p\beta^2-q\alpha\beta+r\alpha^2=0$. For these values of the constants, $\alpha$, $\beta$, $\gamma$ and $\delta$, we have $\tilde p=\tilde r=0$. The coefficient $\tilde q$ is necessarily nonzero and it can be scaled to $1$ using multiplication by an appropriate value of $\nu$. Certain freedom in varying group parameters is preserved even after fixing of the form of the triple $(p,q,r)$ and $(\tilde p,\tilde q,\tilde r)$. This allows us to set constraint for the coefficient $\tilde s$. Thus, transformation $\tilde t=1/t$ alternates the sign of $\tilde s$. So, it can be assumed positive one.
In the case $D=0$ we choose values of $\alpha$, $\beta$, $\gamma$ and $\delta$ for which $p\delta^2-q\gamma\delta+r\gamma^2$ and the pair $(\delta,\gamma)$ is not proportional to the pair $(\alpha,\beta)$. Then we obtain that $\tilde p=0$ and $\tilde q=\nu\beta(q\gamma-2p\delta)+\nu\alpha(\delta q-2r\gamma)=0$. Appropriate choice of the pair $(\alpha,\beta)$ allows us to set $\tilde r=1$. Then the residual constant $\tilde s$ can be scaled to one by choice of $\nu.$
If $D<0$, we have $pr\ne0$ and can set $p>0$. We always can make $\tilde p=\tilde r=1$ and $\tilde q=0$, e.g., this gauge can be set by the transformation $\tilde t=\frac{2pt+q-\sqrt{4pr-q^2}}{2pt+q+\sqrt{4pr-q^2}}.$ In this case the constant $\tilde s$ cannot be scaled.
Therefore, up to $\hat G^\sim_2$-equivalence, we have three cases of $g$ which admit extension of Lie symmetry algebra by one basis operator. These are Cases 6–8 of Table 1.
If $k=2$, and $g=1\bmod \hat G^\sim_2$ we get a five-dimensional Lie symmetry algebra which is $\mathfrak{sl}(2,\mathbb{R}){\mathbin{\mbox{$\lefteqn{\hspace{.77ex}\rule{.4pt}{1.2ex}}{\in}$}}}2A_1$.
The arbitrary constant element $a$ does not affect the results of the group classification problem and can be scaled to any fixed nonzero value by the transformations from the usual equivalence group $G^\sim$. It is convenient to perform the gauge $a=1/n.$
Note that the group classification of subclass was carried out independently in and [@wafo2004d]. The results were presented there without gauging of the obtained forms of $g$ by equivalence transformations.
Solution of a boundary-value problem using Lie symmetries
=========================================================
There exist several approaches exploiting Lie symmetries in reduction of boundary-value problems (BVPs) for PDEs to those for ODEs. The classical technique is to require that both equation and boundary conditions are left invariant under the one-parameter Lie group of infinitesimal transformations. Of course the infinitesimal approach is usually applied (see, e.g., ). The first works in this direction appeared in the late sixties (see, e.g., ). The authors of used the classical approach, namely at first symmetries of a PDE were derived and then boundary conditions were checked whether they are also invariant under the action of the generators of symmetry found. In the case of a positive answer the BVP for the PDE was reduced to a BVP for an ODE. Using this technique a number of boundary-value problems were solved (see, e.g., ).
In [@BiPo2012; @PoBi2012] it was mentioned that for most applications it is not natural to require that appropriate symmetries of a system of DEs preserve a particular BVP for this system but it is enough to impose that these symmetries map BVPs from a certain class of such problems to each other. Hence the induction of well-defined equivalence transformations on a properly chosen class of physically relevant BVPs can serve as a criterion for selecting symmetries to be taken into account, e.g., in the course of invariant parameterization or discretization of the system under consideration.
The method suggested in uses specific one-parameter Lie groups of transformations of the independent and dependent variables of the PDE system as well as of all arbitrary elements which appear in the equations under study and in initial and boundary conditions. Namely, only the groups of scalings and translations are considered which can lead to self-similar or travelling-wave solutions only. After the admitted Lie group of scalings and/or translations is specified, the complete set of absolute invariants has to be found. Then a boundary-value problem for the PDE system is reduced to similar but simpler problem for the ODE system. Such an approach was applied to a number of engineering problems (see, e.g., and references therein). There is also the approach in which the group classification of the PDE system and associated initial and boundary conditions is carried out simultaneously (see, e.g., [@Kovalenko; @Kovalenko_dis] and references therein).
Initial- and boundary-value problems for certain classes of nonlinear PDEs were solved in using the method proposed in . In this Section we demonstrate that the classical approach is much easier, especially taking into account that problems of group classification are solved already for wide classes of nonlinear PDEs.
We look for nonzero solution of the initial- and boundary-value problem $$\begin{aligned}
\label{BV_Eq_GenBurgers}
&&u_t+a(u^n)_x=g(t)u_{xx},~~~ x\in [0,+\infty),~t>0, \nonumber\\
&&\lim_{t\rightarrow+0} u(t,x)=0,~~~x \in (0,+\infty),\\
&&u(t,0)=q(t),~~~t>0, \nonumber\\
&&\lim_{x\rightarrow+\infty}u(t,x)=0,~~~t>0,\nonumber\end{aligned}$$ where $a$ is a nonzero constant, $g$ and $q$ are arbitrary smooth nonvanishing functions and $n\neq0,1$.[^1]
We have derived the Lie symmetries for the variable coefficient equation (\[Eq\_GenBurgers\]) and now we examine which of these symmetries leave the initial and boundary conditions of the problem (\[BV\_Eq\_GenBurgers\]) invariant. The procedure starts by assuming a general symmetry of the form $$\label{general_symmetry}
\Gamma =\sum_{i=1}^n\alpha_i\Gamma_i,$$ where $n$ is the number of Lie point symmetries of the given partial differential equation and $\alpha_i,~i = 1,\dots,n$, are constants to be determined.
Lie symmetries for equation (\[Eq\_GenBurgers\]) appear in Table 1. In Case 2, for which $g(t)=\varepsilon t^{\rho}$, the generator (\[general\_symmetry\]) takes the form $$\Gamma=\alpha_1\partial_x+\alpha_2\Big(2t\partial_t+(\rho+1)x\partial_x+\frac{\rho-1}{n-1}u\partial_u\Big).$$ Application of $\Gamma$ to the first boundary condition which is written as $
x=0$ and $u(t,0)=q(t)
$ gives $
\alpha_1=0$ and $\alpha_2\big(-2t\frac{dq}{dt}+\frac{\rho-1}{n-1} q \big)=0.
$ For nonzero $\alpha_2$, we have $$q(t)=\gamma t^{\frac{\rho-1}{2n-2}},$$ where $\gamma>0$ is a constant. It can be shown that the symmetry $\Gamma$ with $\alpha_1=0$ leaves invariant the second boundary condition and the initial condition. Hence the admitted Lie symmetry can be used to reduce the initial- and boundary-value problem (\[BV\_Eq\_GenBurgers\]) to a problem with the governing equation being an ordinary differential equations. In fact the Lie symmetry $2t\partial_t+(\rho+1)x\partial_x+\frac{\rho-1}{n-1}u\partial_u$ produces the transformation (called usually Ansatz) $$\label{ansatz}
u=t^{\frac{\rho-1}{2n-2}}\phi(\eta),\quad\mbox{where}\quad\eta=xt^{-\frac{\rho+1}2},$$ that reduces (\[BV\_Eq\_GenBurgers\]) into the BVP for ODE $$\begin{aligned}
\label{redeq1}
&&2\varepsilon\phi''+(\rho+1)\eta\phi'-2a(\phi^{n})'-\frac{\rho-1}{n-1}\phi=0,\quad \eta\in[0,+\infty), \\\label{redeq2}
&&\phi(0)=\gamma, \\\label{redeq3}
&&\lim_{\eta\rightarrow+\infty} \phi(\eta)=0.\end{aligned}$$
Let $\rho=(2-n)/n$. Then takes the form $$\varepsilon\phi''+\tfrac1n(\eta\phi'+\phi)-a(\phi^{n})'=0$$ and can be integrated once to get $\varepsilon\phi'+\tfrac1n\eta\phi-a\phi^{n}+c=0.$ When we set $c=0$ this equation becomes the Bernoulli equation that is linearizable to the form $\frac{\varepsilon}{1-n}z'+\frac1n\eta z-a=0$ by the substitution $\phi^{1-n}=z$. The general solution of this equation is of the form $$z=e^{-\frac{1-n}{2n\varepsilon}\eta^2}\left(C+ \frac{a(1-n)}{\varepsilon}\int^{\eta}_0e^{\frac{1-n}{2n\varepsilon}\theta^2}{\rm d}\theta\right),$$ where $C$ is an arbitrary constant. If $\varepsilon n(n-1)>0$ it can be written in terms of the error function as $z=e^{\frac{\eta^2}{\sigma^2}}\left(C+ \frac{a(1-n)\sqrt{\pi}}{2\varepsilon\sigma}\operatorname{erf}(\sigma\eta)\right)$, where $\sigma=\sqrt{\frac{n-1}{2\varepsilon n}}$, ${\rm erf}(\theta)=\frac{2}{\sqrt{\pi}}\int_0^{\theta}\!e^{-s^2}\!{\rm d}s$ is the error function. Therefore, a particular solution of the second-order ODE on the function $\phi$ is $$\label{sol_phi}
\phi=\begin{cases}e^{-\frac{1}{2\varepsilon n}\eta^2}\left(C+ \frac{a(1-n)}{\varepsilon}\int^{\eta}_0e^{\frac{1-n}{2n\varepsilon}\theta^2}{\rm d}\theta\right)^{\frac1{1-n}},\quad \mbox{if}\quad \varepsilon n(n-1)<0,\\
e^{-\frac{1}{2\varepsilon n}\eta^2}\left(C+ \frac{a(1-n)\sqrt{\pi}}{2\varepsilon\sigma}\operatorname{erf}(\sigma \eta)\right)^{\frac1{1-n}}, \quad\mbox{if}\quad \varepsilon n(n-1)>0,
\end{cases}$$ where $\sigma=\sqrt{\frac{n-1}{2\varepsilon n}}.$ This is the solution of BVP – with $\rho=(2-n)/n$, when $C=\gamma^{1-n}$ and $\varepsilon n>0$. It’s typical behavior is shown on Figure 1. Using we can now obtain the solution of the following BVP $$\begin{aligned}
\label{BV_Eq_GenBurgers1}
&&u_t+a(u^n)_x=\varepsilon t^{\frac{2-n}n}u_{xx},~~~ x\in [0,+\infty),~t>0, \nonumber\\
&&\lim_{t\rightarrow+0} u(t,x)=0,~~~ x\in (0,+\infty),\\
&&u(t,0)=\gamma t^{-\frac1n},~~~t>0, \nonumber\\
&&\lim_{x\rightarrow+\infty}u(t,x)=0,~~~t>0.\nonumber\end{aligned}$$ For $\varepsilon>0$ and $n>1$ it is of the form $$\label{sol_u}
u=t^{-\frac1n}\exp\left[-\frac{1}{2\varepsilon n}x^2 t^{-\frac2n}\right]\left(\gamma^{1-n} +\frac{a(1-n)\sqrt{\pi}}{2\varepsilon\sigma}\operatorname{erf}(\sigma xt^{-\frac{1}n})\right)^{\frac1{1-n}},\quad \sigma=\sqrt{\frac{n-1}{2n\varepsilon}}.$$
Note that it satisfies BVP for all values of positive $\gamma$ if $a<0$. If $a>0$, then the parameters have to satisfy inequality $\gamma^{1-n}>\frac{a(n-1)\sqrt{\pi}}{2\varepsilon\sigma}.$ The typical behavior of the latter solution is shown on Figure 2.
{width="60mm"}
{width="60mm"}
The above procedure can be applied to the remaining cases that appear in Table 2. If we omit Case 9 which is the well-known Burgers equation, only in Cases 4 and 6 there exists a Lie symmetry that leaves the boundary and initial conditions invariant. However, the results for this case can be obtained from the above by setting $n=2$.
Conclusion {#conclusion .unnumbered}
==========
The group classification problem for the class of variable-coefficient generalized Burgers equations is firstly performed in the framework of modern group analysis. Namely, as preliminary step we investigated equivalence transformations within the class. It was shown that equivalence group of the subclass of class singled out by the condition $n=2$ is wider than the equivalence group of the whole class. Therefore, the group classification list is presented in Table 1 up to $G^\sim$-equivalence for equations with $n\neq2$ and up to $\hat G^\sim_2$-equivalence for those with $n=2$. The list of Lie symmetries for the class comes to complete the existing results that appear in the literature. Then we have shown that direct application of Lie symmetries for finding solution of an initial and boundary-value problem is more general and easier procedure than that one used, e.g., in .
Acknowledgements {#acknowledgements .unnumbered}
----------------
OV is grateful for the hospitality and financial support by the University of Cyprus and to Prof. Roman Popovych for useful comments. PGLL thanks the University of Cyprus for its kind hospitality and the University of KwaZulu-Natal and the National Research Foundation of South Africa for their continued support.
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[^1]: Similar BVP was solved in , but in our opinion, by the use of a more complicated method.
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