text
stringlengths 0
6.23M
| __index_level_0__
int64 0
419k
|
|---|---|
TITLE: Meaning of modules of finite dimension
QUESTION [0 upvotes]: Let $k$ be an algebraically closed field, let $R=k^{m}$ where $m \geq 1$ and let $M$ be a $R$-bimodule. What shall we understand by a "basis" of $M$? or "finite dimensional over $R$"?
REPLY [2 votes]: Well, since in this case $R$ is commutative, it satisfies the invariant basis number property, so that it is meaningful to speak of the rank of $M$. This is probably what is meant by the "dimension" of $M$ over $R$, i.e., the number $n$ such that $M \cong R^n$. In this case a basis of $M$ means a set of $n$ linearly independent elements that generate $M$. | 92,008 |
\begin{document}
\title{A class of exactly solvable rationally extended non-central potentials
in Two and Three Dimensions}
\author{Nisha Kumari$^{a}$\footnote{e-mail address: [email protected] (N.K)}, Rajesh Kumar Yadav$^{b}$\footnote {e-mail address: [email protected] (R.K.Y)}, Avinash Khare$^{c}$\footnote{e-mail: [email protected] (A.K)} and \\ Bhabani Prasad Mandal$^{a}$\footnote{e-mail address: [email protected] (B.P.M).}}
\maketitle
{$~^a$Department of Physics, Banaras Hindu University, Varanasi-221005, India.\\
$~^b$Department of Physics, S. P. College, S. K. M. University, Dumka-814101, India.\\
$~^c$Department of Physics, Savitribai Phule Pune University, Pune-411007, India.\\
}
\begin{abstract}
We start from a seven parameters (six continuous and one discrete) family of
non-central exactly solvable
potential in three dimensions and construct a wide class of ten parameters
(six continuous and four
discrete) family of rationally extended exactly solvable
non-central real as well as $PT$ symmetric complex potentials.
The energy eigenvalues and the eigenfunctions of these extended non-central
potentials are obtained explicitly and it is shown that the wave eigenfunctions
of these potentials are either associated with the exceptional orthogonal
polynomials (EOPs) or some type of new polynomials which can be further
re-expressed in terms of the corresponding classical orthogonal polynomials.
Similarly, we also construct a wide class of rationally extended exactly
solvable non-central real as well as complex PT-invariant potentials in
two dimensions.
\end{abstract}
\section{Introduction}
In non-relativistic quantum mechanics the exactly solvable (ES) problems play
an important role in the understanding of different quantum mechanical
systems associated with any branch of theoretical
physics. For many of the quantum mechanical systems, whose exact solutions
are unknown, these ES potentials are generally
considered as a starting potential to get their approximate eigenspectrum.
Most of the ES potentials are either one dimensional
or are central potentials which are essentially one dimensional on the half
line.
There are only few examples of exactly solvable non-central potentials such as
anisotropic harmonic oscillator whose solutions are well known, see for example
\cite{fluge}. In a comprehensive study, Khare and Bhaduri \cite{kb} have
discussed a number of exactly solvable non-central potentials
by considering the Schr\"odinger equation in
three dimensional spherical polar and two dimensional polar co-ordinates. The classical orthogonal polynomials (such as
Hermite, Laguerre and Jacobi polynomial etc)
are playing a fundamental role in the construction of the bound state
solutions to these ES potentials. The solution
of most of the ES non-central potentials is connected with the above well known
orthogonal polynomials. It has been observed that the solution
of all the non-central potentials obtained in Ref. \cite{kb} is associated
either with the classical Laguerre or Jacobi orthogonal polynomials.
After the recent discovery of the two new families of orthogonal polynomials
namely the exceptional $X_m$ Laguerre and
$X_m$ Jacobi orthogonal polynomials \cite{dnr1, xm1}, a number of new exactly
solvable potentials have been discovered
\cite{que,bqr,os,hos,hs,qu,cq_12,yg1,yg2,dim}. In most of these cases,
these new potentials are the rational extension of the corresponding
conventional potentials \cite{cks, levai89}. Various properties of these
new extended potentials have also been studied by different groups
\cite{pdm,nfold2,fplank,codext1,qscat,scatpt,gtextd,para_sym,bpm5,nrab}.
It is then natural to consider the rational extension of
the conventional non-central potentials discussed in Ref. \cite {kb} and
discover new exactly solvable non-central potentials whose solutions are in
terms of the rational extension of the Jacobi or Laguerre polynomials.
The purpose of this paper is precisely to address this issue.
In this present manuscript, our aim is to obtain the rational extension of
all the non-central
conventional potentials discussed in Ref. \cite{kb}. Further, one of the major
development after the work of \cite{kb} has been the discovery of $PT$-invariant
complex potentials with real energy eigenvalues.
The concept of $PT$ (combined parity ($P$) and time reversal ($T$))
symmetric quantum mechanics \cite{bender} also plays a crucial role in the understanding of the complex quantum mechanical systems. Bender et.al.,
\cite{bender} have showed that the eigenspectrum of such non-hermitian
$PT$ symmetric complex systems are real provided the PT-symmetry is not
spontaneously broken. The second purpose of this paper is to consider
$PT$-symmetric, noncentral, exactly solvable, potentials and obtain their
rational extensions. It turns out that the bound state eigenfunctions of some of these
potentials are not in the exact form of EOPs rather they are written
in the form of some new types of polynomials which can be further written in
terms of the corresponding classical orthogonal polynomials.
The paper is organized as follows:
In section $2$, we start from the non-central potentials with seven
parameters (six continuous and one discrete)
in spherical polar co-ordinates and explain how one can obtain its
rationally extended solution in terms of ten parameters (six continuous and
four discrete) solutions. Various forms of $r$, $\theta$ and
$\phi$ dependent potential terms are also mentioned. For an illustration,
we discuss in detail in Sec. $2.1$ and $2.2$ respectively two examples (one real
and one complex and $PT$ symmetric) of ten parameters
rationally extended non-central potentials. In particuar, we show that
the corresponding eigenfunctions are product of the exceptional Jacobi,
exceptional Laguerre and/or some type of new orthogonal polynomials.
A list of possible forms of $\theta$ and $\phi$ dependent terms with the
corresponding eigenfunctions as well as the
other parametric relations are given in Tables $I$ and $II$ respectively.
Some examples of RE non-central potentials in two dimensional polar
co-ordinates are also mentioned in brief in Sec. $3$. In particular, we start
from the five parameter (four continuous and one discrete) families of
non-central potentials in polar coordinates and obtain the corresponding
rationally extended non-central potentials with seven parameters (four
continuos and three discrete). Finally, we summarize
our results in section $4$.
\section{Non central potential in $3$-dimensional spherical polar co-ordinates}
In spherical polar coordinates $(r,\theta,\phi)$, consider a non central
potential \cite{lif_76} of the form
as
\be\label{non_form}
V(r,\theta,\phi)=\tilde{U}(r)+\frac{V(\theta)}{r^2}+\frac{U(\phi)}{r^2\sin^2 (\theta)}.
\ee
The Schr\"odinger equation corresponding to this potential i.e.,
\ba\label{sch_non}
\bigg[-\bigg(\frac{\partial^2\Psi}{\partial r^2}+\frac{2}{r}\frac{\partial\Psi}{\partial r}\bigg)- \frac{1}{r^2}\bigg(\frac{\partial^2\Psi}{\partial \theta^2}+\cot\theta \frac{\partial\Psi}{\partial\theta}\bigg)-\frac{1}{r^2\sin^2 \theta}\frac{\partial^2 \Psi}{\partial \phi^2}\bigg]=(E-V(r,\theta,\phi))\Psi,
\ea
has been solved exactly \cite{kb} by using the fact that the eigenfunction
can be written in the product form
\be\label{wf_non}
\Psi(r,\theta,\phi)=\frac{R(r)}{r}\frac{\Theta(\theta)}{(\sin\theta)^{\frac{1}{2}}}\Phi(\phi).
\ee
Using Eq. (\ref{wf_non}) in Eq. (\ref{sch_non}), we obtain three exactly
solvable uncoupled equations given by
\be\label{un_cop_1}
-\frac{\partial^2 \Phi(\phi)}{\partial \phi^2}+U(\phi) \Phi(\phi)=m^2\Phi(\phi),
\ee
\be\label{un_cop_2}
-\frac{\partial^2 \Theta(\theta)}{\partial \theta^2}+\bigg[V(\theta)+\bigg(m^2-\frac{1}{4}\bigg)\cosec^2\theta\bigg] \Theta(\theta)=\ell^2\Theta(\theta)
\ee
and
\be\label{un_cop_3}
-\frac{\partial^2 R(r)}{\partial r^2}+\bigg[\tilde{U}(r)+\frac{(\ell^2-1/4)}{r^2}\bigg]R(r)=ER(r).
\ee
By considering different forms of a seven parameter family of potentials,
the above three Eqs. (\ref{un_cop_1}) - (\ref{un_cop_3}) have been
solved exactly \cite{kb} and in this way one has constructed several
non-central potentials
$V(r,\theta,\phi)$ by considering different forms of $\tilde{U}(r), U(\theta)$
and $U(\phi)$ with the corresponding
eigenvalues being $E, \ell^2$ and $m^2$ respectively. The eigenfunctions corresponding to these three equations (\ref{un_cop_1}) - (\ref{un_cop_3})
are either in terms of classical
Laguerre or Jacobi orthogonal polynomials. The complete eigenfunctions are
obtained by using Eq. (\ref{wf_non}). The
forms of the potential $V(r,\theta,\phi)$ with their solutions can be found
in detail in Ref. \cite {kb}. As an illustration, one of the seven parameter
family of potential considered in \cite{kb} is given by
\ba
V(r,\theta,\phi)&=&\frac{\omega^2 r^2}{4}+\frac{\delta}{r^2}
+\frac{C}{r^2\sin^2\theta}+\frac{D}{r^2\cos^2\theta} \nonumber\\
&+&\frac{G}{r^2\sin^2\theta\sin^2 p\phi}
+\frac{F}{r^2\sin^2\theta \cos^2p \phi}\,,
\ea
In this work, If we change the potential $V(r,\theta,\phi) \Rightarrow V_{m_1,m_2,m_3}(r,\theta,\phi)$
by redefining the extended form $\tilde{U}(r)\Rightarrow \tilde{U}_{m_1,ext}(r)$, $V(\theta)\Rightarrow V^{(h)}_{m_2,ext}(\theta)$, $U(\phi)\Rightarrow U^{(h)}_{m_3,ext}(\phi)$
i.e.,
\ba\label{v_rat}
V(r,\theta,\phi) &\Rightarrow &V_{m_1,m_2,m_3}(r,\theta,\phi)\nonumber\\
&=&\tilde{U}_{m_1,ext}(r)+\frac{1}{r^2}V^{(h)}_{m_2,ext}(\theta)+\frac{1}{r^2\sin^2\theta}U^{(h)}_{m_3,ext}(\phi),
\ea
and the eigenfunction
\ba\label{wf_non_rat}
\Psi(r,\theta,\phi)&\Rightarrow&\Psi_{m_1,m_2,m_3}(r,\theta,\phi)\nonumber\\
&=& \frac{R_{m_1}(r)}{r}\frac{\Theta^{(h)}_{m_2}(\theta)}{(\sin\theta)^{\frac{1}{2}}}\Phi^{(h)}_{m_3}(\phi),
\ea
then Eq. (\ref{sch_non}) becomes
\ba\label{sch_rat}
&&\bigg[-\frac{1}{R_{m_1}}\frac{\partial^2R_{m_1}}{\partial r^2}+\tilde{U}_{m_1,ext}(r)-\frac{1}{4r^2}\bigg]+\frac{1}{r^2}\bigg[-\frac{1}{\Theta^{(h)}_{m_2}}\frac{\partial^2\Theta^{(h)}_{m_2}}{\partial \theta^2}+V^{(h)}_{m_2,ext}(\theta)-\frac{1}{4}\cosec^2\theta\bigg]\nonumber\\
&&+\frac{1}{r^2\sin^2\theta}\bigg[-\frac{1}{\Phi^{(h)}_{m_3}}\frac{\partial^2\Phi^{(h)}_{m_3}}{\partial \phi^2}+U^{(h)}_{m_3,ext}(\phi)\bigg]=E.\nonumber\\
\ea
Here $h=I, II$ correspond to real potential while $h= (PT)_1, (PT)_2$ correspond to the complex and $PT$
symmetric potential in $\theta$ and/or $\phi$.
Similar to Eqs. (\ref{un_cop_1})-(\ref{un_cop_3}), the above Eq. (\ref{sch_rat}) can also be easily uncoupled into three new exactly
solvable equations given by
\be\label{ucext_1}
-\frac{\partial^2 \Phi^{(h)}_{m_3}(\phi)}{\partial \phi^2}+U^{(h)}_{m_3,ext}(\phi)\Phi^{(h)}_{m_3}(\phi)=m^2\Phi^{(h)}_{m_3}(\phi),
\ee
\be\label{ucext_2}
-\frac{\partial^2 \Theta^{(h)}_{m_2}(\theta)}{\partial \theta^2}+\bigg[V^{(h)}_{m_2,ext}(\theta))+\bigg(m^2-\frac{1}{4}\bigg)\cosec^2\theta\bigg] \Theta^{(h)}_{m_2}(\theta)=\ell^2\Theta^{(h)}_{m_2}(\theta)
\ee
and
\ba\label{ucext_3}
-\frac{\partial^2 R_{m_1}(r)}{\partial r^2}+\bigg[\tilde{U}_{m_1,ext}(r)+\frac{(\ell^2-1/4)}{r^2}\bigg]R_{m_1}(r)=ER_{m_1}(r).
\ea
By knowing the solutions of the above Eqs. (\ref{ucext_1})-(\ref{ucext_3}), a complete solution of
the extended non-central potential $V_{m_1,m_2,m_3}(r,\theta,\phi)$ can be obtained by using Eq. (\ref{wf_non_rat}) with the
energy eigenvalues $E$.
We show that there is one choice of $\tilde{U}_{m_1,ext}(r)$, four choices of $V^{(h)}_{m_2,ext}(\theta)$ (two real and two $PT$ symmetric) and three choices of
$U^{(h)}_{m_3,ext}(\phi)$ (two real and one $PT$ symmetric) for which one can
obtain exact solutions of the non-central potential. These choices are:
{\bf {$(a)$ Form of $\tilde{U}_{m_1,ext}(r)$:}}
\be\label{ext_r}
\tilde{U}_{m_1,ext}(r)=\frac{\omega^2 r^2}{4}+\frac{\delta}{r^2}+\tilde{U}_{m_1,rat}(r),
\ee
where
\ba\label{rat1}
\tilde{U}_{m_1,rat}(r)&=&-2m_1\omega-\omega^2r^2\frac{L^{(\tilde{\delta}+1)}_{m_1-2}(-\omega r^2/2)}{L^{(\tilde{\delta}-1)}_{m_1}(-\omega r^2/2)}\nonumber\\
&+&\omega(\omega r^2+2(\tilde{\delta}-1))\frac{L^{(\tilde{\delta})}_{m_1-1}(-\omega r^2/2)}{L^{(\tilde{\delta}-1)}_{m_1}(-\omega r^2/2)}\nonumber\\
&+& 2\omega^2 r^2\bigg(\frac{L^{(\tilde{\delta})}_{m_1-1}(-\omega r^2/2)}{L^{(\tilde{\delta}-1)}_{m_1}(-\omega r^2/2)}\bigg)^2.\nonumber
\ea
Here $L^{(\tilde{\delta})}_{m_1}(-\omega r^2/2)$ is a classical Laguerre polynomial.
{\bf {$(b)$ Forms of $V^{(h)}_{m_2,ext}(\theta)$:}}
$(i)$ For $h=I$
\be\label{ext_theta_1}
V^{(I)}_{m_2,ext}(\theta)=\frac{C}{\sin^2 \theta}+\frac{D}{\cos^2 \theta}+V^{(I)}_{m_2,rat}(\theta),
\ee
where the rational part $V^{(I)}_{m_2,rat}(\theta)$ is given by
\ba\label{rat2}
V^{(I)}_{m_2,rat}(\theta)&=&4\bigg[-2m_2(\alpha -\beta -m_2+1)-(\alpha -\beta -m_2+1)\big(\alpha +\beta +(\alpha -\beta +1)\cos(2\theta)\big)\nonumber\\
&\times & \frac{P_{m_2-1}^{(-\alpha ,\beta) }(\cos (2\theta))}{P_{m_2}^{(-\alpha -1,\beta -1)}(\cos (2\theta))}+\frac{(\alpha -\beta -m_2+1)^{2}\sin^2(2\theta)}{2}
\bigg(\frac{P_{m_2-1}^{(-\alpha ,\beta) }(\cos(2\theta))}{P_{m_2}^{(-\alpha -1,\beta -1)}(\cos(2\theta))}\bigg)^{2}\bigg].\nonumber
\ea
$(ii)$ For $h=II$
\be\label{ext_theta_2}
V^{(II)}_{m_2,ext}(\theta)=\frac{C}{\sin^2 \theta}+\frac{D}{\sin\theta \tan \theta}+V^{(II)}_{m_2,rat}(\theta),
\ee
with the rational part
\ba\label{rat_theta2}
V_{m_2,rat}^{(II)}(\theta ) &=& \bigg[-2 m_{2}(\alpha-\beta - m_{2} + 1 ) -
(\alpha-\beta - m_{2} +1)(\alpha + \beta + (\alpha-\beta + 1 )\cos \theta)\nonumber \\
&\times &\left(\frac{P_{m_{2}-1}^{(-\alpha , \beta)} (\cos \theta)}
{P_{m_{2}}^{(-\alpha-1,\beta -1)}(\cos \theta ) }\right)
+ \frac{(\alpha-\beta -m_{2}+1 )^{2}\sin^{2}\theta}{2}\left(\frac{P_{m_{2}-1}^{(-\alpha,\beta )} (\cos \theta)}
{P_{m_{2}}^{(-\alpha-1, \beta -1)}(\cos \theta) }\right)^{2} \bigg].\nonumber
\ea
$(iii)$ For $h=(PT)_1$
\be\label{ext_theta_pt}
V^{(PT)_1}_{m_2,ext}(\theta)=\frac{C}{\sin^2 \theta}+\frac{iD}{\tan \theta}+V^{(PT)_1}_{m_2,rat}(\theta),
\ee
is a complex and $PT$ symmetric form of $V^{(h)}_{m_2,ext}(\theta)$ with the corresponding complex and $PT$ symmetric rational term
\ba
V^{(PT)_1}_{m_2,rat}(\theta) &=& -2\cosec^2\theta\bigg[ 2i\cot\theta \frac{\dot{q}^{(A,B)}_{m_2}(z)}{q^{(A,B)}_{m_2}(z)}-\cosec^2\theta\nonumber\\
&\times & \Bigg( \frac{\ddot{q}^{(A,B)}_{m_2}(z)}{q^{(A,B)}_{m_2}(z)}-\bigg( \frac{\dot{q}^{(A,B)}_{m_2}(z)}{q^{(A,B)}_{m_2}(z)}\bigg )^2 \Bigg )-m_2\bigg ].\nonumber
\ea
$(iv)$ For $h=(PT)_2$
The potential given in case (ii) can also be made complex and $PT$ symmetric by multiplying the potential parameter $D$ by imaginary number $i$ and get
\be\label{ext_theta_pt2}
V^{(PT)_2}_{m_2,ext}(\theta)=\frac{C}{\sin^2 \theta}+\frac{iD}{\sin\theta \tan \theta}+V^{(PT)_2}_{m_2,rat}(\theta),
\ee
with the complex rational part
\ba\label{rat_theta_pt2}
V_{m_2,rat}^{(PT)_2}(\theta ) &=& \bigg[-2 m_{2}(\alpha-\beta - m_{2} + 1 ) -
(\alpha-\beta - m_{2} +1)(\alpha + \beta + (\alpha-\beta + 1 )\cos \theta)\nonumber \\
&\times &\left(\frac{P_{m_{2}-1}^{(-\alpha , \beta)} (\cos \theta)}
{P_{m_{2}}^{(-\alpha-1,\beta -1)}(\cos \theta ) }\right)
+ \frac{(\alpha-\beta -m_{2}+1 )^{2}\sin^{2}\theta}{2}\left(\frac{P_{m_{2}-1}^{(-\alpha,\beta )} (\cos \theta)}
{P_{m_{2}}^{(-\alpha-1, \beta -1)}(\cos \theta) }\right)^{2} \bigg].\nonumber
\ea
Here the potential parameters $\alpha$ and $\beta$ are complex.
{\bf {$(c)$ Forms of $V^{(h)}_{m_3,ext}(\phi)$:}}
$(i)$ For $h=I$
\be\label{ext_phi_1}
V^{(I)}_{m_3,ext}(\phi)=\frac{G}{\sin^2 (p\phi )}+\frac{F}{\cos^2 (p\phi)}+V^{(I)}_{m_3,rat}(\phi),
\ee
where
\ba\label{rat3}
U^{(I)}_{m_3,rat}(\phi)&=&4p^2\bigg[-2m_3(\tilde{\alpha}-\tilde{\beta} -m_3+1)-(\tilde{\alpha} -\tilde{\beta} -m_3+1)\big(\tilde{\alpha} +\tilde{\beta} +(\tilde{\alpha} -\tilde{\beta} +1)\cos(2p\phi)\big)\nonumber\\
&\times & \frac{P_{m_2-1}^{(-\tilde{\alpha} ,\tilde{\beta}) }(\cos (2p\phi))}{P_{m_2}^{(-\tilde{\alpha} -1,\tilde{\beta} -1)}(\cos (2p\phi))}+\frac{(\tilde{\alpha} -\tilde{\beta} -m_2+1)^{2}}{2}\sin^2(2p\phi)
\bigg(\frac{P_{m_2-1}^{(-\tilde{\alpha} ,\tilde{\beta}) }(\cos(2p\phi))}{P_{m_2}^{(-\tilde{\alpha} -1,\tilde{\beta} -1)}(\cos(2p\phi))}\bigg)^{2}\bigg].\nonumber
\ea
Note that here $p$ is any positive integer.
$(ii)$ For $h=II$
\be\label{ext_phi_2}
V^{(II)}_{m_3,ext}(\phi)=\frac{G}{\sin^2 (p\phi)}+\frac{F}{\sin(p\phi) \tan(p\phi)}+V^{(II)}_{m_3,rat}(\phi),
\ee
where
\ba
U_{m_3,rat}^{(II)}(\phi ) &= & p^{2}\bigg[-2 m_{3}(\tilde{\alpha }- \tilde{\beta } - m_{3} + 1 ) -
(\tilde{\alpha }-\tilde{\beta } - m_{3} +1)(\tilde{\alpha } + \tilde{\beta }+ (
\tilde{\alpha }-\tilde{\beta } + 1 )\cos ( p\phi ) )\nonumber \\
&\times &\left(\frac{P_{m_{3}-1}^{(-\tilde{\alpha }, \tilde{\beta })} (\cos (p\phi ))}
{P_{m_{3}}^{(-\tilde{\alpha }-1, \tilde{\beta } -1)}(\cos ( p\phi )) }\right)
+ \frac{(\tilde{\alpha }-\tilde{\beta } -m_{3}+1 )^{2}\sin^{2}(p\phi )}{2}\nonumber\\
&\times &\left(\frac{P_{m_{3}-1}^{(-\tilde{\alpha }, \tilde{\beta })} (\cos (p\phi ))}
{P_{m_{3}}^{(-\tilde{\alpha }-1, \tilde{\beta } -1)}(\cos ( p\phi )) }\right)^{2} \bigg].\nonumber
\ea
Here $p$ is any odd positive integer.
$(iii)$ For $h=(PT)_1$
The above case (ii) of the $\phi$ dependent terms can be made complex and $PT$ symmetric by replacing the potential parameter $F\rightarrow iF$ and
we get
\be\label{ext_phi_3}
V^{(PT)}_{m_3,ext}(\phi)=\frac{G}{\sin^2 (p\phi)}+\frac{iF}{\sin(p\phi) \tan(p\phi)}+V^{(PT)_1}_{m_3,rat}(\phi),
\ee
where the complex rational term with the complex parameters $\tilde{\alpha}$ and $\tilde{\beta}$ given by
\ba
U_{m_3,rat}^{(PT)_1}(\phi ) &= & p^{2}\bigg[-2 m_{3}(\tilde{\alpha }- \tilde{\beta } - m_{3} + 1 ) -
(\tilde{\alpha }-\tilde{\beta } - m_{3} +1)(\tilde{\alpha } + \tilde{\beta }+ (
\tilde{\alpha }-\tilde{\beta } + 1 )\cos ( p\phi ) )\nonumber \\
&\times &\left(\frac{P_{m_{3}-1}^{(-\tilde{\alpha }, \tilde{\beta })} (\cos (p\phi ))}
{P_{m_{3}}^{(-\tilde{\alpha }-1, \tilde{\beta } -1)}(\cos ( p\phi )) }\right)
+ \frac{(\tilde{\alpha }-\tilde{\beta } -m_{3}+1 )^{2}\sin^{2}(p\phi )}{2}\nonumber\\
&\times &\left(\frac{P_{m_{3}-1}^{(-\tilde{\alpha }, \tilde{\beta })} (\cos (p\phi ))}
{P_{m_{3}}^{(-\tilde{\alpha }-1, \tilde{\beta } -1)}(\cos ( p\phi )) }\right)^{2} \bigg].\nonumber
\ea
Note that here $p$ is any odd positive integer.
Here $P^{(\tilde{\alpha},\tilde{\beta})}_{m_2}(z)$ and $P^{(\alpha,\beta)}_{m_3}(z)$ are classical Jacobi polynomials.
By taking various combinations of these allowed choices, we then have twelve
different, rational, exactly solvable non-central
potentials in three dimensions, each with ten parameters. These choices of
potentials, the
corresponding eigenvalues and eigenfunctions are given in Tables $I$ and $II$.
In particular, in Table $I$ we give the four forms of possible $V^{(h)}_{m_2,ext}(\theta)$
(two real and two complex but $PT$-invariant) with their corresponding eigenvalues
and eigenfunctions. In Table $II$, we similarly give two real
and one complex and $PT$ symmetric forms of $U^{(h)}_{m_3,ext}(\phi)$ and the
corresponding eigenvalues and eigenfunctions.
As an illustration, we discuss two examples in detail, one real and one
$PT$-invariant complex case in Secs. $2.1$ and $2.2$ respectively.
\subsection{Example of Rationally Extended (RE) non-central real potential}
In this section, we discuss an example of the ten parameters (six continuous
and four discrete) RE non-central real potential and its bound state solutions
explicitly. We consider the potential of the form
\ba\label{non_ten}
V_{m_1,m_2,m_3}(r,\theta,\phi)&=&\frac{\omega^2 r^2}{4}+\frac{\delta}{r^2}+\tilde{U}_{m_1,rat}(r)+\frac{C}{r^2\sin^2\theta}+\frac{D}{r^2\cos^2\theta}+\frac{1}{r^2}V^{(I)}_{m_2,rat}(\theta)\nonumber\\
&+&\frac{G}{r^2\sin^2\theta\sin^2 p\phi}
+\frac{F}{r^2\sin^2\theta \cos^2p \phi}+\frac{1}{r^2\sin^2\theta}U^{(I)}_{m_3,rat}(\phi),
\ea
where the six parameters $\omega, \delta, C, D, F$ and $ G $ are continuous
parameters while the rest
four i.e., $p, m_1, m_2$ and $m_3$ are discrete parameters. In particular, each
of them can take any inegral value. The rational terms $\tilde{U}_{m_1,rat}(r), V^{(I)}_{m_2,rat}(\theta)$ and $U^{(I)}_{m_3,rat}(\phi)$
are given by Eqs. (\ref{ext_r}), (\ref{ext_theta_1}) and (\ref{ext_phi_1})
respectively. It is easy to show that the eigenvalues of this extended
non-central potential are the same as that of the conventional
case given by Eq. (\ref{sch_non}) but the eigenfunctions are different
which are obtained in terms of EOPs. The complete eigenfunction is given by Eq. (\ref{wf_non_rat})
which is ultimately a product of these EOPs.
To solve the above extended non-central potential, first we consider
a simple case of $m_1=m_2=m_3=1$ and then we generalize it to any arbitrary
positive integer values of $m_1,m_2$ and $ m_3$.
{\bf Case (i): For $m_1=m_2=m_3=1$}
In this case, the ten parameters RE non-central potential is reduced to
a seven parameters RE non-central potential
\ba\label{pot_extd_1}
V_{1,1,1}(r,\theta,\phi)&=&\frac{\omega^2 r^2}{4}+\frac{\delta}{r^2}+\tilde{U}_{1,rat}(r)+\frac{C}{r^2\sin^2\theta}+\frac{D}{r^2\cos^2\theta}+\frac{1}{r^2}V^{(I)}_{1,rat}(\theta)\nonumber\\
&+&\frac{G}{r^2\sin^2\theta\sin^2 p\phi}
+\frac{F}{r^2\sin^2\theta \cos^2p \phi}+\frac{1}{r^2\sin^2\theta}U^{(I)}_{1,rat}(\phi),
\ea
here $p$ is any positive integer. To get the exact solution of the above
Eq. (\ref{pot_extd_1}), we define the rational terms
$\tilde{U}_{1,rat}(r), V^{(I)}_{1,rat}(\theta)$ and $U^{(I)}_{1,rat}(\phi)$ as (by putting $m_1=m_2=m_3$ in the rational parts of Eqs.
(\ref{ext_r}), (\ref{ext_theta_1}) and (\ref{ext_phi_1}))
\ba\label{prad_1}
\tilde{U}_{1,rat}(r)=\frac{4\omega}{(\omega r^2+2\tilde{\delta})}-\frac{16\omega \tilde{\delta}}{(\omega r^2+2\tilde{\delta})^2},
\ea
\ba\label{ptheta_1}
V^{(I)}_{1,rat}(\theta)=\frac{8(\alpha+\beta)}{\big ((\alpha+\beta )-(\beta-\alpha )\cos ( 2\theta ) \big)}-\frac{8\big((\alpha+\beta)^2-(\beta-\alpha)^2\big)}{\big((\alpha+\beta )-(\beta-\alpha )\cos ( 2\theta ) \big)^2},
\ea
and
\ba\label{pphi_1}
U^{(I)}_{1,rat}(\phi)=4p^2\bigg[\frac{2(\tilde{\alpha}+\tilde{\beta})}{\big ((\tilde{\alpha}+\tilde{\beta} )-(\tilde{\beta}-\tilde{\alpha} )\cos ( 2p\phi ) \big)}-\frac{2\big((\tilde{\alpha}+\tilde{\beta})^2-(\tilde{\beta}-\tilde{\alpha})^2\big)}{\big((\tilde{\alpha}+\tilde{\beta} )-(\tilde{\beta}-\tilde{\alpha} )\cos ( 2p\phi ) \big)^2}\bigg].
\ea
On comparing Eq. (\ref{pot_extd_1}) with Eq. (\ref{v_rat} ) (for $m_1=m_2=m_3=1$ and $h=I$), we get the rationally extended
trigonometric P\"oschl-Teller potential \cite{que,os}
\ba\label{extd_phi_1}
U^{(I)}_{1,ext}(\phi)= U^{(I)}_{con}(\phi)+ U^{(I)}_{1,rat}(\phi),
\ea
where
\be\label{con_rpt}
U^{I}_{con}(\phi) = G \cosec^2 (p\phi) + F\sec^2(p\phi)
\ee
is the corresponding conventional potential. The unnormalized $\phi$ dependent wave function of Eq. (\ref{ucext_1}) (for $m_3=1$)
with the extended potential (\ref{extd_phi_1}) in terms of $X_1$ exceptional orthogonal polynomials $\hat{P}_{n_3+1}^{(\tilde{\alpha} ,\tilde{\beta}) }(z)$ is well known and given by \cite {que,os}
\be\label{ext_phi_sol_1}
\Phi^{I}_{1,n_3}(\phi)\propto \frac{(1-z )^{\frac{\tilde{\alpha} }{2} + \frac{1}{4}}(1 +z )^{\frac{\tilde{\beta} }{2} + \frac{1}{4}}}
{\big ((\tilde{\alpha}+\tilde{\beta} )-(\tilde{\beta}-\tilde{\alpha} )\cos ( 2p\phi ) \big)}\hat{P}_{n_3+1}^{(\tilde{\alpha} ,\tilde{\beta}) }(z);\qquad 0\leq p\phi\leq \pi/2,
\ee
where $n_3=0,1,2,3...$, \quad $z=\cos(2p\phi)$ and the positive constant parameters
\ba\label{al_telda}
\tilde{\alpha} =\frac{1}{2}\sqrt {1+\frac{4G}{p^2}};\quad \tilde{\beta} &=& \frac{1}{2}\sqrt {1+\frac{4F}{p^2}}.
\ea
The eigenvalue spectrum of this extended potential is same (i.e. isospectral)
as that of the conventional
potential $U_{con}(\phi)$ which is given by
\be\label{ms}
m^2=p^2(2n_3+\tilde{\alpha}+\tilde{\beta}+1)^2.
\ee
Again from Eqs. (\ref{pot_extd_1}) and (\ref{v_rat}), the rationally extended $\theta$ dependent potential is given by
\ba\label{extd_theta_1}
V^{(I)}_{1,ext}(\theta)= V^{(I)}_{con}(\theta)+ V^{(I)}_{1,rat}(\theta),
\ea
where the conventional potential
\be\label{con_tpt}
V^{(I)}_{con}(\theta) = C \cosec^2(\theta) + D\sec^2(\theta).
\ee
The Schr\"odinger equation (\ref{ucext_2}) (for $m_2=1$ and $h=I$) becomes
\ba\label{sch_theta_1}
-\frac{\partial^2 \Theta^{(I)}_{1}(\theta)}{\partial \theta^2}+\bigg[\bigg(C+m^2-\frac{1}{4}\bigg)\cosec^2\theta+D\sec^2\theta+ V^{(I)}_{1,rat}(\theta)\bigg] \Theta^{(I)}_{1}(\theta)=\ell^2\Theta^{(I)}_{1}(\theta).
\ea
Using the rational term $V^{(I)}_{1,rat}(\theta)$ from Eq. (\ref{ptheta_1}), the wave function and the eigenspectrum of the above Eq. (\ref{sch_theta_1}) are thus given by
\be\label{ext_theta_sol_1}
\Theta^{(I)}_{1,n_2}(\theta)\propto \frac{(1-z )^{\frac{\alpha}{2} + \frac{1}{4}}(1 +z )^{\frac{\beta}{2} + \frac{1}{4}}}
{\big ((\alpha+\beta )-(\beta-\alpha )\cos ( 2\theta ) \big)}\hat{P}_{n_2+1}^{(\alpha ,\beta) }(z);\qquad 0\leq \theta \leq \pi/2,
\ee
and
\be\label{ls}
\ell^2=(2n_2+\alpha+\beta+1)^2; \qquad n_2=0,1,2,...,
\ee
where $z=\cos(2\theta)$ and the parameters
\ba\label{alpha}
\alpha &=& \sqrt {C+m^2},\nonumber\\
\beta &=& \frac{1}{2}\sqrt {1+4D}.
\ea
Note that the eigenvalue spectrum is unchanged while the eigenfunctions are
different from those of the nonrational case.
Similar to the above cases, from Eqs. (\ref{pot_extd_1}) and (\ref{v_rat}), the radial component of the extended potential is given by
\ba\label{extd_rad_1}
\tilde{U}_{1,ext}(r)=\tilde{U}_{con}(r) + \tilde{U}_{1,rat}(r),
\ea
where the conventional radial oscillator potential
\be\label{con_rad}
\tilde{U}_{con}(r) = \frac{\omega^2 r^2}{4}+\frac{\delta}{r^2}.
\ee
From Eq. (\ref{ucext_3}) (for $m_1=1$), finally, we get the exactly solvable Schr\"odinger equation
\be
-\frac{\partial^2 R_{1}(r)}{\partial r^2}+\bigg[\frac{\omega^2 r^2}{4}+\frac{(\delta+\ell^2-1/4)}{r^2}+\tilde{U}_{1,rat}(r)\bigg]R_{1}(r)=ER_{1}(r),
\ee
with the solution in term of $X_1$ Laguerre EOPs $\hat{L}_{n_1+1}^{(\tilde{\delta}) }\big(\frac{\omega r^2}{2}\big)$ given by \cite{que,os}
\be\label{ext_rad_sol_1}
R_{1,n_1}(r)\propto \frac{r^{(\tilde{\delta}+1/2)}\exp{\big(-\frac{\omega r^2}{4}\big)}}
{(\omega r^2+2\tilde{\delta})}\hat{L}_{n_1+1}^{(\tilde{\delta}) }\big(\frac{\omega r^2}{2}\big);\qquad 0< r < \infty.
\ee
The energy eigenvalue $E$ which depends on $n_1, n_2$ and $n_3$ is given by
\be\label{en123}
E_{n_1,n_2,n_3}=\omega(2n_1+1+\tilde{\delta}),
\ee
where $\tilde{\delta}=\sqrt{\delta+\ell^2}$ and
\be\label{en_1}
\ell^2=\bigg[(2n_2+1)+\sqrt{D+1/4}+\bigg \{ C+\bigg(\sqrt{F+p^2/4}+\sqrt{G+p^2/4}+p(2n_3+1)\bigg)^2\bigg \}^{1/2}\bigg]^2.
\ee
Again note that the energy eigenvalues are unchanged while the corresponding
eigenfunctions are different from the nonrational case.
{\bf Case (ii): For any positive integer values of $m_1, m_2$ and $m_3$}
In this case, we consider a more general form of the ten parameters potential given in Eq. (\ref{non_ten}). Similar to the $X_1$ case, on comparing Eqs. (\ref{non_ten}) and (\ref{v_rat}), we obtain the
rationally extended trigonometric P\"oschl-Teller equation
\ba\label{extd_phi}
U^{(I)}_{m_3,ext}(\phi)= U^{(I)}_{con}(\phi)+ U^{(I)}_{m_3,rat}(\phi),
\ea
where $U^{(I)}_{con}(\phi)$ and $U^{(I)}_{m_3,rat}(\phi)$ are given by Eqs. (\ref{con_rpt}) and (\ref{rat3}).
The unnormalized wavefunction of Eq. (\ref{ucext_1}) with the extended potential (\ref{extd_phi}) in terms of
the $X_{m_3}$ exceptional Jacobi polynomial $\hat{P}_{n_3+m_3}^{(\tilde{\alpha} ,\tilde{\beta}) }(z)$ is given by \cite {os}
\be\label{ext_phi_sol}
\Phi^{(I)}_{m_3,n_3}(\phi)\propto \frac{(1-z )^{\frac{\tilde{\alpha} }{2} + \frac{1}{4}}(1 +z )^{\frac{\tilde{\beta} }{2} + \frac{1}{4}}}
{P_{m_3}^{(-\tilde{\alpha} -1,\tilde{\beta} -1)}(z)}\hat{P}_{n_3+m_3}^{(\tilde{\alpha} ,\tilde{\beta}) }(z);\qquad 0\leq p\phi\leq \pi/2,
\ee
where $n_3=0,1,2,....;$ \quad $m_3=1,2,3...$; \quad $z=\cos(2p\phi)$ and the parameters
$\tilde{\alpha}$ and $\tilde{\beta}$ will be same as obtained in Eq. (\ref{al_telda}).
The eigenvalue of this extended potential is same (i.e. isospectral) as that of the conventional
potential $U^{(I)}_{con}(\phi)$ given by Eq. (\ref{ms}).
Similar to the above case from Eqs. (\ref{non_ten}) and (\ref{v_rat}), the $\theta$ and $r$ dependent
extended potentials which depend on parameters
$m_2$ and $m_3$ respectively are given as
\ba\label{extd_theta}
V^{(I)}_{m_2,ext}(\theta)= V^{(I)}_{con}(\theta)+ V^{(I)}_{m_2,rat}(\theta),
\ea
and
\ba\label{extd_rad}
\tilde{U}_{m_1,ext}(r)=\tilde{U}_{con}(r) + \tilde{U}_{m_1,rat}(r).
\ea
The corresponding conventional potential terms $V^{(I)}_{con}(\theta)$ and $\tilde{U}_{con}(r)$ and the rational terms $V^{(I)}_{m_2,rat}(\theta)$ and $\tilde{U}_{m_1,rat}(r)$ are given in Eqs. (\ref{extd_theta_1}), (\ref{extd_rad_1}) and (\ref{rat2}), (\ref{rat1}).
Following the same procedure as in the $X_1$ case, the solutions of the the Eqs. (\ref{ucext_2}) and (\ref{ucext_3}) with the corresponding
potentials $V^{(I)}_{m_2,ext}(\theta)$ and $\tilde{U}_{m_1,ext}(r)$ can be obtained in a simple way. Thus the bound state wavefunctions
corresponding to these potentials are given by
\be\label{ext_theta_sol}
\Theta^{(I)}_{m_2,n_2}(\theta)\propto \frac{(1-z )^{\frac{\alpha}{2} + \frac{1}{4}}(1 +z )^{\frac{\beta}{2} + \frac{1}{4}}}
{P_{m_2}^{(-\alpha -1,\beta -1)}(z)}\hat{P}_{n_2+m_2}^{(\alpha ,\beta) }(z);\qquad 0\leq \theta \leq \pi/2,
\ee
and
\be\label{ext_rad_sol}
R_{m_1,n_1}(r)\propto \frac{r^{(\sqrt{\delta+\ell^2}+1/2)}\exp{\big(-\frac{\omega r^2}{4}\big)}}
{L_{m_1}^{(\sqrt{\delta+\ell^2}-1)}\big(-\frac{\omega r^2}{2}\big)}\hat{L}_{n_1+m_1}^{(\sqrt{\delta+\ell^2}) }\big(\frac{\omega r^2}{2}\big);\qquad 0< r < \infty,
\ee
where $\hat{P}_{n_2+m_2}^{(\alpha ,\beta) }(z)$ and $\hat{L}_{n_1+m_1}^{(\sqrt{\delta+\ell^2}) }\big(\frac{\omega r^2}{2}\big)$ are
$X_{m_2}$ exceptional Jacobi and $X_{m_1}$ exceptional Laguerre polynomials respectively. The energy eigenspectrum and the other parametric
relations will be same as that of the $X_1$ case.
\subsection{ Example of Rationally Extended $PT$ symmetric complex non-central potential}
Similar to the above example of real case, here we consider an example of RE
non-central potential which is complex but symmetric under the combined
operation of the parity ($P$) ($P: r\rightarrow r, \theta\rightarrow \pi-\theta, \phi\rightarrow \phi+\pi$) and the
time reversal ($T$) ($T: t \rightarrow -t, i\rightarrow -i$) operators and given by
\ba\label{pot_extd_pt}
V^{(PT)}_{m_1,m_2,m_3}(r,\theta,\phi)&=&\frac{\omega^2 r^2}{4}+\frac{\delta}{r^2}+\tilde{U}_{m_1,rat}(r)+\frac{C}{r^2\sin^2\theta}+\frac{iD}{r^2\tan\theta}+\frac{1}{r^2}V^{(PT)}_{m_2,rat}(\theta)\nonumber\\
&+&\frac{G}{r^2\sin^2\theta\sin^2 p\phi}
+\frac{F}{r^2\sin^2\theta \cos^2p \phi}+\frac{1}{r^2\sin^2\theta}U^{(I)}_{m_3,rat}(\phi),
\ea
where $V^{(PT)}_{m_2,rat}(\theta)$ is given by Eq. (\ref{ext_theta_pt}).
In this case, we obtain a complete solution of
this potential by considering the same form of the $\phi$ and $r$ dependent terms (as defined in the first example) with a
new form of $\theta$ dependent term which is now complex but PT-invariant.
{\bf Case (i) For $m_1=m_2=m_3=1$}
For this particular case, on comparing the above Eq. (\ref{pot_extd_pt}) with Eq. (\ref{v_rat}) (by defining $h=PT$ for $\theta$ dependent term and
$h=I$ for $\phi$ dependent term), we get the $PT$ symmetric extended potential
\ba\label{extd_theta_pt}
V^{(PT)}_{1,ext}(\theta)= V^{PT}_{con}(\theta)+ V^{(PT)}_{1,rat}(\theta),
\ea
where
\be\label{con_pt}
V^{(PT)}_{con}(\theta )=C\cosec^2\theta+iD\cot\theta,
\ee
is the conventional $PT$ symmetric trigonometric Eckart potential\footnote{Which is
easily obtained by complex co-ordinate transformation $x\rightarrow ix$ of the
rationally extended hyperbolic Eckart potential given in \cite{cq_12}.} and the associated rational term
\ba
V^{(PT)}_{1,rat}(\theta)&=&\frac{1}{A^2(A-1)^2}\bigg[\frac{-4iB[A^2(A-1)^2-B^2]}{(iB+A(A-1)\cot\theta )}\nonumber\\
&+& \frac{2[A^2(A-1)^2-B^2]^2}{(iB+A(A-1)\cot \theta )^2}\bigg].
\ea
The form of the $\phi$ and the $r$ dependent extended terms will be same as
defined by Eqs. (\ref{extd_phi_1}) and (\ref{extd_rad_1}). The solution of Eq. (\ref{ucext_2}) (for $m_2=1$ and $h=PT$) with the above potential (\ref{extd_theta_pt}) is not in
the exact form of EOPs rather they are written in the form of some types of new polynomials (discussed in detail in Ref. \cite {cq_12}) given as
\be\label{wfpt_eck_1}
\Theta^{PT}_{1,n_2}(\theta)\propto \frac {(z-1)^{\frac{\alpha_{n_2}}{2}}(z+1)^{\frac{\beta_{n_2}}{2}}}{(iB+A(A-1)\cot \theta) )}y^{(A,B)}_{n_2}(z),
\ee
with $z=i\cot \theta $. Here the polynomial function $y^{(A,B)}_{n_2}(z)$
can be expressed in terms of the classical Jacobi
polynomials $P^{(\alpha_{n_2},\beta_{n_2})}_{n_2}(z)$ as
\ba
y^{(A,B)}_{n_2}(z)&=&\frac{2(n_2+\alpha_{n_2})(n_2+\beta_{n_2})}{(2n_2+\alpha_{n_2}+\beta_{n_2})}q^{(A,B)}_1(z)P^{(\alpha_{n_2},\beta_{n_2})}_{n_2-1}(z)\nonumber\\
&-&\frac{2(1+\alpha_1 ) (1+\beta_1 )}{(2+\alpha_1+\beta_1)}P^{(\alpha_{n_2},\beta_{n_2})}_{n_2}(z).
\ea
Here $q^{(A,B)}_1(z)=P^{(\alpha_1,\beta_1)}_{1}(z)$ (Classical Jacobi
polynomial $P^{(\alpha_1,\beta_1)}_{n_2}(z)$ for $n_2=1$). The parameters $\alpha_{n_2}$ and $\beta_{n_2}$ in terms of $A$ and $B$ are
given by
\be
\alpha_{n_2}=-(A-1+n_2)+\frac{B}{(A-1+n_2)}; \quad \beta_{n_2} =-(A-1+n_2)-\frac{B}{(A-1+n_2)}.
\ee
with
\be
A=\frac{1}{2}+\sqrt{C+m^2}, \quad \mbox{and}\quad B=\frac{D}{2}.
\ee
The other two parameters $\alpha_1$ and $\beta_1$ are simply obtained by putting $n_2=1$ in $\alpha_{n_2}$ and $\beta_{n_2}$.
The energy eigenvalues are given by
\be\label{enpt_1}
\ell^2=(A-1+n_2)^2+\frac{B^2}{(A-1+n_2)^2}; \quad n_2=0,1,2,...
\ee
Thus the complete wavefunction associated with the extended $PT$ symmetric complex non-central
potential Eq. (\ref{v_rat}) (for $m_1=m_2=m_3=1$) is obtained by using Eq. (\ref{wf_non_rat}), which is a product of the
$X_1$ Jacobi polynomial (as given by Eq. (\ref{ext_phi_sol_1})), $X_1$ Laguerre polynomial (as given by Eq. (\ref{ext_rad_sol_1})) times
a new polynomial given in Eq. (\ref{wfpt_eck_1}).
{\bf Case (ii) For any positive integer values of $m_1, m_2$ and $m_3$}
Again by considering the same form of the $\phi$ and $r$ dependent terms for any arbitrary values of $m_3$ and $m_1$,
the above complex potential can be generalized
easily for any non-zero positive integer values of $m_2$ by defining
\ba\label{extd_theta_ptm}
V^{(PT)}_{m_2,ext}(\theta)= V^{(PT)}_{con}(\theta)+ V^{(PT)}_{m_2,rat}(\theta),
\ea
where $V^{PT}_{con}(\theta )$ and $V^{(PT)}_{m_2,rat}$ are given by Eqs. (\ref{con_pt}) and (\ref{ext_theta_pt}) respectively.
The wavefunction associated with this potential corresponding to the Eq. (\ref{ucext_2}) is given by
\be\label{wfpt_eckm}
\Theta^{(PT)}_{m_2,n_2}(\theta)\propto \frac{(z-1)^\frac{\alpha_{n_2}}{2}(z+1)^\frac{\beta_{n_2}}{2}}{q^{(A,B)}_{m_2}(z)}y^{(A,B)}_{\nu,m_2}(z); \quad \nu=n_2+m_2-1,
\ee
where $q^{(A,B)}_{m_2}(z)=P^{(\alpha_{m_2},\beta_{m_2})}_{m_2}$ and the polynomial
function $y^{(A,B)}_{\nu,m_2}(z)$ is
\ba\label{y}
y^{(A,B)}_{\nu,m_2}(z)&=&\frac{2(n_2+\alpha_{n_2})(n_2+\beta_{n_2})}{(2n_2+\alpha_{n_2}+\beta_{n_2})}q^{(A,B)}_{m_2}(z)P^{(\alpha_{n_2},\beta_{n_2})}_{n_2-1}(z)\nonumber\\
&-&\frac{2(m_2+\alpha_{m_2})(m_2+\beta_{m_2})}{(2m_2+\alpha_{m_2}+\beta_{m_2})}q^{(A+1,B)}_{m_2-1}(z)P^{(\alpha_{n_2},\beta_{n_2})}_{n_2}(z),
\ea
with the parameters
\ba
\alpha_{m_2}=-(A-1+m_2)+\frac{B}{(A-1+m_2)}; \quad \beta_{m_2} =-(A-1+m_2)-\frac{B}{(A-1+m_2)}.\nonumber\\
\ea
The energy eigenvalues will be same as given by Eq. (\ref{enpt_1}). Thus
the complete wavefunction and the eigenvalues of this complex non-central extended potential
are obtained by using Eqs. (\ref{wf_non_rat}) and (\ref{en123}).
{\footnotesize Table I. In this table all the four forms of
$V^{(h)}_{m_2, ext}(\theta )$ (for $h=I,II, (PT)_1$ and $(PT)_2$) with their
corresponding energy eigenvalues ($\ell^2$) and the eigenfunctions
($\Theta ^{(h)}_{m_2,n_2}(\theta )$) are given. Cases (i) and (iii) are
discussed in detail in the text.}
\begin{tabular}{|l|*3{c|}} \hline
\textbf{$V^{(h)}_{m_2, ext}(\theta )$}&\pbox{30cm}{ \textbf{$\ell^2$}}& $\Theta ^{(h)}_{m_2,n_2}(\theta )$\\ \hline
\pbox{60cm} {$(i) \quad V^{(I)}_{m_2,ext} (\theta )$}
&\pbox{60cm} {$(2n_2 +\alpha +\beta +1)^2$\\
$n_2 = 0,1,2,...$\\
$\alpha =\sqrt{C+m^2}$ \\
$\beta =\frac{1}{2}\sqrt{1+4D}$ }&
\pbox{60cm} {$\frac{(1-z)^{\frac{\alpha }{2}+\frac{1}{4}}(1+z)^{\frac{\beta }{2} +\frac{1}{4}} } {P^{(-\alpha -1, \beta -1)}_{m_2} (z)}
\hat{P}^{(\alpha ,\beta )}_{n_2+m_2} (z) ;$ \\
$z = \cos 2\theta $\\
$m_2=1,2... $}\\ \hline
\pbox{60cm} {$(ii) \quad V_{m_2, ext}^{(II)}(\theta )$}
&\pbox{60cm} {$(n_2 +\frac{\alpha +\beta +1}{2})^2$\\
$\alpha =\sqrt{C+m^2 -D}$ \\
$\beta =\sqrt{C+m^2 +D}$ }&
\pbox{60cm} {$\frac{(1-z)^{\frac{\alpha }{2}+\frac{1}{4}}(1+z)^{\frac{\beta }{2} +\frac{1}{4}}} {P^{(-\alpha -1, \beta -1)}_{m_2} (z)}
\hat{P}^{(\alpha ,\beta )}_{n_2+m_2} (z) ;$ \\
$z = \cos \theta $}\\ \hline
\pbox{60cm} {$(iii) \quad V_{m_2, ext}^{(PT)_1}(\theta )$}
&\pbox{60cm} {$(A-1+n_2)^2 + \frac{B^2}{(A-1+n_2)^2}$\\
$A=\frac{1}{2} +\sqrt{C+m^2}; B =\frac{D}{2}$ \\
$ \alpha _{n_2} = -(A-1+n_2)+ \frac{B}{(A-1+n_2)}$ \\
$\beta _{n_2} = -(A-1+n_2)- \frac{B}{(A-1+n_2)} $ }&
\pbox{60cm} {$\frac{(z-1)^{\frac{\alpha _{n_2}}{2}} (z+1)^{\frac{\beta _{n_2}}{2}}}{q_{m_2}^{(A, B)} (z)}y^{(A,B)}_{\nu,m_2}(z);$ \\
$z=i \cot \theta $ \\
$\nu = n_2 +m_2-1,$}\\ \hline
\pbox{60cm} {$(iv) \quad V_{m_2, ext}^{(PT)_2}(\theta )$}
&\pbox{60cm} {$(n_2 +\frac{\alpha +\beta +1}{2})^2=(n_2+A)^2$\\
$\alpha =\sqrt{C+m^2 -iD}$ \\
$\beta =\sqrt{C+m^2 +iD}$ \\
$C+m^2=-B^2+(A-\frac{1}{2})^2$\\
$D=2AB$\\}&
\pbox{60cm} {$\frac{(1-z)^{\frac{\alpha }{2}+\frac{1}{4}}(1+z)^{\frac{\beta }{2} +\frac{1}{4}}} {P^{(-\alpha -1, \beta -1)}_{m_2} (z)}
\hat{P}^{(\alpha ,\beta )}_{n_2+m_2} (z) ;$ \\
$z = \cos \theta $}\\ \hline
\end{tabular} \pagebreak
{\footnotesize Table II. The three different forms of $\phi$ dependent terms $V^{(h)}_{m_3, ext}(\phi)$ (for $h=I,II,(PT)_1$) with their corresponding energy eigenvalues ($m^2$) and the eigenfunctions ($\Phi ^{(h)}_{m_3,n_3}(\phi)$) are given. Out of these, case (i) is already considered in detail in the text.}
\begin{tabular}{|l|*3{c|}} \hline
\textbf{$U^{(h)}_{m_3, ext}(\phi)$}&\pbox{30cm}{ \textbf{$m^2$}}& $\Phi ^{(h)}_{m_3,n_3}(\phi)$\\ \hline
\pbox{60cm} {$ (i) \quad U^{(I)}_{m_3,ext} (\phi )$}
&\pbox{60cm} {$p^2(2n_3 +\tilde{\alpha } +\tilde{\beta } +1)^2$\\
$n_3 = 0,1,2,...$\\
$\tilde{\alpha } =\frac{1}{2}\sqrt{1+\frac{4G}{p^2}}$ \\
$\tilde{\beta } =\frac{1}{2}\sqrt{1+\frac{4F}{p^2}}$ }&
\pbox{60cm} {$\frac{(1-z)^{\frac{\tilde{\alpha } }{2}+\frac{1}{4}}(1+z)^{\frac{\tilde{\beta } }{2}+\frac{1}{4}}} {P^{(-\tilde{\alpha } -1, \tilde{\beta } -1)}_{m_3} (z)}
\hat{P}^{(\tilde{\alpha } ,\tilde{\beta } )}_{n_3+m_3} (z) ;$ \\
$z = \cos (2p\phi) $ \\
$p=1,2,3,...$\\
$m_3=1,2,3....$}\\ \hline
\pbox{60cm} {$(ii) \quad U_{m_3, ext}^{(II)} (\phi )$}
&\pbox{60cm} {$p^2(n_3 +\frac{\tilde{\alpha } +\tilde{\beta } +1}{2})^2$\\
$\tilde{\alpha } =\frac{1}{2} \sqrt {1+\frac{4G}{p^2} -\frac{4F}{p^2}}$ \\
$\tilde{\beta } =\frac{1}{2} \sqrt {1+\frac{4G}{p^2} +\frac{4F}{p^2}}$ }&
\pbox{60cm} {$\frac{(1-z)^{\frac{\tilde{\alpha } }{2}+\frac{1}{4}}(1+z)^{\frac{\tilde{\beta } }{2}+\frac{1}{4}}} {P^{(-\tilde{\alpha } -1, \tilde{\beta } -1)}_{m_3} (z)}
\hat{P}^{(\tilde{\alpha } ,\tilde{\beta } )}_{n_3+m_3} (z) ;$ \\
$z = \cos (p\phi )$\\
$p=1,3,5,...$}\\ \hline
\pbox{60cm} {$(iii) \quad U_{m_3, ext}^{(PT)_1} (\phi )$}
&\pbox{60cm} {$p^2(n_3 +\frac{\tilde{\alpha } +\tilde{\beta } +1}{2})^2=(n_3p+A)^2$\\
$\tilde{\alpha } =\frac{1}{2} \sqrt {1+\frac{4G}{p^2} -\frac{4iF}{p^2}}$ \\
$\tilde{\beta } =\frac{1}{2} \sqrt {1+\frac{4G}{p^2} +\frac{4iF}{p^2}}$\\
$G=A^2-B^2-Ap$\\
$F=B(2A-p)$ }&
\pbox{60cm} {$\frac{(1-z)^{\frac{\tilde{\alpha } }{2}+\frac{1}{4}}(1+z)^{\frac{\tilde{\beta } }{2}+\frac{1}{4}}} {P^{(-\tilde{\alpha } -1, \tilde{\beta } -1)}_{m_3} (z)}
\hat{P}^{(\tilde{\alpha } ,\tilde{\beta } )}_{n_3+m_3} (z) ;$ \\
$z = \cos (p\phi )$\\
$p=1,3,5,...$}\\ \hline
\end{tabular}
\section{RE non-central potentials in $2$-Dimensions}
In two dimensional polar co-ordinates $(r,\phi)$, the Schr\"odinger equation corresponding
to the non-central potential $V_{m_1,m_3}(r,\phi)$ is given by $(\hbar=2m=1)$
\be\label{sch_2d1}
\bigg[-\frac{d^2}{dr^2}-\frac{1}{r}\frac{d}{dr}-\frac{1}{r^2}
\frac{d^2}{d\phi^2}\bigg]\Psi(r,\phi)+V_{m_1,m_3}(r,\phi)\Psi(r,\phi)= E\psi(r,\Phi)\,.
\ee
The forms of the non-central potential in this co-ordinate system is given by
\be\label{pot_2d}
V^{(h)}_{m_1,m_3}(r,\phi)=\tilde{U}_{m_1,ext}(r)+\frac{1}{r^2}U^{(h)}_{m_3,ext}(\phi ),
\ee
with $h=1,2,(PT)_1$ and $(PT)_2$.
The above Schr\"odinger equation is exactly solvable, if we define the wave function in the form
\be\label{wf_2d1}
\Psi(r,\phi)=\frac{R_{m_1}(r)}{r^{1/2}}\Phi^{(h)}_{m_3}(\phi )\,.
\ee
Using $\Psi(r,\phi)$ in Eq. (\ref{sch_2d1}), the angular component of the wave
function satisfies the equation
\be\label{phi_2d}
\bigg[-\frac{d^2}{dr^2}+U_{m_3,ext}(\phi)\bigg]\Phi^{(h)}_{m_3}(\phi)=m^2\Phi^{(h)}_{m_3}(\phi),
\ee
and the radial component satisfies
\be\label{rad_2d}
\bigg[-\frac{d^2}{dr^2}+\tilde{U}_{m_1,ext}(r)+\frac{\big(m^2-\frac{1}{4}\big)}{r^2}\bigg]R_{m_1}(r)=ER_{m_1}(r),
\ee
where $m^2$ is the eigenvalue of the angular equation.
These two equations (\ref{phi_2d}) and (\ref{rad_2d}) are identical to the
corresponding
equations (\ref{ucext_1}) and (\ref{ucext_3})
obtained in the three dimensional case except the parameter $\ell$ in the
three dimensional case is now replaced by $m$ in the radial
component of the Eq. (\ref{rad_2d}).
In this case, we have one choice of $\tilde{U}_{m_1,ext}(r)$ and four choices of $V^{(h)}_{m_3,ext}(\phi)$ (two real and two $PT$ symmetric).
The form of $\tilde{U}_{m_1,ext}(r)$ will be same as in the case of three dimensions with the parameter $\tilde{\delta}=\sqrt{\delta+m^2}$.
Out of these four choices of $V^{(h)}_{m_3,ext}(\phi)$, three ( two real and one $PT$ symmetric, $h=I,II,(PT)_1$) are already discussed in Table $II$
of the previous section while the fourth form of the potential ($h=(PT)_2$) is
special to the two dimensions.
The form of this complex potential is given by
\be\label{ext_phi_2pt}
V^{(PT)_2}_{m_3,ext}(\phi)=\frac{G}{\sin^2 (p\phi)}+\frac{iF}{\tan (p\phi)}+V^{(PT)_2}_{m_2,rat}(\phi),
\ee
where
\ba
V^{(PT)_2}_{m_3,rat}(\phi) &=& -2p^2\cosec^2(p\phi)\bigg[ 2i\cot(p\phi) \frac{\dot{q}^{(A/p,B/p)}_{m_3}(z)}{q^{(A/p,B/p)}_{m_3}(z)}-\cosec^2(p\phi)\nonumber\\
&\times & \Bigg( \frac{\ddot{q}^{(A/p,B/p)}_{m_3}(z)}{q^{(A/p,B/p)}_{m_3}(z)}-\bigg( \frac{\dot{q}^{(A/p,B/p)}_{m_3}(z)}{q^{(A/p,B/p)}_{m_3}(z)}\bigg )^2 \Bigg )-m_3\bigg ];\quad m_3=1,2,3....\nonumber
\ea
with $z=\cos (p\phi);\quad 0\leq p\phi\leq \pi$ and
\be
q_{m_3}^{(A/p, B/p)}(z)=P^{(\tilde{\alpha}_{m_3},\tilde{\beta}_{m_3})}_{m_3}(z)
\ee
is a classical Jacobi polynomials with the parameters
\ba
\tilde{\alpha} _{m_3} &=& -(A/p-1+m_3)+ \frac{B/p}{(A/p-1+m_3)} \nonumber\\
\tilde{\beta} _{m_3} &=& -(A/p-1+m_3)- \frac{B/p}{(A/p-1+m_3)}.
\ea
Here $p$ is restricted to the positive odd integers only and a dot on
$q_{m_3}^{(A/p, B/p)}(z)$ indicates single derivative with $z$.
An explanation is in order as to why the potential as given by
Eq. (\ref{ext_phi_2pt}) is a PT-symmetric complex potential in two
dimensions but only a complex but not PT-symmetric in three
dimensions. The point is that unlike three space dimensions, parity in two
space dimensions correspond to say $x \rightarrow -x, y \rightarrow +y$, i.e.
it corresponds to ($P: r\rightarrow r, \phi\rightarrow \pi-\phi$). The time
reversal corresponds to ($T: t\rightarrow -t, i\rightarrow -i$) symmetry, the
above potential (\ref{ext_phi_2pt}) is $PT$ symmetric in $2$-dimensions but
not in three space dimensions since in three dimensions $\phi \rightarrow
\pi + \phi$.
Of course one can also consider the potential as given by
Eq. (\ref{ext_phi_2pt}) in three dimensions and there it is merely complex
but non $PT$-symmetric potential. However, the spectrum is still real thereby
confirming the well known fact that $PT$-symmetry is sufficient but not
necessary for the spectrum to be real. Note that we also have considered
this type of $\theta$-dependent potential in case $(iii)$ of Table I (detail
solution is also given in Section $2.2$), since such a potential is indeed
complex and PT-invariant term. This is because, in three dimensions, under
parity, unlike $\phi$, $\theta \rightarrow \pi - \theta$.
Following the same procedure as in the three dimensional case, the solution of
this $2$-dimensional
$PT$ symmetric non-central potential
\be\label{v_2d}
V^{(PT)_2}_{m_1,m_3}(r,\phi)=\tilde{U}_{m_1,ext}(r)+\frac{1}{r^2}U^{(PT)_2}_{m_3,ext}(\phi),
\ee
can also be obtained in a straightforward way. In particular, the solution for
the special case of $m_1=m_3=1$
is straightforward one, therefore we only consider the general case of any
arbitrary positive integers $m_1$ and $m_3$.
Using Eq. (\ref{ext_phi_2pt}) in the angular equation
(\ref{phi_2d}) we get the solution of the form
\be\label{ext_phi_sol_2d}
\Phi^{(PT)_2}_{m_3,n_3}(\phi)\propto \frac{(z-1)^{\frac{\tilde{\alpha} _{n_3}}{2}} (z+1)^{\frac{\tilde{\beta} _{n_3}}{2}}}{q_{m_3}^{(A/p, B/p)} (z)}y^{(A/p,B/p)}_{\nu,m_3}(z),
\ee
where
\ba
\tilde{\alpha} _{n_3} &=& -(A/p-1+n_3)+ \frac{B/p}{(A/p-1+n_3)}\nonumber\\
\tilde{\beta} _{n_3} &=& -(A/p-1+n_3)- \frac{B/p}{(A/p-1+n_3)},
\ea
and the form of $y^{(A/p,B/p)}_{\nu,m_3}(z)$ will be same as given by Eq. (\ref{y}) (where we replace $n_2\rightarrow n_3, m_2\rightarrow m_3, A\rightarrow A/p, B\rightarrow B/p$ and $z\rightarrow \cos(p\phi)$).
The energy eigenvalue $m^2$ is given by
\be
m^2=p^2\bigg[(A/p-1+n_3)^2 + \frac{B^2/p^2}{(A/p-1+n_3)^2}\bigg];\quad n_3=0,1,2,3...
\ee
The parameters $A$ and $B$ in terms of $F$ and $G$ are related as
\be
\frac{A}{p}=\frac{1}{2} +\frac{1}{2}\sqrt{1+4G/p^2}; \quad 2B =F.
\ee
Now using equation (\ref{ext_r}) in the radial part of the Schr\"odinger equation (\ref{rad_2d}) we get
\be\label{ext_rad_sol_2dm}
R_{m_1,n_1}(r)\propto \frac{r^{(\sqrt{\delta+m^2}+1/2)}\exp{\big(-\frac{\omega r^2}{4}\big)}}
{L_{m_1}^{(\sqrt{\delta+m^2}-1)}\big(-\frac{\omega r^2}{2}\big)}\hat{L}_{n_1+m_1}^{(\sqrt{\delta+m^2}) }\big(\frac{\omega r^2}{2}\big);\qquad 0< r < \infty,
\ee
with the energy eigenvalues
\be\label{en2d}
E_{n_1,n_3}=\omega(2n_1+1+\tilde{\delta}),
\ee
where
\be\label{del}
\tilde{\delta}=\sqrt{\delta+m^2}.
\ee
Thus the complete wavefunction and the eigenspectrum for the above seven
parameter (four continuous and three discrete) family of potential
(\ref{v_2d}) are given by Eqs. (\ref{wf_2d1}) and (\ref{en2d}).
In the particular case of $m_1=m_3=0$, these potentials are reduced to their corresponding conventional potentials whose
solutions are associated with the classical orthogonal polynomials.
\section{Summary}
In this work, we have constructed twelve rationally extended non-central real
and $PT$ symmetric complex potentials in three dimensional spherical polar co-ordinates. The solutions of these
potentials are obtained by using
the recently discovered rationally extended potentials whose solutions are
in terms of $X_{m_1}, X_{m_2} $ or $X_{m_3}$ exceptional Laguerre and (or) Jacobi
orthogonal polynomials. The eigenfunctions
and the energy eigenvalues of these twelve extended non-central potentials are
obtained explicitly and shown that the eigenfunctions
of these extended non-central potentials are the product of Laguerre and Jacobi EOPs. It is found that the
three dimensional Schr\"odinger equation is exactly solvable for the one possible choice for
$\tilde{U}_{m_1,ext}(r)$, four possible choices for $V^{(h)}_{m_2,ext}(\theta)$
and three choices for $U^{(h)}_{m_3,ext}(\phi)$.
All possible choices of $\theta$ and $\phi$ dependent potentials and the
corresponding solutions are listed in Tables $I$ and $II$.
The various combinations of $\tilde{U}_{m_1,ext}(r), V^{(h)}_{m_2,ext}\quad (\theta)$ and $U^{(h)}_{m_3,ext}(\phi)$ lead to the
total twelve different forms (four real and eight $PT$ symmetric complex) of
the RE non-central potentials.
In the examples of $PT$ symmetric cases, some of the solutions corresponding
to the $\theta$
dependent term is not in the exact form of EOPs, they are written in the forms of some types of new orthogonal polynomials ($y^{(A,B)}_{\nu,m_2}(z)$) which
are simplified further in the terms of classical Jacobi polynomials.
In this works we have only consider one choice of $r$ dependent extended potential as a RE radial oscillator case.
One can also replace the RE radial oscillator part with the conventional coulomb $U_{con}(r)=-\frac{e^2}{r}+\frac{\delta}{r^2}$
(as shown in Ref. \cite{kb}) then one will
have nine parameters RE non-central potential and the spectrum can also be obtained easily. Few attempts at rational
extension of Coulomb have been done \cite{yg2}, but they are not very general, so we are not mentioning them.
In a particular case of $m_1=m_2=m_3=0$, these potentials are reduced to their
conventional counterparts (which are non-rational) with seven parameters (six continuous and one discrete)
whose solutions are in terms of classical orthogonal polynomials.
Out of these twelve conventional cases, the eight $PT$ symmetric complex non-central seven parameters conventional potentials
are also new and not discussed earlier.
We have also considered the Schr\"odinger equation in two dimensional
polar co-ordinates and constructed four possible forms (two real
and two $PT$ symmetric complex) of the seven parameters (four continuous and
three discrete) RE non-central potentials. The solutions of these potentials
are also obtained in terms of EOPs.
{\bf Acknowledgments}
B.P.M. acknowledges the financial support
from the Department of Science and Technology (DST), Gov. of India under SERC project sanction
grant No. $SR/S2/HEP-0009/2012$. A.K. wishes to thank Indian National Science Academy (INSA) for
the award of INSA senior scientist position at Savitribai Phule Pune University. | 166,111 |
Watching Closely and Singing Fully
On today’s show: Mother Jones senior editor Nick Baumann tells the story of an American expat who was cleared of having ties to terrorism but still feels the effects of the government’s suspicions. Writer and director Michael Maren talks about his film, “A Short History of Decay,” with Bryan Greenberg and Harris Yulin, who star in it. Former Gourmet editor-in-chief Ruth Reichl on her first novel, called Delicious! We’ll talk to two editors of Civil Eats, a daily news source about America’s food system that just won James Beard Award for Publication of the Year. And opera legend Jessye Norman looks back on her childhood in Georgia and her career in music. | 171,683 |
Toll Free 1-855-BH-PULSE (1-855-247-8573)
You haven’t been yourself lately. You can’t put your finger on what’s wrong, but something just doesn’t feel right. It could be nothing—but you, and in less than one minute, you’ll learn how to take your pulse. If your heartbeat is irregular, please see your primary care physician. If you need a doctor, we can help you find one.
Stop that stroke from happening! | 255,300 |
Enthusiasts around the world find some way to the casinos when and as required. Thanks to the internet revolution, it is now possible to be a part of the Vegas fun without even traveling to the casinos there or wasting precious gambling time in making it to the venue. You can now sit in the comfort of your bedroom or living room and either indulge in the slots or tables on your own or get friends along and have club fun at home!
All that you need to learn even as a newbie is now accessible online. Gambling and bankroll strategies are an integral and important part of the arena and getting up-close to the tactics is now but intent away. All that you need to do is get online and click. Search engines are just as enthusiastic about revealing information on these strategies as they are for any other. The best sites are those that give you an education as you play.
What is the use of the access if you cannot learn and interact with the pros! This is exactly what you should be looking for when you want to access online gambling and bankroll strategy information. The maneuvers around the tables and the management of the bankroll are all over the internet. It pays to research and accept information from the right resources prior to hitting the jackpot. dream casino
Your love for gambling sites should also reveal the winning bankrolls strategies to you like:
- Deciding first on how you want to operate – bankroll wager in 2 hours or gambling for 2 days and going strong onward
- Bankroll Management at a level that is justifiable
- Never more than an average bet size of 1.5 or 2% of the whole bankroll at hand
- Banking on the lowest house edges especially at the competitions
To keep the bankroll ticking as you enjoy the payouts it is important to manage the funds at hand. It is very essential to also follow the rules played by some of the most successful people in the industry. The internet is a storehouse of information and all that you really have to do is ask! Information comes to you from professionals who are adept at gambling and have lived most part of their lives on the tables. Ticks and maneuvers are best learned from the professional right?
There are highly interactive forums that you can tap potential on via the internet. They not only allow you to access the FAQs on gambling and bankrolls strategies but also help you to be able to play in your own time and with timely guidance and help flashing all the way long! Make the most of the connectivity not only to play but also to pick up tips and suggestions along the way. After all, which other arena would enable you to learn from the mistakes of others and have so much of fun as you earn and that too from home! Earn while you learn and add to the bankroll with online gambling that is planned and strategized. | 1,002 |
This video is called OUTFOXED: Rupert Murdoch’s War on Journalism — Trailer.
Rupert Murdoch is an immigrant (from Australia) when he is in the USA to tend to
Faux Fox News, the War Street Wall Street Journal, and other parts of his far Right media empire.
Rupert Murdoch, arguably, (using arguments somewhat similar to xenophobes’ for whom especially non white people are “immigrants” even if their ancestors have lived for hundreds of years in a so-called “white” country) is an immigrant in his native Australia, as he descends from people who immigrated to that continent … fair enough … while its Aboriginal people were robbed of their land and massacred … not so fair.
Rupert Murdoch is an immigrant in Britain. That unfortunately does not stop his media empire there from whipping up hysteria against immigrants. Including refugees, who fled to Britain from torture and war.
From British weekly Socialist Worker:
‘The Sun used me to push its own agenda’
Zimbabwean former asylum seeker Urginia Mauluka is shocked after being used in an anti-immigrant Sun article. She spoke to Matthew Cookson
The Sun newspaper is not known for its sympathetic portrayal of asylum seekers—so Urginia Mauluka was a little surprised to get a call saying that it wanted to interview her about her experiences.
Urginia, who fled from her native Zimbabwe in 2002 and became a British citizen earlier this year, nevertheless agreed.
The Zimbabwean state and pro-government independence war veterans had targeted Urginia because of her role as an independent photographer documenting the violence of the regime and its supporters.
She said, “I thought my story could be documented in such a way to inform people and other asylum seekers about the reality of life for us. I thought the Sun would be responsible.”
But when she saw the printed version of the article, the mistakes it contained and the way it was presented left her “shocked” and “angry”.
The Sun printed Urginia’s story as part of a double page spread on immigration on Friday of last week, launching a new anti-immigrant campaign.
“I feel like somebody has used me for their own agenda,” Urginia told Socialist Worker.
“I had been attacked several times before June 2002 as I was a photographer who went onto the farms as well as the war veterans’ demonstrations. My life was exposed because of my job.
“I had also been arrested, detained and beaten by the police and war veterans a number of times. The pain was so intense I couldn’t cry. These occasions come back to me at times.
“My uncle decided to bring me over to Britain and I had a plane ticket in my camera bag when I was attacked by the police in June 2002—not May 2002 as claimed in the Sun.
“The Sun said that my uncle ‘moved to Britain five years ago to escape the torture’ but he had been settled here around 12 years before I arrived. And how could he invite me in 2002 if he didn’t move here until 2004?
“He has been very upset by the article.
Angry
“I am very angry that the headline over the article is ‘I’m Staying’ and the final sentence is ‘I will never go back.’ I did not say this.
“I miss home every day. Like every Zimbabwean who has been forced to move from home, I eat, breathe and live Zimbabwe. All we want to do is go home to rebuild our country, but only when it is safe for us.
“I have always said I want to return, but only when Robert Mugabe is gone as Zimbabwe’s leader.
“After that treatment from my country, now I’m getting this from the Sun.
“The article is irresponsible and full of mistakes. The Sun has printed a letter in response by me, but they edited and cut it.”
Urginia is concerned at the way her experiences were used by the newspaper, which she feels doesn’t understand what many migrants and asylum seekers have gone through.
She said, “I didn’t realise that I was going to be part of an anti-immigrant feature. My story was next to a box on eastern European criminals.
“Asylum seekers want to work, but they aren’t allowed to. It’s hard enough to survive when you flee, leaving your family behind.
“And then you get into this immigration system going through more horrible things.
“I had to live off my uncle when I moved here, even though I was a grown woman and could have worked. I did voluntary work with the Refugee Council to feel like someone again.
“This experience shows that refugees and migrants need to be careful when they are interviewed.”
Britain: The government has been urged to rethink its cruel policy of locking up child asylum-seekers after the Home Office admitted that almost 1,000 children have been detained in the last five years: here.
Former Fox News Host Calls Fox A ‘Right-Wing Partial-News-But-Mostly-Opinion Network': here.
SANTIAGO, Dec 3 : here.
This week, Prince Alwaleed bin Talal al-Saud of Saudi Arabia — the largest shareholder of News Corp outside the Murdoch family — endorsed Rupert Murdoch’s son James to succeed the elder Murdoch when he retires: here.
Murdoch’s UK tabloid ordered to turn over evidence in phone hacking case: here.
Adelaide is Australia’s festival city. Its arts festival is currently in swing. Polite debate, aesthetics and high-octane wine are putting the world to rights. With one exception. Adelaide is where Rupert Murdoch began his empire: here.
Britain: Former deputy prime minister John Prescott, whose spaniel-like loyalty to Tony Blair set new levels of sycophancy, has discovered how awful Rupert Murdoch’s media empire is: here.
Pingback: David Cameron and the Murdoch empire | Dear Kitty. Some blog
Pingback: British neo-nazis, new book | Dear Kitty. Some blog
Pingback: Racism in British media | Dear Kitty. Some blog
Pingback: No free speech in Australia | Dear Kitty. Some blog
Pingback: Tony Blair and other fat cat politicians | Dear Kitty. Some blog
Pingback: Refugees from NATO’s wars drowning in the Mediterranean | Dear Kitty. Some blog
Pingback: Rupert Murdoch media promote racist violence | Dear Kitty. Some blog | 2,775 |
The Mega Sonic cat repeller by Defenders was the first of the motion detector type cat deterrents I purchased. Rightly or wrongly I assumed that spending £20 on a problem would fix it but it doesn’t quite work out like that, not at first anyway.
The model I purchased was the STV610 and I got it from Amazon for just under £17 (RRP is £35). I could have purchased a mains adaptor for another tenner but decided against as I had plenty of rechargeable batteries so thought I would put them to use instead.
The Mega Sonic cat repeller is finished in a deep green colour and blends in well if you don’t use the supplied silver stand. I fixed mine to a decking post at the top of some steps regularly used by visiting cats.
The instructions said the batteries should last up to 2 months in normal use. I found my rechargeable batteries lasted about 2 weeks. I tried Duracell and got 4 weeks out of them.
The Mega Sonic cat repeller claims to use a “sophisticated passive infra red detection system which constantly monitors a fan shaped area of a 98 degree arc covering a range of around 120 square metres.”
If movement is detected within that area it lets out a burst of noise undetectable to us but audible to cats.
First off, that 120 metres cannot be adjusted so any plants that are within its detection range are going to set it off as they sway in the breeze. As my garden is not huge I could not find anywhere I could put it that did not have either my own plants or my neighbours bushes and hedging in its range.
Does it work?
Apart from one cat that looked to be getting on in years so his hearing was probably on the way out, it seemed to be effective after a few weeks and to be fair the instructions do say that it can take from 2 weeks to a month to completely work on your local cat population.
I found though that after about a month, one or two of the local cats had got used to it and started ignoring the noise. Some of the braver ones by this time had even started to approach it and see what the fuss was about.
As I was limited to where I could put it this was a problem but I do believe if the garden was larger and I was able to periodically move it around to keep the cats on their toes, the Mega-Sonic would have been an excellent solution.
As I was limited to where I could put it this was a problem but I do believe if the garden was larger and I was able to periodically move it around to keep the cats on their toes, the Mega-Sonic would have been an excellent solution.
If you buy this cat repeller and have got a mains socket in the garden, get the adapter and forget using the batteries. If you go away for a fortnight or forget to change them, it is not going to be working and you will be back to where you started by the time you get back.
Cons
- May not be suitable for small gardens
- Cannot lower range
- Gobbles batteries if it keeps activating
Pros
- Mains adaptor is available – make sure you get it
- Cannot be heard by humans or birds 2 year warranty
- The top selling cat deterrent in the UK
Consumer Reviews
At the time of writing there are over 2300 consumer reviews for this particular cat repeller on Amazon with an average rating of 3.5 out of 5
I think 3.5 stars is about right and have awarded the Defenders Mega Sonic Cat Repeller the same. | 160,624 |
\section{Stochastic homogenization: corrector equation and RWRE}
\label{s:theo}
\setcounter{equation}{0}
We start with the description of the discrete set of diffusion coefficients, first present the discrete
elliptic point of view, and then turn to the random walk in random
environment viewpoint.
The aim of this section is to introduce a formalism, and give an
intuition on both points of view.
\medskip
The results recalled here are essentially due to Papanicolaou and Varadhan \cite{Papanicolaou-Varadhan-79}, Kozlov \cite{Kozlov-87}, and
Kipnis and Varadhan \cite{Kipnis-Varadhan-86}).
\medskip
We present the results in the case of independent and identically distributed (i.i.d.)\ conductivities, although everything in this section remains valid provided the conductivities lie in a compact set of $(0,+\infty)$,
are stationary, and ergodic.
In particular, we shall apply this theory (and its quantitative counterpart) to coefficients which have finite correlation length. Yet, for the sake of clarity, we stick to i.i.d.\ coefficients in this presentation.
\subsection{Random environment}
We say that $x, y$ in $\Z^d$ are neighbors, and write $x \sim y$, whenever $|y-x| = 1$. This relation turns $\Z^d$ into a graph, whose set of (non-oriented) edges is denoted by $\mb{B}$.
We now define the conductances on $\mb{B}$, and their statistics.
\begin{defi}[environment]\label{2-1:defi:envi}
Let $0 < \alpha \le \beta < + \infty$, and $\Omega = [\alpha,\beta]^{\mb{B}}$. An element $\om = (\om_e)_{e \in \mb{B}}$ of $\Omega$ is called an \emph{environment}.
With any edge $e = (x,y) \in \mb{B}$, we associate the \emph{conductance} $\om_{(x,y)}:=\om_e$ (by construction $\om_{(x,y)} = \om_{(y,x)}$).
Let $\nu$ be a probability measure on $[\alpha,\beta]$.
We endow $\Omega$ with the product probability measure $\P = \nu ^{\otimes \mb{B}}$.
In other words, if $\omega$ is distributed according to the measure $\P$, then $(\omega_e)_{e \in \mb{B}}$ are independent random variables of law $\nu$.
We denote by $L^2(\Omega)$ the set of real square integrable functions on $\Omega$ for the measure $\mathbb{P}$,
and write $\expec{\cdot}$ for the expectation associated with $\P$.
\end{defi}
We then introduce the notion of stationarity.
\begin{defi}[stationarity]\label{2-1:defi:stat}
For all $z\in \Z^d$, we let $\theta_z:\Omega\to \Omega$ be such that for all $\om \in \Omega$ and $(x,y)\in \mb{B}$,
$(\theta_z \ \omega)_{(x,y)}=\omega_{(x+z,y+z)}$. This defines an additive action group $\{\theta_z\}_{z\in \Z^d}$ on $\Omega$ which preserves
the measure $\mathbb{P}$, and is ergodic for $\mathbb{P}$.
We say that a function $f:\Omega \times \Z^d\to \R$ is \emph{stationary} if and only if for all $x,z\in\Z^d$ and
$\mathbb{P}$-almost every $\omega\in \Omega$,
$$
f(x+z,\omega)\,=\,f(x,\theta_z \ \omega).
$$
In particular, with all $f\in L^2(\Omega)$, one may associate the stationary function (still denoted by $f$) : $\Z^d\times\Omega\to \R,
(x,\omega) \mapsto f(\theta_x \ \omega)$.
In what follows we will not distinguish between $f\in L^2(\Omega)$ and its stationary extension on $\Z^d\times\Omega$.
\end{defi}
\subsection{Corrector equation}\label{sec:2-1-1-2}
We associate with the conductances on $\mb B$ a conductivity matrix on $\Z^d$.
\begin{defi}[conductivity matrix]\label{2-1:defi:conduct}
Let $\Omega$, $\mathbb{P}$, and $\{\theta_z\}_{z\in \Z^d}$ be as in Definitions~\ref{2-1:defi:envi} and~\ref{2-1:defi:stat}.
The stationary diffusion matrix $A:\Z^d\times\Omega\to \mathcal{M}_d(\R)$ is defined by
$$
A:(x,\omega) \mapsto \diag(\om_{(x,x+\ee_i)} ,\dots, \om_{(x,x+\ee_d)}).
$$
\end{defi}
For each $\omega\in\Omega$, we consider the discrete elliptic operator $L$ defined by
\begin{equation}\label{2-1:eq:def-op-space}
L=-\nabla^*\cdot A (\cdot,\omega)\nabla,
\end{equation}
where $\nabla$ and $\nabla^*$ are the forward and backward discrete gradients, acting on functions $u:\Z^d\to \R$ by
\begin{equation}\label{2-1:eq:disc-nabla}
\nabla u(x):=\left[
\begin{array}{l}
u(x+\ee_1)-u(x) \\
\vdots\\
u(x+\ee_d)-u(x)
\end{array}
\right],
\
\nabla^* u(x):=\left[
\begin{array}{l}
u(x)-u(x-\ee_1) \\
\vdots\\
u(x)-u(x-\ee_d)
\end{array}
\right],
\end{equation}
and we denote by $\nabla^*\cdot$ the backward divergence.
In particular, for all $u:\Z^d\to \R$,
\begin{equation}\label{2-1:eq:def-op-space2}
Lu:\Z^d\to \R,\ z\,\mapsto\, \sum_{z'\sim z}\omega_{(z,z')} (u(z)-u(z')).
\end{equation}
The standard stochastic homogenization theory for such discrete elliptic operators (see for instance \cite{Kunnemann-83}, \cite{Kozlov-87})
ensures that
there exist homogeneous and deterministic coefficients $A_\ho$ such that the solution operator of the continuum
differential operator $-\nabla \cdot A_\ho\nabla$ describes $\mathbb{P}$-almost surely
the large scale behavior of the solution operator
of the discrete differential operator $-\nabla^*\cdot A (\cdot,\omega)\nabla $.
As for the periodic case, the definition of $A_\ho$ involves the so-called correctors.
Let $\xi\in \R^d$ be a fixed direction. The corrector $\phi:\Z^d\times \Omega \to \R$ in the direction $\xi$ is the unique solution (in a sense made precise below) to
the equation
\begin{equation}\label{2-1:eq:corr-sto}
-\nabla^*\cdot A(x,\omega)(\xi+\nabla \phi(x,\omega))\,=\,0,\qquad x \in \Z^d.
\end{equation}
The following lemma gives the existence and uniqueness of this corrector $\phi$.
\begin{lem}[corrector]\label{2-1:lem:corr}
Let $\Omega$, $\mathbb{P}$, $\{\theta_z\}_{z\in \Z^d}$, and $A$ be as in Definitions~\ref{2-1:defi:envi}, \ref{2-1:defi:stat}, and~\ref{2-1:defi:conduct}.
Then, for all $\xi\in \R^d$, there exists a unique measurable function $\phi:\Z^d\times \Omega\to \R$ such that
$\phi(0,\cdot)\equiv 0$, $\nabla \phi$ is stationary, $\expec{\nabla \phi}=0$, and $\phi$ solves \eqref{2-1:eq:corr-sto} $\mathbb{P}$-almost surely.
Moreover, the symmetric homogenized matrix $A_\ho$ is characterized by
\begin{equation}\label{2-1:eq:hom-coeff}
\xi\cdot A_\ho\xi\,=\,\expec{(\xi+\nabla\phi)\cdot A(\xi+\nabla \phi)}.
\end{equation}
\end{lem}
The standard proof of Lemma~\ref{2-1:lem:corr}
makes use of the regularization of \eqref{2-1:eq:corr-sto} by a zero-order term $\mu>0$:
\begin{equation}\label{2-1:eq:mod-corr-sto}
\mu\phi_\mu(x,\omega)-\nabla^*\cdot A(x,\omega)(\xi+\nabla \phi_\mu(x,\omega))\,=\,0, \qquad x \in \Z^d.
\end{equation}
\begin{lem}[regularized corrector]\label{2-1:lem:mod-corr}
Let $\Omega$, $\mathbb{P}$, $\{\theta_z\}_{z\in \Z^d}$, and $A$ be as in Definitions~\ref{2-1:defi:envi}, \ref{2-1:defi:stat}, and~\ref{2-1:defi:conduct}.
Then, for all $\mu>0$ and $\xi\in \R^d$, there exists a unique stationary function $\phi_\mu \in L^2(\Omega)$ with $\expec{\phi_\mu}=0$
which solves \eqref{2-1:eq:mod-corr-sto} $\mathbb{P}$-almost surely.
\end{lem}
To prove Lemma~\ref{2-1:lem:mod-corr}, we follow \cite{Papanicolaou-Varadhan-79}, and introduce difference operators on $L^2(\Omega)$:
for all $u\in L^2(\Omega)$, we set
\begin{equation}\label{2-1:eq:disc-nabla-sto}
\DD u(\omega):=\left[
\begin{array}{l}
u(\theta_{\ee_1}\omega)-u(\omega) \\
\vdots\\
u(\theta_{\ee_d}\om)-u(\omega)
\end{array}
\right],
\
\DD^*u(\omega):=\left[
\begin{array}{l}
u(\om)-u(\theta_{-\ee_1}\om) \\
\vdots\\
u(\om)-u(\theta_{-\ee_d}\om)
\end{array}
\right].
\end{equation}
These operators play the same
roles as the finite differences $\nabla$ and $\nabla^*$ --- this time for
the variable $\omega$ (in other words, they define a difference calculus on $L^2(\Omega)$). They allow us to define the counterpart on $L^2(\Omega)$ to the operator $L$ of \eqref{2-1:eq:def-op-space}:
\begin{defi}\label{2-1:defi:operator-sto}
Let $\Omega$, $\mathbb{P}$, $\{\theta_z\}_{z\in \Z^d}$, and $A$ be as in Definitions~\ref{2-1:defi:envi}, \ref{2-1:defi:stat}, and~\ref{2-1:defi:conduct}.
We define $\calL:L^2(\Omega)\to L^2(\Omega)$ by
\begin{eqnarray*}
\calL u(\omega)&=&-\DD^*\cdot A(\omega) \DD u(\omega) \\
&=&\sum_{z \sim 0} \om_{0,z} (u(\om) -u(\theta_z \ \om))
\end{eqnarray*}
where $\DD$ and $\DD^*$ are as in \eqref{2-1:eq:disc-nabla-sto}.
\end{defi}
The fundamental relation between $L$ and $\calL$ is the following
identity for stationary fields $u:\Z^d\times \Omega \to \R$: for all
$z\in \Z^d$ and almost every $\omega \in \Omega$,
\begin{equation*}
Lu(z,\omega) \,=\,\calL u(\theta_z\omega).
\end{equation*}
In particular, the regularized corrector $\phi_\mu$ is also the unique weak
solution in $L^2(\Omega)$ to the equation
\begin{equation*}
\mu \phi_\mu(\omega)-\DD^*\cdot A(\omega)(\xi+\DD
\phi_\mu(\omega))\,=\,0, \quad \omega \in \Omega,
\end{equation*}
and its existence simply follows from the Riesz representation theorem on $L^2(\Omega)$.
The regularized corrector $\phi_\mu$ is an approximation of the corrector $\phi$ in the following sense:
\begin{lem}\label{2-1:lem:corr-mod-corr}
Let $\Omega$, $\mathbb{P}$, $\{\theta_z\}_{z\in \Z^d}$, and $A$ be as in Definitions~\ref{2-1:defi:envi}, \ref{2-1:defi:stat}, and~\ref{2-1:defi:conduct}.
For all $\mu>0$ and $\xi\in \R^d$, let $\phi$ and $\phi_\mu$ be the corrector and regularized corrector of Lemmas~\ref{2-1:lem:corr} and~\ref{2-1:lem:mod-corr}.
Then, we have
\begin{equation*}
\lim_{\mu\to 0}\expec{|\DD\phi_\mu-\DD\phi|^2}\,=\,0.
\end{equation*}
\end{lem}
From the elementary a priori estimates
\begin{equation*}
\expec{|\nabla \phi_\mu|^2}\,=\,\expec{|\DD \phi_\mu|^2}\,\leq \, C,
\qquad \expec{\phi_\mu^2}\,\leq \, C \mu^{-1},
\end{equation*}
for some $C$ independent of $\mu$, we learn that $\DD\phi_\mu$ is bounded in $L^2(\Omega,\R^d)$ uniformly in $\mu$, so that
up to extraction it converges weakly in $L^2(\Omega,\R^d)$ to some random field $\Phi$ (which is a gradient).
This allows one to pass to the limit in the weak formulations and obtain
the existence of a field $\Phi=(\Phi_1,\dots,\Phi_d) \in L^2(\Omega,\R^d)$ such that for all
$\psi\in L^2(\Omega)$,
\begin{equation}\label{eq:corr-eq-ant1}
\expec{\DD \psi \cdot A(\xi+\Phi)}\,=\,0.
\end{equation}
Using the following weak Schwarz commutation rule
\begin{equation*}
\forall j,k \in \{1,\dots,d\}, \quad \expec{(\DD_j \psi ) \Phi_k}\,=\,\expec{(\DD_k \psi) \Phi_j}
\end{equation*}
one may define $\phi:\Z^d\times \Omega\to \R$ such that $\nabla \phi$ is stationary, $\Phi=\nabla
\phi$, and $\phi(0,\omega)=0$ for almost every $\omega\in \Omega$. By definition this function $\phi$ is not stationary.
It is a priori not clear (and even wrong in dimension $d \le 2$) whether there exists some
function $\psi\in L^2(\Omega)$ such that $\DD \psi=\Phi$ (this is a major difference with the periodic case).
The uniqueness of $\Phi$ is a consequence of Lemma~\ref{2-1:lem:corr-mod-corr}, which follows from the fact that $\DD \phi_\mu$ is a Cauchy sequence
in $L^2(\Omega)$. To prove this, we shall appeal to spectral theory.
The operator $\calL$ of Definition \ref{2-1:defi:operator-sto} is bounded, self-adjoint, and non-negative on $L^2(\Omega)$. Indeed, for all $\psi,\chi \in L^2(\Omega)$, we have by direct computations
$$
\expec{(\calL \psi)^2}^{1/2} \,\leq\, 4d \sqrt{\beta} \expec{\psi^2}^{1/2}, \quad
\expec{ (\calL \psi) \chi}\,=\,\expec{\psi (\calL \chi)}, \quad \expec{\psi \calL \psi}\geq 2d \alpha \expec{\psi^2} .
$$
Hence, $\calL$ admits a spectral decomposition in $L^2(\Omega)$.
For all $g\in L^2(\Omega)$ we denote by $e_g$ the projection of the spectral measure of $\calL$ on $g$.
This defines the following spectral calculus: for any bounded continuous function $\Psi:[0,+\infty)\to \R_+$,
\begin{equation*}
\expec{(\Psi(\calL)g)g}\,=\,\int_{\R^+}\Psi(\lambda)de_g(\lambda).
\end{equation*}
Let $\xi\in \R^d$ with $|\xi|=1$ be fixed, and define the local drift as $\mathfrak{d}\,=\,-\DD^*\cdot A\xi\in L^2(\Omega)$.
For all $\mu\geq \nu >0$ we have $\phi_\mu=(\mu+\calL)^{-1}\mathfrak{d}$ and
$\phi_\nu=(\nu+\calL)^{-1}\mathfrak{d}$, by the Cauchy-Schwarz inequality,
\begin{eqnarray*}
\expec{|\DD \phi_\mu-\DD\phi_\nu|^2} &\leq& \alpha^{-1} \expec{(\phi_\mu-\phi_\nu)\calL (\phi_\mu-\phi_\nu)} \\
&=& \alpha^{-1} \expec{\phi_\mu\calL\phi_\mu} -2 \alpha^{-1} \expec{\phi_\mu\calL \phi_\nu} +\alpha^{-1} \expec{\phi_\nu\calL\phi_\nu} \\
&=& \alpha^{-1} \expec{\mathfrak{d}(\mu+\calL)^{-1}\calL(\mu+\calL)^{-1}\mathfrak{d}} -2 \alpha^{-1} \expec{\mathfrak{d}(\mu+\calL)^{-1}\calL (\nu+\calL)^{-1}\mathfrak{d}} \\
&&+\alpha^{-1} \expec{\mathfrak{d}(\nu+\calL)^{-1}\calL(\nu+\calL)^{-1}\mathfrak{d}} .
\end{eqnarray*}
By the spectral formula with functions
$$
\Psi(\lambda)\,=\, \frac{\lambda}{(\mu+\lambda)^2}, \frac{\lambda}{(\mu+\lambda)(\nu+\lambda)}, \frac{\lambda}{(\nu+\lambda)^2},
$$
we obtain
\begin{eqnarray}
\expec{|\DD \phi_\mu-\DD\phi_\nu|^2} &\leq& \alpha^{-1}\int_{\R^+} \left(\frac{\lambda}{(\mu+\lambda)^2}-2\frac{\lambda}{(\mu+\lambda)(\nu+\lambda)}+ \frac{\lambda}{(\nu+\lambda)^2}\right) de_{\mathfrak{d}}(\lambda) \nonumber \\
&=& \alpha^{-1}\int_{\R^+} \frac{\lambda(\nu-\mu)^2}{(\mu+\lambda)^2(\nu+\lambda)^2}de_{\mathfrak{d}}(\lambda) \label{eq:ex-sp-calc}\\
&\leq & \alpha^{-1}\int_{\R^+} \frac{\mu^2}{(\mu+\lambda)^2\lambda}de_{\mathfrak{d}}(\lambda)\nonumber ,
\end{eqnarray}
since $0<\nu\leq \mu$. Since the upper bound is independent of $\nu$, we have proved the claim if we can show that it tends to zero as $\mu$ vanishes. This is a consequence of the Lebesgue dominated convergence theorem provided we show that
\begin{equation}\label{2-1:eq:KV-1}
\int_{\R^+}\frac{1}{\lambda} de_{\mathfrak{d}}(\lambda)\,<\,\infty.
\end{equation}
On the one hand, by the a priori estimate of $\DD \phi_\mu$,
$$
\expec{\phi_\mu \calL \phi_\mu} \,\leq \, \beta \expec{|\DD \phi_\mu|^2} \,\leq \, \beta C.
$$
On the other hand, by the same type of spectral calculus as above, we have
$$
\expec{\phi_\mu \calL \phi_\mu} \,=\,\int_{\R^+}\frac{\lambda}{(\mu+\lambda)^2}de_{\mathfrak{d}}(\lambda) .
$$
Estimate \eqref{2-1:eq:KV-1} then follows from the monotone convergence theorem.
This concludes the proof of Lemma~\ref{2-1:lem:corr-mod-corr}.
\subsection{Random walk in random environment}\label{sec:2-1-1-3}
We now turn our attention to the probabilistic aspects of homogenization. This presentation is informal. It aims at being accessible to non-specialists of probability theory, and at highlighting the inner similarities with the corrector approach of subsection~\ref{sec:2-1-1-2}.
\subsubsection{The continuous-time random walk}
\label{ss:ctrw}
Let the environment $\omega$ be fixed for a while (that is, we have picked a realization of the conductivities $\omega_e \in [\alpha,\beta]$, $e\in \mb B$). The random walk we wish to define, that we will denote by $(X_t)_{t\in \R_+}$, is a random process whose behavior is influenced by the environment.
To the specialist, it can be defined by saying that it is the Markov process whose transition rates are the $(\omega_e)_{e \in \mb{B}}$. The Markov property means that given any time $t \ge 0$, the behavior of the process after time $t$ depends on its past only through its location at time $t$. In other words, the process ``starts afresh'' at time $t$ given its current location. In order to give a complete description of the process, it thus suffices to describe its behavior over a time interval $[0, t]$, for some $t > 0$, no matter how small. As $t$ tends to $0$, this behavior is given by
\begin{equation}
\label{nonconstr}
\PPo_z \Ll[X_{t} = z' \Rr] =
\left|
\begin{array}{ll}
t\omega_{z,z'} + o(t) & \text{if } z' \sim z, \\
1-\sum_{y \sim z} t\omega_{z,y} + o(t) & \text{if } z' = z, \\
o(t) & \text{otherwise},
\end{array}
\right.
\end{equation}
where $\PPo_z$ is the probability measure corresponding to the walk started at $z$, that is, $\PPo_z[X_0 = z] = 1$. Equation~\eqref{nonconstr} is what is meant when it is said that $\omega_{z,z'}$ is the jump rate from $z$ to $z'$.
A more constructive way to represent the random walk is as follows. Let the walk be at some site $z \in \Z^d$ at time $t$, and start an ``alarm clock'' that rings after a random time $T$ which follows an exponential distribution of parameter
\begin{equation}
\label{defpomega}
p_\omega(z) := \sum_{z' \sim z} \omega_{z,z'}.
\end{equation}
This means that for any $s \ge 0$, the probability that $T > s$ is equal to $e^{-p_{\omega(z)} s}$. When the clock rings, the walk chooses to move to one (out of $2d$) neighboring site $z'$ with probability
\begin{equation}
\label{def:leadsto}
p(z\leadsto z') :=
\frac{\omega_{z,z'}}{p_\omega(z)},
\end{equation}
and this choice is made independently of the value of $T$.
Note that by the Markov property, the fact that the walk has not moved during the time interval $[t,t+s]$ should give no information on the time of the next jump.
Only exponential distributions have this memoriless property.
Let us see why thus defined, the random walk satisfies \eqref{nonconstr}. The probability that the clock rings during the time interval $[0,t]$ is
$$
1-e^{-p_\omega(z)t} = p_{\omega(z)} t + o(t).
$$
Since $p_{\omega}(z)$ is bounded by $2d\beta$ uniformly over $z$, the probability that the walk makes two or more jumps is $o(t)$. The probability that it ends up at $z' \sim z$ at time $t$ is thus
$$
\Ll(p_{\omega(z)} t + o(t)\Rr) p(z\leadsto z') - o(t) = \omega_{z,z'} t + o(t),
$$
and the probability that it stays still is indeed as in \eqref{nonconstr}.
\medskip
The link between the random walk and the elliptic operators of the previous subsection is as follows.
We let $(P_t)_{t \in \R_+}$ be the semi-group associated with the random walk, that is, for any $t \ge 0$ and any bounded function $f:\Z^d \to \R$, we let
\begin{equation*}\label{eq:defi-semi}
P_tf(z)\,=\,\mathbf E_z^\omega[f(X_t)],
\end{equation*}
where $\mathbf E_z^\omega$ denotes the expectation associated to the probability measure $\PPo_z$, under which the random walk starts at $z\in \Z^d$ in the environment $\omega$. This is a semi-group since $X$ has the Markov property. As we now show, the infinitesimal generator of this semi-group is the elliptic operator $-L$ (where $L$ is defined in \eqref{2-1:eq:def-op-space}).
Recall that the infinitesimal generator of a semi-group, applied to $f$, is given by
\begin{equation*}
\lim_{t\to 0} \left(\frac{P_tf-f}{t} \right),
\end{equation*}
for any $f$ for which the limit exists. From the description \eqref{nonconstr}, this limit is easily computed. Indeed,
\begin{eqnarray}
P_tf(z)&=&\mathbf E_z^\omega[f(X_t)]\nonumber \\
&=&(1-p_\omega(z)t)f(z)+t \sum_{z'\sim z} \omega_{(z,z')} f(z')+o(t)\\
&=&f(z)+t \sum_{z'\sim z}\omega_{(z,z')} (f(z')-f(z))+o(t),\label{2-1:eq:bonne-proba2}
\end{eqnarray}
so that by \eqref{2-1:eq:def-op-space2},
\begin{equation}
\label{limsemigrp}
\lim_{t\to 0} \left(\frac{P_tf(z)-f(z)}{t} \right) \,=\, \sum_{z'\sim z}\omega_{(z,z')} (f(z')-f(z))\,=\,-Lf(z),
\end{equation}
and we have identified the infinitesimal generator to be $-L$, as announced.
\medskip
An important feature of this random walk is that the jump rates are symmetric: the probability to go from $z$ to $z'$ in an ``infinitesimal'' amount of time is equal to the probability to go from $z'$ to $z$, as can be seen on \eqref{nonconstr}. This may be rephrased as saying that the counting measure on $\Z^d$ (which puts mass $1$ to every site) is \emph{reversible} for the random walk.
If we were running the random walk on a finite graph instead of $\Z^d$, one could normalize the counting measure to make it into a probability measure, and then interpret reversibility as follows. Start the random walk at a point chosen uniformly at random on the graph, and let it run up to time $t$. Reversibility (of the uniform probability measure) is equivalent to the fact that the distribution of this trajectory is the same as the distribution of the time-reversed path, running from time $t$ to time $0$. In particular, since the starting point is chosen uniformly at random, the distribution of the walk at any time must be the uniform probability measure. Of course, the problem is that on $\Z^d$, it does not make sense to say that we choose the starting point ``uniformly at random'', or in other words, we cannot normalize the counting measure into a probabability measure.
In the previous section, we moved from the operator $L$ to its ``environmental'' version, the operator $\L$. This takes a very enlightening probabilistic meaning. Instead of considering the random walk itself, we may consider the \emph{environment viewed by the particle}, which is the random process defined as
$$
t \mapsto \omega(t) := \theta_{X(t)}\omega,
$$
where $(\theta_x)_{x \in \Z^d}$ are the translations introduced in Definition~\ref{2-1:defi:stat}. One can convince oneself that $(\omega(t))_{t \in \R_+}$ is a Markov process, and the important point is that its infinitesimal generator is precisely $-\L$. A little computation (see for instance \cite[Proposition~3.1]{these}) shows that the reversibility of the counting measure for the random walk translates into the reversibility of the measure $\P$ for the process $(\omega(t))_{t \in \R_+}$. This reversibility implies that $(\omega(t))_{t \in \R_+}$ is stationary if we take the initial environment according to $\P$ and then let the process run, that is, under the measure
\begin{equation}\label{2-1:eq:def-pdt-measure}
\P_0 :=\P \mathbf P_0^\omega,
\end{equation}
the so-called \emph{annealed} measure.
Stationarity means that for any positive integer $k$ and any $s_1,\ldots, s_k \ge 0$, the distribution of the vector $(\omega(s_1+t),\ldots, \omega(s_k+t))$ does not depend on $t \ge 0$. In particular, under the measure $\P_0$, the distribution of $\omega(t)$ is the measure ${\P}$ for any $t \ge 0$.
To conclude, we argue that from the reversibility follows the fact that the operator $\L$ is self-adjoint in $L^2(\Omega)$ --- which we have already seen more directlty in the previous subsection. Indeed, the invariance under time reversal implies that, under $\P_0$ and for any $t \ge 0$, the vectors $(\omega(0), \omega(t))$ and $(\omega(t), \omega(0))$ have the same distribution. For any bounded functions $f,g :\Omega \to \R$, we thus have
\begin{equation}
\label{presqselfadj}
\E_0\Ll[ f(\om(0))\ g(\om(t)) \Rr] = \E_0\Ll[ f(\om(t))\ g(\om(0)) \Rr],
\end{equation}
where we write $\E_0$ for the expectation associated to $\P_0$. Letting $(\mcl{P}_t)_{t \in \R_+}$ be the semi-group associated to $-\L$, we arrive at
\begin{eqnarray*}
\E_0\Ll[ f(\om(0))\ g(\om(t)) \Rr] & = & \expec{ \EEo_0 \Ll[ f(\om(0)) \ g(\om(t)) \Rr] } \\
& = & \expec{ f(\om) \ \EEo_0 \Ll[ g(\om(t)) \Rr] } = \expec{ f \ \mcl{P}_t g},
\end{eqnarray*}
and \eqref{presqselfadj} thus reads
$$
\expec{ f \ \mcl{P}_t g} = \expec{ \mcl{P}_t f \ g}.
$$
This shows that $\mcl{P}_t$ is self-adjoint for any $t \ge 0$. Passing to the limit as in \eqref{limsemigrp}, we obtain the self-adjointness of $\L$ itself. In fact, reversibility of a Markov process and self-adjointness of its infinitesimal generator are two sides of the same coin.
\subsubsection{Central limit theorem for the random walk}
\label{ss:clt}
The aim of this paragraph is to sketch a probabilistic argument justifying the following result.
\begin{thm}[\cite{Kipnis-Varadhan-86}]
\label{t:kv}
Under the measure $\P_0$ and as $\eps$ tends to $0$, the rescaled random walk $X^{(\eps)} := (\sqrt{\e}X_{t/\e})_{t \in \R_+}$ converges in distribution (for the Skorokhod topology) to a Brownian motion with covariance matrix $2A_\ho$, where $A_\ho$ is as in \eqref{2-1:eq:hom-coeff}. In other words, for any bounded continuous functional $F$ on the space of cadlag functions, one has
\begin{equation}
\label{convergegen}
\E_0\Ll[ F(X^{(\eps)}) \Rr] \xrightarrow[\eps \to 0]{} E[F(B)],
\end{equation}
where $B$ is a Brownian motion started at the origin and with covariance matrix $2 \Ah$, and $E$ denotes averaging over $B$. Moreover, for any $\xi \in \R^d$, one has
\begin{equation}
\label{convergesquare}
t^{-1} \E_0\Ll[ \Ll(\xi \cdot X_t\Rr)^2 \Rr] \xrightarrow[t \to + \infty]{} 2 \xi \cdot \Ah \xi.
\end{equation}
\end{thm}
Note that the convergence of the rescaled square displacement in \eqref{convergesquare} does not follow from \eqref{convergegen}, since the square function is not bounded.
From now on, we focus on the one-dimensional projections of $X$. As in subsection~\ref{sec:2-1-1-2}, we let $\xi$ be a fixed vector of $\R^d$. The idea is to decompose $\xi \cdot X_t$ as
\begin{equation}
\label{decomp}
\xi \cdot X_t = M_t + R_t,
\end{equation}
where $(M_t)_{t \ge 0}$ is a martingale and $R_t$ is a (hopefully small) remainder.
We recall that $(M_t)_{t \ge 0}$ is a martingale under $\mathbf E_0^\omega$ if for all $t\geq 0$ and $s\geq 0$,
\begin{equation}\label{2-1:eq:am-I-martingale?}
\mathbf E^\omega_0[M_{t+s}|\calF_t]\,=\,M_t,
\end{equation}
where $\calF_t$ is the $\sigma$-algebra generated by $\{M_\tau,\tau\in [0,t]\}$. (If $M$ was a gambler's money, one should say that he is playing a fair game when \eqref{2-1:eq:am-I-martingale?} holds, since knowing the history up to time $t$, his expected amount of money at time $t+s$ is the amount he has at time $t$).
We look for a martingale of the form $M_t=\chi^\omega(X_t)$ for some function $\chi^\omega$. By the Markov property of $X$, $(M_t)_{t \ge 0}$ of this form is a martingale if and only if, for any $z \in \Z^d$ and any $t \ge 0$,
\begin{equation}\label{eq:martingale}
\mathbf E^\omega_z[\chi^\omega(X_t)]\,=\,\chi^\omega(z),
\end{equation}
From \eqref{nonconstr}, we learn that
$$
\mathbf E^\omega_z[\chi^\omega(X_t)] = \chi^\om(z) - s L\chi^\omega(z) + O(s^2).
$$
Hence, a necessary condition is that $L\chi^\om(z) = 0$, and in fact, this condition is also sufficient. Keeping in mind that we also want the remainder term to be small, we would like $\chi^\omega$ to be a perturbation of the $z \mapsto \xi \cdot z$, so that a right choice for $\chi^\omega$ should be
\begin{equation*}
\chi^\omega(z)\,=\,\xi \cdot z+\phi(z,\omega)
,
\end{equation*}
where $\phi$ is the corrector of Lemma~\ref{2-1:lem:corr} (compare equation~\eqref{2-1:eq:corr-sto} to $L\chi^\om(z) = 0$). The link between the corrector equation and the RWRE appears precisely there. We thus define
$$
M_t = \xi \cdot X_t + \phi(X_t,\omega) \quad \text{and} \quad R_t = -\phi(X_t,\omega).
$$
We now argue that the martingale $M$ has stationary increments under the measure $\P_0$. For simplicity, let us just see that the distribution of $M_{t+s} - M_t$ under $\P_0$ does not depend on~$s$ (instead of considering vectors of increments). The argument relies on the stationarity of the process of the environment viewed by the particle seen above. Given this stationarity, it suffices to see that $M_{t+s} - M_t$ can be written as a function of $(\omega(t),\omega(t+s))$ only. This is possible because the increment $M_{t+s} - M_t$ depends only on~$\nabla \phi$, which is a function of $\omega$ only (contrary to $\phi$ itself which is a priori a function of $x$ and $\omega$).
\medskip
Martingales are interesting for our purpose since they are ``Brownian motions in disguise''. To make this idea more precise, let us point out that any one-dimensional continuous martingale can be represented as a time-change of Brownian motion (this is the Dubins-Schwarz theorem, see \cite[Theorem~V.1.6]{ry}). If $(B_t)_{t \ge 0}$ is a Brownian motion, a time-change of it is for instance $M_t = B_{t^{7}}$. Note that in this example, the time-change can be recovered by computing $E[M_t^2] = t^7$.
Here, the martingale we consider has jumps (since $X$ itself has), which complicates the matter a little, but let us keep this under the rug. Intuitively, in order to justify the convergence to a Brownian motion, we should show that the underlying time-change grows linearly at infinity. One can check that two increments of a martingale over disjoint time intervals are always orthogonal in $L^2$ (provided integration is possible). In our case, since $M_t$ has stationary increments, it thus follows that letting $\sigma^2(\xi) = \E_0[M_1^2]$, we have
\begin{equation}
\label{meanlinear}
\E_0[M_t^2] = \sigma^2(\xi) \ t,
\end{equation}
so we are in good shape (i.e.\ on a heuristic level, it indicates that the underlying time-change indeed grows linearly). Letting $f:z\mapsto (x\cdot \xi+\phi(x,\omega)\cdot \xi)^2$ and using \eqref{nonconstr}, one can write
\begin{eqnarray*}
\lefteqn{\mathbf E^\omega_0[(M_t\cdot \xi)^2]\,=\,\mathbf E^\omega_0[f(X_t)]}\\
&= & f(0)+t \sum_{z'\sim 0}\omega_{(z,z')}\big(z'\cdot \xi+\phi(z',\omega)\cdot \xi \big)^2+o(t)\\
&=&\underbrace{2t (\xi+\nabla \phi(0,\omega)\cdot \xi)\cdot A(0,\omega) (\xi+\nabla \phi(0,\omega)\cdot \xi)}_{=: t\ v(\omega)} + o(t),
\end{eqnarray*}
since $\phi(0,\omega)=0$. From the definition of $\Ah$ in \eqref{2-1:eq:hom-coeff}, we thus get
\begin{equation}
\label{identifcv}
\sigma^2(\xi) = \expec{v} = 2 \xi \cdot \Ah \xi.
\end{equation}
Provided we can show that the remainder is negligible, this already justifies \eqref{convergesquare}.
In order to prove that $(\sqrt{\eps} M_{\eps^{-1} t})_{t \ge 0}$ converges to a Brownian motion of variance $\sigma^2(\xi)$ as $\eps$ tends to~$0$, knowing \eqref{meanlinear} is however not sufficient: one does not recover all the information about the time-change by computing $\E_0[M_t^2]$ alone. This can be understood from the fact that in the Dubins-Schwarz theorem, the time-change that appears can be random itself, and is in fact the quadratic variation of the martingale. We will not go into explaining what the quadratic variation of a martingale is in general, but simply state that in our case, it is given by
$$
V_t = \int_0^t v(\omega(s)) \ \d s,
$$
and what we should prove is thus that
\begin{equation}
\label{ergodicthm}
t^{-1} V_t = t^{-1} \int_0^t v(\omega(s)) \ \d s\xrightarrow[t \to + \infty]{\text{a.s.}} \sigma^2(\xi) = 2 \xi \cdot \Ah \xi.
\end{equation}
One can show that the process $(\omega(t))_{t \ge 0}$ is ergodic for the measure $\P_0$ (see for instance \cite[Proposition~3.1]{these}), and thus the convergence in \eqref{ergodicthm} is a consequence of the ergodic theorem. Note in passing this surprising fact that the proof of a central limit theorem was finally reduced to a law of large numbers type of statement.
\medskip
In order to obtain a central limit theorem for $\xi \cdot X$ itself (instead of the martingale part), we need to argue that the remainder term is small. In order to do so, and following \cite{Kipnis-Varadhan-86}, we will rely on spectral theory.
We recall that the operator $\calL$ of Definition \ref{2-1:defi:operator-sto} being a bounded non-negative self-adjoint operator on $L^2(\Omega)$, it admits a spectral decomposition in $L^2(\Omega)$.
Moreover, for any $g\in L^2(\Omega)$ the characterizing property of the spectral measure~$e_g$ previously defined is that for any continuous function $\Psi:[0,+\infty)\to \R_+$, one has
\begin{equation*}
\expec{g \ \Psi(\calL)g}\,=\,\int_{\R^+}\Psi(\lambda) \ \d e_g(\lambda).
\end{equation*}
We now argue that
\begin{equation}\label{2-1:eq:KV-2}
\frac{1}{t}\overline{\mb E}[R_t^2]\,=\,2\int_{\R^+}\frac{1-e^{-t\lambda}}{t\lambda^2} de_{\mathfrak{d}}(\lambda) \xrightarrow[t \to + \infty]{} 0,
\end{equation}
where $\mathfrak{d}$ is the local drift in the direction $\xi$, that is,
\begin{equation}
\label{defmfkd}
\mfk{d} = - \nabla^* \cdot A(0,\omega) \xi = -\DD^* \cdot A \xi = \sum_{z \sim 0} \omega_{0,z} \xi \cdot z.
\end{equation}
We start by showing the equality in \eqref{2-1:eq:KV-2}, and will later show that the spectral integral tends to $0$ as $t$ tends to infinity. Recall that
$$
R_t = -\phi(X_t,\om) = \phi(0,\om)-\phi(X_t,\om),
$$
and that $\phi(0,\om)-\phi(x,\om)$ can be obtained as the limit of $\phi_\mu(0,\om)-\phi_\mu(x,\om)$. To make them easier, we do the computations below as if $\phi$ was a stationary $\phi_\mu$. The argument can be made rigorous through spectral analysis as in \cite[Theorem~8.1]{Mourrat-10}, or using Lemma~\ref{2-1:lem:corr-mod-corr}. We expand the square:
$$
\E_0[R_t^2] = \E_0[(\phi(0,\om))^2] + \E_0[(\phi(X_t,\om))^2] - 2 \E_0[\phi(0,\om)\phi(X_t,\om)].
$$
Note that, by the simplifying assumption,
$$
\E_0[(\phi(X_t,\om))^2] = \E_0[(\phi(0,\theta_{X_t}\ \om))^2] = \E_0[(\phi(0,\om(t)))^2].
$$
Since $(\om(t))_{t \in \R_+}$ is stationary under $\P_0$, the last expectation is equal to $\E_0[(\phi(0,\om))^2] = \expec{\phi^2}$. Since $\L \phi = \mfk{d}$, we have
$$
\expec{ \phi^2} =\expec{\L^{-1} \mfk{d} \ \L^{-1} \mfk{d}} = \expec{\mfk{d} \ \L^{-2} \mfk{d}} = \int \lambda^{-2} \ \d e_\mfk{d}(\lambda).
$$
For the cross-product,
\begin{equation}
\label{I-semig}
\E_0[\phi(0,\om)\phi(X_t,\om)] = \expec{ \phi(0,\omega) \ \EEo_0\Ll[ \phi(0,\om(t)) \Rr] },
\end{equation}
and $\EEo_0\Ll[ \phi(0,\om(t)) \Rr]$ is the image $\phi$ by the semi-group associated to $-\L$, that is, $e^{-t\L}$. Using also the fact that $\L \phi = \mfk{d}$, we can rewrite the r.h.s.\ of \eqref{I-semig} as
$$
\expec{\phi \ e^{-t\L} \phi} = \expec{\L^{-1} \mfk{d} \ e^{-t\L} \L^{-1} \mfk{d}} = \int \lambda^{-2} e^{-t\lambda} \ \d e_\mfk{d}(\lambda),
$$
and this justifies the equality \eqref{2-1:eq:KV-2}. The fact that the spectral integral in \eqref{2-1:eq:KV-2} tends to zero follows from the dominated convergence theorem and \eqref{2-1:eq:KV-1}.
\medskip
Before concluding this section, we point out several differences between the argument as we presented it and the original one from \cite{Kipnis-Varadhan-86}. Here, we used the existence of the corrector, borrowed from subsection~\ref{sec:2-1-1-2}, to construct the martingale. In \cite{Kipnis-Varadhan-86}, the martingale is constructed directly, by considering the martingale
$$
\xi \cdot X_t + \phi_\mu(\omega(t)) - \phi_\mu(\omega(0)) - \mu \int_0^t \phi_\mu(\omega(s)) \ \d s,
$$
and showing that for each fixed $t$, it is a Cauchy sequence in $L^2(\P_0)$ (and thus converges) as $\mu$ tends to $0$. This is achieved through spectral analysis, using the estimate \eqref{2-1:eq:KV-1}. This estimate is obtained through a general argument of (anti-) symmetry (see the proof of \cite[Theorem~4.1]{Kipnis-Varadhan-86}), which has been systematized by \cite{DeMasi-Ferrari-89}. Another difference is that the arguments of \cite{Kipnis-Varadhan-86} are developped for general reversible Markov processes. | 198,895 |
TITLE: Two ways of explaining the notation $\mathbb{R}^2$
QUESTION [3 upvotes]: I found two ways to explain the notation $\mathbb{R}^2$.
First is by Cartesian product: $\mathbb{R} \times \mathbb{R}$.
Secondly, regarding $2$ as any set with two elements. suppose it's $\{0,1\}$.
Then the map from $\{0,1\}$ to $\mathbb{R}$ forms a linear space. which can be also denoted as $\mathbb{R}^{\{0,1\}} = \mathbb{R}^2$.
In mathematics, if two things looks similar or the same, there will always be some deep reason behind it. So what's the deep reason of the above statements?
REPLY [2 votes]: I don't understand what you exactly mean by "a deep reason", but here's a very elementary answer that can help understand the connection.
First think of the first definition of $\mathbb R^2$ as $\mathbb R\times \mathbb R$. From the definition of Cartesian product, $\mathbb R\times \mathbb R=\{(a,b):a,b\in \mathbb R\}$ is the collection of all possible tuples of real numbers.
Now, consider a function $f:\{0,1\}\to \mathbb R$ and note the tuple $\left(f(0),f(1)\right)$. From the definition of $f$, it is clear that $\left(f(0),f(1)\right)$ is a tuple of real numbers. Now, consider the set of all such functions, and name it $S_f$. Keep in mind the range of these functions.
Note that $S_f$ is absolutely same as $\mathbb R\times \mathbb R$ (defined using Cartesian product) by considering a bijection $F:\mathbb R\times \mathbb R \to S_f$ defined by $F((a,b))=f_{ab}$ where $f_{ab}:\{0,1\}\to \mathbb R$ is the function that takes $0$ to $a$ and $1$ to $b$, i.e., $f_{ab}(0)=a$ and $f_{ab}(1)=b$.
Hope that helps. | 87,236 |
Let's say theres no jagged rocks on the bottom , how easily do you think a big wave charger can skill going down Niagara falls bareback. If an old lady did it in a wood barrel with her cat.. all the jaws and mavericks guys can do It with steeeez..
What do you think?
Results 1 to 3 of 3
Thread: niagara wipeout
niagara wipeout
Shoot us an email when you get done trying and let us know how things went for ya!
If you got gills ya.
I might be able to help | 355,882 |
\begin{document}
\title{Homogeneous rank one perturbations \\
and inverse square potentials}
\author{Jan Dereziński}
\thanks{The financial support of the National Science
Center, Poland, under the grant UMO-2014/15/B/ST1/00126, is gratefully
acknowledged. The author is grateful to Serge~Richard, Vladimir~Georgescu and Laurent~Bruneau for discussions and collaboration.}
\address{Department of Mathematical Methods in Physics\\
Faculty of Physics\\
University of Warsaw\\
Pasteura 5\\
02-093 Warszawa, Poland}
\email{[email protected]}
\begin{abstract}
Following~\cite{D,BDG,DR}, I describe several exactly solvable families of closed operators on $L^2[0,\infty[$. Some of these families are defined by the theory of singular rank one perturbations.
The remaining families are Schrödinger operators with inverse square potentials and various boundary conditions.
I describe a close relationship between these families. In all of them one can observe interesting ``renormalization group flows'' (action of the group of dilations).
\end{abstract}
\subjclass{34L40, 33C10}
\keywords{ Closed operators, rank one perturbations, one-dimensional Schr\"odinger operators, Bessel functions, renormalization group.}
\maketitle
\section{Introduction}
My contribution consists of an introduction and 3 sections, each describing an interesting family of exactly solvable closed operators on $L^2[0,\infty[$.
The first two sections seem at first unrelated. Only in the third section the reader will see a relationship.
Section~\ref{sec1} is based on~\cite{D}. It is devoted to
two families of operators, $H_{m,\lambda}$ and $H_0^\rho$, obtained by a rank one perturbation of a certain generic self-adjoint operator. The operators
can be viewed as an elementary toy model illustrating some properties of the renormalization group. Note that in this section we do not use special functions. However we use a relatively sophisticated
technique to define an operator, called sometimes {\em singular perturbation theory} or the {\em Aronszajn--Donoghue method}, see e.g.,~\cite{AK,KS,DF}.
Section~\ref{sec2} is based on my joint work with Bruneau and Georgescu~\cite{BDG}, and also with Richard~\cite{DR}. It is devoted to Schrödinger operators with potentials proportional to $\frac{1}{x^2}$. Both $-\partial_x^2$ and $\frac{1}{x^2}$ are homogeneous of degree $-2$. With appropriate homogeneous boundary conditions, we obtain a family of operators $H_m$, which we call {\em homogeneous Schrödinger operators}. They are also homogeneous of degree -2. One can compute all basic quantities for these operators using special functions--more precisely, {\em Bessel-type functions} and the {\em Gamma function}.
The operators $H_m$ are defined only for $\Re{m}>-1$. We conjecture that they cannot be extended to the left of the line $\Re m=-1$ in the sense described in our paper. This conjecture was stated in~\cite{BDG}.
It has not been proven or disproved so far.
Finally, Section~\ref{sec3} is based on my joint work with Richard~\cite{DR}, and also on~\cite{D}. It describes more general Schrödinger operators with the inverse square potentials. They are obtained
by mixing the boundary conditions. These operators in general are no longer homogeneous, because their homogeneity is (weakly) broken by their boundary condition---hence the name {\em almost homogeneous Schrödinger operators}. They can be organized in two families $H_{m,\kappa}$ and $H_0^\nu$.
It turns out that there exists a close relationship between the operators from Section~\ref{sec3} and from Section~\ref{sec1}: they are similar to one another. In particular, they have the same point spectrum.
Almost homogeneous Schrödinger operators in the self-adjoint case have been described in the literature before, see e.g.,~\cite{GTV}. However, the non-self-adjoint case seems to heve been first described in~\cite{DR}. A number of new exact formulas about these operators is contained in~\cite{BDG,PR,DR} and~\cite{D}.
Let us also mention one amusing observation, which seems to be original, about self-adjoint extensions of \[-\partial_x^2+\Big(-\frac14+\alpha\Big)\frac{1}{x^2}.\] The ``renormalization group'' acts on the set of these extensions, as described in a table
after Prop.~\ref{table0}. Depending on $\alpha\in\rr$, we obtain 4 ``phases'' of the problem. Some analogies to the condensed matter physics are suggested.
\section{Toy model of renormalization group}
\label{sec1}
Consider the Hilbert space $\cH= L^2[0,\infty[$ and the operator $X$
\[ Xf(x):=xf(x).\]
Let $m\in\cc$, $\lambda\in\cc\cup\{\infty\}$.
Following~\cite{D}, we consider a family of operators formally given by
\beq H_{m,\lambda}:=X+\lambda |x^{\frac{m}{2}}\rangle\langle
x^{\frac{m}{2}}|.\label{pertu}\eeq
In the perturbation $ |x^{\frac{m}{2}}\rangle\langle
x^{\frac{m}{2}}|$ we use the Dirac ket-bra notation, hopefully self-explanatory. Unfortunately, the function $x\mapsto x^{\frac{m}{2}}$
is never square integrable. Therefore, this perturbation is never an operator. It can be however understood as a quadratic form. We will see below how to interpret
(\ref{pertu}) as an operator.
If $-1<\Re m<0$,
the perturbation
$ |x^{\frac{m}{2}}\rangle\langle
x^{\frac{m}{2}}|$ is form bounded relatively to $X$,
and then $H_{m,\lambda}$ can be defined by the form boundedness technique. The perturbation is formally rank one. Therefore,
\begin{align*} &(z-H_{m,\lambda})^{-1}=(z-X)^{-1}\\
&+\sum_{n=0}^\infty
(z-X)^{-1}|x^{\frac{m}{2}}\rangle
(-\lambda)^{n+1}\langle x^{\frac{m}{2}}|(z-X)^{-1}|x^{\frac{m}{2}}\rangle^n
\langle x^{\frac{m}{2}}
|(z-X)^{-1}\\
&=(z-X)^{-1}\\
&+\Big(\lambda^{-1}-\langle x^{\frac{m}{2}}|(z-X)^{-1}|x^{\frac{m}{2}}\rangle\Big)^{-1}
(z-X)^{-1}|x^{\frac{m}{2}}\rangle
\langle x^{\frac{m}{2}}
|(z-X)^{-1}.
\end{align*}
It is an easy exercise in complex analysis to compute
\begin{equation*}
\langle x^{\frac{m}{2}}|(z-X)^{-1}|x^{\frac{m}{2}}\rangle
=\int_0^\infty x^m(z-x)^{-1}\d x=
(-z)^{m}\frac{\pi}{\sin\pi
m}.\end{equation*}
Therefore, the resolvent of
$H_{m,\lambda}$ can be given in a closed form:
\begin{multline*} (z-H_{m,\lambda})^{-1}=(z-X)^{-1}\\
\hspace{-16ex}+\Big(\lambda^{-1}-(-z)^{m}\frac{\pi}{\sin\pi
m}\Big)^{-1} (z-X)^{-1}|x^{\frac{m}{2}}\rangle\langle x^{\frac{m}{2}}
|(z-X)^{-1}.
\end{multline*}
The rhs of the above formula defines a function with values in bounded operators satisfying the resolvant equation for all
$-1<\Re m<1$ and
$\lambda\in\cc\cup\{\infty\}$. Therefore, the method of pseudoresolvent~\cite{Kato2} allows us to define a holomorphic family of
closed operators $H_{m,\lambda}$.
Note that
$H_{m,0}=X$.
The case $m=0$ is special: $H_{0,\lambda}=X$ for all $\lambda$.
One can however introduce another holomorphic family of operators $H_0^\rho$ for any $\rho\in\cc\cup\{\infty\}$ by
\begin{equation*}
(z-H_{0}^\rho)^{-1}=(z-X)^{-1}
-\big(\rho+\ln (-z)\big)^{-1} (z-X)^{-1}|x^0\rangle\langle
x^0|(z-X)^{-1}.
\end{equation*}
In particular, $H_0^\infty=X$.
Let $\rr\ni\tau\mapsto U_\tau$ be the group of
dilations on $L^2[0,\infty[$, that is \[(U_\tau f)(x)=\e^{\tau/2} f(\e^\tau x).\]
We say that $B$ is homogeneous of degree $\nu$ if
\[U_\tau B
U^{-1}_\tau= \e^{\nu\tau}B.\]
E.g., $X$ is homogeneous of degree $1$ and
$|x^{\frac{m}{2}}\rangle\langle
x^{\frac{m}{2}}|$ is homogeneous of degree $1+m$.
The group of dilations (``the renormalization group'') acts on our
operators in a simple way:
\begin{align*}
U_\tau H_{m,\lambda}U_\tau^{-1}&=\e^\tau H_{m,\e^{\tau
m}\lambda},\\
U_\tau H_{0}^\rho U_\tau^{-1}&=\e^\tau H_0^{\rho+\tau}.
\end{align*}
The essential spectrum of $H_{m,
\lambda}$ and $H_0^\nu$ is $[0,\infty[$.
The point spectrum is more intricate, and is described by the following theorem:\pagebreak
\begin{theorem}\leavevmode \ben\item
$z\in\cc\backslash[0,\infty[$ belongs to the point spectrum of
$H_{m,\lambda}$ iff
\[
(-z)^{-m}=\lambda\frac{\pi}{\sin\pi m}.\]
\item
$H_0^\rho$ possesses an eigenvalue iff $-\pi<\Im\rho<\pi$, and
then it is $z=-\e^\rho$.
\een\end{theorem}
For a given pair $(m,\lambda)$ all eigenvalues form a geometric sequence that
lies
on a logarithmic spiral, which
should be viewed as a curve on the Riemann surface of the logarithm.
Only its ``physical sheet'' gives rise to eigenvalues.
For $m$
which are not purely imaginary, only a finite piece of the spiral is
on the ``physical sheet'' and therefore
the number of eigenvalues is finite.
If $m$ is purely imaginary, this spiral degenerates to a
half-line starting at the origin.
If $m$ is real, the spiral degenerates to a circle. But then the
operator has at most
one eigenvalue.
The following theorem about the number of eigenvalues of $H_{m,\lambda}$ is proven in~\cite{DR}. It describes an interesting pattern of ``phase transitions'' when we vary the parameter $m$. In this theorem, we denote by $\sp_\p(A)$ denotes the set of eigenvalues of an operator $A$ and $\#X$ denotes the number of elements of the set $X$.
\begin{theorem}
Let $m= m_\r+\i m_\i \in \cc\backslash\{0\}$ with $|m_\r|<1$.
\begin{enumerate}
\item[(i)] Let $m_\r=0$.
\begin{enumerate}\item[(a)] If $\frac{\ln(|\varsigma|)}{m_\i}\in ]-\pi,\pi[$,
then $\#\sp_\p(H_{m,\lambda}) = \infty$, \item[(a)] if
$\frac{\ln(|\lambda\frac{\pi}{\sin\pi m}|)}{m_\i}\not \in ]-\pi,\pi[$
then $\#\sp_\p(H_{m,\lambda}) = 0$.\end{enumerate}
\item[(ii)] If $m_\r\neq 0$ and if $N\in \nn$ satisfies
$N<\frac{m_\r^2+m_\i^2}{|m_\r|} \leq N+1$, then
\begin{equation*}
\#\sp_\p(H_{m,\lambda})\in \{N,N+1\}.
\end{equation*}
\end{enumerate}
\end{theorem}
\section{Homogeneous Schrödinger operators}
\label{sec2}
Let $\alpha\in\cc$. Consider the differential expression
\begin{equation*}L_\alpha=-\partial_x^2+\Big(-\frac14+
\alpha\Big)\frac{1}{x^{2}}.\label{1}\end{equation*}
$L_\alpha$ is is homogeneous of
degree $-2$.
Following~\cite{BDG}, we would like to interpret $L_\alpha$ as a closed operator on $L^2[0,\infty[$
homogeneous of degree $-2$.
$L_\alpha$, and closely related operators $H_m$ that we introduce
shortly,
are interesting for many
reasons.
\begin{itemize}
\item They appear
as the radial part of the Laplacian in all dimensions,
in
the decomposition of
Aharonov-Bohm Hamiltonian, in the membranes with conical
singularities, in the theory of many body systems with contact interactions and in the Efimov effect.
\item They have rather subtle and rich properties illustrating various
concepts of the operator theory in Hilbert spaces (eg. the Friedrichs and
Krein
extensions, holomorphic families of closed operators).
\item Essentially all basic objects related to $H_m$, such as their
resolvents, spectral projections, M{\o}ller and scattering operators, can be
explicitly computed.
\item A number of
nontrivial identities involving special functions, especially from the Bessel
family, find an
appealing operator-theoretical interpretation in terms of the operators
$H_m$. E.g. the Barnes identity
leads to the formula for M{\o}ller
operators.
\end{itemize}
We start the Hilbert space theory of the operator $L_\alpha$ by defining
its two naive interpretations on $L^2[0,\infty[$:
\begin{enumerate} \item The minimal operator $L_\alpha^{\min}$: We start from $L_\alpha$ on
$C_\c^\infty]0,\infty[$, and then we take its closure.
\item The maximal operator
$L_\alpha^{\max}$: We consider the domain consisting of all
$f\in L^2[0,\infty[$ such that $L_\alpha f\in L^2[0,\infty[$.
\end{enumerate}
We will see that it is often natural to write $\alpha=m^2$. Let us describe basic properties of $L_{m^2}^{\max}$ and $L_{m^2}^{\min}$:
\begin{theorem} \leavevmode\begin{enumerate}
\item For $1\leq\Re m$, $L_{m^2}^{\min}=L_{m^2}^{\max}$.
\item For $-1<\Re m<1$,
$L_{m^2}^{\min}\subsetneq L_{m^2}^{\max}$, and the codimension of
their domains is $2$.
\item $ (L_{\alpha}^{\min})^*=L_{\bar\alpha}^{\max}$.
Hence, for $\alpha\in\rr$, $L_\alpha^{\min}$ is Hermitian.
\item $L_{\alpha}^{\min}$ and $L_{\alpha}^{\max}$ are homogeneous of
degree $-2$.
\end{enumerate}
\end{theorem}
Let $\xi$ be a compactly supported cutoff equal $1$ around $0$.
Let $-1\leq \Re m $.
It is easy to check that $x^{\frac12+m}\xi$ belongs to $\Dom L_{m^2}^{\max}$.
We
define the operator $H_m$ to be the restriction of $L_{m^2}^{\max}$ to
\[\Dom L_{m^2}^{\min}+\cc x^{\frac12+m}\xi.\]
The operators $H_m$ are in a sense more interesting than $L_{m^2}^{\max}$ and $L_{m^2}^{\min}$:
\begin{theorem}\leavevmode \begin{enumerate}
\item For $1\leq\Re m$, $L_{m^2}^{\min}=H_m=L_{m^2}^{\max}$.
\item For $-1<\Re m<1$,
$L_{m^2}^{\min}\subsetneq H_m
\subsetneq L_{m^2}^{\max}$ and the codimension of the domains is $1$.
\item $ H_m^*=H_{\bar m}$. Hence, for $m\in ]-1,\infty[$, $H_m$ is self-adjoint.
\item $H_m$ is homogeneous of
degree $-2$.
\item $\sp H_m=[0,\infty[$.
\item $\{\Re m>-1\}\ni m\mapsto H_m$ is a holomorphic family of
closed operators. \end{enumerate}
\end{theorem}
The theorem below is devoted to self-adjoint operators within the family~$H_m$.
\begin{theorem}\label{th:imain}
\leavevmode\begin{enumerate}
\item For $\alpha\geq 1$, $L_\alpha^{\min}=H_{\sqrt \alpha}$ is essentially
self-adjoint
on $C_{\rm c}^\infty]0,\infty[$.
\item For $\alpha<1$, $L_\alpha^{\min}$ is Hermitian but not essentially
self-adjoint on $C_{\rm c}^\infty]0,\infty[$. It has deficiency indices $1,1$.
\item
For $0\leq\alpha<1$,
the operator $H_{\sqrt\alpha}$ is the Friedrichs extension
and
$H_{-\sqrt\alpha}$ is the Krein extension of
$L_\alpha^{\min}$.
\item $H_{\frac12}$ is the Dirichlet Laplacian and
$H_{-\frac12}$ is the Neumann Laplacian on halfline.
\item
For $\alpha<0$,
$L_\alpha^{\min}$ has no homogeneous selfadjoint extensions.
\end{enumerate}
\end{theorem}
Various objects related to $H_m$
can be computed with help of functions from the Bessel family.
Indeed, we have the following identity
\begin{equation*}
x^{-\frac12}
\Big(-\partial_x^2+\big(-\frac14+
\alpha\big)\frac{1}{x^{2}}\pm1\Big)x^{\frac12}
=-\partial_x^2-\frac1x\partial_x+\big(-\frac14+
\alpha\big)\frac{1}{x^{2}}\pm1,\end{equation*}
where the rhs defines the well-known (modified) Bessel equation.
One can compute explicitly the resolvent of $H_m$:
\begin{theorem}\label{th:reso}
Denote by $R_m(-k^2;x,y)$ the integral kernel of
the operator $(k^2+H_m)^{-1}$. Then for $\Re k>0$ we have
\[
R_m(-k^2; x,y) =
\begin{cases} \sqrt{xy}I_m(kx)K_m(ky) & \text{ if } x<y, \\
\sqrt{xy}I_m(ky)K_m(kx) & \text{ if } x>y, \end{cases}
\]
where $I_m$ is the modified Bessel function and $K_m$ is the
MacDonald function.
\end{theorem}
The operators $H_m$ are similar to self-adjoint operators. Therefore, they possess the spectral projection onto any Borel subset of their spectrum
$[0,\infty[$. In particular, below we give a formula for the spectral projection of $H_m$ onto the interval $[a,b]$:
\begin{proposition}\label{prop:hmproj} For $0<a<b<\infty$, the integral kernel of $\one_{[a,b]}(H_m)$ is
\begin{equation*}
\one_{[a,b]}(H_m)(x,y)
= \int_{\sqrt a}^{\sqrt b} \sqrt{xy}J_m(k x) J_m(k y)k\d k,\label{hankel2}
\end{equation*}
where $J_m$ is the Bessel function.
\end{proposition}
One can diagonalize the operators $H_m$ in a natural way, using the so-called Hankel transformation $\cF_m$, which is
the operator on $L^2[0,\infty[$ given by
\begin{equation}
\big(\cF_mf\big)(x):= \int_0^\infty J_m(kx)\sqrt{kx}f(x)\d x
\label{hankel}\end{equation}
\begin{theorem} \label{th:hank} $\cF_m$ is a bounded invertible involution on
$L^2[0,\infty[$ diagonalizing $H_m$, more precisely
$$
\cF_mH_m\cF_m^{-1}=X^2.
$$
It satisfies $\cF_mA=-A\cF_m$,
where
\[A=\frac1{2\i} (x\partial_x+\partial_xx)\]
is the self-adjoint generator of dilations.
\end{theorem}
It turns out that the Hankel transformation can be expressed in terms of the generator of dilations. This expression, together with the Stirling formula for the asymptotics of the Gamma function, proves
the boundedness of $\cF_m$.
\begin{theorem}
Set
\[\cI f(x)=x^{-1}f(x^{-1}),\quad
\Xi_m(t)=\e^{\i\ln(2)t}\frac{\Gamma(\frac{m+1+\i t}{2})}{\Gamma(\frac{m+1-\i t}{2})}.
\]Then
\[\cF_m=\Xi_m(A)\cI.\]
Therefore,
we have the identity
\begin{equation}
H_m:=\Xi_m^{-1}(A) X^{-2}\Xi_m(A).\label{ham}
\end{equation}
\end{theorem}
(Result obtained independently by Bruneau, Georgescu,
and myself in~\cite{BDG}, and
by Richard and Pankrashkin in~\cite{PR}).
The operators $H_m$ generate 1-parameter groups of bounded operators. They possess scattering theory and one can explicitly compute their M{\o}ller (wave) operators and the scattering operator.
\begin{theorem}
The M{\o}ller operators associated to the
pair $H_m,H_k$ exist and
\begin{equation*}
\Omega_{m,k}^\pm \,:=\,\lim_{t\to\pm\infty}\e^{\i tH_m}\e^{-\i tH_k}
=\e^{\pm \i(m-k)\pi/2}\cF_m\cF_k
=
\e^{\pm \i(m-k)\pi/2}\frac{\Xi_k(A)}{\Xi_m(A)}.
\label{pankra}\end{equation*}
\end{theorem}
The formula
(\ref{ham}) valid for
$\Re m>-1$
can be used as an alternative definition of the family $H_m$ also beyond this domain.
It defines a family of closed operators for the
parameter $m$ that belongs to
\beq \{m\mid \Re m\neq-1,-2,\dots\}\cup\rr.\label{doma}\eeq
Their spectrum is always equal to $[0,\infty[$
and they are analytic in the interior of (\ref{doma}).
In fact, $\Xi_m(A)$ is a unitary
operator for all real values of $m$.
Therefore, for $m\in\rr$, (\ref{ham}) is well-defined and self-adjoint.
$\Xi_m(A)$ is
bounded and invertible also for all $m$ such that $\Re
m\neq-1,-2,\dots$. Therefore, formula (\ref{ham}) defines an
operator for all such $m$.
One can then pose various questions:
\begin{itemize}
\item
What happens with these operators along the lines $\Re m=-1,-2,\dots$?
\item What is the meaning of these operators to the left of $\Re=-1?$ (They are not differential operators!)
\end{itemize}
Let us describe a certain precise conjecture about the family $H_m$. In order to state it we need to define the concept of a holomorphic family of closed operators.
The definition (or actually a number of equivalent definitions) of a
holomorphic family of bounded operators is quite obvious and
does not need to be recalled. In the case of unbounded operators the
situation is more subtle, and is described e.g., in~\cite{Kato2}, see also~\cite{DW}.
Suppose that $\Theta$ is an open subset of $\cc$, $\cH$ is a Banach
space, and $\Theta\ni z\mapsto H(z)$ is a function whose values
are closed operators on $\cH$. We say that this is a
holomorphic family of closed operators if for each $z_0\in\Theta$
there exists a neighborhood $\Theta_0$ of $z_0$, a Banach space
$\cK$ and a holomorphic family of injective bounded operators $\Theta_0\ni
z\mapsto B(z)\in B(\cK,\cH)$ such that $\Ran B(z)=\cD(H(z))$ and
\begin{equation*}
\Theta_0\ni z\mapsto H(z)B(z)\in B(\cK,\cH)\label{holo2}
\end{equation*}
is a holomorphic family of bounded operators.
We have the following practical criterion:
\begin{theorem}\label{crit} Suppose that $\{H(z)\}_{z\in\Theta}$ is a
function whose values are closed operators on $\cH$. Suppose in
addition that for any $z\in\Theta$ the resolvent set of $H(z)$
is nonempty. Then $z\mapsto H(z)$ is a holomorphic family of
closed operators if and only if for any $z_0\in\Theta$ there exists
$\lambda\in\C$ and a neighborhood $\Theta_0$ of $z_0$ such that
$\lambda$ belongs to the resolvent set of $H(z)$
for $z\in\Theta_0$ and $z\mapsto
(H(z)-\lambda)^{-1}\in B(\cH)$ is holomorphic on $\Theta_0$.
\end{theorem}
The above theorem indicates that it is more difficult to study
holomorphic families of closed operators that for some values of the
complex parameter have an empty resolvent set.
We have the following conjecture (formulated as an open question in~\cite{BDG}), so far unproven:
\begin{conjecture}
It is impossible to extend \[\{\Re m>-1\}\ni m\mapsto H_m\]
to
a holomorphic family of closed operators on a larger connected open subset of
$\cc$.
\end{conjecture}
\section{Almost homogeneous Schrödinger operators}
\label{sec3}
For $-1<\Re m<1$ the codimension of $\Dom(L_{m^2}^{\min})$ in $\Dom(L_{m^2}^{\max})$
is two. Therefore, following~\cite{DR}, one can fit a 1-parameter family of closed operators in between $L_{m^2}^{\min}$ in $L_{m^2}^{\max}$, mixing the boundary condition
$x^{\frac12+m}$ and $x^{\frac12-m}$. These operators in general are no longer homogeneous---their homogeneity is broken by the boundary condition. We will say that they are {\em almost homogeneous}.
More precisely,
for any $\kappa\in \C\cup \{\infty\}$ let $H_{m,\kappa}$
be the restriction of $L_{m^2}^{\max}$ to the domain
\begin{multline*}
\Dom(H_{m,\kappa}) = \big\{f\in \Dom(L_{m^2}^{\max})\mid
\hbox{ for some } c \in \C,\\
f(x)- c
\big(x^{1/2-m} +\kappa
x^{1/2+m}\big)\in\Dom(L_{m^2}^{\min})
\text{ around }
0\big\},\qquad\kappa\neq\infty;
\end{multline*}
\begin{multline*}
\Dom(H_{m,\infty}) =\big\{f\in \Dom(L_{m^2}^{\max})\mid
\hbox{ for some } c \in \C,\\
f(x)- c
x^{1/2+m}\in\Dom(L_{m^2}^{\min})\text{ around } 0\big\}.
\end{multline*}
The case $m=0$ needs a special treatment.
For
$\nu\in\C\cup \{\infty\}$, let $H_0^\nu$ be the restriction of
$L_0^{\max}$ to
\begin{multline*}
\Dom(H_0^\nu) = \big\{f\in \Dom(L_{0}^{\max})\mid
\hbox{ for some } c \in \C,\\
f(x)- c
\big(x^{1/2}\ln x +\nu x^{1/2}\big)\in\Dom(L_0^{\min})
\text{ around } 0\big\},\qquad\nu\neq\infty;
\end{multline*}
\begin{multline*}
\Dom(H_0^\infty) = \big\{f\in \Dom(L_0^{\max})\mid
\hbox{ for some } c \in \C,\\
f(x)- c
x^{1/2}\in\Dom(L_0^{\min})\hbox{ around }
0\big\}.
\end{multline*}
Here are the basic properties of almost homogeneous Schrödinger operators.
\begin{proposition}\leavevmode
\begin{enumerate}
\item For any $|\Re(m)|<1$, $\kappa\in \C\cup \{\infty\}$
\begin{equation*}\label{Eq_duality}
H_{m,\kappa}=H_{-m,\kappa^{-1}}.
\end{equation*}
\item
$H_{0,\kappa}$ does not depend on $\kappa$,
and these operators coincide with $H_0^\infty$.
\item We have
\begin{align*}
U_\tau H_{m,\kappa}U_{-\tau}&=\e^{-2\tau}H_{m,\e^{-2\tau m}\kappa},\\
U_\tau H_0^\nu U_{-\tau}&=\e^{-2\tau}H_0^{\nu+\tau}.
\end{align*}
In particular, only
\begin{equation*}
H_{m,0}=H_{-m},\quad
H_{m,\infty}=H_m,\quad
H_0^\infty=H_0
\end{equation*} are homogeneous.
\end{enumerate}\label{basic}
\end{proposition}
The following proposition describes self-adjoint cases among these operators.
\begin{proposition}.
\begin{equation*}\label{Eq_adjoint}
H_{m,\kappa}^*=H_{\bar m,\bar\kappa}\qquad \hbox{ and }\qquad
H_0^{\nu*} = H_0^{\bar \nu}.
\end{equation*}
In particular,
\begin{enumerate}
\item[(i)] $H_{m,\kappa}$ is self-adjoint for $m\in]-1,1[$ and $\kappa\in\R\cup\{\infty\}$,
and for $m\in \i\R$ and $|\kappa|=1$.
\item[(ii)] $H_0^\nu$
is self-adjoint for
$\nu\in \R\cup \{\infty\}$.
\end{enumerate}\end{proposition}
The essential spectrum of $H_{m,\kappa}$ and $H_0^\nu$ is always $[0,\infty[$. The following proposition describes the point spectrum in the self-adjoint case.
\begin{proposition}\leavevmode
\begin{enumerate}\item If $m\in]-1,1[$ and $\kappa
\geq0$ or $\kappa=\infty$, then $H_{m,\kappa}$ has no eigenvalues.\item
If $m\in]-1,1[$ and $\kappa<0$, then $H_{m,\kappa}$ has a single eigenvalue\\ at
$-4\big(\frac{\Gamma(m)}{\kappa\Gamma(-m)}\big)^{\frac1m}$.
\item If $m\in\i\R$ and
$|\kappa|=1$, then $H_{m,\kappa}$ has an infinite sequence of eigenvalues accumulating at $-\infty$ and $0$. If $m=\i m_\I$ and
$ \e^{\i\alpha}=\frac{\kappa\Gamma(- \i m_\I)}{\Gamma(\i m_\I)}$, then
these eigenvalues are $-4\exp(-\frac{\alpha+2\pi n}{m_\I})$, $n\in\zz$. \end{enumerate}\label{table0}
\end{proposition}
It is interesting to analyze how the set of self-adjoint extensions of
the Hermitian operator
\begin{equation*}L_\alpha^{\min}=-\partial_x^2+\Big(-\frac14+
\alpha\Big)\frac{1}{x^{2}}\end{equation*}
depends on the real parameter $\alpha$.
Self-adjoint extensions form a set isomorphic either to a point or to a circle. The ``renormalization group'' acts on this set by a continuous flow, as described by Proposition \ref{basic}.
This flow may have fixed points.
The following table describes the various ``phases'' of the theory of self-adjoint extensions of $L_\alpha^{\min}$. To each phase I give a name inspired by condensed matter physics. The reader does not have to take these names very seriously, however I suspect that they have some deeper meaning.
\bigskip
\\
\begin{tabular}{llll}\hline\\
$1\leq\alpha$&``gas''&point&\parbox{0.42\linewidth}{Unique fixed point: Friedrichs extension=Krein extension.}\\\\
\hline\\
$0<\alpha<1$&``liquid''&circle&\parbox{0.42\linewidth}
{Two fixed points: Friedrichs and Krein extension.\\
Ren. group flows from Krein to Friedrichs.\\
On one semicircle of non-fixed points all have one bound state; on the other all have no bound states.}\\\\\hline\\
$\alpha=0$&\parbox{0.25\linewidth}{``liquid--solid\\ phase transition''}&circle&\parbox{0.42\linewidth}
{Unique fixed point: Friedrichs extension=Krein extension.\\ Ren. group flows from Krein to Friedrichs.\\
Non-fixed points have one bound state.}\\\\
\hline\\
$\alpha<0$&``solid''&circle&\parbox{0.42\linewidth}
{No fixed points.\\Ren. group rotates the circle.\\ All have infinitely many bound states.}\\\\
\hline\\
\end{tabular}
\noindent The above table can be represented by the following picture, hopefully self-explanatory:
\medskip
\noindent
\begin{tikzpicture}
\draw[very thick,dotted](0.2,1)circle(0.7);
\draw[thick,->](0.2,1.7)--(0.1,1.7) ;
\draw[thick,->](0.2,0.3)--(0.3,0.3) ;
\draw[very thick,dashed](3,1)circle(0.7);
\fill[black](2.3,1)circle(0.1) node[anchor=east] {K=F};
\draw[thick,->](3,1.7)--(2.9,1.7) ;
\draw[thick,->](3,0.3)--(3.1,0.3) ;
\draw[very thick,dashed](5.8,0.3)arc(-90:90:0.7);
\draw[thin](5.8,1.7)arc(90:270:0.7);
\fill[black](5.8,1.7)circle(0.1) node[anchor=south] {F};
\fill[black](5.8,0.3)circle(0.1) node[anchor=south] {K};
\draw[thick,->](5.1,1)--(5.1,1.1) ;
\draw[thick,->](6.5,1)--(6.5,1.1) ;
\fill[black](9,1)circle(0.1) node[anchor=east] {K=F};
\draw[thick,-] (-1,0) -- (3,0) node[anchor=north] {\bf 0};
\fill[black] (3,0) circle (0.4ex);
\draw[thick,-] (3,0) -- (7,0) node[anchor=north] {\bf 1};
\fill[black] (7,0) circle (0.4ex);
\draw[thick,->] (7,0) -- (10,0)
node[anchor=south east] {$\alpha$};
\node at (0.2,-0.9){``solid''};
\node at (3,-0.7){``phase};
\node at (3,-1.1){transition''};
\node at (5.8,-0.9){``liquid''};
\node at (9,-0.9){``gas''};
\draw[thick,dotted,-](3,0) --(3,-1.5);
\draw[thick,dotted,-](7,0) --(7,-1.5);
\end{tikzpicture}
\bigskip
There exists a close link between almost homogeneous Schrödinger operators described in this section and the ``toy model of renormalization group'' described in Section~\ref{sec1}. It turns out that the corresponding operators are similar to one another.
Define the unitary operator
\[ (If)(x):=x^{-\frac14}f(2\sqrt x).\]
Its inverse is
\[
(I^{-1}f)(x):=\Big(\frac{y}{2}\Big)^{\frac12}f\Big(\frac{y^2}{4}\Big).\]
Note that
\begin{equation*}
I^{-1}XI=\frac{X^2}{4},\quad
I^{-1}AI=\frac{A}{2}.
\end{equation*}
We change slightly notation:
the operators $H_m$, $H_{m,\kappa}$ and $H_0^\nu$ of this section
will be denoted
$\tilde H_m$, $\tilde H_{m,\kappa}$ and $
\tilde H_0^\nu$.
Recall that in (\ref{hankel}) we
introduced the Hankel transformation $\cF_m$, which is a
bounded invertible involution satisfying
\begin{eqnarray*}\cF_m\tilde H_m\cF_m^{-1}&=&X^2,\\
\cF_mA\cF_m^{-1}&=&-A.
\end{eqnarray*}
Recall also that in Section~\ref{sec1} we introduced the operators $H_{m,\lambda}$ and $H_0^\rho$.
The following theorem is proven in~\cite{D}:
\begin{theorem}.
\ben\item
If $
\lambda\frac{\pi}{\sin(\pi m)}=\kappa\frac{\Gamma(m)}{\Gamma(-m) },$
then the operators $H_{m,\lambda}$ are similar to $\tilde H_{m,\kappa}$:
\[
\cF_m^{-1}I^{-1} H_{m,\lambda}I\cF_m=\frac14\tilde
H_{m,\kappa},\]
\item
If $\rho=-2\nu$, then the operators $H_0^\rho$ are similar to $\tilde H_0^\nu$:
\[
\cF_m^{-1}I^{-1} H_0^\rho I\cF_m=\frac14\tilde
H_0^\nu,\]
\een
\end{theorem} | 1,581 |
.
Through The Dusty Paths Of Our Lives (2012) | 294,694 |
New Mexico 18 and Up Sports Betting Sites
In October 2018, New Mexico became the 6th state to offer sports betting at the local level following the US Supreme Court repeal of PASPA in May of the same year. Despite the state not passing a bill, the Santa Ana Star Casino and Hotel was able to offer sports betting due to it being included in a tribal pact with the state and became the first to begin accepting wagers in the state.
But this only for land-based sportsbooks. Can NM adults who are at least 18 years old legally place real-money bets on sports online and on mobile devices? We answer that question, provide our recommended list of 18+ online sportsbooks, and rundown pending bills and laws related to sports betting in the state of New Mexico.
Top Rated Online Sportsbooks For 18+ New Mexican Bettors
Can you bet on sports at age 18 in New Mexico?
The minimum sports betting age in New Mexico is age twenty-one and not eighteen years old. While the legal sports betting sites we recommend allow you to sign up at age eighteen, we recommend following the minimum age requirements that the state legislature has deemed appropriate, even though any legal recourse is highly unlikely.
Is it legal to bet at 18+ online sportsbooks in New Mexico?
There is no sports betting laws in New Mexico or at the federal level that prevents you from placing a bet at any of the 18 and up online sportsbooks we recommend for online and mobile wagering for sports. Each one is located offshore and outside the jurisdiction of the United States. We still recommend following the required age of 21 for your state but must be honest in saying that there has never been anyone charged at 18 years of age for betting at a sportsbook located offshore.
Are sports betting sites safe for New Mexico residents?
Not every sports betting site that you’ll find online can be trusted, which is why we created this site. We want to help residents from New Mexico and all states find the safest age 18 and over sports betting sites available online. Each site we recommend is 100% and you’ll have no issues when wagering at any of them. We can’t say the same for all sites you’ll find online but we can safely say that about the sportsbooks above.
Did the PASPA repeal pave the way for legal sports betting in New Mexico?
Yes. Thanks to PASPA being repealed by the US Supreme Court in May 2018, and because of the wording of the state-tribal casino pact, sports betting at the state level became officially legal in tribal casinos throughout New Mexico and those wanting to receive a tribal sportsbook license can begin the application process.
New Mexico Sports Betting Laws/Pending Bills
Currently, New Mexico has only one bill related to sports betting that has been introduced in 2019. Unfortunately, this bill would eliminate any possibility of the state lottery potentially offering state-regulated sports wagering in the future.
State-Licensed Sportsbooks in New Mexico
The Santa Ana Star become the first state-licensed sportsbook located in New Mexico in October 2018, and they were joined by Buffalo Thunder in March 2019. To date, these are the only two sportsbooks located in the NM jurisdiction and the state does not currently offer any state-regulated online wagering.
Land-Based
- Buffalo Thunder
- Santa Ana Star
New Mexico College and Professional Sports Teams
Residents of New Mexico currently root have two colleges competing in sports at the NCAA Division I level but the state does not have any franchises as members of a major professional sports league. Below you’ll find all the teams from New Mexico that will regularly receive betting lines from the sportsbooks we recommend.
NCAA Division I Teams:
- New Mexico Lobos: Moutain West (Albuquerque)
- New Mexico State Aggies: WAC (Las Cruces)
Major Professional Sports Teams:
- None
Legal Minimum Gambling Age in New Mexico
In order to gamble at a tribal casino in New Mexico and play casino games, including slot machines and table games along with poker, you must meet the minimum required gambling age of at least 21 years old.
Other Forms of Legal 18+ Gambling in New Mexico
However, if you are age 18 and over in New Mexico, there are quite a few legal options available to you in regards to gambling. Age 18+ adults can play bingo, buy lottery tickets, and bet on horse races. Additionally, the 18 and over online sportsbooks, we recommend also have 18+ online casinos and poker rooms that are legally available as well.
Additional Resources
If you or someone you know has begun to develop bad gambling habits that are negatively affecting your everyday financial well-being in New Mexico, please contact the New Mexico Council On Problem Gambling for immediate help. | 179,479 |
The Consumer Technology Association (CTA) announced today that its annual conference, CES, will now be held from Jan. 11 to 14 in 2021. The conference was originally scheduled to occur from Jan. 6 to 9. For more information, visit the CES site.
CES Shifts Dates for 2021 Virtual Conference
September 10, 2020
I want to hear from you. Tell me how we can improve.
BNP Media Owner & Co-CEO, Tagg Henderson | 168,172 |
The University of Indonesia (UI) is ready to implement output based research budget expected to improve research productivity, because so far research is more focused on process burdened by administrative issues.
Deputy Rector III of the University of Indonesia in Research and Innovation, Rosari Saleh said it was conducted in accordance with Kemenristek Dikti’s direction in order to carry out a paradigm shift in research which is no longer on process but output from research. The change towards output based research familiarized by Kemenristek Dikti is believed to improve productivity of research in Indonesia.
She said UI was not concerned over the paradigm shift and UI had already started to implement output based research budget (ARBO) in building a research climate here. She also stated a number of new policies in the world of research and innovation of the University of Indonesia have been conducted. She also said every research proposal should be coupled with signing of an agreement to ensure significant output for the improvement of research quality nationwide as well as international.
According to her a set of programs have also been prepared to answer the challenges of improving research quality, including by providing intensive assistance for every research idea that originates from students to lecturers. UI even provides a special symposium space for thesis level students to encompass as many fresh ideas as possible in research and innovation.
Link: | 71,214 |
TITLE: Decompose Poisson random variable as sum of Poisson random variables
QUESTION [2 upvotes]: If $X,Y$ are independent Poisson random variables with parameter $\lambda_1, \lambda_2$, then $X+Y$ is Poisson random variable with parameter $\lambda_1+\lambda_2$. I am wondering whether the converse if true, given a poisson random variable on a probability space $(\Omega, \mathcal{F},P)$, can we always decompose it into independent poisson random variables with parameter $\lambda_1,\lambda_2$ such that there sum is the given random variable?
REPLY [0 votes]: Suppose $W\sim\operatorname{Poisson}(\lambda).$
Suppose $0<\lambda_1 <\lambda,$ and let $\lambda_2 = \lambda - \lambda_1.$
Let $p = \dfrac{\lambda_1}\lambda = \dfrac{\lambda_1}{\lambda_1+\lambda_2}.$
Let $X\mid W \sim\operatorname{Binomial}(W,p),$ i.e. this is the number of successes in $W$ independent trials with probability $p$ of success on each trial. Let $Y= W-X.$ Then $Y\mid W \sim\operatorname{Binomial}(W,1-p).$
Given all of this, one can conclude that
$W=X+Y.$
$X\sim\operatorname{Poisson}(\lambda_1).$
$Y \sim\operatorname{Poisson}(\lambda_2).$
$X,Y$ are independent.
Proving this is a standard exercise. How to do it is a question that has been posted here a number of times. | 127,262 |
TITLE: How to prove every open set is Lebesgue measurable?
QUESTION [1 upvotes]: I am currently using Stein's book to self study measure theory and now I'm stuck on proving this property of Lebesgue measure below.
Property: Every open set in $\mathbb R^d$ is measurable
The book just says this immediately follows from the definition of Lebesgue measure (a subset $E$ of $\mathbb R^d$ is Lebesgue measurable, is for any $\varepsilon>0$ there exists an open set $O$ with $E\subset O$ and $m^*\left(O\smallsetminus E\right)<\varepsilon$), but I'm not sure how the property is derived from the definition.
Thanks in advance.
REPLY [1 votes]: Modern usage is that $E\subset E.$ The author must be using this, and not requiring that $E\subsetneqq O.$ Because in the case $E=\Bbb R^n$ there is no $O$ such that $E\subsetneqq O\subset \Bbb R^n$, but if $E=\Bbb R^n$ then $E $ $ is $ measurable. So we can let $O=E,$ and it should not be hard to prove that $0=m^*(\phi)=m^*(O \backslash E).$ | 11,490 |
In order to get relief from the debts, people are looking out for various ways that are comfortable for them. Whether it is debt of a person who is living a normal economical standard or it may be business person, debt is a debt. And the difficulty in repaying it as well as the level of debt will vary with each other. It is very essential to smartly overcome those problems. But the people who are affected with the debt will be stressed and they cannot able to think what to do next and so on. To help such kind of people, debt relief companies are available and if you are one of people who are suffering with the debt problem as well as do not know what to do next, then you can approach these companies. While picking the service provider you should give preference to the ones who are experienced in the field for years. When the organization has more knowledge in these their business services then they can help you out in the reductions on your total loan amounts with ease. You just need to spend time with the service provider and make them to know about the requirements of yours in detail. You should decide on a reliable debt relief company so that you may even have a good reduction in your interest rates. Before choosing any debt relief services, you have to speak with the business representative and they will have the ability to examine your finances and gives the best settlement strategy.
Consultations are free of cost in majority of the companies. So you should have a clear cut discussion with the company regarding everything to avoid the troubles. The professionals of debt relief will provide suggestions to make you to get rid of the debt usually in twenty four to forty eight weeks and so that the debt amounts might be reduced into half. And also these companies help consumers repay debt on the mentioned schedule. Some credit counseling companies are charitable, but most are for profit. Credit counseling agencies usually put up a debt management strategy that reduces the consumer’s payment obligation. The companies may do this through prearranged contracts with credit card issuers that permit them to reduce interest rates on current debt to your creditor. Moreover, a debt management plan is better than financing or negotiation. Hence it is very important to hire any of the debt relief service providers to meet your requirements.
There are some procedures available to find out the reliable one like freedomdebtrelief. You should visit the website of the company and go through the information that depicts the kind of services they are providing. A genuine service provider will have the good website with the clear details. Then you should also go through the testimonials that are provided by their previous clients. This will make you to know more about the company in detail. In addition to these, you should also check whether there is any online complaint present on the name of the company. If you find any, just stay out. | 409,669 |
Hot Stuff (Hot Zone Book 1)
Información del producto
Presione el siguiente botón para traducir el texto a español
Descripc.
Detalles del producto
- Autor: Carly Phillips
- Editorial: CP Publishing
- Formato: Kindle eBook
- Número de páginas: 333
- Idioma: English (Published)
- Autografiado: No
- Memorabilia: No
| 227,354 |
It started as a patriotic mystery, but now it's a Fourth of July tradition.
You might have spotted the mini-American flags popping up in Plattsburgh.
The Independence Day operation started a couple of years ago and no one was quite sure who was responsible for the red, white and blue display.
Now the secret flag planters are revealing themselves and working to get other community members to help spread the American spirit.
If you'd like to help plant the little Old Glories, organizers say they plan to be out again next year. | 119,466 |
TITLE: Basic Application of Implicit Function Theorem
QUESTION [2 upvotes]: I am studying an old exam for a course in real analysis and came across this problem (not a homework problem).
Let
$f(x, y, z) = (xyz, (z \cos x) + y - 1)$
and observe that $f(2\pi, 1, 0) = (0, 0)$. Only one of the following is true, circle and carefully
justify the correct statement:
(a) There exists an open set $U \subset \mathbb R^1$ containing $2\pi$ and a $C^1$ function
$g = (g_1,g_2) : U \to \mathbb R^2$
such that $f(x, g_1 (x), g_2(x)) = (0, 0)$ for all $x \in U$.
(b) There exists an open set $U \subset \mathbb R^2$ containing $(1, 0)$ and a $C^1$ function
g : $U \to \mathbb R^1$
such that $f(g(y, z), y, z) = (0, 0)$ for all $(y, z) \in U$.
The answer is marked as (b) but it seems that it should be (a) because the implicit function theorem says that given a function $f: \mathbb R ^{m+n} \to \mathbb R^m$ and some conditions on the derivative it is possible to find a function from $ g(x) \mathbb R^n \to \mathbb R^m$ s.t. $(g(x), x) = 0$ for all $x \in \mathbb R^n$. The rest of the problem (verifying that these conditions hold) I am not worried about. Am I just confused about what the implicit function theorem states/how to use it? Or is the answer (a)?
REPLY [1 votes]: I actually agree with you. Discussion:
The implicit function theorem basically says that if you have $m$ independent scalar equations in $n+m$ variables, then you can (locally) solve for $m$ of the variables in terms of $n$ of the variables. (The technical details of the result are about what "independent" and "locally" really mean.) In the process, the number of independent variables drops by the number of scalar constraints. One way to remember this is to note that when the system is linear, the result is exactly the same as the rank-nullity theorem in the case of a full rank matrix.
In your case you start with $3$ independent variables and impose $2$ scalar constraints, so you should have $3-2=1$ independent variable at the end.
The one possible problem is that the Jacobian with respect to your chosen "$y$" variables, which in the first statement are $y$ and $z$, might not be invertible at the point in question. But it looks to me like it is; I find that it is $\begin{bmatrix} 0 & 2 \pi \\ 1 & 0 \end{bmatrix}.$ | 65,538 |
We knowledge, experience, and memories of the struggle for civil rights in the South. Julian Bond led the Civil Rights South Seminar (Civil Rights Tour) for nearly a decade, and made this learning experience an overwhelming success. We were constantly impressed and honored by his attention to detail on these seminars and the rare glimpse he gave us of a difficult time in history. Julian made certain that each excursion consisted of stops at museums, houses, churches and monuments that drove home the stories of the Civil Rights Movement. Participants relished these close encounters with those that worked both on the frontlines and behind the scenes of the American Civil Rights Movement and the demonstrative walks over the Edmund Pettus Memorial Bridge in Selma, Alabama.
We will miss Julian’s smile, calm demeanor, and, most of all, his passion for history and human rights, as we remember the man who was, as said by U.Va. President Teresa A. Sullivan, “a mentor and role model for all of us.” (U.Va. Today)
In honoring Julian, you may find solace in following the suggestions of his family. Julian’s wife, Pam Horowitz, his sons Horace Mann Bond II, Jeffrey, and Michael, his daughters Phyllis Jane Bond McMillan and Julia Louis Bond, his sister Jane, brother James, and his eight grandchildren will be holding a private, family-only service to commit his ashes to the Gulf of Mexico. This service will take place at sea on Saturday, August 22, 2015 at 2:00 PM Central Daylight Time. The family “invite[s] you to gather at a body of water near your home and precisely at 2:00 PM CDT, spread flower petals on the water and join us in bidding farewell to Horace Julian Bond.” (USA Today)
The University seeks to keep Julian’s legacy alive by establishing a professorship in his name: the Julian Bond Professorship in Civil Rights and Social Justice, a permanent position within the College and Graduate School of Arts & Sciences that will continue Julian’s scholarly legacy. You can learn more about the professorship here. | 69,751 |
Thomas Sabo
Thomas Sabo Bracelet Silver D
A0011-001-12
Thomas Sabo Bracelet Silver. A0011-001-12.
The Thomas Sabo Sterling Silver collection offers an extensive and expression range of jewellery for both men and women in a variety of designs. This Thomas Sabo Silver bracelet is a simple but elegant piece crafted from sterling silver fine links and a spring ring clasp. This delicate bracelet can be worn alone or can be stacked alongside other pieces in the collection that can also be found on our website.
Sterling Silver.
Brand Thomas Sabo
Supplier Model No. A0011-001-12
Collection | 374,048 |
TITLE: Question in Hungerford's book
QUESTION [3 upvotes]: I'm trying to solve this question in Hungerford's Algebra
I didn't use the corollary:
And I used this map: $g:S^{-1}R_1\to S^{-1}R_2$, $g(r/s)=f(r)/f(s)$. I'm wondering how to prove using the corollary, is it hard?
Second, the map I have chosen indeed extends $R$? I identified the elements $r/1$ of $S^{-1}R$ as the elements of $r$, is that right?
Any help is welcome
Thanks a lot
REPLY [4 votes]: You actually don't even need to use an explicit construction of the $F_i=\mathrm{Frac}(R_i)$. If $R_1\hookrightarrow R_2$ is an injection of domains, then, by considering the composite $R_1\hookrightarrow R_2\hookrightarrow F_2$, you get a ring map which, by injectivity, sends non-zero elements of $R_1$ to units (since every non-zero element of $F_2$ is a unit). So the universal property of $R_2\hookrightarrow F_2$ yields a unique ring map $\varphi:F_1\rightarrow F_2$ compatible with the original map $R_1\hookrightarrow R_2$. If the map $R_1\hookrightarrow R_2$ is in fact an isomorphism, then, by similar considerations, you get a unique ring map $\psi:F_2\hookrightarrow F_1$ compatible with $R_2\cong R_1$. Now consider $\psi\circ\varphi:F_1\rightarrow F_1$. It is compatible with $R_1\hookrightarrow F_1$ by construction, so, by the uniqueness part of the universal property of $R_1\hookrightarrow F_1$, it must be the identity. Similarly $\varphi\circ\psi$ is the identity, so $\varphi$ and $\psi$ are mutually inverse isomorphisms.
More generally, if $R$ is a ring, $S$ a multiplicative subset of $R$, and $S^{-1}R$ the localization, with $\alpha:R\rightarrow S^{-1}R$ the canonical map, then the only ring map $\varphi:S^{-1}R\rightarrow S^{-1}R$ such that $\varphi\circ\alpha=\alpha$ is $\varphi=\mathrm{id}_{S^{-1}R}$ (this follows from the uniqueness part of the universal property of $\alpha$). Put succinctly, localizations of $R$ have no non-identity $R$-algebra endomorphisms. This is a particular case of the fact that a localization map is an epimorphism in the category of commutative rings with identity (again this is because of the uniqueness clause in the universal property which characterizes $S^{-1}R$).
Of course, to prove the existence of $S^{-1}R$, one uses an explicit construction, most commonly as a set of equivalence class of fractions. Then, for a ring map $f:R\rightarrow R^\prime$ such that $f(S)$ is contained in the units of $R^\prime$, the unique map $\varphi:S^{-1}R\rightarrow R^\prime$ such that $\varphi\circ\alpha=f$ is given by the formula the OP gives: $\varphi(r/s)=f(r)f(s)^{-1}$. It is well-defined by the definition of the equivalence relation used in the construction of $S^{-1}R$ and the fact that $f(s)\in (R^\prime)^\times$. In the case $\varphi$ is an isomorphism of domains, once the maps between the fields of fractions are written down, it can be checked by hand (using fractions) that it is an isomorphism. So you don't have to use the universal property if you don't want to; the explicit construction as a ring of fractions is sufficient. Note also that, in the notation of the first paragraph, you apply the universal property of localization to the map $R_1\hookrightarrow R_2\hookrightarrow F_2$, not directly to $R_1\hookrightarrow R_2$, since there is no reason that non-zero elements in $R_1$ should be mapped to units in $R_2$; they are mapped to non-zero elements by injectivity, and because $R_2$ injects into $F_2$, the image of a non-zero element in $R_1$ under $R_1\hookrightarrow R_2\hookrightarrow F_2$ is a non-zero element of the field $F_2$, hence invertible, so you can apply the universal property. | 9,977 |
1353 Riverstone Pkwy, Suite 120-342, 30114, Canton
(Georgia) - General, Remodeling - Kitchen & Bathroom, Remodeling - Sunrooms & Patio Enclosures
At Stonecrest Homes & Renovations, we are more than just another “Contractor”. We offer a wide array of services for your every construction need. Michael Walker, President and Owner of Stonecrest Homes & Renovations, is a third generation home builder who learned the business from his father and grandfather. Our remodeling and restoration services provide quality work in Kitchen Remodeling, Bathroom Remodeling, Room Additions, Fire Damage Restoration, Water Damage Restoration, and more.
Stonecrest Homes & Renovations in Canton map
Reviews about Stonecrest Homes & Renovations | 263,539 |
Built To Last. Built To Perform. In Extreme Conditions.
We Cut Lead Times, Not Corners.
In order to create high efficiency solutions for our customers, we know we have to do the same for ourselves. With continuous refining of all equipment and processes, our research, development, and manufacturing capabilities are now second to none. And as both the military and private industry increasingly focus on lean processes, we are able to:
- Cut lead times dramatically
- Create a highly efficient supply chain staffed with experienced managers
- Build highly reliable supplier networks
- Upgrade our manufacturing facilities without impacting costs
We live by continuous improvement, and take that spirit into all your programs and assignments.
Manufacturing Capabilities
Breadth and depth are critical to providing our customers with the solutions they need — sometimes at a moment’s notice. Ward Leonard's facilities are outfitted to expertly manufacture and assemble all our motors and control systems.
Our Extensive Capabilities & Machinery Include:
- 25 to 150 ton press capability, single & progressive dies, with coil feeder
- 50 ton Hydraulic assembly press, 72.0” length
- Amada CNC Turret Press
- Balancing
- Banding, taping & undercutting
- Blanking up to 32" in diameter
- Brazing — torch & induction
- CMM inspection capability for all machined parts
- Cylindrical grinding
- Fabrication — coil & core plate
- Hard & soft coils
- Heat treat & stress relieve to 1500ºF
- Machining
- Manufacture of motor cooling blowers, junction boxes, impellers, shafts, custom enclosures, bearing housings, couplings, specialty bolts, pins, & flanges
- Motor varnishing
- Notching machines
- Overhead crane capability up to 10 tons
- Painting
- Performing shaft repairs, bearing journal repairs, & coupling repairs
- Programmable servo indexing machining of laminations
- Sealed insulation procedure per MIL-STD 2037 — NAVSEA approved
- Sheet metal fabrication & repair
- Surface grinding, milling, drilling, keyways, broaching, grit blast, degrease, & paint
- Vertical & horizontal NC machine centers
- Vertical turret lathe
- VPI system — multiple resin capable
- Welding to MIL-STD- 248, 278
- Standard varnish tanks
You Might Also Be Interested In…
Highlights of
Ward Leonard’s
75 Years of Progress
Download > | 176,876 |
\begin{document}
\baselineskip=17pt
\begin{abstract}
Let $H_0$ be a regular element of an irreducible Lie Algebra $\g$,
and let $\mu_{H_0}$ be the orbital measure supported on $O_{H_0}$.
We show that $\hat{\mu}_{H_0}^k\in L^2(\g)$ if and only if
$k>\dim\g / (\dim\g-\rank\g)$.
\end{abstract}
\maketitle
\section{Introduction}
Let $G$ be a compact, connected, simple Lie group and $\g$ its Lie
algebra. It is well known that the non-trivial adjoint orbits in
$\g$ are compact submanifolds of proper dimension, but geometric
properties ensure that they generate $\g$. Consequently, if
$H_0\neq 0$ is in the torus $\mathfrak{t}$ of $\g$, and $\mu
_{H_0}$ is the orbital measure supported on the orbit $O_{H_0}$
containing $H_0$, ie, $\mu_{H_0}$ is the unique (up to
normalization) $G$-invariant measure on $O_{H_0}$, then some
convolution power
of $\mu _{H_0}$ is absolutely continuous to Lebesgue measure on $\g$ and even belongs to $L^{1+\varepsilon }$ for some $
\varepsilon >0$ (see \cite{RS}). In \cite{Ra}, Ragozin showed that
dimension of $\g$ convolution powers sufficed, and this was
improved in a series of papers culminating in \cite{GHGAFA} with the minimal number of convolution powers being $k_{G}=\rank G$ for the classical simple Lie algebras of type $
B_{n}$, $C_{n}$ and $D_{n}$ and $k_{G}=\rank G+1$ for type
$A_{n}$. There it was also shown that if $\mu _{h}$ was the
orbital measure supported on the conjugacy class in $G$ containing
the non-central element $h$, then $\mu _{h}^{k_{G}}\in L^{2}(G)$.
In the simplest case $G=SU(2)$, $\g=\mathbb{R}^{3}$ and the
adjoint orbits are (the two dimensional) spheres centred at the
origin. The sum of two such spheres contains an open set and
consequently the convolution of any two orbital measures is
absolutely continuous \cite{Ra2}. In general, the generic orbits
(the so-called regular orbits defined below) have codimension
$\rank G$ and two convolution powers of such an orbital measure is
absolutely continuous (in either the group or algebra case).
Furthermore, for the generic orbital measure $\mu _{h}$ on the
group, one can use the Weyl character formula to see that $\mu
_{h}^{k_{2}}\in L^{2}(G)$ for $k_{2}=1+\rank G/(\dim G- \rank G)$
(see \cite{HStudia}) and this fact can be transferred to the Lie
algebra setting as well \cite{GHprivate}.
In this note we give a direct proof that if $\mu_{H_0}$ is any generic orbital measure on $\g$, then $\hat{\mu} _{H_0}^{k}\in L^{2}(\g)$ if and only if $
k>1+\rank \g/(\dim \g-\rank \g)$. The novelty of our approach is
our geometric method, involving the root systems, of handling the
singularities which arise in the integral of the Fourier transform
of the measure.
Products of generic orbital measures are also studied in
\cite{DRW} and \cite {RT}; our approach recovers some of what was
proven in \cite{RT}.
\section{Definitions and Lemmas}
Let $T$ be a maximal torus of $G$ and $t$ be the corresponding
subalgebra of $\g$, also called the torus. Let $\Phi $ be the root
system of $\g$ with Weyl group $\W$ and positive roots $\Phi
^{+}$. Choose a base $\Delta =\{\beta _{1},\ldots ,\beta _{n}\}$
for $\Phi$ and let $\T$ be the associated fundamental Weyl
chamber.
\begin{equation*}
\T=\{H\in t:(H,\beta _{j})>0\text{ for }j=1,\ldots n\}
\end{equation*}
Given $H_0\in t$, the adjoint orbit of $H_0$ is given by
\begin{equation*}
O_{H_0}=\{Ad(g)H_0:g\in G\}\subseteq \g.
\end{equation*}
If $H_0\in \T$, then $H_0$ is called regular and $O_{H_0}$ is a
called a regular orbit.
The regular orbital measure, $\mu _{H_0}$, is the $G$-invariant
measure supported on the regular orbit $O_{H_0},$ normalized so
the Harish-Chandra formula gives
\begin{equation*}
\widehat{\mu }_{H_0}(H)=\frac{A_{H_0}(H)}{\prod_{\alpha \in \Phi ^{+}}(\alpha ,H)
}\text{ for }H\in \T
\end{equation*}
where
\begin{equation*}
A_{H_0}(H)=\sum_{\sigma \in \W}\sgn(\sigma)
e^{i\ip{\sigma(H)}{H_0}}.
\end{equation*}
As $\mu _{H_0}$ is $G$-invariant, the Weyl integration formula implies that $
\hat{\mu} _{H_0}^{k}\in L^{2}(\g)$ if and only if
\begin{equation*}
\int_{\T}\frac{\left| A_{H_0}(H)\right| ^{2k}}{\left|
\prod_{\alpha \in \Phi ^{+}}(\alpha ,H)\right| ^{2k-2}}dH<\infty
\text{.}
\end{equation*}
In this integral some of the inner products $(\alpha ,H)$
represent removable singularities on some walls of the Weyl
chamber. This is the primary obstacle in studying this integral
and we are able to deal with these singularities using geometry
and an induction argument.
Specifically, we will relate the integrand near a collection of
walls to the integrand for a subroot system. The power of our
induction is hidden in the fact that the the integrand is
continuous, and so is bounded on any neighborhood of the origin.
Several technical problems arise; in fact they are necessary
adaptations to the proof of a weaker result (Cor. 1), where the
technical results are not necessary. The case of a Lie algebra of
type $A_2$ is surprisingly representative, and the geometric
motivation for the results presented here come exclusively from
this case.
The notation will get slightly tedious, so we list it all here in
advance. Note that from now on we assume $\Phi$ is irreducible,
but we will consider reducible subroot systems of $\Phi$ that are
``simple"; these are simply those subroot systems for which a
subset of $\Delta$ can be chosen as a base.
\begin{center}
\begin{tabular}{| l | l | }
\hline
$\g$ & An irreducible Lie algebra. \\ \hline
$\Phi$ & The root system of $\g$. \\ \hline
$\Delta=\{\beta_1,\ldots ,\beta_n \}$ & The simple roots of
$\Phi$. \\ \hline
$\Phi^+$ & The positive roots of $\Phi$. \\ \hline
$n$ & The rank of $\g$. \\ \hline
$\W$ & The Weyl group of $\Phi$. \\ \hline
$\T=\{H\in\g : \ip{H}{\beta_i}> 0 \text{ for all } i \}$ &
The fund. Weyl chamber of $\g$. \\ \hline
$\Psi$ & A simple subroot system of $\Phi$. \\ \hline
$\V$ & The Weyl group of $\Psi$. \\ \hline
$\{\gamma_1,\gamma_2,\ldots ,\gamma_m \}\subset \Delta$ & A base
for $\Psi$. \\ \hline
$\Psi^+$ & The positive roots of $\Psi$. \\ \hline
$m$ & Number of simple roots in $\Psi$. \\ \hline
$ \s=\{ H\in \Span \Psi :\ip{H}{\gamma_i}> 0 \text{ for } i>1 \}$ & The fund. Weyl
chamber of $\Psi$. \\ \hline
\end{tabular}
\end{center}
Recall that every $H\in\s$ can be written as a non-negative linear
combination of the simple roots $\gamma_i$. This follows from the
fact that, in the irreducible case, all entries of the inverse of
the Cartan matrix are positive numbers. (See \cite{Hum}, section
13.4, exercise 8.)
We will need to break $\s$ up into the regions
$$R_i=\{H\in \s : \|H\|\geq 1, \ip{\gamma_i}{H}\geq
\ip{\gamma_j}{H} \text{ for all } j\}.$$
So $\s\setminus B_1=\cup_{i=1}^m R_i$. Now if
$$\Psi_1 = \Span_\mathbb{Z} \{\gamma_2,\ldots ,\gamma_m\} \cap
\Psi$$
then the roots of $\Psi_1$ will correspond to removable
singularities on the walls of $\cl(R_1)$ when we calculate the
above integral with root system $\Psi$. Now let $\V_1$ be the Weyl
group of $\Psi_1$, and
$$\C=\{H\in \Span \Psi_1 : \ip{H}{\gamma_i} > 0 \text{ for }
i=2\ldots m \}$$
be the fundamental Weyl chamber of $\Psi_1$. Finally, we define
$$P:\Span \Psi\to \Span \Psi : H \mapsto
\frac{1}{|\V_1|}\sum_{\sigma\in\V_1}\sigma(H).$$
\begin{lemma}[1]
Let $P$ be as above. Then
\begin{enumerate}
\item $\sigma(P(H))=P(H)$ for all $\sigma\in\V_1$.
\item $P$ is the projection from $\Span \Psi$ onto
$(\Span\Psi_1)^\perp$. So $I-P$ is the projection from $\Span
\Psi$ onto $\Span \Psi_1$.
\item $I-P$ in fact maps $\s$ to $\C$.
\item There are constants $a, b>0$ so that $\|P(H)\| \geq a
\|H\|$ and $\|(I-P)H\|\leq b \|P(H)\| $ if $H\in R_1$.
\end{enumerate}
\end{lemma}
Before reading the proof of this result, the reader is encouraged
to graphically verify part (ii) for the case $\Phi=A_2$.
\begin{proof}
(i) If $\sigma_1\in \V_1$ then
$$\sigma_1(P(H))=\frac{1}{|\V_1|}\sum_{\sigma\in\V_1}\sigma_1(\sigma(H))=\frac{1}{|\V_1|}\sum_{\sigma\in
\sigma_1\V_1}\sigma(H)=P(H).$$
(ii) Write $H=s+r$, where $s\in\Span\Psi_1$ and
$r\in(\Span\Psi_1)^\perp$. If $\alpha \in \Psi_1$ then
$$\sigma_\alpha(r)=r-\frac{2\ip{r}{\alpha}}{\ip{\alpha}{\alpha}}\alpha=r.$$
Since $\V_1$ is generated by reflections of the form
$\sigma_\alpha$, $\alpha\in\Psi_1$, it follows that $\sigma(r)=r$
for all $\sigma\in\V_1$. Hence $$P(H)=P(r)+P(s)=r+P(s).$$ If
$\alpha\in \Psi_1$ then by (i) $\sigma_\alpha(P(s))=P(s)$. Since
we also have
$$\sigma_\alpha(P(s))=P(s)-\frac{2\ip{P(s)}{\alpha}}{\ip{\alpha}{\alpha}}\alpha
$$ we get that $P(s)\in(\Span\Psi_1)^\perp$. But
$P(s)\in\Span\Psi_1$ so $P(s)=0$. Putting all of this together, we
get that $P(H)=r$ is the projection of $H$ onto $(\Span
\Psi_1)^\perp$.
Of course it follows that $H-P(H)$ is the projection of $H$ onto
$\Span\Psi_1$.
(iii) If $k>1$ and $H\in\s$ then $$\ip{\gamma_k
}{H-P(H)}=\ip{\gamma_k }{H}> 0$$ since
$P(H)\in\Span\{\gamma_2,\ldots ,\gamma_m\}^\perp$.
(iv) Suppose, in order to obtain a contradiction, that $H\in
\cl(R_1)$ and $P(H)=0$. Then $H\in \Span\Psi_1$ and $H \in\cl(\s)$
so we can write $$H=c_2\gamma_2+\ldots +c_m\gamma_m$$ with all
$c_i\geq 0$. Thus
$$\ip{H}{\gamma_1}=c_2\ip{\gamma_2}{\gamma_1}+\ldots
+c_m\ip{\gamma_m}{\gamma_1}.$$
We also have $\ip{\gamma_i}{\gamma_j}\leq 0$ if $i\neq j$, so in
fact $\ip{H}{\gamma_1}\leq 0$. Since $H\in \s$
$\ip{H}{\gamma_1}\geq 0$. Combining these we get
$\ip{H}{\gamma_1}=0$. From the definition of $R_1$ we get, for
each $i=1,\ldots, m$, that
$$0\leq \ip{H}{\gamma_i}\leq \ip{H}{\gamma_1} = 0$$
which contradicts the fact that $\|H\|\geq 1$. Thus $P(H)\neq 0$
on $\cl(R_1)$. In particular, $P(H)$ is nonzero on the compact set
$\cl(R_1) \cap \{H: \|H\|=1 \}$, so there is an $a>0$ such that
$a\leq \|P(H)\|$ if $\|H\|=1$, $H\in R_1$. Thus we see that
$a\|H\|\leq \|P(H)\|$ on $R_1$. Finally, we can take
$b=\frac1{a}+1$.
\end{proof}
We commented earlier that the roots of $\Psi_1$ will cause
problems in $R_1$ when integrating. As it turns out, all the other
roots of $\Psi$ are very well behaved on $R_1$. (It is quite
helpful to think of the roots of $\Psi_1$ as the ``good" roots on
$R_1$, and the roots of $\Psi\setminus \Psi_1$ as the ``bad"
roots.)
\begin{lemma}[2]
There exists $C>0$ such that for all
$\alpha\in\Psi^+\setminus\Psi_1^+$ and for all $H\in R_1$
$$\ip{H}{\alpha}\geq C\|H\|.$$
\end{lemma}
\begin{proof}
Take $\alpha\in\Psi^+\setminus\Psi_1^+$. Write $\alpha=\sum a_i
\gamma_i$ with all $a_i\geq 0$. Since $\alpha\notin \Psi_1$,
$a_1>0$. Now if $H\in \cl(R_1)$
$$ \ip{H}{\alpha}= \sum a_i\ip{H}{\gamma_i} \geq a_1
\ip{H}{\gamma_1}>0.$$
Thus the function $$f(H)=\ip{H}{\alpha}$$ is non zero on the
compact set $\cl(R_1)\cap\{H:\|H\|=1\}$. Hence it attains a
positive minimum $M_\alpha$. We can take
$C=\min_{\alpha\in\Psi^+\setminus\Psi_1^+}M_\alpha$.
\end{proof}
We will be interested in subroot systems of $\Phi$ of the form
$$\{a_1\alpha_1+a_2\alpha_2+\ldots +a_m\alpha_{m}:
a_i\in\mathbb{Z} \text{ for all } i\}\cap \Phi$$ where
$\{\alpha_1, \alpha_2,\ldots \alpha_{m}\}\subset \Delta$. We will
call these \textbf{simple} subroot systems. Note that $\Psi_1$ is
a simple subroot system of $\Phi$. Simple subroot systems are the
only type of subroot systems that will come up in our induction.
Restricting our attention to simple subroot systems makes the
verification of the following technical lemma easier.
\begin{lemma}[3]
Suppose $\Phi$ is an irreducible root system, with simple subroot
system $\Psi$ with $m$ simple roots, where $m<n$. Then
$$\frac{n}{|\Phi|}<\frac{m}{|\Psi|}$$
\end{lemma}
\begin{proof}
When we look at this result for a particular $m$, it is clearly
sufficient to prove it for the largest $\Psi$ with $m$ simple
roots. We list these subroot systems in Appendix A, along with the
ratios in question. See \cite{Hum} for basic facts needed about
subroot systems.
\end{proof}
It is worth noting that this lemma is not true if we allow $\Phi$
to be reducible. For example, consider a subroot system of Lie
type $B_3$ ($\frac{m}{|\Psi|}=\frac16$) in a root system of Lie
type $B_3\times A_1\times A_1\times A_1$
($\frac{n}{|\Phi|}=\frac14$).
We now set $\e>0$ to be any number with
$\e<\frac{m}{|\Psi|}-\frac{n}{|\Phi|}$ for all proper simple
subroot systems $\Psi$ of $\Phi$. We will need this $\e$ later for
technical reasons.
\section{The main result}
\begin{theorem}
Let $\Phi$ be an irreducible root system. Then
$\hat{\mu}_{H_0}^k\in L^2(\g)$ if and only if
$k>1+\frac{n}{|\Phi|}=\dim \g / (\dim \g - \rank\g)$.
\end{theorem}
\begin{cor}[1] If $\mu$ is a regular orbital measure, then
$\hat{\mu}^{\frac32}\in L^2(\g)$.
\end{cor}
\begin{cor}[2] If $\mu$ is a regular orbital measure then $\mu^2\in
L^p(\g)$ for all $p<\frac{\dim(\g)}{\rank(\g)}$.
\end{cor}
\begin{proof}
Our arguments show that $\hat{\mu}^2\in L^{p'}$ for
$p'<1+\frac{n}{|\Phi|}$. By the Hausdorff-Young inequality,
$\mu^2\in L^p$ for all $p<\frac{\dim(\g)}{\rank(\g)}$.
\end{proof}
It is worth noting that Corollary 2 is sharp when
$\g=\mathfrak{su}(2)$ by a result of Ragozin (see \cite{Ra2}, Prop
A.5).
\begin{proof}
(Of main theorem.) We prove a related result for all simple
subroot systems $\Psi$ of $\Phi$. All the notation will be as
before, including the definition of $\e$.
Our induction hypothesis: For all simple proper subroot systems
$\Psi$ of $\Phi$, if $k<1+\frac{n}{|\Phi|}+\e$ then $$\int_{\s\cap
B_r}
\frac{|A_{H_0}(H)|^{2k}dH}{|\prod{_{\alpha\in\Psi^+}\ip{\alpha}{
H}}|^{2k-2}} = O(r^{n-(k-1)|\Psi|}).$$ By this we mean that this
integral is bounded above, as a function of $r$, by
$Cr^{n-(k-1)|\Psi|}$ for some $C>0$.
For $m=1$, $\Psi$ is of Lie type $A_1$ and we get $$\int_1^r
\frac{|e^{itH_0}-e^{-itH_0}|^{2k}}{|t|^{2k-2}}dt.$$ Hence when
$k<1+\frac12$ the integrand is $O(r^{2-2k})$ and $2-2k>-1$. So if
$k<1+\frac12$ the integral is $O(r^{1-2(k-1)})$. Lemma 3 tells us
that
$1+\frac12=1+\frac{n}{|\Phi|}+(\frac12-\frac{n}{|\Phi|})>1+\frac{n}{|\Phi|}
+\e$.
Now we assume the result for all simple subroot systems of rank
$m-1$.
Consider a subroot system $\Psi$ of rank $m$. We will describe the
growth of the integral on $R_1$. Since we have not specified any
particular order among the $R_i$, and the integrand is continuous,
this is sufficient.
Let $\sigma_1, \ldots , \sigma_t$ be representatives from the left
cosets of $\V_1\leq \V$. We break up $|A_{H_0}|$ by cosets of
$\Psi_1$.
\begin{eqnarray*}
&&\int_{R_1\cap B_r} \frac{|
\sum_{j=1}^t\sum_{\sigma\in\V_1}\sgn(\sigma_j\sigma)
e^{i\ip{\sigma_j\sigma(H)}{H_0}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+}\ip{\alpha}{
H}}|^{2k-2}} \\ &\leq& 2^{2k}\sum_{j=1}^t \int_{R_1\cap B_r}
\frac{| \sum_{\sigma\in\V_1}\sgn(\sigma_j\sigma)
e^{i\ip{\sigma_j\sigma(H)}{H_0}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+}\ip{\alpha}{
H}}|^{2k-2}}\\
\end{eqnarray*}
For convenience we forget about the constant, and just write the
term of the $\sigma_j$ coset. We start by factoring out
$\left|\sgn(\sigma_j) e^{i\ip{\sigma_j(P(H))}{H_0}} \right|=1$ to
get $$ \int_{R_1\cap B_r} \frac{| \sum_{\sigma\in\V_1}\sgn(\sigma)
e^{i\ip{\sigma_j\sigma(H)}{H_0}-i\ip{\sigma_j(P(H))}{H_0}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+}\ip{\alpha}{
H}}|^{2k-2}}. $$
Since $P(H)=\sigma(P(H))$ (for $\sigma\in\V_1$) and
$\ip{\sigma(v)}{w}=\ip{v}{\sigma(w)}$ this integral equals $$
\int_{R_1\cap B_r} \frac{| \sum_{\sigma\in\V_1}\sgn(\sigma)
e^{i\ip{\sigma(H-P(H))}{\sigma_j(H_0)}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+\setminus\Psi^+_1}\ip{\alpha}{
H}}|^{2k-2}|\prod{_{\alpha\in\Psi^+_1}\ip{\alpha}{ H}}|^{2k-2}}.
$$
Now we apply Lemma 2 to get the upper bound
$$ \int_{R_1\cap B_r}
\frac{C}{\|H\|^{(k-1)(|\Psi|-|\Psi_1|)}}\frac{|
\sum_{\sigma\in\V_1}\sgn(\sigma)
e^{i\ip{\sigma(H-P(H))}{\sigma_j(H_0)}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+_1}\ip{\alpha}{
H}}|^{2k-2}}.$$
At this point we can safely replace $\sigma_j(H_0)$ with
$H_0'=(I-P)\sigma_j(H_0)$. Since $P(H)$ is orthogonal to
$\Psi_1$, we can change the inner products from $\ip{\alpha}{H}$
to $\ip{\alpha}{H-P(H)}$. If we also recall the bound
$\|P(H)\|\geq a\|H\|$ for all $H\in R_1$ from Lemma 1, this gives
$$ \int_{R_1\cap B_r}
\frac{C}{\|H\|^{(k-1)(|\Psi|-|\Psi_1|)}}\frac{|
\sum_{\sigma\in\V_1}\sgn(\sigma)
e^{i\ip{\sigma(H-P(H))}{H_0'}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+_1}\ip{\alpha}{
H-P(H)}}|^{2k-2}}$$
$$ \leq \int_{R_1\cap B_r}
\frac{C'}{\|P(H)\|^{(k-1)(|\Psi|-|\Psi_1|)}}\frac{|
\sum_{\sigma\in\V_1}\sgn(\sigma)
e^{i\ip{\sigma(H-P(H))}{H_0'}}|^{2k}dH}{|\prod{_{\alpha\in\Psi^+_1}\ip{\alpha}{
H-P(H)}}|^{2k-2}}.$$
$P$ maps onto a one dimensional subspace, say $\Span v_1$,
$\|v_1\|=1$. We can do a change of variables so that we are
integrating first with respect to $H'=H-P(H)\in \C$ and then $s$,
where $P(H)=sv_1$. If $a$ and $b$ are as in Lemma 1 then $s\geq a$
and $\|(I-P)H\|\leq b\|P(H)\|$ for $H\in R_1$ . Note that
$H\mapsto (P(H), (I-P)H)$ is an orthogonal change of variables so
the Jacobian is a constant.
If we now use Fubini's Theorem to rewrite our integral (and forget
the constant) we get
\begin{eqnarray*}
&&
\int_{a}^r \frac{1}{s^{(k-1)(|\Psi|-|\Psi_1|)}}\int_{\C\cap B_{b s}} \frac{| \sum_{\sigma\in\V_1}\sgn(\sigma)
e^{i\ip{\sigma(H')}{H'_0}}|^{2k}dH'}{|\prod{_{\alpha\in\Psi^+_1}\ip{\alpha}{
H'}}|^{2k-2}}ds.
\\
\end{eqnarray*}
Note that no element of $\Psi$
annihilates $\sigma_j(H_0)$, so it is regular. It follows that no
element of $\Phi_1$ annihilates $H_0'$. Thus we can apply the
induction hypothesis. Since $m<n$,
$$1+\frac{m}{|\Psi|}=1+\frac{n}{|\Phi|}+(\frac{m}{|\Psi|}-\frac{n}{|\Phi|})>1+\frac{n}{\Phi}+\e
.$$ So if $k<1+\frac{n}{|\Phi|}+\e$ we have that the above
integral is at most
\begin{eqnarray*}
&&\int_{a}^r
\frac{1}{s^{(k-1)(|\Psi|-|\Psi_1|)}}O(s^{m-1-|\Psi_1|(k-1)})dr
\\
&=&O(r^{m-|\Psi|(k-1)}).
\\
\end{eqnarray*}
At some point in our induction we get that $n=m$ and $\Psi=\Phi$.
At this point our full induction hypothesis does not hold, but we
have that actual integral we are interested in is at most
$$\int_{\delta}^r O(r^{n-1-|\Phi|(k-1)})dr$$ if
$k<1+\frac{n}{|\Phi|}+\e$. This integral converges if
$$1+\frac{n}{|\Phi|}+\e>k>1+\frac{n}{|\Phi|}.$$
Hence $\hat{\mu}_{H_0}\in L^2(\g)$ if $k>1+\frac{n}{|\Phi|}$.
Now we show the necessity of the condition $k>1+\frac{n}{|\Phi|}$.
We can rewrite $$ \frac{|\sum_{\sigma\in\W}\sgn(\sigma)
e^{i\ip{\sigma(H)}{H_0}}|^{2k}}{|\prod{_{\alpha\in\Phi^+}\ip{\alpha}{
H}}|^{2k-2}}$$ as $$\frac1{\|H\|^{|\Phi|(k-1)}}
\frac{|\sum_{\sigma\in\W}\sgn(\sigma)
e^{i\|H\|\ip{\sigma(\frac{H}{\|H\|})}{H_0}}|^{2k}}
{\left|\prod{_{\alpha\in\Phi^+}\ip{\alpha}{\frac{H}{\|H\|}}}\right|^{2k-2}}$$
and consider this as $r^{-|\Phi|(k-1)} f(r, \phi_1, \ldots ,
\phi_{n-1})$, where $f$ is a function in polar coordinates.
$$f(r,\phi_1,\ldots ,
\phi_{n-1})=\frac{|\sum_{\sigma\in\W}\sgn(\sigma)
e^{ir\ip{(1,\phi_1,\ldots ,\phi_{n-1})}{H_0}}|^{2k}}
{|\prod{_{\alpha\in\Phi^+}\ip{\alpha}{(1,\phi_1,\ldots
,\phi_{n-1})}}|^{2k-2}}$$ As before, we will integrate in $\T$
with a ball around the origin removed, so we will always assume
$r\geq 1$.
If we fix $\Phi=(\phi_1,\ldots ,\phi_{n-1})$, we see that
$f_\Phi(r):=f(r,\phi_1,\ldots ,\phi_{n-1})$ is (the absolute
value of) the sum of continuous functions that are periodic in
$r$. Thus $f$ is almost periodic in $r$.
Since $\|\mu_{H_0}^k\|\neq 0$ and $f$ is continuous, we can find a
point $(r_0,\psi_1,\ldots ,\psi_{n-1})$, a $\delta>0$ and a
$\epsilon>0$ so that if
$$U=\{(r, \phi_1,\ldots ,\phi_{n-1}):
|\|\phi_i-\psi_i\|\leq\delta\forall i, |r-r_0|\leq\delta \}\subset
\T$$
then $f>2\epsilon>0$ on $U$.
We will have to change to polar coordinates to use this
observation. The Jacobian of this change of variables is
$$\Delta = r^{n-1}\sin^{n-2}\phi_1 \ldots \sin^{n-2}\phi_{n-1}.$$
If necessary, we can modify $U$ so that $|\Delta|\geq Cr^{n-1}$ on
$U$, for some constant $C$. We then get that our integral greater
or equal to
$$\int_{\psi_1-\delta}^{\psi_1+\delta}\ldots
\int_{\psi_{n-1}-\delta}^{\psi_{n-1}+\delta}\int_{1}^\infty
C\frac1{r^{|\Phi|(k-1)}}f(r, \phi_1,\ldots ,\phi_{n-1}) r^{(n-1)}
dr d\phi_{n-1}\ldots d\phi_1. $$
Say that $f_\Phi$ has an $\epsilon$ almost period in every
interval of size $M$. Pick $N\geq M+2\delta$. We know that
$f_\Phi(r)\geq 2\epsilon$ on $[r_0-\delta, r_0+\delta]$. Pick
$\tau_n$, an $\epsilon$ almost period of $f_\Phi$ in the interval
$[nN-r_0+\delta, (n+1)N-r_0-\delta]$, where $n>0$. Hence
$f_\Phi\geq\epsilon$ on $[r_0+\tau_n-\delta,
r_0+\tau_n+\delta]\subset [nN, (n+1)N]$. If $\chi_F$ is the
indicator function of $F=\bigcup_n [r_0+\tau_n-\delta,
r_0+\tau_n+\delta]$, then the inner integral is at least
$$\int_{1}^\infty C\epsilon \chi_F
r^{-(k-1)|\Phi|+n-1}dr\geq\int_{1}^\infty C\epsilon \chi_E
r^{-(k-1)|\Phi|+n-1}dr$$
where $E=\bigcup_n [nN, nN+2\delta]$. This integral is at least
$$\sum_{n=1}^\infty 2\delta C\epsilon (nN)^{-(k-1)|\Phi|+n-1}.$$
If $k\leq 1+\frac{n}{|\Phi|}$, this diverges. Thus the inner
integral is infinite for all $\phi_1,\ldots ,\phi_{n-1}$ in the
appropriate range. So if $k\leq 1+\frac{n}{|\Phi|}$, our integral
is infinite and $\mu^k\notin L^2(\g)$.
\end{proof}
\begin{remark}[1]
A similar result holds when $\Phi=\Phi_1\times\ldots \times\Phi_m$
is reducible. Say that the number of simple roots in $\Phi_i$ is
$r_i$, and the fundamental Weyl chamber of $\Phi_i$ is $\T_i$. In
this case the integrand splits to give
$$\int_{\T_1}\int_{\T_2}\ldots
\int_{\T_m}\frac{|A_{H_0}(t_1+t_2+\ldots
+t_m)|^{2k}}{|\prod{_{\alpha\in\Phi^+}\ip{\alpha}{ t_1+t_2+\ldots
+t_m}}|^{2k-2}}dt_m\ldots dt_1.$$
This factors as
\begin{eqnarray*}
\int_{\T_1}\frac{A_{H_0}^{\Phi_1}(t_1)}{|\prod_{\alpha\in\Phi_1^+}\ip{\alpha}{t_1}}dt_1
\int_{\T_2}\frac{A_{H_0}^{\Phi_2}(t_2)}{|\prod_{\alpha\in\Phi_3^+}\ip{\alpha}{t_2}}dt_2
\ldots
\int_{\T_m}\frac{A_{H_0}^{\Phi_m}(t_m)}{|\prod_{\alpha\in\Phi_m^+}\ip{\alpha}{t_m}}dt_m.\\
\end{eqnarray*}
Since none of these factors can be zero, this is finite iff all
the integrals converge. Hence $\mu_{H_0}\in L^2(\g)$ iff $$k>\max
\{1+\frac{r_1}{|\Phi_1|}, 1+\frac{r_2}{|\Phi_2|},\ldots ,
1+\frac{r_m}{|\Phi_m|} \}.$$
\end{remark}
\begin{remark}[2] A measure $\mu$ is called $L^p$-improving if there is some
$p<2$ such that the operator $T_\mu: f \mapsto \mu*f$ is bounded
from $L^p(\g)$ to $L^2(\g)$. Using sophisticated arguments Ricci
and Travaglini \cite{RT} prove that for a regular, orbital measure
$\mu$, $T_\mu$ maps $L^p(\g)$ to $L^2(\g)$ if and only if $p\geq
1+\rank(\g)/(2\dim(\g)-\rank(\g))=p(\g)$. The same reasoning as
given in \cite{HStudia} Corollary 12 shows that our arguments give
the weaker result: $T_\mu$ is bounded from $L^p(\g)$ to $L^2(\g)$
for any $p>p(\g)$.
\end{remark}
\section*{Appendix A}
\begin{center}
\begin{tabular}{| c | c | c | c | }
\hline
$\Phi$ & $\frac{n}{|\Phi|}$ & $\Psi$ & $\frac{m}{|\Psi|}$ \\
\hline\hline
$A_n$ & $\frac1{n+1}$ & $A_{m}, m<n$ & $\frac1{m+1}$ \\ \hline
$B_n$ & $\frac1{2n}$ & $B_{m}, m<n$ & $\frac1{2m}$ \\ \hline
$C_n$ & $\frac1{2n}$ & $C_{m}, m<n$ & $\frac1{2m}$ \\ \hline
$D_n$ & $\frac1{2(n-1)}$ & $D_{m}, m<n$ & $\frac1{2(m-1)}$ \\ \hline
$E_6$ & $\frac1{12}$ & $D_{m},m<6$ & $\frac1{2(m-1)}$ \\ \hline
$E_7$ & $\frac1{18}$ & $D_{m},m<7$ & $\frac1{2(m-1)}$ \\ \hline
$E_7$ & $\frac1{18}$ & $E_6$ & $\frac1{12}$ \\ \hline
$E_8$ & $\frac1{30}$ & $D_{m},m<8$ & $\frac1{2(m-1)}$ \\ \hline
$E_8$ & $\frac1{30}$ & $E_6$ & $\frac1{12}$ \\ \hline
$E_8$ & $\frac1{30}$ & $E_7$ & $\frac1{18}$ \\ \hline
$F_4$ & $\frac1{12}$ & $B_{m},m<4$ & $\frac1{2m}$ \\ \hline
$G_2$ & $\frac1{6}$ & $A_1$ & $\frac1{2}$ \\ \hline
\end{tabular}
\end{center} | 105,435 |
Dating books for christian girls, bog girl on dating apps, pansexual dating sites usa, christian teenage dating guidelines for parents
Scripts for adult themed dating sites, free online dating sites country singles, international dating site find love in usa, how many dating site makes is free for women, backpage bbw dating huge
Enjoy what are some free dating sites that work sporting events, movies, dining out. Fyi- there was only one time in the history of our country that we were in balance and had no deficit and that was many years ago sometime shortly after we gained independence as a country? But, mostly, the result of felling the great tree was to sever the link between heaven and earth? I can honestly say that this has erotic women seeking men chattanooga tn been my favorite addition to our 4th grade homeschool curriculum and we will definitely be using it in future years. Monday was the first day on the job for the jacksonville downtown investment authority's new ceo aundra wallace. As you asian dating service northern blvd flushing ny can see from the screenshot below the backup was successful. In fact, to be exact, there are 105 words that are most widely used and, therefore, important for most dating site to send and receive messages in usa students to know. The ancient egyptian goddess, neith, was thought to be in control adult dating sim mobile of weaving together the threads of human destiny. My boyfriend said he loves me but is not ready for a relationship and broke it off with me. I've been dancing here for about 5months and i've been improving so much. It really just depends on the individual and how motivated and committed they are.
Despite the expiry of bagels, the slow pace and infrequency how many dating sites are there of actually connecting with potential matches make it all too easy to be super-passive in the app, which can render it useless. I clearly do not believe gina dating saratoga springs ny that you are a 14 year old girl, just based on the lack of modern verbiage. I asked my friends, aka three twenty-something women, to share multiple screenshots of them using their go-to lines on different matches, and the results are truly nothing short of incredible! Hey, if it looks stupid but it works, it's not stupid. Locke admitted hip-hop was not his first choice in music. Due to its convenient location, miami airport is close to a large number of hotels where travelers can stay and dating sites steven milam rest at after their flights. Matches 1 - 49 after backpage tucson escorts bedpage is the tucson, w4m, dating in dates, enjoy either w4m or women seeking men. Hope all that criticizing brings you so very much peace and happiness? Careful not to let all of your straw blow away new york dating show lauren urasek in the wind, and be sure to use the doggy door when you come back. Every once in a while we all dating a convict as a christian need a little motivation to light a fire under our rear ends. Tiny haines, alaska, is ninety miles north of juneau, accessible philippines online dating chat mainly by water or airand only when the weather is good. Building a lasting peace not so much? The dotted most dating site to send and receive messages in usa eighth sound is more suited to a digital delay but works just as well with an analog pedal. I was informed that having uncontrolled diabetes and hypertension are not qualified illness unless i developed a chronic kidney disease or had stroke.
For violations involving accidents, the fines are doubled. You can take comfort in knowing that your benefits will help take care of your family after your death. One would definitely have gotten us had we not read about it. It looks like you are not supporting this creator yet. I am well aware that there are major financial considerations involved in doing something like this, for teams, players and broadcast partners. But i called the number that other how to get to know a girl on a dating site guy gave u on this on pages it works i got it. Then came they, and laid hands cody f hutchison xxx adult dating on jesus, and took him. Far from being a coward, ive bravely taken a stand as best i could. But then he saw how much money he could make and turned reselling dating spots san diego into a bigger business. We highly advise you to have christian dating advice blogs dumpster to secure your pictures, videos and other data from being lost irretrievably. Develop research speed dating in chicago december 8th capacity in industrial engineering. To know more desi speed dating la about it, you first have to understand the types of steam wallet codes available. So my friend, when you know how to use rewards and punishments in the right way, it will seem as though reaching your goals is an inevitable outcome. I'm looking for a person that knows what theywants out of life and is working on acknowledging goals. This is a less common option, but it can come with a number of benefits, if you can afford the handset.
Dating sites usa visa card, white seeking black dating sites, totally free us dating sites like skout, dating single america women in usa
Encyclopaedias come in many volumes, im 35 dating a 60 year old woman sometimes each volume is dedicated to a certain subject matter. I dare say anyone who argues against liberty women seeking men in modesto ca backpage and freedom is pro authoritarian. Instead of focusing on yourself, you go into overwhelm by trying to online dating sites favored by high income midle aged involve yourself in too many interests and activities.
- Dating over 50 and match advice
- Adult dating chat website
- Dating in canada vs usa
- Online dating waste of time
- Dating a wholesome girl
- Free online chatting dating websites
- Drag queen dating services south florida
- What dating sites actuallywork
Paul vi, apostolic exhortation marialis cultus, 6. It is a sin the way this man was treating his wife, but what is the best over 50 dating site it is equally a sin for her to sexually refuse him because of his sin. We are at odds with this longstanding developed world of biological reactions which suited humans in their past development. The signee free dating site in california is acknowledging that they will not sue if any damage occurs! From foundation stabilization, to repair of structural elements, including strengthening or protection after the repair, cpr should be your first call. With arthur you talked about the first years of middle age in england and with uhtred you continue the same history after the creation of the saxon kingdoms. However, please check back regularly for any updates that may occur! Please do not use this word when you are talking about anything that is true, it has been used incorrectly too many times and alarmists see it as on their side even when it is not used that way. Smoking speeds up your heart rate and increases your blood pressure? With fourteen second left on the clock, magic hit a three pointer, but the game was already well over by that point.
Craigslist women seeking men small dicks, best dating apps costa rica, pensacola women seeking men, woonsocket online dating sites
Anime is the animation of a cartoon show whereas manga is the comic book also called a graphic novel. We don the big brands, the small brands, and everything in between! I want to do whats good for me but not if it means another immature, abusive relationship. I have children, and sometimes they live at home i'm not looking for a how to get responses online dating one night stand or a fling or forum like that. Experienced the same forbidden impulse the joke expresses. Saratoga place - saratoga place, nestled among the tree-lined streets of prestigious palmer ranch yet close to the heart of sarasota, black girl interracial dating florida's downtown and bayfront areas, will bring the perfect balance of energy and serenity to your lifestyle. But women dating women after 50 it's completely up to me if i wish to share. Unlike the other sites featured in the list torrentz does not host any torrent files, it merely how to calculate gross profit online dating redirects visitors to other places on the web. Morgan said no and emphasized that many cubans were committed to democracy. Please shoot me free dating sites kink an email if interested. This essentially means the entire radiocarbon dating 50 years ago network is identified as spam? However, one study found the visuomotor performance on driving simulation to be significantly impaired when kava was consumed with alcohol. Cypress springs high school is located in cypress, texas and educates 9th grade through 12th grade students. After warning her that they'd kill her and her baby if she told anyone, most dating site to send and receive messages in usa the threesome took off. By encouraging individuals to develop to their fullest potential, we will promote growth dating while divorce pending texas and success in life for the youth of the penn hills community.
Christian online dating tips, sophomore boy dating senior girl, lastest free dating sites, best dating site for over 50 canada, muslim girl dating website
Take it as an opportunity to work on yourself and if the lines of communication do open up again, youll be in top 10 free international dating sites a much better spot. I'm constantly approached by young women,' she says? Are christian dating advice for adults you working on anymore sharpe novels! I like many free mormon dating sites of the authors thoughts. Thanks for coming to campbell, we look forward to serving you. Some interesting metaphors on how understanding surfing and how good surfers handle waves and unpredictable ocean conditions, can help you navigate free online dating sites that allow free communication the dating world, improve your dating skills and become a master dater and seducer. Share your story and become part of ours. Now i would like to have nsa dating san antonio a personal page. You ve selected a website builder and your website is posted and ready for the public new york dating sites married to view. Really quiet location thats tucked out of the way. Bluetooth is an open wireless protocol for transferring data between devices!
- Indian dating sites chicago
- What its like dating a girl your height
- How to pick a username for online dating
- Dating girl for months then she stopped texting
- Dating sites for only christian in usa
- Why do so many people look unhappy on dating sites
San diego speed dating yelp, dating a dragon monster girl greentext, dating over 50 in atlanta, dallas texas dating app, best dating sites for over 60 in canada
The car in front ours paid for our meal. Flhealthcharts free online dating sites scotland is your one-stop-site for florida public health statistics and community health data. Returns the multinomial of a set of numbers! Off to a late start, but now online dating sites port st. lucie ive got hardened experience at my back. The couple decides to spend their family vacation with their son and daughter at their beach house, only to be terrorized by a group of doppelgangers that mirror the dating a former bad girl wilson family. But it can be an exception which we get from interpreter or compiler that the class, module or function which we test, is not defined. A famous pastime for college going students, just taking a walk down this road will give you an idea of the pulse of the city and its outgoing culture? A special activation file will be downloaded to your computer. Dithering serves no one when one partner wants relative dating địa chất to split up. We crave having someone by our side who will love us through our moments of imperfection, and share the memories of our lives with us? And when you take a closer look at the reasons why men cheat and dating a christian person the benefits they gain through infidelity, you'll quickly see that i'm right. There is little to see there and women travelers may feel uncomfortable. But few catfishes are women seeking men casual encounter ever quite as entertaining as the absolutely adorable, unfiltered, and ridiculously funny johnny. | 268,737 |
By Sabrina Otero Times-News correspondent [email protected] “They say teachers learn more than their students, and that is completely true,” asserts Caroline Countryman, a 17-year-old dance instructor for Arts Alive in downtown Burlington. Nowadays, while most teenagers are struggling trying to find a job, Caroline is benefiting from skills that’s she been honing since early childhood. [...]
About TeensAndTwenties.comTeensAndTwenties.com is a news site for, about, and produced by Alamance County youth. | 329,669 |
Bulletin of Experimental Treatments for AIDS, April, 1998
Mark Bowers
DNA-based vaccines have recently proven fit for human testing. How do they compare with other vaccine technologies that have proven successful in the past? What promise do they hold in the search for an effective vaccine against HIV?
DNA-based vaccines are the newest technology that may hold the key to controlling the spread of HIV and other infectious diseases. All of an organism's genetic information is stored in DNA. Because of technological advances, it has become possible to find and clone stretches of DNA that contain the instructions for producing specific molecules useful in immunizations. Short stretches of viral DNA are selected because they do not cause disease and because they represent the whole virus to the immune system. These DNA instructions are then inserted into plasmids, small circular constructs of bacterial DNA. These plasmids can enter human cells and immediately begin to produce the target molecules. They are introduced into the person receiving vaccination by either attaching them to gold beads and propelling them through the skin using a "gene gun" or by mixing them in saline and injecting them into muscle through a hypodermic needle. It is human cells that actually make the viral protein and mobilize an immune response.
These molecules stimulate the immune system to produce antibodies and cellular (T-cell) responses that may prevent or control infectious diseases such as HIV disease. DNA technology may provide the answer to the urgent demand for an HIV vaccine that is safe, inexpensive and easy to produce. In animals, DNA vaccines have already achieved protective immunity for diarrhea-causing viruses, bacteria that cause tuberculosis and parasites that cause malaria. Will the new technology be equal to the challenge of HIV?
There are now 5 technologies for making vaccines: live attenuated (weakened) vaccines, killed whole viruses, purified component vaccines (not currently used to make HIV vaccine candidates), genetically engineered vaccines and DNA vaccines. Each technique has produced a small number of successful vaccines, but there are many human diseases, including HIV disease, for which no vaccine exists. Following are brief descriptions of current efforts to harness each method to produce an effective HIV vaccine.
Live attenuated vaccine technology was the first route to successful vaccination, pioneered by Edward Jenner in the 18th century. This technology led to the eventual successful eradication of smallpox and containment of polio in the developed world. The possibility of testing a live attenuated HIV vaccine candidate garnered media attention last summer when the International Association of Physicians in AIDS Care (IAPAC) announced that they had recruited 50 potential volunteers for a vaccine study from among the physician and activist communities. The wisdom of pursuing such a course has been hotly debated, despite the obvious obstacle that no live attenuated HIV vaccine candidate yet exists.
Several companies are seriously considering developing live attenuated HIV as a vaccine candidate because infection with an attenuated strain of HIV is believed to be the reason that some HIV-infected people are long-term non-progressors. Research is currently underway at the Macfarlane Burnet Centre in Australia to develop a vaccine candidate based on a mutant strain of HIV that infected 9 Australians as long as 17 years ago. Six of the original 9 are long-term non-progressors, 2 died in their eighties, presumably of old age, and 1 elderly woman with severe systemic lupus erythematosus (an immune system disorder unrelated to HIV) died of Pneumocystis carinii pneumonia (PCP).
Debate about the underlying cause of this woman's death hinges on the fact that lupus is characterized by general immune activation, not restricted to the loss of T-cells. She had been on prolonged therapy with prednisone, an immunosuppressive drug sometimes used to combat the immune activation and autoimmunity characteristic of systemic lupus. Blood samples from the period prior to her death were not retained for later evaluation, so it is a matter of speculation and debate whether HIV played a role in her development of PCP infection and in her subsequent death, or if prednisone and lupus completely account for her death.
The strain of virus that was transmitted to these 9 individuals lacks a large segment of HIV's nef gene, a lack which apparently limits the damage the virus can inflict on the immune system. A vaccine candidate based on this deletion mutant strain could be available for human testing in 18-24 months, according to John Mills, MD, the Centre's director. Whether this crippled HIV can revert to virulence and cause progressive HIV disease is the wider question raised by the death of the woman described above.
David Baltimore, MD, who heads the AIDS Vaccine Research Committee of the National Institutes of Health, expressed concern about the attenuated virus approach. He cited unpublished results of 2 vaccine studies in monkeys in which some vaccinated animals went on to develop disease, given enough time. For Baltimore, live attenuated viruses have not yet been proven safe enough to consider proceeding to human trials.
Therion Biologics of Massachusetts is currently outlining a plan to develop a live attenuated HIV vaccine candidate that the company will present to the Food and Drug Administration (FDA) later this year. The template for this vaccine candidate was supplied by Ronald Desrosiers, MD, a primate researcher at Harvard Medical School. Desrosiers' live attenuated simian immunodeficiency virus (SIV) vaccine has protected monkeys from SIV infection for more than 7 years. Biostratum of Research Triangle, North Carolina is developing a live attenuated vaccine candidate in which HIV genes other than nef have been deleted. Key questions of safety and liability remain to be worked out before human testing can be contemplated in the U.S., despite the existence of an apparently willing cohort of potential volunteers for a trial of a live attenuated vaccine.
Vaccines made from whole "killed" viruses have been useful in preventing disease since the 19th century. The Salk polio vaccine was "killed" (technically speaking, inactivated) chemically with formaldehyde. Burt Dorman at Acrogen, a small vaccine development company in Oakland, California, is currently updating the technology used to inactivate viruses before seeking permission to test a whole killed HIV vaccine candidate in humans. The strongest argument for a whole killed virus approach is that other strategies do not include the preserved outer coat of HIV (the envelope). The immune system recognizes HIV's outer coat most frequently when it encounters intact, infectious virus, the population of virus that rapidly spreads from cell to cell. Sub-unit vaccines made from parts of the envelope may be insufficient to provoke a broad-based immune response. The immune system may need to see the whole virus in order to respond effectively.
Genetically engineered vaccines have only recently been developed, and none are yet approved for human use. Several HIV vaccine candidates have been developed using recombinant DNA technology. Examples include the gp120 and gp160 sub-unit vaccines that were made of parts of the outer coat of HIV. In 1994, FDA denied the California biotechnology companies that made these candidate vaccines, Chiron and Genentech, permission to progress to Phase III clinical testing. Early clinical study results did not seem promising enough to counterbalance the risks of wide-scale public exposure to the vaccines. Based on strains of HIV that grow well under laboratory conditions, these candidate vaccines failed the crucial test of neutralizing HIV taken from the blood of infected individuals. HIV taken from an infected individual is called a primary isolate. Laboratory strains of HIV have adapted to laboratory cultures and differ from primary isolates in significant ways.
Work on the Genentech candidate vaccine was then moved to a separate but affiliated company, VaxGen, and Donald Francis, MD, continued to seek funding for further testing and for development of a hybrid vaccine candidate called a bivalent vaccine. By early 1998, $20 million was secured from private sources and a 3-year Phase III clinical trial involving 7,500 volunteers is scheduled to begin this year in Thailand and the U.S. Both studies await formal approval from FDA and Thai authorities. In the U.S., confirmatory Phase I and II studies of the bivalent candidate vaccine (AIDSVAX) have already received FDA approval.
A recent series of wire stories and newspaper articles erroneously suggested that FDA had reversed its 1994 decision and was ready to fund wide-scale testing of the VaxGen vaccine candidate. Individual vaccine candidates that VaxGen expects to test soon in the U.S. and Thailand differ from the original candidate and from one another in that a component of each has been created from primary isolates common among infected people in each country. This combined strategy addresses earlier problems with the vaccine, and suggests that human immune responses to these vaccines will be broader and better focused on the different strains of HIV that currently infect people in the 2 target populations.
Immune Response Corporation is one of 2 companies that have developed and tested a treatment vaccine specifically for individuals who are HIV infected. Their vaccine candidate is a sub-unit vaccine made from proteins that are found in the inner core of HIV. Currently in Phase III testing, Remune is an immune-based therapy based on whole killed HIV stripped of envelope proteins. The current study is expected to determine whether Remune can delay HIV disease progression by beefing up immune responses in HIV positive individuals with CD4 cell counts between 300 and 549 cells/mm3. Researchers hope to demonstrate that Remune brings forth strong HIV-specific immune responses and beta chemokine production.
AIDS ReSEARCH Alliance in Los Angeles reported earlier this year that a recent clinical trial of Cel-Sci Corporation's therapeutic vaccine candidate, HGP-30, a sub-unit vaccine based on a different core protein of HIV, has shown mostly disappointing results.
Much interest has been generated by a new, non-traditional approach to vectors -- bacteria, other viruses or plasmids that are used to introduce DNA or recombinant vaccines into individuals (a recombinant is a protein made by genetic engineering). Canarypox virus is the vector for several candidate vaccines now in human testing. The advantage of combining a live virus with a genetically modified protein lies in the ability to provide immunity against viruses that cannot be reliably attenuated or inactivated without distorting them so grossly that they no longer resemble live virus.
DNA vaccines have generated excitement since 1993, when the first reports of immune responses to naked DNA vaccines in mice were reported. The ability of one DNA vaccine to prevent malaria in animals boosted research efforts and led to safety trials in human volunteers. Also in 1993, a DNA vaccine developed at Merck Research Laboratories protected mice against lethal doses of influenza A.
Theoretically, these vaccines are successful where others have failed because they awaken a broad immune response, including not only antibodies but also cytotoxic (killer) T-cells that can then seek out and destroy cells that are already infected. In this respect, DNA vaccines operate like live attenuated vaccines, but without the risk. However, the exact mechanism of DNA vaccines is not known with certainty, and theories -- although based on the best knowledge now available -- remain unproven.
DNA vaccines are easier to make and cheaper than other vaccine technologies. A candidate vaccine for malaria, containing a single gene from the organism that causes the disease, took only 3 months to prepare. The important decision is to select the appropriate gene to make a candidate vaccine. Such decisions are frequently made empirically.
Research has shown that the cells that pick up naked DNA genes are usually dendritic cells, which are cells of the immune system that systematically patrol the body on the lookout for foreign antigens. Dendritic cells are antigen-presenting cells that trap foreign substances on their mop-like surfaces and carry them to lymph nodes for further identification and processing.
Research on a DNA vaccine candidate for HIV is being conducted at the University of Pennsylvania. Widely publicized research completed last year involved attenuated HIV genes inserted into a plasmid, mixed with the local anesthetic bupivicaine and injected into 3 chimpanzees. Two of these chimps were then "challenged" (exposed to massive doses of HIV) but both remained free of infection. The third chimp was kept as a negative control and received no challenge. A fourth control chimp who did not receive any vaccination was challenged and became infected. Sensitive laboratory assays were able to detect HIV in the blood of each of the protected chimps, but only briefly. The implication was that the chimps' immune systems were ultimately able to clear the infection. Neither chimp mounted an antibody response. The importance of killer T-cell responses and the relative unimportance of antibodies were strongly suggested by this study. The study left some questions entirely unanswered. Eight booster shots were given to each vaccinated chimp. Are this many boosters needed? If not, how many is enough?
Research at the University of Pennsylvania now includes a dose-ranging study of a plasmid containing DNA for the HIV genes env and rev that contain codes for the envelope and reverse transcriptase enzyme. Fifteen HIV-positive volunteers in this study were given increasingly larger intramuscular injections of the vaccine candidate (30, 100 and 300 micrograms). After 4-8 months, 12 received a 100 microgram booster. No pattern of changes in CD4 cell counts or viral load was noted. The vaccine and booster injections were well tolerated. Comparisons of the effects of route of administration, whether by jet-injection or needle injection, on antibody and cytotoxic lymphocyte responses are now being evaluated.
The future of vaccines for HIV is dependent on a host of political, economic and scientific variables. National political will to see the development of an effective HIV vaccine by the year 2007 was expressed by President Bill Clinton; unfortunately, the statement was accompanied by no dramatic increase in direct funding for vaccine research and development. Other political pressures have been exerted by the International Association of Physicians in AIDS Care's ongoing advocacy for testing a live attenuated vaccine candidate soon. Political opposition to the IAPAC position is provided by Nobel laureate Baltimore, who says that the development of an AIDS vaccine is at least a decade away. Expressing a view somewhere between these positions is Anthony Fauci, MD, director of HIV research at National Institute of Allergy and Infectious Diseases. He said last December that empirical decisions will have to be made about testing an HIV vaccine without all the scientific answers being available.
The economics of an HIV vaccine are out of balance as well. Biotechnology firms backed away from HIV vaccine research 3 years ago because the return on initial investments seemed disappointingly small. Meanwhile, small non-profits have channeled needed funds to a multinational study of the VaxGen vaccine candidate. Larger organizations such as the International AIDS Vaccine Initiative (IAVI) will award $30,000 to Ronald Desrosiers, MD, at the New England Regional Primate Center to stu;dy the safety of live-attenuated vaccines in monkeys, more than $400,000 to Macfarlane Burnet Center in Australia to study a DNA-based, live-attenuated vaccine in animals, and almost $500,000 to Dan Farber Cancer Research Institute to develop hybrid viruses. A $4 million grant from Starr Foundation to IAVI increased funds available for additional grants. The AIDS Vaccine Research Committee has targeted specific areas of vaccine research, culminating in 49 widely publicized NIH Innovation Grants totaling $11.8 million, awarded last autumn.
Other international efforts include a collaborative Indian-U.S. vaccine initiative, the Vaccine Initiative Program, that has adopted as its mandate the creation of an HIV vaccine that targets the strain of HIV most prevalent in India. Infrastructure is being developed to study and test candidate vaccines at all phases, including selection of testing sites, special diagnostic kits specific to India and basic research. The rapid spread of HIV in India, where 2-5 million people are already infected in a country of more than 900 million, makes this effort a national priority.
Increased collaborations and private interest and investment will eventually speed the discovery of a successful vaccine for HIV. Questions that remain are when to move to wide-scale testing, as directly addressed by Donald Francis in his urgent efforts to see the VaxGen vaccine candidates widely tested. Increased attention to the problem of HIV variation and construction of an HIV vaccine that provides immunity to a wide spectrum of strains is another recent trend in vaccine research. A related question is whether private industry will significantly contribute to research and development of an HIV vaccine. Current interest at Chiron seems high, and the technology of choice, DNA vaccines, may effectively address concerns about wide supply at low cost. Hybrid vaccine approaches are also receiving increased attention, and new vectors such as live viruses genetically engineered to deliver vaccines are generating great interest.
Scientific interest was generated by David Baltimore's remark at the 5th Conference on Retroviruses and Opportunistic Infections in Chicago in February, that an effective vaccine may call forth more than the 2 recognized types of immune response, humoral and cellular. The third type of immune response he characterized as "superinfection," explaining that a infection with benign strain of virus (such as a weakened or attenuated virus administered as a vaccine) can block later infection with a virulent strain by physically excluding the virulent virus, perhaps through the work of an as-yet-unidentified cell. Since Baltimore heads the AIDS Vaccine Research Committee at the NIH, his remarks are bound to reflect the shape of future basic research in the quest for an AIDS vaccine.
Mark Bowers is Managing Editor of treatment publications at the San Francisco AIDS Foundation.
The purpose of this Phase I study is to evaluate the safety and immunogenicity of APL-400-047, an HIV-1 gag-pol DNA vaccine, after administration to adult HIV-negative volunteers. A secondary aim of this study is to compare the intramuscular administration route of needle and syringe to that of a Biojector 2000 needle-free jet injection system. If enrolled in this study, you will receive four immunizations over the course of 6 months; you will have a 20% chance of receiving placebo immunizations. You will either receive injections of the vaccine by needle and syringe or by a Biojector jet gun, depending on when you enter the study. To enroll, you must test negative for HIV within 8 weeks of immunization, and be negative for hepatitis B surface antigen. You cannot join this study if you have a history of immunodeficiency, chronic illness or autoimmune disease, or in some cases, a history of cancer. You also cannot join if you have ever attempted suicide, or if you have a history of suicidal ideation or past psychosis. If you are at higher risk behavior for HIV infection, as determined during screening, you also may not enroll in this study. This trial is enrolling in Seattle WA, Birmingham AL, Rochester NY and Nashville TN. For more information, call 1-800-TRIALS-A. (NIH-00916 or NIAID VEU 031)
This Phase II study is comparing the safety and immunogenicity of two HIV vaccine strategies: canary pox ALVAC-HIV vCP205 vaccine administered alone or administered with SF-2 rgp120, another kind of vaccine. If enrolled in this study, you will be randomly assigned to receive either both vaccines, one vaccine and a placebo, or two placebo vaccines. You will receive 4 immunizations over 6 months, with follow-up visits for up to four years. You can join this study if you are HIV-negative, and if you do not have a history of immunodeficiency, chronic illness, malignancy or autoimmune disease. You also cannot have any medical or psychiatric condition or occupational responsibilities that preclude compliance with the study. The study is enrolling throughout the country; call 1-800-TRIALS-A for more information. (NIH-00869 or NIAID VEU 202)
Clinical trial information compiled by Katy Stephenson of the Community Consortium in San Francisco.
Baltimore D. Lessons from people with nonprogressive HIV infection. The New England Journal of Medicine 332:259-260. January 26, 1995.
Berkley S. The international AIDS vaccine initiative. Journal of the International Association of Physicians in AIDS Care 3:30-34. November 1997.
Bowers M. HIV vaccines. BETA 18-21. September 1996.
Boyer J and others. Protection of chimpanzees from high-dose heterologous HIV-1 challenge by DNA vaccination. Nature Medicine 3:526-532. May 1997.
Butler D and others. Vaccines: a roller-coaster of hopes. Nature 386:537-538. April 10, 1997.
Cao Y and others. Virologic and immunologic characterization of long-term survivors of human immunodeficiency virus type 1 infection. The New England Journal of Medicine 332:201-208. January 26, 1995.
Carter G. Carter Indexes of AIDS Treatments and Infections. 1996.
Emini E. Hurdles in the path to an HIV-1 vaccine. Science and Medicine 2:38-47. May/June 1995.
Esparza J and Piot P. HIV vaccine development: UNAIDS perspective. Joint United Nations Program on HIV/AIDS (UNAIDS).
Farthing C. SIV vaccine for AIDS. Science 279:14. January 2, 1998.
Fauci A. AIDS in 1996: much accomplished, much to do. The Journal of the American Medical Association 276:155-156. July 10, 1996.
First human tests set for an AIDS vaccine. Washington Times A5. January 12, 1998.
Funds earmarked for AIDS vaccine centre. Nature 388:510. August 7, 1997.
Haynes BF and others. Toward an understanding of the correlates of protective immunity to HIV infection. Science 271:324-328. January 19, 1996.
Katongole-Mbidde E. The need for a vaccine against HIV/AIDS. Journal of the International Association of Physicians in AIDS Care 3:26-29. November 1997.
Kennedy RC. DNA vaccination for HIV. Nature Medicine 3:501-502. May 1997.
Krieger L. Vaccine news. San Francisco Examiner A4. January 14, 1998.
Marlink R. Achieving an HIV vaccine: the need for an accelerated campaign. Journal of the International Association of Physicians in AIDS Care 3:35-37. November 1997.
McDonnell W. Immunization. The Journal of the American Medical Association 278:2000-2007. December 10, 1997.
McDonnell W and Askari F. About DNA vaccines. The New England Journal of Medicine 334:1. January 4, 1996.
Nary G. Editorial. Journal of the International Association of Physicians in AIDS Care. August 1997.
Norley S. Anti-HIV vaccines: current status and future developments. Drugs 46:947-960. 1993.
Pantaleo G and others. Studies in subjects with long-term nonprogressive human immunodeficiency virus infection. The New England Journal of Medicine 332:209-216. January 26, 1995.
Robinson H and others. The scientific future of DNA for immunization. American Society for Microbiology. 1997.
Snow B. Monkey trials: animal studies for AIDS vaccines. Bay Area Reporter, May 23, 1997.
Swinbanks D. Vaccine institute treads out a wary path. Nature 389:655. October 16, 1997.
Taubes G. Salvation in a snippet of DNA? Science 278:1711-1714. December 5, 1997.
Therion Biologics. Press release. NIAID commences clinical trial with Therion Biologics' recombinant multi-antigen AIDS vaccine. TBC-3B. June 25, 1997.
U.S. firm's AIDS vaccine set for large study. Reuters. January 11, 1998.
Vaginal DNA vaccines effective against STDs. Fox News Online. January 13, 1998.
Voelker R. Collaboration needed for HIV vaccine success. The Journal of the American Medical Association 277:9. January 1, 1997.
Voelker R. HIV vaccine innovations. The Journal of the American Medical Association 277:1270. April 23/30, 1997.
Waalen J. DNA vaccines: the making of a revolution. Annals of Internal Medicine 126:169-171. January 15, 1997.
Wadman M. US dispute over live AIDS vaccine trials. Nature 389:426. October 2, 1997.
Wasima R. Intermediate-size trials for the evaluation of HIV vaccine candidates: a workshop summary. Journal of Acquired Immune Deficiency Syndromes and Human Retrovirology 16:195-203. 1997.
Young P. Bright outlook on direct DNA immunizations. ASM News 63:659-663. December 1997.
980401
BE980403 | 339,937 |
mens health chicago
- 08-05-2012, 10:22 AM
- 08-05-2012, 03:27 PM
I am not a medical Dr, please keep in mind that this answer is for information purposes only, and is not intended to diagnose, treat or replace sound medical advice from your physician or health care provider.
09-27-2012, 06:05 PM
It's not a "clinic". I use them myself as a primary MD as well as TRT. They are 100% legit and the docs are very knowledgable. Highly recommended and they do accept insurance, but depending what you are looking for there are some downfalls. First; TRT is their SECONDARY specialty as they are indeed a primary care clinic catering only to men. I have gone there for medical issues outside of TRT; including getting my flu shot for work, and when I got bronchitis. As far as TRT... here's what you can expect from them:
1. Regular blood work. Blood work is drawn in house (no going to Labcorp).
2. No script for self dosing- you have to go there weekly. I have found them willing to work with me when I can't come in due to work travel however it is expected that you go in unless there is some reasonable issue which they will help you overcome and still keep up with your regimen. Forget walking out with a script and going back when you need another. I live 5 miles away so I have no problems stopping by once a week and I find it more comfortable that way; but I have done self injections when I have to go out of town for work. They provide the necessary paperwork so flying isn't an issue even overseas that require advance notice by local government (Australia).
3. They are a true TRT clinic- unless you truly need it (per blood work) you won't get it.... some have tried that I personally know and failed.
4. They have studied TRT significantly and do follow up with anti-estrogen, HCG, ect. You can expect a "partial physical" every week when you go in as I call it. Check blood pressure, temp, quick screen of any issues you may be having (both TRT and not TRT issues/concerns). I like this as I know the doc is on top of it every week.
5. If you don't have insurance- they are more expensive than any other place in the area. There is one full time doc and site, and two others part time which practice at local hospitals also in the Chicagoland area.
That's about all I can say. I recommend the place... and no I don't work there, know anyone there outside of my doc, etc. I'm just a patient for both TRT as well as when something else is needed. Think of the place as the same thing as a women's clinic but for men.
Feel free to PM me if you have any questions. I'm in the Schaumburg area.
P.S. They have smoking hot nurses there as a bonus. :-)
Similar Forum Threads
Mens Health: 8 Training MistakesBy YellowJacket in forum Training ForumReplies: 3Last Post: 02-08-2003, 03:44 AM | 102,272 |
The Kievan period The Christian community that developed into what is now known as the Russian Orthodox Church is traditionally the life of. 2) Ivan III himself selected Dmitrii as heir in , most probably for foreign policy revolting against Ivan during the Muscovite-Lithuanian war possibly on the . Simon and before the bishops and before the whole clerical coun connection with . brought up in probably as a Uniate, and hence would probably be less hosti.
Vasily Vasiliyevich known as Vasily II the Blind (Василий II Тёмный), was the Grand Prince of Moscow whose long reign (–) was plagued by the. Vasily II, grand prince of Moscow from to Although the year-old Vasily II was named by his father Vasily I (ruled Moscow –) to succeed .
Zoe Palaiologina (Byzantine Greek: Ζωή Παλαιολογίνα), who later changed her name to . Even while traveling to Russian lands, it became apparent that the Vatican's plans to make Sophia represent Catholicism In particular, the Grand Princess was unable to obtain government posts for her relatives: her brother Andreas. Sophia Alekseyevna ruled as regent of Russia from to She allied herself with a singularly capable courtier and politician, Prince Vasily Golitsyn.
Tsar Ivan the Terrible, a man who greatly increased Russia's power and Ivan's first wife was Anastasia Romanovna, with whom he entered a Anastasia told Ivan to not wed a pagan, which Maria Temryukovna Tsar Ivan IV admires his sixth wife Vasilisa Melentyeva, painting by Grigory Sedov, Anastasia Romanovna, 1st wife of Ivan IV. | Source Marfa Sobakina, 3rd wife of Ivan the Terrible. | Source This may be Anna Koltovskaya with Ivan. | Source Vasilisa Melentyeva, 6th wife of Ivan the Terrible. | Source.
Ivan II Ivanovich the Fair (30 March – 13 November ) was the Grand Prince of Moscow and Grand Prince of Vladimir in Until that date, he had ruled the towns of Ruza and Zvenigorod. He was the second son of Ivan Kalita, and succeeded his brother Simeon the. Ivan III Vasilyevich also known as Ivan the Great, was a Grand Prince of Moscow and "Grand .. Following his second marriage, Ivan developed a complicated court ceremonial on the Byzantine model and began to use the title of "Tsar and.
The grandson of Ivan the Great, Ivan the Terrible, or Ivan IV, acquired vast amounts of land during his long reign (), an era marked by the conquest of the khanates of Kazan, Astrakhan and Siberia. The first tsar of all Russia, Ivan the Terrible, or Ivan IV, had a complex. Ivan IV Vasilyevich commonly known as Ivan the Terrible (Russian: About this sound Ива́н Гро́зный (help·info), Ivan Grozny; "Ivan the Formidable" or "Ivan the. | 61,824 |
It's finally getting a little cooler in Charlotte so I've been working on knitting and crocheting hats. This year I'd like to work specifically on the slouch design, which is in my opinion a very comfortable hat to wear. The problem with wearing beanies is that they mat down and mess up your hair. Slouchy hats on the other hand are loose and I think they are very stylish.
On a recent visit to Hobby Lobby I came across a really neat free pattern by Debbie Stoller to promote the yarn Bamboo Ewe.
My first hat came out really slouchy since I used a Vanna's Choice 4 ply worsted yarn. Vanna's yarn is heavy worsted which is slightly heavier and thicker. The second attempt matched the gauge and I'm really happy with the results. I used Karaoke soy silk yarn from my stash.
I loved making this hat so I purchased a skein of Bamboo Ewe in the Lipstick colorway. This yarn is also considered a 4 ply yarn, but is not as heavy as Vanna's. Red Heart has sure come a long way with their yarn. The feel and softness of this yarn is dreamy.
I told you I was into making slouch hats. This will be the crochet version which definitely goes much faster. I could not resist including the pink butterfly to this picture...so we'll see if the butterfly stays or goes. I'm in the process of creating a video which shows how to make this cute butterfly.
| 99,213 |
TITLE: On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion
QUESTION [4 upvotes]: Consider the polynomial ring $R=\mathbb C[x,y]$.
Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&x^5+y^5&5x^5&10x^5\\10y^5&10y^5&5y^5&x^5+y^5&5x^5\\5y^5&10y^5&10y^5&5y^5&x^5+y^5 \end{pmatrix} \in M_5(R)$.
Also consider the polynomial $p(x,y)=\det (A-I)\in R=\mathbb C[x,y]$.
How to show that $p(x,y) $ can be factored into $5^2=25$ linear polynomials in $x$ and $y$ (over $\mathbb C$) ?
If $x+y=1$, then it can be shown that $1$ is an an eigenvalue of $A$, so the image of $p(x,y)$ in $\mathbb C[x,y]/(x+y-1)$ is $0$; thus $x+y-1$ is a factor of $p(x,y)$; but other than that, I'm unable to say anything else.
I feel I some how have to apply Hilbert Nullstellensatz, but I don't know how.
REPLY [6 votes]: Define the $\, n\times n\,$ matrix $\, A = \{a_{i, j}\}_{i, j=1}^n \,$ where
$\, a_{i, j} = \binom{n}{j-i}x^n + \binom{n}{i-j}y^n. \,$
The matrix $A$ is a special Toeplitz matrix. Let $\, p(x, y) := \det(A-I).\,$ Since $\, x + y - 1 \,$
is a factor, then also $\, z x + w y - 1 \,$ is a factor where $\, z, w \,$ are any pair of $\,n$th roots of unity and there are $\,n^2\,$ pairs. | 85,636 |
\begin{document}
\title[Interpolation by polynomials in convex domains]{Interpolation by multivariate polynomials in convex domains}
\author{Jorge Antezana}
\address{Departamento de Matem\'atica, Universidad Nacional de La Plata, and Instituto Argentino de Matem\'atica ``Alberto P. Calder\'on'' (IAM-CONICET), Buenos Aires, Argentina}
\email{\href{mailto:[email protected]}{\texttt{[email protected]}}}
\author{Jordi Marzo}
\address{Dept.\ Matem\`atica i Inform\`atica,
Universitat de Barcelona and BGSMath,
Gran Via 585, 08007 Bar\-ce\-lo\-na, Spain}
\email{\href{mailto:[email protected]}{\texttt{[email protected]}}}
\author{Joaquim Ortega-Cerd\`a}
\address{Dept.\ Matem\`atica i Inform\`atica,
Universitat de Barcelona and BGSMath,
Gran Via 585, 08007 Bar\-ce\-lo\-na, Spain}
\email{\href{mailto:[email protected]}{\texttt{[email protected]}}}
\thanks{Supported by the the Spanish Ministerio de Econom\'ia y Competividad
(grant MTM2017-83499-P) and the Generalitat de Catalunya (grant 2017 SGR 358)}
\begin{abstract}
Let $\Omega$ be a convex open set in $\mathbb R^n$ and let $\Lambda_k$
be a finite subset of $\Omega$. We find necessary geometric conditions for
$\Lambda_k$ to be interpolating for the space of multivariate polynomials of
degree at most $k$. Our results are asymptotic in $k$. The density conditions
obtained match precisely the necessary geometric conditions that sampling sets
are known to satisfy and they are expressed in terms of the equilibrium
potential of the convex set. Moreover we prove that in the particular case
of the unit ball, for $k$ large enough, there is no family of orthogonal reproducing kernels in the
space of polynomials of degree at most $k$.
\end{abstract}
\date{\today}
\maketitle
\section{Introduction}
Given a measure $\mu$ in $\mathbb R^n$ we consider the space $\mathcal P_k$ of
polynomials of total degree at most $k$ in
$n$-variables endowed with the natural scalar product in $L^2(\mu)$. We assume
that $L^2(\mu)$ is a norm for $\mathcal P_k$, i.e. the support of $\mu$ is not
contained in the zero set of any $p\in \P_k$, $p\ne 0$. In this case the point
evaluation at any given point $x\in \R^n$ is a bounded linear functional and
$(\P_k, L^2(\mu))$ becomes a reproducing kernel Hilbert space, i.e for any
$x\in \R^n$, there is a unique function $K_k(\mu, x,\cdot)\in \mathcal P_k$ such that
\[
p(x) = \langle p, K_k(\mu,x,\cdot)\rangle = \int p(y) K_k(\mu,x, y) \, d\mu(y).
\]
Given a point $x\in \R^n$ the normalized reproducing kernel is denoted by
$\kappa_{k,y}$, i.e.
\[
\kappa_{k,y}(\mu,x) = \frac{K_k(\mu,x,y)}{\| K_k(\mu,x,\cdot) \|_{L^2(\mu)}} = \frac{K_k(\mu,x,y)}{\sqrt{K_k(\mu,x,x)}}.
\]
We will denote by $\beta_k(\mu,x)$ the value of the reproducing kernel in the diagonal
$$\beta_k(\mu,x)=K_k(\mu,x,x).$$
The function $1/\beta_k(\mu,x)$ is the so called Christoffel function.
For brevity we may omit sometimes the dependence on $\mu$.
\medskip
Following Shapiro and Shields in \cite{shsh} we define sampling and interpolating sets:
\begin{definition}\label{definterp}
A sequence $\Lambda = \{\Lambda_{k}\}$ of finite sets of points on $\R^n$ is
said to be
\emph{interpolating} for $(\mathcal P_{k},L^2(\mu))$ if the associated family
of
normalized reproducing
kernels at the points $\lambda\in \Lambda_k$, i.e. $\kappa_{k,\lambda},$
is a Riesz sequence in the Hilbert space $\P_k$, uniformly in $k$, i.e there is
a constant $C>0$ independent of $k$ such that for any linear combination of the
normalized reproducing kernels we have:
\begin{equation} \label{ineq:interp}
\frac 1{C} \sum_{\lambda\in\Lambda_k} |c_\lambda|^2 \le \bigl\|
\sum_{\lambda\in \Lambda_k} c_\lambda
\kappa_{k,\lambda} \bigr\|^2\le C \sum_{\lambda\in\Lambda_k} |c_\lambda|^2,\quad
\forall \{c_\lambda\}_{\lambda\in\Lambda_k}.
\end{equation}
\end{definition}
The definition above is
usually decoupled in two separate conditions.
The left hand side
inequality in (\ref{ineq:interp})
is usually called the \emph{Riesz-Fischer} property for the reproducing kernels
and it is equivalent to the fact that the following moment problem is solvable:
for arbitrary values $\{v_\lambda\}_{\lambda_\in \Lambda_k}$ there exists a
polynomial
$p\in \P_k$ such that $p(\lambda)/\sqrt{\beta_k(\lambda)} = \langle p,
\kappa_{k,\lambda}\rangle = v_\lambda$ for all $\lambda\in \Lambda_k$ and
\[
\|p\|^2 \le C \sum_{\lambda\in \Lambda_k} |v_\lambda|^2 = \sum_{\lambda\in
\Lambda_k} \frac{|p(\lambda)|^2}{\beta_k(\lambda)}.
\]
This is the reason $\Lambda$ is
called an interpolating family.
The right hand side inequality in (\ref{ineq:interp})
is called the Bessel property for the normalized reproducing kernels
$\{\kappa_{k,\lambda}\}_{\lambda\in \Lambda_k}$. The Bessel property is equivalent to have
\begin{equation}\label{Carlesonseq}
\sum_{\lambda\in\Lambda_k} \frac{|p(\lambda)|^2}{\beta_k(\lambda)} \le C
\|p\|^2
\end{equation}
for all $p\in \mathcal P_k.$ That is, if we denote $\mu_k := \sum_{\lambda\in \Lambda_k}
\frac{\delta_{\lambda}}{\beta_k(\lambda)}$, we are requiring that the identity
is a
continuous embedding of $(\P_k, L^2(\mu))$ into $(\P_k, L^2(\mu_k))$.
\medskip
The notion of sampling play a similar but opposed role.
\begin{definition} \label{defsampling}
A sequence $\Lambda = \{\Lambda_{k}\}$ of finite sets of points on $\R^n$ is
said to be
\emph{sampling} or \emph{Marcinkiewicz-Zygmund} for $(\mathcal P_{k},L^2(\mu))$
if the associated family of normalized reproducing
kernels at the points $\lambda\in \Lambda_k$, $\kappa_{k,\lambda}(x)$
is a frame in the Hilbert space $\P_k$, uniformly in $k$, i.e there is
a constant $C>0$ independent of $k$ such that for any polynomial $p\in P_k$:
\begin{equation} \label{def:sampling}
\frac 1{C} \sum_{\lambda\in\Lambda_k} |\langle p, \kappa_{k,\lambda} \rangle|^2 \le
\| p\|^2 \le C \sum_{\lambda\in\Lambda_k} |\langle p,
\kappa_{k,\lambda} \rangle|^2,\quad
\forall p\in \P_k.
\end{equation}
\end{definition}
Observe that the left hand side inequality in (\ref{def:sampling}) is the Bessel condition mentioned above. If we were considering a
single space of polynomials $\mathcal P_{k_0}$ then the notion of interpolating
family amounts to say that the corresponding reproducing kernels are
independent. On the other hand, the notion of sampling family corresponds to the
reproducing kernels span the whole space $\P_{k_0}$.
In this work we will restrict our attention to two classes of measures:
\begin{itemize}
\item The
first is $d\mu(x) = \chi_\Omega(x) dV(x)$ where $\Omega$ is a
smooth bounded convex domain and $dV$ is the Lebesgue measure.
\item The second is of the form $d\mu(x) = (1-|x|^2)^{a-1/2}\chi_\B(x) dV(x)$ where
$a\ge 0$ and $\B$ is the unit ball $\B = \{x\in \R^n : |x| \le 1\}$.
\end{itemize}
In these two cases there are good explicit estimates for the size of the
reproducing kernel on the diagonal $K_k(\mu,x,x),$ and therefore both notions,
interpolation and sampling families, become more tangible.
In \cite{BerOrt} the authors obtained necessary geometric conditions for sampling families in bounded smooth convex sets with weights when
the weights satisfy two technical conditions: Bernstein-Markov and moderate growth. These properties
are both satisfied for the Lebesgue measure in a convex set.
The case of interpolating families in convex sets was not considered, since there were
several technical hurdles to apply the same technique.
Our aim in this paper is to fill this gap and obtain necessary geometric
conditions for interpolating families in the two settings mentioned above.
The geometric conditions that usually appear in this type of problem come into
three flavours:
\begin{itemize}
\item A separation condition. This is implied by the Riesz-Fischer
condition i.e. the left hand side of (\ref{ineq:interp}). The fact that one should be able to interpolate the
values one and
zero implies that different points $\lambda, \lambda' \in \Lambda_k$ with
$\lambda \ne \lambda'$ cannot be too close. The separation conditions in our settings are studied
in Section 3.1.
\item A Carleson type condition. This is a condition that ensures the
continuity of the embedding as in (\ref{Carlesonseq}). A geometric
characterization of the Carleson is given in Theorem \ref{Carleson}
for convex domains and the Lebesgue measure, and in Theorem
\ref{Carleson en bolas} for the ball and the measures $\mu_a$.
\item A density condition. This is a global condition that usually follows from
both the Bessel and the Riesz-Fischer condition. A density necessary condition
for interpolating sequences is provided in Theorem \ref{density} for convex sets
endowed with the Lebesgue measure, and in Theorem \ref{teo:density_ball} for
the ball and the measures $\mu_a$. Moreover, in this last setting we get an extension
of the density results proved in \cite{BerOrt} for sampling sequences.
\end{itemize}
Finally, a natural question is whether or not there exists a family $\{\Lambda_k\}$ that is both sampling and interpolating. To answer this question is very difficult in general \cite{OUbook}. A particular case
is when $\{\kappa_{k,\lambda}\}_{\lambda\in \Lambda_k}$ form an orthonormal basis.
In the last section we study the existence of orthonormal basis of reproducing kernels in the case of the ball
with the measures $\mu_a$. More precisely, if the spaces $\P_k$ endowed with the inner product of $L^2(\mu_a)$, then in Theorem \ref{noONbasis} we prove that for $k$ big enough the space $\P_k$ does
not admit an orthonormal basis of reproducing kernels. To
determine whether or not there exists a family $\{\Lambda_k\}$ that is both sampling and interpolating for $(\P_k,\mu_a)$ remains an open problem.
\section{Technical results}
Before stating and proving our results we will recall the behaviour of the kernel in the diagonal, or equivalently the Christoffel function, we will define an appropriate metric and introduce some needed tools.
\subsection{Christoffel functions and equilibrium measures}
To write explicitly the sampling and interpolating conditions we need an estimate of the Christoffel function. In \cite{BerOrt} it was observed that in the case of the measure
$d\mu(x) = \chi_\Omega(x) dV(x)$ it is possible to obtain precise estimates for
the size of the reproducing kernel on the diagonal:
\begin{thm}\label{thm:diagonalestimate}
Let $\Omega$ be a smoothly bounded convex domain in $\R^n$. Then the
reproducing kernel for $(\mathcal P_{k},\chi_\Omega dV)$ satisfies
\begin{equation}\label{kernelconvex}
\beta_k(x)=K_k(x,x) \simeq \min\Bigl(
\frac{k^n}{\sqrt{d(x,\partial \Omega)}}, k^{n+1}\Bigr)\quad
\forall x\in\Omega.
\end{equation}
where $d(x,\partial \Omega)$ denotes the Euclidean distance of $x\in \Omega$ to the boundary
of $\Omega$.
\end{thm}
For the weight $(1-|x|^2)^{a-1/2}$ in the ball $\mathbb B$ the asymptotic behaviour of the Christoffel is well known.
\begin{prop} \label{prop:Christoffel}
For any $a \ge 0$ and $d\ge 1$ let $$d\mu_a(x)=(1-|x|^2)^{a-1/2}\chi_\B(x) dV(x).$$ Then the reproducing kernel for $(\mathcal P_{k},d\mu_a )$ satisfies
\begin{equation}\label{kernelball}
\beta_k(\mu_a,x)=K_k(\mu_a,x,x) \simeq \min\Bigl(
\frac{k^n}{d(x,\partial \mathbb B)^a}, k^{n+2a}\Bigr)\quad
\forall x\in\Omega.
\end{equation}
\end{prop}
The proof follows from \cite[Prop 4.5 and 5.6]{petxu}, Cauchy–Schwarz inequality and the extremal characterization of the kernel
$$K_k(\mu_a;x,x)=\left\{ |P(x)|^2\;\; : \;\;P\in \mathcal P_k,\int |P|^2 d\mu_a \le 1 \right\}.$$
To define the equilibrium measure we have to introduce a few concepts from pluripotential theory, see \cite{Kli91}. Given a non pluripolar compact set $K\subset \R^n\subset \C^n$ the pluricomplex Green function is the semicontinuous regularization
$$G^*_K(z)=\limsup_{\xi \to z}G_K(\xi),$$
where
$$G_K(\xi)=\sup \left\{ \frac{\log^+|p(\xi)|}{\deg(p)}\; : \; p\in P(\C^n),\; \sup_K |p(\xi)|\le 1\right\}.$$
The pluripotential equilibrium measure for of $K$ is the (probability) Monge-Amp\`ere Borel measure
$$d \mu_{eq}=(d d^c G_K^*)^n.$$
In the general case, when $\Omega$ is a smooth bounded convex domain the equilibrium
measure is very well understood, see \cite{BedTay} and \cite{blmr}. It behaves
roughly as $d\mu_{eq}\simeq 1/\sqrt{d(x,\partial\Omega)}dV$. In particular,
the pluripotential equilibrium measure for the ball $\mathbb B$ is given (up to normalization) by $d \mu_0(x)=\frac{1}{\sqrt{1-|x|^2}}dV(x).$
\subsection{An anisotropic distance}\label{metric}
The natural distance to formulate the separation condition and the
Carleson condition is not the Euclidean distance. Consider in the unit ball
$\B\subset \R^n$ the following distance:
\[
\rho(x,y) = \arccos\left\{\langle x, y\rangle + \sqrt{1-|x|^2}\sqrt{1-|y|^2}\right\}.
\]
This is the geodesic distance of the points $x',\ y'$ in the
sphere $\mathbb S^{n}$ defined as $x'=(x,\sqrt{1-|x|^2})$ and
$y'=(y,\sqrt{1-|y|^2})$.
If we consider anisotropic balls $B(x,\varepsilon) = \{y \in \mathbb B:
\rho(x,y) < \varepsilon\}$,
they are comparable to a box centered at $x$ (a
product of intervals) which are of size $\varepsilon$ in the tangent directions
and
size $\varepsilon^2 +
\varepsilon\sqrt{1-|x|^2}$ in the normal direction. If we want to refer to a
Euclidean
ball of center $x$ and radius $\varepsilon$ we would use the notation
$\B(x,\varepsilon)$.
The Euclidean volume of a ball $B(x,\varepsilon)$ is comparable to
$\varepsilon^{n}\sqrt{1-|x|^2}$ if $(1-|x|^2) > \varepsilon^2$ and
$\varepsilon^{n+1}$ otherwise.
This distance $\rho$ can be extended to an arbitrary smooth convex domain
$\Omega$ by using Euclidean balls contained in $\Omega$ and tangent to the
boundary of $\Omega$. This can be done in the following way. Since $\Omega$ is
smooth, there is a tubular neighbourhood $U\subset \R^n$ of the boundary of
$\Omega$ where each point $x\in U$ has a unique closest point $\tilde x$ in
$\partial \Omega$ and the normal line to $\partial \Omega$ at $\tilde x$ passes
by $x$. There is a fixed small radius $r>0$ such that for any point $x\in U\cap
\Omega$ it is contained in a ball of radius $r$, $B(p, r)\subset \Omega$ and
such that it is tangent to $\partial \Omega$ at $\tilde x$. We define on $x$ a
Riemannian metric which comes from the pullback of the standard metric on
$\partial \tilde B(p,r)$ where $\tilde B(p,r)$ is a ball in $\R^{n+1}$ centered
at $(p,0)$ and of radius $r>0$ by the projection of $\R^{n+1}$ onto the first
$n$-variables. In this way we have defined a Riemannian metric in the domain
$\Omega\cap U$. In the core of $\Omega$, i.e. far from the boundary we use the
standard Euclidean metric. We glue the two metrics with a partition of unity.
The resulting metric $\rho$ on $\Omega$ has the relevant property that the
balls of radius $\epsilon$ behave as in the unit ball, that is a ball
$B(x,\varepsilon)$ of center $x$ and of radius $\varepsilon$ in this metric is
comparable to a box of size
$\varepsilon$ in the tangent directions
and size $\varepsilon^2 + \varepsilon\sqrt{d(x,\partial \Omega)}$ in the normal
direction.
\subsection{Well localized polynomials}
The basic tool that we will use to prove the Carleson condition and the
separation are well localized polynomials. These were studied by Petrushev and
Xu in the unit ball with the measure $d\mu_a=(1-|x|^2)^{a-\frac{1}{2}}dV,$ for $a\ge 0.$
We recall their basic properties:
\begin{thm}[Petrushev and Xu] \label{theorem_PX}
Let $d\mu_a=(1-|x|^2)^{a-\frac{1}{2}}dV$ for $a\ge 0.$
For any $k\ge 1$ entire and any $y\in \B\subset \R^n$ there are polynomials $L_k^a (\cdot , y) \in \mathcal P_k$ that
satisfy:
\begin{enumerate}
\item $L_k^a$ as a variable of $x$ is a polynomial of degree $2k$.
\item $L_k^a(x,y) = L_k^a(y, x)$.
\item $L_k^a$ reproduces all the polynomials of degree $k$, i.e.
\begin{equation} \label{reproducing}
p(y) = b_n^a \int_\B L_k^a(x,y) p(x)\, d\mu_a(x).\qquad \forall p\in \mathcal P_k.
\end{equation}
\item For any $\gamma>0$ there is a $c_\gamma$ such that
\begin{equation} \label{offdiag}
|L_k^a (x, y)| \le c_\gamma \frac{\sqrt{\beta_k(\mu_a,x) \beta_k(\mu_a,y) }}{(1+ k
\rho(x,y))^\gamma}.
\end{equation}
\item The kernels $L_k^a$ are Lispchitz with respect to the metric $\rho$, more
concretely, for all $x\in B(y, 1/k)$:
\begin{equation}\label{integ_weight}
|L_k^a (w, x) - L_k^a (w, y)| \le c_\gamma \frac{k
\rho(x,y) \sqrt{ \beta_k(\mu_a,w) \beta_k(\mu_a,y) }}{ (1+k \rho(w,y))^\gamma}
\end{equation}
\item \label{diagonal_weight}
There is $\varepsilon > 0$ such that $L_k^a (x, y) \simeq
K_k(\mu_a; y, y)$ for all
$x\in B(y, \varepsilon/k)$.
\end{enumerate}
\end{thm}
\begin{proof}
All the properties are proved in \cite[Thm 4.2, Prop 4.7 and 4.8]{petxu} except the behaviour near the diagonal
number~\ref{diagonal_weight}. Let us start by
observing that by the Lipschitz condition \eqref{integ_weight} it is enough to prove
that $L_k^a (x,x)\simeq K_k(\mu_a;x,x)$.
This follows from the definition of $L_k^a$ which is done as follows.
The subspace $V_k\subset L^2(\B)$ are the polynomials of degree $k$ that are
orthogonal to lower degree polynomials in $L^2(\B)$ with respect to the measure $d\mu_a$.
Consider the kernels $P_k(x, y)$
which are the kernels that give the orthogonal projection on $V_k$. If
$f_1,\ldots, f_r$ is an orthonormal basis for $V_k$ then $P_k(x,y) = \sum_{j=
1}^r f_j(x)f_j(y)$. The kernel $L_k^a$ is defined as
\[
L_k^a (x,y) = \sum_{j = 0}^\infty \hat a \left(\frac j k\right) P_j(x,y).
\]
We assume that $\hat a$ is compactly supported, $\hat a \ge 0$, $\hat a \in
\mathcal C^\infty(\mathbb R)$, $\operatorname{supp} \hat a \subset [0, 2]$,
$\hat a(t) = 1$
on $[0,1]$ and $\hat a (t) \le 1$ on $[1,2]$ as in the picture:
\begin{center}
\ifx\XFigwidth\undefined\dimen1=0pt\else\dimen1\XFigwidth\fi
\divide\dimen1 by 3174
\ifx\XFigheight\undefined\dimen3=0pt\else\dimen3\XFigheight\fi
\divide\dimen3 by 2499
\ifdim\dimen1=0pt\ifdim\dimen3=0pt\dimen1=4143sp\dimen3\dimen1
\else\dimen1\dimen3\fi\else\ifdim\dimen3=0pt\dimen3\dimen1\fi\fi
\tikzpicture[x=+\dimen1, y=+\dimen3]
{\ifx\XFigu\undefined\catcode`\@11
\def\temp{\alloc@1\dimen\dimendef\insc@unt}\temp\XFigu\catcode`\@12\fi}
\XFigu4143sp
\ifdim\XFigu<0pt\XFigu-\XFigu\fi
\clip(2913,-4962) rectangle (6087,-3100);
\tikzset{inner sep=+0pt, outer sep=+0pt}
\pgfsetlinewidth{+7.5\XFigu}
\draw (2925,-4725)--(6075,-4725);
\draw (3150,-4950)--(3150,-3200);
\draw (4500,-4770)--(4500,-4680);
\draw (5850,-4770)--(5850,-4680);
\draw (3150,-3600)--(4635,-3600);
\draw
(4635,-3600)--(4636,-3600)--(4640,-3600)--(4649,-3601)--(4664,-3601)--(4684,
-3603)
--(4707,-3604)--(4730,-3606)--(4753,-3609)--(4773,-3612)--(4790,-3616)--(4806,
-3620)
--(4820,-3625)--(4833,-3631)--(4845,-3638)--(4855,-3644)--(4865,-3652)--(4875,
-3661)
--(4886,-3671)--(4896,-3683)--(4907,-3696)--(4918,-3710)--(4929,-3726)--(4940,
-3743)
--(4951,-3761)--(4962,-3780)--(4973,-3799)--(4984,-3820)--(4995,-3841)--(5006,
-3862)
--(5018,-3885)--(5026,-3902)--(5035,-3920)--(5044,-3939)--(5054,-3959)--(5064,
-3980)
--(5074,-4001)--(5085,-4024)--(5095,-4047)--(5107,-4071)--(5118,-4095)--(5129,
-4120)
--(5140,-4144)--(5151,-4169)--(5162,-4193)--(5173,-4217)--(5183,-4240)--(5193,
-4263)
--(5202,-4285)--(5211,-4306)--(5219,-4326)--(5227,-4346)--(5235,-4365)--(5244,
-4388)
--(5253,-4410)--(5261,-4431)--(5269,-4453)--(5278,-4474)--(5286,-4495)--(5294,
-4515)
--(5303,-4535)--(5311,-4554)--(5320,-4572)--(5328,-4589)--(5337,-4605)--(5346,
-4619)
--(5355,-4632)--(5364,-4644)--(5373,-4655)--(5382,-4664)--(5393,-4673)--(5405,
-4681)
--(5418,-4688)--(5432,-4695)--(5449,-4700)--(5468,-4705)--(5489,-4709)--(5513,
-4713)
--(5539,-4716)--(5567,-4719)--(5594,-4721)--(5620,-4722)--(5641,-4724)--(5656,
-4724)
--(5666,-4725)--(5669,-4725)--(5670,-4725);
\pgftext[base,left,at=\pgfqpointxy{4500}{-4590}]
{$k$};
\pgftext[base,left,at=\pgfqpointxy{5850}{-4545}]
{$2k$};
\pgftext[base,left,at=\pgfqpointxy{2970}{-3645}]
{$1$};
\pgftext[base,left,at=\pgfqpointxy{3690}{-3465}]
{$\hat a(x/k)$};
\endtikzpicture
\end{center}
\noindent Then, all the terms are positive in the diagonal. Hence, we get
\[
\beta_k (\mu_a,x) = K_k(\mu_a;x, x) \le L_k^a (x, x) \le K_{2k}(\mu_a;x,x) = \beta_{2k}(\mu_a,x).
\]
Since $\beta_k (\mu_a,x) \simeq \beta_{2k} (\mu_a,x)$ we obtain the desired estimate.
\end{proof}
They also proved the following integral estimate \cite[Lemma 4.6]{petxu}
\begin{lem} \label{lem3_weight}
Let $\alpha>0$ and $a\ge 0.$ If $\gamma>0$ is big enough we have
\[
\int_{\B} \frac{K_k(\mu_a,y,y)^\alpha }{(1+k \rho(x,y))^\gamma}d\mu_a (y) \lesssim
\frac 1 {K_k(\mu_a,x,x)^{1-\alpha}}.
\]
\end{lem}
\section{main results}
\subsection{Separation}
In our first result we prove that for $\Lambda=\{ \Lambda_k \}$ interpolating there exist $\epsilon>0$ such that
$$\inf_{\lambda,\lambda'\in \Lambda_k,\lambda\neq \lambda'}\rho(\lambda,\lambda')\ge \frac{\epsilon}{k}.$$
\begin{thm}\label{separadas}
If $\Omega$ is a smooth convex set and $\Lambda=\{ \Lambda_k \}$ is an interpolating
sequence then there is an $\varepsilon>0$ such that the balls
$\{B(\lambda,\varepsilon/k)\}_{\lambda\in \Lambda_k}$ are pairwise disjoint.
\end{thm}
\begin{proof}
Consider the metric in $\Omega$ defined in section \ref{metric}. We can restrict the argument to a ball, of a fixed radius $r(\Omega),$ in one of the two cases:
tangent to the boundary or at a positive distance to the complement $\R^n\setminus \Omega.$
Let us assume that there is another point from $\Lambda_k$, $\lambda' \in
B(\lambda,\varepsilon/k)$. Since it is interpolating we can build a polynomial
$p\in \mathcal P_k$ such that $p(\lambda') = 0$, $p(\lambda) = 1$ and
$\|p\|^2 \lesssim 1/K_k(\mu_{\frac{1}{2}},\lambda, \lambda)$. Take a ball $\Omega$ such
that it contains $\lambda$ and $\lambda'$ and that it is tangent to $\partial
\Omega$ at a closest point to $\lambda$.
To simplify the notation assume that radius of this ball is one, and it is denoted by $\B.$
In this ball the kernel $L_k^{\frac{1}{2}}$ from Theorem \ref{theorem_PX}, for the Lebesgue measure $a=\frac{1}{2},$ is
reproducing so
\begin{equation} \label{eq1:separation}
1 = \int_\B (L_k^{\frac{1}{2}}(\lambda, w)-L_k^{\frac{1}{2}}(\lambda', w)) p(w) dV(w).
\end{equation}
We can use the estimate $$|p(w)| \le \sqrt{\beta_k(\mu_{\frac{1}{2}},w)} \|p\| \le
\sqrt{\beta_k(\mu_{\frac{1}{2}},w)/\beta_k(\mu_{\frac{1}{2}},\lambda))}$$ and the inequality (\ref{integ_weight})
to obtain
\[
1 \lesssim k\rho(\lambda,\lambda') \int_\B \frac{\beta_k(\mu_{\frac{1}{2}},w)
dV(w)}{(1+k\rho(y,\lambda))^\gamma},
\]
Taking $\alpha = 1$ and $a=\frac{1}{2}$ in Lemma~\ref{lem3_weight} we obtain $1 \lesssim
k\rho(\lambda,\lambda')$
as stated.
\end{proof}
Observe that considering the general case $L_k^a$ in (\ref{eq1:separation}), one can prove the corresponding result
for
interpolating sequences for $\mathcal P_k$ with weight $d\mu_a(x)=(1-|x|^2)^{a-\frac{1}{2}}dV(x)$ in the ball $\mathbb B.$
\subsection{Carleson condition}
Let us deal with condition \eqref{Carlesonseq}. For a convex smooth set $\Omega \subset \R^n$
is a particular instance
of the
following definition.
\begin{definition}
A sequence of measures $\mu_k \in \mathcal M(\Omega)$ are called Carleson
measures for $(\mathcal P_k,d\mu )$ if there is a constant $C>0$ such that
\[
\int_{\Omega} |p(x)|^2\, d\mu_k(x) \le C \|p\|_{L^2(\mu)}^2,
\]
for all $p\in \mathcal P_k$.
\end{definition}
In particular if $\Lambda_k$ is a sequence of interpolating sets then the
sequence of measures
$\mu_k = \sum_{\lambda\in \Lambda_k} \frac{\delta_\lambda}{\beta_k(\lambda)}$
is Carleson.
The geometric characterization of the Carleson measures when $\Omega$ is a
smooth convex domain is in terms of anisotropic balls.
\begin{thm}\label{Carleson} A sequence of measures $\mu_k$ is Carleson for the polynomials
$\P_k$ in a smooth bounded convex domain $\Omega$ if and only if there is a constant
$C$ such that for all points $x\in \Omega$
\begin{equation}\label{geonec}
\mu_k(B(x,1/k)) \le C V(B(x,1/k)).
\end{equation}
\end{thm}
\begin{proof}
We prove the necessity. For any $x\in\Omega$ there is a cube $Q$ that contains
$\Omega$ which is tangent to $\partial\Omega$ at a closest point to $x$ as in
the picture:
\begin{figure}
\includegraphics[width=4cm]{cube.png}
\caption{}
\end{figure}
This cube has fixed dimensions independent of the point $x\in\Omega$. We can
construct a polynomial $Q_k^x$ of degree at most $k n$ taking the product of one
dimensional polynomials $L_k^{\frac{1}{2}}$. We test against these
polynomials that peak at $B(x,1/k)$
$$\int_{B(x,1/k)} |Q_k^x|^2 d\mu_k \le \int_{\Omega} |Q_k^x|^2 d\mu_k\le C \| Q_k^x \|_{L^{2}(Q)},$$
by property $(6)$ in Theorem \ref{theorem_PX} and the estimate (\ref{kernelball}) the necessary condition follows.
For the sufficiency we use the reproducing property of $L_k^{\frac{1}{2}}(z,y)$. That is for
any point $x\in \Omega$ there is a Euclidean ball $\B_x$ contained in $\Omega$
such that $x\in \B_x$ and it is tangent to $\partial \Omega$ in the closest
point to $x$ as in the picture. Moreover since $\Omega$ is a smoothly bounded
convex domain we can assume that the radius $\B$ has a lower bound independent
of $x$. In this ball we can reconstruct any polynomial $p\in \mathcal{P}_k$ using $L_k^{\frac{1}{2}}$. That
is
\[
\int_\Omega |p(x)|^2 \, d\mu_k(x) \le
\int_\Omega \left|\int_{B_x} L_{2k}^{\frac{1}{2}}(x,y) p^2(y)\, dV(y)\right|\, d\mu_k(x).
\]
We use the estimate (\ref{offdiag}) and we get
\[
\int_\Omega |p(x)|^2 \, d\mu_k(x) \lesssim \int_\Omega\int_{B_x}
\frac{\sqrt{\beta_k(x)\beta_k(y)}}{(1+ k
\rho(x,y))^\gamma} |p(y)|^2 dV(y)\, d\mu_k(x).
\]
We break the integral in two regions, when $\rho(x,y) < 1 $ and otherwise. When
$k$ is big enough we obtain:
\[\begin{split}
\int_\Omega |p(x)|^2 \, d\mu_k(x) \le
&\int_\Omega\int_{B_x \cap\rho(x,y) > 1}
|p(y)|^2 dV(y) d\mu_k(x) + \\
&C\int_\Omega\int_{B_x\cap\rho(x,y) <
1}
\frac{\sqrt{\beta_k(x)\beta_k(y)}}{(1+ k
\rho(x,y))^\gamma} |p(y)|^2 dV(y)\, d\mu_k(x)
\end{split}
\]
The first integral in the right hand side is bounded by $\int_\Omega
|p(y)|^2\, dV(y)$ since $\mu_k(\Omega)$ is bounded by hypothesis (it is
possible to cover $\Omega$ by balls $\{B(x_n,1/k)\}$ with controlled overlap).
In the second integral, observe that if $w\in B(x,1/k)$ then $\rho(w,x)\le 1/k$
and therefore
\[
\frac{\sqrt{\beta_k(x)\beta_k(y)}}{(1+ k
\rho(x,y))^\gamma} \lesssim
\frac 1{V(B(x,1/k))} \int_{B(x,1/k)} \frac{\sqrt{\beta_k(w)\beta_k(y)}}{(1+ k
\rho(w,y))^\gamma} dV(w).
\]
We plug this inequality in the second integral and we can bound it by
\[
C\int_\Omega |p(y)|^2 \int_{\rho(w,y) < 2}\frac{\sqrt{\beta_k(w)\beta_k(y)}}{(1+ k
\rho(w,y))^\gamma} \frac {\mu_k(B(w,1/k))}{V(B(w,1/k))}dV(w) dV(y).
\]
We use the hypothesis \eqref{geonec} and Lemma~\ref{lem3_weight} with $\alpha =
1/2$ to bound it finally by $C\int_\Omega |p(y)|^2dV(y)$.
\end{proof}
The weighted case in the unit ball is simpler.
\begin{thm}\label{Carleson en bolas}
Let $d\mu_a (x)=(1-|x|^2)^{a-\frac{1}{2}}dV(x)$ for $a\ge 0$ the weight in the unit ball $\B\subset \R^n.$
A sequence of measures $\{ \mu_k \}$ are Carleson for $(\P_k,\mu_a)$ if there is a constant
$C$ such that for all points $x\in \B$
\begin{equation} \mu_k(B(x,1/k)) \le C\; \mu_a(B(x,1/k)).
\end{equation}
\end{thm}
\begin{proof}
Supose $\{ \mu_k \}$ are Carleson. Then for any $x\in \B$
\begin{equation}
\int_{B(x,1/k)} |L_k^a(x,w)|^2\, d\mu_k(w) \le C \|L_k^a(x,\cdot) \|^2_\mu.
\end{equation}
By property (\ref{diagonal_weight}) in Theorem \ref{theorem_PX} and the estimate
$$K_k(\mu,x,x)\le \|L_k^a(x,\cdot) \|^2_\mu\le K_{2k}(\mu,x,x) ,$$
the result follows. The necessity follows exactly like in the unweighted case with the obvious changes.
\end{proof}
\subsection{Density condition}
In \cite[Theorem 4]{BerOrt} a necessary density condition for
sampling sequences for polynomials in convex domains was obtained. It states
the following:
\begin{thm} \label{densitysamp}
Let $\Omega$ be a smooth convex domain in $\R^n$, and let $\Lambda$ be a
sampling sequence. Then for any $\B(x,r) \subset \Omega$ the following
holds:
\[
\limsup_{k\to\infty} \frac{\#\Lambda_k \cap \B(x,r)}{\dim \mathcal P_k}\ge
\mu_{eq}(\B(x,r)).
\]
Here $\mu_{eq}$ is the equilibrium measure associated to $\Omega$.
\end{thm}
Let us see
how, with a similar technique, a corresponding density condition can be
obtained as well in the case of interpolating sequences.
\begin{thm} \label{density}
Let $\Omega$ be a smooth convex domain in $\R^n$, and let $\Lambda$ be an
interpolating sequence. Then for any $\B(x,r) \subset \Omega$ the following
holds:
\[
\limsup_{k\to\infty} \frac{\#\Lambda_k \cap \B(x,r)}{\dim \mathcal P_k}\le
\mu_{eq}(\B(x,r)).
\]
Here $\mu_{eq}$ is the equilibrium measure associated to $\Omega$.
\end{thm}
\begin{remark}
In the statements of Theorems~\ref{densitysamp} and \ref{density} we could
have replaced $\mathbb B(x, r)$ by any open set, in particular they could have been formulated with balls
$B(x,r)$ in the anisotropic metric.
\end{remark}
\begin{proof}
Let $F_k\subset \P_k$ be the subspace spanned by
\[
\kappa_\lambda(x)=K_k(\lambda,x)/\sqrt{\beta_k(\lambda)}\qquad \forall
\lambda \in \Lambda_k.
\]
Denote by $g_\lambda$ the dual (biorthogonal) basis to $\kappa_\lambda$ in
$F_k$. We
have clearly that
\begin{itemize}
\item We can span any function in $F_k$ in terms of $\kappa_\lambda$, thus:
\[
\sum_{\lambda\in \Lambda_k} \kappa_\lambda(x)g_\lambda(x)=\mathcal K_k(x,x),
\]
where $\mathcal K_k(x,y)$ is the reproducing
kernel of the subspace $F_k$.
\item The norm of $g_\lambda$ is uniformly bounded since $\kappa_\lambda$ was a
uniform Riesz sequence.
\item $g_\lambda(\lambda)=\sqrt{\beta_k(\lambda)}$. This is due to the
biorthogonality and the reproducing property.
\end{itemize}
We are going to prove that the measure $\sigma_k=\frac 1{\dim \mathcal
P_k}\sum_{\lambda\in
\Lambda_k}\delta_\lambda$,
and the measure $\nu_k=\frac 1{\dim \mathcal P_k} \mathcal K_k(x,x)d\mu(x)$
are very
close to each other. This are two positive measures that are not probability
measures but they have the same mass (equal to $\frac{\#\Lambda_k}{\dim
\mathcal P_k}\le 1$). Therefore, there is a way to quantify the closeness
through the Vaserstein $1$-distance. For an introduction to Vaserstein distance
see for instance \cite{Villani}. We want to prove that
$W(\sigma_k,\nu_k)\to 0$ because the Vaserstein distance metrizes the
weak-* topology.
In this case, it is known that $\mathcal K_k(x,x)\le K_k(x,x)$ and
$\frac 1{\dim \mathcal P_k}
\beta_k(x)\,
\to \mu_{eq}$ in the weak-* topology, where $\mu_{eq}$ is the normalized
equilibrium measure associated to $\Omega$ (see for instance \cite{BBN11}). Therefore,
$\limsup_k \sigma_k\le \mu_{eq}$.
In order to prove that $W(\sigma_k,\nu_k)\to 0$ we use a non positive
transport plan as in \cite{LevOrt}:
\[
\rho_k(x,y)=\frac 1{\dim \mathcal P_k} \sum_{\lambda\in \Lambda_k}
\delta_\lambda(y)
\times g_\lambda(x)\kappa_\lambda(x)\,d\mu(x)
\]
It has the right marginals, $\sigma_k$ and $\nu_k$
and we can estimate the integral
\[
W(\sigma_k,\nu_k)\le
\iint_{\Omega \times \Omega} |x-y|d|\rho_k|=O(1/\sqrt{k}).
\]
The only point that merits a clarification is that we need an inequality:
\[
\begin{split} \frac 1{\dim \mathcal P_k} \sum_{\lambda\in \Lambda_k}
\int_\Omega
|\lambda-x|^2
\frac{|K_k(\lambda,x)|^2}{ K_k(x,x)}\,d\mu(x)\le \\
\frac 1{\dim \mathcal P_k}
\iint_{\Omega \times
\Omega} |y-x|^2
|K_k(y,x)|^2\, d\mu(x)d\mu(y).
\end{split}
\]
This is problematic. We know that $\Lambda_k$ is an interpolating
sequence for the polynomials of degree $k$. Thus the normalized reproducing
kernels at $\lambda\in\Lambda_k$ form a Bessel sequence for $\P_k$ but the
inequality that we need is applied to $K_k(x,y)(y_i-x_i)$ for all $i=1,\ldots,
n$. That is to a
polynomial of degree $k+1$. We are going to show that if $\Lambda_k$ is an
interpolating sequence for the polynomials of degree $k$ it is also a Carleson
sequence for the polynomials of degree $k+1$.
Observe that since it is interpolating then it is uniformly separated, i.e.
$B(\lambda, \varepsilon/k)$ are disjoint. That means that in particular
$$\mu_{k} (B(z, 1/(k+1)) \lesssim V(B(z,1/(k+1)).$$ Thus $\mu_k$ is a
Carleson measure for $\P_{k+1}$.
Finally in \cite[Theorem~17]{BerOrt} it was proved that
\[
\frac 1{\dim \mathcal P_k}
\iint_{\Omega \times
\Omega} |y-x|^2
|K_k(y,x)|^2\, d\mu(x)d\mu(y) = O(1/k).
\]
\end{proof}
From the behaviour on the diagonal of the kernel \eqref{kernelball}
its easy to check that the kernel is both Bernstein-Markov (sub-exponential) and has moderate growth, see definitions in \cite{BerOrt}. From the characterization for sampling sequences
proved in \cite[Theorem 1]{BerOrt} and with the obvious changes in the proof of the previous theorem we deduce the following:
\begin{thm} \label{teo:density_ball}
Consider the space of polynomials $\mathcal P_k$ restricted to the ball $\B\subset \R^n$ with the measure $d\mu_a(x)=(1-|x|^2)^{a-\frac{1}{2}}dV.$
Let $\Lambda=\{\Lambda_k \}$ be a sequence sets of points in $\B.$
\begin{itemize}
\item If $\Lambda$ is a sampling sequence
\[
\liminf_{k\to\infty} \frac{\# ( \Lambda_k \cap \B(x,r) )}{\dim \mathcal P_k}\ge
\mu_{eq}(\B(x,r)).
\]
\item If $\Lambda$ is interpolating
\[
\limsup_{k\to\infty} \frac{\# ( \Lambda_k \cap \B(x,r) )}{\dim \mathcal P_k}\le
\mu_{eq}(\B(x,r)).
\]
\end{itemize}
\end{thm}
\begin{remark}
One can construct interpolation or sampling
sequences with density arbitrary close to the critical density with
sequences of points $\{\Lambda_k\}$ such that the corresponding Lagrange interpolating polynomials are uniformly bounded.
In particular de above inequalities are sharp, for a similar construction on the sphere see \cite{MOC10}.
\end{remark}
\subsection{Orthonormal basis of reproducing kernels}
Sampling and interpolation are somehow dual concepts. Sequences which are both sampling and interpolating (i.e. complete interpolating sequences)
are optimal in some sense because they are at the same time minimal sampling sequences and maximal interpolating sequences.
They will satisfy the equality in Theorem \ref{teo:density_ball}.
In general domains, to prove or disprove the existence of such sequences is a difficult problem \cite{OUbook}.
If
$\Lambda=\{ \Lambda_k \}$ is a complete interpolating sequence
the corresponding reproducing kernels $\{ \kappa_{k,\lambda} \}$ is a Riesz basis in the space of polynomials (uniformly in the degree).
An obvious example of complete interpolating sequences would be sequences
providing an orthonormal basis of reproducing kernels. In dimension 1, with the weight $(1-x^2)^{a-1/2},$
a basis of Gegenbauer polynomials $\{ G^{(a)}_j \}_{j=0,\dots , k}$ is orthogonal and the reproducing kernel in $\mathcal P_k$ evaluated at the zeros of the polynomial $G^{(a)}_{k+1}$
gives an orthogonal sequence. In our last result we prove that
for greater dimensions there are no orthogonal basis of $\mathcal P_k$ of reproducing kernels
with the measure $d\mu_a(x)=(1-|x|^2)^{a-1/2}dV(x).$
Our first goal is to show that sampling sequences are dense enough, Theorem \ref{noempty}. Recall that in the bulk (i.e. at a fixed positive distance from the boundary) the Euclidean
metric and the metric $\rho$ are equivalent. In our first result we prove that the right hand side of (\ref{def:sampling}) and the separation imply that
there are points of the sequence in any ball (of the bulk) of big enough radius.
\begin{prop} \label{prop_bound}
Let $d\mu_a (x)=(1-|x|^2)^{a-\frac{1}{2}}dV(x)$ for $a\ge 0$ the weight in the unit ball $\B\subset \R^n.$
Let $\Lambda_k\subset \B$ be a finite subset and $C,\epsilon>0$ be constants such that
\begin{equation}
\int_\B |P(x)|^2 d\mu_a(x) \le C \sum_{\lambda\in\Lambda_k} \frac{|P(\lambda)|^2}{K_{k}(\mu_a;\lambda,\lambda)},
\end{equation}
for all $P\in \mathcal{P}_k$
and
$$\inf_{\substack{\lambda,\lambda'\in \Lambda_k \\ \lambda\neq \lambda' } }\rho(\lambda,\lambda')\ge \frac{\epsilon}{k}.$$
Let $|x_0|=C_0<\frac{1}{4},$ $\epsilon<M$ and $k\ge 1$ be such that
$\Lambda_k \cap \mathbb{B}(x_0,M/k)=\emptyset.$ Then $M<A$ for a certain constant $A$ depending only on $C,\epsilon,n$ and $a.$
\end{prop}
\proof
By the construction of function $L_\ell^a (x,y),$ it is clear that for any $\ell\ge 0$
$$K_\ell (\mu_a; x,x)\le \int_{\B }L_\ell^a (x,y)^2 d\mu_a (y)\le K_{2\ell}(\mu_a; x,x).$$
Let $P(x)=L^a_{[k/2]}(x,x_0)\in \mathcal{P}_k.$ From the property above, the hypothesis and Proposition \ref{prop:Christoffel} we get
\begin{equation}
k^n \sim K_{[k/2]} (\mu_a; x_0,x_0)\le \int_{\B } P(y)^2 d\mu_a (y)\lesssim
\sum_{|\lambda-x_0|>M/k} \frac{|P(\lambda)|^2}{K_k(\mu_a ;\lambda,\lambda)}.
\end{equation}
From \cite[Lemma 11.3.6.]{DX13}, given $x\in \B$ and $0<r<\pi$
\begin{equation}
\mu_a (B(x,r))\sim r^n (\sqrt{1-|x|^2}+r)^{2a},
\end{equation}
and therefore
\begin{equation}
\mu_a(B(x,r))\sim
\begin{cases}
r^{n+2a}& \mbox{if}\;\; 1-|x|^2<r^2, \\
r^n (1-|x|^2)^a & \mbox{otherwise},
\end{cases}
\end{equation}
and
\begin{equation}\label{measure_ball}
\mu_a (B(x,r))\gtrsim
\begin{cases}
r^{n+2 a}& \mbox{if}\;\; |x|>\frac{1}{2}, \\
r^n & \mbox{otherwise}.
\end{cases}
\end{equation}
From (4) in Theorem \ref{theorem_PX}, the separation of the sequence, and the estimate \eqref{measure_ball} we get
\begin{equation}
\begin{split}
0 & <c \le \sum_{|\lambda-x_0|>M/k} \frac{1}{(1+[k/2]\rho(x_0,\lambda))^{2\gamma}}
\\
&
=
\sum_{|\lambda-x_0|>M/k} \frac{1}{ \mu_a (B(\lambda,\epsilon/2k))} \int_{B(\lambda,\epsilon/2k)} \frac{d \mu_a (x)}{(1+[k/2]\rho(x_0,\lambda))^{2\gamma}}
\\
&
\lesssim
\left[ \sum_{\frac{M}{k}<|\lambda-x_0|<\frac{1}{2}}+ \sum_{\frac{1}{2}<|\lambda-x_0|} \right] \frac{1}{ \mu_a (B(\lambda,\epsilon/2k))} \int_{B(\lambda,\epsilon/2k)}
\frac{d \mu_a (x)}{(1+ 2 k \rho(x_0,x))^{2 \gamma}}
\\
&
\lesssim
\left(\frac{k}{\epsilon}\right)^n \int_{\frac{M}{k}}^{\frac{3}{4}} \frac{r^{n-1}}{(k r)^{2\gamma}}dr +
\frac{k^{2a +n-2 \gamma}}{\epsilon^{2 a +n}} \mu_a (B(0,1/2)^c).
\end{split}
\end{equation}
Now, for $\gamma=n+a$ we get
$$0<c\le \frac{1}{k^{n+2a}}\left[ -\frac{1}{r^{n+2a}} \right]_{r=\frac{M}{k}}^{\frac{3}{4}}+ \frac{1}{k^n} ,$$
and then a uniform (i.e. independent of $k$) upper bound for $M <A=A(C,\epsilon,n,a).$
\qed
\begin{prop} \label{noempty}
Let $\Lambda=\{\Lambda_k\}$ be a separated sampling sequence for $\B\subset \R^n.$ Then there exist
$M_0,k_0>0$ such that for any $M>M_0$ and all $k\ge k_0$
$$\# \left( \Lambda_k \cap \mathbb{B}(0,M/k)\right)\sim M^n.$$
\end{prop}
\proof
Let $\epsilon>0$ be the constant from the separation, i.e.
$$\inf_{\substack{\lambda,\lambda'\in \Lambda_k \\ \lambda\neq \lambda' } }\rho(\lambda,\lambda')\ge \frac{\epsilon}{k}.$$
Assume that $M/k\le \frac{1}{2}.$ For $\lambda\in \Lambda_k \cap \mathbb{B}(0,M/k)$ we have
$V(\B(\lambda,\frac{\epsilon}{k}))\sim (\frac{\epsilon}{k})^n$ and therefore
\begin{equation}
\# \left( \Lambda_k \cap \mathbb{B}(0,M/k)\right) \left(\frac{\epsilon}{k}\right)^n \lesssim \left(\frac{M}{k}\right)^n.
\end{equation}
For the other inequality, take the constant $A$ (assume $A>\epsilon$) given in Proposition \ref{prop_bound} depending on the sampling and the separation constants of $\Lambda$ and $n.$ For $M>A$ and $k>0$ such that
$\mathbb{B}(0,\frac{M}{k})\subset \mathbb{B}(0,\frac{1}{4})$ one can find $N$ disjoint balls $\mathbb{B}(x_j,\frac{A}{k})$ for $j=1,\dots N$ included in $\mathbb{B}(0,M/k)$ and
such that
$$N V(\mathbb{B}(0,\frac{A}{k}))>\frac{1}{2} V(\mathbb{B}(0,\frac{M}{k})).$$
Observe that each ball $\mathbb{B}(x_j,\frac{A}{k})$ contains by Proposition \ref{prop_bound} at least one point from $\Lambda_k$ and therefore
$$\# \left( \Lambda_k \cap \mathbb{B}(0,M/k)\right)\ge N \gtrsim \left(\frac{M}{A}\right)^n.$$
\qed
We will use the following result from \cite{Fug01}.
\begin{thm} \label{Fuglede-theorem}
Let $\mathbb{B} \subset \mathbb{R}^{n},$ $n>1,$ be the unit ball. There do not exist infinite subsets
$\Lambda\subset \mathbb{R}^{n}$ such that the exponentials $e^{i\langle x, \lambda \rangle},$ $\lambda \in \Lambda,$ are pairwise
orthogonal in $L^{2}(\B ).$ Or, equivalently, there do not exist infinite subsets
$\Lambda\subset \R^{n}$ such that $|\lambda-\lambda'|$ is a zero of $J_{n/2},$ the Bessel function
of order $n/2,$ for all distinct $\lambda,\lambda'\in \Lambda.$
\end{thm}
Following ideas from \cite{Fug74} we can prove now our main result about orthogonal basis. A similar argument can be used on the sphere to study tight spherical designs.
\begin{thm} \label{noONbasis}
Let $\B\subset \R^n$ be the unit ball and $n>1.$ There is no sequence of finite sets $\Lambda=\{ \Lambda_k \}\subset \B$ such that the reproducing kernels
$\{K_k(\mu: x,\lambda) \}_{\lambda\in \Lambda_k}$ form an orthogonal basis of $\mathcal{P}_k$ with respect to the measure $d\mu_a=(1-|x|^2)^{a-\frac{1}{2}}dV$.
\end{thm}
\proof[Theorem \ref{noONbasis}]
The following result can be easily deduced from \cite[Theorem 1.7]{KL13}:
Given $\{u_k\}_k,\{v_k\}_k$ convergent sequences in $\R^n$ and
$u_k\to u, v_k\to v,$ when $k\to \infty.$ Then
$$\lim_{k\to \infty} \frac{K_k(\mu;\frac{u_k}{k},\frac{v_k}{k})}{K_k(\mu;0,0)}=\frac{J^*_{n/2}(|u-v|)}{J^*_{n/2}(0)}.$$
Let $\Lambda_k$ be such that $\{\kappa_{\lambda}\}_{\lambda \in \Lambda_k}$ is an orthonormal basis of
$\mathcal{P}_k$ with respect to the measure $d\mu_a=(1-|x|^2)^{a-\frac{1}{2}}dV$. Then
$$K_k(\mu;\lambda_{(k)},\lambda_{(k)}')=0,$$
for $\lambda_{(k)}\neq \lambda_{(k)}'\in \Lambda_k.$
We know that $\Lambda_k$ is uniformly separated for some $\epsilon>0$
$$\rho(\lambda_{(k)},\lambda_{(k)}')\ge \frac{\epsilon}{k}.$$
Then the sets $X_k=k(\Lambda_k\cap \mathbb{B}(0,1/2))\subset \R^n$ are uniformly separated
$$|\lambda- \lambda'|\gtrsim \epsilon,\;\;\lambda\neq \lambda'\in X,$$
and
$X_k$ converges weakly to some uniformly separated set $X\subset \R^n.$ The limit is not empty because
by Proposition \ref{noempty} for any $M>0,$
$$\# \left( \Lambda_k \cap \mathbb{B}(0,M/k)\right)\sim M^d.$$
Observe that this last result would be a direct consequence of the necessary density condition for complete interpolating sets if we could
take balls of radius $r/n$ for a fixed $r>0$ in the condition. Finally, we obtain an infinite set $X$ such that for $\lambda\neq \lambda'\in X$
$$J^*_{n/2}(|\lambda- \lambda'|)=0,$$
in contradiction with Theorem \ref{Fuglede-theorem}.
\qed
\begin{remark}
Note that the fact that the interpolating sequence $\{\Lambda_k\}$ is complete was
used only to guarantee that $\# \left( \Lambda_k \cap \mathbb{B}(0,M/k)\right)\sim M^d$. So, the above result could be extended to
sequences $\{\Lambda_k\}$ such that $\{\kappa_{k,\lambda}\}_{\lambda\in \Lambda_k}$ is orthonormal (but not necessarily a basis for $\P_k$) if $\Lambda_k \cap \mathbb{B}(0,M/k)$ contains enough points.
\end{remark} | 72,571 |
07 March 2017, 12:45 | Clarence Schmidt
Cuomo Holds First Public Meeting of His Cabinet in Two-and-a Half Years 02/28/2017 10:39 PM
The rash of antisemitic incidents that has hit the the U.S. is "disgusting" and "reprehensible", New York Governor Andrew Cuomo said Sunday at Yad Vashem during a whirlwind one-day unity trip to Israel's Cuomo said there have been some 100 acts of antisemitism in the USA in recent weeks, including in New York, something he said "violates every tenet of the New York State tradition". The governor made the comments as he paid his respects at Yad Vashem, the Jewish Holocaust memorial in Jerusalem.
Gov. Andrew Cuomo says a recent rash of anti-Semitic acts in the United States is "reprehensible" and his state will have no tolerance for them.
Cuomo left Saturday and will return from Israel Monday.
"The relationship between the Jewish community and the state of NY is built on mutual support and respect", Mr. Cuomo, a Democrat, said in remarks before a tour of Israel's Holocaust memorial. I will be here again.
YouTube launches its own live TV streaming service
Local news programming from network affiliates will also be included, according to Robert Kyncl, YouTube's chief business officer. Since its inception, YouTube in particular has had copyright issues with the TV and movies studios and music labels.
Cuomo was last in israel in 2014, also on a visit aimed at showing solidarity with israel, that time during Operation Protective Edge.
Following the lunch, the Governor announced the launch of the New York-Israel Commission, and the partnership between New York Genome Center and Technion-Israel Institute of Technology. "On behalf of the State of Israel, I would like to express our appreciation for your visit and for the clear and powerful message you have sent". And we are deeply touched by Vice President Pence who went and gave a hand - and a voice - in fixing the broken gravestone.
There have been more than 100 bomb threats against US Jewish organisations since the beginning of the year and three Jewish cemeteries have been vandalised, with some analysts blaming the politics of the Donald Trump era. "It is a sign of great hope and civil courage".
"We hope that the actions being taken by Governor Cuomo will serve as a precedent for other states nationwide", said Moishe Bane, President of the Orthodox Union. The visit also was meant to bolster economic ties between Israel and New York State..
97-Year-Old Twins Found Dead in RI Driveway
John Haley said the women, who were healthy and active, were outside for about 11 hours before they were found. The sisters were immediately transported to Rhode Island Hospital where they later died.
Both sides committed war crimes in Syrian conflict
Last month, the IS-affiliated militants captured several villages and a large town from Syrian rebels in the Daraa countryside. The findings are based on 291 interviews with victims and witnesses, as well as forensic evidence and satellite imagery.
DeVos Slammed as 'Totally Nuts' for Calling HBCUs 'Pioneers of School Choice'
In it, Betsy DeVos said HBCUs are "pioneers" of school choice , and I'm pretty sure the entire education community face-palmed when she did.
Snap Inc. set to price its IPO at $17 per share
Snapchat is expected to grow its US audience to 70.4 million by the end of 2017, according to a new report from eMarketer . Facebook's Instagram last August added Instagram Stories, a close clone of Snapchat Stories.
Google Pixel 2 CONFIRMED for later this year
If you're an iPhone user and if you have made use of Google's Allo app, you might have seen the assistant baked inside the app. However, the company announced earlier this week that smartphones running Android 6 or Android 7 would get Google Assistant.
Philadelphia Flyers: Mark Streit Traded to Tampa
But Filppula, who has a partial-no move clause, rejected a trade this week to Toronto, a bigger deal involving Brian Boyle . The Penguins shed salary just before the trade, sending Eric Fehr to the Toronto Maple Leafs , along with Steve Oleksy.
In2: Film - Win Merchandise from the new Wolverine movie - Logan
You know what I love about him is that he thinks out of the box, he seems to absolutely love what he is doing. Jackman also confessed that he got the idea of Logan at 4am in the night. "Kohli is awesome .
Samsung Galaxy S8 Press Render Surfaces Online
The phone pictured here is the smaller version of Samsung's upcoming Galaxy S8 flagship smartphone duo. The first was to eliminate a legacy port and free up internal space for other components.. | 237,642 |
Caroline Scott Harrison, the wife of U.S. President Benjamin Harrison, died in the fall of 1892, after a trip to the Adirondacks failed to cure her tuberculosis. Her death left the White House without a first lady. Harrison’s daughter, Mary Scott McKee, filled that role for the last few months of Harrison’s term (he lost his bid for re-election that November). In those days, presidential terms ended in March, so Mrs. McKee carried on as first lady for about five months.
She.)
The antics of the children at the White House and elsewhere were widely reported in the press. When the young boy was taken to have a look at Barnum’s circus in 1890, it made the papers. He was, in those days, the “most famous baby in the world.” A rather exciting incident occurred when “Baby McKee,” riding in a cart drawn by a pet goat, was carried off the White House grounds, and along Pennsylvania Avenue before being chased down by the chief executive himself. Poems about him were printed in the humor magazines, Puck and the Harvard Lampoon. It was even suggested that Baby McKee’s image should be featured in the logo for the Daughters of the American Revolution, an organization in which his grandmother was very active.
When young Ben was taken on a vacation to Cape May, New Jersey in 1889, a reporter noted that “During his absence, Washington will be involved in a ‘gob of gloom,’ and the newspaper correspondents will have comparatively nothing to do.” President Harrison jokingly told reporters that they should stop giving so much attention to his grandson, lest people think the boy’s influence was greater than his own. A member of the Senate complained of the pointless coverage the papers gave the lad, and offered to give a new hat to any reporter who could say he had never written such a piece. No hats were awarded.
After leaving office, Harrison went to live in Indianapolis, and his daughter and grandchildren went along to help him get settled in. Harrison’s son-in-law (Baby McKee’s father), James Robert McKee, began an important association with the General Electric Company in 1893, so that he needed to be in New York State much of the time. Though the parents of young Ben were in Manhattan a lot, Ben and his sister mostly lived in Saratoga Springs, a place familiar to their mother as a result of connections with the Walworth family.
The Saratogian recalled in 1921 that “Mrs. McKee is known to many Saratogians and especially to the summer colony,” and observed that her son was also well-known in town. Local friends of the McKees included George Foster Peabody, and the families of James Lee Scott and Senator Edgar T. Brackett. The McKees, while in Saratoga, usually stayed at a cottage owned by Patrick McDonald, located on Union Avenue, very near the racetrack. Other times, they inhabited a cottage on Circular Street, and the United States Hotel was another of their favorite haunts. The McKees, though probably mainly summer visitors, were listed as residents of Saratoga in the 1900 Census.
The former president made trips away from Indiana, and often one or both of his grandchildren went along. Though Harrison tried to stay out of the limelight, that often was not possible. While he was shopping with his daughter and grandson in New York City, the party were spotted when an onlooker shouted “Hey there! There goes Harrison!” and a crowd instantly formed. Harrison was also spotted in 1894 when he took Ben to see his first play in New York. Harrison passed the summer of 1894 at Old Forge, spending a lot of time with his grandchildren. The following summer he bought property on the Fulton Chain of Lakes, and had the rustic Berkeley Lodge built, where he could entertain the grand-kids.
While living in Saratoga, the children’s lives still made it into the pages of the newspapers, though the arrival on the national scene of “Baby Ruth,” a daughter of President Grover Cleveland, took a lot of the spotlight away from the McKee kids. In July 1895, it was reported that “Baby McKee and his sister, who are at the United States Hotel, may be seen pedaling [bicycles] up the avenues every day.”
That summer, it made the news when Ben contracted scarlet fever in Saratoga. The entire family, including the visiting ex-president, were quarantined for a while. Granddad visited again in November, after the boy’s sickness had passed. In May 1896, the Troy Daily Times told how Baby McKee had become ill in Saratoga after he “caught cold on the lawn after a heavy storm.” His parents, who had gone to New York, rushed back to the racing city when they learned that the illness was diagnosed as bronchial pneumonia.
In 1896, Mrs. McKee had a falling out with her father after he married the much younger Mary Dimmick, who was her mother’s niece. The Troy Daily Times ran an item on April 1st saying that once Mrs. McKee had finished packing up her belongings in Indianapolis, her family would be moving into the McDonald cottage in Saratoga. Though the paper said it was expected that Harrison and his soon-to-be bride would spend part of the summer at Saratoga, “and will probably be guests of the McKee family,” the father and daughter remained estranged. Presumably he had little, if any, contact with his grandchildren from then on.
In 1901, when the former president was struck by a sudden illness at his home in Indianapolis, Mrs. McKee did not proceed there right away. Both of her children were sick with measles in Saratoga, and she and her husband stayed there to watch over them. They did not make it to Indianapolis before Harrison had expired.
In 1904, Mrs. McKee spoke to a reporter in Indianapolis, explaining that her son “is not Baby McKee any longer.” He was now a “husky young man of 17” who was preparing to begin college. “In recent years,” said the article, “Benjamin Harrison McKee has not had a picture taken, and at his request his mother will not tell reporters anything about him.” According to his mom: “He thinks he got enough notoriety when a baby to last him for some time.”
Among the few media reports about the grown-up “Baby” were one from 1907, while he was a student at Yale, when he made a brief comment about an uncle who’d suffered a nervous breakdown, and some stories about a confrontation he’d had with a Chicago taxi driver in 1914. Several visits he made to Saratoga, to visit old friends, were noted.
Only brief announcements let the public know what he was up to. He worked for some overseas banks, but more importantly, volunteered as an ambulance driver in France, beginning in 1916, prior to America’s involvement in the world war. (There is also evidence that he became a member of the French Army for a time.) He married an English woman, Constance Magnett in 1917, and seems to have spent much of his time overseas until the 1940s. Perhaps in Europe he could escape the moniker given to him as an infant. His mother passed away in 1930, and news reports said her son was residing in Paris at that time.
In 1932, an item in the Boston Herald noted McKee’s service in the ambulance corps in France during the war, but observed that he was no longer in the media’s eye. As a baby, it said: “There were songs about him. Colors were named for him. Multitudes were interested in everything the papers printed about the boy, and they printed plenty.”
The 1940 Census shows McKee as a resident of Manhattan. In 1942, according to his draft registration card, his address was Berkeley Lodge, Greenwich, Connecticut. He probably was staying with his father, who, upset about his declining health, took his own life in October that year.
McKee made the news one last time, in 1958, when his death in Nice, France was reported. Papers gave his residence as Field Point Park, near Greenwich, Connecticut. | 76,032 |
Hi, I'm trying to get answer to the question
Suppose a homogeneous array with 8 rows and 6 columns is stored in column major order starting at address 20 (base ten). If each entry in the array requires only one memory cell, what is the address of the entry in the third row and fourth column? Whatif each entry requires two memory cells? To complete this Assignment: Submit an answer to the posed problem and address the following: Identify the address of the entry in the third row and fourth column of the array, in the given problem. Identify the addressof the entry in the third row and fourth column of the array, in the given problem if each entry requires two memory cells. Explain how you determined your answers..
Please help me understand how you came to this answer would you mind sketching a hypothetical array, in the given problem, to help visualise it?
Yes, I have. version 2013 | 17,040 |
\begin{document}
\begin{abstract} We prove special decay properties of solutions to the initial value problem associated to the $k$-generalized Korteweg-de Vries equation.
These are related with persistence properties of the solution flow in weighted Sobolev spaces and with sharp unique
continuation properties of solutions to this equation. As application of our method we also obtain results concerning the decay behavior of perturbations of the traveling wave solutions as well as results for solutions corresponding to special data.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\section{Introduction}
In this work we shall study special decay properties of real solutions to the initial value problem (IVP)
associated to the $k$-generalized Korteweg-de Vries ($k$-gKdV) equation
\begin{equation}
\label{kgKdV}
\begin{cases}
\partial_t u +\partial_x^3u +u^k\partial_x u = 0,\;\;\;\;\;\; x\in\rr,\;\;k\in\mathbb Z^+,\\
u(x,0) = u_0(x).
\end{cases}
\end{equation}
These decay properties of the solution $u(t)$ will be measured in appropriate weighted
$L^2(w dx)$-spaces.
First, we shall be concerned with asymmetric (increasing) weights, for which the result
will be restricted to forward times $t>0$. In this regard we find the result in \cite{Ka} for the KdV equation, $k=1$ in \eqref{kgKdV}, in the space $L^2(e^{\beta x}dx),\,\beta>0$. There it was shown that the persistence property holds for $L^2$-solutions in $L^2(e^{\beta x}dx),\,\beta>0$, for $t>0$ (persistence property in the function space $X$ means that the solution $u(\cdot)$ describes a continuous curve on $X$, $u\in C([0,T]:X)$). Moreover, formally in this space the operator $\partial_t+\partial_x^3$ becomes $\partial_t+(\partial_x-\beta)^3$ so the solutions of the equation exhibit a parabolic behavior. More precisely, the following result for the KdV equation was proven in \cite{Ka} (Theorem 11.1 and Theorem 12.1)
\begin{TA} \label{Ka}
Let $ u\in C([0,\infty)\,:\,H^2(\rr))$ be a solution of the IVP \eqref{kgKdV} with $k=1$ and
\begin{equation}
\label{1.3}
u_0\in H^2(\rr)\cap L^2(e^{\beta x}dx),\;\;\;\;\text{for some}\;\;\beta>0,
\end{equation}
then
\begin{equation}
\label{1.4}
e^{\beta x}u\in C([0,\infty)\,:\,L^2(\rr))\cap C((0,\infty)\,:\,H^{\infty}(\rr)),
\end{equation}
with
\begin{equation}
\label{1.5}
\|u(t)\|_2=\|u_0\|_2,\;\;\;\;\;\;\;\|u(t)-u_0\|_{-3,2}\leq Kt,\;\;\;\;\;\;t>0,
\end{equation}
\begin{equation}
\label{1.6}
\|e^{\beta x}u(t)\|_2\leq e^{Kt}\,\| e^{\beta x}u_0\|_2,\;\;\;\;\;t>0,
\end{equation}
and
\begin{equation}
\label{1.7}
\int_0^{\infty} \,e^{-Kt}\,\| e^{\beta x}\partial_xu(t)\|_2^2dt \leq \frac{1}{4\beta} \|e^{\beta x}u_0\|_2^2,
\end{equation}
where $K=K(\beta,\|u_0\|_2)$.
Moreover, the map data-solution $u_0\to u(t)$ is continuous from $L^2(\rr)\cap L^2(e^{\beta x}dx)$ to $C([0,T]\,:\,L^2(e^{\beta x}dx))$, for any $T>0$.
\end{TA}
On a similar regard, in \cite{EKPV06} a unique continuation result was established. This gives an upper bound for the possible space decay of solutions to the IVP \eqref{kgKdV}:
\begin{TB} \label{EKPV06}
There exists $c_0>0$ such that for any pair
$$
u_1,\,u_2\in C([0,1]:H^4(\rr)\cap L^2(|x|^2dx))
$$
of solutions of \eqref{kgKdV}, if
\begin{equation}
\label{3:2}
u_1(\cdot,0)-u_2(\cdot,0),\,\;\, u_1(\cdot,1)-u_2(\cdot,1)\in L^2(e^{c_0x_{+}^{3/2}}dx),
\end{equation} then $u_1\equiv u_2$.
\end{TB}
Above we used the notation: $ x_{+}=\max\{x;\,0\}$. Similarly, we will use later on $ x_{-}=\max\{-x;\,0\}$
The power $3/2$ in the exponent in \eqref{3:2} reflects the asymptotic behavior of the Airy function.
The solution
of the initial value problem (IVP)
\begin{equation}
\begin{aligned}
\begin{cases}
\partial_t v + \partial_x^3 v=0,\\
v(x,0)=v_0(x),
\end{cases}
\end{aligned}
\end{equation}
is given by the group $\{U(t)\,:\,t\in \rr\}$
$$
U(t)v_0(x)=\frac{1}{\root{3}\of{3t}}\,Ai\left(\frac{\cdot}{\root {3}\of{3t}}\right)\ast v_0(x),
$$
where
$$
Ai(x)=c\,\int_{-\infty}^{\infty}\,e^{ ix\xi+i \xi^3/3 }\,d\xi,
$$
is the Airy function which satisfies the estimate
$$
|Ai(x)|\leq c\,\frac{e^{-c x_{+}^{3/2}}}{(1+x_{-})^{1/4}}.
$$
Observe that Theorem B gives a restriction on the possible decay
of a non-trivial solution of
\eqref{kgKdV}
at two different times. More precisely,
taking $u_2\equiv 0$ one has that if $u_1(t)$ is a solution of the IVP \eqref{kgKdV} such that
\begin{equation}\label{a1}
|u_1(x,t)|\leq e^{-a_0x_{+}^{3/2}}\;\;\text{at}\;\;t=0,\,1,\;\;\text{for}\;\;a_0>>1, \;\text{then\hskip10pt }u_1\equiv 0.
\end{equation}
Our first theorem shows that the above result is close to be optimal. By rescaling, the persistence property can not hold in the space $L^2(e^{a_0 x_{+}^{3/2}}dx)$ in an arbitrary large time interval. However, it does it with a factor $a(t)$ in front of the exponential term $x_{+}^{3/2}$ which measures how this exponential property decreases with time.
To simplify the exposition, we shall first state our result in the case of the KdV, i.e. $k=1$ in \eqref{kgKdV} :
\begin{theorem} \label{A1}
Let $a_0$ be a positive constant. For any given data
\begin{equation}
\label{0.1}
u_0\in L^2(\rr)\cap L^2(e^{a_0x_{+}^{3/2}}dx),
\end{equation}
the unique solution of the IVP \eqref{kgKdV} provided by Theorem C below satisfies that for any $T>0$
\begin{equation}
\label{0.2}
\sup_{t\in [0,T]}\,\int_{-\infty}^{\infty} e^{a(t)x_{+}^{3/2}}|u(x,t)|^2dx \leq C^*=C^*( a_0, \|u_0\|_2, \|e^{a_0x_{+}^{3/2}/2}u_0\|_2, T),
\end{equation}
with
\begin{equation}
\label{0.3}
a(t)=\frac{a_0}{(1+27 a_0^2t/4)^{1/2}}.
\end{equation}
\end{theorem}
This result extends to the difference of two appropriate solutions of the IVP \eqref{kgKdV} with $k=1$.
\begin{theorem} \label{B1}
Let $a_0$ be a positive constant. Let $u(t),\,v(t)$ be solutions of the IVP \eqref{kgKdV} with $k=1$ such that
\begin{equation}
\label{4.1}
\begin{aligned}
& u\in C([0,\infty)\,:\,H^1(\rr)\cap L^2(|x|dx)) \cap\dots, \\
& v\in C([0,\infty)\,:\,H^1(\rr)) \cap\dots
\end{aligned}
\end{equation}
If
\begin{equation}
\label{4.2}
\int_{-\infty}^{\infty} e^{a_0x_{+}^{3/2}}|u(x,0)-v(x,0)|^2dx <\infty,
\end{equation}
then for any $T>0$
\begin{equation}
\label{4.3}
\sup_{t\in[0,T]}\,\int_{-\infty}^{\infty} e^{a(t)x_{+}^{3/2}}|u(x,t)-v(x,t)|^2dx \leq C^*,
\end{equation}
with
\begin{equation}
\label{4.41}
C^*=C^*( a_0, \|u_0\|_{1,2}, \|v_0\|_{1,2}, \| |x|^{1/2}u_0\|_2, \|u_0-v_0\|_{1,2}, \|e^{a_0x_{+}^{3/2}/2}(u_0-v_0)\|_2, T),
\end{equation}
and
\begin{equation}
\label{4.42}
a(t)=\frac{a_0}{(1+27 a_0^2t/4)^{1/2}}.
\end{equation}
\end{theorem}
The results in Theorem \ref{A1} and Theorem \ref{B1} apply to other powers $k$ in the IVP \eqref{kgKdV}:
\begin{theorem} \label{BB1}
Let $a_0$ be a positive constant. For any given data
$$
u_0\in H^{s_k}(\rr)\cap L^2(e^{a_0x_{+}^{3/2}})
$$
with
$\,s_2=1/4$ if $k=2$, $s_3=0$ if $k=3$, and $s_k>(k-4)/2k,\;k\geq 4$, then the unique solution of the IVP \eqref{kgKdV} provided
by Theorem D below satisfies \eqref{0.2} with $a(t)$ as in \eqref{0.3} for any $T>0$ if $k=2,3$ and for $T=T(\|u_0\|_{s_k,2})>0$ for $k\geq 4$.
\end{theorem}
Similarly, for Theorem \ref{B1}.
Remarks:
(a) Following the argument in the proof of Theorem 1.4 in \cite{EKPV06} one has that given $a_0>0,\,\epsilon>0$, there exist $u_0\in \mathbb S(\rr)$, $c_1,\,c_2>0$ and $\Delta T>0$ such that the corresponding solution $u(x,t)$ of the IVP \eqref{kgKdV} with $k=1$
satisfies
$$
c_1\,e^{-(a_0+\epsilon)x^{3/2}}\leq u(x,t)\leq c_2\,e^{-(a_0-\epsilon)x^{3/2}},\;\;\;\;\;\,x>>1,\;\;t\in [0,\Delta T].
$$
(b) For different powers $k$ in \eqref{kgKdV} one has that the same statement of Theorem \ref{A1} is valid except for
the last part concerning the continuity of the map data-solution $u_0\to u(t)$ which will be continuous
from $H^{s_k}(\rr)\cap L^2(e^{\beta x}dx)$ to $C([0,T]: L^2(e^{\beta x}dx))$ with $s_k$ and $T$ as in the statement of Theorem \ref{BB1}.
(c) For other results involving asymmetric weights of a polynomial type we refer to \cite{KrFa} and \cite{GiTs}.
(d) Consider the 1-D semi-linear Schr\"odinger equation
\begin{equation}
\label{NLS}
\partial_t v =i( \partial_x^2 v + F(v,\overline v)),
\end{equation}
with $F:\cc^2\to \cc$, $F\in C^{2}$ and $F(0)=\partial_uF(0)=\partial_{\bar
u}F(0)=0$.
As far as we are aware it is unknown whether or not there exist non-trivial solutions $v(t)$ of \eqref{NLS} satisfying
$$
v(t)\in L^2(e^{a_0x_+^{1+\epsilon}}\,)\;\;\;\;\;\,t\in[0,T],\;\;\;\;\;\;\;\text{for some}\;\;T>0,\;\,\epsilon>0.
$$
Next, we consider weighted spaces with symmetric weight of the form
$$
L^2(\langle x\rangle^bdx)=L^2((1+x^2)^{b/2}dx)
$$
for which persistent properties should hold regardless of the time direction considered, i.e. forward $t>0$ or backward $t<0$.
In this setting our second result establishes that for a solution of the IVP \eqref{kgKdV} to satisfy the persistent property in $L^2(\langle x\rangle^bdx)$ it needs to have a similar property in an appropriate Sobolev space $H^s(\rr)$, i.e. decay in $L^2$ can only hold if $u(t)$ is regular enough in $L^2$ :
\begin{theorem} \label{C1}
Let $u\in C(\rr : L^2(\rr))$ be the global solution of the IVP \eqref{kgKdV} with $k=1$ provided by Theorem C
below. If there exist $\alpha>0$ and two different times $t_0,\,t_1\in\rr$ such that
\begin{equation}
\label{00.1}
|x|^{\alpha} u(x,t_0),\;|x|^{\alpha}u(x,t_1)\in L^2(\rr),
\end{equation}
then
\begin{equation}
\label{00.2}
u\in C(\rr : H^{2\alpha}(\rr)).
\end{equation}
\end{theorem}
\begin{theorem} \label{D1}
Let $u,\,v\in C(\rr : H^1(\rr))$ be global solutions of the IVP \eqref{kgKdV} with $k=1$ provided by Theorem C.
If there exist $\alpha>1/2$ and two different times $t_0,\,t_1\in\rr$ such that
\begin{equation}
\label{000.1}
|x|^{\alpha} (u(x,t_0)-v(x,t_0)),\;|x|^{\alpha}(u(x,t_1)-v(x,t_1))\in L^2(\rr),
\end{equation}
then
\begin{equation}
\label{000.2}
u-v\in C(\rr : H^{2\alpha}(\rr)).
\end{equation}
\end{theorem}
Combining Theorem \ref{A1} and Theorem \ref{C1} and taking an initial data $u_0\in L^2(\rr)$ with compact support such that
$$
u_0\notin H^s(\rr),\;\;\;\forall \,s>0,
$$
one gets:
\begin{corollary}
\label{corollary1}
There exists a solution
\begin{equation}
\label{cc1}
u\in C(\rr : L^2(\rr)) \cap\dots .
\end{equation}
of the IVP \eqref{kgKdV} with $k=1$ provided by Theorem C such that
\begin{equation}
\label{cc2}
u(\cdot,0)=u_0(\cdot)
\end{equation}
has compact support and
\begin{equation}
\label{cc3}
\begin{aligned}
&u(\cdot,t)\in C^{\infty}(\rr),\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\forall \,t \neq 0,\\
&u(\cdot,t)\notin L^2(|x|^{\epsilon}dx),\;\;\;\;\;\;\;\forall \,\epsilon>0,\;\;\forall \,t \neq 0,\\
&u(t)\in L^2(e^{a(t)x_{+}^{3/2}/2}dx),\;\;\;\;\;\;\;\,\forall \,t>0,\\
&u(t)\in L^2({e^{a(t)x_{-}^{3/2}/2}dx}),\;\;\;\;\;\;\;\,\forall \,t<0,
\end{aligned}
\end{equation}
\end{corollary}
with
\begin{equation}
\label{cc4}
a(t)=\frac{a_0}{(1+27 a_0^2|t|/4)^{1/2}}.
\end{equation}
The results in Theorem \ref{C1} and Corollary \ref{corollary1} extend to the other powers $k=2,3,4...$ in \eqref{kgKdV} with the appropriate modifications, accordingly to the value of $k$, on the regularity
and on the length of the time interval $[0,T]$ as it was described in the statement of Theorem \ref{BB1}.
The result in Theorem \ref{D1} holds for any power $k$ in \eqref{kgKdV} and any pair of solutions
$$
u,\;v\in C([-T,T]:H^1(\rr))
$$
for any $T>0$ if $k=2,3$ and for $T=T(\|u_0\|_{1,2}, \|v_0\|_{1,2})$ for $k\geq 4$.
As a second consequence of our results above one has that there exist compact perturbations
of the travelling wave solution with speed $c$
\begin{equation}
\label{solsol}
u_{k,c}(x,t)=\phi_{k,c}(x-ct),
\end{equation}
with
\begin{equation}
\label{soliton}
\phi_{k,c}(x)=(c_k \,c\,{\rm sech}^2(k\sqrt{c}\,x/2))^{1/k}
\end{equation}
for the equation in \eqref{kgKdV} which destroy its exponential decay character:
\begin{corollary}
\label{corollary2}
For a given data of the form
\begin{equation}
\label{c5}
u_0(x)=\phi_{k,c}(x)+v_0(x),
\end{equation}
with $v_0\in H^1(\rr)$ compactly supported such that $v_0\notin H^{1+\epsilon}(\rr)$ for any $\epsilon>0$,
then the corresponding solution of the IVP provided by Theorem D
\begin{equation}
\label{c6}
u\in C([-T,T] : H^1(\rr)) \cap\dots
\end{equation}
of the IVP \eqref{kgKdV} satisfies that
\begin{equation}
\label{c7}
u(\cdot,t)\notin L^2(|x|^{1+\epsilon}dx),\;\;\;\;\;\;\;\forall \,\epsilon>0,\;\;\forall \,t \in [-T,T]-\{0\}.
\end{equation}
\end{corollary}
Remarks:
(a) As in Corollary \ref{corollary1}, for $t>0$ the loss of decay is in the left hand side of $\rr$, and for $t<0$ in the right hand side of $\rr$.
(b) Combining the results in Theorem \ref{C1}, and its extension for all the equations in \eqref{kgKdV} commented above, with those found in \cite{NP12} one can also conclude that for $k=2,4,5,...$
$$
|x|^{\alpha}u(\cdot, t)\in L^2(\rr),\;\;\;\;\;\;\;\forall \,t\in[-T,T],
$$
and that for $k=1,3$ for any $\epsilon>0$
$$
|x|^{\alpha-\epsilon}u(\cdot, t)\in L^2(\rr),\;\;\,t\in[-T,T]-[t_0,t_1],\;\;\;\;\;|x|^{\alpha}u(\cdot, t)\in L^2(\rr),\;\;\;t\in[t_0,t_1].
$$
The main difference between the cases $k = 2, 4,5,...$ and $k = 1, 3$ is that for the later
the best available well-posedness results require the use of the spaces $X_{s,b}$ defined in \eqref{xsb}, which makes fractional weights difficult to handle.
(c) The equivalent of Theorem \ref{C1} for the semi-linear Schr\"odinger equation \eqref{NLS} in all dimension $n$ was obtained in \cite{NP09}.
We need to recall some results concerning the well-posedness (local and global) of the IVP \eqref{kgKdV}. First, we remember the definition of the space $X_{s,b}$ introduced in the context of dispersive equations in \cite{Bo}.
For $s, b\in\rr$, $X_{s,b}$ denotes the completion of the Schwartz space $\mathcal S(\rr^2)$ with respect to the norm
\begin{equation}
\label{xsb}
\|F\|_{X_{s,b}}=(\int^{\infty}_{-\infty} \int^{\infty}_{-\infty} (1+|\tau-\xi^3|)^{2b}(1+|\xi|)^{2s}|\widehat{F}(\xi,\tau)|^2d\xi d\tau)^{1/2}.
\end{equation}
The following result was established in \cite{Bo}, see also \cite{KPV96}:
\begin{TC} \label{Bo}
There exists $b\in(1/2,1)$ such that for any $u_0\in L^2(\rr)$ there exists a unique solution $u(t)$ of the IVP \eqref{kgKdV} with $k=1$ satisfying
\begin{equation}
\label{1.09}
u \in C([0,T]:L^2(\rr)),\;\;\;\;\;\text{for any}\;\;T>0,
\end{equation}
\begin{equation}
\label{1.10}
u\in X_{0,b},\;\;\;\partial_x(u^2)\in X_{0,b-1},\;\;\;\partial_t u\in X_{-3,b-1}.
\end{equation}
Moreover, the map data-solution from $L^2(\rr)$ into the class defined in \eqref{1.09}-\eqref{1.10} is Lipschitz for any $T>0$.
\end{TC} | 43,415 |
Hotel Narvil Conference & SPA
It is an extraordinary spot located in the forest and at
the banks of the Narew River, only 40 minutes from the centre of Warsaw.
It is a combination of an elegant hotel, a modern conference centre and an exclusive SPA near Warsaw. The hotel was designed in full harmony with nature, which served as an inspiration for its name and interiors. Discover the discreet, tranquil atmosphere of this area existing in harmony with an extensive range of attractions and opportunities for relaxation.
Modern rooms and top-standard apartments inspired by nature create the unique oasis of serenity and a perfect place to rest.
Achieve internal harmony thanks to a combination of proximity to nature and relaxation. Find respite at this luxurious SPA hotel near Warsaw — Hotel Narvil. Enjoy soothing body and facial care rituals, specialist and relaxing massages, calming baths, as well as a number of special treatments.
Exquisite cuisine, harmony with nature and return to natural qualities and flavours will satisfy the taste of even the most demanding gourmets.
When developing the Aruana Restaurant, we were driven by harmony with nature and return to natural qualities and flavours.
3300 m2 of space
1500 conference participants
33 conference rooms
Thirty-three comfortable conference rooms are equipped with modern technological solutions. There is also a possibility to combine and divide individual rooms, which allows for even more flexible adjustment of space inside this hotel near Warsaw to the requirements of a given event.
An experienced team of professionals will present you with an offer adopted to all needs of the specific event.
Beautiful green surroundings and perfect location make for an ideal place for business events, as well as spending free time with family and friends. Discover the possibilities offered by Narvil Conference & Spa Hotel, located near Warsaw.
Złap oddech i zwolnij tempo. Zrelaksuj się w sposób holistyczny. W towarzystwie doskonałej kuchni, zachwycającej przyrody Mazowsza, różnorodnej strefy Wellness, zabiegów kojących ciało i duszę oraz niezwykłej architektury. Narvil to hotel pod Warszawą, który dba o wszystkie potrzeby Gości, a szeroka oferta Spa zabierze ich w nowy wymiar odprężenia. | 230,668 |
The Drobo is currently back online and available for use.
The Grad Lab's Documentation Space, Sound Tracking and Sound Control will remain offline as we work to finish repairs and updates in these spaces. Please continue to watch your MICA Mail and the front page of the Grad Lab Website for updates. | 228,160 |
Details
- Type:
Bug
- Status: Closed
- Priority:
Blocker
- Resolution: Fixed
- Affects Version/s: 9.1.2
-
- Component/s: Old Core - PDF export
- Labels:
- Difficulty:Unknown
- Documentation:N/A
- Documentation in Release Notes:N/A
- Similar issues:
Description
I'm using the xwiki v9.1.2 and I found that when exporting a page to PDF, the non-English characters will be replaced by "#".
I spent a lot of time searching in jira.xwiki.org and found that there's similar issues like
XWIKI-4724 and XWIKI-1609. However, non of the solutions work on me.
Finally I found this commit log in github - and I manually create a fop-config.xml file in xwiki\WEB-INF\classes folder, with the "<auto-detect/>" not commented out. Then the problem solved.
Attachments
Issue Links
- is duplicated by
XWIKI-14174 PDF Export custom fonts
- Closed | 366,471 |
Our nest is not quite empty, but with Son-Three having set up housekeeping across town, it certainly is emptier. The funny thing is that I think I've actually been seeing a lot more of him since he moved out than I had been before he left. We had scheduling issues when he was still at home since he worked the overnights and I worked second shift, he was rarely out of bed before I left for work and gone when I got home.
Now, he has been over mid-afternoon several times to do laundry and, of course, raid the fridge. He was over here yesterday, poking around the kitchen and called out to me, "Hey, Mom, how come there's so much more food in the house now that I'm gone?"
Hmmmm, I wonder.
I Am Not a Wimp
55 minutes ago
My husband just said tonight, "In a few years, these kids will be grown," but your post gives me so much hope that they won't actually be gone!
Here's some more hope, Michelle - our son got married last October and my husband were enjoying the empty nest. Next thing we know our married son and wife have temporarily moved in with us! I had always heard they come back - just didn't think that soon ;)
Yes, and , Michelle, our 24 year old daughter went off to college but returned to go to the local college. For several years she lived with friends in an apt about half hour from here but then got so stretched for cash, that she moved into the studio cottage that we have on our property. But, really? that meant she moved back with us - for about 18 months. Overall, it was sweet to have her here again -
She is now living again about 30 minutes away and her visits are frequent and, yes, about using the washing machine and raiding the fridge! oh, and playing with my ipad - which she loves.
I look forward to all this- especially the 'more food in the fridge'! :)
I'm both looking forward to and dreading these years...
When my oldest is gone, I'm sure it'll take me months to stop buying milk by the cow. | 387,132 |
TITLE: $\inf\limits_{x \in [a,b]}|f^{\prime} (x)| \geq \frac{1}{b-a}$
QUESTION [3 upvotes]: Let $f \in C^1([0,1])$ be a non-decrease function such that $f(0)=0, f(1)=1$.
Does there exist $[a,b] \subset [0,1]$ such that $\inf\limits_{x \in [a,b]}|f^{\prime} (x)| \geq \frac{1}{b-a}$?
REPLY [1 votes]: Depends on $f$.
$f(x)=x^2$ is a counter example.
If $a>1/2$, the infimum is at most 2 and $b-a<1/2$ so the inequality is false.
If $a=0$, the infimum is zero so the inequality is definitely false.
If $0<a\le 1/2$, the infimum is at most 1 whereas $b-a<1$, so inequality is again false.
—
Edit: in the mean time rtybase found the key: this is true only if $f$ stays constant equal zero, then increases straight to one, and then remains constant at 1. In this case $[a,b]$ will be exactly the interval in which the function grows linearly from 0 to 1. I let you work the details from rtybase’s comment until you prove this claim. | 143,559 |
TITLE: How to prove via induction.
QUESTION [0 upvotes]: How would you prove (using induction) that:
(If $f(1) = 1996$)
$f(n) = \frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdot ... \cdot \frac{(n-1)^2}{n^{2} - 1} \cdot f(1)$ given that $f(1) + f(2) + f(3) + f(4) + ... + f(n)= n^2 f(n)$ and that $f$ is defined for all positive integers $n>1$?
REPLY [0 votes]: Given that:
$f(1) = 1996$ and
$f(1) + f(2) + f(3) + f(4) + ... + f(n)= n^2 f(n)$
Prove by induction:
$f(n) = \frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdot ... \cdot \frac{(n-1)^2}{n^{2} - 1} \cdot f(1)$
base case $n=2$
$1996 + f(2) = 2^2 f(2)\\
(2^2 - 1) f(2) = 1996\\
f(2) = \frac 1{2^2+1} f(2)$
Inductive hypothesis:
Assume
$f(n) = \frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdot ... \cdot \frac{(n-1)^2}{n^{2} - 1} \cdot f(1)$
We must show that
$f(n+1) = \frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdot ... \cdot \frac{(n-1)^2}{n^{2} - 1}\cdot \frac{(n)^2}{(n+1)^{2} - 1} \cdot f(1)$
Based on the inductive hypothesis.
$f(1) + f(2) + f(3) + f(4) + ... + f(n) + f(n+1)= (n+1)^2 f(n+1)\\
f(1) + f(2) + f(3) + f(4) + ... + f(n)= ((n+1)^2 -1) f(n+1)\\
n^2 f(n)= ((n+1)^2 -1) f(n+1)\\
n^2\frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdot ... \cdot \frac{(n-1)^2}{n^{2} - 1} \cdot f(1)=((n+1)^2 -1) f(n+1)\\
f(n+1) = \frac{(n)^2}{(n+1)^{2} - 1} \cdot\frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdot ... \cdot \frac{(n-1)^2}{n^{2} - 1} \cdot f(1)$
QED
REPLY [0 votes]: Induction base holds, let's prove induction step.
Assume that holds $$f(n) = \frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdots \frac{(n-1)^2}{n^{2} - 1} \cdot f(1).$$
We have by our formula that $$(n+1)^2f(n+1)=f(1)+\dots+f(n)+f(n+1)=n^2f(n)+f(n+1),$$ from where we get
$$((n+1)^2-1)f(n+1)=n^2f(n)$$ and $$f(n+1)=\frac{n^2}{(n+1)^2-1}f(n).$$
By using our assumption, we get $$f(n+1)=\frac{1}{(2^{2} - 1)} \cdot \frac{2^2}{(3^{2} - 1)} \cdot \frac{3^2}{(4^2 - 1)} \cdots\frac{(n-1)^2}{n^{2} - 1}\frac{(n+1-1)^2}{(n+1)^2-1} \cdot f(1).$$
REPLY [0 votes]: Suppose it is true for $n \in \Bbb N$. Then
$$n^2f(n)+f(n+1)=f(1)+...+f(n)+f(n+1)=(n+1)^2f(n+1) \iff f(n+1) = \frac{n^2}{(n+1)^2-1}f(n)$$ | 98,606 |
\begin{document}
\maketitle
\begin{abstract}
Traditional linear subspace reduced order models (LS-ROMs) are able to
accelerate physical simulations, in which the intrinsic solution space falls
into a subspace with a small dimension, i.e., the solution space has a small
Kolmogorov $n$-width. However, for physical phenomena not of this type, such
as advection-dominated flow phenomena, a low-dimensional linear subspace
poorly approximates the solution. To address cases such as these, we have
developed an efficient nonlinear manifold ROM (NM-ROM), which can better
approximate high-fidelity model solutions with a smaller latent space
dimension than the LS-ROMs. Our method takes advantage of the existing
numerical methods that are used to solve the corresponding full order models
(FOMs). The efficiency is achieved by developing a hyper-reduction technique
in the context of the NM-ROM. Numerical results show that neural networks can
learn a more efficient latent space representation on advection-dominated data
from 2D Burgers' equations with a high Reynolds number. A speed-up of up to
$11.7$ for 2D Burgers' equations is achieved with an appropriate treatment of
the nonlinear terms through a hyper-reduction technique.
\end{abstract}
\section{Introduction}\label{sec:intro}
Physical simulations are influencing developments in science, engineering, and
technology more rapidly than ever before. However, high-fidelity, forward
physical simulations are computationally expensive and, thus, make intractable
many decision-making applications, such as design optimization, inverse
problems, optimal controls, and uncertainty quantification. These applications
require many forward simulations to explore the parameter space in the outer
loop. To compensate for the computational expense issue, many surrogate models
have been developed: from simply using interpolation schemes for specific
quantity of interests to physics-informed surrogate models. This paper focuses
on the latter because a physics-informed surrogate model is more robust in
predicting physical solutions than the simple interpolation schemes.
Among many types of physics-informed surrogate models, the projection-based
linear subspace reduced order models (LS-ROMs) take advantage of both the known
governing equation and data with linear subspace solution representation
\cite{benner2015survey}. Although LS-ROMs have been successfully applied to many
forward physical problems \cite{ghasemi2015localized, jiang2019implementation,
yang2016fast, yang2017efficient, zhao2014pod, cstefuanescu2013pod,
mordhorst2017pod, dimitriu2013application, antil2012reduced} and partial
differential equation(PDE)-constrained optimization problems
\cite{amsallem2015design, choi2020gradient, choi2019accelerating, fu2018pod},
the linear subspace solution representation suffers from not being able to
represent certain physical simulation solutions with a small basis dimension,
such as advection-dominated or sharp gradient solutions. This is because LS-ROMs
work only for physical problems, in which the intrinsic solution space falls
into a subspace with a small dimension, i.e., the solution space has a small
Kolmogorov $n$-width. Although there have been many attempts to resolve these
shortcomings of LS-ROMs with various methods, \cite{abgrall2016robust,
reiss2018shifted, carlberg2013gnat, carlberg2018conservative, choi2020sns,
xiao2014non, constantine2012reduced, choi2020space, choi2019space,
taddei2020space, carlberg2015adaptive, rim2018transport, parish2019windowed,
peherstorfer2018model, welper2020transformed}, all these approaches are still
based on the linear subspace solution representation. We transition to a
nonlinear, low-dimensional manifold to approximate the solution better than
linear methods.
There are many works available in the current literature that looked into the
nonlinear manifold solution represenation, using neural networks (NNs) as
surrogates for physical simulations \cite{lagaris1998artificial,
dissanayake1994neural, van1995neural, meade1994numerical, raissi2019physics,
chen2018neural, khoo2017solving, long2018pde, beck2019machine, zhu2019physics,
berg2018unified, sirignano2018dgm, han2018solving, lu2019deeponet,
lu2019deepxde, pang2019fpinns, zhang2019quantifying, weinan2018deep,
he2018relu}. However, these methods do not take advantage of the existing
numerical methods for high-fidelity physical simulations. Recently, a neural
network-based ROM is developed in \cite{lee2020model}, where the weights and
biases are determined in the training phase and the existing numerical methods
are utilized in their models. The same technique is extended to preserve the
conserved quantities in the physical conservation laws \cite{lee2019deep}.
However, their approaches do not achieve any speed-up because the nonlinear
terms that still scale with the corresponding FOM size need to be updated
every time step or Newton step.
We present a fast and accurate physics-informed neural network ROM with a
nonlinear manifold solution representation, i.e., the nonlinear manifold ROM
(NM-ROM). We train a shallow masked autoencoder with solution data from the
corresponding FOM simulations and use the decoder as the nonlinear manifold
solution representation. Our NM-ROM is different from the aformentioned
physics-informed neural networks in that we take advantage of the existing
numerical methods of solving PDE in our approach and a considerable speed-up is
achieved.
\section{Full order model}\label{sec:FOM}
A parameterized nonlinear dynamical system is considered, characterized by
a system of nonlinear ordinary differential equations (ODEs), which can be
considered as a resultant system from semi-discretization of Partial
Differential Equations (PDEs) in space domains
\begin{equation} \label{eq:fom}
\frac{d\sol}{dt} = \flux(\sol,t; \param),\quad\quad
\sol(0;\param) = \solArg{0}(\param),
\end{equation}
where $t\in \timeDomain$ denotes time with the final time
$\totaltime\in\RRplus{}$, and $\sol(t;\param)$ denotes the time-dependent,
parameterized state implicitly defined as the solution to
problem~\eqref{eq:fom} with $\sol:\timeDomain\times \paramDomain\rightarrow
\RR{\nspacedof}$. Further, $\flux: \RR{\nspacedof} \times \timeDomain
\times \paramDomain \rightarrow \RR{\nspacedof}$ with
$(\solDummy,\timeDummy;\paramDummy)\mapsto\flux(\solDummy,\timeDummy;\paramDummy)
$ denotes the time derivative of $\sol$, which we assume to be nonlinear
in at least its first argument. The initial state is denoted by
$\solArg{0}:\paramDomain\rightarrow \RR{\nspacedof}$, and $\param \in
\paramDomain$ denotes parameters in the domain
$\paramDomain\subseteq\RR{\nparam}$.
A uniform time discretization is assumed throughout the paper, characterized
by time step $\dt\in\RRplus{}$ and time instances $\timeArg{n} =
\timeArg{n-1} + \dt$ for $n\in\nat{\ntimedof}$ with $\timeArg{0} = 0$,
$\ntimedof\in\natNo$, and $\nat{N}\defeq\{1,\ldots,N\}$. To avoid
notational clutter, we introduce the following time discretization-related
notations: $\solArg{n} \defeq \solFuncArg{n}$, $\solapproxArg{n} \defeq
\solapproxFuncArg{n}$, $\redsolapproxArg{n} \defeq \redsolapproxFuncArg{n}$,
and $\fluxArg{n} \defeq \flux(\solFuncArg{n},t^{n}; \param)$, where $\sol$, $\solapprox$, $\redsolapprox$ and $\flux$ are defined in.
The implicit backward Euler (BE)\footnote{Other time integrators can be used in
our NM-ROMs.} time integrator numerically solves Eq.~\eqref{eq:fom}, by solving
the following nonlinear system of equations, i.e.,
$\solArg{n} - \solArg{n-1} = \dt\fluxArg{n}$,
for $\solArg{n}$ at $n$-th time step. The corresponding residual function is
defined as
\begin{align}\label{eq:residual_BE}
\begin{split}
\resn_{\BE}(\solArg{n};\solArg{n-1},\param) &\defeq
\solArg{n} - \solArg{n-1} -\dt\fluxArg{n}.
\end{split}
\end{align}
\section{Nonlinear manifold reduced order model (NM-ROM)}\label{sec:NM-ROM}
The NM-ROM applies solution representation using a nonlinear manifold
$\spatialSubspace \defeq \{\scaledDecoder\left(\reddummy\right)|\reddummy \in
\RR{\nbasisspace}\}$, where $\scaledDecoder: \RR{\nbasisspace} \rightarrow
\RR{\nspacedof}$ with $\nbasisspace\ll\nspacedof$ denotes a nonlinear function
that maps a latent space of dimension $\nbasisspace$ to the full order model
space of dimension, $\nspacedof$. That is, the NM-ROM approximates the solution
in a trial manifold as
\begin{equation}\label{eq:spatialNMROMsolution}
\sol \approx \solapprox= \solArg{ref} + \scaledDecoder \left(\redsolapprox
\right).
\end{equation}
The construction of the nonlinear function, $\scaledDecoder$, is explained in
Section~\ref{sec:NN}. By plugging Eq.~\eqref{eq:spatialNMROMsolution} into
Eq.~\eqref{eq:residual_BE}, the residual function at $n$th time step becomes
\begin{align}\label{eq:trialManifold_residual_BE}
\begin{split}
\resRedApproxArg{n}_{\BE}(\redsolapproxArg{n};\redsolapproxArg{n-1},\param)
&\defeq
\resn_{\BE}(\solArg{ref}+\scaledDecoder\left(\redsolapproxArg{n}\right);
\solArg{ref}+\scaledDecoder\left(\redsolapproxArg{n-1}\right),\param) \\ &=
\scaledDecoder\left(\redsolapproxArg{n}\right)-\scaledDecoder\left(\redsolapproxArg{n-1}\right)
-\dt\flux(\solArg{ref}+\scaledDecoder\left(\redsolapproxArg{n}\right),t_n;\param),
\end{split}
\end{align}
which is an over-determined system that we close with the least-squares
Petrov--Galerkin (LSPG) projection. That is, we minimize the squared norm of the
residual vector function at every time step:
\begin{align} \label{eq:manifoldOpt1}
\begin{split}
\redsolapproxArg{n} = \argmin{\reddummy\in\RR{\nbasisspace}} \quad&
\frac{1}{2} \left \|\resRedApproxArg{n}_{\BE}(\reddummy;\redsolapproxArg{n-1},\param)
\right \|_2^2.
\end{split}
\end{align}
The Gauss--Newton method with the starting point $\redsolapproxArg{n-1}$ is
applied to solve the minimization problem~\eqref{eq:manifoldOpt1}. However,
the nonlinear residual vector, $\resRedApproxArg{n}_{\BE}$, scales with FOM size
and it needs to be updated every time the argument of the function changes,
which occurs either at every time step or Gauss--Newton step. More
specifically, if the backward Euler time integrator is used,
$\scaledDecoder\left(\redsolapproxArg{n}\right)$,
$\flux(\solArg{ref}+\scaledDecoder\left(\redsolapproxArg{n}\right),t;\param)$,
and their Jacobians need to be updated whenever $\redsolapproxArg{n}$
changes. Without any special
treatment on the nonlinear residual term, no speed-up can be expected. Thus, we
apply a hyper-reduction to eliminate the scale with FOM size in the nonlinear
term evaluations (see Section~\ref{sec:HR}). Finally, we denote this
non-hyper-reduced NM-ROM as NM-LSPG.
\section{Shallow masked autoencoder}\label{sec:NN}
The nonlinear function, $\scaledDecoder$, is the decoder
$\unscaledDecoder$ of an autoencoder in the form of a feedforward neural
network. The autoencoder compresses FOM solutions of Eq.~\eqref{eq:fom} with an
encoder $\unscaledEncoder$ and decompresses back to reconstructed FOM solution
with an decoder $\unscaledDecoder$. The autoencoder is trained to reconstruct
the FOM solutions of Eq.~\eqref{eq:fom} by minimizing the mean square error
between original and reconstructed FOM solutions. Therefore, the dimension of
the encoder input and the decoder output is $\nspacedof$ and the dimension of
the encoder $\unscaledEncoder$ output and the decoder $\unscaledDecoder$ input
is $\nbasisspace$.
We intentionally use a non-deep neural network, i.e., three-layer autoencoder,
for the decoder to achieve an efficiency that is requried by the hyper-reduction
(see Section~\ref{sec:HR}). More specifically, the first layers of the encoder
$\unscaledEncoder$ and decoder $\unscaledDecoder$ are fully-connected layers,
where the nonlinear activation functions are applied and the last layer of the
encoder $\unscaledEncoder$ is fully-connected layer with no activation
functions. The last layer of the decoder $\unscaledDecoder$ is
sparsely-connected layer with no activation functions. The sparsity is
determined by a mask matrix. These network architectures are shown in
Fig.~\ref{fg:threeLayerAE}.
\begin{figure}[!htbp]
\centering
\subfigure[Without masking]{
\includegraphics[width=0.45\textwidth]{ThreeLayerAEDense.png}
}
\subfigure[With masking]{
\includegraphics[width=0.45\textwidth]{ThreeLayerAESparse.png}
}
\caption{Three-layer encoder/decoder architectures: (a) unmasked and (b) after
the sparsity mask is applied. Nodes and edges in orange color represent
active path in the subnet that stems from the sampled outputs that are marked
as the orange disks. Note that the masked shallow neural network has a sparser
structure than the unmasked one. A sparser structure leads to a more
efficient model.}
\label{fg:threeLayerAE}
\end{figure}
There is no way to determine hidden layer sizes \textit{a priori}. If the
number of learnable parameters is not enough, the decoder is not able to
represent the nonlinear manifold well. On the other hand, too many learnable
parameters may result in over-fitting, so the decoder is not able to generalize
well, which means the trained decoder cannot be used for problems whose data is
unseen, i.e., the predictive case.
To avoid over-fitting, we first divide the training
data into train and validation datasets. Then, the autoencoder is trained
using the train dataset and tested for the generalization using the
validation dataset. If the mean squared errors on the validation and train datasets are very
different, then over-fitting has occurred. We then reduce the size of the hidden
layer and re-train the model \cite{kramer1991nonlinear}.
\section{Hyper-reduction}\label{sec:HR}
The hyper-reduction techniques are developed to eliminate the FOM scale
dependecy in nonlinear terms \cite{chaturantabut2010nonlinear,
drmac2016new, drmac2018discrete, carlberg2013gnat, choi2020sns}, which is
essential to acheive an efficiency in our NM-ROM. We follow the gappy POD
approach \cite{everson1995karhunen}, in which the nonlinear residual term is
approximated as
\begin{equation}\label{eq:ResApprox}
\resRedApproxArg{} \approx \basismatres\redres,
\end{equation}
where $\basismatres \defeq
[\basisresvecArg{1},\ldots,\basisresvecArg{\nbasisres} ] \in
\RR{\ndof\times\nbasisres}$, $\nbasisspace \leq \nbasisres \ll \ndof$, denotes
the residual basis matrix and $\redres \in \RR{\nbasisres}$ denotes the
generalized coordinates of the nonlinear residual term. Here,
$\resRedApproxArg{}$ represents a residual vector function, e.g., the backward
Euler residual, $\resRedApproxArg{n}_{BE}$, defined in
Eq.~\eqref{eq:trialManifold_residual_BE}. We use the singular value
decomposition of the FOM solution snapshot matrix to construct $\basismatres$,
which is justified in \cite{choi2020sns}. In order to find $\redres$, we apply
a sampling matrix $\samplemat \defeq [\unitvecArg{p_1}, \ldots,
\unitvecArg{p_{\nressample}}]^T \in \RR{\nressample\times\ndof}$, $\nbasisspace
\leq \nbasisres \leq \nressample \ll \ndof$ on both sides of
\eqref{eq:ResApprox}. The vector, $\unitvecArg{p_i}$, is the $p_i$th column of
the identity matrix $\identity{\ndof}\in\RR{\ndof\times\ndof}$. Then the
following least-squares problem is solved:
\begin{align} \label{eq:ResApprox_least-squares}
\begin{split}
\redres := \argmin{\reddummy\in\RR{\nbasisres}} \quad&
\frac{1}{2} \left \|\samplemat(\resRedApproxArg{} - \basismatres\reddummy)
\right \|_2^2.
\end{split}
\end{align}
The solution to Eq.~\eqref{eq:ResApprox_least-squares} is given as $ \redres =
(\samplemat\basismatres)^\dagger\samplemat\resRedApproxArg{}$, where the
Moore--Penrose inverse of a matrix $\weightmat \in \RR{\nressample \times
\nbasisres}$ with full column rank is defined as $\weightmat^{\dagger} :=
(\weightmat^T\weightmat)^{-1}\weightmat^T$. Therefore, Eq.~\eqref{eq:ResApprox}
becomes $\resRedApproxArg{} \approx \obliqueprojector \resRedApproxArg{}$,
where $\obliqueprojector:= \basismatres
(\samplemat\basismatres)^\dagger\samplemat$ is the oblique projection matrix.
We do not construct the sampling matrix $\samplematNT$. {\it Instead, it
maintains the sampling indices $\{p_1,\ldots,p_{\nbasisflux}\}$ and
corresponding rows of $\basismatres$ and $\resRedApproxArg{}$.} This enables
hyper-reduced ROMs to achieve a speed-up.
The sampling indices (i.e., $\samplematNT$) can be determined by Algorithm 3 of
\cite{carlberg2013gnat} for computational fluid dynamics problems and Algorithm
5 of \cite{carlberg2011efficient} for other problems.
The hyper-reduced residual, $\obliqueprojector\resRedApproxArg{n}_{BE}$, is
used in the minimization problem in Eq.~\eqref{eq:manifoldOpt1}:
\begin{align} \label{eq:NM-LSPG-HR-spOpt1}
\begin{split}
\redsolapproxArg{n} = \argmin{\reddummy\in\RR{\nbasisspace}} \quad&
\frac{1}{2} \left \|\ (\samplemat\basismatres)^\dagger\samplemat
\resRedApproxArg{n}_{\BE}(\reddummy;\redsolapproxArg{n-1},\param)
\right \|_2^2.
\end{split}
\end{align}
Note that the pseudo-inverse $(\samplemat\basismatres)^\dagger$ can be
pre-computed. Due to the definition of $\resRedApproxArg{n}_{\BE}$
in Eq.~\eqref{eq:trialManifold_residual_BE}, the sampling matrix $\samplematNT$
needs to be applied to the following two terms:
$\scaledDecoder\left(\redsolapproxArg{n}\right) -
\scaledDecoder\left(\redsolapproxArg{n-1}\right)$ and
$\flux(\solArg{ref}+\scaledDecoder\left(\redsolapproxArg{n}\right),t;\param)$ at
every time step. The first term,
$\samplemat(\scaledDecoder\left(\redsolapproxArg{n}\right) -
\scaledDecoder\left(\redsolapproxArg{n-1}\right))$, requires that only selected
outputs of the decoder be computed. Furthermore, for the second term, the nonlinear residual elements that are selected by the sampling matrix need
to be computed. This implies that we have to keep track of the outputs of
$\scaledDecoder$ that are needed to compute the selected nonlinear residual
elements by the sampling matrix, which is usually a larger set than the outputs
that are selected solely by the sampling matrix. Therefore, we
build a subnet that computes only the outputs of the decoder that are
required to compute the nonlinear residual elements. Such outputs are demonstrated as the solid oragne disks and the corresponding subnet is depicted in Fig.~\ref{fg:threeLayerAE}(b). Finally, we denote
this hyper-reduced NM-ROM as NM-LSPG-HR.
\section{2D Burgers' equation}\label{sec:2dburgers}
We demonstrate the performance of our NM-ROMs (i.e., NM-LSPG and NM-LSPG-HR) by
comparing it with LS-ROMs (i.e., LS-LSPG and LS-LSPG-HR) that was first
introduced in \cite{carlberg2011efficient}. We solve the following parameterized
2D viscous Burgers' equation:
\begin{equation}\label{eq:2dburgers_eq}
\begin{split}
\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} &= \frac{1}{Re}\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right) \\
\frac{\partial v}{\partial t} + u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} &= \frac{1}{Re}\left(\frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}\right) \\
(x,y) &\in \spaceDomain = [0,1] \times [0,1] \\
t &\in[0,2],
\end{split}
\end{equation}
with the boundary condition
\begin{equation}
u(x,y,t;\mu)=v(x,y,t;\mu)=0 \quad \text{on} \quad \Gamma = \left\{(x,y)|x\in\{0,1\},y\in\{0,1\}\right\}
\end{equation}
and the initial condition
\begin{align}
u(x,y,0;\mu) = \left\{
\begin{array}{ll}
\mu \sin{(2\pi x)}\cdot \sin{(2\pi y)} \quad &\text{if } (x,y) \in [0,0.5]\times [0,0.5] \\
0 \quad &\text{otherwise}
\end{array}
\right. \\
v(x,y,0;\mu) = \left\{
\begin{array}{ll}
\mu \sin{(2\pi x)}\cdot \sin{(2\pi y)} \quad &\text{if } (x,y) \in [0,0.5]\times [0,0.5] \\
0 \quad &\text{otherwise}
\end{array}
\right.
\end{align}
where $\mu\in \paramDomain=[0.9,1.1]$ is a parameter and $u(x,y,t;\mu)$ and
$v(x,y,t;\mu)$ denote the $x$ and $y$ directional velocities, respectively, with
$u:\spaceDomain \times [0,2] \times \paramDomain \rightarrow \RR{}$ and
$v:\spaceDomain \times [0,2] \times \paramDomain \rightarrow \RR{}$ defined as
the solutions to Eq.~\eqref{eq:2dburgers_eq}, and $Re$ is a Reynolds number
which is set $Re=10,000$. For this case, the FOM solution snapshot shows slowly
decaying singular values. We observe that a sharp gradient, i.e., a shock,
appears in the FOM solution (e.g., see Fig.~\ref{fg:2dburgersFOMROMS}(a)).
We use $60\times 60$ uniform mesh with the backward difference scheme for the
first spatial derivative terms and the central difference scheme for the second spatial derivative terms. Then, we use the backward Euler scheme with time
step size $\Delta t = \frac{2}{\nt}$, where $\nt = 1,500$ is the number of time
steps.
For the training process, we collect solution snapshots associated with the
parameter $\mu \in \paramDomain_{train}=\{0.9,0.95,1.05,1.1\}$,
such that $\ntrain=4$, at which the FOM is solved. Then, the number of train
data points is $\ntrain\cdot(\nt+1)=6,004$ and $10\%$ of the train data are used
for validation purposes. We employ the Adam optimizer \cite{kingma2014adam} with
the SGD and the initial learning rate of $0.001$, which decreases by a factor of
$10$ when a training loss stagnates for $10$ successive epochs. We set
the encoder and decoder hidden layer sizes to $6,728$ and $33,730$, respectively
and vary the dimension of the latent space from $5$ to $20$. The weights and
bias of the autoencoder are initialized via Kaiming initialization
\cite{he2015delving}. The batch size is $240$ and the maximum number of
epochs is $10,000$. The training process is stopped if the loss on the
validation dataset stagnates for $200$ epochs.
After the training is done, the NM-LSPG and LS-LSPG solve the
Eq.~\eqref{eq:2dburgers_eq} with the target parameter $\mu=1$, which is not
included in the train dataset. Fig.~\ref{fg:2dErrVSredDim} shows the relative
error versus the reduced dimension $\nbasisspace$ for both LS-LSPG and NM-LSPG.
It also shows the projection errors for LS-ROMs and NM-ROMs, which are the lower
bounds that any LS-ROMs and NM-ROMs can reach, respectively. As
expected the relative error for the NM-LSPG is lower than the one for the
LS-LSPG. We even observe that the relative errors of NM-LSPG are even lower
than the LS projected error.
\begin{figure}[!htbp]
\centering
\subfigure[Relative errors vs reduced
dimensions]{
\includegraphics[width=0.48\textwidth]{MaxRelErrVSRedDim2dU2.pdf}
}
\subfigure[Relative errors vs $\mu$]{
\includegraphics[width=0.48\textwidth]{MaxRelErrVSparam2dU.pdf}
}
\caption{The comparison of the NM-LSPG-HR and NM-LSPG on the maximum relative
errors. A maximum relative error that is $1$ means the ROM failed to solve the problem.}
\label{fg:2dErrVSredDim}
\end{figure}
We vary the number of residual basis and residual samples, with the fixed
number of training parameter instances $\ntrain=4$ and the reduced
dimension $\nbasisspace=5$, and measure the wall-clock time. The results are
shown in Table~\ref{tb:2dDEIMtest}. Although the LS-LSPG-HR can achieve better
speed-up than the NM-LSPG-HR, the relative error of the LS-LSPG-HR is too large
to be reasonable, e.g., the relative errors of around $37 \%$. On the other
hand, the NM-LSPG-HR achieves much better accuracy, i.e., a relative error of
around $1 \%$, and a factor $11$ speed-up.
\begin{table}[!htbp]
\caption{The top 6 maximum relative errors and wall-clock times at different
numbers of residual basis and samples which range from $40$ to
$60$.}\label{tb:2dDEIMtest}
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{6}{|c|}{NM-LSPG-HR} & \multicolumn{6}{|c|}{LS-LSPG-HR}\\
\hline
Residual basis, $\nbasisres$ & 55 & 56 & 51 & 53 & 54 & 44 & 59 & 53 & 53 & 53 & 53 & 53\\
\hline
Residual samples, $\nressample$ & 58 & 59 & 54 & 56 & 57 & 47 & 59 & 58 & 59 & 56 & 55 & 53\\
\hline
Max. rel. error (\%) & 0.93 & 0.94 & 0.95 & 0.97 & 0.97 & 0.98 & 34.38 & 37.73 & 37.84 & 37.95 & 37.96 & 37.97\\
\hline
Wall-clock time (sec) & 12.15 & 12.35 & 12.09 & 12.14 & 12.29 & 12.01 & 5.26 & 5.02 & 4.86 & 5.05 & 4.75 & 7.18\\
\hline
Speed-up & 11.58 & 11.39 & 11.63 & 11.58 & 11.44 & 11.71 & 26.76 & 28.02 & 28.95 & 27.83 & 29.61 & 19.58 \\
\hline
\end{tabular}}
\end{table}
Fig.~\ref{fg:2dburgersFOMROMS} shows FOM solutions at the last time and
absolute differences between FOM and other approaches, i.e., NM-LSPG-HR and LS-LSPG-HR with the reduced dimension being $\nbasisspace=5$ and a black-box NN
approach (BB-NN). The BB-NN approach is similar to the one described in \cite{DBLP:journals/corr/abs-1806-02071}. The main difference is that $L1$-norms and physics constraints were not used in our loss function. This approach gave a maximum relative error of $38.6\%$ and has a speed-up of $119$. While this
approach is appealing in that it does not require access to the PDE solver, the
errors are too large for our application. For NM-LSPG-HR, $55$
residual basis dimension and $58$ residual samples are used and for LS-LSPG-HR,
$59$ residual basis dimension and $59$ residual samples are used. Both FOM and
NM-LSPG-HR show good agreement in their solutions, while the LS-LSPG-HR is not
able to achieve a good accuracy. In fact, the NM-LSPG-HR is able to achieve an
accuracy as good as the NM-LSPG for some combinations of the small number of
residual basis and residual samples.
Finally, Fig.~\ref{fg:2dErrVSredDim}(b) shows the maximum relative error over
the test range of the parameter points. Note that the NM-LSPG and NM-LSPG-HR
are the most accurate within the range of the training points, i.e., $[0.9,
1.1]$. As the parameter points go beyond the training parameter domain, the
accuracy of the NM-LSPG and NM-LSPG-HR start to deteriorate gradually. This
implies that the NM-LSPG and NM-LSPG-HR have a trust region. Its trust region
should be determined by each application.
\begin{figure}[!htbp]
\centering
\subfigure[FOM]{
\includegraphics[width=0.25\textwidth]{FOM2dU.png}
}~~~~~~~ \hspace{-22pt}
\subfigure[NM-LSPG-HR]{
\includegraphics[width=0.25\textwidth]{AbsDiffNMROMDEIM2dU.png}
}~~~~~~~ \hspace{-22pt}
\subfigure[LS-LSPG-HR]{
\includegraphics[width=0.25\textwidth]{AbsDiffLSROMDEIM2dU.png}
}~~~~~~~ \hspace{-22pt}
\subfigure[BB-NN]{
\includegraphics[width=0.25\textwidth]{AbsDiffBlackBoxNN2dU.png}
}~~~~~~~
\caption{(a) Solution snapshots, $u$, of FOM and absolute differences of
(b) NM-LSPG-HR,
(c) LS-LSPG-HR, and (d) BB-NN with respect to FOM solution at $t=2$.}
\label{fg:2dburgersFOMROMS}
\end{figure}
\section{Discussion \& conclusion}\label{sec:discussion-conclusion}
In this work, we have successfully developed an accurate and efficient NM-ROM.
We demonstrated that both the LS-ROM and BB-NN are not able to represent
advection-dominated or sharp gradient solutions of 2D viscous Burgers' equation
with a high Reynolds number. However, our new approach, NM-LSPG-HR, solves such
problem accurately and efficiently. The speed-up of the NM-LSPG-HR is achieved
by choosing the shallow masked decoder as the nonlinear manifold and applying
the efficient hyper-reduction computation. Because the difference in the
computational cost of the FOM and NM-LSPG-HR increases as a function of the
number of mesh points, we expect more speed-up as the number of mesh points
becomes larger.
Compared with the deep neural networks for computer vision and natural language
processing applications, our neural networks are shallow with a small number of
parameters. However, these networks were able to capture the variation in our 2D
Burgers' simulations. A main future work for transferring this work to more
complex simulations, will be to find the right balance between a shallow network
that is large enough to capture the data variance and yet small enough to run
faster than the FOM. Another future work will be to find an efficient way of
determining the proper size of the residual basis and the number of sample
points \textit{a priori}. To find the optimal size of residual basis and the
number of sample points for hyper-reduced ROMs, we relied on test results. This
issue is not just for NM-LSPG-HR, but also for LS-LSPG-HR.
\section*{Broader Impact}
The broader impact of this work will be to accelerate physics simulations to
improve design optimization and control problems, which require
thousands of simulation runs to learn an optimal design or viable control
strategy. While this is not computationally feasible with high-fidelity FOMs,
the development of the NM-ROMs is an important step in this direction.
\begin{ack}
This work was performed at Lawrence Livermore National Laboratory and was
supported by the LDRD program (project 20-FS-007). Youngkyu was also
supported for this work through generous funding from DTRA. Lawrence Livermore
National Laboratory is operated by Lawrence Livermore National Security,
LLC, for the U.S. Department of Energy, National Nuclear Security
Administration under Contract DE-AC52-07NA27344 and LLNL-CONF-815209. We
declare that there were no conflicting interests of any type during the
production of this research.
\end{ack}
\bibliographystyle{unsrt}
\bibliography{references}
\end{document} | 69,543 |
TITLE: When $n\geq2$, let $a_n = \lceil \frac{n}{\pi}\rceil$ and let $b_n = \lceil{\csc({\frac{\pi}{n}})}\rceil$.
QUESTION [2 upvotes]: When $n\geq2$, let $a_n =\left \lceil \frac{n}{\pi}\right\rceil$ and let $b_n = \left\lceil{\csc({\frac{\pi}{n}})}\right\rceil$.
The terms of the sequences starting with $n = 2$ are:
{$a_n$} = $1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, ...$ and
{$b_n$} = $1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, ...$
Note that the sequences differ when $n=3$. Is it true that $a_n = b_n$ for all $n>3$?
REPLY [3 votes]: No. When $n = 80143857$,
$$\begin{align}
n / \pi &= 25510581.9999999952976568107626972575226719258409876014\cdots, \\
\csc(\pi/n) &= 25510582.0000000018308933581165258478077895828199645771\cdots.
\end{align}$$
This counterexample was found with a small JavaScript code and verified by Wolfram|Alpha. | 186,452 |
TITLE: How to show $P_U $ is linear for the following condition?
QUESTION [0 upvotes]: Let $V$ be a vector space and $U, W$ are subspaces such that $V=U\bigoplus W$. Then we know $v\in V$ can be uniquely written as $v=u+w$ for $u\in U$ and $w\in W$. Define the function $P_U:V\to V$ by $P_U (v)=P_U (u+w)=u$.
How to show $P_U $ is linear for the following condition?
My try:
$P_U (x+y)=P_U (u_1+u_2+w_2+w_2)=u_1+u_2=($something here I don't know$)=P_U (x)+P_U (y)$
$P_U (ax)=au=$something here I don't know$=aP_U (x)$.
Someone have any suggestion? In addition, how to show that Ker$(P_U)=W,$ Im$(P_U)=U$?
REPLY [1 votes]: You are very close for the first part. One approach to solving problems of this form is to work forwards and backwards until you meet in the middle. You worked the "forwards" part perfectly. Working backwards,
$P_U(x) + P_U(y) = P_U(u_1 + w_1) + P_U(u_2+w_2) = u_1 + u_2$. You have arrived at the same expression where you ended the forwards part. Now you glue both parts together reversing the order of the part you did backwards like below:
If $x=u_1 + w_1$ and $y=u_2 + w_2$ where $u_i \in U$ and $w_i \in W$ then
\begin{align*}
P_U(x+y) & = P_U(u_1 + w_1 + u_2 + w_2) \\
& = P_U( (u_1 + u_2) + (w_1 + w_2)) \\
& = u_1 + u_2 \\
& = P(u_1 + w_1) + P(u_2 + w_2) \\
& = P(x) + P(y)
\end{align*}
For the second part,
\begin{align*}
P_U(ax) &= P_U(au_1 + aw_1) \\
&= au_1 \\
&= aP_U(u_1 + w_1) \\
&= aP_U(x)
\end{align*}
To show the $Ker(T)=W$ and $Im(T)=U$ you could start out by writing the definition of $Ker(T)$ and $Im(T)$. Then one standard approach to prove that two sets are equal is to show that $Ker(T) \subset W$ and $W \subset Ker(T)$. | 193,649 |
\begin{document}
\bibliographystyle{amsalpha}
\title{Seshadri's criterion and openness of projectivity}
\author{J\'anos Koll\'ar}
\begin{abstract}
We prove that projectivity is an open condition for deformations of algebraic spaces with rational singularities.
\end{abstract}
\maketitle
Many projective varieties have deformations that are not projective, not even algebraic in any sense; K3 and elliptic surfaces furnish the best known examples.
In these cases the non-algebraic deformations are also very far from being algebraic.
A typical non-projective K3 surface does not contain any compact curves and
a non-projective K3 surface is not even bimeromorphic to an algebraic surface. The latter property is not an accident.
Let $g: X\to \dd$ be a smooth, proper morphism of a complex manifold to a disc.
If the central fiber $X_0$ is projective, then it is K\"ahler, and the $X_s$ are also K\"ahler for $|s|\ll 1$
by \cite{MR0112154}. A K\"ahler variety that is bimeromorphic to an algebraic variety is projective by \cite{Moi-66}. Thus if $X_s$ is bimeromorphic to an algebraic variety, then it is projective for $|s|\ll 1$. We can summarize this somewhat imprecisely as:
\medskip
$\bullet$
Projectivity is an open condition in the category of Moishezon manifolds.
\medskip
Note, however, that even if all fibers of $g: X\to \dd$ are projective,
$g$ need not be projective over any smaller disc $\dd_\epsilon$; see
\cite{Atiyah58} or \cite[Exmp.4]{k-defgt}.
Examples of Hironaka---reproduced in \cite[App.B]{hartsh}---show that
projectivity is not a closed condition.
The aim of this paper is to prove that projectivity is an open condition for deformations of algebraic spaces with rational singularities.
A result of this type, with rather strong restrictions on the singularities, is in \cite[12.2.10]{km-flips}; see Paragraph~\ref{corr.12.2.10} for some comments.
I have been trying to remove the restrictions, and a key problem that emerged was that the argument used Kleiman's criterion for ampleness, which is not known to apply to algebraic spaces.
By contrast, Seshadri's criterion does hold for algebraic spaces, and
leads to a quite general openness result.
First I state the 1-parameter version. This is clearer and its proof uses all the essential ideas.
The general version, Theorem~\ref{12.2.10.thm.S}, is substantially stronger.
\begin{thm} \label{12.2.10.thm.dd}
Let $g:X\to \dd$ be a proper, flat morphism of complex analytic spaces.
Assume that
\begin{enumerate}
\item $X_0$ is projective,
\item the fibers $X_s$ have rational singularities for $s\neq 0$, and
\item $g$ is bimeromorphic to a projective morphism $g^{\rm p}:X^{\rm p}\to \dd$.
\end{enumerate}
Then $g$ is projective over a smaller punctured disc $\dd^\circ_\epsilon\subset \dd$.
\end{thm}
{\it Remarks} (\ref{12.2.10.thm.dd}.4) Note that we make no assumptions on the singularities of $X_0$.
(\ref{12.2.10.thm.dd}.5) Even if $g$ is smooth, usually $g$ is not projective over the disc $\dd_\epsilon$. So, although there is a line bundle $L$ on $X$ such that
$L$ is ample over $\dd^\circ_\epsilon$, it can not be chosen to be also ample on $X_0$. This is quite typical for simultaneous resolutions of deformations of surfaces with Du~Val singularities; see \cite{Atiyah58} or \cite[Exmp.4]{k-defgt}.
(\ref{12.2.10.thm.dd}.6) Assumption (\ref{12.2.10.thm.dd}.2) can be relaxed to $X_s$ having 1-rational singularities (see Definition~\ref{rtl.1.rtl.defn}), but a similar openness property does not hold for algebraic spaces with slightly worse singularities.
There are flat, proper families of normal, compact surfaces
where all fibers are bimeromorphic to $\p^2$, yet
the projective fibers correspond to a countable, dense set on the base; see Example~\ref{bad.fams.exmp}.
In this example each surface has a single singular point, which is simple elliptic (that is, biholomorphic to a cone over an elliptic curve).
These are among the mildest surface singularities, but they are not rational.
(\ref{12.2.10.thm.dd}.7) Assumption (\ref{12.2.10.thm.dd}.3) is automatic if $g:X\to \dd$ is obtained by base change from a morphism of algebraic spaces, and it implies that every fiber of $g$ is an algebraic space.
A converse is conjectured in \cite[Conj.2]{k-moi}, and proved in
\cite[Thm.1.4]{rao-tsa-1} and \cite[Thm.1.2]{rao-tsa} for smooth morphisms.
(\ref{12.2.10.thm.dd}.8) Instead of assumption (\ref{12.2.10.thm.dd}.3) we could assume that all fibers are algebraic spaces and the irreducible components of the Chow-Barlet space parametrizing 1-cycles on $X/\dd$ are proper over $\dd$; see \cite{bar-mag} or Proposition~\ref{bir.chow.prop}
\medskip
{\it Idea of proof.} After shrinking $\dd$ we may assume that $X$ retracts to $X_0$. Since $X_0$ is projective, it has an ample line bundle $L$.
Let $\Theta\in H^2(X, \q)$ be the pull-back of
$c_1(L)$ to $X$. Note that $\Theta$ is a topological cohomology class that is usually not the Chern class of a holomorpic line bundle.
Fix a very general $s\in \dd$ and let $p_s\in C_s\subset X_s$ be a pointed curve.
We show in Proposition~\ref{bir.chow.prop} that it has a specialization to $p_0\in C_0\subset X_0$ such that
$\mult_{p_0}C_0\geq \mult_{p_s}C_s$. Thus
$$
\Theta\cap [C_s]=\Theta\cap [C_0]
\geq \epsilon\cdot \mult_{p_0}C_0\geq \epsilon\cdot\mult_{p_s}C_s,
$$
where the first $\geq $ is the easy direction of Seshadri's criterion \ref{sesh.crit.thm} on $X_0$. Thus $\Theta_s:=\Theta|_{X_s}\in H^2(X_s, \q)$ satisfies the assumption of
Seshadri's criterion on $X_s$.
Thus we need to show that Seshadri's criterion works for (possibly non-algebraic) cohomology classes. This is done in Proposition~\ref{sesh.top.ver.prop}, using some foundational work comparing numerical and homological equivalence in Proposition~\ref{N1.to.H2.prop}. This is where rationality of the singularities of $X_s$ is used.
Going from very general fibers to all fibers over a smaller disc uses Proposition~\ref{RT.vb.Zd.M}.
\medskip
The general version is the following, whose proof is completed in Paragraph~\ref{12.2.10.thm.S.pf}.
\begin{thm} \label{12.2.10.thm.S}
Let $g:X\to S$ be a proper morphism of complex analytic spaces and
$S^*\subset S$ a dense, Zariski open subset such that $g$ is flat over $S^*$.
Assume that
\begin{enumerate}
\item $X_0$ is projective for some $0\in S$,
\item the fibers $X_s$ have rational singularities for $s\in S^*$, and
\item $g$ is bimeromorphic to a projective morphism $g^{\rm p}:X^{\rm p}\to S$.
\end{enumerate}
Then there is a Zariski open neighborhood
$0\in U\subset S$ and a locally closed, Zariski stratification
$U\cap S^*=\cup_i S_i$ such that each
$$
g|_{X_i}:X_i:=g^{-1}(S_i)\to S_i \qtq{is projective.}
$$
\end{thm}
{\it Remark \ref{12.2.10.thm.S}.4.} Note that $g$ is not assumed flat and
the comments in (\ref{12.2.10.thm.dd}.6) also apply to $X\to S$.
\medskip
The Lefschetz principle then gives the analogous result for algebraic spaces.
\begin{cor} \label{12.2.10.thm.S.cor} Let $k$ be a field of characteristic 0, and $g:X\to S$ a proper morphism of algebraic spaces that are of finite type over $k$. Let
$S^*\subset S$ be a dense, Zariski open subset such that $g$ is flat over $S^*$.
Assume that
\begin{enumerate}
\item $X_0$ is projective for some $0\in S$, and
\item the fibers $X_s$ have 1-rational singularities for $s\in S^*$.
\end{enumerate}
Then there is a Zariski open neighborhood
$0\in U\subset S$ and a locally closed, Zariski stratification
$U\cap S^*=\cup_i S_i$ such that each
$$
g|_{X_i}:X_i:=g^{-1}(S_i)\to S_i \qtq{is projective.}\hfill\qed
$$
\end{cor}
\begin{ques} Is Corollary~\ref{12.2.10.thm.S.cor} also true in positive or mixed characteristic? Our proof uses topological cohomology groups in an essential way; it is not clear what would replace them.
\end{ques}
Theorem~\ref{12.2.10.thm.dd} was developed to complete the proof of the second part of the following; for details see the original paper.
\begin{thm} \cite[Thm.1]{k-defgt} \label{mmp.extends.conj.thm.c.cor}
Let $g:X\to S$ be a flat, proper morphism of complex analytic spaces.
Fix a point $0\in S$ and assume that the fiber
$X_0$ is projective, of general type, and with canonical singularities.
Then there is an open neighborhood $0\in U\subset S$ such that
\begin{enumerate}
\item the plurigenera $h^0(X_s, \omega_{X_s}^{[r]})$ are independent of $s\in U$ for every $r$, and
\item the fibers $X_s$ are projective for every $s\in U$.
\qed
\end{enumerate}
\end{thm}
\begin{exmp} \label{bad.fams.exmp}
Let $E\subset \p^2$ be a smooth cubic. Fix $m\geq 10$ and let $X\to S$ be the universal family of surfaces obtained by blowing up $m$ distinct points $p_i\in E$, and then contracting the birational transform of $E$. (So $S$ is an open subset in $E^m$.) Such a surface is projective iff there are positive $n_i$ such that
$\sum_i n_i[p_i]\sim nL|_E$ where $L$ is the line class on $\p^2$ and $n=\frac13 \sum_i n_i$.
Thus the projective fibers
correspond to a countable union of hypersurfaces
$H(n_1,\dots, n_m)\subset S$. All fibers have simple elliptic singularities and trivial canonical class.
(For $m=12$ the singularities are biholomorphic to cones over smooth plane cubics.)
\end{exmp}
\begin{say}[Correction to {\cite[Chap.12]{km-flips}}]
\label{corr.12.2.10}
Statements \cite[12.2.6 and 12.2.10]{km-flips} are incorrect.
The applications of these results concern families of 3-folds with $\q$-factorial, terminal singularities. With these additional assumptions,
the proofs given there are correct. However, trying to formulate them with minimal sets of assumptions led to errors.
First, in \cite[12.2.6]{km-flips} one should also assume that the fibers $X_s$ have rational singularities. This is necessary since the proof uses
\cite[12.1.5]{km-flips}. No other changes needed.
The bigger problem is with \cite[12.2.10]{km-flips}. During the proof we claim to apply Kleiman's ampleness criterion to
algebraic spaces. However \cite{Kleiman66b} states and proves
the criterion for quasi-divisorial schemes. This class includes all varieties that are either projective, or proper and $\q$-factorial, but it is not clear that the fibers $X_s$ are quasi-divisorial.
A proof for
$\q$-factorial, 3-dimensional, algebraic spaces is explained in \cite[5.1.3]{k-etc}, this is enough for the applications in \cite{km-flips}.
A higher dimensional Kleiman criterion needed for \cite[12.2.10]{km-flips}
was recently established by Villalobos-Paz \cite{dvp}. With this in place, the rest of the proof of \cite[12.2.10]{km-flips} is correct.
\end{say}
\begin{ack} I thank D.~Abramovich, D.~Barlet, F.~Campana, J.-P.~Demailly, O.~Fujino, S.~Kleiman, S.~Mori, T.~Murayama, V.~Tosatti, D.~Villalobos-Paz, C.~Voisin and C.~Xu
for helpful comments, corrections and V.~Balaji for a serendipitous e-mail.
Partial financial support was provided by the NSF under grant number
DMS-1901855.
\end{ack}
\section{Seshadri's criterion and variants}
The criterion---proved by Seshadri but first published in \cite[Sec.I.7]{Ha70}---is the following.
\begin{thm}\label{sesh.crit.thm} Let $X$ be a proper algebraic space and $L$ a line bundle on $X$. Then $L$ is ample iff there is an $\epsilon>0$ such that
$$
\deg L|_C\geq \epsilon \cdot \mult_pC
\eqno{(\ref{sesh.crit.thm}.1)}
$$
for every integral curve $C\subset X$ and every $p\in C$.
\end{thm}
By linearity then the same holds for all 1-cycles $Z$ on $X$.
\medskip
An important observation is that while the criterion is frequenty stated for schemes only, it in fact holds for proper algebraic spaces, even if they are reducible and non-reduced.
This is in marked contrast with Kleiman's criterion, which is still not known for algebraic spaces in general, though a recent result of Villalobos-Paz \cite{dvp}
proves Kleiman's criterion for $\q$-factorial algebraic spaces with log terminal singularities over a field of characteristic 0.
In algebraic geometry Seshadri's criterion is usually used to show that a line bundle $L$ is ample. Here I focus on a reformulation of Theorem~\ref{sesh.crit.thm}.
\begin{cor}\label{sesh.crit.thm.as} Let $X$ be a proper algebraic space. Then $X$ projective iff there is a line bundle $L$ and an $\epsilon>0$ such that
$$
\deg L|_C\geq \epsilon \cdot \mult_pC
\eqno{(\ref{sesh.crit.thm.as}.1)}
$$
for every integral curve $C\subset X$ and every $p\in C$. \qed
\end{cor}
Now we make a slight twist and replace the line bundle $L$ by
a cohomology class $\Theta\in H^2\bigl(X(\c), \q\bigr)$. The key point is that in our applications we will be able to find such a cohomology class $\Theta$, but it usually will not be a $(1,1)$ class.
\begin{prop}\label{sesh.top.ver.prop}
Let $X$ be a proper algebraic space over $\c$ with 1-rational singularities (see Definition~\ref{rtl.1.rtl.defn}). Then $X$ is projective iff there is a cohomology class $\Theta\in H^2\bigl(X(\c), \q\bigr)$ and an $\epsilon>0$ such that
$$
\Theta\cap [C]\geq \epsilon \cdot \mult_pC
\eqno{(\ref{sesh.top.ver.prop}.1)}
$$
for every integral curve $C\subset X$ and every $p\in C$.
\end{prop}
Here $[C]\in H_2\bigl(X(\c), \q\bigr)$ denotes the homology class of $C$, and
we tacitly use the identification $H_0\bigl(X(\c), \q\bigr)\cong \q$ to view
$ \Theta\cap [C]$ as a number.
The proof relies on an injection $N_1(X,\q)\into H_2(X, \q)$ which we define next.
\begin{defn} \label{cone.of.curves}
Let $X$ be a proper, complex analytic space.
For $K=\q$ or $\r$,
let $N_1(X,K)$ denote the $K$-vectorspace generated by
compact complex curves $C\subset X$, modulo numerical equivalence. That is,
$$
\tsum\ a_iA_i\equiv \tsum\ b_jB_j\qtq{iff}
\tsum\ a_i\deg L|_{A_i}= \tsum\ b_j\deg L|_{B_j}
$$
for every holomorphic line bundle $L$ on $X$.
Let $H_2^{\rm alg}(X, K)\subset H_2(X, K)$ denote the vector subspace generated
by homology classes of compact complex curves. Sending an algebraic homology class to a numerical equivalence class gives a natural surjection
$$
\res^{\rm hom}_{\rm num}: H_2^{\rm alg}(X, \q)\onto N_1(X,\q).
\eqno{(\ref{cone.of.curves}.1)}
$$
We see in Proposition~\ref{N1.to.H2.prop} that $\res^{\rm hom}_{\rm num} $ is an isomorphism if $X$ has rational singularities. In this case the inverse map gives an injection
$$
N_1(X,\q)\into H_2(X, \q).
\eqno{(\ref{cone.of.curves}.2)}
$$
Nore, however, that the natural map is (\ref{cone.of.curves}.1),
and (\ref{cone.of.curves}.2) exists only if $X$ has 1-rational singularities.
\end{defn}
\begin{defn} \label{rtl.1.rtl.defn} Let $X$ be a normal, complex analytic space and $g:Y\to X$ a resolution of singularities. $X$ has {\it rational singularities} iff $R^ig_*\o_Y=0$ for every $i>0$. The $R^ig_*\o_Y$ are independent of the choice of $Y$, so this is a property of $X$ only.
If $R^1g_*\o_Y=0$, then $X$ is said to have {\it 1-rational singularities.}
\end{defn}
\begin{say}[Proof of Proposition~\ref{sesh.top.ver.prop}]\label{sesh.top.ver.prop.pf}
If $C\mapsto [C]$ gives an injection $N_1(X, \q)\into H_2\bigl(X(\c), \q\bigr)$, then we can view $C\mapsto \Theta\cap [C]$ as a linear map
$$
\Theta\cap : N_1(X, \q)\to \q.
$$
By definition, line bundles span the dual space of $ N_1(X, \q)$, so
there is a line bundle $L$ on $X$ and an $m>0$ such that
$\deg(L|_C)=m\cdot \Theta\cap [C]$
for every integral curve $C\subset X$. Thus
$$
\deg(L|_C)=m\cdot \Theta\cap [C]\geq m\epsilon \cdot \mult_pC
$$
for every integral curve $C\subset X$ and every $p\in C$.
Then $L$ is ample by Theorem~\ref{sesh.crit.thm}, so $X$ is projective.
It remains to show that $C\mapsto [C]$ gives an injection $N_1(X, \q)\into H_2\bigl(X(\c), \q\bigr)$ if $X$ has 1-rational singularities only. This is proved in Proposition~\ref{N1.to.H2.prop}.
\qed
\end{say}
As a consequence of Lefschetz's theorems, sending a curve $C$ to its homology class $[C]$ descends to an injection $N_1(X,\q)\to H_2(X, \q)$ for smooth projective varieties, see \cite[p.161]{gri-har}.
Next we show that the same holds if $X$ has 1-rational singularities; the key ingredient is \cite[12.1.4]{km-flips}.
\begin{prop}\label{N1.to.H2.prop}
Let $X$ be a proper, complex analytic space
that is bimeromorphic to a projective variety.
Assume that $X$ has 1-rational singularities. Then sending a curve to its homology class gives an injection $N_1(X,\q)\into H_2(X, \q)$.
\end{prop}
Proof. Take a projective resolution $g:Y\to X$. Let $Z$ be a numerically trivial 1-cycle. We need to show that $[Z]\in H_2(X, \q)$ is the 0 class.
For every closed, irreducible, analytic curve
$C\subset X$ there is a closed, irreducible, analytic curve
$C_Y\subset Y$ such that $g(C_Y)=C$. (In general $C_Y\to C$ may have degree $>1$.) Thus every $Z\in N_1(X, \q)$ lifts to $Z_Y\in N_1(Y,\q)$ such that
$g_*(Z_Y)=Z$ (as cycles).
Let $N_1(Y/X,\q)\subset N_1(X, \q)$ denote the vector subspace generated by
compact complex curves $C\subset Y$ that map to a point in $X$.
If there is a $Z'_Y\in N_1(Y/X,\q)$
such that $Z_Y\equiv Z'_Y$, then
$[Z]=g_*[Z_Y]=g_*[Z'_Y]=0$ in $H_2(X,\q)$.
Otherwise there is a line bundle $L_Y$ on $Y$ that is
trivial on $N_1(Y/X,\q)$ but $(L_Y\cdot Z_Y)\neq 0$.
The key point is that, by \cite[12.1.4]{km-flips},
some (positive) power $L_Y^m$ descends to a line bundle $L$ on $X$, and then $(L\cdot Z)=m(L_Y\cdot Z_Y)\neq 0$ gives a contradiction.
(This is where we use that $X$ has 1-rational singularities.) \qed
\section{Chow-Barlet spaces with marked multiplicities}
Let $g:X\to S$ be a proper morphism. We are interested in the
set of proper 1-cycles $Z\subset X$ with a marked point $p\in Z$ such that $g(Z)\subset S$ is a single point and $\mult_pZ=m$.
The corresponding coarse moduli space is usually called a
{\it Chow variety} in the algebraic case
and a {\it Barlet space} in the complex analytic setting; see
\cite{bar-mag} for their theory.
We need only a rough approximation of these by a countable union of projective morphisms. This can be easily derived from the classical theory of Chow varieties for projective spaces.
\begin{prop}\label{bir.chow.prop}
Let $g:X\to S$ be a proper morphism of complex analytic spaces that is bimeromorphic to a projective morphism. Fix $m\in \n$. Then there are countably many diagrams of complex analytic spaces over $S$
$$
\begin{array}{ccc}
C_i & \into & W_i\times_SX\\
w_i \downarrow \uparrow{\sigma_i} &&\\
W_i&&
\end{array}
\eqno{(\ref{bir.chow.prop}.1)}
$$
indexed by $i\in I$, such that
\begin{enumerate}\setcounter{enumi}{1}
\item the $w_i:C_i\to W_i$ are proper, of pure relative dimension 1 and flat over a dense, Zariski open subset $W_i^\circ\subset W_i$,
\item the fiber of $w_i$ over any $p\in W_i^\circ$ has multiplicity $m$ at $\sigma_i(p)$,
\item the $W_i$ are irreducible, the structure maps $\pi_i:W_i\to S$ are projective, and
\item the fibers over all the $W_i^\circ$ give all irreducible curves that have multiplicity $m$ at the marlked point.
\end{enumerate}
\end{prop}
Proof. By assumption there is a bimeromorphic morphism $r:Y\to X$ such that
$Y$ is projective over $S$.
The Chow variety of curves on $Y/S$ exists and its irreducible components are projective over $S$ (cf.\ \cite[Sec.I.5]{rc-book}). The universal curve over it parametrizes all pointed curves on $Y$.
If we have a family of pointed curves
$$
\begin{array}{ccc}
C_Y & \into & W\times_SY\\
w_Y \downarrow \uparrow{\sigma_Y} &&\\
W&&
\end{array}
\eqno{(\ref{bir.chow.prop}.6)}
$$
taking its image on $X$ gives
$$
\begin{array}{ccc}
C & \into & W\times Y\\
w \downarrow \uparrow{s} &&\\
W&&
\end{array}
\eqno{(\ref{bir.chow.prop}.7)}
$$
Here $w:C\to W$ is proper, of relative dimenson 1 and flat over
a dense, Zariski open subset $W^\circ\subset W$.
The multiplicity of a fiber at a section is an upper semicontinuous function on $W^\circ$. For each $m\in \n$ let $W^m\subset W$ denote the closure of the
set of points $p\in W^\circ$ for which $\mult_{\sigma(p)}C_p= m$.
We repeat this for all irreducible components of $W\setminus W^\circ$.
At the end we get countably many diagrams as in (\ref{bir.chow.prop}.1)
that satisfy (\ref{bir.chow.prop}.2--4) but not yet (\ref{bir.chow.prop}.5).
Let $X^\circ\subset X$ be the largest open set over which $r$ is an isomorphism. The above procedure gives all irreducible pointed curves that have nonempty intersection with $X^\circ$. Equivalently, all curves that are not contained in $X\setminus X^\circ$. (We may also get some curves contained in $X\setminus X^\circ$.)
We can now use dimension induction to get countably many diagrams that give us all curves on $g: (X\setminus X^\circ)\to S$. The union of these families with the previous ones gives
countably many diagrams
that satisfy (\ref{bir.chow.prop}.2--5). \qed
\section{Projectivity of very general fibers}
Here we prove that, under the assumptions of Theorem~\ref{12.2.10.thm.S}, there are many projetive fibers.
\begin{prop}\label{12.2.10.vgen.prop}
Notation and assumptions as in Theorem~\ref{12.2.10.thm.S}.
Then there is a Euclidean open neighborhood $0\in U\subset S$ and
countably many nowhere dense, closed, analytic subsets $\{ H_j\subset U: j\in J\}$, such that $X_s$ is projective for every $s\in U\setminus \cup_j H_j$.
\end{prop}
Proof. First choose $0\in U\subset S$ such
that $X_U$ retracts to $X_0$.
Since $X_0$ is projective, it has an ample line bundle $L$.
Let $\Theta\in H^2(X_U, \q)$ be the pull-back of
$c_1(L)$ to $X_U$. Note that $\Theta$ is a topological cohomology class that is usually not the Chern class of a holomorpic line bundle.
Consider now the diagrams (\ref{bir.chow.prop}.1) indexed by the contable set $I$. Let $J\subset I$ index those diagrams for which $H_i:=\pi_i(W_i)\subset S$ is nowhere dense in $U$.
We aim to show that
the restriction $\Theta_s:=\Theta|_{X_s}$ satisfies (\ref{sesh.top.ver.prop}.1) for $s\in U\setminus \cup_{j\in J} H_j$.
Indeed, pick a curve $C_s\subset X_s$ and a point $p_s\in C_s$.
Set $m=\mult_{p_s}C_s$.
By assumption there is an $i\in I\setminus J$ and a diagram as in (\ref{bir.chow.prop}.1)
$$
\begin{array}{ccc}
C_i & \into & W_i\times_SX\\
w_i \downarrow \uparrow{\sigma_i} &&\\
W_i&&
\end{array}
\eqno{(\ref{12.2.10.vgen.prop}.1)}
$$
with a dense, Zariski open subset $W^\circ_i\subset W_i$, such that
\begin{enumerate}\setcounter{enumi}{1}
\item $(C_s, p_s)$ is one of the fibers of $w_i$ over $W^\circ_i$,
\item $\mult_{\sigma(p)}C_p= m$ for all $p\in W^\circ_i$, and
\item $\pi_i:W_i\to S$ is projective and its image contains $0\in S$.
\end{enumerate}
Thus there is a disc $D$ (say of radius $>1$) and a holomorphic map
$\tau:D\to W_i$ such that $\pi_i(\tau(0))=0\in S$ and
$\pi_i(\tau (1))=s\in S$.
After pulling back and discarding embedded points we get
$$
\begin{array}{ccc}
C^D & \into & D\times_SX\\
w \downarrow \uparrow{\sigma} &&\\
D&&
\end{array}
\eqno{(\ref{12.2.10.vgen.prop}.2)}
$$
where $w$ is flat. Let $C^D_t$ denote the fiber over $t\in D$ and
$p_t=\sigma(t)$ the marked point. Thus $C^D_0$ lies over $0\in S$ and
$C^D_1$ is indentified with our original curve $C_s$.
Note that
$$
\mult_{p_t} C^D_t=\mult_{p_1} C^D_1=\mult_{p_s}C_s \qtq{for all}t\in D^\circ.
$$
Since multiplicity is an upper semi-continous function, we conclude that
$$
\mult_{p_0}C^D_0\geq \mult_{p_t}C^D_t=\mult_{p_s}C_s.
$$
Here $C^D_0 $ is a 1-cycle on the projective scheme $X_0$,
and $\Theta_0$ is the Chern class of an ample line bundle on $X_0$.
Thus
$$
\Theta\cap [C^D_0]\geq \epsilon\cdot \mult_{p_0}C^D_0,
$$
by the easy direction of Theorem~\ref{sesh.crit.thm}, where $\epsilon$ depends only on $X_0$ and $\Theta_0$.
Putting these together gives that
$$
\Theta_s\cap [C_s]=\Theta\cap [C^D_1]=\Theta\cap [C^D_0]
\geq \epsilon\cdot \mult_{p_0}C^D_0\geq \epsilon\cdot\mult_{p_s}C_s.
$$
Thus $X_s$ is projective by Proposition~\ref{sesh.top.ver.prop}.
\qed
\section{Projectivity of general fibers}
We prove a result about the set of projective fibers of proper analytic maps.
To formulate it, let $g:X\to S$ be a proper morphism of complex analytic spaces and set
$$
\operatorname{PR}_S(X):=\{s\in S: X_s \mbox{ is projective}\}.
$$
We prove in Proposition~\ref{RT.vb.Zd.M} that
$\operatorname{PR}_S(X) $ either contains a dense, Zariski open subset or
it is contained in a countable union of
Zariski closed, nowhere dense subsets.
Results of this type appear in many places, for example
\cite{iitaka, lie-ser, ueno-83, rao-tsa-1, rao-tsa}. All the arguments below can be found in them.
The next lemma is frequently stated in the algebraic case as in
\cite[p.43]{gm-book}, but the proof works for proper analytic maps as well. See, for example
\cite[p.10]{parusinski}.
\begin{lem}\label{top.of.prop.maps.1}
Let $g:X\to S$ be a proper morphism of complex analytic spaces.
Then there is a dense, Zariski open subset $S^\circ\subset S$ such that
$g^\circ:X^\circ\to S^\circ$ is a topologically locally trivial fiber bundle.
\qed
\end{lem}
Applying this inductively we get the following.
\begin{cor}\label{top.of.prop.maps.2}
Let $g:X\to S$ be a proper morphism of complex analytic spaces.
Then the sheaves $R^ig_*\z_X$ are constructible in the analytic Zariski topology. \qed
\end{cor}
\begin{prop} \label{RT.vb.Zd.M} Let $g:X\to S$ be a proper morphism of normal, irreducible analytic spaces.
Then there is a dense, Zariski open subset $S^\circ\subset S$ such that
\begin{enumerate}
\item either $X$ is locally projective over $S^\circ$,
\item or $\operatorname{PR}_{S}(X)\cap S^\circ$ is locally contained in a countable union of Zariski closed, nowhere dense subsets.
\end{enumerate}
\end{prop}
{\it Remark \ref{RT.vb.Zd.M}.3.} The following example clarifies (\ref{RT.vb.Zd.M}.2).
Let $L\subset \c^2$ be a general line and $Z$ its image in a complex torus
$\c^2/\z^4$. Then $Z$ is irreducible and everywhere dense in the Euclidean topology. However, if $U\subset \c^2/\z^4$ is contractible, then
$Z\cap U$ is a countable union of Zariski closed, nowhere dense subsets of $U$.
\medskip
Proof. By passing to a Zariski open subset, we may assume that
$R^2g_*\o_X$ is locally free and $R^2g_*\z_X$ is locally constant.
By passing to the universal cover, we may also assume that
$R^ig_*\z_X$ is a constant sheaf. The exponential sequence now shows that the holomorphic line bundles on $X$ are given by the kernel of
$$\partial: R^2g_*\z_X\to R^2g_*\o_X,$$ and for $s\in S$, the holomorphic line bundles on $X_s$ are given by the kernel of
$$
H^2(X_s, \z_{X_s})=R^2g_*\z_X|_s\to R^2g_*\o_X|_s= H^2(X_s, \o_{X_s}).
$$
Let $\Theta$ be a global section of $R^2g_*\z_X$. If $\partial\Theta\equiv 0$
then there is a global line bundle $L_\Theta$ corresponding to it.
If such an $L_\Theta$ is ample on some $X_s$, then it is ample over a
Zariski open neighborhood of $s$ and we are in case (\ref{RT.vb.Zd.M}.1). Assume now that this is not the case.
If $\partial\Theta$ is not identically $0$, then $\partial\Theta=0$ defines a Zariski closed, nowhere dense subset $H_\Theta\subset S$.
We claim that $\operatorname{PR}_{S}(X)$ is contained in the union of these
$H_\Theta$. Indeed, pick $s\in S$ in their complement. Then all line bundles on $X_s$ are restrictions of a line bundle on $X$ (up to numerical equivalence) and these are not ample by assumption. \qed
\begin{complem} \label{RT.vb.Zd.M.c} Notation and assumptions as in Proposition~\ref{RT.vb.Zd.M}. Assume in addition that $g$ is bimeromorphic to a projective morphism and we are in case (\ref{RT.vb.Zd.M}.1). Then $X$ is projective over $S^\circ$.
\end{complem}
Proof. Choose a projective modification $\pi:X'\to X$. Set $g':=g\circ \pi$. By passing to a
Zariski open subset, we may assume that
$R^2g_*\o_X$ and $R^2g'_*\o_{X'}$ are locally free, and $R^2g_*\z_X$ and $R^2g'_*\z_{X'}$ are locally constant. Since $g'$ is projective, the monodromy on the algebraic classes of $R^2g'_*\z_{X'}$ is finite.
Since $X$ is normal, a
holomorphic line bundle is trivial on $X$ iff its pull-back to $X'$ is trivial,
hence the natural map $R^2g_*\z_X\to R^2g'_*\z_{X'}$ is injective on algebraic classes. Therefore
the monodromy on the algebraic classes of $R^2g_*\z_{X}$ is also finite.
Once the monodromy on the kernel of $\partial: R^2g_*\z_X\to R^2g_*\o_X$
is finite, we can trivialize it by a finite cover of $S$.
We thus find a relatively ample line bundle $L_\Theta$ after a finite cover of $S$, which then gives a relatively ample line bundle on $X$. \qed
\begin{say}[Proof of Theorem~\ref{12.2.10.thm.S}]\label{12.2.10.thm.S.pf}
By Proposition~\ref{12.2.10.vgen.prop}, $\operatorname{PR}_{S}(X)$ contains the
complement of a countable union of Zariski closed, nowhere dense subsets.
Therefore, by the Baire category theorem, $\operatorname{PR}_{S}(X)$ is not contained in a countable union of closed, nowhere dense subsets. Thus we are in case (\ref{RT.vb.Zd.M}.1) and $g$ is locally projective over a
dense, Zariski open subset $S^\circ\subset S$. Global projectivity over $S^\circ$ is given by Complement~\ref{RT.vb.Zd.M.c}.
We finish the proof by induction applied to $S\setminus S^\circ$. \qed
\end{say}
\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\def\cprime{$'$} \def\dbar{\leavevmode\hbox to 0pt{\hskip.2ex
\accent"16\hss}d} \def\cprime{$'$} \def\cprime{$'$}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def\cprime{$'$}
\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\def\polhk#1{\setbox0=\hbox{#1}{\ooalign{\hidewidth
\lower1.5ex\hbox{`}\hidewidth\crcr\unhbox0}}} \def\cdprime{$''$}
\def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$} \def\cprime{$'$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2} | 210,922 |
Someone once observed that a faux pas is when someone accidentally blurts out the truth.
The short-fingered vulgarian did that today, and it caused such a reaction that he had to say he didn’t mean it.
It was a discussion of abortion, and particularly whether the extreme right could trust Trump’s commitment to the anti-abortion cause. After all, he is on record years ago being pro-choice, right? So he was being interviewed by Chris Matthews, and Matthews kept pushing the anti-abortion to the logical conclusion: if abortion is illegal, should women who get abortions be punished for it?
It makes total sense. Anti-choicers claim to believe that aborting a fetus is exactly the same as killing a living human being. If it is, then anyone who does it should be prosecuted for murder, right?
And what’s more, even if you don’t pull the trigger but you hire someone to do it you also get prosecuted for murder.
And Trump went along with the whole thing. For someone who is ” very smart, really very, very smart, believe me,” he apparently wasn’t smart enough to see where this was leading, or the likely consequences of this argument (kind of a habit with him, no?), so he plunged on ahead.
“The answer is there has to be some form of punishment,” Trump said.
“For the woman?” Matthews said.
Trump said, “Yes,” and nodded. Matthews pressed further: 10 days or 10 years? Trump said he didn’t know, and that it’s “complicated.”
“It will have to be determined,” Trump said.
Of course, by the end of the day he was walking back his statements because the anti-choicers had called him to heel. They say that they never supported punishment for the woman who obtains an abortion, and I suspect that this is true for several reasons.
First, it’s bad PR. I continue to believe that the anti-choice movement is composed primarily of people who think they don’t know anyone who has had an abortion. Still, they realize they’re out there, and they know they would seem heartless if they were calling for women who get abortions to go to prison, so they have decided not to pursue that remedy.
Second, and this is one area in which they are actually telling the truth, they consider the women who obtain abortions to be victims. Patronizing doesn’t even cover it. What they are really saying is that women do not have moral agency, so they are not responsible for their actions. Therefore, why prosecute them?
Finally, and they will never tell you this, deep down they really don’t consider fetuses full human beings the way they claim. They say they do, but they recognize that even when it’s a painful choice it’s not the same as murder. If they did, to be morally consistent they would have to push to prosecute the women for murder, just as they would like to prosecute the doctors.
So at the end of the day we owe Trump something. It won’t happen often, but on Wednesday he blurted out the truth and exposed the malevolent core of the anti-choice movement.
So thanks, Donald. You probably won’t hear it from me again.
4 thoughts on “Thank you, Donald Trump!”
Trump and the truth? Whatever do they see in each other?
I don’t agree – I think if the fanatics did push to make all abortions illegal, they would institute punishment for women who seek or get an abortion. Do you really think Ted Cruz would show any compassion for someone involved in an ‘illegal’ activity? In my jaded view, I don’t think he would.
Ironic, isn’t it? A pathological liar done in by his own inadvertent truth-telling.
I wouldn’t go so far to say that pro-lifers think that women have no moral agency. Patronizing, yes. But what they’re mostly afraid of is losing the 1950’s status quo, where no one had sex outside of marriage (as if that was ever true) and women were second-class citizens.
I also appreciate Donald Trump for pointing out the absolute hypocrisy, however inadvertent. | 41,189 |
Page 1 of 4
1
2
3
>
»
Show 40 post(s) from this thread on one page
Firearm & Gun Forum - FireArmsTalk.com
(
)
-
General Rifle Discussion
(
)
- -
Choosing an M1A style rifle
(
)
Ruzai
08-19-2011 07:47 AM
Choosing an M1A style rifle
I have a rare chance to buy a M1A style rifle at a good price and get to have it soon. The conditions are it has to be from Springfield Armory. There is one in-particular M1A style rifle that they sell that I cant buy for the good price but dont know what iit is (at the moment), but if that happens I'll just go with the better model just under that one and I cant have anything from the custom shop. (I'd be doing any custom work myself anyhow).
I want to know some of you guy's experiences with the M1A style rifles to help me decide. Should I get the Stainless Steel option for the barrel? Do the fiberglass stocks feel flimsy and a bit plasticy as some people keep saying? What kind of accuracy can I expect out of different models realisticly?
I'll be using this gun for a target gun and a fun-factor gun as well as a replacement for my FAL (I wanted something that's a bit more of an American Classic battle rifle). I wont be lugging the gun around in the woods since I dont want to mess up a good stock and a the potiential custom work on the stock I have planned.
I'm eyeballing the SuperMatch M1A and the M21 Tactical, any gripes, complaints, comments on any of the M1A rifles would be much appreciated.
Troy Michalik
08-19-2011 03:07 PM
I've got a SA M1A. I bought it back in '94 before the gun bill passed and prices got completely stupid. I bought a basic model with the synthetic stock. The stock is not flimsy at all and the parkerizing has held up great.
My experience with this rifle has been outstanding. The factory magazines have performed flawlessly, it is a breeze to take down and clean once you get the trigger group broke loose (mine was in really tight). It shot better than I did for a long time and probably still does half the time. :D
CA357
08-19-2011 04:01 PM
I
had
a synthetic stocked Scout Squad and will buy another if I ever get the dough again.
Txhillbilly
08-19-2011 04:28 PM
Ruzai,You ought to send M14sRock a PM.
He is the "Know it all" source for these rifles,and has quite a few in his arsenal.I'm sure he'd be more than willing to advise you on getting one.
dunerunner
08-19-2011 04:28 PM
I shot a very nice M1A belonging to mrm14. It was sweet!! Shot perfectly at 100 yards, offhand, with the bipod attached! I, however; was all over the target! :D
mrm14
08-19-2011 04:30 PM
I'd go with the Super Match M1A over the M-21 varient for accuracy. I beleive the Super Match has the barrel with 1 in 11 twist rifleing and that should allow you to run the heavier grain weight target bullets better. My Loaded has the 1 in 11 twist barrel and I have the best accutacy with the 175 SMK's. The 168"s run pretty good as well. I have an older "Loaded" M1A and when it was new it would lay out groups just at 1" @ 100 yards. Now with near 40,000 rounds down the pipe the grouping has opened up a bit to about 1-3/4" or so at 100 yards. Still pretty darn good for a side op rod rifle. Probably will re-barrel this one with a Kreiger barrel when I get around to it.
mrm14
08-19-2011 04:31 PM
Quote:
Originally Posted by
Txhillbilly
(Post 566060)
Ruzai,You ought to send M14sRock a PM.
He is the "Know it all" source for these rifles,and has quite a few in his arsenal.I'm sure he'd be more than willing to advise you on getting one.
+1 on asking M14s Rock.
mrm14
08-19-2011 04:32 PM
Quote:
Originally Posted by
dunerunner
(Post 566061)
I shot a very nice M1A belonging to mrm14. It was sweet!! Shot perfectly at 100 yards, offhand, with the bipod attached! I, however; was all over the target! :D
Hey dune, I was a little more than sloppy that day of shooting myself.:D We should have put Don behind it and let him have a go at it. I think he was shooting better than both of us that day.
lonyaeger
08-19-2011 05:02 PM
I have an M1A Socom with the synthetic stock. It is not flimsy at all.
Ruzai
08-19-2011 08:36 PM
The factory website says both the Super Match and the M-21 have a 1:10 twist. I like the heavy weight bullets out of whatever I shoot, so I'm pretty sure I'm gonna need that 1:10 twist. I dont think I NEED the Krieger Barrel on it but I wouldnt mind having it. I really like the adjustable stock on the M-21 while still having that traditional rifle stock feel.
mrm14 you might want to try some Outers Foul-Out to see if it'll close up the grouping some. I can imagine with that much copper and lead you're bound to have some build up, even with regular cleanings. I know a couple guys that shoot regularly and they swear by the stuff for keeping their barrels near factory clean.
I'll have to send M14sRock a PM and ask him about the specs and all that good stuff. I'm in the middle of that "hurry up and wait" stage of getting the proper identification for the rifle. I moved to Denver this last month and I need a Colorado ID card to buy the gun with the good deal, still waiting on a copy of my birth certficate to get here so I can go get one.
All times are GMT. The time now is
06:08 AM
.
Page 1 of 4
1
2
3
>
»
Show 40 post(s) from this thread on one page | 193,652 |
Cover Letter Examples For Dental Receptionists | Resume …
Best IT Cover Letter Examples | LiveCareer
Sample Cover Letters For Job Openings | The Letter Sample
Example Of To Whom It May Concern Cover Letter – Best …
General Cover Letter Examples | Letters – Free Sample Letters
Police Officer Cover Letter Example | Cover Letter …
Typical Cover Letter Format – Best Template Collection
Cover letter examples Uk ~ Document Blogs
Sample Cover Letter | Fotolip.com Rich image and wallpaper
Search Results for “Nursing Cover Letter Example …
Best Cover Letters For Resume | Resume Examples 2017
Information Technology (IT) Cover Letter Examples …
Dear Sir Or Madam Cover Letter Sample – Guamreview.Com
Examples Of Good Cover Letters | whitneyport-daily.com
SIMPLE COVER LETTER Easy Template PixSimple Cover Letter …
Resume Examples Templates: What is a Cover Letter For job …
Examples Of Cover Letter For Resume Template – Resume Builder
Cover Letter For Computer Technician | The Letter Sample
Cover Letter Format : Creating an Executive Cover Letter …
Sample Format Of Cover Letter For Job Application …
Cover Letter Examples Short | Resume Template || Cover Letter
Sample Cover Letter For An It Professional – Guamreview.Com
IT Cover Letter Examples
7+ easy cover letter examples | mail clerked
it cover letter sample for it project manager position …
professional cover letter | apa example
10 Relocation Cover Letter Examples for Resume – Writing …
Best Technical Support Cover Letter Examples | LiveCareer
Business Proposal Cover Letter Sample | The Letter Sample
sample of cover letter | Administrative Supervisor Cover …
Customer Service Skills For Cover Letter | Resume Template …
10+ cover letter samples to whom it may concern | memo heading
Free Cover Letter Examples for Every Job Search | LiveCareer
Great executive assistant cover letters. A Resume Sample …
basic cover letter sample | apa example
Cover Letter Examples Computer Science | The Letter Sample
project manager cover letter examples it project manager …
Cover Letter Sample Doc | Resume Template || Cover Letter
Cover Letter Examples – Resume Cv
10+ cover letter samples to whom it may concern | memo heading
Cover Letter Recent Graduate sample cover letter for …
Best Cover Letters | Letters – Free Sample Letters
Cover Letter Title Examples | Resume Template || Cover Letter
Resume Examples Templates: Cover Letter for Food Service …
Examples Of Cover Letter For Resume Template …
Cover Letter Examples Entry Level | The Letter Sample
Resume Cover Letter Template 2017 – Resume Builder
Simple Email Cover Letter Sample | The Letter Sample
Resume Cover Letter Examples For High School Students …
10 Unique Teaching assistant Cover Letter | letterideas.info
10+ Resume Cover Letter Examples – PDF
Cover Letter Examples Office Assistant office assistant …
First Job Cover Letter Examples | The Letter Sample
95 best images about Cover letters on Pinterest
6+ short cover letter examples | resume type
Best Cover Letters Examples – Best Letter Sample
Ceo Cover Letter Examples – Best Letter Sample
The Best Cover Letter One Executive – Writing Resume …
Best Management Cover Letter Examples | LiveCareer
Example Of Cover Letter For Resume Template – Resume Builder
Human Resources Internship Cover Letter | The Letter Sample
Cover Letter Samples. Sample Covering Letter For Cv Sample …
Project Coordinator Cover Letter Sample | The Letter Sample
Examples Of Cover Letter For Resume Template …
8+ basic cover letter samples | letter adress
Network Engineer Cover Letter Example | Example Cover Letter
sample–cover–letters-7 – Resume Cv
Example Of A Cover Letter For A Job Resume | Letters …
Student Cover Letter Sample | Resume Badak
Samples Of Cover Letters | | ingyenoltoztetosjatekok.com
Finest Cover Letter Resume Examples | Resume Examples 2018
Resume Examples Templates: Free Medical Office Assistant …
Resume Examples Templates: Resume Cover Letter …
Cover Letter Help | whitneyport-daily.com
Cover Letter Examples For Manager Position | Resume …
School Administrator Cover Letter Sample – Guamreview.Com
Effective Cover Letters and Templates
sample short cover letter | apa example
Cover Letter Sample For It Professional Image collections …
Cover Letter Examples Word Document | Resume Template …
Effective Cover Letter Format – Best Template Collection
10 General Cover Letter Sample – SampleBusinessResume.com …
11+ cover letters examples for students | mail clerked
Cover Letter Examples for It Jobs Example Cover Letter …
Examples Of Cover Letters | | jvwithmenow.com
Analyze Cover Letter Samples & Create Professional Ones
sample short cover letter | apa example
Examples of Cover Letters | Fotolip.com Rich image and …
Resume Examples Templates: How To Write Basic Cover Letter …
Job Application Cover Letter | | jvwithmenow.com
Entry Level Cover Letter Sample | Sample Cover Letters
Resume Examples Templates: Phlebotomist Cover Letter No …
Cover Letter Sample Job Vacancy Copy Resume Letter For …
Information Technology (IT) Cover Letter Sample | Resume …
Sample General Cover Letters – jantaraj.com
Free Sample Cover Letter For Entry Level Position – Cover …
Art Director Cover Letter Sample – Guamreview.Com
Cover Letter Sample | UVA Career Center
basic cover letter sample | apa example
Internship Application Letter Sample …
Cover Letter Not Qualified For Job | Resume Template …
Resume Examples Templates: Nursing Cover Letter Example …
Bartender Cover Letter No Experience | The Letter Sample
Sample Of Cover Letters | Michael Resume
Resume Examples Templates: Job Cover Letter Free Download …
12 Construction Manager Cover Letter Sample | Job and …
Cover Letter Format : Creating an Executive Cover Letter …
Example Of A Cover Letter | | jvwithmenow.com
Executive Cover Letter Examples | Resume Badak
Cover Letter Format Examples Template | Resume Builder
Receptionist Cover Letter Example – …
cover letter internship examples resume sample job …
Best Customer Service Cover Letter Examples | LiveCareer
general cover letter example general resume cover letter …
Best Healthcare Cover Letter Examples | LiveCareer
Cover Letter Model Sample It Job Formal Format For Request …
Cover Letters to whom It May Concern Examples …
Best IT Cover Letter Examples | LiveCareer
Admin cover letter examples – Business Proposal Templated …
12+ cover letter sample to whom it may concern | memo heading
Online Cover Letter Examples – Best Letter Sample
Cover Letter Samples : How to make it perfect?
Resume Cover Letter Template 2017 | Resume Builder
Cover Letter Examples For Medical Assistant | Cover Letter …
Cover Letter Sample For Trader Best Cover Letter Sample …
11+ sales cover letter examples | applicationleter.com
8+ consulting cover letter examples | memo heading
7+ financial analyst cover letter | precis format
10 Relocation Cover Letter Examples for Resume – Writing …
Sample Cover Letter For Resume Information Technology …
Cover Letter For Job No Experience | Resume Template …
16 Best Cover Letter Samples for Internship – WiseStep
Cover Letter Format : Creating an Executive Cover Letter …
Cover Letter Standard Format – Best Template Collection
Proper Sample Cover Letters For Resumes – Letter Format …
Cover Letter For Office Assistant – Resume Builder
| 333,024 |
DESIGN that’s smart!
Our uniform has been created in consultation with local trade and service industries. It incorporates Workplace Health and Safety requirements as well as a professional dress for the business or trade workplace.
All of the uniform items below are available through:
Joprim Uniform Shop
The rear of St Agnes Primary School carpark
Boronia Street, Port Macquarie
Phone: 02 6584 1076
Boys Uniform
- College knee length shorts or pants bearing the College’s initials on the waist band
- College polo shirt
- College jacket
- Covered polishable black shoes & white or black socks
- Caps – plain black or navy colours only
Girls Uniform
- Knee length black skirt, black pants, black knee length shorts (all must be purchased through Joprim bearing NSTC initials on the waist band).
- College polo shirt
- College blouse
- College jacket
- Covered polishable black shoes and white or black socks. ‘Slipper’ style shoes are not acceptable.
- Caps – plain black or navy colours only
Orders for Trade and Hospitality Clothing
The College has arranged suppliers, a list will be provided at time of enrolment.
Hospitality Uniform
The following items are essential for Hospitality students
Trade uniform
The following items are essential for Trade students
- College trade shirt, navy cotton drill (compulsory for Automotive and Metal & Engineering)
- College long sleeve trade shirt, high visibility (compulsory for Construction & Electrotechnology)
- Furniture Making students may wear either the College trade shirt, navy cotton drill or the College long sleeve trade shirt, high visibility.
- Leather work boots
- A Safety Pack is essential for all Trade students, including safety glasses, hearing protection and dust mask. Cost is approximately $20. This is available from the College or local suppliers
- The College provides specialist personal protective equipment including: welding gloves/masks and all other safety equipment required for workshop use.
Dress for Out of Uniform Days
Parents and students are reminded that all clothing worn on these days must be appropriate.
If a student arrives at school on one of these days in clothing deemed to be inappropriate, parents may be contacted and asked to collect the students in order to rectify the matter. It is inappropriate for brief and revealing clothing to be worn to school. This includes low-cut, strapless, backless or midriff tops, miniskirts and brief shorts. Students are also asked to consider the appropriateness of all writing and graphics on T-shirts. Due to Workplace Health & Safety regulations, thongs and other similar footwear may not be worn. All students must wear either joggers or some other kind of enclosed footwear | 403,799 |
Copyright (C) 2009 - 2013 Societá Agricola Verde Vita, All Rights Reserved
di Andrea e Luca Frosini via Loreto e Carraiola, 7/E 51100 Masiano - PISTOIA - ITALY - P.Iva 01293600472
Societ? Agricola VERDE VITA s.s.
Versione 2.0.5 del 20/09/2013 by
Elle
Gi
Software
GPS
+43 59' 29.59"
49,891553
10 55' 29.08"
10,924745. | 409,862 |
Chanda Tresvant is a singer born and raised in CALIFORNIA with 3 siblings a twin sister Claudia , older sister Taryn and younger brother Derek Tresvant.Who mother is Donna Walker and Father Derek Tresvant . Chanda Tresvant comes from a musical Background who father is a radio promoter who was in billboard magazine. Chanda Tresvant is also related to Ralph Tresvant known as the lead singer to the legendary group New Edition! Growing up in the valley of California Chanda Tresvant went to performing arts Schools .She was in school talent shows when she was in elementary and middle school and joined the school play . Growing up in a middle class family there was its ups and down just like any other family. Chanda Tresvant grew up singing from the time she can talk but didn’t decide to take it seriously until she graduated High School and decided to pursue the career. During high school every sat she would take acting classes with Famous actor Todd Bridges mother to help her perfect being a actress . After high school she went being a video girl for rappers such as Father,Young Dolph,Yg,Nipsey Hussle,Decadez ,and ext…….. in 2018 Chanda Tresvant still resides in California in the valley and came out with a new single “Why they Actin?” that debuted on Feb 14 on spotify and itunes !
Q:First of all tell us about the start of your professional career?
The beginning of my career was me wanting to be a artist but me not knowing how to go about doing it so I had to get in where I fit in ! When I finished high school I received a message on social media asking can I be apart of a music video ! So I went and did it but I was scared because I didn’t know who the guy was ! After I shot the music video it ended up on MTV and I would have to say that’s where it all started ! I just kept on going and more people started to ask me to be in there music video !Deep inside it’s not what I was trying to do but it was my only way to make connections in the music industry .
Q:What advice would you give to beginners who are nervous?
The best advice I would give to beginners who is nervous , is that confidence is the key ! if you don’t believe in yourself nobody else will! When your nervous it would show through your music and your stage performance. You have to make people believe your number 1.
Q:When you decide it’s time to make a new record, is that more exciting or stressful?
When it’s time to make a new record for me it’s exciting because I’m always curious on what new sound me and my producer would make .
Q:How easily do songs tend to come to you?
A song may come to me in 1 day but will take me a week to perfect it .
Q: What’s your motto or the advice you live by?
The advice that I live buy is to ignore negativity, don’t respond because silence is Golden
Q: What is you favorite song to belt out at the bar/in the car right now?
So many good songs are out I don’t really have a favorite
Q :What are your future goals as a music artist?
My future goal as a music artist is to leave my mark in the music industry as a multi talented artist . And to not only sing/rap also go into acting and more modeling because I love it all.
Q: What was your most memorable performance to date?
As if today I haven’t started touring yet but it’s in progress very soon !
Q:How can we follow you on social media ??
You can follow me on all social medias instagram,twitter,and Snapchat I’m the “trapanesechik” and Facebook I’m “Chanda Tresvant”.
Chanda Tresvant social media links | 97,117 |
TITLE: Classifying singularities of an entire function at infinity
QUESTION [1 upvotes]: In preparing for an examination, I have run across the following problem that has me stumped:
Let $f$ be an entire function, where $f(0)=\alpha, \alpha\in\mathbb{R}$, and for all $z\in\mathbb{C}\backslash\{0\},\ |f(z)f(1/z)|\leq 1$. Classify the singularities at $z=\infty$ of $f$ for all values of $\alpha$.
In taking limits, I believe that it is clear that if $\alpha\neq 0$, $f$ has an removable singularity at $z=\infty$. However, I am having trouble seeing what happens if $\alpha=0$.
I know that if $f$ is entire, $f$ is a polynomial if and only if $f$ has a pole at $z=\infty$. Also, I have attempted using Schwarz's Lemma, but was not able to come to a conclusion as to whether $f$ has a pole or an essential singularity at $\infty$.
What would be the proper path to classify the infinite singularities given a polynomial with no constant term?
In advance, thank you for any help provided.
REPLY [2 votes]: The relevant bit of technique here is to factor out the zero when $\alpha = 0$. Note in the case where $\alpha \neq 0$, $f$ is an entire function with a removable singularity at $\infty$, and hence is a constant.
For $\alpha = 0$, provided that $f$ is not uniformly $0$, then $f(z) = z^k h(z)$ where $h(0) \neq 0$ is entire. Then for any $z \in \mathbb{C} - 0$, $$|h(z) h(1/z)| = \left| h(z) z^k h(1/z) (1/z)^k \right| = |f(z) f(1/z)| \leq 1.$$
So by previous analysis, $h(z)$ is an entire function satisfying our condition with $h(0) \neq 0$, and hence must in fact be a constant. So $f(z) = c z^k$ for some $c \in \mathbb{C}$, and in particular $f(z)$ has a pole at $\infty$ or is the $0$ function. | 127,348 |
Warning: include() [function.include]: failed to open stream: HTTP request failed! HTTP/1.1 404 Not Found
in /home/viralvis/public_html/vv_thirty_thirty.php on line 231
Warning: include() [function.include]: Failed opening '' for inclusion (include_path='.:/usr/lib/php:/usr/local/lib/php') in /home/viralvis/public_html/vv_thirty_thirty.php on line 231
61260 members
Since the introduction of the first visitor exchange in the late 90s, and the launch of our first visitor exchange, ViralVisitors.com in 2001, thousands of visitor exchanges have popped up- and continue to do so.
Since launching our first visitor exchange, we have introduced 4 additional visitor exchanges and have
continually been a leader in the evolution of effective visitor exchange advancements.
The majority of visitors exchanges on the 'Net today, are based on encouraging a small percentage to surf
hundreds if not thousands of member pages per day. This has proven to be a less effective method of producing
positive results for the majority of members. One of the biggest problems with this type of visitor exchange
activity is the majority of people surfing this method usually become brain dead by the time they surf a
couple hundred sites in one day.
Other problems arising with the visitor exchange industry, are those unscrupulous people who are using scams
and ponzis, camouflaged as a visitor exchange, to illegally relieve unsuspecting Cyber Citizens of their money.
To counter-act these measures, we have introduced stringent guidelines on what is acceptable to submit to our
visitor exchange programs, and we are leaning/leading new membership activity towards more average Cyber Citizens,
including but not limited too, the wide range of Blog Publishers on the internet today.
The fact is, a majority of people using the Internet are busy, active people, who have limited time available to
do anything on the Internet or elsewhere. To accommodate these 'average' Internet users, we have re-configured
our visitor exchange software programming, to accommodate and give MAXIMUM exposure to people who surf 30 sites
per day.
Any surfer on our exchanges, may surf 30 members sites on all 5 exchanges simultaneously in an average of
15 or 20 minutes (see our FREE MoneyLegs.com Morphing Newbies Series for details). By doing this activity
for 15 or 20 minutes a day, the average surfer can build up to 1,000+++ visitors to their website, per program,
in any given month. That's a total of 5000+++ VISITORS a MONTH from our 5 visitor exchanges alone!
The dual purpose of having 1000's of people surfing 30 visitor's sites a day, rather than having a smaller
percentage surf 100's of sites a day, is we will grow into a stronger community AND, the majority of these
people will actually be alert to what they are seeing, producing more positive results for visitors coming to
your site.
The second part of the '30/30 PLAN' is related to a number showing in the top of your members area on each
of our exchanges. This number will range between 30 and 0 with a '30' view ranking getting priority viewing
over page view rankings decreasing to 0. By surfing '30' sites per day you are guaranteed to keep your
priority at '30' and your page in the top of our member surf viewing queue. No worries if you are unable to
surf everyday. On any given day a member surfs 100 visitor sites, their ranking will automatically be reset
to '30'. Since members are only required to surf 100 members sites per month to activate their account, this
will automatically set your account to '30', allowing for holidays and unexpected absences from the system.
We realize that people cannot be expected to surf 7! days a week, 365 days a year, so we have allowed the
opportunity to reset the count to '30' simply by surfing 100 visitor sites at any given time.
SO to KEEP IT SUPER SIMPLE! Just keep your account's page view ranking at '30' as much as you can.
We are focusing our growth on the Blogger Population because there are currently between 3.5 and 5 MILLION
Blog Publishers on the Internet. The vast majority of these publishers are maintaining a Blog with a serious
interest in having other people read their entries. Currently, the majority of these Blog Publishers are
unfamiliar with the potential to expand their readership using quality visitor exchange programs. On June
1st 2005, we have introduced our own Blog Community, BLOGestates.com. BLOGestates in a hybrid of many months
of research and tracking to develop the most secure, easy to use, Blog system on the Internet. BLOGestates.com has multiple aspects that will attract both regular r! eadership and Blog Publishers. We have reconfigured all
our visitor exchanges to be easily integrated with Blog Publishers and Blog Readership development. BLOGestates.com is designed to Always BE FREE to private publishers in any genre, but for commercial related Blogs, there is a
membership fee of $5. 99 a month or $59.99 a year. The commercial membership includes several additional benefits.
Please visit BLOGestates.com for details.
Thank you for taking the time to read this page. Please join MoneyLegs.com free to read through our Morphing Newbies Series. These 10 courses are designed to help anyone with a legitimate website get effective long
term visitor traffic and results.
NOTE: One of the key perks of having a Commercial Blog account on BLOGestates is one of our unique innovations
that has been a part of all of our programs since the late 90s, called the 'Follow ME' Link directory.
Understanding and filling out the Link Directory is very critical in expanding your traffic potential.
Any person that joins our visitor exchanges from your links guarantees you 30% Bonus Credits on all surfing
activity generated by anyone that you have personally referred to our visitor exchanges. Since links to all
of our programs are included in each of our programs, filling in your 'Follow ME' Link directories you can
easily begin earning 1,000+++ free visitors a month to your site - even if you are not actively referring
to our visitor exchanges. Our 'Follow ME' Link directory system is also included on BLOGestates, so simply
by having a registered membership with our exchanges, surfing 30 sites a day will also enhance your BLOG
rankings on search engines. As your readership expands, other publishers will come into our system,including
our visitor exchanges.
As this cause and effect evolves, Bonus Credits will see visitors to your site turn into a
Viral Traffic MONSTER.
The ViralVisitors GUARANTEE will assure you one of the most
refreshing and unique Click Xchange surfing and response
experiences on the 'Net. We are committed to take Click
Xchange/Startpage programs to a new level of Guaranteed
FREE Visitor Traffic to your Website. | 252,960 |
\begin{document}
\title[$t$--analogs of $q$--characters]
{Quiver varieties and
$t$--analogs of $q$--characters of quantum affine algebras
}
\author{Hiraku Nakajima}
\address{Department of Mathematics, Kyoto University, Kyoto 606-8502,
Japan
}
\email{[email protected]}
\urladdr{http://www.kusm.kyoto-u.ac.jp/\textasciitilde nakajima}
\thanks{Supported by the Grant-in-aid
for Scientific Research (No.11740011), the Ministry of Education,
Japan.}
\subjclass{Primary 17B37;
Secondary 14D21, 14L30, 16G20}
\begin{abstract}
Let us consider a specialization of an untwisted quantum affine
algebra of type $ADE$ at a nonzero complex number, which may or may
not be a root of unity.
The Grothendieck ring of its finite dimensional representations has
two bases, simple modules and standard modules.
We identify entries of the transition matrix with special values of
``computable'' polynomials, similar to Kazhdan-Lusztig polynomials.
At the same time we ``compute'' $q$-characters for all simple modules.
The result is based on ``computations'' of Betti numbers of
graded/cyclic quiver varieties.
(The reason why we put `` " will be explained in the end of the
introduction.)
\end{abstract}
\maketitle
\tableofcontents
\section*{Introduction}
Let $\g$ be a simple Lie algebra of type $ADE$ over $\C$, $\Lg = \g
\otimes \C[z,z^{-1}]$ be its loop algebra, and $\Ul$ be its quantum
universal enveloping algebra, or the quantum loop algebra for short.
It is a subquotient of the quantum affine algebra $\Ua$, i.e., without
central extension and degree operator. Let $\Ule$ be its
specialization at $q=\ve$, a nonzero complex number. (See
\secref{sec:qloop} for definition.)
It is known that $\Ule$ is a Hopf algebra. Therefore the category
$\mathscr Rep\Ule$ of finite dimensional representations of $\Ule$ is
a monoidal (or tensor) abelian category. Let $\operatorname{Rep}\Ule$
be its Grothendieck ring. It is known that $\operatorname{Rep}\Ule$ is
commutative (see e.g., \cite[Corollary~2]{FR}).
The ring $\operatorname{Rep}\Ule$ has two natural bases, simple
modules $L(P)$ and standard modules $M(P)$, where $P$ is the Drinfeld
polynomial. The latters were introduced by the author \cite{Na-qaff}.
The purpose of this article is to ``compute'' the transition matrix
between two bases.
(The reason why we put `` " will be explained in the end of the
introduction.)
More precisely, we define certain ``computable'' polynomials
$Z_{PQ}(t)$, which are analogs of Kazhdan-Lusztig polynomials for
Weyl groups. Then we show that the multiplicity $[M(P):L(Q)]$ is equal
to $Z_{PQ}(1)$.
This generalizes a result of Arakawa~\cite{Arakawa} who expressed the
multiplicities by Kazhdan-Lusztig polynomials when $\g$ is of type
$A_n$ and $\ve$ is not a root of unity.
Furthermore, coefficients of $Z_{PQ}(t)$ are equal to multiplicities of
simple modules of subquotients of standard module with respect to a
Jantzen filtration if we combine our result with \cite{Gr}, where the
transvesal slice used there was given in \cite{Na-qaff}.
Since there is a slight complication when $\ve$ is a root of unity, we
assume $\ve$ is {\it not\/} so in this introduction.
Then the definition of $Z_{PQ}(t)$ is as follows.
Let $\bfR_t \defeq \operatorname{Rep}\Ule\otimes_\Z\Z[t, t^{-1}]$,
which is a $t$--analog of the representation ring.
By \cite{Na-qaff}, $\bfR_t$ is identified with the dual of the
Grothendieck group of a category of perverse sheaves on affine graded
quiver varieties (see \secref{sec:quiver} for the definition) so that
(1) $\{ M(P)\}$ is the specialization at $t=1$ of the dual base of
constant sheaves of strata, extended by $0$ to the complement,
and
(2) $\{ L(P)\}$ is that of the dual base of intersection cohomology
sheaves of strata.
A property of intersection cohomology complexes leads to the
following combinatorial definition of $Z_{PQ}(t)$:
Let $\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$
be the involution on $\bfR_t$, dual to the Grothendieck-Verdier duality.
We denote the two bases of $\bfR_t$ by the same symbols $M(P)$, $L(P)$
at the specialization at $t=1$ for simplicity.
Let us express the involution in the basis $\{ M(P) \}_P$, classes of
standard modules:
\begin{equation*}
\overline{M(P)} = \sum_{Q: Q\le P} u_{PQ}(t) M(Q),
\end{equation*}
where $\le$ is a certain ordering $<$ among $P$'s. We then define an
element $L(P)$ by
\begin{equation}\label{eq:can}
\overline{L(P)} = L(P), \qquad
L(P) \in M(P) + \sum_{Q: Q<P} t^{-1}\Z[t^{-1}] M(Q).
\end{equation}
The above polynomials $Z_{PQ}(t)\in \Z[t^{-1}]$ are given by
\begin{equation*}
M(P) = \sum_{Q: Q\le P} Z_{PQ}(t) L(Q).
\end{equation*}
The existence and uniqueness of $L(P)$ (and hence of $Z_{PQ}(t)$) is
proved exactly as in the case of the Kazhdan-Lusztig polynomial.
In particular, it gives us a combinatorial algorithm computing
$Z_{PQ}(t)$, once $u_{PQ}(t)$ is given.
In summary, we have the following analogy:
\begin{center}
\begin{tabular}{|c|c|}
\hline
$\bfR_t$ & the Iwahori-Hecke algebra $H_q$
\\
standard modules $\{ M(P)\}_P$ & $\{ T_w \}_{w\in W}$
\\
simple modules $\{ L(P)\}_P$ & Kazhdan-Lusztig basis $\{ C'_w \}_{w\in W}$
\\
\hline
\end{tabular}
\end{center}
See \cite{KL} for definitions of $H_q$, $T_w$,
$C'_w$.
The remaining task is to ``compute'' $u_{PQ}(t)$.
For this purpose we introduce a $t$--analog $\widehat\chi_{\ve,t}$
of the $q$--character, or $\ve$--character.
The original $\ve$--character $\chi_\ve$, which is a specialization of
our $t$--analog at $t=1$, was introduced by Knight~\cite{Kn} (for
Yangian and generic $\ve$) and Frenkel-Reshetikhin~\cite{FR} (for
generic $\ve$) and Frenkel-Mukhin~\cite{FM2} (when $\ve$ is a root of
unity).
It is an injective ring homomorphism from $\operatorname{Rep}\Ule$ to
$\Z[Y_{i,a}^\pm]_{i\in I,a\in\C^*}$, a ring of Laurent polynomials of
infinitely many variables.
It is an analog of the ordinary character homomorphism of the finite
dimensional Lie algebra $\mathfrak g$.
Our $t$--analog is an injective $\Z[t,t^{-1}]$-linear map
\begin{equation*}
\widehat\chi_{\ve,t} \colon
\bfR_t
\to \widehat{\mathscr Y}_t \defeq
\Z[t, t^{-1}, V_{i,a}, W_{i,a}]_{i\in I,a\in\C^*}.
\end{equation*}
We have a simple, explicit definition of an involution
$\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$ on
$\widehat{\mathscr Y}_t$ (see \eqref{eq:bar}). The involution on
$\bfR_t$ is the restriction. Therefore the matrix $(u_{PQ}(t))$ can be
expressed in terms of values of $\widehat\chi_{\ve,t}(M(P))$ for all
$P$.
We define $\widehat\chi_{\ve,t}$ as the generating function of
Betti numbers of nonsingular graded/cyclic quiver varieties.
We axiomize its properties. The axioms are purely combinatorial
statements in $\widehat{\mathscr Y}_t$, involving no geometry nor
representation theory of $\Ule$. Moreover, the axioms uniquely
characterize $\widehat\chi_{\ve,t}$, and give us an algorithm for
computation. Therefore the axioms can be considered as a definition of
$\widehat\chi_{\ve,t}$. When $\g$ is not of type $E_8$, we can
directly prove the existence of $\widehat\chi_{\ve,t}$ satisfying the
axioms without using geometry or representation theory of $\Ule$.
Two of the axioms are most important. One is the characterization of
the image of $\widehat\chi_{\ve,t}$. Another is the multiplicative
property.
The former is a modification of Frenkel-Mukhin's result~\cite{FM}.
They give a characterization of the image of $\chi_\ve$, as an analog
of the Weyl group invariance of the ordinary character homomorphism.
And they observed that the characterization gives an algorithm
computing $\chi_\ve$ at {\it l\/}--fundamental representations. This
property has no counterpart in the ordinary character homomorphism for
$\g$, and is one of the most remarkable feature of $\chi_\ve$.
We use a $t$--analog of their characterization to ``compute''
$\widehat\chi_{\ve,t}$ for {\it l\/}--fundamental representations.
A standard module $M(P)$ is a tensor product of {\it l\/}--fundamental
representations in $\operatorname{Rep}\Ule$ (see \corref{cor:product}
or \cite{VV-std}). If $\widehat\chi_{\ve,t}$ would be a ring
homomorphism, then $\widehat\chi_{\ve,t}(M(P))$ is just a product of
$\widehat\chi_{\ve,t}$ of {\it l\/}--fundamental representations.
This is {\it not\/} true under usual ring structures on $\bfR_t$ and
$\widehat{\mathscr Y}_t$. We introduce `twistings' of multiplications
on $\bfR_t$, $\widehat{\mathscr Y}_t$ so that $\widehat\chi_{\ve,t}$
is a ring homomorphism. The resulted algebras are {\it not\/}
commutative.
We can add another column to the table above by \cite{Lu:can}.
\begin{center}
\begin{tabular}{|c|}
\hline
$\mathbf U^-_q$: the $-$ part of the quantized enveloping algebra
\\
PBW basis
\\
canonical basis
\\
\hline
\end{tabular}
\end{center}
In fact, when $\mathfrak g$ is of type $A$, affine graded quiver
varieties are varieties used for the definition of the canonical base
\cite{Lu:can}.
Therefore it is more natural to relate $\bfR_t$ to the dual of
$\mathbf U^-_q$.
In this analogy, $\widehat\chi_{\ve,t}$ can be considered as an analog
of Feigin's map from $\mathbf U^-_q$ to the skew polynomial ring
(\cite{IM,Jo,Be,Re}).
We also have an analog of the monomial base. ($E((\mathbf c))$ in
\cite[7.8]{Lu:can}. See also \cite{CX,Re}.)
This article is organized as follows. In \secref{sec:qloop} we recall
results on quantum loop algebras and their finite dimensional
representations.
In \secref{sec:rings} we introduce a twisting of the multiplication on
$\widehat{\mathscr Y}_t$.
In \secref{sec:axiom} we give axioms which $\widehat\chi_{\ve,t}$
satisfies and derive their consequences. In particular,
$\widehat\chi_{\ve,t}$ is uniquely determined from the axioms.
In \secref{sec:quiver} we introduce graded and cyclic quiver
varieties, which will be used to prove the existence of
$\widehat\chi_{\ve,t}$ satisfying the axioms.
In \S\ref{sec:proof1}, \ref{sec:proof2}, \ref{sec:proof3} we check
that a generating function of Betti numbers of nonsingular
graded/cyclic quiver varieties satisfies the axioms.
In \secref{sec:perverse} we prove that the characterization of simple
modules mentioned above.
In \secref{sec:pm 1} we study the case $\ve = \pm 1$ in detail.
In \secref{sec:conjecture} we state a conjecture concerning finite
dimensional representations studied in the literatures~\cite{NT,HKOTY}.
In this introduction and also in the main body of this article, we put
`` '' to the word `compute'. What we actually do in this article is to
give a purely combinatorial algorithm to compute somethings. The
author wrote a computer program realizing the algorithm for computing
$\widehat\chi_{\ve,t}$ for {\it l\/}--fundamental representations when
$\g$ is of type $E$. Up to this moment (2001, April), the program
produces the answer except two {\it l\/}--fundamental representations
of $E_8$. It took 3 days for the last successful one, and the remaing
ones are inaccessible so far. In this sense, our character formula is
{\it not\/} computable in a strict sense.
The result of this article for generic $\ve$ was announced in
\cite{Na-ann}.
\subsection*{Acknowledgement}
The author would like to thank D.~Hernandez and E.~Frenkel for pointing out
mistakes in an earlier version of this paper.
\section{Quantum loop algebras}\label{sec:qloop}
\subsection{Definition}
Let $\g$ be a simple Lie algebra of type $ADE$ over $\C$. Let $I$ be
the index set of simple roots.
Let $\{\alpha_i\}_{i\in I}$, $\{ h_i\}_{i\in I}$, $\{ \Lambda_i
\}_{i\in I}$ be the sets of simple roots, simple coroots and
fundamental weights of $\g$ respectively. Let $P$ be the weight
lattice, and $P^*$ be its dual. Let $P^+$ be the semigroup of dominant
weights.
Let $q$ be an indeterimate. For nonnegative integers $n\ge r$, define
\begin{equation*}
[n]_q \defeq \frac{q^n - q^{-n}}{q - q^{-1}}, \quad
[n]_q ! \defeq
\begin{cases}
[n]_q [n-1]_q \cdots [2]_q [1]_q &(n > 0),\\
1 &(n=0),
\end{cases}
\quad
\begin{bmatrix}
n \\ r
\end{bmatrix}_q \defeq \frac{[n]_q !}{[r]_q! [n-r]_q!}.
\end{equation*}
Later we consider another indeterminate $t$. We define a $t$-binomial
coefficient
\(
\left[\begin{smallmatrix}
n \\ r
\end{smallmatrix}\right]_t
\)
by replacing $q$ by $t$.
Let $\Ul$ be the quantum loop algebra associated with the loop algebra
$\Lg = \g\otimes \C[z,z^{-1}]$ of $\g$. It is an associative
$\Q(q)$-algebra generated by $e_{i,r}$, $f_{i,r}$ ($i\in I$,
$r\in\Z$), $q^h$ ($h\in P^*$), $h_{i,m}$ ($i\in I$, $m\in
\Z\setminus\{0\}$) with the following defining relation:
{\allowdisplaybreaks[4]
\begin{subequations}
\begin{gather*}
q^0 = 1, \quad q^h q^{h'} = q^{h+h'}, \quad
[q^h, h_{i,m}] = 0, \quad
[h_{i,m}, h_{j,n}] = 0,
\label{eq:relHH2}\\
q^h e_{i,r} q^{-h} = q^{\langle h,\alpha_i\rangle} e_{i,r},
\quad
q^h f_{i,r} q^{-h} = q^{-\langle h,\alpha_i\rangle} f_{i,r},
\label{eq:relHE'}
\\
( z - q^{\pm \langle h_j,\alpha_i\rangle} w)
\psi_i^s(z) x_j^\pm(w) =
( q^{\pm\langle h_j,\alpha_i\rangle} z - w)
x_j^\pm(w) \psi_i^s(z), \quad
\label{eq:relHE}
\\
\left[x_{i}^+(z), x_{j}^-(w)\right] =
\frac{\delta_{ij}}{q - q^{-1}}
\left\{\delta\left(\frac{w}{z}\right)\psi^+_i(w) -
\delta\left(\frac{z}{w}\right)\psi^-_i(z)\right\},
\label{eq:relEF}
\\
(z - q^{\pm 2\langle h_j,\alpha_i\rangle} w)
x_{i}^\pm(z) x_{j}^\pm(w)
= (q^{\pm 2\langle h_j,\alpha_i\rangle} z - w
) x_{j}^\pm(w) x_{i}^\pm(z),
\\
\sum_{\sigma\in S_b}
\sum_{p=0}^{b}(-1)^p
\begin{bmatrix} b \\ p\end{bmatrix}_{q}
x_{i}^\pm(z_{\sigma(1)})\cdots x_{i}^\pm(z_{\sigma(p)})
x_{j}^\pm(w)
x_{i}^\pm(z_{\sigma(p+1)})\cdots x_{j}^\pm(z_{\sigma(b)}) = 0,
\quad \text{if $i\neq j$,}
\label{eq:relDS}
\end{gather*}
\end{subequations}
where}
$s = \pm$,
$b = 1-\langle h_i, \alpha_j\rangle$,
and
$S_b$ is the symmetric group of $b$ letters.
Here $\delta(z)$, $x_i^+(z)$, $x_i^-(z)$, $\psi^{\pm}_{i}(z)$ are
generating functions defined by
{\allowdisplaybreaks[4]
\begin{gather*}
\delta(z) \defeq \sum_{r=-\infty}^\infty z^{r}, \qquad
x_i^+(z) \defeq \sum_{r=-\infty}^\infty e_{i,r} z^{-r}, \qquad
x_i^-(z) \defeq \sum_{r=-\infty}^\infty f_{i,r} z^{-r}, \\
\psi^{\pm}_i(z)
\defeq q^{\pm h_i}
\exp\left(\pm (q-q^{-1})\sum_{m=1}^\infty h_{i,\pm m} z^{\mp m}\right).
\end{gather*}
We} also need the following generating function
\begin{equation*}
p_i^\pm(z) \defeq
\exp\left(
- \sum_{m=1}^\infty \frac{h_{i,\pm m}}{[m]_{q}} z^{\mp m}
\right).
\end{equation*}
We have
\(
\psi^{\pm}_i(z) = q^{\pm h_i}
p_i^\pm(q z)/p_i^\pm(q^{-1} z).
\)
Let $e_{i,r}^{(n)} \defeq e_{i,r}^n / [n]_{q}!$,
$f_{i,r}^{(n)} \defeq f_{i,r}^n / [n]_{q}!$.
Let $\Uli$ be the $\Z[q,q^{-1}]$-subalgebra generated by
$e_{i,r}^{(n)}$, $f_{i,r}^{(n)}$ and $q^h$
for $i\in I$, $r\in \Z$, $h\in P^*$.
Let $\Uli^+$ (resp.\ $\Uli^-$) be $\Z[q,q^{-1}]$-subalgebra generated
by $e_{i,r}^{(n)}$ (resp.\ $f_{i,r}^{(n)}$) for $i\in I$, $r\in \Z$,
$n\in Z_{> 0}$.
Let $\Uli^0$ be the $\Z[q,q^{-1}]$-subalgebra generated by $q^h$, the
coefficients of $p_i^\pm(z)$ and
\begin{equation*}
\begin{bmatrix}
q^{h_i}; n \\ r
\end{bmatrix}
\defeq
\prod_{s=1}^r \frac{q^{h_i} q^{n-s+1} - q^{-h_i} q^{-n+s-1}}
{q^s - q^{-s}}
\end{equation*}
for all $h\in P$, $i\in I$, $n\in \Z$, $r\in \Z_{> 0}$. We have
$\Uli = \Uli^+\cdot \Uli^0\cdot \Uli^-$ (\cite[6.1]{CP-roots}).
Let $\ve$ be a nonzero complex number.
The specialization $\Uli\otimes_{\Z[q,q^{-1}]}\C$ with respect to the
homomorphism $\Z[q,q^{-1}]\ni q\mapsto \varepsilon\in\C^*$ is denoted
by $\Ule$. Set
\begin{equation*}
\Ule^\pm \defeq \Uli^\pm\otimes_{\Z[q,q^{-1}]}\C, \qquad
\Ule^0 \defeq \Uli^0\otimes_{\Z[q,q^{-1}]}\C.
\end{equation*}
It is known that $\Ul$ is isomorphic to a subquotient of the quantum
affine algebra $\Ua$ defined in terms of Chevalley generators $e_i$,
$f_i$, $q^h$ ($i\in I\cup \{0\}$, $h\in P^*\oplus\Z c$). (See
\cite{Dr,Be}.)
Using this identification, we define a coproduct on $\Ul$ by
\begin{equation*}\label{eq:comul}
\begin{gathered}
\Delta q^h = q^h \otimes q^h, \quad
\Delta e_i = e_i\otimes q^{-h_i} + 1 \otimes e_i,
\\
\Delta f_i = f_i\otimes 1 + q^{h_i} \otimes f_i.
\end{gathered}
\end{equation*}
Note that this is different from one in \cite{Lu-book}, although there
is a simple relation between them \cite[1.4]{Kas}.
The results in \cite{Na-qaff} hold for either comultiplication
(tensor products appear in (1.2.19) and (14.1.2)).
In \cite[\S2]{Na-ann} another comultiplication was used.
It is known that the subalgebra $\Uli$ is preserved under
$\Delta$. Therefore $\Ule$ also has an induced coproduct.
For $a\in\C^*$, there is a Hopf algebra automorphism $\tau_a$ of $\Ul$
given by
\begin{equation*}
\tau_a(e_{i,r}) = a^r e_{i,r}, \quad
\tau_a(f_{i,r}) = a^r f_{i,r}, \quad
\tau_a(h_{i,m}) = a^{m} h_{i,m}, \quad
\tau_a(q^h) = q^h.
\end{equation*}
It preserves $\Uli\otimes_{\Z[q,q^{-1}]}\C[q,q^{-1}]$ and induces an
autormorphism of $\Ule$, which is denoted also by $\tau_a$.
We define an algebra homomorphism from ${\mathbf U}_{\ve}(\g)$ to $\Ule$ by
\begin{equation}\label{eq:subalg}
e_i\mapsto e_{i,0},\quad f_i\mapsto f_{i,0}, \quad
q^h \mapsto q^h, \qquad (i\in I, h\in P^*).
\end{equation}
(See \cite[\S1.1]{Na-qaff} for the definition of ${\mathbf U}_{\ve}(\g)$.)
\subsection{Finite dimensional representation of $\Ule$}
Let $V$ be a $\Ule$-module. For $\lambda\in P$, we define
\begin{equation*}
V_\lambda \defeq \left\{ v\in V\left|\,
q^{h} v = \ve^{\langle h,\lambda\rangle} v,
\begin{bmatrix}
q^{h_i}; 0 \\ r
\end{bmatrix} v =
\begin{bmatrix}
\langle h_i, \lambda\rangle \\ r
\end{bmatrix}_{\ve} v \right\}\right..
\end{equation*}
The module $V$ is said to be of {\it type $1$\/} if $V =
\bigoplus_\lambda V_\lambda$.
In what follows we consider only modules of type $1$.
By \eqref{eq:subalg} any $\Ule$-module $V$ can be considered as a
${\mathbf U}_{\ve}(\g)$-module. This is denoted by
$\operatorname{Res}V$. The above definition is based on the definition
of type $1$ representation of ${\mathbf U}_{\ve}(\g)$, i.e., $V$ is of
type $1$ if and only if $\operatorname{Res}V$ is of type $1$.
A $\Ule$-module $V$ is said to be an {\it l--highest weight module\/}
if there exists a vector $v$ such that
\(
\Ule^+\cdot v = 0
\),
\(
\Ule^0\cdot v \subset \C v
\)
and
\(
V = \Ule\cdot v
\).
Such $v$ is called an {\it l--highest weight vector}.
\begin{Theorem}[\cite{CP-roots}]
A simple {\it l\/}--highest weight module $V$ with an {\it
l\/}--highest weight vector $v$ is finite dimensional if and only
if there exists an $I$--tuple of polynomials $P = (P_i(u))_{i\in I}$
with $P_i(0) = 1$ such that
\begin{gather*}
q^h v = \ve^{\langle h, \sum_i \deg P_i \Lambda_i\rangle} v,
\quad
\begin{bmatrix}
q^{h_i}; 0 \\ r
\end{bmatrix} v =
\begin{bmatrix}
\deg P_i \\ r
\end{bmatrix}_{\ve} v,
\\
p_i^+(z) v = P_i(1/z) v, \quad
p_i^-(z) v = c_{P_i}^{-1} z^{\deg P_i} P_i(1/z) v,
\end{gather*}
where $c_{P_i}$ is the top term of $P_i$, i.e., the coefficient of
$u^{\deg P_i}$ in $P_i$.
\end{Theorem}
The $I$-tuple of polynomials $P$ is called the {\it l--highest
weight}, or the Drinfeld polynomial of $V$.
We denote the above module $V$ by $L(P)$ since it is
determined by $P$.
For $i\in I$ and $a\in\C^*$, the simple module $L(P)$ with
\begin{equation*}
P_i(u) = 1 - au, \qquad
P_j(u) = 1 \quad\text{if $j\neq i$}
\end{equation*}
is called an {\it l--fundamental representation\/} and denoted by
$L(\Lambda_i)_a$.
Let $V$ be a finite dimensional $\Ule$-module with the weight space
decomposition $V = \bigoplus V_\lambda$. Since the commutative
subalgebra $\Ule^0$ preserves each $V_\lambda$, we can further
decompose $V$ into a sum of generalized simultaneous eigenspaces of
$\Ule^0$.
\begin{Theorem}[{\protect \cite[Proposition 1]{FR}, \cite[Lemma
3.1]{FM2}, \cite[13.4.5]{Na-qaff}}]
Simultaneous eigenvalues of $\Ule^0$ have the following forms:
\begin{gather*}
\ve^{\langle h, \deg Q^1_i - \deg Q^2_i\rangle}\quad \text{for $q^h$},
\qquad
\begin{bmatrix}
\deg Q^1_i - \deg Q^2_i \\ r
\end{bmatrix}_\ve \quad \text{for $\begin{bmatrix}
q^{h_i}; 0 \\ r
\end{bmatrix}$},
\\
\frac{Q^1_i(1/z)}{Q^2_i(1/z)}\quad \text{for $p_i^+(z)$},
\qquad
\frac{c_{Q^1_i}^{-1}z^{\deg Q^1_i} Q^1_i(1/z)}
{c_{Q^2_i}^{-1}z^{\deg Q^2_i}Q^2_i(1/z)}\quad \text{for $p_i^-(z)$},
\end{gather*}
where $Q^1_i$, $Q^2_i$ are polynomials with $Q^1_i(0) = Q^2_i(0) = 1$
and $c_{Q^1_i}$, $c_{Q^2_i}$ are as above.
\end{Theorem}
We simply write the $I$-tuple of rational functions
$(Q^1_i(u)/Q^2_i(u))$ by $Q$.
A generalized simultaneous eigenspace is called an {\it l--weight
space}. The corresponding $I$-tuple of rational functions is called
an {\it l--weight}. We denote the {\it l\/}--weight space by $V_Q$.
The $q$--character, or $\ve$--character \cite{FR,FM2} of a finite
dimensional $\Ule$-module $V$ is defined by
\begin{equation*}
\chi_\ve(V) = \sum_Q \dim V_Q\; e^Q.
\end{equation*}
The precise definition of $e^Q$ will be explained in the next section.
\subsection{Standard modules}
We will use another family of finite dimensional {\it l\/}--highest weight
modules, called standard modules.
Let $\bw\in P^+$ be a dominant weight. Let $w_i = \langle
h_i,\bw\rangle\in\Z_{\ge 0}$.
Let $G_{\bw} = \prod_{i\in I} \GL(w_i,\C)$.
Its representation ring
$R(G_{\bw})$ is the invariant part of the Laurant polynomial ring:
\begin{equation*}
R(G_{\bw})
= \Z[x_{1,1}^\pm,\dots, x_{1,w_1}^\pm]^{\mathfrak S_{w_1}}
\otimes \Z[x_{2,1}^\pm,\dots,
x_{2,w_2}^\pm]^{\mathfrak S_{w_2}}
\otimes\cdots\otimes
\Z[x_{n,1}^\pm,\dots, x_{n,w_n}^\pm]^{\mathfrak S_{w_n}},
\end{equation*}
where we put a numbering $1,\dots,n$ to $I$.
In \cite{Na-qaff}, we constructed a $\Uli\otimes_{\Z} R(G_\bw)$-module
$M(\bw)$ such that it is free of finite rank over
$R(G_\bw)\otimes\Z[q,q^{-1}]$ and has a vector $[0]_{\bw}$ satisfying
\begin{subequations}
\begin{gather*}
e_{i,r}[0]_\bw = 0\quad\text{for any $i\in I$, $r\in \Z$},
\\
M(\bw) = \left(\Uli^-\otimes_{\Z} R(G_\bw)\right)[0]_\bw,
\label{eq:span}
\\
q^h [0]_\bw = q^{\langle h,\bw\rangle} [0]_\bw,
\\
p_i^+(z)[0]_\bw
= \prod_{p=1}^{w_i} \left(1-\frac{x_{i,p}}{z}\right)[0]_\bw,
\\
p_i^-(z)[0]_\bw
= \prod_{p=1}^{w_i} \left(1-\frac{z}{x_{i,p}}\right)[0]_\bw,
\end{gather*}
\end{subequations}
If an $I$-tuple of monic polynomials $P(u) =
(P_i(u))_{i\in I}$ with $\deg P_i = w_i$ is given,
then we define a {\it standard module\/} by the specialization
\begin{equation*}
M(P) = M(\bw)\otimes_{R(G_\bw)[q,q^{-1}]} \C,
\end{equation*}
where the algebra homomorphism $R(G_\bw)[q,q^{-1}]\to \C$ sends
$q$ to $\ve$ and $x_{i,1},\dots, x_{i,w_k}$ to roots of $P_i$.
The simple module $L(P)$ is the simple quotient of $M(P)$.
The original definition of the universal standard module
\cite{Na-qaff} is geometric. However, it is not difficult to give an
algebraic characterization. Let $M(\Lambda_i)$ be the universal
standard module for the dominant weight $\Lambda_i$. It is a
$\Uli[x,x^{-1}]$-module. Let $W(\Lambda_i) =
M(\Lambda_i)/(x-1)M(\Lambda_i)$. Then we have
\begin{Theorem}[{\protect\cite[1.22]{Na-tensor}}]
Put a numbering $1,\dots,n$ on $I$. Let $w_i = \langle
h_i,\bw\rangle$. The universal standard module $M(\bw)$ is the
$\Uli\otimes_\Z R(G_\lambda)$-submodule of
\[
W({\Lambda_1})^{\otimes w_1}
\otimes \cdots\otimes
W({\Lambda_n})^{\otimes w_n}
\otimes
\Z[q,q^{-1},x_{1,1}^\pm,\dots, x_{1,w_1}^\pm,
\cdots,
x_{n,1}^\pm,\dots, x_{n,w_n}^\pm]
\]
\rom(the tensor product is over $\Z[q,q^{-1}]$\rom)
generated by
\(
\bigotimes_{i\in I} [0]_{\Lambda_i}^{\otimes \lambda_i}
\).
\rom(The result holds for the tensor product of any order.\rom)
\end{Theorem}
It is not difficult to show that $W(\Lambda_i)$ is isomorphic to a
module studied by Kashiwara~\cite{Kas2} ($V(\lambda)$ in his
notation). Since his construction is algebraic, the standard module
$M(\bw)$ has an algebraic construction.
We also prove that $M(P^1P^2)$ is equal to $M(P^1)\otimes M(P^2)$ in
the representation ring $\operatorname{Rep}\Ule$ later. (See
\corref{cor:product}.) Here the $I$-tuple of polynomials $(P_i
Q_i)_i$ for $P = (P_i)_i$, $Q = (Q_i)_i$ is denoted by $PQ$ for
brevity.
\section{A modified multiplication on $\widehat{\mathscr Y}_t$}
\label{sec:rings}
We use the following polynomial rings in this article:
{\allowdisplaybreaks[4]
\begin{equation*}
\begin{split}
& \widehat{\mathscr Y}_t \defeq
\Z[t, t^{-1}, V_{i,a}, W_{i,a}]_{i\in I,a\in\C^*},
\\
& \mathscr Y_t \defeq
\Z[t,t^{-1},Y_{i,a}, Y_{i,a}^{-1}]_{i\in I, a\in\C^*},
\\
& \mathscr Y \defeq
\Z[Y_{i,a}, Y_{i,a}^{-1}]_{i\in I, a\in\C^*},
\\
& \overline{\mathscr Y} \defeq
Z[y_i,y_i^{-1}]_{i\in I}.
\end{split}
\end{equation*}
}
We consider $\widehat{\mathscr Y}_t$ as a polynomial ring in
infinitely many variables $V_{i,a}$, $W_{i,a}$ with coefficients in
$\Z[t,t^{-1}]$. So a {\it monomial\/} means a monomial only in
$V_{i,a}$, $W_{i,a}$, containing no $t$, $t^{-1}$. The same
convention applies also to $\mathscr Y_t$.
For a monomial $m\in\widehat{\mathscr Y}_t$, let $w_{i,a}(m)$,
$v_{i,a}(m)\in\Z_{\ge 0}$ be the degrees in $V_{i,a}$, $W_{i,a}$,
i.e.,
\begin{equation*}
m = \prod_{i,a} V_{i,a}^{v_{i,a}(m)}\, W_{i,a}^{w_{i,a}(m)}.
\end{equation*}
We also define
\begin{equation*}
u_{i,a}(m) \defeq
w_{i,a}(m) - v_{i,a\ve^{-1}}(m) - v_{i,a\ve}(m)
+ \sum_{j:C_{ji} = -1} v_{j,a}(m).
\end{equation*}
When $\ve$ is not a root of unity, we define $(\tilde
u_{i,a}(m))_{i\in I, a\in\C^*}$ for a monomial $m$ in
$\widehat{\mathscr Y}_t$, as the solution of
\begin{equation*}
u_{i,a}(m) = \tilde u_{i,a\ve^{-1}}(m) + \tilde u_{i,a\ve}(m)
- \sum_{j: a_{ij} = -1} \tilde u_{j,a}(m).
\end{equation*}
To solve the system, we may assume that $u_{i,a}(m)=0$ unless $a$ is a
power of $q$. Then the above is a recursive system, since $q$ is not a
root of unity. So it has a unique solution such that $\tilde
u_{i,q^s}(m) = 0$ for sufficiently small $s$. Note that $\tilde
u_{i,a}(m)$ is nonzero for possibly infinitely many $a$'s, although
$u_{i,a}(m)$ is not.
If $m^1$, $m^2$ are monomials, we set
\begin{equation}\label{eq:def_d}
\begin{split}
d(m^1, m^2) & \defeq
\sum_{i,a} \left( v_{i,a\ve}(m^1) u_{i,a}(m^2)
+ w_{i,a\ve}(m^1) v_{i,a}(m^2)\right)
\\
&= \sum_{i,a} \left( u_{i,a}(m^1) v_{i,a\ve^{-1}}(m^2)
+ v_{i,a}(m^1) w_{i,a\ve^{-1}}(m^2) \right).
\end{split}
\end{equation}
From the definition, $d(\ ,\ )$ satisfies
\begin{equation}\label{eq:shift}
d(m^1 m^2, m^3) = d(m^1, m^3) + d(m^2, m^3), \qquad
d(m^1, m^2 m^3) = d(m^1, m^2) + d(m^1, m^3).
\end{equation}
When $\ve$ is not a root of unity, we also define
\begin{equation*}
\tilde d(m^1, m^2) \defeq
- \sum_{i,a} u_{i,a}(m^1) \tilde u_{i,a\ve^{-1}}(m^2)
.
\end{equation*}
Since $u_{i,a}(m^2) = 0$ except for finitely many $a$'s, this is
well-defined. Moreover, we have
\begin{equation*}
\tilde d(m^1, m^2) = d(m^1, m^2) + \tilde d_W(m^1, m^2),
\end{equation*}
where $\tilde d_W$ is defined as $\tilde d$ by replacing $u_{i,a}$ by
$w_{i,a}$. Here we have used $\tilde u_{i,a}(m) = \tilde w_{i,a}(m) -
v_{i,a}(m)$.
We define an involution
$\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$ on
$\widehat{\mathscr Y}_t$ by
\begin{equation}
\label{eq:bar}
\overline{t} = t^{-1}, \quad
\overline{m} = t^{2d(m,m)} m,
\end{equation}
where $m$ is a monomial in $V_{i,a}$, $W_{i,a}$.
We define an involution
$\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$ on
${\mathscr Y}_t$ by $\overline{t} = t^{-1}$,
$\overline{Y_{i,a}^\pm} = Y_{i,a}^\pm$.
We define a new multiplication $\ast$ on $\widehat{\mathscr Y}_t$ by
\begin{equation*}
m^1 \ast m^2 \defeq t^{2d(m^1,m^2)} m^1 m^2,
\end{equation*}
where $m^1$, $m^2$ are monomials and $m^1 m^2$ is the usual
multiplication of $m^1$ and $m^2$. By \eqref{eq:shift} it is
associative.
({\bf NB}: The multiplication in \cite{Na-ann} was $m^1 \ast m^2
\defeq t^{2d(m^2,m^1)} m^1 m^2$. This is because the coproduct is
changed.)
From definition we have
\begin{equation}\label{eq:antihom}
\overline{m^1\ast m^2} = \overline{m^2}\ast\overline{m^1}.
\end{equation}
Let us give an example which will be important later. Suppose that
$m$ is a monomial with $u_{i,a}(m) = 1$, $u_{i,b}(m) = 0$ for
$b\neq a$. Then
\begin{equation}\label{eq:example}
\begin{split}
\left[m(1 + V_{i,a\ve})\right]^{\ast n}
& \defeq \underbrace{
m(1 + V_{i,a\ve})\ast \cdots
\ast m(1 + V_{i,a\ve})}_{\text{$n$ times}}
\\
& = m^n \sum_{r=0}^n
t^{r(n-r)}
\begin{bmatrix}
n \\ r
\end{bmatrix}_t V_{i,a\ve}^r.
\end{split}
\end{equation}
When $\ve$ is not a root of unity, we define another multiplication
$\tilde\ast$ by
\begin{equation*}
m^1 \tilde\ast\, m^2
\defeq t^{\tilde d(m^1,m^2)-\tilde d(m^2,m^1)} m^1 m^2.
\end{equation*}
We define a $\Z[t,t^{-1}]$-linear homomorphism
\(
\widehat\Pi\colon\widehat{\mathscr Y}_t\to \mathscr Y_t
\)
by
\begin{equation}\label{eq:YY}
m = \prod_{i,a} V_{i,a}^{v_{i,a}(m)}\, W_{i,a}^{w_{i,a}(m)}
\longmapsto t^{-d(m,m)} \prod_{i,a} Y_{i,a}^{u_{i,a}(m)}.
\end{equation}
This is not a ring homomorphism with respect to either the ordinary
multiplication or $\ast$. However, when $\ve$ is not a root of unity,
we can define a new multiplication on $\mathscr Y_t$ so that the above
is a ring homomorphism with respect to this multiplication and $\tilde\ast$.
It is because $\ve(m^1, m^2)$ involves only $u_{i,a}(m^1)$, $u_{i,a}(m^2)$.
We denote also by $\tilde\ast$ the new multiplication on $\mathscr
Y_t$. We have
\begin{equation}\label{eq:use}
\begin{gathered}
\widehat\Pi(m^1 \ast m^2)
= t^{\tilde d_W(m^1,m^2) - \tilde d_W(m^2, m^1)}
\widehat\Pi(m^1) \tilde\ast \widehat\Pi(m^2),
\\
\widehat\Pi\circ
\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}
= \setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}
\circ\widehat\Pi.
\end{gathered}
\end{equation}
Further we define homomorphisms $\Pi_t\colon\mathscr Y_t\to\mathscr Y$,
$\overline{\Pi}\colon\mathscr Y \to \Z[y_i^\pm]$ by
\begin{equation*}
\Pi_t\colon \mathscr Y_t \ni
\begin{aligned}[c]
t & \longmapsto 1
\\
Y_{i,a} & \longmapsto Y_{i,a}
\end{aligned}
\in \mathscr Y, \qquad
\overline\Pi\colon
\mathscr Y\ni Y_{i,a} \longmapsto y_i\in \Z[y_i, y_i^{-1}]_{i\in I}.
\end{equation*}
The composition $\widehat{\mathscr Y}_t\to\mathscr Y$ or
$\widehat{\mathscr Y}_t\to\Z[y_i^\pm]$ is a ring homomorphism with
respect to both the usual multiplication and $\ast$.
\begin{Definition}
A monomial $m\in\widehat{\mathscr Y}_t$ is said {\it $i$--dominant\/} if
$u_{i,a}(m)\ge 0$ for any $i\in I$.
A monomial $m\in\widehat{\mathscr Y}_t$ is said {\it l--dominant\/} if
it is $i$--dominant for all $i\in I$, i.e., $\widehat\Pi(m)$ contains
only nonnegative powers of $Y_{i,a}$.
Similarly a monomial $m\in\mathscr Y$ is said {\it l--dominant\/} if
it contains only nonnegative powers of $Y_{i,a}$.
Note that a monomial $m\in\Z[y_i, y_i^{-1}]_{i\in I}$ contains only
nonnegative powers of $y_{i}$ if and only if it is dominant as a
weight of $\g$.
\end{Definition}
Let
\[
m = \prod_{i,a} Y_{i,a}^{u_{i,a}}
\]
be a monomial in $\mathscr Y$ with $u_{i,a}\in\Z$. We associate to $m$
an $I$-tuple of rational functions $Q = (Q_i)$ by
\begin{equation*}
Q_i(u) = \prod_a \left(1 - au\right)^{u_{i,a}}.
\end{equation*}
Conversely an $I$-tuple of rational functions $Q = (Q_i)$ as above
determines a monomial in $\mathscr Y$. We denote it by $e^Q$.
This is the $e^Q$ mentioned in the previous section. Note that $e^Q$
is {\it l\/}--dominant if and only if $Q$ is an $I$-tuple of
polynomials.
We also use a similar identification between an $I$-tuple of
polynomials $P = (P_i)$ and a monomial $m$ in $W_{i,a}$ ($i\in I$,
$a\in \C^*$):
\begin{equation*}
m = \prod_{i,a} W_{i,a}^{w_{i,a}} \longleftrightarrow
P = (P_i);\quad P_i(u) = \prod_a (1 - au)^{w_{i,a}}.
\end{equation*}
We denote $m$ also by $e^P$, hoping that it makes no confusion.
\begin{Definition}
Let $m$, $m'$ be monomials in $\widehat{\mathscr Y}_t$.
We say $m\le m'$ if $m/m'$ is a monomial in $V_{i,a}$ ($i\in I$,
$a\in\C^*$). We say $m < m'$ if $m \le m'$ and $m\neq m'$. It defines
a partial order among monomials in $\widehat{\mathscr Y}_t$.
Similarly for monomials $m$, $m'$ in $\mathscr Y$, we say
$m\le m'$ if $m/m'$ is a monomial in
\(
{\widehat\Pi}(V_{i,a})
\)
($i\in I$, $a\in\C^*$).
For two $I$-tuple of rational functions $Q$, $Q'$, we say $Q\le Q'$ if
$e^Q \le e^{Q'}$.
Finally for monomials $m$, $m'$ in $\Z[y_i,y_i^{-1}]_{i\in I}$, we say
$m\le m'$ if $m/m'$ is a monomial in
\(
{\overline{\Pi}\circ\Pi_t\circ\widehat\Pi}(V_{i,a})
\)
($i\in I$, $a\in\C^*$). But this is nothing but the usual order on
weights.
\end{Definition}
\section{A $t$--analog of the $q$--character: Axioms}\label{sec:axiom}
A main tool in this article is a $t$--analog of the $q$--character:
\begin{equation*}
\widehat\chi_{\ve,t}\colon \bfR_t
= \operatorname{Rep}\Ule\otimes_\Z \Z[t,t^{-1}]
\to \widehat{\mathscr Y}_t.
\end{equation*}
For the definition we need geometric constructions of standard
modules, so we will postpone it to \secref{sec:quiver}.
In this section, we explain properties of $\widehat\chi_{\ve,t}$ as
axioms. Then we show that these axioms uniquely characterize
$\widehat\chi_{\ve,t}$, and in fact, give us an algorithm for
``computation''.
Thus we may consider the axioms as the definition of
$\widehat\chi_{\ve,t}$.
Our first axiom is the highest weight property:
\begin{Axiom}
The $\widehat\chi_{\ve,t}$ of a standard module $M(P)$ has a form
\begin{equation*}
\widehat\chi_{\ve,t}(M(P)) = e^P + \sum a_m(t) m,
\end{equation*}
where each monomial $m$ satisfies $m < e^P$.
\end{Axiom}
Composing maps $\widehat{\mathscr Y}_t\to\mathscr Y_t$,
$\mathscr Y_t\to\mathscr Y$, $\mathscr Y\to \Z[y_i^\pm]$ in
\secref{sec:rings}, we define maps
\begin{gather*}
\chi_{\ve,t}=\widehat\Pi\circ\widehat\chi_{\ve,t}
\colon \bfR_t \to \mathscr Y_t,
\\
\chi_\ve=\Pi_t\circ\chi_{\ve,t}
\colon\operatorname{Rep}\Ule \to \mathscr Y,
\qquad
\chi=\overline{\Pi}\circ\chi_\ve
\colon\operatorname{Rep}\Ule \to \Z[y_i, y_i^{-1}]_{i\in I}.
\end{gather*}
The $\widehat\chi_{\ve,t}$ is a homomorphism of
$\Z[t,t^{-1}]$-modules, not of rings.
Frenkel-Mukhin \cite[5.1, 5.2]{FM} proved that the image of $\chi_\ve$ is
equal to
\begin{equation*}
\bigcap_{i\in I}
\left( \Z[Y_{j,a}^\pm]_{j:j\neq i, a\in\C^*}
\otimes\Z[Y_{i,b}(1+V_{i,b\ve})]_{b\in\C^*}\right).
\end{equation*}
We define its $t$--analog, replacing $(1+V_{i,b\ve})^n$ by
\begin{equation*}
\sum_{r=0}^n t^{r(n-r)}\begin{bmatrix} n \\ r
\end{bmatrix}_t V_{i,b\ve}^{r}.
\end{equation*}
More precisely, for each $i\in I$, let $\widehat{\mathscr K}_{t,i}$ be
the $\Z[t,t^{-1}]$-linear subspace of $\widehat{\mathscr Y}_t$
generated by elements
\begin{equation}\label{eq:form}
E_i(m) \defeq
m\, \prod_a
\sum_{r_a=0}^{u_{i,a}(m)}
t^{r_a(u_{i,a}(m)-r_a)}
\begin{bmatrix}
u_{i,a}(m) \\ r_a
\end{bmatrix}_t V_{i,a\ve}^{r_a},
\end{equation}
where $m$ is an $i$--dominant monomial, i.e., $u_{i,a}(m) \ge 0$ for
all $a\in\C^*$.
Let
\[
\widehat{\mathscr K}_t \defeq
\bigcap_i \widehat{\mathscr K}_{t,i}, \qquad
\mathscr K_t \defeq \widehat\Pi(\widehat{\mathscr K}_t)
\subset \mathscr Y_t.
\]
\begin{Axiom}
The image of $\widehat\chi_{\ve,t}$ is contained in $\widehat{\mathscr
K}_t$.
\end{Axiom}
Next axiom is about the multiplicative property of
$\widehat\chi_{\ve,t}$. As explained in the introduction, it is not
multiplicative under the usual product structure on $\bfR_t$.
\begin{Axiom}
Suppose that two $I$-tuples of polynomials $P^1 = (P^1_i)$, $P^2 =
(P^2_i)$ satisfy the following condition:
\begin{equation}
\label{eq:Z}
\begin{minipage}[m]{0.75\textwidth}
\noindent
$a/b\notin\{ \ve^n \mid n\in\Z, n \ge 2\}$ for any
pair $a$, $b$ with $P^1_i(1/a) = 0$, $P^2_j(1/b) =
0$ \textup($i,j\in I$\textup).
\end{minipage}
\end{equation}
Then we have
\begin{equation*}
\widehat\chi_{\ve,t}(M(P^1P^2)) = \widehat\chi_{\ve,t}(M(P^1))\ast
\widehat\chi_{\ve,t}(M(P^2)).
\end{equation*}
\end{Axiom}
We have the following special case
\begin{equation*}
\widehat\chi_{\ve,t}(M(P^1P^2))
= \widehat\chi_{\ve,t}(M(P^1))\widehat\chi_{\ve,t}(M(P^2))
\end{equation*}
under the stronger condition $a/b\notin \ve^\Z$ by the
definition of $\ast$.
The last axiom is about a specialization at a root of unity.
Suppose that $\ve$ is a primitive $s$-th root of unity.
We choose and fix $q$, which is not root of unity. The axiom will say
that $\widehat\chi_{\ve,t}(M(P))$ can be written in terms of
$\widehat\chi_{q,t}(M(P_q))$ for some $P_q$.
By Axiom~3, more precisely the sentence following Axiom~3, we may
assume that inverse of roots of $P_i(u) = 0$ ($i\in I$) is contained
in $a\ve^\Z$ for some $a\in\C^*$. Therefore
\begin{equation*}
P_i(u) = \prod_{n=0}^{s-1} (1 - a \ve^n u)^{N_{i,n}}.
\end{equation*}
with $N_{i,n}\in\Z_{\ge 0}$. We define $P_q = ((P_q)_i)$ by
\begin{equation*}
(P_q)_i(u) = \prod_{n=0}^{s-1} (1 - a q^n u)^{N_{i,n}}.
\end{equation*}
We set $N_{i,n} = 0$ if $n\notin \{0,\dots,s-1\}$.
Let
\[
\widehat\chi_{q,t}(M(P_q)) = \sum a_m(t) m.
\]
By previous axioms, each $m$ is written as
\begin{equation}\label{eq:m}
m = e^{P_q} \prod_{i\in I, n\in \Z} V_{i,a q^n}^{M_{i,n}}
= \prod_{i\in I, n\in \Z} W_{i,a q^n}^{N_{i,n}}
V_{i,a q^n}^{M_{i,n}}
\end{equation}
with $M_{i,n}\in \Z_{\ge 0}$.
By previous axioms $M_{i,n}$ is independent of $q$ (cf.\
\thmref{thm:cons}(4)).
We define monomials $m|_{q = \ve}$, $m[k]$ by
\begin{equation}\label{eq:m_shift}
\begin{split}
m|_{q = \ve} & \defeq
\prod_{i\in I, n\in \Z} W_{i,a\ve^n}^{N_{i,n}}
V_{i,a\ve^n}^{M_{i,n}},
\\
m[k] & \defeq
\prod_{i\in I, n\in \Z} W_{i,a q^n}^{N_{i,n+k}}
V_{i,a q^n}^{M_{i,n+k}}.
\end{split}
\end{equation}
Note that $m|_{q=\ve} = m[k]|_{q=\ve}$ if $k\equiv 0\mod s$.
We define
\begin{equation*}
D^-(m) \defeq \sum_{k < 0} d_q(m,m[ks]),
\end{equation*}
where we define $d_q$ as $d$ in \eqref{eq:def_d} replacing $\ve$ by
$q$.
\begin{Axiom}
We have
\begin{equation*}
\widehat\chi_{\ve,t}(M(P))
= \sum t^{2D^-(m)}\, a_m(t)\, m|_{q = \ve}.
\end{equation*}
\end{Axiom}
We can consider similar axioms for
\(
\chi_\ve = \Pi_t\circ\widehat\Pi\circ\widehat\chi_{\ve,t}
\).
Axioms~3,4 are simplified when $t=1$. Axiom~3 is
\(
\chi_\ve(M(P^1P^2)) = \chi_\ve(M(P^1))\chi_\ve(M(P^2))
\).
Axiom~4 says
\(
\chi_\ve(M(P)) = \chi_q(M(P))|_{q=\ve}
\).
The original $\chi_\ve$ defined in \cite{FR,FM2} satisfies those
axioms: Axioms~1,2 were proved in \cite[Theorem~4.1, Theorem~5.1]{FM}.
Axiom~3 was proved in \cite[Lemma~3]{FR}. Axiom~4 was proved in
\cite[Theorem~3.2]{FM2}.
Let us give few consequences of the axioms.
\begin{Theorem}\label{thm:cons}
\textup{(1)} The map $\chi_{\ve,t}$ \textup(and hence
also $\widehat\chi_{\ve,t}$\textup) is injective.
The image of $\chi_{\ve,t}$ is equal to ${\mathscr K}_t$.
\textup{(2)} Suppose that a $\Ule$-module $M$ has the following
property: $\widehat\chi_{\ve,t}(M)$ contains only one {\it
l\/}--dominant monomial $m_0$.
Then $\widehat\chi_{\ve,t}(M)$ is uniquely determined from $m_0$ and
the condition $\widehat\chi_{\ve,t}(M)\in\widehat{\mathscr K}_t$.
\textup{(3)} Let $m$ be an {\it l\/}--dominant monomial in $\mathscr
Y_t$, considered as an element of the dual of $\bfR_t$ by taking the
coefficient of $\chi_{\ve,t}$ at $m$. Then $\{ m \mid \text{$m$ is
{\it l\/}--dominant}\}$ is a base of the dual of $\bfR_t$.
\textup{(4)} The $\widehat\chi_{\ve,t}$ is unique, if it exists.
\textup{(5)} $\widehat\chi_{\ve,t}(\tau_a^*(V))$ is obtained from
$\widehat\chi_{\ve,t}(V)$ by replacing $W_{i,b}$, $V_{i,b}$ by
$W_{i,ab}$, $V_{i,ab}$.
\textup{(6)} The coefficients of a monomial $m$ in
$\widehat\chi_{\ve,t}(M(P))$ is a polynomial in $t^2$. \textup(In
fact, it will become clear that it is a polynomial in $t^2$ with
nonnegative coefficients.\textup)
\end{Theorem}
\begin{proof}
These are essentially proved in \cite{FR,FM}. So our proof is sketchy.
(1) Since $\chi_{\ve,t}(M(P))$ equals $\widehat\Pi(e^P)$ plus the sum
of lower monomials, the first assertion follows by induction on $<$.
The second assertion follows from the argument in \cite[5.6]{FM},
where we use the standard module $M(P)$ instead of simple modules.
(2) Let $m$ be a monomial appearing in $\widehat\chi_{\ve,t}(M)$,
which is not $m_0$. It is not {\it l\/}--dominant by the
assumption. By Axiom~2, $m$ appears in $E_i(m')$ for some
monomial $m'$ appearing in $\widehat\chi_{\ve,t}(M)$. In particular,
we have $m < m'$. Repeating the argument for $m'$, we have $m < m_0$.
The coefficient of $m$ in $\widehat\chi_{\ve,t}(M)$ is equal to the
sum of coefficients of $m$ in $E_i(m')$ for all possible $m'$'s. ($i$
is fixed.) Again by induction on $<$, we can determine the coefficient
inductively.
(3) By Axiom~1, the transition matrix between $\{ M(P)\}$ and the dual
base of $\{ m\}$ above is uppertriangular with diagonal entries $1$.
(4) By Axiom~4, we may assume that $\ve$ is not a root of unity.
Consider the case $P_i(u) = 1 - au$, $P_j(u) = 1$ for $j\neq i$ for
some $i$. By \cite[Corollary 4.5]{FM}, Axiom~1 implies that the
$\widehat\chi_{\ve,t}(M(P))$ for $P$ does not contains {\it
l\/}--dominant terms other than $e^P$.
(See \propref{prop:nodom} below for a geometric proof.)
In particular, $\widehat\chi_{\ve,t}(M(P))$ is uniquely determined by
above (2) in this case.
We use Axiom~3 to ``calculate'' $\widehat\chi_{\ve,t}(M(P))$ for
arbitrary $P$ as follows.
We order inverses of roots (counted with multiplicities) of $P_i(u) =
0$ ($i\in I$) as $a_1$, $a_2$, \dots, so that
$a_p/a_q\neq \ve^n$ for $n\ge 2$ if $p < q$. This is
possible since $\ve$ is not a root of unity. For each $a_p$, we
define a Drinfeld polynomial $Q^p$ by
\begin{equation*}
Q^p_{i_p}(u) = (1 - a_p u), \qquad
Q^p_{j}(u) = 1 \quad (j\neq i_p),
\end{equation*}
if $1/a_p$ is a root of $P_{i_p}(u) = 0$. Therefore we have $P_i =
\prod_p Q^p_i$. By our choice, we have
\begin{equation*}
\widehat\chi_{\ve,t}(M(P))
= \widehat\chi_{\ve,t}(M(Q^1))\ast \widehat\chi_{\ve,t}(M(Q^2))
\ast\cdots
\end{equation*}
by Axiom~3. Each $\widehat\chi_{\ve,t}(M(Q^p))$ is uniquely determined
by the above discussion. Therefore $\widehat\chi_{\ve,t}(M(P))$ is
also uniquely determined.
(5) It is enough to check the case $V = M(P)$. In this case,
$\tau_a^*(M(P))$ is the standard module with Drinfeld polynomial
$P(au)$. The assertion follows from the axioms.
(6) This also follows from the axioms. By Axiom~4, we may assume $\ve$
is a root of unity. By Axiom~3, we may assume $M(P)$ is an {\it
l\/}--fundamental representation. In this case, the assertion
follows from Axiom~2, since
\(
t^{r(n-r)}\left[
\begin{smallmatrix}
n \\ r
\end{smallmatrix}\right]_t
\)
is a polynomial in $t^2$.
\end{proof}
In \cite[\S5.5]{FM}, Frenkel-Mukhin gave an explict combinatorial
algorithm to ``compute'' $\widehat\chi_{\ve,t}(M)$ for $M$ as in (2).
We will give a geometric interpretation of their algorithm in
\secref{sec:proof1}.
By the uniqueness, we get
\begin{Corollary}\label{cor:coincide}
The $\chi_{\ve}$ coincides with the $\ve$-character
defined in \cite{FR,FM2}.
\end{Corollary}
By \cite[Theorem~3]{FR}, $\chi$ is the ordinary character of the
restriction of a $\Ule$-module to a $\Ue$-module.
As promised, we prove
\begin{Corollary}\label{cor:product}
In the representation ring $\operatorname{Rep}\Ule$, we have
\begin{equation*}
M(P^1 P^2) = M(P^1)\otimes M(P^2)
\end{equation*}
for any $I$-tuples of polynomials $P^1$, $P^2$.
\end{Corollary}
\begin{proof}
Since $\chi_\ve$ is injective, it is enough to show that
$\chi_\ve(M(P^1 P^2)) = \chi_\ve(M(P^1))\chi_\ve(M(P^2))$.
In fact, it is easy to prove this equality directly from the geometric
defintion in \eqref{eq:geomdef}. However, we prove it only from Axioms.
By Axiom~4, we may assume $\ve$ is not a root of unity. We order
inverses of roots (counted with multiplicities) of $P^1_iP^2_i(u) = 0$
($i\in I$) as in the proof of \thmref{thm:cons}(4). Then we have
\begin{equation*}
\chi_\ve(M(P^1P^2)) = \prod_p \chi_\ve(M(Q^p))
\end{equation*}
by Axiom~3. The product can be taken in any order, since
$\operatorname{Rep}\Ule$ is commutative. Each $a_p$ is either the
inverse of a root of $P^1_i(u) = 0$ or $P^2_i(u)=0$. We divide
$a_p$'s into two sets accordingly. Then the products of
$\chi_\ve(M(Q^a))$ over groups are equal to $\chi_\ve(M(P^1))$ and
$\chi_\ve(M(P^2))$ again by Axiom~3. Therefore we get the assertion.
\end{proof}
We also give another consequence of the axioms.
\begin{Theorem}\label{thm:bar}
The $\widehat{\mathscr K}_t$ is invariant under the multiplication
$\ast$ and the involution
$\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$ on
$\widehat{\mathscr Y}_t$.
Moreover, $\bfR_t$ has an involution induced from one on
$\widehat{\mathscr Y}_t$. When $\ve$ is not a root of unity, it also
has a multiplication induced from that on $\mathscr Y_t$.
\end{Theorem}
The following proof is elementary, but less conceputal. We will give
another geometric proof in \secref{sec:proof2}.
\begin{Remark}
The multiplication on $\bfR_t$ in an earlier version was not
associative, although it works for the computation of tensor product
decompositions of two simple modules.
A modification of the multiplication here was inspired by a paper of
Varagnolo-Vasserot \cite{VV2}.
\end{Remark}
\begin{proof}
For simplicity, we assume that $\ve$ is not a root of unity. The proof
for the case when $\ve$ is a root of unity can be given by a
straightforward modification.
Let us show
\(
f\ast g\in \widehat{\mathscr K}_t
\)
for $f$, $g\in \widehat{\mathscr K}_t$.
By induction and \eqref{eq:example} we may assume
that $f$ is of form
\begin{equation*}
m'\left(1 + V_{i,b\ve}\right),
\end{equation*}
where $m'$ is a monomial with $u_{i,b}(m') = 1$, $u_{i,c}(m') = 0$ for
$c\neq b$, and that $g = E_i(m)$ is as \eqref{eq:form}.
By a direct calculation, we get
\begin{multline*}
t^{-2d(m',m)} f\ast g - E_i(mm')
\\
= \left(t^{2n} - 1 \right) m m'
\prod_{a\neq b\ve^{-2}}
\sum_{r_a=0}^{u_{i,a}(m)}
t^{r_a(u_{i,a}(m)-r_a)}
\begin{bmatrix}
u_{i,a}(m) \\ r_a
\end{bmatrix}_t
\,
V_{i,a\ve}^{r_a}
\sum_{s=0}^{n-1} t^{s(n-s)}
\begin{bmatrix}
n-1 \\ s
\end{bmatrix}_t
V_{i,b\ve^{-1}}^{s+1}
\end{multline*}
where $n=u_{i,b\ve^{-2}}(m)$. If $n=0$, then the right hand side is
zero, so the assertion is obvious.
If $n\neq 0$, then we have
\begin{equation*}
u_{i,a}\left(mm'V_{i,b\ve^{-1}}\right)
=
\begin{cases}
u_{i,b\ve^{-2}}(m) - 1 & \text{if $a = b\ve^{-2}$},
\\
u_{i,a}(m) & \text{otherwise}.
\end{cases}
\end{equation*}
Therefore the above expression is equal to
\(
\left(t^{2n} - 1 \right)E_i\left(mm'V_{i,b\ve^{-1}}\right)
\).
Next we show the closedness of the image under the involution. By
\eqref{eq:antihom} and the above assertion, we may assume
\(
f = m'\left(1 + V_{i,b\ve}\right)
\)
as above. We further assume $m'$ does not contain $t$, $t^{-1}$. Then
we get
\begin{equation*}
\overline{f} = t^{2d(m',m')} f.
\end{equation*}
This is contained in $\widehat{\mathscr K}_t$.
Now we can define $\tilde\ast$ and
$\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$
on $\bfR_t$ so that
\begin{gather*}
\chi_{\ve,t}(\overline{V})
= \widehat\Pi\left(\overline{\widehat\chi_{\ve,t}(V)}\right)
= \overline{\chi_{\ve,t}(V)},
\\
\chi_{\ve,t}(V_1\tilde\ast V_2)
= \chi_{\ve,t}(V_1)\tilde\ast\chi_{\ve,t}(V_2),
\end{gather*}
where we have assumed that $\ve$ is not a root of unity for the second
equality.
By the above discussion together with \eqref{eq:use}, the right hand
sides are contained in $\mathscr K_t$, and therefore in the image of
$\chi_{\ve,t}$ by \thmref{thm:cons}(1).
Since $\chi_{\ve,t}$ is injective by \thmref{thm:cons}(1),
$\overline{V}$, $V_1\ast V_2$ are well-defined.
\end{proof}
\begin{Remark}
In this article, the existence of $\widehat\chi_{\ve,t}$ satisfying the
axioms is provided by a geometric theory of quiver varieties. But the
author conjectures that there exists purely combinatorial proof of the
existence, independent of quiver varieties or the representation
theory of quantum loop algebras.
When $\mathfrak g$ is of type $A$ or $D$, such a combinatorial
construction is possible \cite{Na-AD}.
When $\mathfrak g$ is $E_6$, $E_7$, an explict construction of
$\widehat\chi_{\ve,t}$ is possible with the use of a computer.
\end{Remark}
\section{Graded and cyclic quiver varieties}\label{sec:quiver}
Suppose that a finite graph $(I,E)$ of type $ADE$ is given. The set
$I$ is the set of vertices, while $E$ is the set of edges.
Let $H$ be the set of pairs consisting of an edge together with its
orientation. For $h\in H$, we denote by $\vin(h)$ (resp.\ $\vout(h)$)
the incoming (resp.\ outgoing) vertex of $h$.
For $h\in H$ we denote by $\overline h$ the same edge as $h$ with the
reverse orientation.
We choose and fix a function $\varepsilon\colon H \to \C^*$ such that
$\varepsilon(h) + \varepsilon(\overline{h}) = 0$ for all $h\in H$.
Let $V$, $W$ be $I\times \C^*$-graded vector spaces such that
its $(i\times a)$-component, denoted by $V_i(a)$, is finite dimensional
and $0$ at most finitely many $i\times a$.
In what follows we consider only $I\times\C^*$-graded vector spaces
with this condition.
For an integer $n$, we define vector spaces by
\begin{equation}
\label{eq:LE}
\begin{gathered}[m]
\HomL(V, W)^{[n]} \defeq
\bigoplus_{i\in I, a\in\C^*}
\Hom\left(V_i(a), W_i(a\ve^{n})\right),
\\
\HomE(V, W)^{[n]} \defeq
\bigoplus_{h\in H, a\in\C^*}
\Hom\left(V_{\vout(h)}(a), W_{\vin(h)}(a\ve^{n})\right).
\end{gathered}
\end{equation}
If $V$ and $W$ are $I\times\C^*$-graded vector spaces as above, we
consider the vector spaces
\begin{equation}\label{def:bM}
\bM \equiv \bM(V, W) \defeq
\HomE(V, V)^{[-1]} \oplus \HomL(W, V)^{[-1]}
\oplus \HomL(V, W)^{[-1]},
\end{equation}
where we use the notation $\bM$ unless we want to specify $V$, $W$.
The above three components for an element of $\bM$ is denoted by $B$,
$\alpha$, $\beta$ respectively.
({\bf NB}: In \cite{Na-qaff} $\alpha$ and $\beta$ were denoted by $i$,
$j$ respectively.)
The
$\Hom\left(V_{\vout(h)}(a),V_{\vin(h)}(a\ve^{-1})\right)$-component of
$B$ is denoted by $B_{h,a}$. Similarly, we denote by $\alpha_{i,a}$,
$\beta_{i,a}$ the components of $\alpha$, $\beta$.
We define a map $\mu\colon\bM\to \HomL(V,V)^{[-2]}$ by
\begin{equation*}
\mu_{i,a}(B,\alpha,\beta)
= \sum_{\vin(h)=i} \ve(h)
B_{h,a\ve^{-1}} B_{\overline{h},a} +
\alpha_{i,a\ve^{-1}}\beta_{i,a},
\end{equation*}
where $\mu_{i,a}$ is the $(i,a)$-component of $\mu$.
Let $G_V \defeq \prod_{i,a} \GL(V_i(a))$. It acts on $\bM$ by
\begin{equation*}
(B, \alpha, \beta) \mapsto g\cdot (B, \alpha, \beta)
\defeq \left(g_{\vin(h),a\ve^{-1}} B_{h,a} g_{\vout(h),a}^{-1},\,
g_{i,a\ve^{-1}}\alpha_{i,a},\,
\beta_{i,a} g_{i,a}^{-1}\right).
\end{equation*}
The action preserves the subvariety $\mu^{-1}(0)$ in $\bM$.
\begin{Definition}\label{def:stable}
A point $(B, \alpha, \beta) \in \mu^{-1}(0)$ is said to be {\it stable\/} if
the following condition holds:
\begin{itemize}
\item[] if an $I\times\C^*$-graded subspace $S$ of $V$ is
$B$-invariant and contained in $\Ker \beta$, then $S = 0$.
\end{itemize}
Let us denote by $\mu^{-1}(0)^{\operatorname{s}}$ the set of stable points.
\end{Definition}
Clearly, the stability condition is invariant under the action of
$G_V$. Hence we may say an orbit is stable or not.
We consider two kinds of quotient spaces of $\mu^{-1}(0)$:
\begin{equation*}
\N_0(V,W) \defeq \mu^{-1}(0)\dslash G_V, \qquad
\N(V,W) \defeq \mu^{-1}(0)^{\operatorname{s}}/G_V.
\end{equation*}
Here $\dslash$ is the affine algebro-geometric quotient, i.e., the
coordinate ring of $\N_0(V,W)$ is the ring of $G_V$-invariant
functions on $\mu^{-1}(0)$. In particular, it is an affine variety. It
is the set of closed $G_V$-orbits.
The second one is the set-theoretical quotient, but coincides with a
quotient in the geometric invariant theory (see \cite[\S3]{Na:1998}).
The action of $G_V$ on $\mu^{-1}(0)^{\operatorname{s}}$ is free thanks
to the stability condition (\cite[3.10]{Na:1998}).
By a general theory, there exists a natural projective morphism
\begin{equation*}
\pi\colon \N(V,W) \to \N_0(V,W).
\end{equation*}
(See \cite[3.18]{Na:1998}.)
The inverse image of $0$ under $\pi$ is denoted by $\NLa(V,W)$.
We call these varieties {\it cyclic quiver varieties\/} or {\it graded
quiver varieties}, according as $\ve$ is a root of unity or {\it not}.
Let $\Nreg(V,W)\subset\N_0(V,W)$ be a possibly empty open subset of
$\N_0(V,W)$ consisting of free $G_V$-orbits. It is known that
$\pi$ is isomorphism on $\pi^{-1}(\Nreg(V,W))$ \cite[3.24]{Na:1998}.
In particular, $\Nreg(V,W)$ is nonsingular and is pure dimensional.
A $G_V$-orbit though $(B,\alpha,\beta)$, considered as a point of
$\N(V,W)$ is denoted by $[B,\alpha,\beta]$.
We associate polynomials $e^{W}$, $e^{V}\in\widehat{\mathscr
Y}_t$ to graded vector spaces $V$, $W$ by
\begin{equation}\label{eq:rule}
e^{W} = \prod_{i\in I,a\in\C^*} W_{i,a}^{\dim W_i(a)},
\quad
e^{V} = \prod_{i\in I,a\in\C^*} V_{i,a}^{\dim V_i(a)}.
\end{equation}
Suppose that we have two $I\times\C^*$-graded vector spaces $V$, $V'$
such that $V_i(a) \subset V'_i(a)$ for all $i$, $a$. Then $\N_0(V,W)$
can be identified with a closed subvariety of $\N_0(V',W)$ by the
extension by $0$ to the complementary subspace (see
\cite[2.5.3]{Na-qaff}). We consider the limit
\begin{equation*}
\N_0(\infty,W) \defeq \bigcup_{V} \N_0(V,W).
\end{equation*}
It is known that the above stabilizes at some $V$ (see \cite[2.6.3,
2.9.4]{Na-qaff}).
The complement $\N_0(V,W)\setminus\Nreg(V,W)$ consists of a finite
union of $\Nreg(V',W)$ for smaller $V'$'s \cite[3.27,
3.28]{Na:1998}. Therefore we have a decomposition
\begin{equation}\label{eq:stratum}
\N_0(\infty,W) = \bigsqcup_{[V]} \N_0(V,W),
\end{equation}
where $[V]$ denotes the isomorphism class of $V$. The transversal
slice to each stratum was constructed in \cite[\S3.3]{Na-qaff}.
Using it, we can check
\begin{align}
&\begin{minipage}[m]{0.75\textwidth}
\noindent
If $\Nreg(V,W)\neq\emptyset$, then
$e^{V} e^{W}$ is {\it l\/}--dominant.
\end{minipage}\label{eq:l-dom}
\\
&\begin{minipage}[m]{0.75\textwidth}
\noindent
If $\Nreg(V,W)\subset\overline{\Nreg(V',W)}$, then
$e^{V'} \le e^V$.
\end{minipage}\label{eq:closure}
\end{align}
On the other hand, we consider the disjoint union for $\N(V,W)$:
\[
\N(W) \defeq \bigsqcup_{[V]} \N(V,W).
\]
Note that there are no obvious morphisms between $\N(V,W)$ and
$\N(V',W)$ since the stability condition is not preserved under the
extension.
We have a morphism $\N(W)\to \N_0(\infty,W)$, still denoted by $\pi$.
The original quiver varieties \cite{Na:1994,Na:1998} are the special
case when $\ve = 1$ and $V_i(a) = W_i(a) = 0$ except $a=1$.
On the other hand, the above varieties $\N(W)$, $\N_0(\infty,W)$ are
fixed point set of the original quiver varieties with respect to a
semisimple element in a product of general linear groups. (See
\cite[\S4]{Na-qaff}.) In particular, it follows that $\N(V,W)$ is
nonsingular, since the corresponding original quiver variety is so.
This can be also checked directly.
Since the action is free, $V$ and $W$ can be considered as
$I\times\C^*$-graded vector bundles over $\N(V,W)$. We denote them by the
same notation. We consider $\HomE(V,V)$, $\HomL(W,V)$, $\HomL(V, W)$
as vector bundles defined by the same formula as in \eqref{eq:LE}. By
the definition, $B$, $\alpha$, $\beta$ can be considered as sections of those
bundles.
We define a three term sequence of vector bundles over $\N(V,W)$ by
\begin{equation}
\label{eq:taut_cpx_fixed}
C_{i,a}^\bullet(V,W):
V_i(a\ve)
\xrightarrow{\sigma_{i,a}}
\displaystyle{\bigoplus_{h:\vin(h)=i}}
V_{\vout(h)}(a)
\oplus W_i(a)
\xrightarrow{\tau_{i,a}}
V_i(a\ve^{-1}),
\end{equation}
where
\begin{equation*}
\sigma_{i,a} = \bigoplus_{\vin(h)=i} B_{\overline h,a\ve}
\oplus \beta_{i,a\ve},
\qquad
\tau_{i,a} = \sum_{\vin(h)=i} \varepsilon(h) B_{h,a} + \alpha_{i,a}.
\end{equation*}
This is a complex thanks to the equation $\mu(B,\alpha,\beta) = 0$.
We assign the degree $0$ to the middle term.
By the stability condition, $\sigma_{i,a}$ is injective.
We define the rank of complex $C^\bullet$ by $\sum_p (-1)^p \rank
C^p$. Then we have
\begin{equation*}
\rank C_{i,a}^\bullet(V,W) = u_{i,a}(e^{V}e^{W}).
\end{equation*}
We denote the right hand side by $u_{i,a}(V,W)$ for brevity.
We define a three term complex of vector bundles over
$\N(V^1,W^1)\times \N(V^2,W^2)$ by
\begin{equation}\label{eq:hecke_complex}
\HomL(V^1,V^2)^{[0]}
\xrightarrow{\sigma^{21}}
\begin{matrix}
\HomE(V^1, V^2)^{[-1]} \\
\oplus \\
\HomL(W^1, V^2)^{[-1]} \\
\oplus \\
\HomL(V^1, W^2)^{[-1]}
\end{matrix}
\xrightarrow{\tau^{21}}
\HomL(V^1,V^2)^{[-2]},
\end{equation}
where
\begin{equation*}
\begin{split}
\sigma^{21}(\xi) & = (B^2 \xi - \xi B^1) \oplus
(-\xi \alpha^1) \oplus \beta^2 \xi, \\
\tau^{21}(C\oplus I\oplus J)
&= \varepsilon B^2 C + \varepsilon C B^1 + \alpha^2 J + I \beta^1.
\end{split}
\end{equation*}
We assign the degree $0$ to the middle term.
By the same argument as in \cite[3.10]{Na:1998}, $\sigma^{21}$ is
injective and $\tau^{21}$ is surjective. Thus the quotient
$\Ker\tau^{21}/\Ima\sigma^{21}$ is a vector bundle over
$\N(V^1,W^1)\times \N(V^2, W^2)$. Its rank is given by
\begin{equation}\label{eq:rank}
d(e^{V^1}e^{W^1}, e^{V^2}e^{W^2}).
\end{equation}
If $V^1=V^2$, $W^1=W^2$, then the restriction of
$\Ker\tau^{21}/\Ima\sigma^{21}$ to the diagonal is isomorphic to the
tangent bundle of $\N(V,W)$ (see \cite[Proof of 4.1.4]{Na-qaff}). In
particular, we have
\begin{equation}\label{eq:dim}
\dim\N(V,W) = d(e^{V}e^{W},e^{V}e^{W}).
\end{equation}
Let us give the definition of $\widehat\chi_{\ve,t}$.
We define $\widehat\chi_{\ve,t}$ for all standard modules
$M(P)$.
Since $\{ M(P)\}_P$ is a basis of $\operatorname{Rep}\Ule$, we
can extend it linearly to any finite dimensional $\Ule$-modules.
The relation between standard modules and graded/cyclic quiver
varieties is as follows (see \cite[\S13]{Na-qaff}):
Choose $W$ so that $e^W = e^P$, i.e.,
\[
P_i(u) = \prod_a (1-au)^{\dim W_{i}(a)}.
\]
Then a standard module $M(P)$ is defined as $H_*(\NLa(W),\C)$, which
is equipped with a structure of $\Ule$-module by the convolution
product. Moreover, its {\it l\/}--weight space $M(P)_Q$ is
\[
\bigoplus_{V: e^Ve^W = e^Q} H_*(\NLa(V,W),\C).
\]
Here $H_k(\ ,\C)$ denotes the Borel-Moore homology with complex
coefficients.
If $\ve$ is not a root of unity, then $V$ is determined from $Q$. So
the above has only one summand.
Let
\begin{equation}\label{eq:geomdef}
\widehat\chi_{\ve,t}(M(P))
\defeq \sum_{[V]} (-t)^{k} \dim H_k(\NLa(V,W),\C)\, e^{V}e^{W}.
\end{equation}
Since $H_k(\NLa(V,W),\C)$ vanishes for odd $k$ \cite[\S7]{Na-qaff}, we
may replace $(-t)^k$ by $t^k$. In particular, it is clear that
coefficients of $\widehat\chi_{\ve,t}(M(P))$ are polynomials in $t^2$
with positive coefficients.
In subsequent sections we prove that the above $\widehat\chi_{\ve,t}$
satisfies the axioms.
By definition, it is clear that $\widehat\chi_{\ve,t}$ satisfies Axiom~1.
Remark that \corref{cor:coincide} directly follows from this geometric
definition (\cite[13.4.5]{Na-qaff}).
We give a simple consequence of the definition:
\begin{Proposition}\label{prop:nodom}
Assume $\ve$ is not a root of unity.
Suppose that all roots of $P_i(u) = 0$ have the same value
\textup(e.g., $P_i(u) = 1 - au$, $P_j(u) = 1$ for $j\neq i$
for some $i$\textup).
Then $M(P)$ has no {\it l\/}--dominant term other than $e^P$.
\end{Proposition}
This was proved in \cite[Corollary 4.5]{FM}. But we give a geometric
proof.
\begin{proof}
Take $W$ so that $e^W = e^P$. It is enough to show that $u_{i,a}(V,W)
< 0$ for some $i$, $a$ if $\N(V,W)\neq\emptyset$ and $V\neq 0$.
By the assumption, there is a nonzero constant $a$ such that $W_i(b) =
0$ for all $i$, $b\neq a$. By the stability condition, we have
\(
V_i(b) = 0
\)
if $b\neq a\ve^n$ for some $n\in\Z_{>0}$. Let $n_0$ be the maximum of
such $n$, and suppose $V_i(a\ve^{n_0})\neq 0$. Since
$W_i(a\ve^{n_0+1}) = V_i(a\ve^{n_0+1}) = V_i(a\ve^{n_0+2}) = 0$, we
have
\begin{equation*}
u_{i,a\ve^{n_0+1}}(V,W) = \rank C_{i,a\ve^{n_0+1}}^\bullet(V,W) < 0.
\end{equation*}
\end{proof}
\section{Proof of Axiom~2: Analog of the Weyl group invariance}
\label{sec:proof1}
For a complex algebraic variety $X$, let $e(X;x,y)$ denote the virtual
Hodge polynomial defined by Danilov-Khovanskii~\cite{DK} using a mixed
Hodge strucuture of Deligne~\cite{De}. It has the following properties.
\begin{enumerate}
\item $e(X;x,y)$ is a polynomial in $x$, $y$ with integer coefficients.
\item If $X$ is a nonsingular projective variety, then $e(X;x,y) =
\sum_{p,q} (-1)^{p+q} h^{p,q}(X)x^p y^q$, where the $h^{p,q}(X)$ are the
Hodge numbers of $X$.
\item If $Y$ is a closed subvariety in $X$, then $e(X;x,y) = e(Y;x,y) +
e(X\setminus Y;x,y)$.
\item If $f\colon Y\to X$ is a fiber bundle with fiber $F$ which is
locally trivial in the Zarisky topology, then $e(Y;x,y) =
e(X;x,y)e(F;x,y)$.
\end{enumerate}
We define the virtual Poincar\'e polynomial of $X$ by $p_t(X) \defeq
e(X;t,t)$. (In fact, this reduction does not loose any
information. The following argument shows that $e(X;x,y)$ appearing
here is a polynomial in $xy$.)
The actual Poincar\'e polynomial is defined as
\begin{equation*}
P_t(X) = \sum_{k=0}^{2\dim X} (-t)^k \dim H_k(X,\C),
\end{equation*}
where $H_k(X,\C)$ is the Borel-Moore homology of $X$ with complex
coefficients.
\begin{Remark}
In stead of virtual Poincar\'e polynomials, we can use numbers of
rational points in the following argument, if we define graded/cyclic
varieties over an algebraic closure of a finite field $\mathbf k$. As
a consequence, those numbers are special values of ``computable''
polynomials $P(t)$ at $t=\sqrt{\#\mathbf k}$.
\end{Remark}
\begin{Lemma}\label{lem:vir}
The virtual Poincar\'e polynomial of $\NLa(V,W)$ is equal to the actual
Poincar\'e polynomial. Moreover, it is a polynomial in $t^2$.
The same holds for $\N(V,W)$.
\end{Lemma}
\begin{proof}
In \cite[\S7]{Na-qaff} we have shown that $\NLa(V,W)$ has
an partition into locally closed subvarieties $X_1,\dots, X_n$ with
the following properties:
\begin{enumerate}
\item $X_1\cup X_2\cup\dots \cup X_i$ is closed in $\NLa(V,W)$ for
each $i$.
\item each $X_i$ is a vector bundle over a nonsingular projective
variety whose homology groups vanish in odd degrees.
\end{enumerate}
A partition satisfying the property (1) is called an {\it
$\alpha$-partition}.
(More precisely, it was shown in [loc.\ cit.] that $X_i$ is a fiber
bundle with an affine space fiber over the base with the above
property. The above statement was shown in \cite{Na-tensor}.)
By the long exact sequence in homology groups, we have
$P_t(\NLa(V,W)) = \sum_i P_t(X_i)$. On the other hand, by the
property of the virtual Poincar\'e polynomial, we have
$p_t(\NLa(V,W)) = \sum_i p_t(X_i)$. Since $X_i$ satisfies the
required properties in the statement, it follows that $\NLa(V,W)$
satisfies the same property.
We have an $\alpha$-partition with the property (2) also for
$\N(V,W)$, so we have the same assertion.
\end{proof}
Recall the complex \eqref{eq:taut_cpx_fixed}. For a $\C^*$-tuple of
nonnegative integers $(n_a)\in\Z_{\ge 0}^{\C^*}$, let
\begin{equation*}
\N_{i;(n_a)}(V,W)
\defeq
\left\{ [B,\alpha,\beta]\in \N(V,W) \Biggm| \text{$\codim_{V_i(\ve^{-1}a)}
\Ima \tau_{i,a}
= n_a$ for each $a\in\C^*$}\right\}.
\end{equation*}
This is a locally closed subset of $\N(V,W)$.
We also set
\begin{equation*}
\NLa_{i;(n_a)}(V,W) \defeq \N_{i;(n_a)}(V,W)\cap\NLa(V,W).
\end{equation*}
We have partitions
\begin{equation*}
\N(V,W) = \bigsqcup_{(n_a)} \N_{i;(n_a)}(V,W), \qquad
\NLa(V,W) = \bigsqcup_{(n_a)} \NLa_{i;(n_a)}(V,W).
\end{equation*}
Let $Q_{i,a}(V,W)$ be the middle cohomology of the
complex $C_{i,a}^\bullet(V,W)$ \eqref{eq:taut_cpx_fixed}, i.e.,
\begin{equation*}
Q_{i,a}(V,W) \defeq \Ker\tau_{i,a}/\Ima\sigma_{i,a}.
\end{equation*}
Over each stratum $\N_{i;(n_a)}(V,W)$ it defines a vector bundle. In
particular, over the open stratum $\N_{i;(0)}(V,W)$, i.e., points
where $\tau_{i,a}$ is surjective for all $i$, its rank is equal to
\begin{equation}\label{eq:rankC}
\rank C^\bullet_{i,a}(V,W) = u_{i,a}(V,W).
\end{equation}
Suppose that a point $[B,\alpha,\beta]\in\N_{i;(n_a)}(V,W)$ is given.
We define a new graded vector space $V'$ by $V'_i(\ve^{-1}a) \defeq
\Ima\tau_{i,a}$. The restriction of $(B,i,j)$ to $V'$ also satisfies
the equation $\mu = 0$ and the stability condition. Therefore it
defines a point in $\N(V',W)$. It is clear that this construction
defines a map
\begin{equation}
\label{eq:projection}
p\colon \N_{i;(n_a)}(V,W)\to \N_{i;(0)}(V',W).
\end{equation}
Let $G(n_a,Q_{i,\ve^{-2}a}(V',W)|_{\N_{i;(0)}(V',W)})$ denote the
Grassmann bundle of $n_a$-planes in the vector bundle obtained by
restricting $Q_{i,\ve^{-2}a}(V',W)$ to $\N_{i;(0)}(V',W)$. Let
\[
{\displaystyle \prod_a}
G(n_a,Q_{i,\ve^{-2}a}(V',W)|_{\N_{i;(0)}(V',W)})
\]
be their fiber product over $\N_{i;(0)}(V',W)$.
By \cite[5.5.2]{Na-qaff} there exists a commutative diagram
\begin{equation*}
\begin{CD}
{\displaystyle \prod_a}
G(n_a,Q_{i,\ve^{-2}a}(V',W)|_{\N_{i;(0)}(V',W)})
@>\pi>> \N_{k;(0)}(V',W) \\
@VV{\cong}V @| \\
\N_{i;(n_a)}(V,W) @>p>> \N_{i;(0)}(V',W),
\end{CD}
\end{equation*}
where $\pi$ is the natural projection. (The assumption $\ve\neq\pm 1$
there was unnecessary. See \secref{sec:pm 1}.)
Since the projection~\eqref{eq:projection} factors through $\pi$, it
induces
\[
p\colon \NLa_{i;(n_a)}(V,W)\to \NLa_{i;(0)}(V',W).
\]
Therefore we have
\begin{equation*}
\begin{split}
&
p_t\left(\NLa_{i;(n_a)}(V,W)\right)
\\
=\; &\prod_{a}
t^{n_a(\rank C^\bullet_{i,\ve^{-2}a}(V',W)-n_a)}
\begin{bmatrix}
\rank C^\bullet_{i,\ve^{-2}a}(V',W)
\\
n_a
\end{bmatrix}_t
p_t\left(\NLa_{i;(0)}(V',W)\right).
\end{split}
\end{equation*}
Using \eqref{eq:rankC}, we get
\begin{equation*}
\begin{split}
\widehat\chi_{\ve,t}(M(P)) = \sum_{[V']}\;
& p_t\left(\NLa_{i;(0)}(V',W)\right) e^{V'}e^{W}
\\
& \times \prod_a
t^{r_a(u_{i,a}(V',W)-r_a)}
\sum_{r_a=0}^{u_{i,a}(V',W)}
\begin{bmatrix}
u_{i,a}(V',W) \\ r_a
\end{bmatrix}_t V_{i,a\ve}^{r_a}.
\end{split}
\end{equation*}
This shows that $\widehat\chi_{\ve,t}(M(P))$ is contained
$\widehat{\mathscr K}_t$. This completes the proof of
Axiom~2.
As we promised, we give an algorithm computing
$\widehat\chi_{\ve,t}(M)$ for $M$ as in \thmref{thm:cons}(2).
Although we will explain it only when $M$ is a standard module $M(P)$,
a modification to the general case is straightforward, if we interpret
$p_t(\NLa_{i;(n_a)}(V,W))$ suitably.
Moreover, we use the Grassmann bundle \eqref{eq:projection} instead of
the condition $\widehat\chi_{\ve,t}(M)\in\widehat{\mathscr K}_t$.
We ``compute'' the virtual Poincar\'e polynomials
$p_t(\NLa_{i;(n_a)}(V,W))$ by induction.
The first step of the induction is the case $V = 0$.
In this case, $\NLa(0,W)$ is a single point, so $p_t(\NLa(0,W)) = 1$.
Moreover, $\NLa(0,W) = \NLa_{i;(0)}(0,W)$ for all $i$.
Suppose that we already ``compute'' all
$p_t(\NLa_{i;(n'_\lambda)}(V',W))$ for $\dim V' < \dim V$.
If $(n_a)\neq (0)$, then by the Grassmann bundle
\eqref{eq:projection} and the induction hypothesis, we can ``compute''
$p_t(\NLa_{i;(n_a)}(V,W))$.
By the assumption in \thmref{thm:cons}(2), $e^{V}e^{W}$ is {\it not\/}
{\it l\/}--dominant, hence there exists $i$, $a$ such that
$u_{i,a}(V,W) < 0$.
Then $\NLa_{i;(0)}(V,W)$ is the empty set by
\cite[5.5.5]{Na-qaff}. Therefore we have
\begin{equation*}
p_t\left(\NLa(V,W)\right)
= \sum_{(n_a)\neq (0)} p_t\left(\NLa_{i;(n_a)}(V,W)\right).
\end{equation*}
We have already ``computed'' the right hand side. We, of course, have
\begin{equation*}
p_t\left(\NLa_{i;(0)}(V,W)\right) = 0.
\end{equation*}
For $j\neq i$, we have
\begin{equation*}
p_t\left(\NLa_{j;(0)}(V,W)\right)
= p_t\left(\NLa(V,W)\right)
- \sum_{(n_a)\neq (0)} p_t\left(\NLa_{j;(n_a)}(V,W)\right).
\end{equation*}
The right hand side is already ``computed''.
\begin{Remark}
(1) Note that the above argument shows that $p_t(\NLa(V,W))$ is a
polynomial in $t^2$ without appealing to \cite[\S7]{Na-qaff} as in
\lemref{lem:vir}.
In fact, nonsingular quasi-projective varieties appearing in
\lemref{lem:vir} are examples of graded quiver varieties such that the
above argument can be applied, i.e., the corresponding standard
modules satisfy the condition in \thmref{thm:cons}(2). Therefore, the
above gives a new proof of the vanishing of odd homology groups.
(2) If the reader carefully compares our algorithm with
Frenkel-Mukhin's one \cite{FM}, he/she finds a difference. The {\it
coloring\/} $s_i$ of a monomial $m = e^{V}e^{W}$ is
\[
\sum_{(n_a)\neq (0)}
p_{t=1}\left(\NLa_{i;(n_a)}(V,W)\right)
\]
in our algorithm. This might be possibly {\it negative\/} integer,
while it is assumed to be nonnegative in [loc.\ cit.]. Therefore, we
must modify their definition of the {\it admissibility\/} of a
monomial $m$.
Let us consider all values $s_i$ such that $m$ is not
$i$--dominant. We say $m$ is {\it admissible\/} if all values are the
same.
In our case, $s_i$ is
\(
p_{t=1}\left(\NLa(V,W)\right)
\)
if $m$ is not $i$--dominant, hence $\NLa_{i;(0)}(V,W)=\emptyset$.
Therefore it is independent of $i$.
\end{Remark}
\section{Proof of Axiom~3: Multiplicative property}
\label{sec:proof2}
Since \cite{Na-hom} it has been known that Betti numbers of arbitrary
quiver varieties are determined by special cases corresponding to
fundamental weights. We will use the same idea in this section.
Let $W^1$, $W^2$, $W$ be $I\times\C^*$-graded vector spaces such that
$W_i(a) = W^1_i(a)\oplus W^2_i(a)$ for $i\in I$, $a\in\C^*$.
Let $P^1$, $P^2$ be $I$-tuples of polynomials corresponding to $W^1$,
$W^2$. Then $W$ corresponds to $P^1P^2$.
We define a map $\N(W^1)\times \N(W^2)\to \N(W)$ by
\begin{equation}\label{eq:directsum}
\left([B^1,\alpha^1,\beta^1], [B^2,\alpha^2,\beta^2]\right)
\longmapsto
[B^1\oplus B^2, \alpha^1\oplus\alpha^2, \beta^1\oplus \beta^2].
\end{equation}
We define a $\prod_{i,a} \GL(W_i(a))$-action on $\N(W)$ by
\begin{equation*}
s \ast [B,\alpha,\beta] \defeq [B, \alpha s^{-1}, s\beta].
\end{equation*}
We define a one-parameter subgroup $\lambda\colon\C^*\to \prod_{i,a}
\GL(W_i(a))$ by
\begin{equation*}
\lambda(t) = \bigoplus_{i,a}
\id_{W^1_i(a)} \oplus\, t\id_{W^2_i(a)}
.
\end{equation*}
Then \eqref{eq:directsum} is a closed embedding and the fixed point
set $\N(W)^{\lambda(\C^*)}$ is its image by \cite[3.2]{Na-tensor}. We
identify $\N(W^1)\times \N(W^2)$ with its image hereafter.
The fixed point set $\N(V,W)^{\lambda(\C^*)}$ is the union of
$\N(V^1,W^1)\times\N(V^2,W^2)$ with $V\cong V^1\oplus V^2$.
Let
\begin{multline*}
\Zm(V^1,W^1;V^2,W^2) \defeq
\\
\left\{ [B,\alpha,\beta]\in\N(W) \left|\,
\lim_{t\to 0}\lambda(t)\ast[B,\alpha,\beta]\in
\N(V^1,W^1)\times\N(V^2,W^2)
\right\}\right..
\end{multline*}
We also define $\Zl(V^1,W^1;V^2,W^2)$ by replacing
$\N(V^1,W^1)\times\N(V^2,W^2)$ by $\NLa(V^1,W^1)\times\NLa(V^2,W^2)$.
These are studied in \cite{Na-tensor}. These are nonsingular locally
closed subvarieties of $\N(W)$ [loc.\ cit., 3.7].
Let $\Zm(W^1;W^2)$, $\Zl(W^1,W^2)$ be their union over $[V^1]$,
$[V^2]$ respectively. These are closed subvarieties of
$\N(W)$ [loc.\ cit., 3.6].
By [loc.\ cit., 3.7, 3.13], the partition
\[
\Zm(W^1;W^2) = \bigsqcup_{[V^1],[V^2]} \Zm(V^1,W^1;V^2,W^2)
\]
is an $\alpha$-partition such that each stratum $\Zm(V^1,W^1;V^2,W^2)$
is isomorphic to the total space of the vector bundle
$\Ker\tau^{21}/\Ima\sigma^{21}$ over $\N(V^1,W^1)\times\N(V^2,W^2)$
in \eqref{eq:hecke_complex}. (More precisely, we restrict the result
of [loc.\ cit.] to the fixed point set.)
Similarly
\[
\Zl(W^1;W^2) = \bigsqcup_{V^1,V^2} \Zl(V^1,W^1;V^2,W^2)
\]
is an $\alpha$-partition such that each stratum $\Zm(V^1,W^1;V^2,W^2)$
is isomorphic to the restriction of $\Ker\tau^{21}/\Ima\sigma^{21}$ to
$\NLa(V^1,W^1)\times\NLa(V^2,W^2)$.
\begin{Proposition}\label{prop:mul}
\textup{(1)} The virtual Poincar\'e polynomial of $\Zl(W^1;W^2)$
\textup(more precisely that of each connected component of
$\Zl(W^1;W^2)$\textup) is equal to its actual Poincar\'e
polynomial. Moreover, it is a polynomial in $t^2$.
The same holds for $\Zm(W^1;W^2)$.
\textup{(2)} We have
\begin{equation}\label{eq:generalize}
\widehat\chi_{\ve,t}(M(P^1)) \ast \widehat\chi_{\ve,t}(M(P^2)) =
\sum_{[V^1],[V^2]}
P_t(\Zl(V^1,W^1;V^2,W^2))\; e^{V^1} e^{V^2} e^{W^1} e^{W^2}.
\end{equation}
\textup{(3)} The above expression is contained in $\widehat{\mathscr
K}_t$.
\end{Proposition}
\begin{proof}
(1) This can be shown exactly as in \lemref{lem:vir}.
(2) The rank of the vector bundle $\Ker\tau^{21}/\Ima\sigma^{21}$ is
equal to $d(e^{W^1}e^{V^1},e^{W^2}e^{V^2})$ (see \eqref{eq:rank}). By
the property of virtual Poincar\'e polynomials, we get the assertion.
(3) Exactly the same as \secref{sec:proof1}.
\end{proof}
Axiom~3 follows from above and the following assertion proved in
\cite[6.12]{Na-tensor}:
\begin{equation*}
\Zl(W^1;W^2) = \NLa(W)
\end{equation*}
under the condition \eqref{eq:Z}.
As we promised, we give
\begin{proof}[A different proof of \thmref{thm:bar}]
We only prove the second statement. In fact, it is not difficult to
show that the following argument also implies the first statement.
We will prove that our $\widehat\chi_{\ve,t}$ satisfies Axiom~4 in the
next section. Therefore, it is enough to check the assertion for
$\widehat\chi_{\ve,t}$ given by the geometric
definition~\eqref{eq:geomdef}.
By \thmref{thm:cons}(1) and \propref{prop:mul}, we get the statement
for the multiplication.
Similarly, for the proof of the statement for the involution, it is
enough to show
\begin{equation*}
\overline{\widehat\chi_{\ve,t}(M(P))} \in \widehat{\mathscr K}_t.
\end{equation*}
This follows from
\begin{equation}\label{eq:Claim}
\overline{\widehat\chi_{\ve,t}(M(P))}
= \displaystyle\sum_{[V]} P_t(\N(V,W))\; e^{V}e^{W}.
\end{equation}
In fact, the same argument as in the proof of \secref{sec:proof1}
shows that the right hand side is contained in $\widehat{\mathscr
K}_t$.
Let us prove \eqref{eq:Claim}.
Since $\NLa(V,W)$ is homotopic to $\N(V,W)$ \cite[4.1.2]{Na-qaff}, its
usual homology group is isomorphic to that of $\N(V,W)$. Since
$\NLa(V,W)$ is compact, the usual homology group is isomorphic to the
Borel-Moore homology. Therefore, the Poincar\'e duality for $\N(V,W)$,
which is applicable since $\N(V,W)$ is nonsingular, implies
\begin{equation*}
t^{2\dim \N(V,W)}
P_{1/t}\left(\NLa(V,W)\right)
= P_{t}\left(\N(V,W)\right).
\end{equation*}
Since $\dim\N(V,W) = d(e^{V}e^{W},e^{V}e^{W})$
(see \eqref{eq:dim}), we get \eqref{eq:Claim}.
\end{proof}
\section{Proof of Axiom~4: Roots of unity}\label{sec:proof3}
In this section, we use a $\C^*$-action on $\N(V,W)$ to calculate
Betti numbers. This idea was originally appeared in \cite{Na-hom} and
\cite[\S5]{Na:1994}.
We assume that $\ve$ is a primitive $s$-th root of unity
($s\in\Z_{>0}$).
We may assume $\alpha = 1$ in the setting of Axiom~4.
We consider $V$, $W$ as $I\times(\Z/s\Z)$-graded vector
spaces.
We define a $\C^*$-action on $\bM(V,W)$ by
\[
t\star (B,\alpha,\beta) = (tB,t\alpha s(t)^{-1},t s(t)\beta),
\qquad (t\in \C^*),
\]
where $s(t)\in \prod \GL(W_i(a))$ is defined by
\begin{equation*}
s(t) = \bigoplus_{i\in I,\; 0\le n < s} t^n\id_{W_i(\ve^n)}.
\end{equation*}
It preserves the equation $\mu(B,\alpha,\beta) = 0$ and commutes with
the action of $G_V$. Therefore it induces an action on the affine
cyclic quiver variety $\N_0(V,W)$. The action preserves the stability
condition. Therefore it induces action on $\N(V,W)$. These induced
actions are also denoted by $\star$. The map
$\pi\colon\N(V,W)\to\N_0(V,W)$ is equivariant.
\begin{Lemma}
Let $[B,\alpha,\beta]\in\N(V,W)$. Consider the flow $t\star
[B,\alpha,\beta]$ for $t\in\C^*$. It has a limit when $t\to 0$.
\end{Lemma}
\begin{proof}
By a generaly theory, it is enough to show that
$t\star[B,\alpha,\beta]$ stays in a compact set. Since $\pi$ is
proper, it is enough to show that $\pi(t\star[B,\alpha,\beta]) =
t\star\pi([B,\alpha,\beta])$ stays in a compact set.
By \cite{Lu:qv} the coordinate ring of $\N_0(V,W)$ is generated by
functions of forms
\begin{equation*}
\langle \chi, \beta_{i,\ve^{n+1}}
B_{h_1,\ve^{n+2}}\dots B_{h_N,\ve^{n+N+1}}
\alpha_{j,\ve^{n+N+2}}\rangle
\end{equation*}
where $\chi$ is a linear form on $\Hom\left(W_{i,\ve^n},
W_{j,\ve^{n+N+2}}\right)$, and $i = \vin(h_1)$, $\vout(h_1) =
\vin(h_2)$, \dots, $\vout(h_N) = j$. By the $\C^*$-action, this
function is multiplied by
\begin{equation*}
t^{n+1} t^N t^{-r+1} = t^{n+N+2-r}
\end{equation*}
where we assume $0\le n < s$ and $r$ is the integer such that
$0\le r < s$ and $r \equiv n+N+2\mod s$. Then $n+N+2-r$ is
nonnegative. Therefore $t\star\pi([B,\alpha,\beta])$ stays in a
compact set for any $[B,\alpha,\beta]$.
\end{proof}
We want to identify a fixed point set in $\N(V,W)$ with some quiver
variety $\N(V_q,W_q)$ defined for $q$ which is {\it not\/} a root of
unity. We first explain a morphism from $\N(V_q,W_q)$ to $\N(V,W)$.
Vector spaces corresponding $\N(V_q,W_q)$ are $I\times\Z$-graded
vector spaces.
Suppose that $V_q, W_q$ are $I\times\Z$-graded vector space such that
$(W_q)_i(q^k) = 0$ unless $0\le k < s$ (no condition for $V_q$).
We consider $W_q$ as an $I\times(\Z/s\Z)$-graded vector space simply
identifying $\Z/s\Z$ with $\{ 0, 1, \dots, s-1\}$. Let us denote by
$W$ the resulting $I\times(\Z/s\Z)$-graded vector space.
We define an $I\times(\Z/s\Z)$-graded vector space $V$ by
\[
V_i(\ve^n) \defeq \bigoplus_{k\not\equiv n\mod s} (V_q)_i(q^k).
\]
If a point in $\bM(V_q,W_q)$ is given, it defines a point in
$\bM(V,W)$ in obvious way. The map $\bM(V_q,W_q)\to \bM(V,W)$
preserves the equation $\mu = 0$ and the stability condition. It is
equivariant under the $G_{V_q}$ action, where $G_{V_q}\to G_V$ is an
obvious homomorphism. Therefore, we have a morphism
\begin{equation}
\label{eq:fixed}
\N(V_q,W_q) \to \N(V,W).
\end{equation}
Note that $W_q$ is uniquely determined by $W$, while $V_q$ is {\it
not\/} determined from $V$.
\begin{Lemma}
A point $[B,\alpha,\beta]\in\N(V,W)$ is fixed by the $\C^*$-action if
and only if it is contained in the image of
\eqref{eq:fixed} for some $V_q$.
Moreover, the map \eqref{eq:fixed} is a closed embedding.
\end{Lemma}
\begin{proof}
Fix a representatitve $(B,\alpha,\beta)$ of $[B,\alpha,\beta]$.
Then $[B,\alpha,\beta]$ is a fixed point if and only if there
exists $\lambda(t)\in G_V$ such that
\begin{equation*}
t\star (B,\alpha,\beta) = \lambda(t)^{-1}\cdot (B,\alpha,\beta).
\end{equation*}
Such $\lambda(t)$ is unique since the action of $G_V$ is free. In
particular, $\lambda\colon\C^*\to G_V$ is a group homomorphism.
Let $V_i(\ve^n)[k]$ be the weight space of $V_i(\ve^n)$ with
eigenvalue $t^k$. The above equation means that
\begin{gather*}
B_{h,\ve^{n+1}}\left(V_{\vout(h)}(\ve^{n+1})[{k+1}]\right)
\subset V_{\vin(h)}(\ve^{n})[{k}],
\qquad
\alpha_{i,\ve^{n+1}}(W_i(\ve^{n+1})) \subset V_i(\ve^{n})[{n}],
\\
\beta_{i,\ve^n}\left(V_i(\ve^n)[k]\right) = 0\quad
\text{if $k\neq n$}.
\end{gather*}
Let us define an $I\times\C^*$-graded subspace $S$ of $V$ by
\begin{equation*}
S_i(\ve^n) \defeq \bigoplus_{k\not\equiv n\mod s} V_i(\ve^n)[k].
\end{equation*}
The above equations imply that $S$ is contained in $\Ker\beta$ and
$B$-invariant. Therefore $S = 0$ by the stability condition.
This means that $[B,\alpha,\beta]$ is in the image of \eqref{eq:fixed}
if we set
\begin{equation*}
(V_q)_i(q^k) \defeq V_i(\ve^k)[k].
\end{equation*}
Conversely, a point in the image is a fixed point. Since $\lambda$ is
unique, the map \eqref{eq:fixed} is injective.
Let us consider the differential of \eqref{eq:fixed}. The tangent
space of $\N(V,W)$ at $[B,i,j]$ is the middle cohomology groups of
the complex
\begin{equation}\label{eq:tangent}
\HomL(V,V)^{[0]}
\xrightarrow{\sigma^{21}}
\begin{matrix}
\HomE(V, V)^{[-1]} \\
\oplus \\
\HomL(W, V)^{[-1]} \\
\oplus \\
\HomL(V, W)^{[-1]}
\end{matrix}
\xrightarrow{\tau^{21}}
\HomL(V,V)^{[-2]}.
\end{equation}
Similarly the tangent space of $\N(V_q,W_q)$ is the middle cohomology
of a complex with $V$, $W$ are replaced by $V_q$, $W_q$.
We have a natural morphism between the complexes so that
the induced map between cohomology groups is the differential of
\eqref{eq:fixed}. It is not difficult to show the injectivity by using
the stability condition.
\end{proof}
Let us consider the tangent space $T$ of $\N(V,W)$ at
$[B,\alpha,\beta]\in\N(V_q,W_q)\subset\N(V,W)$, which is the middle
cohomology of \eqref{eq:tangent}. Let $V = \bigoplus_k V[k]$ be the
weight space decomposition as in the proof of the above lemma. The
tangent space $T$ has a weight decomposition $T = \bigoplus_k T[k]$,
where $T[k]$ is the middle cohomology of
{\footnotesize
\begin{equation*}
\bigoplus_n \HomL(V[n],V[{n+k}])^{[0]}
\xrightarrow{\sigma^{21}}
\begin{matrix}
\bigoplus_n \HomE(V[n], V[{n+k-1}])^{[-1]} \\
\oplus \\
\bigoplus_n \HomL(W[n], V[{n+k-1}])^{[-1]} \\
\oplus \\
\bigoplus_n \HomL(V[n], W[{n+k-1}])^{[-1]}
\end{matrix}
\xrightarrow{\tau^{21}}
\bigoplus_n \HomL(V[n],V[{n+k-2}])^{[-2]},
\end{equation*}}
\noindent
where $W[n] = W(\ve^n)$ if $0\le n < s$ and $0$ otherwise.
The rank of the complex is equal to
\[
\begin{cases}
d_q(e^{V_q} e^{W_q}, e^{V_q} e^{W_q}[k])
& \text{if $k\equiv 0\mod s$},
\\
0 & \text{otherwise}.
\end{cases}
\]
Here $e^{V_q} e^{W_q}[k]$ is defined as in \eqref{eq:m_shift}.
We consider the Bialynicki-Birula decomposition of $\N(V,W)$:
\begin{equation*}
\begin{split}
& \N(V,W) = \bigsqcup_{[V_q]} S(V_q,W_q),
\\
& \qquad S(V_q,W_q) \defeq
\left\{ x\in \N(V,W) \left|\; \lim_{t\to 0} t\star x
\in \N(V_q,W_q) \right\}\right..
\end{split}
\end{equation*}
By a general theory, each $S(V_q,W_q)$ is a locally closed subvariety
of $\N(V,W)$, and the natural map $S(V_q,W_q)\to \N(V_q,W_q)$ is a
fiber bundle whose fiber is an affine space of dimension equal to
$\sum_{k > 0} \dim T[k]$. By the above formula, it is equal to
\[
\sum_{k > 0}
d_q(e^{V_q}e^{W_q}, e^{V_q}e^{W_q}[ks]).
\]
We write this number by $D^+(e^{V_q}e^{W_q})$.
By the property of virtual Poincar\'e polynomials, we have
\begin{equation*}
P_t(\N(V,W)) = \sum_{[V_q]} t^{2D^+(e^{V_q}e^{W_q})}
P_t(\N(V_q,W_q)).
\end{equation*}
(Recall that the virtual Poincar\'e polynomials coincide with the
actual Poincar\'e polynomials for these varieties.)
Combining with an argument in the proof of \eqref{eq:Claim}, we have
\begin{equation*}
\begin{split}
P_t(\NLa(V,W)) &
= t^{2d(e^{V}e^{W},e^{V}e^{W})}
P_{1/t}(\N(V,W))
\\
& = \sum_{[V_q]} t^{2d(e^{V}e^{W},e^{V}e^{W})
- 2D^+(e^{V_q}e^{W_q})} P_{1/t}(\N(V_q,W_q))
\\
& = \sum_{[V_q]} t^{2d(e^{V}e^{W},e^{V}e^{W})
- 2D^+(e^{V_q}e^{W_q})
- 2d_q(e^{V_q}e^{W_q},e^{V_q}e^{W_q})}
P_{t}(\NLa(V_q,W_q)).
\end{split}
\end{equation*}
Since
\begin{equation*}
d(e^{V}e^{W},e^{V}e^{W})
= \dim T = \sum_k \dim T[k]
= \sum_{k} d_q(e^{V_q}e^{W_q},e^{V_q}e^{W_q}[ks]),
\end{equation*}
we have
\begin{equation*}
\begin{split}
& d(e^{V}e^{W},e^{V}e^{W})
- D^+(e^{V_q}e^{W_q})
- d_q(e^{V_q}e^{W_q},e^{V_q}e^{W_q})
\\
=\; & \sum_{k < 0}
d_q(e^{V_q}e^{W_q},e^{V_q}e^{W_q}[ks])
= D^-(e^{V_q}e^{W_q}).
\end{split}
\end{equation*}
Thus we have checked Axiom~4.
\begin{Remark}
When $\ve = 1$, there is a different $\C^*$-action so that the index
$D^-(m)$ can be read off from $a_m(t)$. See \cite[\S7]{Na-ann}.
\end{Remark}
\section{Perverse sheaves on graded/cyclic quiver varieties}
\label{sec:perverse}
The following is the main result of this article:
\begin{Theorem}\label{thm:character}
\textup{(1)} There exists a unique base $\{ L(P)\}$ of $\bfR_t$ such
that
\begin{equation*}
\overline{L(P)} = L(P), \qquad
L(P) \in M(P) + \sum_{Q: Q<P} t^{-1}\Z[t^{-1}] M(Q).
\end{equation*}
\textup{(2)} The specialization of $L(P)$ at $t=1$ coincides with the
simple module with Drinfeld polynomial $P$.
\end{Theorem}
As we mentioned in the introduction, the relation between $M(P)$ and
$L(P)$ in $\bfR_t$ (not in its specialization) can be understood by
a Jantzen filtration \cite{Gr}.
For a later purpose we define matrices in the Laurent polynomial ring
of $t$:
\begin{gather*}
c_{PQ}(t) \defeq
\text{the coefficient of $e^Q$ in $\chi_{\ve,t}(M(P))$},
\\
(c^{PQ}(t)) \defeq (c_{PQ}(t))^{-1},
\\
M(P) = \sum_Q Z_{PQ}(t) L(Q).
\end{gather*}
When $\ve$ is not a root of unity, there is an isomorphism between
$\bfR_t$ and the dual of the Grothendieck group of a category of
perverse sheaves on affine graded quiver varieties in
\cite[\S14]{Na-qaff}. And the full detailed proof of the above theorem
was already explained in \cite{Na-ann}.
However, the latter group becomes larger when $\ve$ is a root of
unity.
So we modify $\bfR_t$ to $\widetilde\bfR_t$, and give a proof of the
above theorem in this $\widetilde\bfR_t$.
Let us fix an $I$-tuple of polynomials $P$ throughout this section.
Let $\mathcal I$ be the set of {\it l\/}--dominant monomials
$m\in\widehat{\mathscr Y}_t$ such that $m \le e^P$. We consider
$\Z[t,t^{-1}]$-module with basis $\mathcal I$, and denote it by
$\widetilde\bfR_t$.
For each monomial $m\in\mathcal I$, let $P_m$ be an $I$-tuple of
polynomials given by
\(
(P_m)_i(u) \defeq \prod_a (1 - ua)^{u_{i,a}(m)}
\).
In other words, $P_m$ is determined so that $\widehat\Pi(e^{P_m}) =
\widehat\Pi(m)$.
If $\ve$ is not a root of unity, then $P_m = P_{m'}$ implies $m = m'$
by the invertibility of the $\ve$-analog of the Cartan matrix. But it
is not true in general. This is the reason why we need a modification.
We modify $\widehat\chi_{\ve,t}$ of the standard module $M(P_m)$ so
that it has the image in $\widetilde\bfR_t$ as follows: If
\begin{equation*}
\widehat\chi_{\ve,t}(M(P_m)) = \sum_{n} a_{n,m}(t) e^{P_m} n,
\end{equation*}
then, we define
\begin{equation*}
M_m \defeq
\sum_{n^*}
a_{n^*,m}(t)\, t^{-d(e^{P_m}n^*, e^{P_m}n^*)} m n^*,
\end{equation*}
where the summation runs {\it only\/} over $n^*$ such that $mn^*$ is {\it
l\/}--dominant.
The $M_m$ is contained in $\widetilde\bfR_t$ by Axiom~1. And
$\widehat\chi_{\ve,t}(M(P_m))$ is recoved from $M_m$.
Let us denote the coefficient of $mn^{*}$ by $c_{mn}(t)$, where
$n = mn^{*}\in\mathcal I$, that is
\begin{equation}\label{eq:c}
M_m = \sum_n c_{mn}(t) n.
\end{equation}
We have $c_{mm} = 1$ and $c_{mn}(t) = 0$ for $n\nleq m$. In
particular, $\{ M_m \}_m$ is a base of $\widetilde\bfR_t$.
We define an involution
$\setbox5=\hbox{A}\overline{\rule{0mm}{\ht5}\hspace*{\wd5}}$ on
$\widetilde\bfR_t$ by
\begin{equation*}
\overline{t} = t^{-1}, \quad \overline{m} = m.
\end{equation*}
We define a map $\widetilde\bfR_t\to\bfR_t$ by $M_m\mapsto M(P_m)$.
When $\ve$ is not a root of unity, this map is injective and the image
is the submodule spanned by $M(P')$'s such that
$\widehat\Pi(e^{P'})\le\widehat\Pi(e^P)$.
The map intertwines the involutions.
We have
\begin{equation*}
\overline{M_m} = \sum_n c_{mn}(t^{-1}) n
= \sum_{n,s} c_{mn}(t^{-1})c^{ns}(t) M_s,
\end{equation*}
where $(c^{ns}(t))$ is the inverse matrix of $(c_{mn}(t))$.
Let
\begin{equation}\label{eq:u}
u_{mn}(t) \defeq \sum_s c_{ms}(t^{-1}) c^{sn}(t),
\qquad\text{or equivalently }
\overline{M_m} = \sum_n u_{mn}(t) M_n.
\end{equation}
By axioms, $u_{mm}(t) = 1$ and $u_{mn}(t) = 0$ if $n\nleq m$.
\begin{Lemma}
There exists a unique element $L_m\in\widetilde\bfR_t$ such that
\begin{equation*}
\overline{L_m} = L_m, \qquad
L_m \in M_m + \sum_{n: n<m} t^{-1}\Z[t^{-1}] M_n.
\end{equation*}
\end{Lemma}
Although the proof is exactly the same as \cite[7.10]{Lu:can}, we give
it for the sake of the reader.
\begin{proof}
Let
\[
M_m = \sum_{n\le m} Z_{mn}(t) L_n.
\]
Then the condition for $\{ L_m\}$ is equivalent to the following
system:
\begin{subequations}\label{eq:ZZ}
\begin{align}
& Z_{mm}(t) = 1, \quad
\text{$Z_{mn}(t)\in t^{-1}\Z[t^{-1}]$ for $n < m$}, \label{eq:Z1}
\\
& Z_{mn}(t^{-1}) = \sum_{s: n \le s\le m} u_{ms}(t) Z_{sn}(t).
\label{eq:Z2}
\end{align}
\end{subequations}
The equation can be rewritten as
\begin{equation*}
Z_{mn}(t^{-1}) - Z_{mn}(t)
= \sum_{s: n \le s < m} u_{ms}(t) Z_{sn}(t).
\end{equation*}
Let $F_{mn}(t)$ be the right hand side.
We can solve this system uniquely by induction: If $Z_{sn}(t)$'s are
given, $Z_{mn}(t)$ is uniquely determined by the above equation and
$Z_{mn}(t)\in t^{-1}\Z[t^{-1}]$, provided $F_{mn}(t^{-1}) = -
F_{mn}(t)$. We can check this condition by the induction hypothesis:
\begin{equation*}
\begin{split}
F_{mn}(t^{-1}) & =
\sum_{s: n\le s < m} u_{ms}(t^{-1}) Z_{sn}(t^{-1})
\\
& = \sum_{s: n\le s < m} \sum_{t:n\le t\le s}
u_{ms}(t^{-1}) u_{st}(t) Z_{tn}(t)
\\
& = - \sum_{t: n\le t < m} u_{mt}(t) Z_{tn}(t)
= - F_{mn}(t),
\end{split}
\end{equation*}
where we have used
\(
\sum_{s:t\le s\le m} u_{ms}(t^{-1}) u_{st}(t) = 0
\)
for $t < m$.
\end{proof}
The proof of \thmref{thm:character}(1) is exactly the same. Since the
map $\widetilde\bfR_t\to\bfR_t$ intertwines the involution, the image
of $L_m$ is equal to $L(P_m)$. Therefore \thmref{thm:character}(2) is
equivalent to the following statement:
\begin{Theorem}\label{thm:character2}
The multiplicity $[M(P):L(Q)]$ is equal to
\begin{equation*}
\sum_n Z_{e^P, n}(1),
\end{equation*}
where the summation is over the set $\{ n\mid e^Q = \widehat\Pi(n) \}$.
\end{Theorem}
The following proof is just a modification of that given in
\cite{Na-ann}.
We choose $W$ so that $e^P = e^W$ as before.
Let $D^b(\N_0(\infty,W))$ be the bounded derived category of complexes
of sheaves whose cohomology sheaves are constant along each connected
component of a stratum $\Nreg(V,W)$ of \eqref{eq:stratum}.
(The connectedness of $\Nreg(V,W)$ is {\it not\/} known.)
If $\Nreg(V,W)^\alpha$ is a connected component of $\Nreg(V,W)$, then
$IC(\Nreg(V,W)^\alpha)$ be the intersection homology complex
associated with the constant local system $\C_{\Nreg(V,W)^\alpha}$ on
$\Nreg(V,W)^\alpha$.
Then $D^b(\N_0(\infty,W))$ is the category of complex of sheaves which
are finite direct sums of complexes of the forms
$IC(\Nreg(V,W)^\alpha)[d]$ for various $V$, $\alpha$ and $d\in\Z$,
thanks to the existence of transversal slices \cite[\S3]{Na-qaff}.
We associate a monomial $m = e^V e^W$ to each $[V]$. It gives us a
bijective correspondence between the set of monomials $m$ with $m\le
e^P$ and the set of isomorphism classes of $I\times\C^*$-graded vector
spaces. If $\Nreg(V,W)\neq\emptyset$, the corresponding monomial $m$
is {\it l\/}--dominant, i.e., $m\in\mathcal I$.
We choose a point in $\Nreg(V,W)$ and denote it by $x_m$.
Let $\C_{\N(V',W)}$ be the constant local system on $\N(V',W)$.
Then $\pi_*\C_{\N(V',W)}$ is an object of $D^b(\N_0(\infty,W))$ again
by the transversal slice argument. From the decomposition theorem of
Beilinson-Bernstein-Deligne, we have
\begin{equation}\label{eq:decomp}
\pi_*(\C_{\N(V',W)}[\dim_\C \N(V',W)])
\cong \bigoplus_{V,\alpha,k}
L_{V,\alpha,k}(V',W)\otimes IC(\Nreg(V,W)^\alpha)[k]
\end{equation}
for some vector space $L_{V',\alpha,k}(V,W)$ \cite[14.3.2]{Na-qaff}.
We set
\begin{equation}\label{eq:L}
L_{mn}(t) \defeq \sum_k \dim L_{V,\alpha,k}(V',W)\, t^{-k},
\end{equation}
where $V$, $V'$ are determined so that $m = e^{V}e^{W}$, $n =
e^{V'}e^W$. By the description of the transversal slice
\cite[\S3]{Na-qaff}, $\dim L_{V,\alpha,k}$ is independent of $\alpha$.
So $\alpha$ can disappear in the left hand side.
Applying the Verdier duality to the both hand side of
\eqref{eq:decomp} and using the self-duality of
$\pi_*(\C_{\N(V',W)}[\dim_\C\N(V',W)])$ and $IC(\Nreg(V,W)^\alpha)$,
we find $L_{m'm}(t) = L_{m'm}(t^{-1})$.
By our definition of $M_m$, we have
\begin{equation*}
M_m = \sum_{[V_n]} t^{-\dim\N(V_n,W_m)} P_t(\NLa(V_n,W_m))\, m e^{V_n},
\end{equation*}
where $W_m$ is given by $\dim (W_m)_{i,a} = u_{i,a}(m)$.
By \cite[\S3]{Na-qaff}, this is equal to
\begin{equation}\label{eq:C_RQ}
\begin{split}
M_m & = \sum_{[V_n]} t^{-\dim\N(V_n,W_m)}
P_t\left(\pi^{-1}(x_m)\cap\N(V_n\oplus V_m,W)\right)\, m e^{V_n}
\\
& = \sum_{[V']} \sum_k t^{\dim\N(V_m,W)-k}
\dim H^k(i_{x_m}^! \pi_* \C_{\N(V',W)}[\dim \N(V',W)])
\, e^{V'}e^{W},
\end{split}
\end{equation}
where $V_m$ is given so that $e^{V_m} e^{W} = m$.
Let
\begin{equation*}
Z_{mn}(t) \defeq
\sum_{k,\alpha}
\dim H^k(i_{x_m}^! IC(\Nreg(V,W)^\alpha))\, t^{\dim\Nreg(V_m,W)-k},
\end{equation*}
where $n = e^V e^W$. By the defining property of the intersection
homology, we have \eqref{eq:Z1} and
$Z_{mn}(t) = 0$ if $n \nleq m$.
Substituting \eqref{eq:decomp} into \eqref{eq:C_RQ}, we get
\begin{equation}\label{eq:C=ZL}
c_{mn}(t) = \sum_{s} Z_{ms}(t) L_{sn}(t).
\end{equation}
Now $L_{sn}(t) = L_{sn}(t^{-1})$ and \eqref{eq:u} imply \eqref{eq:Z2}.
Let $Z^\bullet(W)$ be the fiber product
$\N(W)\times_{\N_0(\infty,W)}\N(W)$.
Let $\mathcal A = H_*(Z^\bullet(W),\C)$ be its Borel-Moore homology
group, equipped with an algebra structure by the convolution (see
\cite[14.2]{Na-qaff}).
Taking direct sum with respect to $V'$ in \eqref{eq:decomp}, we have a
linear isomorphism (forgetting gradings)
\begin{equation*}
\pi_*(\C_{\N(W)}) =
\bigoplus_{V,\alpha} L_{V,\alpha}\otimes IC(\Nreg(V,W)^\alpha),
\end{equation*}
where $L_{V,\alpha} = \bigoplus_{[V'],k} L_{V,\alpha,k}(V',W)$.
By a general theory (see \cite{Gi-book} or \cite[14.2]{Na-qaff}), $\{
L_{V,\alpha}\}$ is a complete set of mutually nonisomorphic simple
$\mathcal A$-modules.
Moreover, taking $H^*(i_{x_m}^!\ )$ of both hand sides, we have
\begin{equation*}
H(\pi^{-1}(x_m),\C)) =
\bigoplus_{V,\alpha}
L_{V,\alpha}\otimes H^*(i_{x_m}^!IC(\Nreg(V,W)^\alpha)),
\end{equation*}
which is an equality in the Grothendieck group of $\mathcal A$-modules.
Here the $\mathcal A$-module structure on the right hand side is given by
$a:\xi\otimes\xi'\mapsto a\xi\otimes\xi'$.
By \cite[\S13]{Na-qaff}, there exists an algebra homomorphism
$\Ule\to\mathcal A$. Moreover \cite[\S14.3]{Na-qaff}, each
$L_{V,\alpha}$ is a simple {\it l\/}--highest weight $\Ule$-module.
Its Drinfeld polynomial is $Q$ such that $\widehat\Pi(e^V e^W) = e^Q$.
(It is possible to have two different $V$, $V'$ give isomorphic
$\Ule$-modules.)
Combining with above discussions, we get \thmref{thm:character2}.
\begin{Remark}
If one enlarges the commutative subalgebra $\Ule^0$ of $\Ule$, then
he/she can recover a bijective correspondence between simple
$\Ule$-modules, and strata of affine quiver varieties. When $\g$ is of
type $A_n$, such the enlargement is ${\mathbf
U}_{\varepsilon}({\mathbf L}\mathfrak{gl}_{n+1})$. (cf. \cite{FM3})
\end{Remark}
\section{The specialization at $\ve = \pm 1$}\label{sec:pm 1}
When $\ve = \pm 1$, simple modules can be described explicitly
\cite[\S4.8]{FM2}. We study their $\widehat\chi_{\ve,t}$ in this
section.
Let $P$ be an $I$-tuple of polynomials. We choose $I\times\C^*$-graded
vector space $W$ so that $e^W = e^P$ as before.
First consider the case $\ve = 1$. The $W$ can be considered as a
collection of $I$-graded vector spaces $\{ W^a\}_{a\in\C^*}$, where
$W^a_i \defeq W_{i}(a)$. Then from the definition of cyclic quiver
varieties, it is clear that we have
\begin{equation*}
\N(W) \cong \prod_a \M(W^a),\qquad
\N_0(\infty,W) \cong \prod_a \M_0(\infty,W^a),
\end{equation*}
Here $\M(W^a)$, $\M_0(\infty,W^a)$ are the original quiver variety
corresponding to $W^a$.
Let $P^a$ be an $I$-tuple of polynomial defined by
$P^a_i(u) = (1 - au)^{\dim W_i(a)}$. The $P^a$ is, of course,
determined directly from $P$. From above description, we have
\begin{equation}\label{eq:prod}
M(P) = \bigotimes_a M(P^a), \qquad
\widehat\chi_{\ve,t}(M(P)) = \prod_a \widehat\chi_{\ve,t}(M(P^a)).
\end{equation}
The latter also follows directly from Axiom~3.
Next consider the case $\ve = -1$. We choose and fix a funcion
$o\colon I\to \{\pm 1\}$ such that $o(i) = -o(j)$ if $a_{ij}\neq 0$,
$i\neq j$. We define an $I$-graded vector space $W^a$ by $W^a_i \defeq
W_i(o(i)a)$. Then we have
\begin{equation*}
\N(W) \cong \prod_a \M(W^a),\qquad
\N_0(\infty,W) \cong \prod_a \M_0(\infty,W^a).
\end{equation*}
More precisely, $\N(V,W) = \prod_a \M(V^a,W^a)$ with $V^a \defeq
\bigoplus_i V_i(-o(i)a)$.
Let $P^a$ be an $I$-tuple of polynomial defined by
$P^a_i(u) = (1 - o(i)au)^{\dim W_i(o(i)a)}$. The $P^a$ is again
determined directly from $P$. We have \eqref{eq:prod} also in this case.
Recall that we have an algebra homomorphism ${\mathbf U}_{\ve}(\g)\to
\Ule$ \eqref{eq:subalg}. By \cite[\S33]{Lu-book}, ${\mathbf
U}_{-1}(\g)$ is isomorphic to ${\mathbf U}_{1}(\g)$. Moreover, the
universal enveloping algebra ${\mathbf U}(\g)$ of $\g$ is isomorphic
to the quotient of ${\mathbf U}_1(\g)$ by the ideal generated by $q^h -
1$ ($h\in P^*$) \cite[9.3.10]{CP-book}. In particular, the category of
type $1$ finite dimensional ${\mathbf U}_{\ve}(\g)$-modules is
equivalent to the category of finite dimensional $\g$-modules.
Therefore we consider $\operatorname{Res}M(P)$ as a $\g$-module.
Thanks to the fact that $\pi\colon\M(W)\to\M_0(\infty,W)$ is
semismall, we have the following \cite[\S15]{Na-qaff}:
\begin{Theorem}
\textup{(1)}
\(
L(P) = \bigotimes_a L(P^a).
\)
\textup{(2)} For each $a$, $\operatorname{Res}(L(P^a))$ is simple as a
$\g$-module. Its highest wight is $\Lambda^a = \sum_i \deg P^a_i
\Lambda_i$
\end{Theorem}
We want to interpret this result from $\widehat\chi_{\ve,t}$. We
may assume that there exists only one nontrivial $P^a$. All other
$P^b$'s are $1$.
We identify an $I$-tuple of polynomials $P$ whose roots are $1/a$ with
a dominant weight by
\begin{equation*}
P \mapsto \deg P \defeq \sum_i \deg P_i\, \Lambda_i, \qquad
\lambda = \sum_i \lambda_i\Lambda_i \mapsto
P_\lambda; (P_\lambda)_i(u) = (1 - au)^{\lambda_i}.
\end{equation*}
We give an explicit formula of $Z_{PQ}(t)$, not based on
inductive procedure:
\begin{Theorem}
\textup{(1)} $Z_{PQ}(1)$ is equal to the multiplicity of the
simple $\g$-module $L(\deg Q)$ of highest weight $\deg Q$ in
$\operatorname{Res}M(P)$.
\textup{(2)} $c_{PQ}(t)$ is a polynomial in $t^{-1}$, so
$c_{PQ}(\infty)$ makes sense.
\textup{(3)} We have
\begin{equation*}
\begin{split}
& \chi_{\ve,t}(L(P)) = \sum_Q c_{PQ}(\infty)\, e^Q
+ \text{non {\it l\/}--dominant terms},
\\
&\qquad\text{or equivalently }
Z_{PQ}(t) = \sum_R c_{PR}(t)c^{RQ}(\infty).
\end{split}
\end{equation*}
\textup{(4)} The coefficient $c_{PQ}(\infty)$ is equal to the
weight multiplicity of the dominant weight $\deg Q$ in $L(\deg P)$.
\end{Theorem}
\begin{proof}
(1) is clear.
(2),(3) By the fact that $\pi\colon\M(W^a)\to\M_0(\infty,W^a)$ is
semismall, we have $L_{V,\alpha,k} = 0$ (in \eqref{eq:decomp}) for
$k\neq 0$, hence $L_{PQ}(t)$ (in \eqref{eq:L}) is a constant. Then
$c_{PQ}(t)$ is a polynomial in $t^{-1}$ by \eqref{eq:C=ZL} and
\eqref{eq:Z1}. Therefore $c_{PQ}(\infty)$ makes sense. We have
$c_{PQ}(\infty) = L_{PQ}(t) = L_{PQ}(0)$ again by \eqref{eq:C=ZL} and
\eqref{eq:Z1}. Thus we get the assertion.
(4) By (3), we have
\begin{equation*}
\chi(L(P)) = \sum_Q c_{PQ}(\infty)e^{\deg Q} + \text{non dominant terms}.
\end{equation*}
Since $\chi$ is the ordinary character, the assertion is clear.
\end{proof}
Note that the multiplicity of a simple (resp.\ Weyl) ${\mathbf
U}_\ve(\g)$-module in $\operatorname{Res}M(P)$ for generic (resp.\ a
root of unity) $\ve$ is independent of $\ve$. Therefore $Z_{PQ}(1)$
gives it. (See \cite[\S7]{Na-ann}.)
\section{Conjecture}\label{sec:conjecture}
There is a large amount of literature on finite dimensional
$\Ule$-modules.
Some special classes of simple finite dimensional $\Ule$-modules are
studied intensively: tame modules \cite{NT} and Kirillov-Reshetikhin
modules \cite{HKOTY} (see also the references therein). For tame
modules, there are explicit formulae of $\chi_\ve$ in terms of Young
tableaux. For Kirillov-Reshetikhin modules, there are conjectural
explicit formula of $\chi$ (i.e., decomposition numbers of
restrictions to ${\mathbf U}_\ve(\g)$-modules).
Although our computation applies to {\it arbitrary\/} simple modules,
our polynomials $Z_{PQ}(t)$ are determined recursively, and it is
difficult to obtain explicit formulae in general. Thus those modules
should have a very special feature among arbitrary modules.
For Kazhdan-Lusztig polynomials, a special classes is known to have
explicit formulae. Those are Kazhdan-Lusztig polynomials for
Grassmannians studied \cite{LS}. A geometric interpretation was given
in \cite{Ze}. Based on an analogy between Kazhdan-Lusztig polynomials
and our polynomials, we propose a class of finite dimensional
$\Ule$-modules. It is a class of {\it small\/} standard modules.
\begin{Definition}
(1) A finite dimensional $\Ule$-module $M$ is said {\it special\/} if
it satisfies the condition in \thmref{thm:cons}(2),
i.e., $\chi_\ve(M)$ contains only one {\it l\/}--dominant monomial.
(2) Let $M(P)$ be a standard module with {\it l\/}--highest weight
$P$. We say $M(P)$ is {\it small\/} if $c_{QR}(t)\in t^{-1}\Z[t^{-1}]$
for any $Q$, $R \le P$ with $Q\neq R$.
Similarly $M(P)$ is called {\it semismall\/} if
$c_{QR}(t)\in \Z[t^{-1}]$ for any $Q,R\le P$.
\end{Definition}
\begin{Remark}
(1) By the geometric definition of $\widehat\chi_{\ve,t}$
\eqref{eq:geomdef}, $M(P)$ is (semi)small if and only if
\(
\pi\colon\N(V,W) \to \N_0(V,W)
\)
is (semi)small for any $V$ such that $e^V e^W$ is {\it l\/}--dominant.
(2) By definition, $M(Q)$ is (semi)small if $M(P)$ is (semi)small and
$Q\le P$.
(3) The (semi)smallness of $M(P)$ is related to (semi)tightness of
monomials in $\mathbf U^-_q$ \cite{Lu:tight}.
\end{Remark}
Since a finite dimensional simple $\Ule$-module contains at least one
{\it l\/}--dominant monomial, namely the one corresponding to the {\it
l\/}--highest weight vector, a special module is automatically
simple. The converse is {\it not\/} true in general. For example, if
$\g = \algsl_2$, $P = (1-u)^2(1-\ve^2 u)$, then one can compute (say,
by our algorithm)
\[
\chi_\ve(L(P)) = Y_{1,1}^2 Y_{1,\ve^2} + Y_{1,1}
+ Y_{1,1}^2 Y_{1,\ve^4}^{-1}
+ 2 Y_{1,1}Y_{1,\ve^2}^{-1}Y_{1,\ve^4}^{-1}
+ Y_{1,\ve^2}^{-2}Y_{1,\ve^4}^{-1}.
\]
This has two {\it l\/}-dominant monomial terms.
\begin{Theorem}
Suppose $M(P)$ is small. Then for any $I$-tuple of polynomials $Q\le
P$, corresponding simple module $L(Q)$ is special.
\end{Theorem}
\begin{proof}
By the characterization of $Z_{QR}(t)$ in \eqref{eq:ZZ}, we have
$Z_{QR}(t) = c_{QR}(t)$ for all $Q,R\le P$. Therefore
\begin{equation*}
\sum_R Z_{QR}(t) \chi_{\ve,t}(L(R)) = \chi_{\ve,t}(M(Q))
= \sum_R Z_{QR}(t) e^R + \text{non {\it l\/}--dominant terms}.
\end{equation*}
Hence we have $\chi_{\ve,t}(L(R))$ is $e^R$ plus non {\it
l\/}--dominant terms.
\end{proof}
\begin{Conjecture}
Standard modules corresponding to tame modules and
Kirillov-Reshetikhin modules are small.
\end{Conjecture} | 164,675 |
Hi All--Average. In trying to decide what to say about Yellow Splash Rim, I can't get the word "average" out of my head. Since being added to our garden in 2002 there hasn't been much wrong, or exceptional, about the plant. It even gets an average amount of sun, about 3 hours or so. It has had a good growth rate, with measurements of 13x5, 24x14, 36x16, 42x18, 46x19 and 45x18. But even there it is like so many other varieties in that it surpasses the registration size, 32x16. And, as another variety with green coloring and a white edge, it has trouble with the 10 feet rule. Even slugs have a normal interest.
Registry -
MyHostas -
Hosta Library -
The pix are from 2002, 2004, and 2007. | 327,029 |
DeWalt DWST17824 Tstak Water Resistant Mobile Cooler
DeWalt DWST17824 Tstak Water Resistant Mobile Cooler. Up to 3 days ice retention (Commercial grade foamed PU insulation). 2 lid can holders. Integrated lid soft drink bottle opener. Compatible with ALL TSTAK modules. What’s in The Box. Tstak Mobile Cooler – DWST17824. 30 Day Satisfaction Guaranteed. WHY FACTORY AUTHORIZED OUTLET? SECURE & HASSLE FREE SHOPPING. All our sales are backed by a 30-Day Satisfaction Guarantee. Products shown as available are normally stocked but inventory levels cannot be guaranteed. The item “DeWalt DWST17824 Tstak Water Resistant Mobile Cooler” is in sale since Tuesday, January 14, 2020. This item is in the category “Home & Garden\Tools & Workshop Equipment\Tool Boxes & Storage\Tool Boxes”. The seller is “factory_authorized_outlet” and is located in Ontario,.
- Brand: DEWALT
- Model: DWST17824
- MPN: DWST17824
| 208,013 |
![if !IE]> <![endif]> <![if !IE]> <![endif]>
480-461-7575
This program is part of the Music Department.
Picture yourself tweaking the board during sound check at Red Rocks. Or signing an indie band you discovered to a three-album contract for your label. Imagine arranging an orchestra to perform a symphony you composed. Or sweetening the final mix of a record you produced.
This is the music industry. And it’s as full of career opportunity as it is glamorous. Whether you’re interested in the recording and production side or in music composition, live performance, and teaching, the music industry includes hundreds of careers that require expert skills and talent from its professionals.
The Music Business program at MCC helps you gain the knowledge you need for a career in the music industry. It helps you build your technical skills through classroom instruction, written assignments, business plan design, and presentations. The program provides you with hands-on experience in real-world situations through the production and administration of actual performing arts projects. And ultimately you develop an aesthetic and creative understanding of performance as both an art and a profession.
The music industry is comprised of many different aspects including: recording, manufacturing and production, marketing and promotion, advertising, retail, wholesale, repair, rental, lessons, songwriting, publishing, licensing, merchandising, live performance, event production, broadcast production, and technology development.
While, careers in the music business are competitive, the live performance sector and the publishing and licensing sectors have seen steady growth over the last years, even as traditional record labels continue to decrease their operations and labor force. One in five Americans owns a musical instrument and this provides for stable job growth in the instrument development, manufacturing, retail, rental and repair sectors of the music industry. The consumption of digital music products including music, software, and hardware is booming and steady growth is projected over the next five years and beyond.
Music business professionals:
Before signing up for classes, you’ll want to meet with an academic advisor. Together, you’ll lay out a program and career path that suits you best. This step is essential to your academic success.
Advising for the Music Business. | 315,632 |
Currently Active Discounts on StartLocal: 45 Coupons | 11058 Discounts | 29 Free Stuff | 79 Sales
Find the most popular Auxiliary Insurance Services from Lara here at StartLocal®.
More results: Expand my results to match businesses within: 5kms(3), 10kms(3), 25kms(20), 50kms(21)
Search this category in another suburb by entering your postcode:
Re-sort Businesses by Alphabetical Order
Address: 24 Hicks St, Lara, VIC, 3212
Phone number: 0352824911
Address: 18 Hicks St, Lara, VIC, 3212
Phone number: 0352822257
Address: 18 Hicks St, Lara, VIC, 3212
Phone number: 0352822257
In total there were 3 businesses found in Lara 3212. Similar sub categories to Auxiliary Insurance Services include Insurance Brokers, Insurance Assessment & Loss Adjusters and Mortgage Brokers of which there are 2,. | 235,599 |
NYC Marathon: Race IQ 2015 w/Tailwind Endurance and FinishLine PT
Description
If you’re racing or just watching, you won’t want to miss this!
Please join Tailwind Endurance and Finish Line Physical Therapy on Monday, October 19, as we welcome an incredible panel of experts to discuss the ins, outs and secrets to success at the TCS New York City Marathon.
WHAT: Race IQ: New York City Marathon
WHEN: Monday, October 19,. | 81,549 |
Rocky Patel ALR Second Edition is another line of Rocky Patel Cigars that coming with three vitola, each limited to 40,000 sticks only. The vitolas are:
- Rocky Patel ALR Second Edition Robusto (5 1/2 Inches long by 52 ring gauge)
- Rocky Patel ALR Second Edition Toro (6 1/2 inches long by 52 ring gauge)
- Rocky Patel ALR Second Edition Gordo or Sixty (6 inches long by 60 ring gauge)
Chief among the differences are that these cigars are box-pressed and come with different packaging.
The blend uses a Dominican Republic, Honduras and Nicaragua long-leaf filler, Honduran binder, and dark, oily Mexican san Andrés wrapper, makes the cigar theme purple and gold.
For the filler the tobacco leaf taken from Rocky Patel’s farms in Estelí and Condega, Nicaragua.
Rocky Patel ALR Second Edition History
Rocky blended this Rocky Patel ALR second edition cigar uses an exquisite San Andrés wrapper; this blend would develop even further with age. He only ordered quantities for a limited production of 100,000 sticks.
Then the cigars hand-rolled and keep in the aging room (aged additional two years in fragrant cedar bins for the utmost in taste and aroma) of the TaviCusa factory in Estelí.
This extensive aging result in the medium to-full bodied cigar delivers notes and graham cracker intertwined unique creamy smooth notes of earth, wood, black cherry, nuts and hint of espresso.
According to Rocky Patel that he told to Cigar Aficionado, that this cigar made by a small group of 10 master hand-rollers in Nicaragua.
That is why this Rocky Patel ALR second edition cigar produced in limited quantity.
This batch was hand-rolled in 2017 and now is ready for all cigar lovers who yearn to smoke the best cigars available in the market.
Due to the type of this cigar is box-pressed, so they can burn longer and with more consistent flavor, that make the smoker more enjoyed.
Back with a whole new bag of tricks.
Rocky Patel ALR Second Edition
The ALR Second Edition released this year, extended of the ALR first edition that released in 2018.
The first edition also produced with the same process that the blend and hand rolled in 2016, that the production also limited 120,000 sticks.
The ALR First Edition is silky and complex medium-bodied that truly decadent. The wrapper leaf used is a rare Ecuadorian grown Habano seed.
The cigar presents with the classic Nicaraguan peppery spice, along with notes of cocoa and coffee, with warm moods notes to boot.
ALR means is Aged-Limited-Rare, it means that the cigar has an extended aged by two years aging after the cigars hand-rolled,
Limited in the quantity, and rare because of the complexity of the taste and flavor, and also the process of making. These fine cigars are complete – silky, complex, medium-bodied.
This Second Edition released at the 2019 IPCPR Trade Show in Las Vegas, Nevada, USA.
If you are interested with this premium cigar, it is may you visit your tobacconist or order to the only shops that available….
Pics: Rocky Patel (RP) | 66,137 |
Industry acceptable Data Science Training for all learner’s training which help fresher/Experienced to up-skill at corporate.
Industry Expert Sr. Lead Analyst / Technical Analyst With 10+ Years provide workshop session @ SLA
After completion of 70% of Data Science training our dedicated placement team arrange interview till placement.
Data Science Training help to gain exposure like corporate level with technical test series
Real time projects and best case study makes SLA workshop very unique and lively for learners.
For Learner’s, Our admin team fresh batch schedule/re-scheduling classes/arrange doubt classes.
Data Science refers to a quantitative and qualitative method and process which is used to increase the productivity and business profitability. It is a technique of extracting, acknowledging and analyzing information such as behavioral data, business patterns, and techniques which are dynamic and necessary for business. Every business organization needs to perform Data Science which can provide various benefits such as increased customer satisfaction, enhancing the productivity and performance of the organization and can also provide the companies with the biggest growth opportunities. Data science is also considered an internal function of any business organization which deals with numbers and figures. Intercourse deep knowledge of recording and analyzing along with dissecting information and presenting the findings to make better decisions making for the management. However, in order to become a Data Science professional, one should also have knowledge in various Data Science tools such as Python, Power BI R-Programming, MS Excel and Access, Visual Basic for Application and Macros(VBA/Macros), SQL, Tableau and Business Intelligence tools. The best way to gain relevant skills and masteries Business Intelligence and Data Science tools is by attending quality Data Science Training in Delhi NCR, Noida and Gurgaon provided by SLA Consultants India. The Data Science Training Program is designed by industry experts that provides extensive and comprehensive expertise in different Data Science techniques and tools, allowing the participant to become an expert quickly. The Data Science with Python and R-Programming Training Program will help the learners acquire expertise in predicting customer Trends and behaviour, analysing, interpreting and delivering information in meaningful ways, driving effective decision making along with enhancing the business productivity. Anyone with a graduation degree is eligible to attend High-quality Data Science Training Course in Delhi NCR which is specifically targeted towards both freshers and working professionals who want to enter into the field of Data Science or become fluent in Data science techniques. The Data Science Certification Training is conducted by highly certified subject matter experts with the word 10 to 15 years of experience in the relevant field.
The Data Science Training Course in Delhi NCR is a dedicated and intelligently designed Data Science Training Course that is targeted towards an individual who is good with the numbers and figures and wants to lead a successful career in the Data Science field. The entire Data Analyst Training is divided into five different models which can be completed within 150 hours. The major highlight of the Data Science Training Course is the Data Science course content which is highly updated as per the industrial standards. It covers a wide range of advanced topics in Data Science which include Excel and VBA macros Analytics, SQL and MS Access, and Advanced, Tableau and MS Power Business Intelligence(BI), R-Programming and Python. Practical Training in Data Science will be provided to the learner to gain experience in Business Data Science and acquiring technical expertise in data models, segmentation techniques, data mining techniques, database design development, etc. It will also help them to develop strong analytical skills allowing them to easily collect, organize and analyze a large amount of information with accuracy and efficiency. Upon completion of the Data Science Training Course in Delhi NCR, Noida and Gurgaon, the participants will be able to interpret Data and Analyse results using Statistical Techniques and provide reports, develop and implement databases, Data Science and data collection system along with other strategies to increase the efficiency and quality, gathering information from primary and secondary data sources, identifying and analysing latest trends and patterns, filtering the information by reviewing the computer reports and performance indicators to identify any issues and much more.
The participants will also be provided with in depth understanding of Data Science using live projects and assignments on Real world cases so that they would not require additional Data Science Training after getting placement. Proper workshops and handouts will be conducted by the teachers to offer them industry relevant knowledge and resolve any of their issues. There are various other job profiles which an individual can pursue after completing the Data Science Training Course in Delhi NCR, Noida and Gurgaon which include Data Analyst, Senior Analyst, Data Manager, Data Scientist, Business Intelligence professional, etc. Upon completion of the Data Science training, valid certification in Data Science will be awarded to all the learners to help them gain the competitive edge over other candidates during the interview. 100% Job Placement Assistance will also be provided to the candidates who successfully cleared the Data Science Training Course by providing them resume building services along with mock interviews to make them ready for entering into the same market.
Over the last few years, Data Science Science and enter into the field. Some of these reasons are mentioned below:
Therefore, if you have made up your mind to learn Data Science, then you can visit our Data Science Training institute in Delhi, Gurgaon or Noida for registration or submit your quotation on our official website.
Course Duration: 150 (Hours) with Highly Skilled Corporate Trainers
(Data Science Training) for 5 Modules – Weekdays / Weekend
This module introduces you to some of the important keywords in R like Business | 300,306 |
TITLE: Rim hook decomposition and volume of moduli spaces
QUESTION [24 upvotes]: I did some computer experiments, counting the number of rim-hook decompositions (aka border-strip decompositions) of rectangles of shape $2n \times n$, where each strip has size $n$.
Here are 12 of the 1379 such decompositions for $n=4$:
Let $a_n$ be the number of such decompositions.
I conjecture that $a_n$ is given by A115047 in OEIS,
that is, $a(0)=1$ and
$$
a(n) = \sum_{i=1}^n
\binom{n - 1}{i - 1} \binom{n + 3}{i + 1} a(i - 1) a(n - i) \frac{ i(n - i + 1)}{2 (n + 2)},
$$
or $1, 5, 61, 1379, 49946, 2648967,...$.
Edit(2018-02-22): My student have checked that the sequences agree up to $n=12$, which is quite compelling.
The interesting part is that there is no such reference in OEIS. This entry regards Weil-Petersson volumes of muduli spaces of an $n$-punctured Riemann sphere, which is quite far from my field.
Q: Can we prove the recursion above?
I think this (conjectured) connection is interesting, giving a combinatorial interpretation of moduli space volumes.
Perhaps this can be extended to other genus?
EDIT: Using the strategy in the formula given in the answer below,
this conjecture is now proved in this preprint.
Edit II Paper is now published in the Journal of Integer sequences.
The answer to the original question is therefore, YES.
REPLY [7 votes]: Abacus is quite useful to count objects related to hooks. For partition $((2n)^n)$, the $n$-abacus is $\{2n, \dots, 3n-1\}$ on $n$ runners. Thus, the number of labelled rim hook decompositions (labelled by the order of removal) equals to the number of permutations of $x_1, \dots, x_n, y_1, \dots, y_n$ such that $x_i$ appears before $y_i$ for every $i$.
In order to count unlabelled rim hook decompositions, we need to find out when the order of two consecutive rim hook removal can be swapped. Then, we can add some constraint that allows one type of removal, but forbids the other type of removal. All labelled rim hook decompositions with the new constraint are in bijection with unlabelled rim hook decompositions.
If we translate the new constraint we get to permutations of $x_i$'s and $y_i$'s, that means: the permutation does not have consecutive $x_i$ and $y_j$ such that $i \gt j$.
It is then straightforward to use inclusion-exclusion to count the number of permutations satisfying both requirements. I get a complicated formula for $a_n$, and the formula matches A115047 for $n$ up to $60$. I believe that it should not be too hard to prove that the formula for $a_n$ satisfies the recursion.
Edit (Mar. 3, 2018)
The $n$-abacus of a partition $\lambda = (\lambda_1, \dots, \lambda_m)$ consists of $n$ runners and $m$ beeds located at $\{\lambda_i + m - i\}$, and the map from beeds to runners is given by $\mathbb{N} \to \mathbb{N} / n \mathbb{N}$. Removing an $n$-hook corresponds to the move of a beed along its runner to a smaller adjacent unoccupied location. A labelled rim hook decomposition is just a sequence of such moves so that no further moves can be made.
For the partition $((2n)^n)$, let $x_i$ be the move of the beed at $2 n + i - 1$ to $n + i - 1$, and let $y_i$ be the move of the beed at $n + i - 1$ to $i - 1$. Two hook removal at $b$ and $b'$ can be swapped if and only if $|b - b'| \gt n$. For this particular partition, that means $x_i y_j$ or $y_j x_i$ such that $i \gt j$. Therefore, we can reformulate the problem of finding $a_n$ to a problem of counting permutations of $x_i$'s and $y_i$'s. For general partitions, the same method works, and we just need to consider possibly larger "alphabet" and longer "forbidden words".
Below is one method of counting such permutations. There must be a much better way to count them.
Suppose that we know some consecutive occurrences of $x_i y_j$'s. Consider the bipartite graph with vertices $x_i$'s and $y_i$'s and edges $x_i y_i$ for all $i$ and $x_i y_j$ for all known consecutive $x_i y_j$ with $i \gt j$. This bipartite graph is a disjoint union of paths, otherwise it would not be legitimate. Let $p$ be the partition such that the parts of $2p$ are the sizes of the connected components of the bipartite graph mentioned above. Therefore, by inclusion-exclusion, we have the formula
$$ a_n = \sum_{p \vdash n} (-1)^{|p - 1|} \frac{1}{|m|!} {|m| \choose m} {|p| \choose p} {|p + 1| \choose p + 1},$$
where $p$ runs over partitions of $n$, $m := (m_1, \dots, m_n)$ with $m_i$ being the multiplicity of $i$ in $p$, $p + c$ is the addition of $c$ to each part of $p$, $|p|$ is the sum of parts in $p$ and ${|p| \choose p}$ is the multinomial coefficient. | 138,143 |
Menu Template:Red Slide Down Menu
Web Button Image by Vista-Buttons.com v4.3.0
Pull Down Menu Box In Javascript
This menu is generated by Flash Menu Builder.
Create your own menu now!
Or follow on Twitter :Css Menu Bar , Css Menu Bar Pull Down Menu Box In Javascript
Pull Down Menu Box Css Tree Menus
Export graphic picture
Using Vista Buttons you can save menu graphic picture as gif-files (*.gif).Menus Horizontal Desplegables En Html
Insert button script into the existing HTML page
You can insert your button script into the existing HTML page. To do so, click "Page insert" button on the Toolbar.Javascript Populate Multiple Menus
Integration with popular web authoring software.
Vista Buttons integrates with Dreamweaver, FrontPage, and Expression Web as an extension/add-in. Create, insert, modify a menu without leaving your favorite web design framework!Menu Css2 Generator
>?"Xp Taskbar Menu Dhtml
- "... "
- "..How can I set up Vista Buttons dreamweaver extension?" | 192,134 |
Way back when, I noticed this too. It would happen with my favorite little test utility, fact. I would get only the first/last calls being "traced" if the function was compiled, but would get the full trace when compiling was disabled. This behavior appeared with 1.3, but was not present with 1.2. (This all from memory...) --Frank | 163,324 |
Bobby & Amelia
United States
Start for free
Total users: 351814
Online: 17673
Total users: 351814
Online: 17673
New Personals on March 15, 2022
Woman, 28 Years, 155 cm, looking for man in age 30 - 40
Sagittarius, Construction, Repair and Maintenance, Arizona / Sun LakesChat now
Man, 45 Years, 173 cm, looking for woman in age 42 - 52
Pisces, Education, Arizona / SaddlebrookeChat now
Woman, 32 Years, 163 cm, looking for man in age 30 - 40
Cancer, Trades, Arizona / DouglasChat now
Man, 30 Years, 194 cm, looking for woman in age 24 - 34
Pisces, Leisure, Sport and Tourism, Arizona / SomertonChat now
Woman, 20 Years, 158 cm, looking for man in age 18 - 24
Cancer, Market Research, Arizona / New RiverChat now
Man, 26 Years, 194 cm, looking for woman in age 19 - 29
Taurus, Restaurant and Hospitality, Arizona / Catalina FoothillsChat now
Man, 29 Years, 189 cm, looking for woman in age 19 - 29
Scorpio, Hospitality and Events Management, Arizona / Fountain HillsChat now
Man, 28 Years, 190 cm, looking for woman in age 26 - 36
Capricorn, Energy and Utilities, Arizona / Catalina FoothillsChat now
Woman, 20 Years, 187 cm, looking for man in age 18 - 26
Libra, Science and Pharmaceuticals, Arizona / GlendaleChat now
Man, 21 Years, 178 cm, looking for woman in age 18 - 24
Aquarius, Sales, Arizona / FlagstaffChat now
Woman, 41 Years, 175 cm, looking for man in age 38 - 48
Aquarius, Retail, Arizona / EloyChat now
Woman, 34 Years, 163 cm, looking for man in age 32 - 42
Cancer, Engineering and Manufacturing, Arizona / Drexel HeightsChat now
Man, 21 Years, 185 cm, looking for woman in age 18 - 28
Gemini, Recruitment and HR, Arizona / PhoenixChat now
Man, 20 Years, 170 cm, looking for woman in age 18 - 28
Pisces, Energy and Utilities, Arizona / Paradise Ari Ari | 344,865 |
TITLE: For $x, y \in \mathbb{R}$, prove that $\max(x, y) = \frac{x + y + |x - y|}{2},$ and $\min(x, y) = \frac{x + y - |x - y|}{2}$.
QUESTION [2 upvotes]: Prove that for all real numbers $x$ and $y$,
$$\max(x, y) = \dfrac{x + y + |x - y|}{2},$$
and
$$\min(x, y) = \dfrac{x + y - |x - y|}{2}.$$
For any real number $x$, the absolute value of $x$, denoted $|x|$ is defined as follows:
\begin{equation}
|x| = \begin{cases} x; & \text{ if } x \geq 0 \\
-x; & \text{ if } x< 0
\end{cases}
\end{equation}
What I understand from this is that $|x| = x$ if $x \geq 0$ or $|x| = −x$ if $x<0$. Other than that I don't really know how to start this.
REPLY [2 votes]: \begin{align}
\max\{x,y\} + \min\{x,y\} &= x + y \\
\max\{x,y\} - \min\{x,y\} &= |x-y| \\
\text{Adding we get} \\
2\max\{x,y\} &= x + y + |x-y| \\
\text{Subtracting we get} \\
2\min\{x,y\} &= x + y - |x-y| \\
\end{align} | 219,538 |
I Am a Hoovering Housewife Me Too for all you accidental housewives! 9 People Report Group Sort popular recent Compromises I never thought I'd be a housewife... well I'm not really, now that I'm a student and will have a profession in the future, but in one way I am... I'm the one who's in charge of keeping the house tidy, making sure there's food on a plate and that the clothes are clean. Sure it's... Terrorita 26-30, F 3 Responses 1 Jul 26, 2007 Cleaning Well, i've been cleaning the apartment for 3 days and finally it's ready!!! That's what you get for being a perfectionist :D Terrorita 26-30, F 0 Jul 27, 2007 More Stories | 66,474 |
The.
The. While there have been many presidential assassination attempts, this musical features eight of them: John Wilkes Booth shot Lincoln , Charles J.
The play features 18 scenes, many containing one or more of the assassins and their background. This show is very ensemble heavy and features music with some or all of the assassins, as well as the co-narrators, the balladeer and the proprietor. The scenes take place over different time periods, but eventually the assassins merge into one union. They want their fame and they speak to Lee Harvey Oswald, who is about to kill himself, about shooting John F.
Kennedy for the fame. In the end, the assassins come out and proudly sing that they had the right to complete their actions, while the ensemble stands behind them in anger at the assassins for ruining the lives of the ensemble. This story is told as a musical. As Stephen Sondheim has commonly done, the music emphasizes the lyrics and contains difficult and uninteresting melodies.
The musical features only 11 songs, unlike most musicals which contain around This is sung at the carnival and features the assassins buying their guns, the first step to even thinking about any kind of assassination. Ballad of Booth, Ballad of Czolgosz, and Ballad of Guiteau are led by the balladeer and they tell the stories of each of the successful assassins. Gun Song is very informational in explaining how much damage a gun causes and how easy it is to shoot a gun.
Another National Anthem features the entire cast. The assassins are justifying why their actions were so necessary and complaining about their absence of positive attention for their doings. The balladeer is responding how he, as an American felt about their doings, representing both himself personally and speaking for the whole county.
You can be what you choose This is historically accurate, as most people who were alive in remember where they were and what they were doing upon hearing about this shocking event. Each character speaks in a way very accurate to his or her native living area. The story is very fast paced. The balladeer covers the stories of the successful assassins in five minute songs each.
Some scenes are funny, such as a Moore and Fromme conversation in the park. However, the show mostly contains very serious scenes that revolve around one general topic: gun violence. Unlike many shows, this show does not feature a hero. Because the brain always feels the need to root for someone, a viewer might find him or herself rooting for a crazy madman.
This character could really be any of the assassins. As I was a cast member of this show over the summer, I identify with Billy, the character I played. They genderbent the character and called her Milly. The way Milly and Moore interact negatively is a lot like the way my mom and I interact negatively. I love the song, Ballad of Czolgosz and it was the only part of the show that there was dancing involved. My least favorite part was when Moore and Fromme were throwing bullets at Gerald Ford.
I found this scene poorly written and completely historically inaccurate. Weidman should have written in Gerald Ford to the show in some other way. Weidman did do a great job executing the theme. He proves that violence does not fix problems by showing nine violent acts and what the outcomes were of each of them.
The writing was also very smart to only feature presidential assassins, as opposed to assassins of other types of famous figures. This worked because of the large number of presidential assassination attempts and the universal awareness of the names of the presidents, despite the universal unawareness of some of the names of the assassins. The author was also very informational in terms of the historical events and I learned a lot of names and settings from this play.
I would recommend this play to anyone over 11 years of age. Anyone younger does not yet understand the emotional destruction that gun violence causes, but anyone older would enjoy this show because it is very interesting. The sets are great and the song lyrics are brilliant.
BLACK AND DECKER ELECTROMATE 400 MANUAL PDF
Assassins Script
.
ALPINE KTX 100EQ PDF
ASSASSINS SONDHEIM SCRIPT PDF
Synopsis[ edit ] This synopsis reflects the current licensed version of the show. The published script of the Off-Broadway production is slightly different. The show opens in a fairground shooting gallery where, amid flashing lights, human figures trundle past on a conveyor belt. John Wilkes Booth enters last and the Proprietor introduces him to the others as their pioneer before he begins distributing ammunition.
C106D DATASHEET PDF
. | 314,053 |
\begin{document}
\maketitle
\begin{abstract} In
this paper we study the branching problems for Hecke algebra
$\H(D_n)$ of type $D_n$. We explicitly describe the decompositions
of the socle of the restriction of each irreducible
$\H(D_n)$-representation to $\H(D_{n-1})$ into irreducible modules
by using the corresponding results for type $B$ Hecke algebras. In
particular, we show that any such restrictions are always
multiplicity free.
\end{abstract}
\section{Preliminaries}
Let $W(B_{n})$ be the Weyl group of type $B_{n}$. It is a finite
group with generators $\{s_{0}, s_{1},\cdots,s_{n-1}\}$ and
relations $$ \aligned
&s_i^2=1,\quad\text{for\, $0\leq i\leq n-1$,}\\
&s_0s_1s_0s_1=s_1s_0s_1s_0,\\
&s_is_{i+1}s_i=s_{i+1}s_{i}s_{i+1},\quad\text{for\, $1\leq i\leq n-2$,}\\
&s_is_j=s_js_i,\quad\text{for\, $0\leq i<j-1\leq n-2$.}\endaligned
$$
Let $u:=s_0s_1s_0$. The subgroup of $W(B_{n})$ generated by $\{u,
s_{1},\cdots,s_{n-1}\}$ is a Weyl group of type $D_{n}$. We denote
it by $W(D_{n})$. It has a presentation with generators $\{u,
s_{1},\cdots,s_{n-1}\}$ and relations $$ \aligned
&u^2=1=s_i^2,\quad\text{for\, $1\leq i\leq n-1$,}\\
&us_2u=s_2us_2,\quad s_is_{i+1}s_i=s_{i+1}s_{i}s_{i+1},\quad\text{for\, $1\leq i\leq n-2$,}\\
&us_1=s_1u,\quad us_i=s_iu,\quad\text{for \,$3\leq i\leq n-1$,}\\
&s_is_j=s_js_i,\quad\text{for\, $1\leq i<j-1\leq n-2$.}\endaligned
$$
Let $\BS_n$ be the symmetric group on $n$ letters. It is
well-known that $W(B_{n})\cong (\Z/2\Z)^{n}\rtimes{\BS}_{n},\,
W(D_{n})\cong (\Z/2\Z)^{n-1}\rtimes{\BS}_{n}$, and the subgroup
generated by $s_{1}, s_{2}, \cdots$ $,s_{n-1}$ (respectively $u,
s_{2}, \cdots, s_{n-1}$) can be identified with the symmetric
group $\BS_n$.
Let $K$ be a field. {\it Throughout this paper we assume that
$\ch{K}\neq 2$}. Let $q, Q$ be two invertible elements in $K$.
There is a Hecke algebra $\H_{q, Q}(B_{n})$ with parameters $q, Q$
associated to $W(B_{n})$ (see \cite{DJ}). In this paper we will
only be concerned with the special case where $Q=1$, i.e.,
$\H(B_n):=\H_{q, 1}(B_{n})$. By definition, $\H(B_n)$ is an
associative algebra with generators $T_0, T_1,\cdots, T_{n-1}$ and
relations
$$\aligned
&T_0^2=1,\quad (T_i+1)(T_i-q)=0,\quad\text{for\, $1\leq i\leq n-1$,}\\
&T_0T_1T_0T_1=T_1T_0T_1T_0,\\
&T_iT_{i+1}T_i=T_{i+1}T_{i}T_{i+1},\quad\text{for\, $1\leq i\leq n-2$,}\\
&T_iT_j=T_jT_i,\quad\text{for\, $0\leq i<j-1\leq n-2$.}\endaligned
$$
Let $T_u:=T_0T_1T_0$. The subalgebra of $\H(B_{n})$ generated by
$\{T_u,T_{1},\cdots,T_{n-1}\}$ is isomorphic to a Hecke algebra of
type $D_n$, i.e., the Hecke algebra associated to the Weyl group
$W(D_{n})$. We denote it by $\H(D_{n})$. It has a presentation
with generators $\{T_u, T_{1},\cdots, T_{n-1}\}$ and relations
$$\aligned
&(T_u+1)(T_u-q)=0,\quad (T_i+1)(T_i-q)=0,\quad\text{for\, $1\leq i\leq n-1$,}\\
&T_uT_{2}T_u=T_{2}T_{u}T_{2},\quad
T_iT_{i+1}T_i=T_{i+1}T_{i}T_{i+1},
\quad\text{for\, $1\leq i\leq n-2$,}\\
&T_uT_1=T_1T_u,\quad T_uT_i=T_iT_u,\quad\text{for\, $3\leq i\leq n-1$,}\\
&T_iT_j=T_jT_i,\quad\text{for\, $1\leq i<j-1\leq n-2$,}
\endaligned
$$
By \cite{DJMu}, $\H(B_n)$ is a cellular algebra in the sense of
\cite{GL}. For each bipartition $\lam=(\lam^{(1)},\lam^{(2)})$ of
$n$, there is a {\it Specht module} $\ts^{\lam}$, and a naturally
defined bilinear form on $\ts^{\lam}$. Let $\td^{\lam}$ be the
quotient of $\ts^{\lam}$ modulo the radical of that form. We have
that
\begin{lem} [\rm \cite{DJMu}] 1) Every simple $\H(B_{n})$ module is a composition factor of some
$\ts^{\lam}$. When $\H(B_{n})$ is semi-simple, each $\ts^{\lam}$
is absolutely irreducible and they form a complete set of pairwise
non-isomorphic simple $\H(B_{n})$-modules.
2) If $\td^{\mu}\neq 0$ is a composition factor of $\ts^{\lambda}$
then $\lam\trianglerighteq\mu$, and every composition factor of
$\ts^{\lam}$ is isomorphic to some $\td^{\mu}$ with
$\lam\trianglerighteq\mu$, where $\trianglerighteq$ is the dominance order defined in \cite{DJMu}.
If $\td^{\lam}\neq 0$ then the
composition multiplicity of $\td^{\lam}$ in $\ts^{\lam}$ is one.
3) The set $\Bigl\{\td^{\lam}\Bigm|\text{$\lam$ is a bipartition
of $n$ and $\td^{\lam}\neq 0$}\Bigr\}$ forms a complete set of
pairwise non-isomorphic simple $\H(B_{n})$-modules.
4) $\H(B_{n})$ is semi-simple if and only if $\,\,
2\biggl(\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)\biggr)\biggl(\prod_{i=1}^{n}\bigl(1+q+q^2+
\cdots+q^{i-1}\bigr)\biggr)\neq 0$. In that case, it is also split
semi-simple.
\end{lem}
Let $\tau$ be the involutive $K$-algebra automorphism of
$\H(B_{n})$ which is defined on generators by
$\tau(T_{1})=T_{0}T_{1}T_{0}, \tau(T_i)=T_i,\,\,\forall\, i\neq
1$. Then $\tau$ maps $\H(D_{n})$ isomorphically onto $\H(D_{n})$.
Let $\sigma$ be the involutive $K$-algebra automorphism of
$\H(B_{n})$ which is defined on generators by
$\sigma(T_{0})=-T_{0}, \sigma(T_i)=T_i,\,\,\forall\, i\neq 0$.
Then $\sigma\downarrow_{\H(D_n)}=\id$. Let $\widetilde{\mathcal
P}_n$ be the set of all the bipartitions of $n$.
\addtocounter{defs}{1}
\begin{defs} For each $\lam=(\lam^{(1)},\lam^{(2)})\in\widetilde{\mathcal{P}}_n$, we
define
$\widehat{\lam}=(\lam^{(2)},\lam^{(1)})\in\widetilde{\mathcal{P}}_n$.
\end{defs}
Let ${\mathcal
P}_n:=\bigl\{\lam\in\widetilde{\mathcal{P}}_n\bigm|\td^{\lam}\neq
0\bigr\}$. There is an involution $\bH$ defined on ${\mathcal
P}_n$ such that
$\Bigl(\td^{\lam}\Bigr)^{\sigma}\cong\td^{\bH(\lam)}$ for any
$\lam\in{\mathcal P}_n$. In particular,
$\td^{\lam}\!\!\downarrow_{\H(D_n)}\cong\td^{\bH(\lam)}\!\!\downarrow_{\H(D_n)}$.
We define an equivalence relation on ${\mathcal P}_n$ by
$\lam\approx\mu$ if and only if $\mu=\bH(\lam)$. By the results in
\cite{P}, \cite{Hu1}, \cite{Hu2} and \cite{Hu3}, we have that
\addtocounter{lem}{1}
\begin{lem} [\rm \cite{Hu1}] Suppose that $\ch K\neq
2$ and $\H(D_n)$ is split over $K$. If $\lam\neq\bH(\lam)$, then
$\td^{\lam}\!\!\downarrow_{\H(D_n)}$ is irreducible; if
$\lam=\bH(\lam)$, then $\td^{\lam}\!\!\downarrow_{\H(D_n)}$ splits
into a direct sum of two $\H(D_n)$-submodules, say $D_{+}^{\lam}$
and $D_{-}^{\lam}$. Moreover, the set $$
\Bigl\{\td^{\lam}\!\!\downarrow_{\H(D_n)}\Bigm|\lam\in{\mathcal
P}_n/{\approx},\,\,\lam\neq\bH(\lam)\Bigr\}\bigcup
\Bigl\{D_{+}^{\lam}, D_{-}^{\lam}\Bigm|\lam\in{\mathcal
P}_n/{\approx},\,\,\lam=\bH(\lam)\Bigr\} $$ forms a complete set
of pairwise non-isomorphic irreducible $\H(D_n)$-modules.
\end{lem}
Let $e>1$ be a fixed integer. Let $\lam$ be a bipartition
of $n$. For each node $\gamma=(i,j)$ of $\lam$, we define the residue of $\gamma$
to be $j-i+e\mathbb{Z}\in\mathbb{Z}/e\mathbb{Z}$. Then we have
the notion of $e$-good (removable) nodes of $\lam$
(see \cite{AM} and \cite{Hu2}). For each integer $m\in\mathbb{N}$,
the set $\mathcal{K}_m$ of Kleshchev bipartitions of $m$ with
respect to $(\sqrt[e]{1};1,-1)$ is defined inductively by
\smallskip
(1)
$\mathcal{K}_0:=\Bigl\{\underline{\emptyset}:=\bigl(\emptyset,\emptyset\bigl)\Bigr\}$;
(2)
$\mathcal{K}_{m}:=\Bigl\{\lam\in\widetilde{\mathcal{P}}_{m}\Bigm|\begin{matrix}\text{$\lam$
is obtained from some $\mu\in\mathcal{K}_{m-1}$ by}\\
\text{adding an $e$-good node}\end{matrix}\Bigr\}$.
\medskip
The {\it Kleshchev's good lattice} with respect to
$(\sqrt[e]{1};1,-1)$ is the infinite graph whose vertices are the
Kleshchev bipartitions with respect to
$(\sqrt[e]{1};1,-1)$ and whose arrows are given by $$
\text{$\mu\overset{x}{\rightarrow}\lam$\quad$\Longleftrightarrow$\quad
$\lam$ is obtained from $\mu$ by adding an $e$-good $x$-node}.
$$ By a result of S. Ariki (see \cite{A}),
$\mathcal{P}_n=\mathcal{K}_n$ when $e$ is the smallest positive integer satisfying
$1+q+q^2+\cdots+q^{e-1}=0$.
\begin{lem} [\rm \cite{P}, \cite{Hu1}, \cite{Hu2}, \cite{Hu3}]
1) If $2\prod_{i=1}^{n-1} \bigl(1+q^i\bigr)\neq 0$ in $K$, then
$\H(D_n)$ is split over $K$. In this case, for each
$\lam=(\lam^{(1)},\lam^{(2)})\in\mathcal{P}_n$, we have that
$\bH(\lam)=\widehat{\lam}$. In particular,
$\td^{\lam}\!\!\downarrow_{\H(D_{n})}\cong\td^{\widehat\lam}\!\!\downarrow_{\H(D_{n})}$,\smallskip
2) If $\,\,\prod_{i=1}^{n-1} \bigl(1+q^i\bigr)=0$, $\,\ch K\neq 2$
and $\H(D_n)$ is split over $K$, then $q$ is a primitive $2\l$-th
root of unity in $K$ for some integer $1\leq\l<n$. In this case,
$\bH$ can be described as follows: if $\lam\in\mathcal{P}_n$ is a
Kleshchev bipartition with respect to $(\sqrt[2\l]{1};1,-1)$, and
$$
\underline{\emptyset}\stackrel{i_1}{\longrightarrow}\cdot\overset{i_2}{\longrightarrow}\cdot
\cdots\stackrel{i_n}{\longrightarrow}\lam, $$ is a path from
$\underline{\emptyset}:=(\emptyset,\emptyset)$ to $\lam$ in
Kleshchev's good lattice with respect to $(\sqrt[2\l]{1};1,-1)$.
Then, the sequence $$
\underline{\emptyset}\stackrel{i_1+\l}{\longrightarrow}\cdot\overset{i_2+\l}{\longrightarrow}\cdot
\cdots\stackrel{i_n+\l}{\longrightarrow}\bH(\lam), $$ also
defines a path in Kleshchev's good lattice with respect to
$(\sqrt[2\l]{1};1,-1)$, and it connects $\underline{\emptyset}$ to
$\bH(\lam)$.\footnote{This result was proved in \cite{Hu2} only in
the case where $K=\mathbb{C}$. For more general $K$, see appendix
of this paper.}
3) If $\,\ch K\neq 2$, then $\H(D_{n})$ is semi-simple if and
only if $$
\biggl(\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)\biggr)\biggl(\prod_{i=1}^{n}\bigl(1+q+q^2+
\cdots+q^{i-1}\bigr)\biggr)\neq 0.$$ In that case, it is also
split semi-simple.
\end{lem}
In this paper, we shall give the modular branching rule for
$\H(D_n)$. That is, for each irreducible $\H(D_n)$-module $D$, we
describe $\soc\bigl(D\!\!\!\downarrow_{\H(D_{n-1})}\bigr)$. The
discussion will be divided into two cases: the case where
$2\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)\neq 0$ and the case where
$\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)=0$ and $2\cdot 1_K\neq 0$. The
main results are presented in Theorem 2.5, Theorem 2.6, Corollary
2.8, Theorem 3.7, Theorem 3.9 and Corollary 3.11. It turns out that
our situation here bears much resemblance to the situation of
representations of the alternating group $A_n$ (which is a normal
subgroup in $\BS_n$ of index $2$), see \cite{B}, \cite{FK},
\cite{LLT} and \cite{Hu2}, and our results are largely motivated
by those in \cite{BO}, where the branching rules for the
representations of the alternating groups were deduced.
\medskip
Finally, in the appendix of this paper, we include a proof (which
is essentially due to Professor S. Ariki) of the fact that the
involution $\bH$ is independent of the base field $K$ as long as
$\ch K\neq 2$ and $\H(D_n)$ is split over $K$.
\smallskip\bigskip
\section{The case where
$2\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)\neq 0$}
Let $\lam$ be a bipartition, $[\lam]$ be its Young diagram. To
simplify notation, we shall identify $\lam$ with $[\lam]$. Recall
that a {\it removable} node is a node of the boundary of $[\lam]$
which can be removed, while an {\it addable} node is a concave
corner on the rim of $[\lam]$ where a node can be added. {\it
Throughout this section, we shall assume that
$2\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)\neq 0$ in $K$.} In
particular, Lemma 1.4(1) applies to both the Hecke algebra
$\H(D_n)$ and the Hecke algebra $\H(D_{n-1})$.
Let $l$ be the smallest positive integer $a$ such that
$1+q+q^2+\cdots+q^{a-1}=0$. If such an integer does not exist,
then we set $l=\infty$. A partition $\lam=(\lam_1,\lam_2,\cdots)$
is said to be $l$-restricted if $\lam_i-\lam_{i+1}<l$ for all $i$.
\begin{lem} [\rm \cite{DJ}, \cite{DR}] Suppose that
$\,2\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)\neq 0$ in $K$. Then
1) for each bipartition $\lam=(\lam^{(1)},\lam^{(2)})$ of $n$, we
have that $\lam\in\mathcal{P}_n$ if and only if both $\lam^{(1)}$
and $\lam^{(2)}$ are $l$-restricted.
2) for each bipartition $\lam\in\mathcal{P}_n$, $\,\,
\soc\Bigl(\td^{\lam}\!\!\!\downarrow_{\H(B_{n-1})}\Bigr)\cong\bigoplus_{\mu\rightarrow\lam}
\td^{\mu}$, where $\mu\rightarrow\lam$ means that $\mu$ is
a bipartition of $\,\,n-1$ such that the Young diagram $[\mu]$ is obtained from the Young diagram
$[\lam]$ by removing an $l$-good node.
\end{lem}
Note that in this case the Kleshchev's good lattice with respect to
$(\sqrt[l]{1};1,-1)$ is well-understandood. Namely, for any Kleshchev bipartition $\lam$, a removable node
$\gamma$ is an $l$-good node of the bipartition $\lam$ if and only if $\gamma$ is an $l$-good node
of the partition $\lam^{(1)}$ or of the partition $\lam^{(2)}$. Here the notion of $l$-good nodes of
partitions is defined in a similar way as $l$-good nodes of bipartitions, see \cite{Kl}, \cite{LLT} and
\cite{AM} for details.\smallskip
We want to describe the decomposition of the socle of
$D\!\!\downarrow_{\H(D_{n-1})}$ into irreducible modules for each
irreducible $\H(D_n)$-module $D$. For each bipartition $\lam$ and
each removable node $A$ of $[\lam]$, we shall denote by
$\lam\setminus{A}$ the Young diagram (or equivalently,
bipartition) obtained by removing the node $A$ from $[\lam]$.
\addtocounter{defs}{1}
\begin{defs} Let $\lam\in\mathcal{P}_n$.
Suppose that $\lam$ has an $l$-good node $A$ such that $$
\lam\setminus{A}=\widehat{(\lam\setminus{A})}=\widehat\lam\setminus{A'}
$$
for some removable node $A'$ in $\widehat\lam$. Then in this case
the node $A'$ is uniquely determined by $A$ and is also an $l$-good
node of $\widehat\lam$. We say that $\lam$ is almost symmetric and
$A'$ is the conjugate node of $A$.
\end{defs}
\addtocounter{example}{2}
\begin{example} Suppose that $n=5$, $l=\infty$.
$\lam=((2,1), (1^2)),\, \mu=((2), (1^3))$ are two bipartitions of
$5$. Let $A$ be the node which is in the first row and the second
column of the first component of $\lam$. Then $\lam$ is almost
symmetric with $\lam\setminus{A}=\widehat{(\lam\setminus{A})}$,
but $\mu$ is not almost symmetric.
\end{example}
\addtocounter{lem}{2}
\begin{lem} Let $\lam=(\lam^{(1)},\lam^{(2)})$ be a
bipartition of $n$. Suppose that $\lam$ is almost symmetric with
$\lam\setminus{A}=\widehat{(\lam\setminus{A})}$ for some removable
node $A$ of $\lam$. Then for any pairs of removable nodes $B, C$
of $\lam$ satisfying $C\neq A$, we have that $
\lam\setminus{B}\neq\widehat{(\lam\setminus{C})}$. In particular,
for any removable nodes $C$ of $\lam$ satisfying $C\neq A$, we
have that $$ \lam\setminus{C}\neq\widehat{(\lam\setminus{C})}. $$
\end{lem}
\noindent {Proof:} \, This is obvious. \hfill\qed
\addtocounter{thm}{4}
\begin{thm} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam$ is almost symmetric with
$\lam\setminus{A}=\widehat{(\lam\setminus{A})}$ for some $l$-good node
$A$ of $\lam$. Then $\lam\neq\widehat\lam$ and $$
\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
D_{+}^{\lam\setminus{A}}\bigoplus D_{-}^{\lam\setminus{A}}\bigoplus
\bigoplus_{\substack{C\in [\lam],\,\, C\neq A\\ \text{$C$ is
$l$-good}}}\td^{\lam\setminus{C}}\!\!\downarrow_{\H(D_{n-1})}.
$$
In particular,
$\,\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ is
multiplicity free.
\end{thm}
\noindent {Proof:}\, This follows directly from Lemma 1.3,
Lemma 1.4, Lemma 2.1, Lemma 2.4 and the fact that
$$\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)=\Bigl\{\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\Bigr\}\!\!\downarrow_{\H(D_{n-1})}.\eqno(2.5.1)$$
We have to prove (2.5.1). It is clear that $$
\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\supseteq\Bigl\{\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\Bigr\}\!\!\downarrow_{\H(D_{n-1})}. $$ It suffices to show that $$
\dim\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\leq\dim\Bigl\{\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\Bigr\}\!\!\downarrow_{\H(D_{n-1})}. $$ Since every simple $\H(D_{n-1})$-module occurs as a direct summand of
$\td^{\mu}\!\!\downarrow_{\H(D_{n-1})}$ for some $\mu\in\mathcal{P}_{n-1}$, we divide the proof into two cases:
\smallskip
\noindent {\it Case 1.} Let $\mu\in\mathcal{P}_{n-1}$ be such that
$\mu\neq\widehat{\mu}$ and
$$\Hom_{\H(D_{n-1})}\Bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})},\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\neq 0. $$
Then by Frobenius Reciprocity (\cite[(11.13)]{CR}),
$$\aligned
&\quad\,
\Hom_{\H(D_{n-1})}\biggl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})},\Bigl\{\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\Bigr\}\!\!\downarrow_{\H(D_{n-1})}\biggr)\\
&\cong\Hom_{\H(B_{n-1})}\biggl(\bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})}\bigr)\!\!\uparrow^{\H(B_{n-1})},
\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\biggr)\\
&\cong\Hom_{\H(B_{n-1})}\biggl(\td^{\mu}\oplus\bigl(\td^{\mu}\bigr)^{\sigma},
\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\biggr)\\
&\cong\Hom_{\H(B_{n-1})}\Bigl(\td^{\mu}\oplus\bigl(\td^{\mu}\bigr)^{\sigma},\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\\
&\cong
\Hom_{\H(B_{n-1})}\Bigl(\bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})}\bigr)\!\!\uparrow^{\H(B_{n-1})},
\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\\
&\cong\Hom_{\H(D_{n-1})}\Bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})},\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr),
\endaligned$$
as required.
\smallskip
\noindent {\it Case 2.} Let $\mu\in\mathcal{P}_{n-1}$ be such that
$\mu=\widehat{\mu}$ and
$$\Hom_{\H(D_{n-1})}\Bigl(D_{+}^{\mu},\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\neq 0. $$
Then by Frobenius Reciprocity (\cite[(11.13)]{CR}),
$$\aligned &\quad\,
\Hom_{\H(D_{n-1})}\biggl(D_{+}^{\mu},\Bigl\{
\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\Bigr\}\!\!\downarrow_{\H(D_{n-1})}\biggr)\\
&\cong\Hom_{\H(B_{n-1})}\biggl(D_{+}^{\mu}\!\!\uparrow^{\H(B_{n-1})},\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\biggr)\\
&\cong\Hom_{\H(B_{n-1})}\biggl(\td^{\mu},\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\biggr)\\
&\cong\Hom_{\H(B_{n-1})}\Bigl(\td^{\mu},\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\\
&\cong
\Hom_{\H(B_{n-1})}\Bigl(D_{+}^{\mu}\!\!\uparrow^{\H(B_{n-1})},\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\\
&\cong\Hom_{\H(D_{n-1})}\Bigl(D_{+}^{\mu},\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr),\endaligned
$$
as required. This completes the proof of (2.5.1).\hfill\qed
\medskip
Let $\lam\in\mathcal{P}_n$. Suppose that $\lam\neq\widehat{\lam}$
and $\lam$ is not almost symmetric. We claim that for any two $l$-good
nodes $B, C$ of $\lam$,
$\lam\setminus{B}\neq\widehat{(\lam\setminus{C})}$. In fact, it is
enough to show that if $B\neq C$, then
$\lam\setminus{B}\neq\widehat{(\lam\setminus{C})}$. Otherwise, if
$B, C$ both lie in the same component of $\lam$, say,
$\lam^{(1)}$, then we have that
$\lam^{(1)}\setminus{B}=\lam^{(2)}=\lam^{(1)}\setminus{C}$, which
is impossible; while if $B, C$ lie in different components of $\lam$,
say, $B\in\lam^{(1)}, C\in\lam^{(2)}$, then
$\lam\setminus{B}=\widehat{(\lam\setminus{C})}$ implies that
$\lam^{(1)}=\lam^{(2)}$, which is again impossible. This proves
our claim. It follows from this and Lemma 1.3 and Lemma 1.4
and Lemma 2.1 and (2.5.1) that
\begin{thm} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam\neq\widehat{\lam}$ and $\lam$ is not almost symmetric. Then
$$ \soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
\bigoplus_{\substack{C\in [\lam]\\ \text{$C$ is $l$-good}}}
\td^{\lam\setminus{C}}\!\!\downarrow_{\H(D_{n-1})}.
$$
In particular,
$\,\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ is
multiplicity free.
\end{thm}
Let $\lam\in\mathcal{P}_n$. Now suppose that $\lam=\widehat\lam$.
It remains to describe the decompositions of the socle of
$D_+^{\lam}\!\!\downarrow_{\H(D_{n-1})}$ and of
$D_-^{\lam}\!\!\downarrow_{\H(D_{n-1})}$ into irreducible
$\H(D_{n-1})$-modules.
\begin{thm} Let $\lam\in\mathcal{P}_n$ be such that
$\lam=\widehat\lam$. Then there is a $\H(D_{n-1})$-module
isomorphism $$
\soc\Bigl({D_+^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong\soc\Bigl({D_-^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\Bigr).
$$
\end{thm}
\noindent {Proof:}\, By assumption $n$ is even. Hence $n-1$ is
odd. In particular, for any bipartition $\mu\in\mathcal{P}_{n-1}$,
$\mu\neq\widehat{\mu}$. By \cite[Corollary 2.4]{Hu3},
$\td^{\mu}\cong\Bigl(\td^{\mu}\Bigr)^{\tau}$. We have that
$$\aligned
&\quad\,\Hom_{\H(D_{n-1})}\Bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})},{D_+^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\\
&\cong
\Hom_{\H(D_{n-1})}\Bigl(\bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})}\bigr)^{\tau},\bigl({D_+^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\bigr)^{\tau}\Bigr)\\
&\cong\Hom_{\H(D_{n-1})}\Bigl(\bigl(\td^{\mu}\bigr)^{\tau}\!\!\downarrow_{\H(D_{n-1})},\bigl({D_+^{\lam}}\bigr)^{\tau}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\\
&\cong\Hom_{\H(D_{n-1})}\Bigl(\td^{\mu}\!\!\downarrow_{\H(D_{n-1})},{D_-^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\Bigr).
\endaligned$$ Now using Lemma 1.3 and Lemma 1.4, the
theorem follows at once. \hfill\qed\medskip
We define an equivalence relation $\sim$ on ${\mathcal P}_{n-1}$
by $\lam\sim\mu$ if and only if $\mu=\widehat\lam$. Then Lemma
2.1 and (2.5.1) implies the following:
\addtocounter{cor}{7}
\begin{cor} Let $\lam\in\mathcal{P}_n$ be such that
$\lam=\widehat\lam$. Then $$
\soc\Bigl({D_+^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
\soc\Bigl({D_-^{\lam}}\!\!\downarrow_{\H(D_{n-1})}\Bigr)
\cong\bigoplus_{\mu} \td^{\mu}\!\!\downarrow_{\H(D_{n-1})},
$$
where the sum $\mu$ is taken over a fixed set of representatives
of equivalence classes in $\mathcal{P}_{n-1}/{\sim}$ such that
$\mu\rightarrow\lam$. In particular,
$\,\soc\Bigl(D_+^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ and
$\soc\Bigl(D_-^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ are both
multiplicity free.
\end{cor}
\begin{cor} For any irreducible $\H(D_n)$-module $D$,
$\,\soc\Bigl(D\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ is multiplicity
free.
\end{cor}
\noindent Now Theorem 2.5, Theorem 2.6 and Corollary 2.8
completely determine the decomposition of
$\soc\Bigl(D\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ into irreducible
$\H(D_{n-1})$-modules for every irreducible $\H(D_n)$-module $D$.
\addtocounter{example}{6}
\begin{example} Suppose $\H(D_5)$ is semi-simple. With
the notations in Example 2.3, we have that $$ \begin{aligned}
\ts^{\lam}\!\!\downarrow_{\H(D_4)}&\cong\ts_+^{((1^2),(1^2))}\bigoplus
\ts_-^{((1^2),(1^2))}\bigoplus\ts^{((2),(1^2))}\!\!\downarrow_{\H(D_4)}\\
&\qquad\qquad
\bigoplus\ts^{((2,1),(1))}\!\!\downarrow_{\H(D_4)},\\
\ts^{\mu}\!\!\downarrow_{\H(D_4)}&\cong\ts^{((1),(1^3))}\!\!\downarrow_{\H(D_4)}\bigoplus
\ts^{((2),(1^2))}\!\!\downarrow_{\H(D_4)}.\end{aligned}$$
\end{example}
\begin{example} Suppose $\H(D_6)$ is semi-simple. Let
$n=6$, $\nu=((2,1), (2,1))$. Then we have that $$
\ts_+^{\nu}\!\!\downarrow_{\H(D_5)}\cong\ts_-^{\nu}\!\!\downarrow_{\H(D_5)}\cong
\ts^{((2,1),(2))}\!\!\downarrow_{\H(D_5)}\bigoplus
\ts^{((2,1),(1^2))}\!\!\downarrow_{\H(D_5)}.
$$
\end{example}
\begin{example} Suppose that $q=1$. Then, as long as
$\ch K\neq 2$, Theorem 2.5, Theorem 2.6 and Corollary 2.8
completely determine the decomposition of
$\soc\Bigl(D\!\!\downarrow_{KW(D_{n-1})}\Bigr)$ into irreducible
$KW(D_{n-1})$-modules for every modular irreducible
$KW(D_n)$-module $D$.
\end{example}
\section{The case where
$\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)=0$ and $2\cdot 1_K\neq 0$}
Throughout this section, we assume that
$\prod_{i=1}^{n-1}\bigl(1+q^i\bigr)=0$, $2\cdot 1_K\neq 0$ and
$K$ is a field such that $\H(D_n)$ is split over $K$. It follows
that $q$ is a primitive $2\l$-th root of unity for some integer
$1\leq\l<n$. In this case $\mathcal{P}_n$ can be identified with
the set of Kleshchev bipartitions of $n$ with respect to
$(\sqrt[2\l]{1};1,-1)$ (see \cite{A}).
Let $v$ be an indeterminate over $\mathbb{Q}$. Let $\mathfrak{h}$
be a $(2\l+1)$-dimensional vector space over $\mathbb{Q}$ with
basis $\{h_0,h_1,\cdots,$ $h_{2\l-1},d\}$. Denote by
$\{\Lambda_0,\Lambda_1,\cdots,\Lambda_{2\l-1},\delta\}$ the dual
basis of $\mathfrak{h}^*$, and we set
$\alpha_i=2\Lambda_i-\Lambda_{i-1}-\Lambda_{i+1}+\delta_{i,0}\delta$
for $i\in\Z/{2\l\Z}$. Assume that the $2\l\times 2\l$ matrix
$(\langle\alpha_i,h_j\rangle)$ is the generalized Cartan matrix
associated to $\ksl_{2\l}$. Let $U_v(\ksl_{2\l})$ be the quantum
affine algebra corresponding to $\ksl_{2\l}$ (see \cite[\S2]{Hu2}
for its definition). Let $\Lambda=\Lambda_0+\Lambda_{\l}$. Let
$\mathcal{F}(\Lambda):=\oplus_{n\geq
0}\oplus_{\lam\in\widetilde{\mathcal{P}}_n}\mathbb{Q}(v)\lam$,
{\it a level $2$ Fock space}, on which $U_v(\ksl_{2\l})$ acts. The
submodule $L(\Lambda)$ generated by the empty bipartition $\underline{\emptyset}$ is the
irreducible integrable highest weight module with highest weight
$\Lambda$. By a well-known result of S. Ariki (\cite{A2}), the dual of the
Grothendieck group $K(\oplus_{k\geq 0}\H(B_k))$ can be made into a
$U_v(\ksl_{2\l})$-module. Ariki introduced the functors of $i$-restriction
and $i$-induction, which plays the role of Chevalley generators $e_i, f_i$.
It is a remarkable fact (see \cite{MM},
\cite[(2.11)]{AM}) that the crystal graph of $L(\Lambda)$ can be realized as
the Kleshchev's good lattice with respect to $(\sqrt[2\l]{1};1,-1)$
if one uses the embedding $L(\Lambda)\subset\mathcal{F}(\Lambda)$.
The operators of removing and adding $2\l$-good nodes play the role of the Kashiwara operators
$\tilde{e}_i,\tilde{f}_i$. On the other hand, Grojnowski (see \cite[Theorem 14.2, 14.3]{G})
gives another realization of the crystal graph of $L(\Lambda)$ on the set of all the simple modules of
$\H(B_k)$ for all $k\geq 0$, where the functors of taking socle (resp., taking cosocle) of the $i$-restriction
(resp. of the $i$-induction) of simple modules play the role of the Kashiwara operators
$\tilde{e}_i,\tilde{f}_i$. These two crystal structures are isomorphic to each other.
By the definition of the second realization, there are the following results (see \cite{Ma}).
\begin{lem}\text{(\cite{G},\cite{GV})} There is an isomorphism $\pi$ between the above two realizations of the crystal structure, such that, if we write $\pi=\oplus_{k\geq 0}\pi_k$, where
$\pi_k$ is a permutation defined on $\mathcal{P}_k$, then for each
bipartition $\lam\in\mathcal{P}_n$, $$
\soc\Bigl(\td^{\pi_n(\lam)}\!\!\downarrow_{\H(B_{n-1})}\Bigr)\cong\bigoplus_{\mu\rightarrow\lam}
\td^{\pi_{n-1}(\mu)}, $$ where $\mu\rightarrow\lam$ means
that $\mu$ is a bipartition of $n-1$ such that the Young diagram
$[\mu]$ is obtained from the Young diagram $[\lam]$ by removing a
$2\l$-good node. In particular, for any bipartition
$\lam\in\mathcal{P}_n$, the socle of
$\td^{\lam}\!\!\downarrow_{\H(B_{n-1})}$ is a direct sum of pairwise
non-isomorphic irreducible $\H(B_{n-1})$-modules, i.e., it is
multiplicity free.
\end{lem}
We remark that both Ariki's and Grojnowski's results are stated in the context of general cyclotomic Hecke algebras of type $G(r,1,n)$, though we only use the special type $B$ case in this paper. It has been conjectured for a long time that the above $\pi$ can be chosen as identity (compare \cite{Kl}, \cite{B}). In a recent preprint \cite{A3}, Ariki proved this conjecture. As a consequence, we have the following.
\begin{lem} [\rm \cite{A3}] With the above
notations, $\pi_n$ can be chosen as the identity map for any $n$.
\end{lem}
To describe the decomposition into irreducible modules of the
socle of the module $D\!\!\downarrow_{\H(D_{n-1})}$ for each
irreducible $\H(D_n)$-module $D$ in this case, we need a better
understanding of the set of fixed-points under the involution
$\bH$.
\begin{lem} Let $\lam\in\mathcal{P}_n$ be a Kleshchev
bipartition with respect to $(\sqrt[2\l]{1};1,-1)$. Suppose that
$\lam=\bH(\lam)$. Then $n$ is even. Moreover, if
$\emptyset\stackrel{i_1}{\longrightarrow}\cdot\overset{i_2}{\longrightarrow}\cdot
\cdots\stackrel{i_n}{\longrightarrow}\lam $ is a path in
Kleshchev's good lattice with respect to $(\sqrt[2\l]{1};1,-1)$, then for any
integer $k$, we have $$ \#\bigl\{j\bigm| \text{$i_j\equiv
k\!\!\!\mod{2\l}$}\bigr\}=\#\bigl\{j\bigm|\text{$i_j\equiv{k+\l}\!\!\!\mod{2\l}$}\bigr\}.
$$
\end{lem}
\noindent {Proof:} \, Let $$
\emptyset\stackrel{i_1}{\longrightarrow}\cdot\stackrel{i_2}{\longrightarrow}\cdot
\cdots\stackrel{i_n}{\longrightarrow}\lam $$ be a path in
Kleshchev's good lattice with respect to $(\sqrt[2\l]{1};1,-1)$. By assumption
and \cite[3.3]{Hu2}, $$
\emptyset\stackrel{i_1+\l}{\longrightarrow}\cdot\stackrel{i_2+\l}{\longrightarrow}\cdot
\cdots\stackrel{i_n+\l}{\longrightarrow}\lam
$$
is also a path in Kleshchev's good lattice with respect to
$(\sqrt[2\l]{1};1,-1)$.
It follows that there is an automorphism defined on the set of
nodes of $\lam$, say $\psi$, such that
$\res\gamma=\res\psi(\gamma)+\l$ in $\Z/{2\l\Z}$. We claim that
this is enough for deducing our theorem by using induction on $n$.
In fact, we first pick a node $A_1\in\lam$ and define
$B_1:=\psi(A_1)$. As $\res A_1=\res B_1+\l\neq\res B_1$, it is
clear that $A_1\neq B_1$. Now if $\psi(B_1)=A_1$, then we can
remove $A_1, B_1$ and use induction on $n$; otherwise, we define
$A_2:=\psi(B_1)$, which is different from $A_1$. Since $\psi$ is
bijective, it follows that $B_2:=\psi(A_2)$ is different from
$B_1=\psi(A_1)$. Repeating this procedure, and noting that $\lam$
has only finitely many nodes, one would get $2k$ pairwise
different nodes of $\lam$, $A_1, A_2,\cdots, A_k$ and
$B_1,B_2,\cdots, B_k$ such that
$B_i=\psi(A_i),\,\,\forall\,\,1\leq i\leq k$ and
$A_{j}=\psi(B_{j-1}),\,\,\forall\,2\leq j\leq k$ and
$\psi(B_{k})=A_{1}$. Then we can remove these $2k$ nodes and use
induction hypothesis on $n$. This completes the proof of our
theorem. \hfill\qed
\addtocounter{defs}{3}
\begin{defs} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam$ has a $2\l$-good node $A$ such that $$
\lam\setminus{A}=\bH(\lam\setminus{A}).
$$
Then we say that $\lam$ is almost $\l$-symmetric.
\end{defs}
\addtocounter{example}{4}
\begin{example} Suppose that $n=5$ and $\l=2$. Let
$\lam=((1), (2^2)),\, \mu=((2), (1^3))$ are two Kleshchev
bipartitions of $5$ with respect to $(\sqrt[4]{1};1,-1)$. Let $A$ be the
node which is in the second row and the second column of the
second component of $\lam$. Then $\lam$ is almost $2$-symmetric
with $\lam\setminus{A}=\bH(\lam\setminus{A})$, but $\mu$ is not
almost $2$-symmetric.
\end{example}
\addtocounter{lem}{2}
\begin{lem} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam$ is almost $\l$-symmetric with
$\,\lam\setminus{A}=\bH(\lam\setminus{A})$ for some $2\l$-good node $A$
of $\,\lam$. Then for any pairs of $2\l$-good nodes $B, C$ of $\,\lam$
satisfying $C\neq A$, we have that $$
\lam\setminus{B}\neq\bH(\lam\setminus{C}).
$$
In particular, for any $2\l$-good nodes $C$ of $\,\lam$ satisfying
$C\neq A$, we have that $$
\lam\setminus{C}\neq\bH(\lam\setminus{C}). $$
\end{lem}
\noindent {Proof:} \, Suppose that
$\lam\setminus{B}=\bH(\lam\setminus{C})$. Since $\bH$ is an
involution, $\,\lam\setminus{A}=\bH(\lam\setminus{A})$ and $C\neq
A$, it follows that $B\neq A$. Note that for $2\l$-good nodes $A, C$
(resp. $A, B$), $A\neq C$ (resp. $A\neq B$) implies that their
residues $\res A, \res C$ (resp. $\res A, \res B$) are different.
Write $\res A=i$, $\res B=j$, $\res C=k$. Then $j\neq i\neq k$.
For each bipartition $\mu$ and each $s\in\Z/{2\l\Z}$, we denote by
$N_s^{\mu}$ the number of $s$-nodes in the Young diagram $[\mu]$
of $\mu$. If $j\neq k+\l$, then by Lemma 1.4(2),
$$N_{k+\l}^{\lam\setminus{B}}=N_{k+\l}^{\lam}\geq
N_{k+\l}^{\lam\setminus{A}}=N_{k+\l}^{\bH(\lam\setminus{A})}=N_{k}^{\lam\setminus{A}}=N_{k}^{\lam\setminus{C}}+1=
N_{k+\l}^{\bH(\lam\setminus{C})}+1, $$ a contradiction. If $j=
k+\l$, then by Lemma 1.4(2),$$
N_{i}^{\lam\setminus{B}}=N_{i}^{\lam}=
N_{i}^{\lam\setminus{A}}+1=N_{i}^{\bH(\lam\setminus{A})}+1=N_{i+\l}^{\lam\setminus{A}}+1=N_{i+\l}^{\lam\setminus{C}}+1=
N_{i}^{\bH(\lam\setminus{C})}+1,
$$
a contradiction. This proves our lemma. \hfill\qed
\addtocounter{thm}{6}
\begin{thm} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam$ is almost $\l$-symmetric with
$\lam\setminus{A}=\bH(\lam\setminus{A})$ for some $2\l$-good node $A$ of
$\lam$. Then $\lam\neq\bH(\lam)$ and $$
\soc\Bigl(D^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
D_{+}^{\lam\setminus{A}}\bigoplus
D_{-}^{\lam\setminus{A}}\bigoplus
\bigoplus_{\substack{C\in [\lam],\,\, C\neq A\\ \text{$C$
is $2\l$-good}}}
\td^{\lam\setminus{C}}\!\!\downarrow_{\H(D_{n-1})}.
$$
In particular, $\,\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}$ is
multiplicity free.
\end{thm}
\noindent {Proof:}\, This follows directly from Lemma 1.3,
(2.5.1), Lemma 3.1, Lemma 3.2, Lemma 3.3 and Lemma 3.6. \hfill\qed
\begin{thm} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam\neq\bH(\lam)$ and $\lam$ is not almost $\l$-symmetric. Then
for any two $2\l$-good nodes $B, C$ of $\lam$,
$\lam\setminus{B}\neq\bH(\lam\setminus{C})$.
\end{thm}
\noindent {Proof:} \, In fact, it suffices to show that, if $B\neq
C$, then $\lam\setminus{B}\neq\bH(\lam\setminus{C})$.
Otherwise, suppose that $B\neq C$ and
$\lam\setminus{B}=\bH(\lam\setminus{C})$. We call two nodes
$\gamma, \gamma'$ $\l$-conjugate, if $\res\gamma=\res\gamma'+\l$
in $\Z/{2\l\Z}$. We deduce that $B$ is not $\l$-conjugate to $C$
(otherwise it would follow that $\lam=\bH(\lam)$). Therefore, we
have that $\res B\neq\res C$ (as $B\neq C$ are both $2\l$-good nodes),
and $\res B\neq\res C+\l$. Now the condition that
$\lam\setminus{B}=\bH(\lam\setminus{C})$ implies that there is a
bijection, say $\varphi$, from the set of the nodes of
$\lam\setminus{B}$ onto the set of the nodes of
$\lam\setminus{C}$, such that for any
$\gamma\in{\lam\setminus{B}}$, $\gamma$ is $\l$-conjugate to
$\varphi(\gamma)$.
We define $C_1=C_0:=C$, $D_1:=\varphi(C_1)$. Since $D_1$ is
$\l$-conjugate to $C_1$ and hence different from $B$, we have that
$D_1\in\lam\setminus{B}$. Hence we can define
$C_2:=\varphi(D_1)\in\lam\setminus{C}$, then $\res C_2=\res C_1$
and hence $C_2\neq B$, then we can still define
$D_2:=\varphi(C_2)$. Since $\varphi$ is bijective, $C_1\neq C_2$
implies that $D_1\neq D_2$. In general, suppose that for integer
$k\geq 2$, $D_i\in\lam\setminus{B},\,\, C_{i}\in\lam\setminus{C}$,
for $1\leq i\leq k$, are already well-defined, such that
$C_{j}=\varphi(D_{j-1}),\,\, D_{j}=\varphi(C_{j}),\,\,
\forall\,2\leq j\leq k$ and $$ C_i\neq C_j,\,\, D_i\neq D_j,
\,\,\res C_i=\res C_j=\res C,\,\, \res D_i=\res D_j=\res C+\l, $$
for any $1\leq i\neq j\leq k$, then we define
$C_{k+1}:=\varphi(D_k)\in\lam\setminus{C}$. As $D_1,\cdots,D_k$
are pairwise different and $\varphi$ is bijective, we get that
$C_1,\cdots,C_{k+1}$ are also pairwise different. Moreover, $\res
C_{k+1}=\res C$, and hence $C_{k+1}\neq B$ and we can still define
$D_{k+1}:=\varphi(C_{k+1})\in\lam\setminus{B}$. It is clear that
$\res D_{k+1}=\res C+\l$, and since $C_1,\cdots,C_{k+1}$ are
pairwise different and $\varphi$ is bijective, we get that
$D_1,\cdots,D_{k+1}$ are also pairwise different. As a
consequence, we get infinitely many pairwise different nodes
$C_1,C_2,\cdots$ in $\lam$, which is impossible. This proves the
theorem. \hfill\qed
It follows from this theorem, Lemma 1.3, Lemma 3.1, Lemma 3.2 and (2.5.1) that
\begin{thm} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam\neq\bH(\lam)$ and $\lam$ is not almost $\l$-symmetric. Then
$$
\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
\bigoplus_{\substack{C\in [\lam]\\ \text{$C$ is $2\l$-good}}}
\td^{\lam\setminus{C}}\!\!\downarrow_{\H(D_{n-1})}.
$$
In particular,
$\,\soc\Bigl(\td^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ is
multiplicity free.
\end{thm}
Now let $\lam\in\mathcal{P}_n$ be such that $\lam=\bH(\lam)$. It remains to
describe the decompositions into irreducible $\H(D_{n-1})$-modules
of $\soc\Bigl(D_+^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ and
$\soc\Bigl(D_-^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$.
\begin{thm} Let $\lam\in\mathcal{P}_n$. Suppose that
$\lam=\bH(\lam)$. Then there is a $\H(D_{n-1})$-module isomorphism
$$ \soc\Bigl(D_+^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
\soc\Bigl(D_-^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr).
$$
\end{thm}
\noindent {Proof:}\, By assumption and Lemma 3.3, we know that $n$ is even.
Hence $n-1$ is odd. In particular, for any bipartition
$\mu\in\mathcal{P}_{n-1}$, $\mu\neq\bH(\mu)$. By \cite[Corollary
2.4]{Hu3}, $\td^{\mu}\cong\Bigl(\td^{\mu}\Bigr)^{\tau}$.
Now using the same argument as in the proof of Theorem
2.7, we prove the theorem. \hfill\qed
\addtocounter{cor}{10}
\begin{cor} Let $\lam\in\mathcal{P}_n$ be such that
$\lam=\bH(\lam)$. In particular, $n$ is even. Then $$
\soc\Bigl(D_+^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)\cong
\soc\Bigl(D_-^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)
\cong\bigoplus_{\mu} \td^{\mu}\!\!\downarrow_{\H(D_{n-1})},
$$
where the subscript $\mu$ is taken over a fixed set of
representatives of equivalence classes in
$\mathcal{P}_{n-1}/{\approx}$ such that
$\mu\rightarrow\lam$. In particular,
$\,\soc\Bigl(D_+^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ and
$\,\soc\Bigl(D_-^{\lam}\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ are
both multiplicity free.
\end{cor}
\noindent {Proof:} \, Since $\lam=\bH(\lam)$, by Lemma 3.3, it
follows that for any $2\l$-good node $C$ of $\lam$, $\lam\setminus{C}\neq\bH(\lam\setminus{C})$.
Now using Lemma 3.1, lemma 3.2, (2.5.1) and Theorem 3.10, we prove
the corollary. \hfill\qed
\begin{cor} For any irreducible $\H(D_n)$-module
$D$, the socle of $D\!\!\downarrow_{\H(D_{n-1})}$ is always multiplicity
free.
\end{cor}
Now Theorem 3.7, Theorem 3.9 and Corollary 3.11
completely determine the decomposition of
$\soc\Bigl(D\!\!\downarrow_{\H(D_{n-1})}\Bigr)$ into irreducible
$\H(D_{n-1})$-modules for every irreducible $\H(D_n)$-module $D$.
\section{Appendix}
In this appendix, we give a proof to show that the involution
$\bH$ (and hence the main result of \cite{Hu2}) is independent of
the base field $K$ as long as $\ch K\neq 2$ and $\H(D_n)$ is split
over $K$. The proof is essentially due to Professor S. Ariki.
Throughout this section, we assume that $\ch K\neq 2$ and
$\H(D_n)$ is split over $K$. By Lemma 1.4(1), it suffices to
consider the case where $q$ is a primitive $2\l$-th root of unity
in $K$ for some integer $1\leq\l<n$. To emphasize the base field
$K$, we denote by $\H_K(D_n)$ the Hecke algebra of type $D_n$ over
$K$, and by $\ts_K^{\lam}$ (resp. $\td_K^{\lam}$) the
corresponding $\H_K(D_n)$-modules. Note that by \cite{A}, the set
$\bigl\{\lam\in\widetilde{\mathcal{P}}_n\bigm|\td_K^{\lam}\neq
0\bigr\}$ depends only on $\l$, but not on the choice of the base
field $K$. So we denote it by $\mathcal{P}_n$ as before. There is
an involution $\bH_K$ defined on the set $\mathcal{P}_n$ such
that, for each $\lam\in\mathcal{P}_n$,
$\Bigl(\td_K^{\lam}\Bigr)^{\sigma}\cong\td_K^{\bH_K(\lam)}$. We
have that
\begin{thm} Suppose that $\ch K=p\neq 2$ and
$\H_K(D_n)$ is split over $K$. Then $\bH_{K}=\bH_{\mathbb{C}}$.
\end{thm}
\noindent {Proof:} \, By \cite{DJMu}, we know that each
irreducible $\H(B_n)$-module remains irreducible under field
extension. By definition of the automorphism $\sigma$ and the
involution $\bH$, it is easy to see that if
$\bH_{F}=\bH_{\mathbb{C}}$ for some splitting field $F$ of
$\H(D_n)$ with $\ch F=p\neq 2$, then for any splitting field $K'$
of $\H(D_n)$ with $\ch K'=p\neq 2$, we have that
$\bH_{K'}=\bH_{\mathbb{C}}$. In particular, for any characteristic
$0$ splitting field $E$ of $\H(D_n)$, we have that
$\bH_{E}=\bH_{\mathbb{C}}$.
Therefore, it suffices to consider the characteristic $p>2$ case.
To ensure the existence of a primitive $2\l$-th root of unity, we
further assume that $(p,\l)=1$. Let $q\in\overline{\mathbb{F}_p}$
(resp. $q_0\in\mathbb{C}$) be a primitive $2\l$-th root of unity,
where $\overline{\mathbb{F}_p}$ is the algebraic closure of the
finite field $\mathbb{F}_p$. Let $X$ be an indeterminate over
$\mathbb Z$. For each polynomial $f\in\mathbb{Z}[X]$, let
$\overline{f}$ be its canonical image in $\mathbb{F}_p[X]$. For
each $m\in\mathbb N$, let $\Phi_m(X)$ be the $m$-th cyclotomic
polynomial over $\mathbb Z$. Then $X^m-1=\prod_{d\mid
m}\Phi_d(X)$. It follows that $\overline{\Phi_{2\l}}(q)=0$. Hence
the map which sends $q_0$ to $q$ extends naturally to a surjective
ring homomorphism from $\mathbb{Z}[q_0]$ onto $\mathbb{F}_p[q]$.
Let $\Phi_{2\l,p}(X)$ be a monic polynomial in $\mathbb{Z}[X]$
such that $\overline{\Phi_{2\l,p}}(X)$ is the minimal polynomial
of $q$ over $\mathbb{F}_p$.
Recall that every finite dimensional algebra becomes split after a
finite field extension. Therefore, there exist some algebraic
integers $\alpha_1,\cdots,\alpha_s\in\mathbb{C}$, some elements
$\overline{\alpha_1},\cdots,\overline{\alpha_s}\in
\overline{\mathbb{F}_p}$, and some monic polynomials, say
$f_i(X_i)\in\mathbb{Z}[q_0,\alpha_1,\cdots,$ $\alpha_{i-1}][X_i]$,
$\overline{f_i}(X_i)\in\mathbb{F}_p[q,\overline{\alpha_1},\cdots,
\overline{\alpha_{i-1}}][X_i]$, where $X_i$ is an indeterminate
over $\mathbb{Z}[q_0,\alpha_1,\cdots,\alpha_{i-1}]$ (resp. over
$\mathbb{F}_p[q,\overline{\alpha_1},\cdots,
\overline{\alpha_{i-1}}]$), $i=1,\cdots,s$, such that
\smallskip
1) $\overline{f_i}$ (respectively, $f_i$) is irreducible over
$\mathbb{F}_p[q,\overline{\alpha_1},
\cdots,\overline{\alpha_{i-1}}]$ (respectively, over
$\mathbb{Z}[q_0,\alpha_1,\cdots,$ $\alpha_{i-1}]$),
2) $f_i(\alpha_i)=0$, $\overline{f_i}(\overline{\alpha_i})=0$, and
$\H_{{\mathbb{F}_p}[q,\overline{\alpha_1},\cdots,\overline{\alpha_s}]}(D_n)$
is split over the field
${\mathbb{F}_p}[q,\overline{\alpha_1},\cdots,\overline{\alpha_s}]$.
\smallskip
Note that (see \cite[Chapter IV, \S1, Theorem 4]{L})
$\mathbb{Z}[q_0]$ is the integral closure of $\mathbb Z$ in
$\mathbb{Q}[q_0]$. Let $R'$ be the integral closure of $\mathbb
Z[q_0]$ in $\mathbb{Q}[q_0,\alpha_1,\cdots,\alpha_{s}]$. By
\cite[Chapter 5, Exercise 2]{AtM}, the natural surjective morphism
from $\mathbb Z[q_0,\alpha_1,\cdots,\alpha_{s}]$ onto
${\mathbb{F}_p}[q,\overline{\alpha_1},\cdots,\overline{\alpha_s}]\subseteq
\overline{\mathbb{F}_p}$ can be extended to a ring homomorphism
from $R'$ to the field $\overline{\mathbb{F}_p}$. We denote it by
$\pi$. Then (by \cite[Chapter I, \S4]{S}) $R'$ is a Dedekind
domain and the field of fractions of $R'$ is
$\mathbb{Q}[q_0,\alpha_1,\cdots,\alpha_{s}]$.
Similarly, let $E$ be a finite extension of
$\mathbb{Q}[q_0,\alpha_1,\cdots,\alpha_{s}]$ such that $\H_E(D_n)$
is split over the field $E$. Let $R$ be the integral closure of
$R'$ in $E$. By \cite[Chapter I, \S4]{S}, $R$ is a Dedekind domain
and the field of fractions of $R$ is $E$. By \cite[Chapter 5,
Exercise 2]{AtM}, the homomorphism $\pi$ can be extended to a ring
homomorphism from $R$ to the field $\overline{\mathbb{F}_p}$. It
follows that the ideal of $R$ generated by $p$ and
$\Phi_{2\l,p}(q_0)$ should be a proper ideal. Let $\mathfrak{m}$
be the kernel of the homomorphism, which is a maximal ideal of $R$
containing $p$ and $\Phi_{2\l,p}(q_0)$. Let
$\mathcal{O}:=R_{\mathfrak{m}}$,
$F:=R_{\mathfrak{m}}/\mathfrak{m}R_{\mathfrak{m}}$. It is clear
that
${\mathbb{F}_p}[q,\overline{\alpha_1},\cdots,\overline{\alpha_s}]\subseteq
F$. Therefore we get a $p$-modular system $(\mathcal{O}, E, F)$,
where $E$ (resp. $F$) is a field of characteristic $0$ (resp.
characteristic $p$) such that $\H_E(D_n)$ (resp. $\H_F(D_n)$) is
split over $E$ (resp. over $F$), and $q_0\in\mathcal{O}\subset E$
is a primitive $2\l$-th root of unity in $E$ which is in the
pre-image of $q$. By results of \cite{GR} and \cite[(2.3)]{Ge},
the decomposition map from the Grothendieck group $K_0(\H_E(D_n))$
to the Grothendieck group $K_0(\H_F(D_n))$ is well-defined.
Let $\lam\in\mathcal{P}_n$. Recall that there is a well-defined
bilinear form $\langle,\rangle_{\mathcal{O}}$ over
$\ts_{\mathcal{O}}^{\lam}$. Let
$M_{\mathcal{O}}:=\rad\langle,\rangle_{\mathcal{O}}$,
$M:=\rad\langle,\rangle_{E}$. Then $M\cong
M_{\mathcal{O}}\otimes_{\mathcal{O}}E$ is the unique maximal
$\H_E(B_n)$-submodule of $\ts_E^{\lam}$, and $\td_E^{\lam}\cong
\ts_E^{\lam}/M$. Let $0\neq a\in\mathcal{O}$, $z\in M_{\mathcal{O}}$.
If $ax=z$ for some $x\in\ts_{\mathcal{O}}^{\lam}$, then $a\langle x,y\rangle=\langle ax,y\rangle=0$
for any $y\in\ts_{\mathcal{O}}^{\lam}$. Since $\mathcal{O}$ is an
integral domain, it follows that $\langle x,y\rangle=0$ for any
$y\in\ts_{\mathcal{O}}^{\lam}$, hence $x\in M_{\mathcal{O}}$. This
shows that $M_{\mathcal{O}}$ is a
pure $\mathcal{O}$-submodule of $\ts_{\mathcal{O}}^{\lam}$. Since
$\mathcal{O}$ is a principal integral domain,
it follows\footnote{Recall that over a principal integral domain,
any pure submodule of a finitely generated module must be its direct
summand.} that $M_{\mathcal{O}}$ is a
direct $\mathcal{O}$-summand of $\ts_{\mathcal{O}}^{\lam}$. In
particular,
$\td_{\mathcal{O}}^{\lam}:=\ts_{\mathcal{O}}^{\lam}/M_{\mathcal{O}}$
and hence $\Bigl(\td_{\mathcal{O}}^{\lam}\Bigr)^{\sigma}$
are both free $\mathcal{O}$-modules.
Note that
$$ \td_E^{\lam}\cong\ts_E^{\lam}/M\cong (\ts_{\mathcal{O}}^{\lam}\otimes_{\mathcal{O}}
E)/(M_{\mathcal{O}}\otimes_{\mathcal{O}}
E)\cong\td_{\mathcal{O}}^{\lam}\otimes_{\mathcal{O}} E,
$$
and hence
$$\td_E^{\bH_{\mathbb{C}}(\lam)}\cong\Bigl(\td_E^{\lam}\Bigr)^{\sigma}\cong\Bigl(\td_{\mathcal{O}}^{\lam}\otimes_{\mathcal{O}}
E\Bigr)^{\sigma} \cong
\Bigl(\td_{\mathcal{O}}^{\lam}\Bigr)^{\sigma}\otimes_{\mathcal{O}}
E,\eqno{(4.2)}$$ which implies that
$\Bigl(\td_{\mathcal{O}}^{\lam}\Bigr)^{\sigma}$ is a full
$\H_{\mathcal{O}}(B_n)$-lattice in
$\td_{E}^{\bH_{\mathbb{C}}(\lam)}$.
By the fact that
$\td_E^{\bH_{\mathbb{C}}(\lam)}\cong\td_{\mathcal{O}}^{\bH_{\mathbb{C}}(\lam)}\otimes_{\mathcal{O}}
E$, we know that $\td_{\mathcal{O}}^{\bH_{\mathbb{C}}(\lam)}$ is
also a full $\H_{\mathcal{O}}(B_n)$-lattice in
$\td_{E}^{\bH_{\mathbb{C}}(\lam)}$. Therefore, the module
$\Bigl(\td_{\mathcal{O}}^{\lam}\Bigr)^{\sigma}\otimes_{\mathcal{O}}
F$ has the same set of composition factors as that of
$\td_{\mathcal{O}}^{\bH_{\mathbb{C}}(\lam)}\otimes_{\mathcal{O}}
F$.
It is clear that the natural homomorphism from
$\ts_F^{\lam}\cong\ts_{\mathcal{O}}^{\lam}\otimes_{\mathcal{O}} F$
to $\td_{\mathcal{O}}^{\lam}\otimes_{\mathcal{O}} F$ is
surjective. Since $\td_F^{\lam}$ is the unique simple head of
$\ts_F^{\lam}$, it follows that $\td_F^{\lam}$ is also the unique simple head of
$\td_{\mathcal{O}}^{\lam}\otimes_{\mathcal{O}} F$. Hence
$\td_F^{\bH_F(\lam)}\cong\Bigl(\td_F^{\lam}\Bigr)^{\sigma}$ is
also the unique simple head of $
\Bigl(\td_{\mathcal{O}}^{\lam}\Bigr)^{\sigma}\otimes_{\mathcal{O}}
F$. Therefore, $\td_F^{\bH_F(\lam)}$ must also be a composition
factor of
$\td_{\mathcal{O}}^{\bH_{\mathbb{C}}(\lam)}\otimes_{\mathcal{O}}
F$, and hence be a composition factor of
$\ts_F^{\bH_{\mathbb{C}}(\lam)}\cong\ts_{\mathcal{O}}^{\bH_{\mathbb{C}}(\lam)}\otimes_{\mathcal{O}}
F$. Hence $\bH_F(\lam)\trianglelefteq\bH_{\mathbb{C}}(\lam)$.
Using induction on the dominance order $\,\trianglelefteq\,$, it
is easy to see that $\bH_F(\lam)=\bH_{\mathbb{C}}(\lam)$ for any
$\lam\in\mathcal{P}_n$, as required. This completes the proof of
the theorem. \hfill\qed
\bigskip\bigskip\bigskip | 38,576 |
TITLE: How does Implicit Function Theorem guarantee curves through a surface?
QUESTION [1 upvotes]: So when proving that the gradient vector of some function $g(x,y,z)$ is always orthogonal to a level set of $g$, my book uses differentiable curves which lie on this level set. As far as I understand, these curves are guaranteed to exist by the Implicit Function Theorem, but I'm having trouble understanding why.
For a level set in $\textbf{R}^2$ I understand, because the implicit function theorem says we can solve for $y$ as a function of $x$ or the other way around. So we can get a function $y=f(x)$ and then we can just define the curve $\varphi(t) = \langle t, f(t) \rangle$. But how does the same conclusion follow for a level set in $\textbf{R}^3$?
REPLY [0 votes]: The implicit function theorem shows that there is a local map $z : \Bbb R^2 \to \Bbb R$ such that $f(x, y, z(x, y))$ is constant. Any curve in the domain of $z$ in $\Bbb R^2$ is carried by the map $(x, y) \mapsto (x, y, z(x, y))$ to a curve in the level surface. | 194,733 |
TITLE: Interpretation of a combinatorial expression
QUESTION [0 upvotes]: I came across this expression :
$ \frac {(2n)!} {2^n n!} $
I know that this evaluates to an odd integer. More specifically :
$ \frac {(2n)!} {2^n n!} = (2n -1) \cdot (2n - 3)\cdots 5\cdot3\cdot1$
I want to know where this expression comes in a real-life scenario. For example $\sum_{k=0}^{n}{n\choose k}$ represents the number of ways to choose a subset from a set of cardinality $n$.
REPLY [3 votes]: $(2n-1)!!=(2n-1)\cdot(2n-3)\cdot(2n-5)\cdots 5\cdot 3\cdot 1$ is an answer to various combinatorial questions such as the following:
In how many ways can $2n$ distinct people split up into $n$ unlabelled teams each of size two such that everyone is on exactly one team each.
To see the answer, place an arbitrary order on the $2n$ people. Without loss of generality, let the order we use be age.
Take whoever is youngest in the group. Pick who will be partnered with the youngest person. Remove both people from the pool of available people.
Take whoever is youngest from those remaining. Pick who will be partnered with that person. Remove both people from the pool of available people.
Repeat the previous step until everyone is paired off. | 152,447 |
\begin{document}
\title{The Homotopy Theory of Simplicially Enriched Multicategories}
\author[M. Robertson]{Marcy Robertson}
\address{Departament of Mathematics, University of Western Ontario, Canada}
\email{[email protected]}
\keywords{Colored operad; multicategory}
\begin{abstract}In this paper we construct a cofibrantly generated model category structure on the category of all small symmetric multicategories enriched in simplicial sets.\end{abstract}
\maketitle
Operads are combinatorial objects that encode a variety of algebraic structures in a particular symmetric monoidal category of interest. Many usual categories of algebras (i.e. categories of commutative and associative algebras, associative algebras, Lie algebras, Poisson algebras, etc.) can be considered categories of operad representations. At the same time, operads are ``algebras'' themselves, or rather monoids in the monoidal category of (symmetric) sequences. A multicategory, or colored operad, is simply an operad with ``many objects,'' analogous to the way a category is a monoid with ``many objects.'' The precise definition and some important examples of operads, multicategories and their algebras will be reviewed in Section $1$ of this paper.\\
The purpose of this paper is the construction of a Quillen model category structure on the category of small multicategories enriched in simplicial sets. Our model structure is a blending of the Bergner model structure on the category of small simplicial categories~\cite{Bergner}, and the Berger-Moerdijk model structure on $S$-colored operads~\cite{BM07}.\\
\subparagraph{Acknowledgments:}The model category structure presented here for simplicially enriched symmetric multicategories was independently obtained by the author as part of her thesis~\cite{Me2} work and Ieke Moerdijk~\cite{MoerdijkUnpubllished} as part of a larger project on $(\infty, 1)$-operads~\cite{MC1,MC2,MC3}. The author is greatly indebted to her thesis advisor, Brooke Shipley, and to Ieke Moerdijk for the many helpful discussions, and to the later for showing her his unpublished manuscript of which she has made liberal use in the preparation of this paper.
\section[Multicategories]{Multicategories}The basic idea of a multicategory is very like the idea of a category, it has objects and morphisms, but in a multicategory the source of a morphism can be an arbitrary sequence of objects rather than just a single object.\\
A \emph{multicategory}, $\mathcal{P},$ consists of the following data:
\begin{itemize}
\item a set of objects $\obj(\cP)$;
\item for each $n\ge0$ and each sequence of objects $x_{1},...,x_{n},x$ a \emph{set} $\mathcal{P}(x_{1},...,x_{n};x)$ of $n$-ary operations which take $n$ inputs (the sequence $x_{1},...,x_{n}$) to a single output (the object $x$).\end{itemize}
These operations are equipped with structure maps for units and composition. Specifically, if $I=\{*\}$ denotes the one-point set, then for each object $x$ there exists a unit map $\eta_{x}:I\rightarrow\cP(x;x)$ taking $*$ to $1_{x}$, where $1$ denotes the unit of the symmetric monoidal structure on the category $\Sets$. The composition operations are given by maps$$\mathcal{P}(x_{1},...,x_{n};x)\times\mathcal{P}(y_{1}^{1},...,y_{k_{1}}^{1};x_{1})\times\cdots\times\mathcal{P}(y_{1}^{n},...,y_{k_{n}}^{n};x_{n})\longrightarrow\mathcal{P}(y_{1}^{1},...,y_{k_{n}}^{n};x)$$which we denote by $$p,q_{1},...,q_{n}\mapsto p(q_{1},...,q_{n}).$$The structure maps satisfy the associativity and unitary coherence conditions of monoids. A \emph{symmetric multicategory} is a multicategory with the additional property that the $n$-ary operations are equivariant under the permutation of the inputs. Explicitly, for $\sigma\in\Sigma_{n}$ and each sequence of objects $x_{1},...,x_{n},x$ we have a right action of $\Sigma_{n}$, i.e., a morphism$$\sigma^{*}:\mathcal{P}(x_{1},\cdots,x_{n};x)\longrightarrow\mathcal{P}(x_{\sigma(1)},...,x_{\sigma(n)};x).$$The action maps are well behaved, in the sense that all composition operations are invariant under the $\Sigma_n$-actions, and $(\sigma\tau)^{*}=\tau^{*}\sigma^{*}$.\\
In practice one often uses the following, equivalent, definition of the composition operations, given by:$$\xymatrix{\mathcal{P}(c_{1},\cdots,c_{n};c)\times\mathcal{P}(d_{1},\cdots,d_{k};c_{i})\ar[r]^{{\circ_{i}\,\,\,\,\,\,\,\,\,\,\,}} & \mathcal{P}(c_{1},\cdots,c_{i-1},d_{1},\cdots,d_{k},c_{i+1},\cdots,c_{n};c).}$$ \\
All of our definitions will still make sense if we replace $\Sets$ by any co-complete symmetric monoidal category $(\C,\otimes,\unit)$. Multicategories whose operations take values in $\C$ are called \emph{multicategories enriched in $\C$} or \emph{$\C$-multicategories}. In particular, the strong monoidal functor $\Sets\longrightarrow\mathcal{C}$ that sends a set $S$ to the $S$-fold coproduct of copies of the unit of $\C$ takes every multicategory to a $\mathcal{C}$-enriched multicategory.\\
\begin{example}[Enriched Categories]Let $S$ be a set and let $(\C,\otimes,\unit)$ be a symmetric monoidal category. There exists a (non-symmetric) $S$-colored operad $\cCat_{S}$ whose algebras are the $\C$-enriched categories with $S$ as set of objects and where the maps between algebras (i.e. functors between the $\C$-enriched categories with object set $S$) are the functors which act by the identity on objects. One puts$$\cCat_{S}((x_1,x_1'),\dots,(x_n,x_n');(x_0',x_{n+1}))=\unit$$whenever $x_i'=x_{i+1}$ for $i=0,\dots,n$, and zero in all other cases. In particular, for $n=0$ we have $\cCat_S(;(x,x))=\unit$ for each $x \in S$, providing the $\cCat_S$-algebras with the necessary identity arrows.\end{example}
\begin{example}[Operad Homorphisms]Let $\cP$ be an arbitrary operad. There exists a colored operad $\cP^1$ on a set $\{0,1\}$ of two colors, whose algebras are triples $(A_0,A_1,f)$ where $A_0$ and $A_1$ are $\cP$-algebras, and $f:A_0 \rightarrow A_1$ is a map of $P$-algebras. Explicitly,$$\cP^1(i_1,\dots,i_n;i)=\begin{cases}\cP(n)&\text{if }\max(i_1,\dots,i_n)\leq i;\\0&\text{otherwise}.\end{cases}$$The structure maps of $\cP^1$ are induced by those of $\cP$ (for $n=0$, we agree that $\max(i_1,\dots,i_n)=-1)$. Given a $\cP^1$-algebra on two objects $A_0$ and $A_1$, the objects $\cP(0,\dots,0;0)$ and $\cP(1,\dots,1;1)$ give $A_0$, respectively $A_1$, their $\cP$-algebra structure; furthermore, $\unit \rightarrow \cP(1)$ corresponds to a map $\alpha:\unit \rightarrow \cP^1(0;1)$ giving a map of $\cP$-algebras $f:A_0\rightarrow A_1$. This colored operad has been discussed extensively in the context of chain complexes by Markl \cite{Markl} and plays a key role in describing the derived mapping spaces between multicategories in our paper~\cite{Me2}. \end{example}
A morphism between two $\C$-enriched, symmetric multicategories $F:\cP\longrightarrow\cQ$, or \emph{multifunctor}, consists of a \emph{set map} of objects $F_{0}:\obj(\cP)\longrightarrow\obj(\cQ)$ together with a family of $\Sigma_n$-equivariant $\C$-morphisms$$\{F:\cP(d_{1},...,d_{n};d)\longrightarrow\cQ(F(d_{1}),...,F(d_{n});F(d))\}_{d_{1},...,d_{n},d\in\cP}$$which are compatible with the composition structure maps. We denote the category of all small symmetric multicategories enriched in $\C$ by $\multi(\C)$. When $\C=\Sets$ we will write $\multi$ rather than $\multi(\Sets)$.\\
\subsection{Enrichment of $\multi$ Over $\cCat$}Multicategories are often called colored operads, or just operads(cf.~\cite{BM06, BV, May, CGMV}, etc.), but we use the term multicategory in this paper because we want to emphasize the relationship between multicategory theory with classical category theory. Informally, we can say that inside every multicategory lies a category which makes up the linear part (i.e. the $1$-operations) of that multicategory. We make this explicit by assigning to each multicategory $\cP$ a category $[\cP]_{1}$ with the same object set as $\cP$ and with morphisms given by $[\cP]_{1}(p,p')=\cP(p;p')$ for any two objects $p,p'$ in $\cP$ (i.e. just look at the operations of $\cP$ which have only one input). The functor $[-]_{1}$ takes all higher operations, i.e. $\cP(p_1,...,p_n;p)$, to be trivial. Composition and identity operations are induced by $\cP$.\\
This relationship with category theory is useful in making sense of ideas which do not have obvious meaning in the multicategory setting. As an example, we will want to identify the ``components'' of a multicategory $\cP$ and we will need a way to say that an $n$-ary operation $\phi$ is ``an isomorphism in $\cP$.'' This is where the relationship between categories and multicategories can be useful, we can say that $\phi$ is an \emph{isomorphism} in $\cP$ if $[\phi]_{1}$ is an isomorphism in the category $[\cP]_{1}$.\\
\begin{definition}\label{fullyfaithful}Let $\cP$ and $\Q$ be two multicategories. A multifunctor $F:\cP\rightarrow\Q$ is \emph{essentially surjective} if $[F]_{1}$ is essentially surjective as a functor of categories. We say that $F$ is \emph{full} if for any sequence $p_1,...,p_n,p$ the function $F:\cP(p_1,...,p_n;p)\rightarrow\Q(Fp_{1},...,Fp_{n};Fp)$ is surjective. We say that $F$ is \emph{faithful} if for any sequence $p_1,...,p_n,p$ the function $F:\cP(p_1,...,p_n;p)\rightarrow\Q(Fp_{1},...,Fp_{n};Fp)$ is injective. The multifunctor $F$ is called \emph{fully faithful} if it is both full and faithful.\end{definition}
\begin{definition}Let $F:\cP\rightarrow\Q$ be a functor between two symmetric multicategories. We say that $F$ is an \emph{equivalence of multicategories} if, and only if, $F$ is both fully faithful and essentially surjective.\end{definition}
\begin{definition}Given a multicategory $\mathcal{P}$ with $\obj(\cP):=S$ and a set map $F_0:T\longrightarrow S$, we construct a \emph{pullback} multicategory $F^{*}(\mathbb{\mathcal{P}})$ with object set $T$ whose operations are given by\begin{equation*}\label{pullback multicategory}F^{*}(\mathcal{P})(d_{1},\cdots,d_{n};d):=\mathcal{P}(Fd_{1},...,Fd_{n};Fd).\end{equation*}\end{definition}
The functor $[-]_{1}$ admits a left adjoint, denoted by $\Xi(-)$, which takes a category $\C$ to a multicategory $\Xi(\C)$ with $\obj(\Xi\C):=\obj(\C)$. The linear operations are just the composition maps of $\C$, i.e. $\Xi\C(c; c'):=\C(c, c'),$ and the higher operations are all trivial, i.e. $\Xi\C(c_1,...,c_n; c) =0.$ Composition and units are induced from $\C$ in the obvious way, and it is an easy exercise to check the necessary axioms. We apply the \emph{symmetrization functor}, $\underline{S},$ (see next paragraph) to make $\Xi\C$ into a symmetric multicategory, but since all non-trivial operations in $\Xi\C$ are unary (they have only one input), the symmetric groups $\Sigma_n$ can have only trivial actions. So really $\Xi\C$ is both a symmetric and a non-symmetric multicategory.\\
The \emph{symmetrization functor} is the left adjoint to the functor which forgets symmetric group actions, and is given as follows:\begin{equation*}\underline{S}(\Xi\C)(x_1,\ldots,x_n;x) =\coprod_{\sigma\in\Sigma_n}\Xi\C(x_{\sigma^{-1}(1)},\ldots, x_{\sigma^{-1}(n)};x),\end{equation*} so a $n$-operation in $\underline{S}(\Xi\C)(x_1,\ldots,x_n;x)$ consists of an ordered pair $(\phi,\sigma)$ where $\sigma\in\Sigma_n$ and$$\phi\in\cP(x_{\sigma^{-1}(1)},\ldots, x_{\sigma^{-1}(n)};x).$$ We always let $\Sigma_n$ act on the right via the natural group action on the symmetric group coordinate and define composition via the equivariance requirements for a multicategory. We leave it as an exercise to check that $\underline{S}(\Xi\C)$ satisfies the requirements for a multicatgory and that $\underline{S}$ is a left adjoint to the functor which forgets symmetric group actions.\\
\begin{remark}Given categories $\C$ and $\D$, that $F:\C\longrightarrow\D$ is an equivalence of categories if, and only if, $\Xi(F):\Xi\C\longrightarrow\Xi\D$ is an equivalence of multicategories. It is also an easy fact to check that if $F:\cP\longrightarrow\Q$ is an equivalence of multicategories then $[F]_{1}:[\cP]_{1}\longrightarrow[\cQ]_1$ is an equivalence of categories. The opposite statement, however, is not always true (See, for example,~\cite{St}).\end{remark}
\section[Multicategories As Monoids]{Multicategories As Monoids}Operads are monoids in the category of symmetric sequences. A \emph{symmetric sequence} in a symmetric monoidal category $\C$ is a sequence of objects in $\C$, $K = \{K_0, K_1,...,K_n,...\}_{n\in\mathbb{N}}$ where each of the $K_n$ in $\C$ is equipped with an action of the symmetric group $\Sigma_n$. A morphism of $\Sigma_*$-objects $f: K\rightarrow L$ is a sequence of morphisms $f(n): K(n)\rightarrow L(n)$ in the category $\C$ which commute with the group actions. The category of symmetric sequences in $\C$ is denoted by $\C^{\Sigma_*}$.\\
There are two monoidal products on $\C^{\Sigma}$. The first is a symmetric monoidal tensor product given by:\begin{equation*}\label{tensorofsequences}(K\otimes L)(n) = \bigoplus_{p+q=n} (\unit[\Sigma_n]\otimes K(p)\otimes L(q))_{\Sigma_p\times\Sigma_q},\end{equation*}where here $\unit[\Sigma_n]$ is our notation for the sum of $\Sigma_n$-copies of the unit object of~$\C$ and $\Sigma_p\times\Sigma_q\subset\Sigma_{p+q}$ is the group embedding which sends permutations $\sigma\in\Sigma_p$ (respectively, $\tau\in\Sigma_q$) to permutations of the subset $\{1,\dots,p\}\subset\{1,\dots,p,p+1,\dots,p+q\}$ (respectively, $\{p+1,\dots,p+q\}\subset\{1,\dots,p,p+1,\dots,p+q\}$). The group $\Sigma_p\times\Sigma_q$ acts on $\unit[\Sigma_n]$ by translations on the right and the co-invariant quotient makes this right $\Sigma_p\times\Sigma_q$-action agree with the left $\Sigma_p\times\Sigma_q$-action on $K(p)\otimes L(q)$. Thus the object $K(p)\otimes L(q)$ is a $\Sigma_p\times\Sigma_q$-object in $\C$ for all $p,q\ge0$. The group $\Sigma_n$ acts also on $\unit[\Sigma_n]$ by translation on the left. This left $\Sigma_n$-action induces a left $\Sigma_n$-action on $(K\otimes L)(n)$ and determines the $\Sigma_*$-object structure of the collection $\{(K\otimes L)(n)\}_{n\in\mathbf{N}}$.\\
A symmetric sequence is called \emph{trivial}, or \emph{constant}, if $K(r) = 0$ for all $r>0$. The sequence, $\unit,$ given by\begin{equation*}\unit(n) = \begin{cases} \unit, & \text{if $n=0$}, \\ 0, & \text{otherwise}, \end{cases}\end{equation*}defines the unit for the tensor product of symmetric sequences. The associativity of the tensor product of $\Sigma_*$-object is inherited from the base category, and the symmetry isomorphism $\tau(C,D): C\otimes D\rightarrow D\otimes C$ uses the symmetric from $\C$, $C(p)\otimes D(q)\simeq D(q)\otimes C(p),$ and translations by block transposition in $\unit[\Sigma_n]$. We leave the details as an exercise.\\
This symmetric monoidal product is closed, i.e. there is an external hom-object, denoted by $\Hom_{\C^{\Sigma_*}}(K,L)$, for all $K,L\in\C^{\Sigma_*}$ defined by a product\begin{equation*}\Hom_{\C^{\Sigma_*}}(K,L) = \prod_{n=0}^{\infty}\Hom_{\C}(K(n),L(n))^{\Sigma_n}.\end{equation*}Here $\Hom_{\C}(C(n),D(n))^{\Sigma_n}$ denotes the invariant sub-object of $\Hom_{\C}(C(n),D(n))$ under the action of the symmetric group.
The category $\C^{\Sigma_{*}}$ also admits a \emph{non-symmetric} monoidal product, which we will call the \emph{composition product}, or the \emph{circle product.} The composition product $K\circ L$ is a generalized symmetric tensor construction given by considering the co-invariants of the tensor products $K(r)\otimes L^{\otimes r}$ under the action of the symmetric groups $\Sigma_r$: \begin{equation*}\label{circleproduct} K\circ L = \bigoplus_{r=0}^{\infty} (K(r)\otimes L^{\otimes r})_{\Sigma_r}\end{equation*} where we use the internal tensor product of $\C^{\Sigma_*}$ to form the tensor power $L^{\otimes r}$, and the external tensor product to form the object $K(r)\otimes M^{\otimes r}.$ We should mention that we are assuming the existence of colimits in $\C$ to form the co-invariant object $(K(r)\otimes L^{\otimes r})_{\Sigma_r}$.\\
The category of symmetric sequences with the circle product forms a monoidal category $(\C^{\Sigma_*},\circ, I)$. An \emph{operad} in $\C$ is a monoid in $(\C^{\Sigma_*},\circ, I)$.
\subsection[Collections]{Collections}Informally, a collection is a symmetric sequence with ``many objects.'' There is not a direct algebraic analogy, but a collection is similar in concept to that of a quiver in Lie theory, in the sense that a quiver is a directed graph serving as the domain of a representation (defined as a functor).\\
Let $S$ be a set. A \emph{collection in $\C$ on the set $S$} is a family of $\C$-objects $K(x_1,...,x_n; x)$ for each sequence $x_1,...x_n;x\in S$ and each $n\ge0$. A morphism of collections, $F:(S,K)\rightarrow(S',K')$, consists of a map of sets $f:S\rightarrow S'$ and for each $n\ge 0$ and each sequence $x_1,...x_n;x$ in $S$ a family of $\C$-morphisms $\{f_{n}:K(x_1,...,x_n;x)\rightarrow K'(fx_1,..., fx_n;fx)\}_{n\ge0}$. We denote by $\coll(\C)$ the category of all collections in $\C$. If we fix the object set $S$, we can consider the category of all collections with $S$-objects, which we denote by $\coll(\C)_{S}$.\footnote{Note that in the category of $\C$-collections with fixed sets of objects, a morphism $F:(S,K)\longrightarrow (S,K')$ is the identity map on objects.} A collection $K$ is called \emph{pointed} if it is equipped with unit maps $1_x:\unit\to K(x;x)$, for all $x\in S$. We have a similar notion of a \emph{collection with symmetric action}, where each $K(x_1,...,x_n;x)$ comes equipped with a $\Sigma_{n}$-action. If $K$ is a collection with symmetric action in $\C$, and $S=\{\star\}$, then $K$ is just a symmetric sequence in $\C$. We denote the category of collections (with or without symmetric action) with $S$-objects by $\coll(\C)_{S}$ and the category of pointed collections with $S$ objects by $\coll(\C)^*_{S}$.\\
As with symmetric sequences, the (symmetric) monoidal product of the enriching category $(\C, \otimes, \unit)$ induces a pointwise \emph{tensor product} on $\coll(\C)_{S}.$ The category $\coll(\C)_{S}$ with this pointwise tensor product forms a closed symmetric monoidal category over $\C$. There is an additional monoidal product on $\coll(\C)_{S}$, called the \emph{circle product}, also known as the \emph{composition product}, which is associative, but (highly) non-commutative. This monoidal structure defines a \emph{right closed} monoidal structure over $\C$. For explicit descriptions of this product, see Berger-Moerdijk~\cite[Appendix]{BM07}. For a set of objects $S$, we define the category of \emph{$\C$-enriched multicategories with fixed object set $S$}, or \emph{$S$-colored operads}, as the category of unitary, associative monoids in $\coll(\C)_{S}$ with respect to the composition product.\\
\section{The Homotopy Theory of $\multi$}One of the most basic examples of a model structure is the standard model category structure on $\cCat$, the category of all small (non-enriched) categories (See~\cite{RezkCat}).
\begin{theorem}The category $\cCat$ admits a cofibrantly generated model category structure where:
\begin{itemize}
\item the weak equivalences are the categorical equivalences;
\item the cofibrations are the functors $F:\cC\rightarrow\cD$ which are injective on objects;
\item the fibrations are the functors $F:\cC\rightarrow\cD$ with the property that for each object $c$ in $\cC$ and each isomorphism $f:Fc\rightarrow d$ in $\cD$ there exists a $c'$ in $\C$ and an isomorphism $g:c\rightarrow c'$ in $\cC$ such that $F(g)=f$.
\end{itemize}\end{theorem}
One can use the relationship between $\multi$ and $\cCat$ to construct a cofibrantly generated model category structure on the category of all small symmetric multicategories. The proof we present below appears in several places, but we believe the first occurrence is in the thesis of Weiss~\cite{Weiss}.
\begin{theorem}The category $\multi$ admits a model category structure where:
\begin{itemize}
\item the weak equivalences are the equivalences of multicategories;
\item the cofibrations are those functors of multicategories $F:\cP\rightarrow\cQ$ which are injective on objects;
\item the fibrations are those functors of multicategories $F:\cP\rightarrow\cQ$ with the property that for each object $x$ in $\cP$ and for each isomorphism $\phi:F(x)\rightarrow q$ in $\cQ$, there exists an isomorphism $\psi: x\rightarrow x'$ for which $F(\psi)=\phi.$\end{itemize}\end{theorem}
A multifunctor $F:\cP\rightarrow\cQ$ is a fibration (respectively, cofibration) of multicategories if, and only if, the functor $[F]_{1}$ is a fibration (respectively, cofibration) of categories. In addition, a multifunctor $F:\cP\rightarrow\cQ$ is a trivial fibration if, and only if, the function $\obj(F):\obj(\cP)\rightarrow\obj(\cQ)$ is surjective and $F$ is fully faithful (as a multifunctor). One can think of the fibration condition as being a path-lifting condition for the ``paths'' in $\cQ$ which are isomorphisms.
\begin{proof}Like the model structure for the category of all small categories, we will prove the existence of the model structure on $\multi$ by directly verifying the axioms.
M1: In their paper~\cite{EM}, Elmendorf-Mandell show that $\multi$ is cocomplete.
M2 and M3: It is easy to verify that if two out of three multi-functors $F$, $G$, $FG$ are equivalences of multicategories than the third is as well. It is also any easy verify that the weak equivalences and cofibrations are closed under retracts. We can then use the fact that fibrations are characterized by a lifting property to conclude that fibrations are also closed under retracts.\\
M4: (Lifting) Consider the following square:
$$\begin{CD}
\cP @>F>> \cR\\
@VIVV @VVPV\\
\cQ @> G >> \cS
\end{CD}$$where $I$ is a cofibration and $P$ is a fibration. We need to prove the existence of a lift $H$ which makes the diagram commute whenever either $I$ or $P$ is also a weak equivalence. Let us first assume that $P$ is a trivial fibration. Because the function $I_{0}$ is injective (since $I$ is a cofibration) and the function $P_{0}$ is surjective (since $P$ is a fibration) we can define a lift $H_{0}$ at the level of objects. In order to extend this map to the higher arities, we now choose an element $\phi$ in the set $\cQ(x_1,...,x_n;x)$ and consider its image $G(\phi)$ in $\cS(Gx_1,...,Gx_n;Gx)$. Since $P$ is a fully faithful multifunctor and $Hx=G$ on the level of objects, we know that the function $P:\cR(Hx_1,...,Hx_n;Hx)\longrightarrow\cS(Gx_1,...,Gx_n;Gx)$ is an isomorphism. We now define $H(\phi)=x^{-1}(G(\phi))$. One can check that this extends $H$ into the desired lift, and that this extension is unique.\\
Let us now assume that $I$ is a trivial cofibration. Since $I$ is an equivalence of multicategories we may construct a functor $I'$ such that $I'\circ I=id_{\cP}$ together with a natural isomorphism $\alpha: I\circ I'\rightarrow id_{\cQ}$. If we restrict ourselves to the image of $I$ then we can choose $\alpha$ in such a way so that for each object $x$ in $\obj(\cP)$ the component at $IP$ is $\alpha_{Ix}=id_{Ix}$.\\
For any object $x$ in $\cQ$ we have a corresponding object $GII'x$ in in the image of $P$, i.e. $GII'x=PFI'x$, and $\alpha_{x}:GII'x\rightarrow Gx$ an isomorphism. Since the multifunctor $P$ is a fibration, we know that there exists an object $Hx$ and an isomorphism $\beta_{x}:FI'x\rightarrow Hx$ in the image of $H$ such that $PHx=Gx$ and $P(\beta_x)=G(\alpha_{x})$. If we restrict to objects in $\Q$ which are in the image of $I$, then $HIx=Fx$ and $\beta_{Ix}=id_{Fx}.$ It follows that $H$ gives a lift on objects.\\
Now, let $\phi$ be an $n$-operation in $\cQ(x_1,...,x_n;x)$. Define $H$ as\begin{equation*}\cR(Hx_1,...,Hx_n;FI'x)\buildrel{\beta^{-1}_{x_1},...,\beta^{-1}_{x_n}}\over\longrightarrow\cR(FI'x_1,...,FI'x_n;FI'x)\buildrel{\beta_{x}}\over\longrightarrow\cR(FI'x;Hx).\end{equation*} It is now easily checked that $H$ gives the desired lift.\\
MC5: Given a multifunctor $F:\cP\longrightarrow\cQ$ we want to factor $F$ as $I\circ P$, with $I$ a cofibration and $P$ a trivial fibration. We can construct a multicategory $\cQ'$ with objects $\obj(\cP)\times\obj(\cQ)$ and operations$$\cQ'(x_1,...,x_n;x):=\cQ(\delta(x_1),...,\delta(x_n);\delta(x))$$where $\delta(x)=x$ for $x$ in $\cQ$ and $\delta(x)=Fx$ for $x$ in $\cP.$ A multifunctor $I:\cP\longrightarrow\cQ'$ by letting $I$ be the identity on objects, i.e $Ix=x.$ For $\phi$ an element in $\cP(x_1,...,x_n;x)$ let $I(\phi)=F(\phi)$. One can check that this is a well defined multifunctor, and a cofibration of multicategories. Define the multifunctor $P:\cQ'\longrightarrow\cQ$ on objects by taking $Px=\delta(x)$ and by letting $P$ be the identity on arrows, i.e. for $\phi$ in $\cQ'(x_1,...,x_n;x)$ let $P(\phi)=\phi$. This multifunctor is clearly fully faithful and surjective on objects, and thus a trivial fibration. The case where $F$ factors as a fibration followed by a trivial cofibration is similar. \end{proof}
Notice that all multicategories are both fibrant and cofibrant under this model structure.
\begin{prop}The adjunction $[-]_1:\multi\leftrightarrows\cCat:\Xi$ is a Quillen adjunction.\end{prop}
\begin{proof} It is straightforward to verify the (much stronger) property that both $\Xi$ and $[-]_{1}$ preserve fibrations, cofibrations, and weak equivalences.\end{proof}
We will now show that the $\multi$ is a cofibrantly generated model category, and the explicit description of the generating cofibrations will be important for our description of the generating cofibrations of simplicially enriched multicategories. Let $\emptyset$ be the initial category with no objects, let $I$ be the category with two objects and a single identity map between them, and let $I'$ be the maximal subcategory of $I$ with the same objects but excluding the map between them. Let $\cH=I\coprod_{I'}I$. The following proposition, together with the fact that fibrations are characterized by a right lifting property, implies that the model category structure on $\multi$ is cofibrantly generated.
\begin{prop}A multifunctor $F:\cP\longrightarrow\Q$ is a trivial fibration if, and only if, $F$ has the right lifting property with respect to the set of multifunctors\begin{equation*}\{\Xi(\emptyset)\hookrightarrow\Xi(I)\}\bigcup\{G_n[I']\longrightarrow G_n[I] | n\ge 0\}\bigcup\{G_n[\cH]\longrightarrow G_n[I] | n \ge0\}\end{equation*}\end{prop}
\begin{proof}\label{proofmodelstructurenonenriched}For each $n\ge1$ consider the multicategory $G_{n}[I]$ that has $n+1$ objects $\{0,1,...,n\}$ and operations generated by a single arrow from $(1,...,n)\mapsto 0$. Notice that a functor $G_{n}\longrightarrow\cP$ is just a choice of an $n$-operation in $\cP$.\\
Let $G_n[I']$ be the sub-operad of $G_n[I]$ which contains the same objects of $G_n[I]$ but only takes the identity operations.\footnote{At $n=1$ the operads $G_n[I]$ and $G_n[I']$ are just $I$ and $I'$, respectively.} It follows that a map $F:\cP\longrightarrow\Q$ has the right lifting property with respect to the inclusion $G_n[I']\hookrightarrow G_n[I]$ if, and only if, the function $F:\cP(x_1,...,x_n;x)\longrightarrow\Q(Fx_1,...,Fx_n;Fx)$is surjective for all $x_1,...,x_n;x$ in $\cP$.
Now, consider the operad $G_n[\cH]$ which has $n+1$ objects $\{0,1,...,n\}$ generated by \emph{two} different arrows from $(1,...,n)\mapsto 0$. There is an obvious map $G_n[\cH]\longrightarrow G_n[I]$ identifying the two generating arrows of $G_n[\cH]$ with the generating arrow of $G_n[I]$.\footnote{In other words, the map induced by applying the free symmetric sequences construction~\ref{freesymmetricsequences} to the fold map $I\coprod_{I'}I\longrightarrow I$.} A multifunctor $F:\cP\longrightarrow\Q$ has the RLP with respect to $G_n[\cH]\longrightarrow G_n[I]$ if, and only if, the map $F:\cP(x_1,...,x_n;x)\longrightarrow \Q(Fx_1,...,Fx_n;Fx)$ is injective for any sequence $x_1,...,x_n;x$ in $\cP$. In other words, a multifunctor $F:\cP\longrightarrow\Q$ has the RLP (right lifting property) with respect to the set \begin{equation*}\{\Xi(\emptyset)\hookrightarrow\Xi(I)\}\bigcup\{G_n[I']\longrightarrow G_n[I] | n\ge 0\}\bigcup\{G_n[\cH]\longrightarrow G_n[I] | n \ge0\}\end{equation*} if, and only if, $F$ is a trivial fibration.\end{proof}
\begin{prop}The category of all small multicategories together is a monoidal model category with respect to the Boardman-Vogt tensor product.\end{prop}
\section{The Homotopy Theory of $\C$-Multicategories}The enrichment of $\multi$ over $\cCat$ extends to the enriched case with only minor modifications. Let $\C$ be the category of simplicial sets with the standard model structure. Given a simplicial category $\cA$, we can form a genuine category $\pi_0(\cA)$ which has the same set of objects as $\cA$ and whose set of morphisms $\pi_{0}(\cA)(x,y):=[\unit, \cA(x,y)]$. This induces a functor $\pi_{0}(-):\cCat(\C)\rightarrow \cCat,$ with values in the category of small categories and, moreover, a functor $\Ho(\cCat(\C))\longrightarrow\Ho(\cCat).$ In other words, any $F:\cC\longrightarrow\cD$ in $\Ho(\cCat(\C))$ induces a morphism $\pi_{0}(\C)\longrightarrow\pi_{0}(\D)$ which is well defined up to a non-unique isomorphism. This lack of uniqueness will not be an issue for the purposes of this paper since we study properties of functors which are invariant up to isomorphism. The \emph{essential image} of a simplicial functor $F:\cA\rightarrow\mathcal{B}$ is the full simplicial subcategory of $\mathcal{B}$ consisting of all objects whose image in the component category $\pi_{0}(\mathcal{B})$ are in the essential image of the functor $\pi_{0}(F)$. As with the non-enriched case, we can consider the linear part of a $\C$-enriched multicategory $\cP$, $[\cP]_{1}$, which is in this case a simplicial category. Applying the functor $\pi_{0}$ to the simplicial category $[\cP]_{1}$ gives us the \emph{underlying category} of the multicategory $\cP$. In order to cut back on notation, we will just denote this category by $[\cP]_{1}$ rather than $\pi_{0}([\cP]_{1})$.
\subsection{The Berger-Moerdijk Model Structure on $S$-colored Operads}As we mentioned in the first section, an \emph{$S$-colored operad}, or \emph{multicategory enriched in $\C$ with object set $S$}, is a monoid in $\coll(\C)_{S}$ with respect to the composition product(see~\cite[Appendix]{BM07}). As long as our enriching category,~$\C$ satisfies a set of technical conditions (described below), the model category structure on $\coll(\C)^{*}_{S}$ can be lifted along a free-forgetful adjunction to a model structure on $\multi(\C)_{S}$. Intuitively speaking, an $n$-operation of the free multicategory with object set $S$ generated by the pointed collection $K\in\coll(\C)^{*}_{S}$ is a tree with inputs labeled by $1,\dots,n$, edges labeled by objects of $K$, and vertices labeled by elements of $K$. We will not include the explicit construction here, but one can find this construction in~\cite{BM06}, or~\cite{BV}.
\begin{theorem}\cite{BM07}\label{forget}The forgetful functor from $\multi(\C)_{S}$ to \emph{pointed} collections has a left adjoint$$F^{\star}:\coll(\C)^{*}_{S}\rightarrow\multi(\C)_{S}.$$\end{theorem}The category of symmetric sequences, the category of $\C$-collections with fixed set of objects $S$, $\coll(\C)_{S}$, and the category of pointed $\C$-collections with fixed set of objects $S$ are all $\C$-model categories. This follows from the standard argument that given a cofibrantly generated monoidal model category $\C$ with cofibrant unit and given any finite group $G$, there is an induced monoidal model category structure on the category of objects with right $G$-action, $\C^G$ where the forgetful functor $\C^G \longrightarrow\C$ preserves and reflects weak equivalences and fibrations~\cite[Hovey]{Hov99}. In particular, a morphism of collections $K\longrightarrow L$ is a weak equivalence (respectively, fibration) if for each $n\ge0$ and each sequence of objects $x_1,\ldots, x_n;x$ in $S$ the morphism$$K(x_1,\ldots, x_n;x)\longrightarrow L(x_1,\ldots, x_n;x)$$is a weak equivalence (respectively, fibration) in $\C$.\\
A \emph{symmetric monoidal fibrant replacement functor} is a fibrant replacement functor which is symmetric monoidal and for every $X$ and $Y$ in $\C$ the following diagram commutes\begin{equation*}\xymatrix{X\otimes Y\ar[r]^{r_{X\otimes Y}}\ar[d]_{r_X\otimes r_Y} & F(X\otimes Y) \\
FX\otimes F(Y),\ar[ur] &}\end{equation*}where $r\colon\id_{\C}\longrightarrow F$ is the natural transformation coming from fibrant replacement.\\
\begin{theorem}~\cite{BM07}Let $\C$ be a cofibrantly generated monoidal model category with cofibrant unit and a symmetric monoidal fibrant replacement functor. Let $S$ be a fixed set of objects (or colors). If $\C$ has a co-algebra interval, then the category of all non-symmetric $\C$-multicategories with $S$-objects (equivalently, the category of non-symmetric $S$-colored operads) admits a cofibrantly generated model category structure where a morphism $F:\cQ\longrightarrow\cP$ is a weak equivalence (respectively, fibration) if and only if, for each $n\ge0$, and each sequence $x_1,\dots,x_n,x$ of objects in $S$, the map $F:\cQ(x_1,\dots,x_n;x)\longrightarrow\cP(Fx_1,\dots,Fx_n;Fx)$ is a weak equivalence (respectively, fibration) of $\C$-objects. If the interval is moreover cocommutative, the same is true for the category of symmetric $\C$-multicategories with $S$-objects (equivalently, the category of symmetric $S$-colored operads).\end{theorem}
\begin{example}The category of simplicial sets is a Cartesian closed, cofibrantly generated, monoidal model category that admits a co-associative, co-commutative interval. As symmetric monoidal fibrant replacement functor, we can choose either the $Ex^\infty$functor or the singular chain complex of the geometric realization functor, since both are product-preserving. Therefore, for a fixed set $S$, the category of $S$-colored operads form a model category.\end{example}
\section{The Bergner Model Structure on Simplicial Categories}The following theorem is due to Bergner~(\cite{Bergner}).\begin{theorem}The category of all small simplicial categories, $\cCat(\mathcal{C}),$ supports a right proper, cofibrantly generated, model category structure. The weak equivalences (respectively, fibrations) are the $\C$-enriched functors$$F:\mathcal{A}\longrightarrow\mathcal{B}$$ such that:
\begin{description}
\item[W1] for all objects $x,y$ in $\mathcal{A}$, the $\C$-morphism $F_{x,y}:\mathcal{A}(x,y)\longrightarrow\mathcal{B}(Fx,Fy)$ is a weak equivalence (respectively, fibration) in the model structure on $\C$ and
\item[W2]the induced functor $\pi_0(F):\pi_{0}(\mathcal{A})\longrightarrow \pi_{0}(\mathcal{B})$ is a weak equivalence (respectively, fibration) of categories.\end{description}\end{theorem}
Bergner also gives an explicit description of the generating (acyclic) cofibrations of this model structure, which is worth describing here. We let $\emptyset$ denote the empty category and $I=\{*\}$ for the category which has one object and one identity arrow (viewed as a simplicial category by applying the strong monoidal functor $\Sets\longrightarrow\sSets$). For any simplicial set $K$ we define a simplicial category $G_{1}[K]$ which has two objects, arbitrarily called $0$ and $1$, and $\Hom(0,1)=K$ as the only non-zero function complex.
Note that if $F:\cA\rightarrow\cB$ satisfies condition $W1$, then checking that $F$ satisfies condition $W2$ is equivalent to checking that the induced functor$$\pi_0(F):\pi_0(\cA)\longrightarrow\pi_0(\mathcal{B})$$is essentially surjective.
\begin{prop}\cite{Bergner}\label{generatingcofibrationsbergner}A functor of simplicial categories $F\cA\rightarrow\cB$ is an acyclic fibration if, and only if, $F$ has the right lifting property (RLP) with respect to all the maps\begin{itemize}
\item $G_1[K]\hookrightarrow G_1[L]$ where $K\hookrightarrow L$ is a generating cofibration of $\sSets$ and
\item the maps $\emptyset\hookrightarrow I$.\end{itemize}\end{prop}
\begin{prop}\cite{Bergner}\label{generatingtrivialcofibrationsbergner}A functor of simplicial categories $F\cA\rightarrow\cB$ is a fibration if, and only if, $F$ has the right lifting property (RLP) with respect to all the maps\begin{itemize}
\item $G_1[K]\hookrightarrow G_1[L]$ where $K\hookrightarrow L$ is a generating cofibration of $\sSets$ and
\item the maps $I\hookrightarrow\mathcal{H}$ where $\obj(\mathcal{H})$ is a set of representatives for the isomorphism classes of simplicial categories on two objects. Each function complex of $\mathcal{H}$ is weakly contractable and has countably many simplices. Furthermore, we require that $\mathcal{H}$ be a cofibrant object in the Dwyer-Kan model category structure on $\cCat(\sSets)_{\{x,y\}}$~\cite{DK1}.\end{itemize}\end{prop}
\subsection{The Proof of The Main Theorem} The main theorem of this paper is the following.
\begin{theorem}The category of small $\C$-enriched symmetric multicategories admits a right proper cofibrantly generated model category structure in which a multifunctor $$F:\cP\rightarrow\Q$$ is a weak equivalence if:
\begin{description}
\item[W1] for any $n\ge 0$ and for any signature $x_1,...,x_n;x$ in $\cP$ the map of $\C$-objects $$F:\cP(x_1,...,x_n;x)\longrightarrow\cQ(Fx_1,...,Fx_n;Fx)$$ is a weak equivalence in the model category structure on $\C$.
\item[W2] the induced functor $[F]_{1}$ is a weak equivalence of categories.
\end{description} A simplicial multifunctor $F:\cP\longrightarrow\Q$ is a fibration if:\begin{description}\label{fibrations}
\item[F1] for any $n\ge 0$ and for any signature $x_1,...,x_n;x$ in $\cP$ the map of $\C$-objects $$F:\cP(x_1,...,x_n;x)\longrightarrow\cQ(Fx_1,...,Fx_n;Fx)$$ is a fibration in the model category structure on $\C$.
\item[F2] the induced functor $[F]_{1}$ is a fibration of categories.
\end{description} The cofibrations (respectively, acyclic cofibrations) are the multifunctors which satisfy the left lifting property (LLP) with respect to the acyclic fibrations (respectively, fibrations).\end{theorem}
We have the following useful characterization of acyclic fibrations.
\begin{lemma} If a multifunctor $F:\cP\longrightarrow\Q$ fixes objects, and satisfies conditions $[F1]$ and $[W1]$, then $F$ is a acyclic fibration.\end{lemma}We can give an explicit description of the generating cofibrations and generating acyclic cofibrations.
\begin{definition}[Generating Cofibrations]The set $\bar{I}$ of generating cofibrations consists of the following $\C$-multifunctors closed under pushouts, transfinite composition, and retracts:
\begin{description}
\item[C1] Given a generating cofibration $K\hookrightarrow L$ in the model structure on $\C$ the induced multifunctors $G_{n}[K]\longrightarrow G_{n}[L]$ for each $n\ge 0$.\footnote{i.e. a cofibration of $\multi(\C)_{\{0,1,...,n\}}$}
\item[C2] the $\C$-functors $\emptyset\longrightarrow\cI$ viewed as $\C$-multifunctors via $\Xi.$\end{description}\end{definition}\bigskip
\begin{definition}[Generating Acyclic Cofibrations]\label{generatingacycliccofibrations}The set $\bar{J}$ of generating acyclic cofibrations consists of the following $\C$-multifunctors closed under pushouts, transfinite composition, and retracts:
\begin{description}
\item[A1] Given a generating acyclic cofibration $K\hookrightarrow L$ of the model structure on $\C$ the induced multifunctors $G_{n}[K]\longrightarrow G_{n}[L]$ for each $n\ge 0$.
\item[A2] The $\C$-functors $\cI\longrightarrow\cH$ viewed as multifunctors via $\Xi.$
\end{description}\end{definition}
The multicategories $G_{n}[K]$ are represented by the ``corolla'' $$\xymatrix{*{1}\ar@{-}[dr] & *{2...}\ar@{-}[d] & *{n}\ar@{-}[dl]\\
& *{K}\ar@{-}[d]\\
& & *{\,}}$$where $K$ is a simplicial set, and each simplex in $K$ is an operation $1,...,n\mapsto 0$ and there are no other non-identity operations.
\subsection[Generating Collections]{Generating Collections}The multicategories $G_{n}[-]$ are the free multicategories generated by the generating collections $G_{n}$. The $G_{n}$ form a set of small projective generators for the category $\coll(\C)_{S}$. In order to give a precise definition of these generating collections we must first understand an alternate, but equivalent, description of $\C$-collections. This section is unfortunately abstract, and the reader may want to skip this section and return to it at a later time.\\
Let $S$ be a finite set. We define a $\C$-collection on $S$-objects as a pre-sheaf $K:\mathbb{F}(S)^{op}\rightarrow\C$, where $\mathbb{F}(S)$ is the category whose objects are triples $(X,x_0,\alpha)$, with $X$ a finite set, $x_0\in X$ a chosen base point, and $\alpha:X\rightarrow S$ a chosen function. The morphisms in $\mathbb{F}(S)$ $$(X,x_0,\alpha)\stackrel{f}{\longrightarrow}(X',x'_0,\alpha')$$ are basepoint preserving bijections $f:X\rightarrow X'$ which are compatible with the chosen functions to $S$, i.e. $\alpha'\circ f = \alpha$.\\
Since we know that every finite pointed set is isomorphic to a set of the form $[n+1]=\{1,\dots,n;0\}$ with $0$ viewed as the basepoint, we can always consider the category $\mathbb{F}^\circ(S)$, which is the full subcategory of $\mathbb{F}(S)$ whose objects are finite pointed sets isomorphic to $[n+1]$. The category $\mathbb{F}^\circ(S)$ is just direct sum of translation groupoids $(S^n\times S)\rtimes \Sigma_n$ where $\Sigma_n$ acts on $S^n\times S$ by permuting the first $n$ coordinates, $(x_1,\dots,x_n;x_{0})^{\sigma}=(x_{\sigma(1)},\dots,x_{\sigma(n)};x_{0})$, and that the inclusion $\mathbb{F}^\circ(S)\longrightarrow\mathbb{F}(S)$ is an equivalence of categories. We can simplify the picture even further by assuming that the object set, $S,$ is equipped with a \emph{linear order}, $\leq$. The category $\mathbb{F}^\leq(S)$ is also equivalent to $\mathbb{F}(S)$, but only has objects $([n+1],0,\alpha)$ for which $\alpha(1)\leq \dots \leq \alpha(n)$ and $\alpha(x_0)=0$. In other words, the category $\mathbb{F}^\leq(S)$ is just $$\coprod_c \coprod_{c_1\leq \dots \leq c_n}\textstyle \sum_{c_1\dots c_n}$$ where $\Sigma_{c_1\dots c_n}\subseteq \Sigma_n$ is the subgroup of permutations $\sigma\in\Sigma_n$ which preserve the order of the sequence, i.e. $c_{\sigma(1)}\leq \dots \leq c_{\sigma(n)}$.\footnote{This just means that $(c_1,\dots,c_n)=(c_{\sigma(1)},\dots,c_{\sigma(n)})$.} Since any finite $S$ can be given a linear order, the category $\mathbb{F}(S)$ is equivalent to one of the form $\mathbb{F}^\leq(S)$.\\
Now, if we consider $\C$-collections to be pre-sheaves $K:(\mathbb{F}^\leq(S))^{op}\longrightarrow\C,$ then for every object $Y$ in $\C$, and for each integer $n\ge 0$ we can define a $\C$-collection where $G_{n}(Y)(x_{1},...,x_{k};x_0)=0$ except in the special case where $n=k$ and $a_{i}=\alpha(i)$ and $x_0=\alpha_0$, in which case we define $G_{n}(Y)(x_{1},...,x_{k};x_0)=Y$. We consider the $G_{n}$ as symmetric multicategories by applying the free multicategory functor $\mathfrak{F}(G_{n})$ (for more on the free multicategory functor see \cite{BM07}, or\cite{EM}).\\
\begin{example}It can be helpful to understand this construction in the one-object case. Let $S=\{\star\}$. Then\begin{equation*}G_n(r) = \begin{cases} \unit[\Sigma_n], & \text{if $r = n$}, \\
0, & \text{otherwise}. \end{cases}\end{equation*} Where $\unit[\Sigma_n]$ denotes the symmetric sequence in~$\C$ formed by the sum over $\Sigma_n$ copies of the tensor unit $\unit\in\C$. The symmetric group $\Sigma_n$ acts on $G_n(n) = \unit[\Sigma_n]$ by translations on the right, and hence acts on $G_n$ on the right by automorphisms of $\Sigma_*$-objects. The symmetric sequences $G_n$, $n\in\mathbb{N}$, are characterized by the following property:\begin{prop}There exists a natural $\Sigma_n$-equivariant isomorphism \begin{equation*}\omega_n(Y): Y(n)\longrightarrow\Hom_{\C^{\Sigma_*}}(G_n,Y),\end{equation*}for all $C\in\C^{\Sigma_*}$.\end{prop}The proof follows immediately from the definitions, namely we have$$\Hom_{\C^{\Sigma_*}}(G_n,Y)\simeq\Hom_{\C}(\unit[\Sigma_n],Y(n))^{\Sigma_n}$$and$$\Hom_{\C}(\unit[\Sigma_n],Y(n))^{\Sigma_n}\simeq\Hom_{\C}(\unit,Y(n))\simeq Y(n).$$ The $\Sigma_n$-action by right translations on $\unit[\Sigma_n]$ corresponds to the internal $\Sigma_n$-action of $Y(n)$ under the isomorphisms in the second line. Hence, we obtain a $\Sigma_n$-equivariant isomorphism $\omega_n(Y):Y(n)\longrightarrow\Hom_{\C^{\Sigma_*}}(G_n,Y)$.\end{example}
We will delay the proof of the model structure momentarily, to prove the following lemmas.
\begin{lemma}[Classifying Fibrations]\label{classifyingfibrations}A $\C$-multifunctor is a fibration if and only if it has the right lifting property with respect to the class of generating acyclic cofibrations.\end{lemma}
\begin{proof} Maps of multicategories are defined ``locally,''i.e. given the multifunctor $F:\cP\longrightarrow\Q$, it is an easy observation that the map of $\C$ objects $$F:\cP(x_1,...,x_n;x)\longrightarrow\Q(Fx_1,...,Fx_n;Fx)$$ is a fibration, if and only if $F$ satisfies the right lifting property (RLP) with respect to multifunctors which are locally acyclic cofibrations, i.e. the set $[A1]$. Furthermore, since the functors $\Xi$ and $[-]_{1}$ form an adjoint pair, we can observe that the multifunctor $F$ satisfies the RLP with respect to the set $[A2]$ if and only if the $\C$-functor $[F]_{1}$satisfies the RLP with respect to the inclusions $\cI\longrightarrow \cH$ which we know to be true by the classification of fibrations in $\cat(\C)$. Putting this together, we conclude that $F$ has the RLP with respect to the set $J$ if and only if it $F$ is a fibration of multicategories. \end{proof}
\begin{lemma}[Classifying Trivial Fibrations]\label{classifying trivial fibrations}A $\C$-multifunctor is a trivial fibration if and only if it has the right lifting property with respect to the class of generating cofibrations.\end{lemma}
\begin{proof} The proof is nearly identical to the proof of the previous lemma. \end{proof}
\begin{lemma} Every acyclic cofibration is a weak equivalence.\end{lemma}
\begin{proof} We break the proof into several smaller claims.
\begin{claim}[Claim 1:]If $m:S\longrightarrow T$ is an injective map of sets, then the induced functor $\cCat(\C)_{S}\longrightarrow\cCat(\C)_{T}$ preserves cofibrant objects.\end{claim}
\begin{claim}[Claim 2:]\label{claim 2} Consider the following pushout square of simplicial multicategories$$\begin{CD}
\cP @>F>> \cP'\\
@VVJV @VVGV\\
\cQ @>K>> \cQ'
\end{CD}.$$If $J:\cP\longrightarrow\cQ$ is a trivial cofibration which is bijective on objects, and if $F:\cP\longrightarrow\cP'$ is a multifunctor which is injective on objects. Then $G\cP'\longrightarrow\cQ'$ is also a trivial cofibration which is bijective on objects.\end{claim}
\begin{proof}[Proof of Claim 2:]\label{proofofclaim2}We may assume that the multifunctor $J$ is the identity on objects, and is therefore a member of the class of generating acyclic cofibrations. To ease notation we will denote the object set of $\cP$ by $S$ and the object set of $\cP'$ by $S'=S\coprod T$ where we identify the effect of $F$ on objects with the coproduct of inclusions $S\hookrightarrow S'$. We can adjoin the set $T:=S/F(0)$ of objects to $\cP$, adding no new operations other than identities. This defines a new multicategory, denoted $\cP\coprod(\coprod_{T}\unit)$. In a similar manner, we construct $\cQ\coprod(\coprod_{T}\unit)$, and decompose the pushout$$\begin{CD}
\cP @>F>> \cP'\\
@VVJV @VVGV\\
\cQ @>K>> \cQ'
\end{CD}$$ as the composition of two pushouts
$$\begin{CD}\cP @>>>\cP\coprod(\coprod_{T}\unit) @>>> \cP'\\
@VVJV @VVJ\coprod(\coprod_{T}id) V @VVGV\\
\cQ @>>> \cQ\coprod(\coprod_{T}\unit) @>>> \cQ'.
\end{CD}.$$ Now, by assumption, $J$ is a trivial cofibration of similical multicategories, and thus a trivial cofibration in the model structure on $\multi(\C)_{S}$. It follows that $J\coprod(\coprod_{T}id)$ is a trivial cofibration in the model structure on $\multi(\C)_{S'}$. This implies that $G$ is a weak equivalence in the model structure on $\multi(\C)_{S'}$. Since $G$ is also the pushout of a coifbraion of simplicial multicategories, we have that $G$ is a trivial cofibration of $\multi(\C)$ which is bijective on objects.\end{proof}
\begin{claim}[Claim 3:]\label{claim3}Let $S$ denote a set, and let $s:\{0,...,n\}\twoheadrightarrow S$ be a surjection of sets. Then for any map $X\longrightarrow Y$ of simplicial sets, the square$$\begin{CD}
G_{n}[X] @>>> s_*(G_{n}[X])\\
@VVV @VVV\\
G_{n}[Y] @>>> s_*(G_{n}[Y])
\end{CD}$$is a pushout of simplicial multicategories. Moreover, if $X\longrightarrow Y$ is a generating trivial cofibration of simplicial sets, then the right vertical map is a weak equivalence in the model category structure on $\multi(\C)_{S}$.\end{claim}
\begin{proof}[Proof of Claim 3]Given $\cP$, a simplicial multicategory with object set $T$, a map $G_n[X]\longrightarrow\cP$ consists of a set map $f:\{0,...,n\}\longrightarrow T$ together with a map $G_n[X]\longrightarrow f^*(\cP)$. Recall that the map $G_{n}[X]\longrightarrow f^{*}(\cP)$ is just a map of simplicial sets $X\longrightarrow\cP(fx_1,...fx_n;fx_0)$. Moreover, a map $s_{*}(G_n[X])\longrightarrow\cP$ is equivalent to a map $f$ which factors as $f=g\circ s$, together with a map $s_*(G_n[X])\longrightarrow g^*(\cP)$\footnote{equivalently, a map $G_n[X]\longrightarrow f^*(\cP)$}. So it is clear that the square is a pushout. If $X\longrightarrow Y$ is a generating acyclic cofibration of simplicial sets, then $G_n[X]\longrightarrow G_n[Y]$ is a generating acyclic cofibration of $\multi(\C)_{S}$. \end{proof}
\begin{remark}Is is essential that our enriching category $\C$ be a monoidal model category since the induced operad $s_*(G_n[X])$ is trivial if $s$ identifies two numbers $i$ and $j$ with $0<i<j$. If, however, $s$ identifies $0$ and $i>0$ we get something more complicated. For example, if $n=1$ and $s$ identifies $0$ and $1$ then $s_*(G_1[X])$ is the free simplicial monoid on $X$.\end{remark}
\begin{claim}\label{NAFA} Given a multicategory $\cP$ enriched in $\C$, a multifunctor $F:\Xi(\cI)\rightarrow\cP$ and the pushout square:
$$\begin{CD}
\Xi(\cI) @>F>> \cP\\
@VVJV @VVGV\\
\Xi(\cH) @>K>> \cQ
\end{CD}$$ then the multifunctor $G$ is fully faithful. \end{claim}
\begin{proof}[Proof of Claim 4] We will proceed by giving an explicit construction of a simplicial multicategory $\cQ$ from the maps $F$ and $J$ and prove that $\cQ$ is the pushout of these maps. It will then be clear from the construction that $G$ is fully faithful.
As usual, we denote the object set of $\cP$ by $\obj(\cP)$, and the $\C$-enriched category with a single object $*$ by $\cI$. We will fix an object $x_{*}$ of $\cP$ which is in the image of $*$ under $F$. The object set of the multicategory $\cQ$ will be denoted by $\obj(\cQ)=\obj(\cP)\coprod *$.
We want to understand all possible $n$-operations of $\Q$, and to make this easier we start by considering the underlying non-symmetric multicategory, which we also denote by $\Q$. The generating $n$-operations of the non-symmetric multicategory $\Q$ are the following:
\begin{enumerate}
\item $\cQ(x_1,...,x_n;x)=\cP(x_1,...,x_n;x).$
\item The $\cQ$ morphisms are given by the $\cP$-morphisms with an action on the left from $\Xi(\cH).$ Explicitly, let $\phi\in\cH(0,1)$, $\psi\in\cP(x_1,...,x_n;x_{*})$, and $\theta\in\cI(*,*)$. Then $\cQ(x_1,...,x_n;*)$ can be given by $$\cH(0,1)\otimes \cP(x_1,...,x_n;x_{*})/\{\phi\circ\ J(\theta)\otimes\psi=\phi\otimes F(\theta)\circ \psi\}.$$
\item There are two instances where we have both a left and a right action coming from $\Xi(\cH)$, which represent $\Q(x_1,...,x_n,\underbrace{*,...,*}_{m};*)$ and $\Q(x_1,...,x_n,\underbrace{*,...,*}_{m};x)$. The $n+m$-operations $\Q(x_1,...,x_n,\underbrace{*,...,*}_{m};*)$ are the $\cP$-morphisms $\cP(x_1,...,x_n, x_*,...,x_*;x_*)$ equipped both left and right actions from $\Xi(\cH)$. The left action from $\phi\in\Xi(\cH)(0,1)$ (as above) and the right actions from $\phi'\in\Xi(\cH)(1,0)$. Explicitly, the operations $$\Xi(\cH)(0,1)\otimes\cP(x_1,...,x_n,\underbrace{x_*,...,x_*}_{m};x_*)\otimes (\underbrace{\Xi(\cH)(1,0),...,\Xi(\cH)(1,0)}_{m})$$ $$\phi\otimes\psi\otimes(\phi^{-1},...,\phi^{-1})\Big/\begin{cases}(\phi\circ J(\theta))\otimes \psi\otimes (\underbrace{\phi^{-1},...,\phi^{-1}}_{m})=\phi\otimes F(\theta)\circ\psi\otimes (\phi^{-1},...,\phi^{-1}),\cr \phi\otimes\psi\otimes(\phi^{-1},...,J(\theta)\circ\phi^{-1},...,\phi^{-1})=\phi\otimes\psi\circ_{i}F(\theta)\otimes(\underbrace{\phi^{-1},...,\phi^{-1}}_{m})\end{cases}\Biggr\}$$ The other case is similar and we leave it as an exercise.
\item The addition of a unit morphism for the object $*$ in $\obj(\cQ)$. For this we just let $$\phi\in\Xi(\cH)(1,1)=\cQ(*;*).$$
\end{enumerate}
Composition in $\Q$ can be informally described as composition in $\cP$ tensored with composition in $\Xi(\cH)$ modulo relations coming from $\Xi(\cI).$\\
We can check that the multicategory we constructed fits into a commutative diagram
$$\begin{CD}
\Xi(\cI) @>F>> \cP\\
@VVJV @VVGV\\
\Xi(\cH) @>K>> \cQ
\end{CD}$$where the map $G$ takes operations in $\cP$ to the $\Q$-operations given by $\cQ(x_1,...,x_n;x)=\cP(x_1,...,x_n;x)$ and the map $K$ can be described as follows:
\begin{equation*}
K(\phi) = \left\{
\begin{array}{rl}
\phi & \text{if } \phi\in\cH(0,0)\\
\phi & \text{if } \phi\in\cH(1,1)\\
\phi\otimes\unit_{x_0} \in\Q(x_0,*)& \text{if } \phi\in\cH(0,1),\\
\unit_{x_0}\otimes\phi \in\Q(*,x_0)& \text{if } \phi\in\cH(1,0),\\
\end{array} \right.\end{equation*} If we are given another multicategory $\R$ and two multifunctors $F':\cP\rightarrow\R$ and $F'':\Xi(\cH)\rightarrow\R$ which satisfy $F''\circ J=F'\circ F$, then we can define a multifunctor $G':\Q\rightarrow\R$ as follows. On objects $G'_0(x_i)=F'_0(x_i)$ and $G'_0(*)=F''_0(1)$. On operations we have
\begin{equation*}
\begin{array}{lc}
G'(\psi)=F'(\psi) \\
G'(\phi\otimes\psi)=F''(\phi)\circ F'(\psi) \\
G'(\phi\otimes\psi\otimes(\phi_1,...,\phi_m))=F''(\phi)\circ F'(\psi)\circ(F''(\phi_1),...,F''(\phi_m))\\
G'(\psi\otimes(\phi_1,...,\phi_m))= F'(\psi)\circ(F''(\phi_1),...,F''(\phi_m))\\
G'(\phi)=G(\phi)\end{array} \end{equation*} The reader can now check that $G'$ is a well-defined multifunctor and that $G'$ is the \emph{unique} multifunctor which satisfies $G'\circ G=F''$ and $G'\circ F=F'$. This proves that the commutative diagram given above is a pushout, and is now clear that $G$ is fully faithful.\end{proof}
\begin{proof}[Proof of Lemma]\label{proofclassifyingacyliccof} It is enough to show that the pushout of a multifunctor from either the set $A1$ or $A2$ is a weak equivalence, i.e. that if $J:\cA\rightarrow\cB$ is in either the set $A1$ or the set $A2$ and the square
$$\begin{CD}
\cA @>F>> \cP\\
@VVJV @VVPV\\
\cB @>G>> \cQ
\end{CD}$$ is a pushout square in $\multi(\C)$, then $G$ is a weak equivalence.
We will split the proof into two cases. Let's first assume that $J$ is an acyclic cofibration from the set $A1$, and consider the following pushout diagram:
$$\begin{CD}
G_{n}[K] @>F>> \cP\\
@VVJV @VVPV\\
G_{n}[L] @>G>> \Q
\end{CD}.$$
Recall that the multifunctor $F:G_{n}[K]\rightarrow\cP$ consists of no other data except a set map $F_{0}:\{0,...,n\}\rightarrow \obj(\cP)$ together with an $\C$-morphism $K\rightarrow\cP(F(1),...,F(n);F(0)).$ The set map $F_0$ can be factored into and injection followed by a surjection, $$\{0,...,n\}\hookrightarrow S'\twoheadrightarrow S$$. Now, since $K\hookrightarrow L$ is a generating acyclic cofibration of $\C$, we factor the pushout square into two pushouts
$$\begin{CD}G_n[K] @>>> s_*(G_n[K] @>>> \cP\\
@VVJV @VVV @VVGV\\
G_n[L] @>>> s_*(G_n[L]) @>>> \cQ'\end{CD}.$$ Then claim 3~\ref{claim3} implies that the middle vertical map is a weak equivalence, and the claim~\ref{claim2} implies that $G$ is a weak equivalence.
Now we assume that $J$ is an acyclic cofibration from set $A2$ and consider the following pushout square: $$\begin{CD}
\Xi(\cI) @>>F> \cP \\
@VVJV @VVPV\\
\Xi(\cH) @>>G> \Q
\end{CD}$$. \\
Recall that $\Xi(I)$ denotes the $\C$-enriched category $I$, viewed as a multicategory via the functor $\Xi(-)$. The category $I$ has one object, $\{0\}$, and endomorphism monoid $I(0,0)=\unit$. The $\C$-enriched category $H$ is has two objects $\{0,1\}$ whose only non-trivial operation is $\cH(0,1)=\unit$. At the level of objects, our map $J$ is the inclusion map $\{0\}\hookrightarrow\{0,1\}$. We pullback along $J$ to factor the diagram into two pushouts:
$$\begin{CD}
\Xi(\cI) @>>F> \cP \\
@VVJ'V @VVP'V\\
J^{*}(\Xi\cH) @>>> \R\\
@VVJ''V @VVP''V\\
\Xi(\cH) @>>G> \Q
\end{CD}$$. \\
In the top pushout, we claim that $J'$ is a weak equivalence in the model structure on $\multi(\C)_{\{0\}}$ (i.e. the model structure on simplicial operads). It is clear that $F$ is bijective on objects, and so $P'$ is a weak equivalence in $\multi(\C)_{S}$ (where, as usual, $S:=\obj(\cP)$).
Now, the map $J''$ is fully faithful in each simplicial degree and bijective on objects, so by applying claim $4$ degreewise, we conclude that $P''$ is fully faithful. By composition, $P:\cP\longrightarrow\cQ$ satisfies condition $W1$. Since $\pi_0$ preserves pushouts and $\pi_0(J)$ is a trivial cofibration, it follows from the model structure on $\multi$, that condition $W2$ is satisfied. \end{proof}We have now proved the lemma.\end{proof}
We can now prove the existence of the model category structure on the category of all small simplicial multicategories.\\
\begin{proof}[Proof of Theorem]\label{proofofmaintheorem}The category $\multi(\C)$ is co-complete (see,~\cite{EM}), and one can quickly check that all three classes of multifunctors are closed under retracts and that the class of weak equivalences satisfies the ``2-out-3'' property. \\
Given an arbitrary multifunctor we can apply the small object argument to produce a factorization $F=P\circ I$ where $I$ is in $\bar{I}$ and $P$ has the right lifting property with respect to $I$. The our lemma~\ref{classifying trivial fibrations} implies that $P$ is an acyclic fibration. In a similar manner, we factor $F=Q\circ J$, where $J$ is in $\bar{J}$ and $Q$ has the right lifting with respect to $Q$. The lemma~\ref{classifyingfibrations} implies that $Q$ is a fibration of multicategories.\\
Finally, we check that given the square
$$\begin{CD}
\cA @>F>> \cP\\
@VVIV @VVPV\\
\cB @>G>> \Q
\end{CD}$$ with $I$ a cofibration and $P$ a fibration of multicategories. If $P$ is also a weak equivalence, then we find a lift by the classification of cofibrations. If $I$ is a weak equivalence, then we factor $I=Q\circ J:\cA\hookrightarrow\widetilde{\cB}\twoheadrightarrow\cB$ where $Q$ is a fibration and $J$ is an acyclic cofibration. Since we have shown that every acyclic cofibration is a weak equivalence, we know that $J$ is a weak equivalence. The ``2-out-of-3'' property for weak equivalences now implies that $Q$ is an acylic fibration.\\
Since $J$ is an acyclic cofibration, and $P$ is a fibration, we know that $P$ has the RLP with respect to $J$. In other words, we have a lift $H:\widetilde{\cB}\longrightarrow\cP$ so that $J\circ H =F$ and $Q\circ G=P\circ H$. \\
Now, since $I$ is an acyclic cofibration and $Q$ is a trivial fibration, there exists a retract $S$ of $Q$ with $S\circ I = J$. The composite $H\circ S$ provides the desired lift.\end{proof} | 183,488 |
The latest version of its presentation software that supports the BlackBerry. BlackBerry Presenter enables users to wirelessly present Microsoft PowerPoint files directly from their BlackBerry smartphones..
Features :
-.
Compatible BlackBerry smartphones
BlackBerry Bold 9000
BlackBerry Bold 9700
BlackBerry Curve 8500 Series
BlackBerry Curve 8900
BlackBerry Storm
BlackBerry Tour
BlackBerry Presenter is not supported by the BlackBerry Curve 8300 Series and/or BlackBerry Pearl Flip 8200 Series smartphones.
Download
BlackBerry Presenter 1.0.0.30 or Visit from your smartphone to download. | 107,016 |
INTRODUCTIONI was eight when my mother went off to live with her boyfriend. My grandmother promised me she’d come back as soon as the “old man” threw her out. I turned ten and she still wasn’t back. I didn’t care because my mother’s parents, Gustave and Philomene Soublet, took better care of me than she ever could. I never wanted to be at her mercy ever again.I called my grandparents Big Mama and Big Papa. I don’t remember knowing why they, like other grandparents, were called those names nor did I ever ask. Eventually I used my own interpretation as to how the names might have originated. The words seemed to literally translate from the French, grandmère and grandpère, to big or maybe older mother and father.Mr. Joe, my mother’s boyfriend, was an old man - as old as Big Papa - and he didn’t like kids. I wasn’t allowed to visit my mother at his house; she always came to my grandparents’ home to see me. I called her by her first name, Iris, because she never acted like a mother to me. She was more like an older sister.Mr. Joe and Iris drank a lot, mostly beer. When somebody else was buying, they drank whiskey. They hung out in a notorious bar on St. Bernard Avenue where my mother had met Mr. Joe. The bar was in the heart of my neighborhood, the Seventh Ward in downtown New Orleans, where most of the Creoles of color lived. Everybody knew everybody, and everybody knew everybody else’s business. And due to segregation in the 1940s, you did most of whatever it was you did in the Seventh Ward. So there was no place to hide; someone always saw you.I often wondered why Iris had chosen such an old man. One night when Big Mama thought I’d fallen asleep on the sofa, I heard her whispering to Big Papa, “Gustave, Iris will never meet anybody decent right now. There aren’t any young men her age left in town. They’ve all gone to war.”“Well, Philly,” Big Papa told her, “New Orleans is a port city; it’s always gonna be a wide-open town. There’s a war in the Pacific and one in the Atlantic, so boocoo soldiers and sailors from all the bases nearby are flockin to the city night and day lookin for some real bazah. And you know what bazah I’m talkin about. You oughta be glad she’s latched on to Mr. Joe, cuz if not, she might be draggin the street all night foolin around with some of those troops passin through. Some of them are real no-count and down-right mean. You know what, bay, I heard they beat up a lot of the women they fool around with and then hightail it out of town.”My grandmother reluctantly agreed Mr. Joe was the better choice, but she said she thought he, as well as all the others, might be dangerous.“I pray every night for Iris, Gustave. I ask God to straighten her out and keep her safe while He’s doing it,” Big Mama cried. She stumbled out of the front room with her shoulders heaving. Big Papa shook me. “Claire,” he said, “it’s time to go to bed.”What seemed strange to me was in spite of all the gossip that floated around, we never found out what Mr. Joe did for a living. My grandfather thought he might be a gambler or a dope dealer because he always seemed to have money and was usually at home during the day. Whatever he did, he hid it well…and so did Iris. He rarely came to our house, and when he did, he stayed only a few minutes.Sometimes when Mr. Joe got drunk, he’d put a handful of paper money in my mother’s hand and tell her to buy something special for her little girl. She never did. She’d tell me about the money, but I never saw a cent of it, nor did she ever buy me anything, let alone something special. I guess she drank up all the money because she didn’t spend any of it on clothes for herself. Iris seemed to have only two outfits, and as Big Mama put it, they were always crotay. We went to Sunday Mass together a few times and I always walked into church with my head down. I was ashamed to be seen with her.My grandparents and I lived across the street from an old three-story, brick building. It took up one square block and was surrounded by a lacy, cast-iron fence. It looked as if it had been plucked right out of France and dropped down into the center of New Orleans. A nursing home for elderly Whites, it was owned and operated by The Little Sisters of the Poor, an order of Catholic nuns.On our side of the street there were three houses; we lived in one side of the last house, a double-shotgun that sat in the middle of the block. From our house to the corner, there was a beautiful garden shaped like an obtuse triangle. It was bordered by giant, sycamore trees. All year long the garden was filled with vegetables and flowers. It, too, belonged to The Little Sisters of the Poor. My grandparents were the garden’s caretakers. Big Papa built a gate so that we could enter it from our yard.My grandfather was a first-class carpenter and a boat builder until he suffered a heart attack and could no longer do strenuous work. He did light, odd jobs to make ends meet and somehow we managed to squeak by. He even found a way to make sure I went to a Catholic elementary school within walking distance from our house. Big Mama always waited for me on the front porch swing when I’d come home from school, and when I’d come home for lunch, she’d have my favorite meal waiting: a bowl of vegetable soup and a ham sandwich. | 408,634 |
TITLE: Prove that $ \sum_{n=0}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}=\frac{z}{1-z} $
QUESTION [0 upvotes]: Show that for $|z|<1$, one have
$$
\frac{z}{1-z^2}+\frac{z^2}{1-z^4}+\cdots+\frac{z^{2^n}}{1-z^{2^{n+1}}}+\cdots = \frac{z}{1-z}
$$
and
$$
\frac{z}{1+z}+\frac{2z^2}{1+z^2}+\cdots+\frac{2^k z^{2^k}}{1+z^{2^k}}+\cdots = \frac{z}{1-z}
$$
Use the dyadic expansion of an integer and the fact that $2^{k+1}-1=1+2+2^2+\cdots+2^k$.
REPLY [2 votes]: HINT: Note that
$$\begin{align*}
\frac{z^{2^k}}{1-z^{2^{k+1}}}&=z^{2^k}\sum_{n\ge 0}\left(z^{2^{k+1}}\right)^n\\
&=z^{2^k}\sum_{n\ge 0}z^{2^{k+1}n}\\
&=\sum\left\{z^n:1\le n\equiv 2^k\pmod{2^{k+1}}\right\}\;,
\end{align*}$$
so
$$\begin{align*}
\sum_{k\ge 0}\frac{z^{2^k}}{1-z^{2^{k+1}}}&=\sum_{k\ge 0}\sum\left\{z^n:1\le n\equiv 2^k\pmod{2^{k+1}}\right\}\\
&=\sum\bigcup_{k\ge 0}\left\{z^n:1\le n\equiv 2^k\pmod{2^{k+1}}\right\}\\
&=\sum\left\{z^n:\exists k\ge 0\left(n\equiv 2^k\pmod{2^{k+1}}\right)\right\}\;.
\end{align*}$$
Moreover,
$$\frac{z}{1-z}=\sum_{n\ge 1}z^n\;.$$
Now show that for each positive integer $n$ there is a unique non-negative integer $k$ such that $n\equiv 2^k\pmod{2^{k+1}}$. The dyadic expansion of $n$ may help you here.
For the second question, note that
$$\frac{2^kz^{2^k}}{1+z^{2^k}}=2^kz^{2^k}\sum_{n\ge 0}(-1)^nz^{2^kn}=2^k\sum_{n\ge 0}(-1)^nz^{2^k(n+1)}\;,$$
so
$$\sum_{k\ge 0}\frac{2^kz^{2^k}}{1+z^{2^k}}=\sum_{k\ge 0}2^k\sum_{n\ge 0}(-1)^nz^{2^{k(n+1)}}\;.\tag{1}$$
Show that if if $n=2^km$, where $k\ge 0$ and $m$ is odd, then the coefficient of $z^n$ in $(1)$ is $$\sum_{\ell=0}^k(-1)^{2^{k-\ell}m-1}2^\ell\;,$$ and evaluate this finite sum to find the coefficient of $z^n$ in $(1)$. | 181,874 |
ANTWERP, Belgium (WOMENSENEWS)–Scientists alarmed by the limited options women have to protect themselves against AIDS are hedging their bets on a variety of vaginal gels, creams and tablets they say could one day protect women not only against HIV, but against other sexually transmitted diseases and common gynecological infections–even unwanted pregnancy.
Some 650 scientists, economists and public policy experts attended the Microbicides 2002 conference here last week to discuss their progress in developing a viable microbicide, which researchers estimate could become available as early as 2007. With an estimated 40 million people living with HIV worldwide–45 percent of them women–and microbicides endorsed last year for inclusion in the United Nations’ five-point plan to fight AIDS, their challenge is an urgent one.
In sub-Saharan Africa, which has been hardest hit by the epidemic, 55 percent of those infected with HIV are women and girls. An additional 15,000 people are infected each day around the world, say officials from UNAIDS, the United Nations agency responsible for monitoring the epidemic.
‘We Have to Create Options’
The push for microbicides reflects an expanded approach to HIV prevention, which until recently had centered largely on promotion of condoms and hope that vaccine research will bear fruit. An effective vaccine is a decade or more away, however, according to Dr. Anthony Fauci, director of the National Institute of Allergy and Infectious Diseases. Even if scientists develop a vaccine that works, they say it’s unlikely to be 100 percent effective or work equally well against the various strains of HIV and the virus’ ability to mutate. The more prevention options available, the thinking now goes, the greater the chance of slowing the disease’s spread.
Mindful of these obstacles, the talk here focused on developing new methods for women to protect themselves.
"People have not given up on promoting condoms," said Sharon Hillier, a microbiologist and self-described "vaginal ecologist" who made several presentations during the conference.
"All of us recognize we have a prevention method that is very inexpensive, that is available all over, that is technically easy to use and quite effective. Given we have something that works, we are going to keep promoting it," said Hillier, director of reproductive infectious disease research at Magee-Women’s Hospital in Pittsburgh. "Having said that, we understand many people will never use condoms, so we have to create options."
Microbicides could work in several ways: They might coat virus particles or the mucosal cells of the vagina through which women are infected, preventing HIV from infiltrating those cells; they might mobilize the body’s innate immune defenses to create an unfriendly environment for the virus; they might prevent HIV from replicating; or they might prevent the virus from fusing with mucosal cells. Some microbicides could employ several of these strategies and some might be used in combination with familiar contraceptives such as condoms or diaphragms. Others may do double duty as AIDS protection and contraception.
Dr. Peter Piot, executive director of UNAIDS, cautioned the researchers here that they need to "be wary of a backlash from men" who may perceive microbicides as a reason to avoid using condoms. "It’s not an issue yet, but one could imagine that as a result of . . . microbicides that the burden will be put again on women," Piot said after opening the conference last Sunday.
Lori Heise, director of the Washington-based Global Campaign for Microbicides, agreed, noting that microbicides need to be promoted as "something that becomes part of a healthy routine rather than an emblem of distrust."
"I’m not advocating throwing away condoms," Heise said Tuesday. But "we have certain advantages with this product. It holds the potential to make sex more fun."
Lack of Funding Slows Research Efforts
At least two of the microbicides being tested are derived from naturally occurring elements in the environment. Carraguard, the only product currently in the second-to-last stage of testing required by the U.S. Food and Drug Administration before it approves a product for sale, is extracted from seaweed and is formulated as a gel that would work by preventing HIV from invading cells in the vagina. Early-stage trials found Carraguard to be safe and scientists are now testing its efficacy against HIV.
Another naturally derived microbicide, Hillier’s lactobaccillus capsule, not only appears to be safe, but seems to reduce the risk of bacterial vaginosis, a common infection among women. Hillier plans another safety trial to determine proper dosage of the capsules before testing their effectiveness against HIV, which she theorizes would work by making the vagina’s environment hostile to HIV.
Other microbicide candidates include BufferGel, which would work similarly to the lactobacillus capsules by strengthening the body’s defenses, and Pro2000, which would work similarly to Carraguard. The Invisible Condom, which is still in the early testing stages, is a "thermo-reversible" liquid that turns into an odorless, tasteless gel at body temperature and has the benefit of mimicking vaginal lubrication.
The enthusiasm of microbicide researchers for these products, however, has failed to catch on with wealthy pharmaceutical companies that typically fund drug development. Several studies, most notably a February report commissioned by the Rockefeller Foundation, found that microbicides would eventually tap a projected $1.8 billion annual market worldwide and avert 2.5 million infections over three years. But the report also found that sales of the first generation of microbicides wouldn’t cover companies’ cost of investing in their development. The task has consequently fallen to foundations and governments, but funds available for clinical trials fall far short of the amount needed, says Arnon Mishkin of the Boston Consulting Group, who studied the problem.
This year’s estimated microbicides budget for the U.S. National Institutes of Health, for example, is $55.7 million and is expected to grow to $68.2 million next year.
New Jersey Democratic Sen. Jon Corzine, noting that the funds represent less than 2 percent of the institutes’ total AIDS research budget, says the government isn’t spending enough on the research. He has proposed a bill that would increase the institutes’ research dollars with the goal of getting a product to market in five years.
The federally funded health research center invests heavily in developing an AIDS vaccine that is more than 10 years away from productive use, Corzine said, but "it has no program dedicated to researching microbicides that could be developed much sooner."
Corzine, who took an interest in microbicides after learning that his state had the highest prevalence of female HIV infections in the country, said that "developing a vaccine is an important long-term project, but in the short term, we need new methods to prevent the spread of AIDS."
Fulvia Veronese, who oversees the institutes’ microbicide work, said that the agency doesn’t place a higher priority on vaccines than on microbicides, even though it officially formalized programs on microbicides and on preventing HIV in women and girls only last year.
"We think that both are high priorities," Veronese said, adding that the agency was actively trying to recruit investigators to work on microbicides. Vaccine research gets "a larger budget because it started as a larger budget. There was a period when they also had to struggle. Microbicides are following the same type of process."
Veronese was asked whether the Bush administration’s stance on reproductive health–including a nine-month freeze of funds that would aid international reproductive health efforts and advocacy of abstinence-only sex education–might threaten the proposed expansion of investigators’ work.
"I can’t comment on that," she said.
Jordan Lite is assistant managing editor of Women’s Enews.
For more information:
Microbicides 2002 conference:
National Institutes of Health
Microbicides HIV/AIDS-Related Research:
Global Campaign for Microbicides: | 263,941 |
October 27, 2014 09:00 ET
SURREY, BRITISH COLUMBIA--(Marketwired - Oct. 27, compilation of.
This new target 60-90% of intersected widths in the Premier area. (*) samples cut to 1opt or 34.29 g/t Au.
This release covers results for the fourth set of 24 holes for 2014 (P-14-678 to 703), as well as a preliminary portion for P-14-707, assays for holes P-14-690, P-14-699 are still pending. To date Ascot has drilled 31,675 meters in 153 holes, three drill rigs are currently drilling the Premier mine area. | 345,800 |
\begin{document}
\date{}
\maketitle
\begin{abstract}
Optimization methods (optimizers) get special attention for the efficient training of neural networks in the field of deep learning.
In literature there are many papers that compare neural models trained with the use of different optimizers.
Each paper demonstrates that for a particular problem an optimizer is better than the others but as the problem changes this type of result is no longer valid and we have to start from scratch.
In our paper we propose to use the combination of two very different optimizers but when used simultaneously they can overcome the performances of the single optimizers in very different problems.
We propose a new optimizer called MAS (Mixing ADAM and SGD) that integrates SGD and ADAM simultaneously by weighing the contributions of both through the assignment of constant weights.
Rather than trying to improve SGD or ADAM we exploit both at the same time by taking the best of both.
We have conducted several experiments on images and text document classification, using various CNNs, and we demonstrated by experiments that the proposed MAS optimizer produces better performance than the single SGD or ADAM optimizers.
The source code and all the results of the experiments are available online at the following link \textit{{{https://gitlab.com/nicolalandro/multi\_optimizer}}}
\end{abstract}
\section{Introduction}
Stochastic Gradient Descent~\cite{robbins1951stochastic} (SGD) is the dominant method for solving optimization problems.
SGD iteratively updates the model parameters by moving them in the direction of the negative gradient calculated on a mini-batch scaled by the step length, typically referred to as the learning rate.
It is necessary to decay this learning rate as the algorithm proceeds to ensure convergence. Manually adjusting the learning rate decay in SGD is not easy. To address this problem, several methods have been proposed that automatically reduce the learning rate.
The basic intuition behind these approaches is to adaptively tune the learning rate based only on recent gradients; therefore, limiting the reliance on the update to only a few past gradients.
ADAptive Moment estimation~\cite{kingma2014adam} (ADAM) is one of several methods based on this update mechanism~\cite{zaheer2018adaptive}.
On the other hand, adaptive optimization methods such as ADAM, even though they have been proposed to achieve a rapid training process, are observed to generalize poorly with respect to SGD or even fail to converge due to unstable and extreme learning rates~\cite{luo2019adaptive}.
To try to overcome the problems of both of these types of optimizers and at the same time try to exploit their advantages, we propose an optimizer that combines them in a new optimizer.
\begin{figure}
\centering
\includegraphics[width=1.0\columnwidth]{images/egiptian_optimizer.pdf}
\caption{Intuitive representation of the idea behind the proposed MAS (Mixing ADAM and SGD) optimizer: the weights are modified simultaneously by both the optimizers.}
\label{fig:proposed}
\end{figure}
As depicted in Figure~\ref{fig:proposed}, the basic idea of the MAS optimizer here proposed, is to combine two different known optimizers and automatically go quickly towards the direction of both on the surface of the loss function when the two optimizers agree (see geometric example in Figure~\ref{fig:intuitive}a). While when the two optimizers used in the combination do not agree, our solution always goes towards the predominant direction between the two but slowing down the speed (see example of Figure~\ref{fig:intuitive}b).
Analyzing the literature there are many papers that compare neural models trained with the use of different optimizers~\cite{bera2020analysis,graves2013generating,duchi2011adaptive,zeiler2012adadelta} or that propose modifications for existing optimizers~\cite{luo2018adaptive,kobayashi2020scw,zhang2018improved}, always aimed at improving the results on a subset of problems.
Each paper demonstrates that an optimizer is better than the others but as the problem changes this type of result is no longer valid and we have to start from scratch.
In our paper we propose to combine simultaneously two different optimizers like SGD and ADAM, to overcome the performances of the single optimizers in very different problems.
\textbf{Contributions}.
In light of the above, we set out the main contributions of our paper.
\begin{itemize}
\item We demonstrate experimentally that the combination of different optimizers in a new optimizer that incorporates them leads to a better generalization capacity in different contexts.
\item We apply the proposed solution to well-known computer vision and text analysis problems and in both these domains, we obtain excellent results, demonstrating that our solution exceeds the generalization capacity of the starting ADAM and SGD optimizers.
\item We open the way to this new type of optimizers with which it will be possible to exploit the positive aspects of many existing optimizers, combining them with each others to build new and more efficient optimizers.
\item To facilitate the understanding of the MAS optimizer and to allow other researchers to be able to run the experiments and extend this idea, we release the source code and setups of the experiments at the following URL~\cite{torch_code}
\end{itemize}
\section{Related Work}
In the literature, there aren't many papers that try to combine different optimizers together.
In this section, we report some of the more recent papers that in some ways use different optimizers in the same learning process.
In~\cite{keskar2017improving} the authors investigate a hybrid strategy, called \textbf{SWATS}, which starts training with an adaptive optimization method and switches to SGD when appropriate.
This idea starts from the observation that despite superior training results, adaptive optimization methods such as ADAM generalize poorly compared to SGD because they tend to work well in the early part of the training, but are overtaken by SGD in the later stages of training.
In concrete terms, SWATS is a simple strategy that goes from Adam to SGD when an activation condition is met.
The experimental results obtained in this paper are not so different from ADAM or SGD used individually, so the authors concluded that using SGD with perfect parameters is the best idea.
In our proposal, we want to combine two well-known optimizers to create a new one that uses simultaneously two different optimizers from the beginning to the end of the training process.
\textbf{ESGD} is a population-based Evolutionary Stochastic Gradient Descent framework for optimizing deep neural networks~\cite{cui2018evolutionary}.
In this approach, individuals in the population optimized with various SGD-based optimizers using distinct hyper-parameters are considered competing species in a context of coevolution.
The authors experimented with optimizer pools consisting of SGD and ADAM variants where it is often observed that ADAM tends to be aggressive early on, but stabilizes quickly, while SGD starts slowly but can reach a better local minimum. ESGD can automatically choose the appropriate optimizers and their hyper-parameters based on the fitness value during the evolution process so that the merits of SGD and ADAM can be combined to seek a better local optimal solution to the problem of interest.
In the method we propose, we do not need another approach, such as the evolutionary one, to decide which optimizer to use and with which hyper-parameters, but it is the same approach that decides at each step what is the contribution of SGD and that of ADAM.
In this paper, we also compare our MAS optimizer with
\textbf{ADAMW}~\cite{loshchilov2017decoupled}~\cite{loshchilov2018fixing}, which is a version of ADAM in which weight decay is decoupled from $L_2$ regularization.
This optimizer offers good generalization performance, especially for text analysis, and since we also perform some experimental tests on text classification, then we also compare our optimizer with ADAMW.
In fact, ADAMW is often used with BERT~\cite{devlin2018bert} applied to well-known datasets for text classification.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{images/vectors.pdf}
\caption{Graphical representation of the basic idea for the proposed MAS optimizer. In (a), if the two translations $\vec{w}_1$ and $\vec{w}_2$ obtained from two different optimizers are similar, then the resulting translation $\vec{w}_1 + \vec{w}_2$ is boosted. In (b), if the translations $\vec{w}_1$ and $\vec{w}_2$ go in two different directions, then the resulting translation is smaller. We also use two hyper-parameters $\lambda_1$ and $\lambda_2$ to weigh the contribution of the two optimizers.}
\label{fig:intuitive}
\end{figure}
\begin{algorithm} \scriptsize
\caption{Stochastic Gradient Descent (SGD)}
\textbf{Input:} the weights $w_k$, learing rate $\eta$, weight decay $\gamma$, dampening $d$, boolean $nesterov$
\begin{algorithmic}[1]
\State{$v_0=0$}
\Function{$\Delta_\text{SGD}$}{$w_{k}$, $\nabla$, $\gamma$, $\mu$, $d$, $nesterov$}
\State{$\widehat{\nabla} = \nabla + w_{k} \cdot \gamma$}
\If{$m \neq 0$}
\If{$k = 0$}
\State{$v_k=\widehat{\nabla}$}
\Else
\State{$v_k = v_{k-1} \cdot \mu + \widehat{\nabla} \cdot (1 - d)$}
\EndIf
\If{$nesterov = True$}
\State{$v_k = \widehat{\nabla} + v_k \cdot \mu$}
\EndIf
\EndIf
\State{\Return{$v_k$}}
\EndFunction
\For{\texttt{batches}}
\State{$w_{k+1} = w_{k} - \eta_s \cdot \Delta_\text{SGD}(w_{k}, \nabla, \gamma, \mu, d, nesterov)$}
\EndFor
\end{algorithmic}
\label{alg:sgd}
\end{algorithm}
\section{Preliminaries}
Training neural networks is equivalent to solving the following optimization problem:
\begin{equation}
\min_{w \in \mathbb{R}^n} \mathcal{L}(w)
\end{equation}
where $\mathcal{L}$ is a loss function and $w$ are the weights.
The iterations of an \textbf{SGD}~\cite{robbins1951stochastic} optimizer can be described as:
\begin{equation}
\label{eq:sgd}
w_{k+1} = w_k - \eta \cdot \nabla\mathcal{L}(w)
\end{equation}
where $w_k$ denotes the weights $w$ at the $k$-th iteration, $\eta$ denote the learning rate and $\nabla\mathcal{L}(w)$ denotes the stochastic gradient calculated at $w_k$.
To propose a stochastic gradient calculated as generic as possible, we introduce the \textbf{weight decay}~\cite{krogh1992simple} strategy, often used in many SGD implementations.
The weight decay can be seen as a modification of the $\nabla\mathcal{L}(w)$ gradient, and in particular, we can describe it as follows:
\begin{equation}
\label{eq:wd}
\widehat{\nabla}\mathcal{L}(w_k) = \nabla \mathcal{L}(w_k)+ w_k \cdot \gamma
\end{equation}
where $\gamma$ is a small scalar called weight decay.
We can observe that if the weight decay $\gamma$ is equal to zero then $\widehat{\nabla}\mathcal{L}(w) = \nabla\mathcal{L}(w)$.
Based on the above, we can generalize the Eq.~\ref{eq:sgd} to the following one that include weight decay:
\begin{equation}
w_{k+1} = w_k - \eta \cdot \widehat{\nabla}\mathcal{L}(w)
\end{equation}
The SGD algorithm described up to here is usually used in combination with \textbf{momentum}, and in this case, we refer to it as \textbf{SGD(M)}~\cite{sutskever2013importance} (Stochastic Gradient Descend with Momentum).
SGD(M) almost always works better and faster than SGD because the momentum helps accelerate the gradient vectors in the right direction, thus leading to faster convergence.
The iterations of SGD(M) can be described as follows:
\begin{equation}
\label{eq:sgd_vk}
v_{k} = \mu \cdot v_{k-1} + \widehat{\nabla}\mathcal{L}(w)
\end{equation}
\begin{equation}
w_{k+1} = w_k - \eta \cdot v_{k}
\end{equation}
where $\mu \in [0, 1)$ is the momentum parameter and for $k=0$, $v_0$ is initialized to $0$.
The simpler methods of momentum have an associated \textbf{damping} coefficient~\cite{damaskinos2018asynchronous}, which controls the rate at which the momentum vector decays.
The dampening coefficient changes the momentum as follow:
\begin{equation}
\label{eq:sgd_vk_dampening}
v_{d_k} = m \cdot v_{k-1} + \widehat{\nabla}\mathcal{L}(w) \cdot (1 - d)
\end{equation}
where $0 \leq d <1$, is the dampening value, so the final SGD with momentum and dampening coefficients can be seen as follow:
\begin{equation}
w_{k+1} = w_k - \eta \cdot v_{d_k}
\end{equation}
\textbf{Nesterov} momentum~\cite{liu2018accelerating} is an extension of the moment method that approximates the future position of the parameters that takes into account the movement.
The SGD with nesterov transforms again the $v_k$ of Eq.~\ref{eq:sgd_vk}, more precisely:
\begin{equation}
\label{eq:sgd_vk_nesterov}
v_{n_k} = \widehat{\nabla}\mathcal{L}(w) + v_{d_k} \cdot m
\end{equation}
\begin{equation}
\label{eq:sgd_vk_nesterov_w_update}
w_{k+1} = w_k - \eta \cdot v_{n_k}
\end{equation}
The complete SGD algorithm, used in this paper, is showed in Alg.~\ref{alg:sgd}.
\begin{algorithm} \scriptsize
\caption{ADaptive Moment Estimation (ADAM)}
\textbf{Input:} the weights $w_k$, learing rate $\eta$, weight decay $\gamma$, $\beta_1$, $\beta_2$, $\epsilon$, boolean $amsgrad$
\begin{algorithmic}[1]
\State{$m_0=0$}
\State{$v^a_0=0$}
\State{$\widehat{v}_0=0$}
\Function{$\Delta_\text{ADAM}$}{$w_{k}$, $\nabla$, $\eta$, $\gamma$, $\beta_1$, $\beta_2$, $\epsilon$, $amsgrad$}
\State{$\widehat{\nabla} = \nabla + w_{k} \cdot \gamma$}
\State{$m_k = m_{k-1} \cdot \beta_1 + \widehat{\nabla} \cdot (1-\beta_1)$}
\State{$v^a_k = v^a_{k-1} \cdot \beta_2 + \widehat{\nabla} \cdot \widehat{\nabla} \cdot (1-\beta_2)$}
\If{$amsgrad = True$}
\State{$\widehat{v}_k = \text{max}(\widehat{v}_{k-1}, v^a_k )$}
\State{$denom = \frac{\sqrt{\widehat{v}_k }}{\sqrt{1 - \beta_2} + \epsilon}$}
\Else
\State{$denom = \frac{\sqrt{v_k^a}}{\sqrt{1 - \beta_2} + \epsilon}$}
\EndIf
\State{$\eta_a = \frac{\eta}{1 - \beta_1}$}
\State{$d_k = \frac{m_k}{denom}$}
\State{\Return{$d_k, \eta_a$}}
\EndFunction
\For{\texttt{batches}}
\State{$d_k, \eta_a = \Delta_\text{ADAM}(w_{k}, \nabla, \eta, \gamma, \beta_1, \beta_2, \epsilon, amsgrad)$}
\State{$w_{k+1} = w_{k} - \eta_a \cdot d_k$}
\EndFor
\end{algorithmic}
\label{alg:adam}
\end{algorithm}
\textbf{ADAM}~\cite{kingma2014adam} (ADAptive Moment estimation) optimization algorithm is an extension to SGD that has recently seen broader adoption for deep learning applications in computer vision and natural language processing.
ADAM's equation for updating the weights of a neural network by iterating over the training data can be represented as follows:
\begin{equation}
\label{eq:adam_mk}
m_{k} =
\beta_1 \cdot m_{k-1}
+ (1 - \beta_1)
\cdot
\widehat{\nabla}\mathcal{L}(w_k)
\end{equation}
\begin{equation}
\label{eq:adam_vk}
v_{k}^a =
\beta_2 \cdot v_{k-1}
+ (1 - \beta_2)
\cdot
\widehat{\nabla}\mathcal{L}(w_k)^2
\end{equation}
\begin{equation}
w_{k+1} = w_k - \eta \cdot
\frac{
\sqrt{1-\beta_2}
}{
1- \beta_1
} \cdot
\frac{
m_{k}
}{
\sqrt{v_{k}^a} + \epsilon
}
\end{equation}
where $m_k$ and $v_k^a$ are estimates of the first moment (the mean) and the second moment (the non-centered variance) of the gradients respectively, hence the name of the method.
$\beta_1$, $\beta_2$ and $\epsilon$ are three new introduced hyper-parameters of the algorithm.
\textbf{AMSGrad}~\cite{chen2018closing} is a stochastic optimization method that seeks to fix a convergence issue with Adam based optimizers.
AMSGrad uses the maximum of past squared gradients $v_{k-1}$ rather than the exponential average to update the parameters:
\begin{equation}
\widehat{v}_k = \text{max}(\widehat{v}_{k-1}, v_{k}^a)
\end{equation}
\begin{equation}
w_{k+1} = w_k - \eta \cdot
\frac{
\sqrt{1-\beta_2}
}{
1- \beta_1
} \cdot
\frac{
m_{k}
}{
\sqrt{\widehat{v}_k} + \epsilon
}
\end{equation}
The complete ADAM algorithm, used in this paper, is showed in Alg.~\ref{alg:adam}.
\section{Proposed Approach}
In this section, we develop the proposed new optimization method called MAS.
Our goal is to propose a strategy that automatically combines the advantages of an adaptive method like ADAM, with the advantages of SGD, throughout the entire learning process.
This combination of optimizers is summed as shown in Fig.~\ref{fig:intuitive} where $w_1$ and $w_2$ represent the displacements on the ADAM and SGD on the surface of the loss function, while $w1 + w2$ represents the displacement obtained thanks to our optimizer.
Below we explain each line of the MAS algorithm represented in Alg.~\ref{alg:proposed}.
The MAS optimizer has only two hyper-parameters which are $\lambda_a$ and $\lambda_s$ used to balance the contribution of ADAM and SGD respectively.
In our experiments, we use only one learning rate $\eta$ for both ADAM and SGD, but it is still possible to differentiate between the two learning rates.
In addition to the hyper-parameters typical of the MAS optimizer, all the hyper-parameters of SGD and ADAM are also needed.
In this paper, we assume to use the most common implementation of gradient descent used in the field of deep learning, namely the mini-batch gradient descent which divides the training dataset into small batches that are used to calculate the model error and update the model coefficients $w_k$.
For each mini-batch, we calculate the contribution derived from the two components ADAM and SGD and then update all the coefficients as described in the three following subsections.
\subsection{ADAM component}
The complete ADAM algorithm is defined in Alg.~\ref{alg:adam}.
In order to use ADAM in our optimizer, we have extracted the $\Delta_{ADAM}$ function which calculates and returns the increments $d_k$ for the coefficients $w_k$, as defined in Eq.~\ref{eq:adam_d}.
\begin{equation}
\label{eq:adam_d}
d_k = \frac{
\sqrt{1-\beta_2} \cdot m_{k}
}{
\sqrt{\widehat{v}_{k}} + \epsilon
}
\end{equation}
The same $\Delta_{ADAM}$ function also returns the new learning rate $\eta_a$ defined in Eq.~\ref{eq:adam_eta}, useful when a variable learning rate is used.
In this last case, MAS uses $\eta_a$ to calculate a new learning rate at each step.
\begin{equation}
\label{eq:adam_eta}
\eta_a=\frac{\eta}{1- \beta_1}
\end{equation}
Now, having $\eta_a$ and $d_k$, we can directly modify the weights $w_k$ exactly as done in the ADAM optimizer and described in Eq.~\ref{eq:adam_update_simple}.
\begin{equation}
\label{eq:adam_update_simple}
w_{k+1} = w_k - \eta_a \cdot d_{k}
\end{equation}
However, we just skip this last step and use $eta_a$ and $d_k$ for our MAS optimizer.
\subsection{SGD component}
As for the ADAM component, also the SGD component, defined in Alg.~\ref{alg:sgd}, has been divided into two parts: the $\Delta_{SGD}$ function which returns the increment to be given to the weight $w_k$, and the formula to update the weights as defined in Eq.~\ref{eq:sgd_vk_nesterov_w_update}.
The $v_{n_k}$ value returned by the $\Delta_{SGD}$ function is exactly the value defined in Eq.~\ref{eq:sgd_vk_nesterov}, which we will use directly for our MAS optimizer.
\subsection{The MAS optimizer}
The proposed approach can be summarized with the following Eq.~\ref{eq:proposed}
\begin{equation}
\label{eq:proposed}
w_{k+1} = w_k -
( \lambda_s \cdot \eta + \lambda_a \cdot \eta_a)
\cdot ( \lambda_s \cdot v_{n_k}
+
\lambda_a \cdot d_k)
\end{equation}
where $\lambda_s$ is a scalar for the SGD component and $\lambda_a$ is another scalar for the ADAM component used for balancing the two contribution of the two optimizers.
$\eta$ is the learning rate of the proposed MAS optimizer, while $\eta_a$ is the learning rate of ADAM defined in Eq.~\ref{eq:adam_eta}.
$d_{k}$ and $v_{n_k}$ are the two increments define in Eq.~\ref{eq:adam_d} and Eq.~\ref{eq:sgd_vk_nesterov} respectively.
Eq.~\ref{eq:proposed} can be expanded in the following Eq.~\ref{eq:proposed_extensive} to make explicit what are the elements involved in the weights update step used by our MAS optimizer.
\begin{equation}
\label{eq:proposed_extensive}
\begin{split}
w_{k+1} = w_k -
( \lambda_s \cdot \eta + \lambda_a \cdot \frac{\eta}{1- \beta_1})
\cdot \\
\cdot ( \lambda_s \cdot v_{n_{k}}
+
\lambda_a \cdot
\frac{
\sqrt{1-\beta_2} \cdot m_{k}
}{
\sqrt{\widehat{v}_{k}} + \epsilon
})
\end{split}
\end{equation}
where $\beta_1$ and $\beta_2$ are two parameters of the ADAM optimizer, $v_{k}^a$ is defined in Eq.~\ref{eq:adam_vk}, and $m_{k}$ is defined in Eq.~\ref{eq:adam_mk}.
\begin{algorithm} \scriptsize
\caption{ Mixing ADAM and SGD (MAS)}
\textbf{Input:} the weights $w_k$, $\lambda_a$, $\lambda_s$, learing rate $\eta$, weight decay $\gamma$, other SGD and ADAM paramiters \dots
\begin{algorithmic}[1]
\For{\texttt{batches}}
\State {$d_{k}, \eta_a = \Delta_\text{ADAM}(w_{k}, \nabla, \eta, \gamma, \dots)$}
\State {$v_{n_k} = \Delta_\text{SGD}(w_{k}, \nabla, \gamma, \dots)$}
\State {$merged = \lambda_s \cdot v_{n_k} + \lambda_a \cdot d_{k}$}
\State {$\eta_{m} = \lambda_s \cdot \eta + \lambda_a \cdot \eta_a$}
\State{$w_{k+1} = w_{k} - \eta_{m} \cdot merged$}
\EndFor
\end{algorithmic}
\label{alg:proposed}
\end{algorithm}
The MAS algorithm can be easily implemented by following the pseudo code defined in Alg.~\ref{alg:proposed} and by calling the two functions $\Delta_{ADAM}$ defined in Alg.~\ref{alg:adam} and $\Delta_{SGD}$ defined in Alg.~\ref{alg:sgd}.
We can also show that convergence is guaranteed for the MAS optimizer if we assume that convergence has been guaranteed for the two optimizers SGD and ADAM.
\begin{theorem}[MAS Cauchy necessary convergence condition]
\label{th:MAS_cauchy}
If ADAM and SGD are two optimizers whose convergence is guaranteed then the Cauchy necessary convergence condition is true also for MAS.
\end{theorem}
\begin{proof}
Under the conditions in which the convergence of ADAM and SGD is guaranteed~\cite{reddi2019convergence,lee2016gradient}, we can say that $\sum_{k=0}^{p}{\eta \cdot v_{n_k}}$ and $\sum_{k=0}^{p}{\eta_a \cdot d_k}$ converge at $\infty$ .
That imply the following:
\begin{equation}
\lim\limits_{p \to \infty} \eta \cdot v_{n_p}
= \lim\limits_{p \to \infty} \eta_a \cdot d_p
= 0
\end{equation}
We can observe that $\lim\limits_{p \to \infty} \sum_{k=0}^p\eta = \lim\limits_{p \to \infty} \sum_{k=0}^p\eta_a = \infty$ so we can obtain the following:
\begin{equation}
\label{eq:dp_limit}
\lim\limits_{p \to \infty} v_{n_p}
= \lim\limits_{p \to \infty} d_p
= 0
\end{equation}
The thesis is that
$\sum_{k=0}^{p} (\lambda_s \cdot \eta + \lambda_a \cdot \eta_a) \cdot (\lambda_s \cdot v_{n_k} + \lambda_a \cdot d_k)$ respect the Cauchy necessary convergence condition, so:
$\lim\limits_{p \to \infty} (\lambda_s \cdot \eta + \lambda_a \cdot \eta_a) \cdot (\lambda_s \cdot v_{n_p} + \lambda_a \cdot d_p) = 0$
and for Eq.~\ref{eq:dp_limit}, this last equality is trivially true:
\begin{align}
\label{eq:theorem}
\lim\limits_{p \to \infty} (\lambda_s \cdot \eta + \lambda_a \cdot \eta_a) \cdot (\lambda_s \cdot v_{n_p} + \lambda_a \cdot d_p) = \nonumber \\
(\lambda_s \cdot \eta + \lambda_a \cdot \eta_a) \cdot \lim\limits_{p \to \infty} (\lambda_s \cdot 0 + \lambda_a \cdot 0)
= 0
\end{align}
\end{proof}
\begin{theorem}
(MAS convercence)
If for $p \to \infty$ is valid that $\sum_{k=0}^{p}{\eta \cdot v_{n_k}}=\eta \cdot m_1$ and $\sum_{k=0}^{p}{\eta_a \cdot d_k}= \eta_a \cdot m_2$ where $m_1 \in \mathbb{R}$ and $m_2 \in \mathbb{R}$ are two finite real values,
then $MAS = \sum_{k=0}^{p} (\lambda_s \cdot \eta + \lambda_a \cdot \eta_a) \cdot (\lambda_s \cdot v_{n_k} + \lambda_a \cdot d_k) = \lambda_s^2 \cdot \eta \cdot m_1 + \lambda_a^2 \cdot \eta_a \cdot m_2 + \lambda_s \cdot \lambda_a \cdot \eta_a \cdot m_1 + \lambda_s \cdot \lambda_a \cdot \eta \cdot m_2$
\end{theorem}
\begin{proof}
We can write MAS series as:
\begin{equation}
\begin{split}
MAS = \lambda_s^2 \cdot \eta \cdot \sum_{k=0}^p v_{n_k}
+ \lambda_a^2 \cdot \eta_a \cdot \sum_{k=0}^p d_k
+\\+ \lambda_s \cdot \lambda_a \cdot \eta_a \cdot \sum_{k=0}^p v_{n_k}
+ \lambda_s \cdot \lambda_a \cdot \eta \cdot \sum_{k=0}^p d_{k}
\end{split}
\end{equation}
This can be rewritten for $p \to \infty$ as:
\begin{equation}
\label{eq:converg_numb}
MAS = \lambda_s^2 \cdot \eta \cdot m_1 + \lambda_a^2 \cdot \eta_a \cdot m_2 + \lambda_s \cdot \lambda_a \cdot \eta_a \cdot m_1 + \lambda_s \cdot \lambda_a \cdot \eta \cdot m_2
\end{equation}
\end{proof}
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{images/toy_f.pdf}
\caption{
Behavior of the three optimizers MAS, ADAM and SGD on the surface defined in Eq.\ref{eq:toy_surface}.
For better visualization the SGD was shifted on X axis of $0.1$.
}
\label{fig:toy}
\end{figure}
\subsection{Geometric explanation}
We can see optimizers as two explorers $w_1$ and $w_2$ who want to explore an environment (the surface of a loss function).
If the two explorers agree to go in a similar direction, then they quickly go in that direction ($w_1 + w_2$).
Otherwise, if they disagree and each prefers a different direction than the other, then they proceed more cautiously and slower ($w_1 + w_2$).
As we can see in Fig.~\ref{fig:intuitive}a, if the directions of the displacement of $w_1$ and $w_2$ are similar then the amplitude of the resulting new displacement $w_1 + w_2$ is increased, however, as shown in Fig.~\ref{fig:intuitive}b, if the directions of the two displacements $w_1$ and $w_2$ are not similar then the amplitude of the new displacement $w_1 + w_2$ has decreased.
In our approach, the sum $w_1+w_2$ is weighted (see red vectors in Fig.~\ref{fig:intuitive}a) so one of the two optimizers SGD or ADAM can become more relevant than the other in the choice of direction for MAS, hence the direction resultant may tend towards one of the two.
In MAS we set the weight of the two contributions so as to have a sum $\lambda_1 + \lambda_2 = 1$ in order to maintain a learning rate of the same order of magnitude.
Another important component that greatly affects the MAS shift module at each training step is its learning rate defined in Eq.~\ref{eq:proposed} which combines $\eta$ and $\eta_a$.
The shifts are scaled using the learning rate, so there is a situation where MAS gets more thrust than the ADAM and SGD starting shifts.
In particular, we can imagine that the displacement vector of ADAM has a greater magnitude than SGD and the learning rate of SGD is greater than that of ADAM.
In this case, the MAS shift has a greater vector magnitude than SGD and a higher ADAM learning rate which can cause a large increase in the MAS shift towards the search of a minimum.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{images/toy_rosenbrook.pdf}
\caption{
Behavior of the three optimizers MAS, ADAM and SGD on the Rosenbrook's surface with $a=1$ and $b=100$}
\label{fig:toy_rosenbrook}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=.8\columnwidth]{images/toy_figd.png}
\caption{
Behavior of the three optimizers MAS, ADAM and SGD on the surface
$z = \frac{\lvert x \lvert} {10} + \lvert y \lvert$ }
\label{fig:toy_another}
\end{figure}
\subsection{Toy examples}
To better understand our proposal, we built a toy example where we highlight the main behaviour of MAS.
More precisely we consider the following example:
\begin{equation}
x = [1,2],~y = [2,4]
\end{equation}
\begin{equation}
p_i = w_1 \cdot(w_2 \cdot x_i)
\end{equation}
\begin{equation}
\label{eq:toy_surface}
\mathcal{L}(w_1, w_2) = \sum_{i=0}^1 (p_i - y_i^2)
\end{equation}
We set $\beta_1=0.9$, $\beta_2=0.999$, $\epsilon=10^{-8}$, $amsgrad=False$, dampening $d=0$, $nesterov=False$ and $\mu=0$.
As we can see in Fig.~\ref{fig:toy} our MAS optimizer goes faster towards the minimum value after only two epochs, SGD is fast at the first epoch, however, it decreases its speed soon after and comes close to the minimum after 100 epochs, ADAM instead reaches its minimum after 25 epochs.
Our approach can be fast when it gets a large $v_k$ from SGD and a large $\eta_a$ from ADAM.
Another toy example can be done with the benchmark Rosenbrook~\cite{rosenbrock1960automatic} function:
\begin{equation}
z = (a - y)^2 + b \cdot (y-x^2)^2
\end{equation}
We set $a=1$ and $b=100$, weight $x=3$ and weight $y=1$, $lr=0.0001$, $epochs=1000$, and default paramiter for ADAM and SGD.
The MAS optimizer sets $\lambda_s = \lambda_a = 0.5$.
The comparative result for the minimization of this function is shown in Fig.~\ref{fig:toy_rosenbrook}.
In this experiment, we can see how by combining the two optimizers ADAM and SGD we can obtain a better result than the single optimizers.
For this function, going from the starting point towards the direction of the maximum slope means moving away from the minimum, and therefore it takes a lot of training epochs to approach the minimum.
Let's use a final toy example to highlight the behavior of the MAS optimizer.
In this case we look for the minimum of the function $z = \frac{\lvert x \lvert} {10} + \lvert y \lvert$.
We set the weights $x = 3$ and $y = 2$, $lr = 0.01$, $epochs = 400$ and use all the default parameters for ADAM and SGD.
MAS assigns the same value $0.5$ for the two lamdas hyper-parameters.
In Fig.~\ref{fig:toy_another} we can see how not all the paths between the paths of ADAM and SGD are the best choice.
Since MAS, as shown in Fig.\ref{fig:intuitive}, goes towards an average direction with respect to that of ADAM and SGD, then in this case ADAM arrives first at the minimum point.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{images/cifar-10-100.pdf}
\caption{On the left some sample images for each of the 10 classes of CIFAR-10 (one class for each row). On the right 10 classes randomly selected from the set of 100 classes of CIFAR-100.}
\label{fig:cifar10-100}
\end{figure}
\section{Datasets}
In this section, we briefly describe the datasets used in the experimental phase.
The \textbf{Cifar10}~\cite{cifar} dataset consists of 60,000 images divided into 10 classes (6000 per class) with a training set size and test set size of 50000 and 10000 respectively.
Each input sample is a low-resolution color image of size $32 \times 32$.
The 10 classes are airplanes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks.
\begin{table}
\caption{Accuracy results on Cifar10, after 6 runs of 200 epochs.}
\label{tab:accuracyCifar10}
\begin{center}
\begin{tabular}{llccccc}
\hline
Name & $\lambda_a$ & $\lambda_s$ & avg. acc & acc max\\
\hline
Resnet18 & & & \\
\hline
Adam & 1 & 0 & 84.68 & 86.24 \\
SGD & 0 & 1 & 78.87 & 79.19 \\
MAS & 0.5 & 0.5 & 85.36 & 85.80 \\
MAS & 0.4 & 0.6 & 85.64 & 86.59 \\
MAS & 0.6 & 0.4 & \textbf{85.89} & 86.56 \\
MAS & 0.7 & 0.3 & 85.39 & 86.09 \\
MAS & 0.3 & 0.7 & 85.57 & \textbf{86.85} \\
\hline
Resnet34 & & & \\
\hline
Adam & 1 & 0 & 82.98 & 83.52 \\
SGD & 0 & 1 & 82.92 & 83.25 \\
MAS & 0.5 & 0.5 & 84.99 & 85.69 \\
MAS & 0.4 & 0.6 & \textbf{85.75} & 86.12\\
MAS & 0.6 & 0.4 & 84.63 & 85.27 \\
MAS & 0.7 & 0.3 & 84.46 & 84.80 \\
MAS & 0.3 & 0.7 & 85.71 & \textbf{86.14} \\
\hline
\end{tabular}
\end{center}
\end{table}
The \textbf{Cifar100} \cite{cifar} dataset consist of 60000 images divided in 100 classes (600 per classes) with a training set size and test set size of 50000 and 10000 respectively.
Each input sample is a $32\times 32$ colour images with a low resolution.
In Fig.~\ref{fig:cifar10-100} we report some representative examples of the two datasets, extracted from a subset of classes.
The \textbf{Corpus of Linguistic Acceptability} (CoLA)~\cite{warstadt2018neural} is another dataset which contains 9594 sentences belonging to training and validation sets, and excludes 1063 sentences belonging to a set of tests kept out. In our experiment, we only used the training set and the test set.
The \textbf{AG’s news corpus}~\cite{agnews,zhang2015character} is the last dataset used in our experiments. It is a dataset that contains news articles from the web subdivided into four classes. It has 30,000 training samples and 1900 test samples.
\begin{table}
\caption{
Accuracy results on Cifar100, after 7 runs of 200 epochs.}
\label{tab:accuracyCifar100}
\begin{center}
\begin{tabular}{llccccc}
\hline
Name & $\lambda_a$ & $\lambda_s$ & avg acc. & acc max\\
\hline
Resnet18 & & & \\
\hline
Adam & 1 & 0 & 49.56 & 50.28 \\
SGD & 0 & 1 & 49.48 & 50.43 \\
MAS & 0.5 & 0.5 & 55.08 & 56.68 \\
MAS & 0.4 & 0.6 & 56.23 & 56.83 \\
MAS & 0.6 & 0.4 & 53.82 & 54.44 \\
MAS & 0.7 & 0.3 & 52.92 & 54.07 \\
MAS & 0.3 & 0.7 & \textbf{58.01} & \textbf{58.48} \\
\hline
Resnet34 & & & \\
\hline
Adam & 1 & 0 & 50.66 & 51.92 \\
SGD & 0 & 1 & 52.81 & 53.45 \\
MAS & 0.5 & 0.5 & 51.52 & 53.26 \\
MAS & 0.4 & 0.6 & 51.73 & 53.48 \\
MAS & 0.6 & 0.4 & 52.15 & 53.95 \\
MAS & 0.7 & 0.3 & \textbf{53.06} & \textbf{54.50} \\
MAS & 0.3 & 0.7 & 51.96 & 53.32 \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Experiments}
The optimizer MAS proposed is a generic solution not oriented exclusively to image analysis, so we conduct experiments on both image classification and text document classification.
By doing so, we are able to give a clear indication of the behavior of the proposed optimizer in different contexts, also bearing in mind that many problems, such as audio recognition, can be traced back to image analysis.
In all the experiments $\beta_1=0.9$, $\beta_2=0.999$, $\epsilon=10^{-8}$, $amsgrad=False$, dampening $d=0$ and $nesterov=False$.
\subsection{Experiments with images}
In this first group of experiments we use two well-known image datasets for: (1) conduct an analysis of the two main parameters of MAS, $\lambda_a$ and $\lambda_s$; (2) compare the performance of MAS with respect to the two starting optimizers SGD and ADAM; (3) analyze the behavior of MAS with different neural models.
The datasets used in this first group of experiments are Cifar10 and Cifar100.
The neural models compared are two ResNet and in particular, we use Resnet18 and Resnet34~\cite{He2015,targ2016resnet}.
We analyzed $\lambda_a$ and $\lambda_s$ when they assume values from the set $\{0.3, 0.4, 0.5, 0.6, 0.7\}$ so that $\lambda_a + \lambda_s = 1$.
The numerical results are all grouped in the two Tabs.~\ref{tab:accuracyCifar10} and ~\ref{tab:accuracyCifar100}.
For Cifar10 we report the average accuracies calculated on 6 runs of each experiment and for 200 epochs.
We set $\eta = 0.001$, momentum $\mu = 0.95$, and batch size equal to 1024.
From the results reported in Tab.~\ref{tab:accuracyCifar10}, we can observe that the best average results are obtained in correspondence with $\lambda_s = 0.6$ and $\lambda_a = 0.4$ but we can also note that all the other settings used for MAS allow obtaining better results than ADAM and SGD for both Resnet18 and Resnet34.
To better understand what happens during the training phase, in Fig.~\ref{fig:cifar10-test-accuracies} we represent the accuracy of the test and the corresponding loss values of the experiments that produced the best results with Resnet18 on Cifar10.
As we can see even though the best execution of ADAM has the lowest loss value, our optimizer with $\lambda_a = 0.3$ and $\lambda_s = 0.7$ offers better testing accuracy.
The other combination represented with $\lambda_a = 0.5 $ and $\lambda_s = 0.5$ is also very similar to the better accuracy of ADAM and in any case always better than the average accuracy produced by ADAM.
So in general we can say that the MAS optimizer leads to a better generalize than the other optimizers used.
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{images/test_accuracies.pdf}\\
\includegraphics[width=0.9\columnwidth]{images/test_loss.pdf}
\caption{Resnet18 test accuracies and test loss of the best results obtained on Cifar10.}
\label{fig:cifar10-test-accuracies}
\end{figure}
For Cifar100 we use the same settings as the Cifar10 experiment except for $\eta = 0.008$ and batch size set to 512.
We can see the results in Tab.~\ref{tab:accuracyCifar100}.
Also for this dataset, we can see that for Resnet18 all MAS configurations work better than SGD and ADAM.
Instead, looking at the results obtained with the Resnet34, we can say that only one configuration of MAS exceeds the average accuracies of SGD and ADAM, but if we look at the maximum accuracy values, more than one configuration of MAS is better than the results available with ADAM and SGD.
In conclusion, as we have seen from the results shown in Tab.~\ref{tab:accuracyCifar10} and Tab.~\ref{tab:accuracyCifar100}, the proposed method leads to a better generalization than the other optimizers used in each experiment.
We get better results both by setting $\lambda_a$ and $\lambda_s$ well, and also even when we don't use the best set of parameters.
\subsection{Experiments with text documents}
In this last group of experiments, we use the two datasets of text documents: CoLA and AG's News.
As neural model, we use a model based on BERT~\cite{devlin2018bert} which is one of the best techniques for working with text documents.
To run fewer epochs we use a pre-trained version~\cite{bert_pretrained} of BERT.
In these experiments, we also introduce the comparison with the AdamW optimizer which is usually the optimizer used in BERT-based models.
For the CoLA dataset we set $\eta = 0.0002$, momentum $\mu = 0.95$, and batch size equal to 100. We ran the experiments 5 times for 50 epochs.
For the AG's News dataset we set the same parameters used for CoLA, but we only run it for 10 epochs because it gets good results in the firsts epochs and also because the dataset is very large and therefore takes more time.
We can see all the results in Tab.~\ref{tab:accuracyCoLA}.
Even for text analysis problems, we can confirm the results of the experiments done on images: although AdamW sometimes has better performances than ADAM, our proposed optimizer performs better than other optimizers used in this paper.
\begin{table}
\caption{Accuracy results of BERT pre-trained on CoLA (50 epochs) and AG's news (10 epochs), after 5 runs.}
\label{tab:accuracyCoLA}
\begin{center}
\begin{tabular}{llccccc}
\hline
Name & $\lambda_a$ & $\lambda_s$ & avg acc. & acc max\\
\hline
CoLA & & & \\
\hline
AdamW & - & - & 78.59 & 85.96 \\
Adam & 1 & 0 & 79.85 & 83.30 \\
SGD & 0 & 1 & 81.48 & 81.78 \\
MAS & 0.5 & 0.5 & 85.92 & 86.72 \\
MAS & 0.4 & 0.6 & 86.18 & \textbf{87.66} \\
MAS & 0.6 & 0.4 & 85.45 & 86.34 \\
MAS & 0.7 & 0.3 & 84.66 & 85.78 \\
MAS & 0.3 & 0.7 & \textbf{86.34} & 86.91 \\
\hline
AG's News & & & \\
\hline
AdamW & - & - & 92.62 & 92.93 \\
Adam & 1 & 0 & 92.55 & 92.67 \\
SGD & 0 & 1 & 91.28 & 91.39\\
MAS & 0.5 & 0.5 & 93.72 & 93.80 \\
MAS & 0.4 & 0.6 & 93.82 & 93.98 \\
MAS & 0.6 & 0.4 & 93.55 & 93.67 \\
MAS & 0.7 & 0.3 & 93.19 & 93.32 \\
MAS & 0.3 & 0.7 & \textbf{93.86} & \textbf{93.99} \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}
In this paper, we introduced MAS (Mixing ADAM and SGD) a new Combined Optimization Method that combines the capability of two different optimizers into one.
We demonstrate by experiments the capability of our proposal to overcome the single optimizers used in our experiments and achieve better performance.
To balance the contribution of the optimizers used within MAS, we introduce two new hyperparameters $\lambda_a$, $\lambda_s$ and show experimentally that in almost all configurations of these parameters, the results are better than the results obtained with the other single optimizers.
In future work, it is possible to change ADAM and SGD and try to mix different optimizers also without the limitations of using only two optimizers.
Another significant future work is to try to change dynamically during the training the influence (lambda hyper-parameters) of the two combined optimizers, in the hypothesis that this can improve further the generalization performance.
{\small
\bibliographystyle{ieeetr}
\bibliography{main}
}
\end{document} | 126,446 |
Looking To Improve Your Wine Knowledge? Start Here With These Tips
Are you so wine savvy that you know everything about it? Nobody does and everyone, including you, can always learn more. Where do your interests lie? This article can help you!
You can preserve the taste and aroma of a wine if you store it properly. Any wine can be negatively affected by temperatures outside its optimal range. Store your wine at 50-55 degrees to allow the flavor time to develop. Specialized wine fridges can be used, or wines can be kept in cool basement spaces.
Reds and whites must be served in correct glassware. Whites need a more fragile glass, while red wine can be served in normal wine glasses. Reds are better in a wide glass. A wider glass will let more air get into the glass, awakening the flavor as it reaches the air. the regions that are growing the wine. Your appreciation of wine will grow as you experience the process of growing and harvesting wine grapes. This will help give you the right understanding, and the right language, to explain these distinctive tastes and aromas to others. Additionally, visiting wine regions can make for spectacular vacations.
If you love wine, plan your next vacation in wine country. Wine country is a beautiful place to visit, and you will gain new appreciation for your favorite wine, as well as insight into its origins.
Keep your wine cellar well stocked. After all, if you only have reds, you won’t be ready for guests. Store sparkling and sweet wines in addition to rich reds and whites so that you always have something your guests will enjoy.
Try different wines when dining out. Choose a unique wine, which can be both fun and exhilarating. This will create a fun atmosphere when it is time to taste the wine.
It is helpful to learn how to get the label of a bottle of wine. A simple approach is to set the wine bottle in a hot oven and, using oven mitts, begin at the corner to peel back the label..
Select the right stemware for the wine you are serving. The stemware should look the part and sparkle. Keep your stemware clean and prepared for the next tasting. If your stemware has chips or is out of date, it’s time to go shopping.
Spanish wines are easy to keep fresh, but the specific method will vary with the type. For instance, Rioja is still great up to seven years after it is bottled. Keep it in a dark, cool location and pop it open when you’re ready for a tasty treat.
Don’t just drink your wine; try cooking with it, as well. You could, for instance, have a nice steak dinner cooked with some red wine. White wine pairs well with such seafood items as scallops or fish. A little wine is a fantastic complement to the meal you are consuming.
If you are drinking wine at an event, there may be a toast. Because of this, you may be clinking your glass often..
Wine is great for helping you relax or to complement a great meal. Understanding wine is a very useful, a fact which this article has illustrated. Apply the tips in this piece, and begin maximizing your experience with wine.
Understanding has not boundaries, if you want to know more
Click on right here | 28,012 |
\begin{document}
\maketitle
\begin{abstract}
We couple the mixed variational problem for the generalized Hodge-Helmholtz or Hodge-Laplace equation posed on a bounded three-dimensional Lipschitz domain with the first-kind boundary integral equation arising from the latter when constant coefficients are assumed in the unbounded complement. Recently developed Calder\'on projectors for the relevant boundary integral operators are used to perform a symmetric coupling. We prove stability of the coupled problem away from resonant frequencies by establishing a generalized G{\aa}rding inequality (T-coercivity). The resulting system of equations describes the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic body possibly having a ``rough" surface. The low-frequency robustness of the potential formulation of Maxwell's equations makes this model a promising starting point for Galerkin discretization.
\iffalse
We consider the scattering of monochromatic electromagnetic waves at a bounded inhomogeneous isotropic simply connected body with a Lipschitz boundary. We couple the mixed formulation of the weak variational problem obtained from the potential formulation of Hodge-Helmholtz and Hodge-Laplace equations with the first-kind boundary integral equations arising from the latter in the unbounded homogeneous region outside the scatterer. Recently developed Calder\'on projectors for the Hodge-Helmholtz and Hodge-Laplace operators are used to perform the symmetric coupling. We prove stability by establishing a generalized G{\aa}rding inequality.
\fi
\end{abstract}
\begin{keywords}
Maxwell, electromagnetism, scattering, Hodge-laplace, Hodge-Helmholtz, Hodge decomposition, Helmholtz decomposition, Calder\'on projector, symmetric coupling, T-coercivity
\end{keywords}
\begin{AMS}
35Q61, 35Q60, 65N30, 65N38, 78A45, 78M10, 78M15
\end{AMS}
\section{Introduction}\label{sec: Introduction}
\par Inside a bounded inhomogeneous isotropic physical body $\Omega_s$, the potential formulation of Maxwell's equations in frequency domain driven by a source current $\mathbf{J}$ with angular frequency $\omega>0$ reads \cite{chew2014vector}
\begin{align}
&\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}} +i\omega\epsilon(\mathbf{x}) \nabla V - \omega^2 \epsilon(\mathbf{x})\,\mathbf{U} = \mathbf{J}\nonumber\\
&\text{div}\,\bra{\epsilon(\mathbf{x})\mathbf{U}}+i\omega V =0,\label{eq: Lorentz gauge}
\end{align}
where the Lorentz gauge relates the scalar potential $V$ to the vector potential $\mathbf{U}$ in \eqref{eq: Lorentz gauge}. Elimination of $V$ using this relation leads to the Hodge-Helmholtz equation
\begin{equation*}
\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}} - \epsilon(\mathbf{x})\,\nabla\,\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{U}} - \omega^2\epsilon(\mathbf{x})\,\mathbf{U} = \mathbf{J}. \label{eq: div Maxwell elimination}
\end{equation*}
\par Away from the source current, in the unbounded region $\Omega':=\mathbb{R}^3 \backslash \overline{\Omega}_s$ outside the scatterer, where we assume a homogeneous material with scalar constant permeability $\mu_0$ and dielectric permittivity $\epsilon_0$, equation \eqref{eq: div Maxwell elimination} reduces to
\begin{equation*}
\mathbf{curl}\,\mathbf{curl}\,\mathbf{U} - \eta\,\nabla\,\text{div}\,\mathbf{U} - \kappa^2\mathbf{U} = 0,
\end{equation*}
with constant coefficients $\eta = \mu_0\epsilon_0^2$ and $\kappa^2 = \mu_0\epsilon_0\omega^2$.
\par The material coefficients are assumed to be bounded in $\mathbb{R}^3$, i.e. $\mu,\epsilon\in L^{\infty}(\mathbb{R}^3)$. In a non-dissipative medium, the functions $\mu$ and $\epsilon$ are real-valued and uniformly positive. Dissipative effects are captured by adding non-negative imaginary parts to the coefficients \cite[Sec. 1.1.3]{assous2018mathematical}. We follow \cite{hazard1996solution} and explicitly suppose that
\begin{align*}
&0<\mu_{\text{min}}\leq\mathfrak{Re}(\mu)\leq\mu_{\text{max}},& &0\leq\mathfrak{Im}(\mu) ,\\
&0<\epsilon_{\text{min}}\leq\mathfrak{Re}(\epsilon)\leq\epsilon_{\text{max}},& &0\leq\mathfrak{Im}(\epsilon).\\
&0 \leq \mathfrak{Re}\bra{\kappa^2}, & &0\leq\mathfrak{Im}\bra{\kappa^2}.
\end{align*}
\par Let $\Omega_s\subset\mathbb{R}^3$ be a bounded domain with Lipschitz boundary $\Gamma=:\partial\Omega$ \cite[Def. 2.1]{steinbach2007numerical}. We suppose for simplicity that its de Rham cohomology is trivial. For given data $\mathbf{J}\in\mathbf{L}^{2}(\Omega_s)$, $\mathbf{g}_R\in\mathbf{H}^{-1/2}(\text{div}_{\Gamma})$, $g_n\in H^{-1/2}(\Gamma)$, $\zeta_D\in H^{1/2}(\Gamma)$ and $\bm{\zeta}_t\in\mathbf{H}^{-1/2}(\text{curl}_{\Gamma})$, we are interested in the following PDEs in $\mathbb{R}^3$:
\newline\newline
\begin{greenFrameTransmission}
\begin{subequations}\label{eq: transmission problem}
\textbf{\color{charcoal}Volume equations}
\begin{align}
&\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}} - \epsilon(\mathbf{x})\,\nabla\,\text{div}\, \bra{\epsilon(\mathbf{x})\,\mathbf{U}} - \omega^2\epsilon(\mathbf{x})\,\mathbf{U} = \mathbf{J}, &\text{in }\Omega_s,\label{eq: strong from coupled sys a}\\
&\mathbf{curl}\,\mathbf{curl}\,\uext - \eta\,\nabla\text{div}\,\uext - \kappa^2\,\uext = 0, &\text{in }\Omega',\label{eq: strong from coupled sys b}
\end{align}
\end{subequations}
\textbf{\color{charcoal}Transmission conditions}
\begin{subequations}
\begin{align}
\gamma^-_{R,\mu}(\mathbf{U}) = \gamma^{+}_R\uext+\mathbf{g}_R,\quad \gamma^{-}_{n,\epsilon}(\mathbf{U}) = \gamma^+_n(\mathbf{U}^{\text{ext}})&+g_n, &\text{on }\Gamma,\label{eq: strong from coupled sys c}\\
\gamma^-_{D,\epsilon}(\mathbf{U}) = \eta\, \gamma^{+}_D \uext + \zeta_D,\quad \left[\gamma_t\mathbf{U}\right]_{\Gamma} = \bm{\zeta}_t, \quad\quad\quad\quad & &\text{on }\Gamma, \label{eq: strong from coupled sys d}
\end{align}
\end{subequations}
\end{greenFrameTransmission}
\newline\newline
where the traces are defined for a smooth vector-field $\mathbf{U}$ by
\begin{align*}
&\gamma^-_{R,\mu}(\mathbf{U}):= -\gamma^-_{\tau}\bra{\mu^{-1}\bra{\mathbf{x}}\,\mathbf{curl}(\mathbf{U})}, & &
\gamma^+_{R}(\uext):= -\gamma^+_{\tau}\bra{\mathbf{curl}\bra{\uext}},\\
&\gamma^-_{D,\epsilon}(\mathbf{U}):=\gamma^-\bra{\text{div}\bra{\epsilon\bra{\mathbf{x}}\,\mathbf{U}}}, & &
\gamma^+_{D}(\uext):=\gamma^+\bra{\text{div}\bra{\uext}},\\
&\gamma^{-}_{n,\epsilon}(\mathbf{U}):= \gamma^-_n(\epsilon\bra{\mathbf{x}}\,\mathbf{U})& &
\gamma^\pm_{t}\bra{\mathbf{U}} := \mathbf{n}\times\bra{\gamma^\pm_{\tau}\bra{\mathbf{U}}}.
\end{align*}
\par We used in those definitions the classical traces
\begin{align*}
&\gamma\bra{\mathbf{U}} := \mathbf{U}\big\vert_{\Gamma},
&& \gamma_n\bra{\mathbf{U}} := \gamma\bra{\mathbf{U}}\cdot \mathbf{n},
&&\gamma_{\tau}\bra{\mathbf{U}} := \gamma\bra{\mathbf{U}}\times \mathbf{n}.
\end{align*}
\par Each of these traces can be extended by continuity to larger Sobolev spaces. The more detailed functional analysis setting in which they must be considered will be reviewed in the next section.
The equations \eqref{eq: strong from coupled sys c} and \eqref{eq: strong from coupled sys d} are transmission conditions for Hodge--Helmholtz and Hodge--Laplace problems \cite[Sec. 2.1.2]{hazard1996solution}. In literature, condition \eqref{eq: strong from coupled sys c} is labeled as ``magnetic'', while \eqref{eq: strong from coupled sys d} is referred to as ``electric'' (simply because one recovers the magnetic field by taking the curl of the potential $\mathbf{U}$). It is very well possible to ``guess" these transmission conditions either by glancing at equation \eqref{eq: strong from coupled sys a} or by translating the classical boundary conditions for the electric and magnetic fields to the potential formulation \cite{chew2014vector}. However, it is emphasized in \cite{claeys2017first} and \cite{claeys2018first} that the traces also appear formally in Green's formula
\begin{multline*}
\int_{\Omega'}\mathbf{U}\cdot\bra{-\Delta_{\eta}\mathbf{V}}\dif\mathbf{x} -\mathbf{V}\cdot\bra{-\Delta_{\eta}\mathbf{U}}\dif\mathbf{x} = -\eta\langle\gamma^+_n\mathbf{U},\gamma^+_D\mathbf{V}\rangle +\eta\langle\gamma^+_D\mathbf{U},\gamma^+_n\mathbf{V}\rangle\\
+\langle\gamma^+_R\mathbf{V},\gamma^+_t\mathbf{U}\rangle
-\langle\gamma^+_R\mathbf{U},\gamma^+_t\mathbf{V}\rangle,
\end{multline*}
where $-\Delta_{\eta} := \mathbf{curl}\,\mathbf{curl}-\eta\,\text{div}\,\nabla$ is the Hodge-Laplace operator.
\par For positive frequencies $\omega>0$, we supplement \eqref{eq: strong from coupled sys a}-\eqref{eq: strong from coupled sys d} with the variants of the Silver-Muller's radiation condition imposed at infinity provided in \cite{hazard1996solution}. In the static case where $\kappa=\omega=0$, we seek a solution in an appropriate weighted Sobolev space that accounts for decay conditions \cite[Sec. 2.5]{schwarz2006hodge}.
\begin{remark}
When derived from Maxwell's equations stated in terms of the magnetic and electric fields, the classical wave equation for an electric field $\mathbf{E}$ reads
\begin{equation}\label{eq: classical electric wave}
\mathbf{curl}\bra{\mu^{-1}\bra{\mathbf{x}}\,\mathbf{curl}\,\mathbf{E}} - \kappa^2\,\epsilon\bra{\mathbf{x}}\,\mathbf{E} = \mathbf{J}.
\end{equation}
The regularizing term $\epsilon\nabla\text{div}\bra{\epsilon \mathbf{U}}$ which appears in \eqref{eq: strong from coupled sys a}, but not in \eqref{eq: classical electric wave}, makes for a significant structural difference \cite{hazard1996solution}. For suitable boundary conditions, the zero-order term $\omega^2\epsilon\mathbf{U}$ in \eqref{eq: transmission problem} is a compact perturbation in the weak formulation of the Hodge-Helmholtz equation. Ergo, coercivity of the associated boundary value problem is preserved in the low frequency limit $\omega\rightarrow 0$. This is not the case for the ``Maxwell operator" found on the left hand side of \eqref{eq: classical electric wave}, whose associated scattering equation is characterized by an ``incessant conversion" between electric and magnetic energies that play symmetric roles \cite{buffa2003galerkin}. Functionally, the infinite dimensional kernel of the curl operator thwart compactness of the embedding $\mathbf{H}\bra{\mathbf{curl},\Omega_s}\hookrightarrow\mathbf{L}^2\bra{\Omega_s}$. This is different from the weak variational formulation of the scalar Helmholtz equation $-\Delta u -\kappa^2 u = f$. In that model of acoustic scattering, potential energy turns out to be a compact perturbation of the kinetic energy due to by Rellich's compact embedding $H^1\bra{\Omega_s}\hookrightarrow L^2\bra{\Omega_s}$.
\end{remark}
\begin{remark}
It is stressed in \cite{chew2014vector} that from the rapid development in quantum optics emerged the need for electromagnetic models valid in both classical and quantum regimes. Robustness of the potential formulation of Maxwell's equations in the low frequency limit makes it a promising candidate for bridging physical scales.
\end{remark}
\begin{remark}
The terminology used above is rooted in geometry. The equations \eqref{eq: strong from coupled sys a}-\eqref{eq: strong from coupled sys b} contain generalized instances of the Hodge--Helmholtz operator $-\Delta - \kappa^2\id = \delta\dif+ \dif\,\delta - \kappa^2\id$ as it applies to differential 1-forms defined over 3D differentiable manifolds. When $\omega=\kappa=0$, the left hand sides reduce to applications of the Hodge-Laplace operator. We refer to \cite{Hiptmair2015} and \cite{hiptmair2002finite} for a thorough introduction to the formulation of Maxwell's equations in terms of differential/integral forms.
\end{remark}
\begin{remark}
Boundary integral operators of the \emph{second-kind} were extensively studied in the literature devoted to the Hodge-Laplace and Hodge-Helmholtz operators acting on differential forms over smooth manifolds (e.g. \cite{mitrea2016hodge}, \cite{mitrea2001layer}, \cite{schwarz2006hodge} and \cite{marmolejo2012transmission}). However, little attention was paid to the formulation of Hodge-Helmholtz/Laplace boundary value problems as \emph{first-kind} boundary integral equations. Only recently, a boundary integral representation formula for Hodge-Helmholtz/Laplace equation in three-dimensional Lipschitz domains was derived in \cite{claeys2017first} which leads to boundary integral operators of the first-kind inducing bounded and coercive sesquilinear forms in the natural energy spaces for that equation. These innovative investigations are particularly relevant to the numerical analysis community. Operators admitting natural variational formulations in well-known energy trace spaces via duality are appealing for the development and numerical analysis of new Galerkin discretizations. For the case $\kappa^2=0$ of the Hodge--Laplace operator in 3D, a thorough \emph{a priori} analysis of a Galerkin BEM was already proposed in \cite{claeys2018first} with additional experimental evidence.
\end{remark}
\par In the following, we couple the mixed formulation of the weak variational problem associated to \eqref{eq: strong from coupled sys a} with the first-kind boundary integral equation arising from \eqref{eq: strong from coupled sys b} using these recently developed Calder\'on projectors for the Hodge--Helmholtz and Hodge-Laplace operators. This paves the way for the design of finite element methods discretizing Hodge-Laplace and Hodge-Helmholtz transmission problems. The proof of the well-posedness of the coupled problem relies on T-coercivity (c.f. \cite{ciarlet2012t}).
\section{Preliminaries} In the subsequent analysis, we will make use of various spaces that have become classical in the literature concerned with electromagnetism. Development of the trace-related theory for Lipschitz domains can be followed in \cite{buffa2001traces_a}, \cite{buffa2001traces_b} and \cite{buffa2002traces}. We summarize its details to fix notation and recall important results. In the next subsections, we slightly generalize the traces to account for the varying coefficients of \eqref{eq: strong from coupled sys a} and adapt them to the system of equations at hand. In Section \ref{sec: Integral operators}, we extend the analysis performed in \cite{hiptmair2003coupling} for the classical electric wave equation to the boundary integral operators arising from Hodge-Helmholtz and Hodge-Laplace problems. In this section, $\Omega$ can denote either $\Omega_s$ or $\Omega'$.
\subsection{Volume function spaces}
\par As usual, $L^2(\Omega)$ and $\mathbf{L}^2(\Omega)$ denote the Hilbert spaces of square integrable scalar and vector-valued functions defined over $\Omega$. We denote their inner products using round brackets, e.g. $(\cdot,\cdot)_{\Omega}$. Similarly, for $k\in \mathbb{N}$, $H^k(\Omega)$ and $\mathbf{H}^k(\Omega)$ refer to the corresponding Sobolev spaces. We write $C^{\infty}_0(\Omega)$ for the space of smooth compactly supported function in $\Omega$, but denote by $\mathscr{D}(\Omega)^3$ the analogous space of vector fields to simplify notation. Their closures $H^1_0(\Omega)$ and $\mathbf{H}^1_0(\Omega)$ in the norms of $H^1(\Omega)$ and $\mathbf{H}^1(\Omega)$, respectively, are the kernels of the scalar and vector-valued Dirichlet traces, which we both denote $\gamma$ alike. $C^{\infty}(\overline{\Omega})$ is defined as the space of uniformly continuous functions over $\Omega$ that have uniformly continuous derivatives of all order. The Banach spaces
\begin{align*}
\mathbf{H}(\text{div}, \Omega) &:=\{\mathbf{U}\in L^2(\Omega) \,\vert\, \text{div}(\mathbf{U})\in L^2(\Omega)\},\\
\mathbf{H}(\epsilon;\text{div}, \Omega) &:=\{\mathbf{U}\in L^2(\Omega) \,\vert\, \epsilon (\mathbf{x})\,\mathbf{U}\in \mathbf{H}(\text{div}, \Omega)\},\\
\mathbf{H}(\mathbf{curl}, \Omega) &:=\{\mathbf{U}\in L^2(\Omega) \,\vert\, \boldcurl(\mathbf{U})\in L^2(\Omega)\},\\
\mathbf{H}\bra{\nabla \text{div},\Omega} &:= \{\mathbf{U}\in \mathbf{H}\bra{\text{div},\Omega} \,\vert\, \text{div}(\mathbf{U})\in H^1(\Omega)\},\\
\mathbf{H}\bra{\epsilon;\nabla \text{div},\Omega_s} &:= \{\mathbf{U}\in \mathbf{L}^2(\Omega)\,\vert\,\epsilon(\mathbf{x})\,\mathbf{U}\in \mathbf{H}\bra{\nabla \text{div},\Omega} \},\\
\mathbf{H}(\mathbf{curl}^2,\Omega)&:=\{\mathbf{U}\in \mathbf{H}(\mathbf{curl}, \Omega) \,\vert\, \mathbf{curl}(\mathbf{U})\in \mathbf{H}(\mathbf{curl}, \Omega)\},\\
\mathbf{H}(\mu^{-1};\mathbf{curl}^2,\Omega)&:=\{\mathbf{U}\in \mathbf{H}(\mathbf{curl}, \Omega) \,\vert\, \mu^{-1}\,\mathbf{curl}(\mathbf{U})\in \mathbf{H}(\mathbf{curl}, \Omega)\},
\end{align*}
equipped with the obvious graph norms will prove to be important.
The variational space for the primal variational formulation of the classical and generalized Hodge--Helmholtz/Laplace operator is given by
\begin{align*}
\mathbf{X}(\Delta,\Omega)&:=\mathbf{H}(\mathbf{curl}^2,\Omega)\cap \mathbf{H}\bra{\nabla \text{div},\Omega}.
\end{align*}
A subscript is used to identify spaces of locally integrable functions/vector fields, e.g. $U\in L^2_{\text{loc}}(\Omega)$ if and only if $\phi U$ is square-integrable for all $\phi\in C^{\infty}_0(\mathbb{R}^3)$. We denote with an asterisk the spaces of functions with zero mean, e.g. $H^1_*(\Omega)$.
\subsection{Trace spaces}
\par Rademacher's theorem \cite[Thm. 3.1.6]{federer2014geometric} guarantees that the boundary $\Gamma=:\partial\Omega$ of a Lipschitz domain $\Omega$ admits a surface measure $\sigma$ and an essentially bounded unit normal vector field $\mathbf{n}\in \mathbf{L}^{\infty}(\Gamma)$ directed toward the exterior of $\Omega$. These ingredients warrant Gau{\ss}' formulae \cite[Thm. 3.34]{mclean2000strongly}.
\subsubsection{Classical traces}\label{sec: classical traces}
\par For $\mathbf{U},\mathbf{V}\in\mathbf{C}^{\infty}(\overline{\Omega})$ and $P\in C^{\infty}(\overline{\Omega})$, the two identities $\text{div}(\mathbf{U}P)=\text{div}(\mathbf{U})\,P+\mathbf{U}\cdot\nabla P$ and $\text{div}(\mathbf{U}\times\mathbf{V})=\mathbf{curl}(\mathbf{U})\cdot\mathbf{V}-\mathbf{curl}(\mathbf{V})\cdot\mathbf{U}$ hold; therefore, the divergence theorem yields ---whenever the integrals are defined--- Green's formulae ($+$ for $\Omega=\Omega_s$)
\begin{subequations}
\begin{gather}
\pm\int_{\Omega} \text{div}(\mathbf{U})\, P + \mathbf{U}\cdot\nabla P \, \dif \mathbf{x} = \int_{\Gamma}\gamma \bra{P} \gamma_n\bra{\mathbf{U}}\dif \sigma,\label{IBP div}\\
\pm\int_{\Omega}\mathbf{U}\cdot\mathbf{curl\,}(\mathbf{V}) -\mathbf{curl\,}(\mathbf{U})\cdot\mathbf{V}\dif \mathbf{x}= \int _{\Gamma}\gamma\bra{\mathbf{V}} \cdot \gamma_{\tau}\bra{\mathbf{U}}\dif \sigma\label{IBP curl},
\end{gather}
\end{subequations}
where \eqref{IBP curl} is valid since $\gamma\bra{\mathbf{U}\times\mathbf{V}}\cdot\mathbf{n} = -(\gamma\bra{\mathbf{U}}\times\mathbf{n})\cdot\gamma\bra{\mathbf{V}}$ is defined almost everywhere. Since the unique extension $\gamma:H_{\text{loc}}^1(\Omega)\rightarrow H^{1/2}(\Gamma)$ of the Dirichlet trace is a bounded operator with a continuous right-inverse $\mathcal{E}$ \cite[Thm. 3.37]{mclean2000strongly}, these traces can be extended by continuity to bounded operators $\gamma_{n}:\mathbf{H}_{\text{loc}}\bra{\text{div},\Omega}\rightarrow H^{-1/2}(\Gamma)$ and $\gamma_{\tau}:\mathbf{H}_{\text{loc}}\bra{\mathbf{curl},\Omega}\rightarrow \mathbf{H}^{-1/2}(\Gamma)$ with null spaces $\ker(\gamma_n)=\mathbf{H}_0(\text{div},\Omega):=\overline{\mathscr{D}(\Omega)^3}^{\mathbf{H}(\text{div},\Omega)}$ and $\ker(\gamma_\tau)=\mathbf{H}_0(\mathbf{curl},\Omega):=\overline{\mathscr{D}(\Omega)^3}^{\mathbf{H}(\mathbf{curl},\Omega)}$ \cite[Chap. 2]{girault2012finite}. Evidently, these extensions generalize \eqref{IBP div} to functions $\mathbf{U}\in\mathbf{H}(\text{div},\Omega)$, $P\in H^1(\Omega)$ \cite[Thm. 3.24]{monk2003finite} and \eqref{IBP curl} to $\mathbf{U}\in\mathbf{H}(\mathbf{curl},\Omega)$, $\mathbf{V}\in \mathbf{H}^1(\Omega)$ \cite[Thm. 3.29]{monk2003finite}, where the boundary terms are to be understood as duality parings $\langle\cdot,\cdot\rangle_{\Gamma}$ with pivot spaces $L^2(\Gamma)$ and ${\mathbf{L}^2}(\Gamma)$, respectively.
\par While the normal component trace $\gamma_n$ described thus is seen to be surjective \cite[Cor. 2.8]{girault2012finite}, it is evident from $\bra{\mathbf{v}\times \mathbf{n}}\cdot\mathbf{n} = 0$ $ \forall \mathbf{v}\in \mathbf{L}^2(\Gamma)$ that the image of the tangential trace $\gamma_{\tau}$ acting on $\mathbf{H}_{\text{loc}}\bra{\mathbf{curl},\Omega}$ is a tangential \textit{proper} subspace of $\mathbf{H}^{-1/2}(\Gamma)$. Naturally, the same holds true for the extension $\gamma_t:\mathbf{H}_{\text{loc}}\bra{\mathbf{curl},\Omega}\rightarrow \mathbf{H}^{-1/2}(\Gamma)$ of the tangential components trace $\gamma_t(\mathbf{U}):=\mathbf{n} \times \gamma_\tau(\mathbf{U})$.
\par Tangential differential operators are required to remedy this problem. For $\bm{\xi}\in \mathbf{H}^{1/2}(\Gamma)$, let $\text{curl}_{\Gamma}(\bm{\xi})\in H^{-1/2}(\Omega)$ be uniquely determined by
\begin{equation*}
\langle\gamma(V),\text{curl}_{\Gamma}(\bm{\xi})\rangle_{\Gamma}=\langle \gamma_\tau(\nabla V), \bm{\xi}\rangle_{\Gamma},\quad \forall V \in C^{\infty}(\overline{\Omega}).
\end{equation*}
As $\mathbf{curl}\circ\nabla = 0$, $\nabla\bra{\mathbf{H}_{\text{loc}}^1(\Omega)}\subset\mathbf{H}_{\text{loc}}\bra{\mathbf{curl},\Omega}$, and the operator $\gamma_{\tau}\circ\nabla:H_{\text{loc}}^1(\Omega)\rightarrow \mathbf{H}^{-1/2}(\Omega)$ is bounded accordingly. In that sense, $\text{curl}_{\Gamma}:\mathbf{H}^{1/2}(\Gamma)\rightarrow H^{-1/2}(\Gamma)$ is adjoint to the vectorial tangential curl operator $\mathbf{curl}_{\Gamma}:=\gamma_{\tau}\circ\nabla\circ\mathcal{E}:H^{1/2}(\Gamma)\rightarrow \mathbf{H}^{-1/2}(\Gamma)$, whose definition is independent of the choice of right-inverse since $\mathbf{H}^1_0(\Omega)\subset \ker(\gamma_{\tau}\circ \nabla)$. Concretely, Green's formulae show that $\text{curl}_{\Gamma}= \gamma_n\,\circ\, \mathbf{curl}\,\circ \mathcal{E}$. Independence of this expression from the choice of lifting $\mathcal{E}$ is guaranteed by the inclusion of $\mathbf{H}^1_0(\Omega)$ in $\ker\bra{\gamma_{n}\circ\mathbf{curl}}$. Similarly, the tangential divergence $\text{div}_{\Gamma}:\mathbf{H}^{1/2}(\Gamma)\rightarrow H^{-1/2}(\Omega)$, defined as the rotated operator $\text{div}_{\Gamma}(\mathbf{p}):=\text{curl}_{\Gamma}\bra{\mathbf{n}\times\mathbf{p}}$, is adjoint to the negative surface gradient $\nabla_{\Gamma}:=\gamma_t\circ\nabla\circ\mathcal{E}$, that is $\langle \nabla_{\Gamma}\,q ,\mathbf{p} \rangle_{\tau} = -\langle\, q, \text{div}_{\Gamma}\bra{\mathbf{p}}\rangle$.
\par The space of traces $\mathbf{H}_T^{1/2}(\Gamma):=\gamma_t(\mathbf{H}_{\text{loc}}^1(\Omega))$ and of rotated traces $\mathbf{H}^{1/2}_R(\Gamma):=\mathbf{n}\times \mathbf{H}^{1/2}_T(\Gamma)$ are complete when equipped with the norms
\begin{align*}
\norm{\mathbf{v}}_{\mathbf{H}^{1/2}_T(\Gamma)}&:=\inf\{\mathbf{U}\in\mathbf{H}_{\text{loc}}^1(\Omega)\,\vert\, \gamma_t(\mathbf{U})=\mathbf{v}\}, &
\norm{\mathbf{u}}_{\mathbf{H}^{1/2}_R(\Gamma)}&:= \norm{\mathbf{n}\times\mathbf{u}}_{\mathbf{H}^{1/2}_T(\Gamma)},
\end{align*}
that enforce continuity of the traces \cite[def. 2.2]{buffa2002traces}.
\begin{lemma}[{See \cite[Lem. 3.2]{hiptmair2003coupling}}]\label{lem: Rellich for boundary}
The embeddings $H^{1/2}_T(\Gamma),\,H^{1/2}_R(\Gamma)\hookrightarrow \mathbf{L}_{t}^2(\Gamma)$ are compact.
\end{lemma}
Based on these intermediate spaces, we define subspaces
\begin{subequations}
\begin{align}
\mathbf{H}^{-1/2}(\text{curl}_\Gamma,\Gamma) &:= \{\bm{\xi}\in \mathbf{H}^{-1/2}_T(\Omega)\,\vert\, \text{curl}_{\Gamma}(\bm{\xi})\in H^{-1/2}(\Gamma)\},\label{H(curl) trace space}\\
\mathbf{H}^{-1/2}(\text{div}_\Gamma,\Gamma) &:= \{\mathbf{p}\in \mathbf{H}^{-1/2}_R(\Omega)\,\vert\, \text{div}_{\Gamma}(\mathbf{p})\in H^{-1/2}(\Gamma)\},\label{H(div) trace space}
\end{align}
\end{subequations}
onto which $\gamma_t$ and $\gamma_\tau$ are continuous and surjective \cite[Thm. 4.1]{buffa2002traces}, respectively. These subspaces can be put in duality with a pairing $\langle\cdot,\cdot\rangle_{\tau}$ for which $\mathbf{L}_{t}^2(\Gamma)$, the space of square integrable tangent vector fields on $\Gamma$, act as pivot space \cite[Sec. 5]{buffa2002traces}. Then, the extension of \eqref{IBP curl} to any
$\mathbf{U}, \mathbf{V}\in \mathbf{H}(\mathbf{curl},\Omega)$ reads \cite[Thm. 3.31]{monk2003finite}
\begin{equation}
\pm\int_{\Omega}\mathbf{U}\cdot\mathbf{curl\,}(\mathbf{V}) -\mathbf{curl\,}(\mathbf{U})\cdot\mathbf{V}\dif \mathbf{x}= \langle\gamma_{t}\bra{\mathbf{V}}, \gamma_{\tau}\bra{\mathbf{U}}\rangle_{\tau}. \label{IBP curl ext}
\end{equation}
\par Finally, upon defining $\gamma_R:=-\gamma_{\tau}\circ\textbf{curl}:\mathbf{H}_{\text{loc}}(\textbf{curl}^2,\Omega)\rightarrow \mathbf{H}^{-1/2}(\text{div}_\Gamma,\Gamma)$, which satisfies for all $\mathbf{V}\in\mathbf{H}(\mathbf{curl},\Omega)$ the crucial integral identity
\begin{equation}
\pm\int_{\Omega}\mathbf{curl}\,\mathbf{curl}\,\mathbf{U}\cdot\mathbf{V} - \mathbf{curl}\,\mathbf{U}\cdot \mathbf{curl}\,\mathbf{V}\dif\mathbf{x} = \langle \gamma_t(\mathbf{V}),\gamma_R(\mathbf{U}) \rangle_{\tau},\label{curl curl integral identity}
\end{equation}
and $\gamma_D:=\gamma\circ\text{div}:\mathbf{H}_{\text{loc}}(\nabla\text{div},\Omega)\rightarrow H^{1/2}(\Gamma)$, we are equipped with a full set of traces to tackle Hodge--Laplace and Hodge--Helmholtz problems.
We indicate with curly brackets the average
\begin{equation*}
\{\gamma_{\bullet}\}:= \frac{1}{2}\bra{\gamma_{\bullet}^+ + \gamma_{\bullet}^-}
\end{equation*}
of a trace and with square brackets its jump
\begin{equation*}
[\gamma_{\bullet}]:=\gamma_{\bullet}^- - \gamma_{\bullet}^+
\end{equation*}
over the interface $\Gamma$, $\bullet=R$, $D$, $t$, $\tau$, or $n$. Analogous notation will be used for the compounded traces introduced in the next section.
\begin{warning}
Notice the sign in the jump $[\gamma]=\gamma^- - \gamma^+$, which is often taken to be the opposite in the literature!
\end{warning}
\begin{lemma}[{See \cite[Lem. 6.4]{claeys2017first}}]\label{lem: properties of surface div and curl}
The surface divergence extends to a continuous surjection $\text{\emph{div}}_{\Gamma}:\mathbf{H}^{-1/2}(\text{\emph{div}}_\Gamma,\Gamma)\rightarrow H^{-1/2}_*(\Gamma)$, while $\mathbf{curl}_{\Gamma}:H^{1/2}_*\rightarrow\mathbf{H}^{-1/2}(\text{\emph{div}}_\Gamma,\Gamma)$ is a bounded injection with closed range such that $\mathbf{curl}_{\Gamma}(\xi)=\nabla_{\Gamma}(\xi)\times\mathbf{n}$ for all $\xi\in H^{1/2}(\Gamma)$. They satisfy $\text{\emph{div}}_{\Gamma}\circ\mathbf{curl}_{\Gamma} = 0$.
\end{lemma}
\iffalse
\subsubsection{Hilbert spaces for inhomogeneous isotropic bodies}\label{subsec: traces for isotropic}
The domain of the generalized Hodge-Helmholtz/Laplace operator needs to be specialized to a functional space accounting for the lack of regularity of the material coefficients.
Solutions of the classical electric wave equation are usually interpreted as generalized functions \cite{buffa2003galerkin}. This means that the quantity $\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}}$ is viewed as the linear functional satisfying
\begin{equation*}
\int_{\Omega_s\cup\Omega'}\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}}\cdot\mathbf{V}\dif\mathbf{x} = \int_{\Omega_s\cup\Omega'}\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}\cdot\mathbf{curl}\,\bra{\mathbf{V}}\dif\mathbf{x}
\end{equation*}
for all $\mathbf{V}\in \mathscr{D}(\Omega_s)^3$, where the integrals are understood as duality pairing. In particular, when $\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}\in\mathbf{H}\bra{\mathbf{curl},\Omega_s\cup\Omega'}$, Green's formula \eqref{curl curl integral identity} states that \emph{in the sense of distribution}
\begin{equation}\label{eq: curl curl in the sense of distribution}
\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}} =\, \mathsf{T}\bra{\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\,\mathbf{curl}\,\mathbf{U}}\Big\vert_{\Omega_s}} + \,\gamma_{R,\mu}\,\bra{\mathbf{U}}\bm{\delta}_{\Gamma}\circ\gamma_t,
\end{equation}
where $\mathsf{T}:\mathbf{L}(\Omega_s)^2\rightarrow \bra{\mathbf{L}(\Omega_s)^2}'$ is the Riesz map and the restriction is simply a convenient notation to distinguish the left hand side from its incarnation as a square-integrable vector-field. Here, the boundary term is defined for $\mathbf{p}\in\mathbf{H}^{-1/2}(\text{div}_\Gamma,\Gamma)$ as the dual element satisfying
\begin{equation*}
\int_{\Gamma}\mathbf{p}\,\bm{\delta}_{\Gamma}\cdot\bm{\xi}\dif\mathbf{x} := \langle \mathbf{p}, \bm{\xi} \rangle_{\tau}
\end{equation*}
for all $\bm{\xi}\in \mathbf{H}^{-1/2}(\text{curl}_\Gamma,\Gamma)$. This justifies the introduction of the Hilbert space
\begin{equation*}
\mathbf{H}\bra{\mathbf{curl}^2;\mu^{-1},\Omega_s}=\{\mathbf{U}\in\mathbf{H}\bra{\mathbf{curl},\Omega_s} \vert\, \mu^{-1}\mathbf{curl}\,\mathbf{U} \in \mathbf{H}\bra{\mathbf{curl},\Omega_s} \}
\end{equation*}
over which \eqref{eq: curl curl in the sense of distribution} holds.
The regularizing term $\epsilon(\mathbf{x})\,\nabla\,\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{U}}$ in \eqref{eq: strong from coupled sys a} cannot be similarly interpreted in the sense of distribution unless $\epsilon(\mathbf{x})$ is assumed regular enough. If $\epsilon(\mathbf{x})$ was taken to be Lipschitz continuous, then the troublesome quantity could be defined for all $\mathbf{V}\in \mathscr{D}(\Omega_s)^3$ by the relation \cite[Sec. 2.3.3]{hazard1996solution}
\begin{equation*}
\int_{\Omega_s} \epsilon(\mathbf{x})\,\nabla\,\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{U}}\cdot\mathbf{V}\dif\mathbf{x} = -\int_{\Omega_s}\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{U}}\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{V}}\dif\mathbf{x}.
\end{equation*}
However, in the current work, we rather depart commit to viewing $\nabla\,\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{U}}$ independently as a square-integrable vector-field obtained by weak differentiation. This is possible for vector-fields lying in
\begin{equation}
\mathbf{H}\bra{\nabla\text{div};\epsilon,\Omega_s}:=\{\mathbf{U}\in \mathbf{L}^2(\Omega_s)\vert\,\text{div}\,\bra{\epsilon\mathbf{U}}\in\mathbf{H}^1(\Omega_s)\}.
\end{equation}
The
can be interpreted in the sense of distributions, this is not immediately possible for the regularizing term $\epsilon(\mathbf{x})\,\nabla\,\text{div}\, \bra{\epsilon(\mathbf{x})\mathbf{U}}$ in \eqref{eq: strong from coupled sys a} unless $\epsilon(\mathbf{x})$ is assumed regular enough. The first solution to this issue is to assume that $\epsilon$ is Lipschitz continuous \cite[Sec. 2.3.3]{hazard1996solution}, then the latter can be defined by the relation
Both the trace $\gamma^-_{R,\mu}(\mathbf{U})$ and the term $\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\mathbf{curl}\,\mathbf{U}}$ are well-defined when $\mu^{-1}\,\mathbf{curl}(\mathbf{U})\in\mathbf{H}\bra{\mathbf{curl},\Omega_s}$.
Therefore, a ``magnetic boundary condition'' entails the introduction of
$$
\mathbf{H}\bra{\mathbf{curl}^2;\mu^{-1},\Omega_s}=\{\mathbf{U}\in\mathbf{H}\bra{\mathbf{curl},\Omega_s} \vert\, \mu^{-1}\mathbf{curl}\,\mathbf{U} \in \mathbf{H}\bra{\mathbf{curl},\Omega_s} \}
$$
in the functional setting of the primal variational formulation of the problem. That is, the quantity $\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\mathbf{curl}\,\mathbf{U}}$, which also appears in the classical electric wave equation, can be interpreted in the sense of distributions, a ``magnetic boundary condition'' entails the introduction of
$$
\mathbf{H}\bra{\mathbf{curl}^2;\mu^{-1},\Omega_s}=\{\mathbf{U}\in\mathbf{H}\bra{\mathbf{curl},\Omega_s} \vert\, \mu^{-1}\mathbf{curl}\,\mathbf{U} \in \mathbf{H}\bra{\mathbf{curl},\Omega_s} \}
$$
in the functional setting of the primal variational problem.
In particular, the second term in \eqref{eq: strong from coupled sys a} cannot be interpreted in the sense of distributions unless $\epsilon(\mathbf{x})\mathbf{U}\in\mathbf{H}\bra{\text{div},\Omega_s}$ \cite[Sec. 2.3.3]{hazard1996solution}.
The domain of the generalized Hodge-Helmholtz/Laplace operator is
\begin{equation*}
\mathbf{X}\bra{\epsilon,\mu;\Delta,\Omega_s}:=\mathbf{H}\bra{\nabla\text{div};\epsilon,\Omega_s}\cap\mathbf{H}\bra{\mathbf{curl}^2;\mu^{-1},\Omega_s},
\end{equation*}
where
\begin{align*}
&\mathbf{H}\bra{\nabla\text{div};\epsilon,\Omega_s}:=\{\mathbf{U}\in \mathbf{L}^2(\Omega_s)\vert\,\text{div}\,\bra{\epsilon\mathbf{U}}\in\mathbf{H}^1(\Omega_s)\},\\
&\mathbf{H}\bra{\mathbf{curl}^2;\mu^{-1},\Omega_s}:=\{\mathbf{U}\in\mathbf{H}\bra{\mathbf{curl},\Omega_s} \vert\, \mu^{-1}\mathbf{curl}\,\mathbf{U} \in \mathbf{H}\bra{\mathbf{curl},\Omega_s} \}.
\end{align*}
\\
\\
\\
The transmission conditions \eqref{eq: strong from coupled sys c} and \eqref{eq: strong from coupled sys d} suggest that the domains of the traces should be extended to functional spaces accounting for the lack of regularity of the material coefficients. In particular, the second term in \eqref{eq: strong from coupled sys a} cannot be interpreted in the sense of distributions unless $\epsilon(\mathbf{x})\mathbf{U}\in\mathbf{H}\bra{\text{div},\Omega_s}$ \cite[Sec. 2.3.3]{hazard1996solution}. However, the necessity of explicitly introducing
$$
\mathbf{H}\bra{\text{div};\epsilon,\Omega_s}:=\{\mathbf{U}\in \mathbf{L}^2(\Omega_s)\vert\,\text{div}\,\bra{\epsilon\mathbf{U}}\in\mathbf{L}^2(\Omega_s)\}
$$ in this work disappears upon choosing a mixed variational formulation for \eqref{eq: strong from coupled sys a} in which the term $-\text{div}\bra{\epsilon(\mathbf{x})\mathbf{U}}$ is replaced by a sufficiently regular unknown. This substitution also resolves the issue regarding the regularity requirements for a vector-valued function to be in the domain of $\gamma^-_{D,\epsilon}$.
The trace $\gamma^-_{R,\mu}(\mathbf{U})$ is well-defined whenever $\mu^{-1}\,\mathbf{curl}(\mathbf{U})\in\mathbf{H}\bra{\mathbf{curl},\Omega_s}$. Therefore, while the quantity $\mathbf{curl}\,\bra{\mu^{-1}(\mathbf{x})\mathbf{curl}\,\mathbf{U}}$, which also appears in the classical electric wave equation, can be interpreted in the sense of distributions, a ``magnetic boundary condition'' entails the introduction of
$$
\mathbf{H}\bra{\mathbf{curl}^2;\mu^{-1},\Omega_s}=\{\mathbf{U}\in\mathbf{H}\bra{\mathbf{curl},\Omega_s} \vert\, \mu^{-1}\mathbf{curl}\,\mathbf{U} \in \mathbf{H}\bra{\mathbf{curl},\Omega_s} \}
$$
in the functional setting of the primal variational problem. But evidently, the transmission conditions \eqref{eq: strong from coupled sys c} will ultimately eliminate the dependency of this trace on the variable material coefficient.
Similar considerations apply to the modified trace $\gamma^{-}_{n,\epsilon}(\mathbf{U})$.
Note that where the material properties are constant, the extension of the divergence trace consists of a simple scaling in the vector space $H^{1/2}(\Gamma)$.
\fi
\subsubsection{Compounded trace spaces}\label{subsec: compound trace spaces}
As explained in \cite[Sec. 3]{claeys2017first}, a theory of differential equations for the Hodge--Helmholtz/Laplace problem in three dimensions entails partitioning our collection of traces into two ``dual" pairs. Accordingly, we now introduce new mappings $\mathcal{T}^{-}_{D,\epsilon}:\mathbf{H}_{\text{loc}}(\mathbf{curl},\Omega_s)\cap\mathbf{H}_{\text{loc}}(\epsilon;\nabla\text{div},\Omega_s)\rightarrow\mathcal{H}_D(\Gamma)$ and $\mathcal{T}^{-}_{N,\mu}:\mathbf{H}_{\text{loc}}(\mu^{-1};\mathbf{curl}^2,\Omega_s)\cap\mathbf{H}_{\text{loc}}(\epsilon;\text{div},\Omega_s) \rightarrow\mathcal{H}_N(\Gamma)$ which we define by
\newline \newline
\begin{greyTracesDef}
\begin{align*}
\mathcal{T}^{-}_{D,\epsilon}\bra{\mathbf{U}}&:= \colvec{2}{\gamma^-_t(\mathbf{U})}{\gamma_{D,\epsilon}^-(\mathbf{U})},
&
\mathcal{T}^{-}_{N,\mu}\bra{\mathbf{U}}&:= \colvec{2}{\gamma_{R,\mu}(\mathbf{U})}{\gamma^-_{n,\epsilon}(\mathbf{U})},
\end{align*}
\end{greyTracesDef}
\newline \newline
respectively, where
\begin{align*}
\mathcal{H}_D&:=\mathbf{H}^{-1/2}(\text{curl}_{\Gamma},\Gamma)\times H^{1/2}(\Gamma), &
\mathcal{H}_N&:=\mathbf{H}^{-1/2}(\text{div}_{\Gamma},\Gamma)\times H^{-1/2}(\Gamma).
\end{align*}
The related classical compounded traces are defined similarly by
\newline\newline
\begin{greyTracesDef}
\begin{align*}
\mathcal{T}^+_D(\mathbf{U})&:=\colvec{2}{\gamma_t^+(\mathbf{U})}{\gamma^+_{D,\eta}(\mathbf{U})}, &
\mathcal{T}^+_N(\mathbf{U})&:=\colvec{2}{\gamma_R(\mathbf{U})}{\gamma_n(\mathbf{U})}.
\end{align*}
\end{greyTracesDef}
\newline
\par The choice of subscripts is motivated by the analogy between this pair of traces and the classical Dirichlet and Neumann boundary conditions for second-order elliptic BVPs. The trace spaces $\mathcal{H}_D$ and $\mathcal{H}_N$ are put in duality using the sum of the inherited component-wise duality parings. That is, for $\vec{\mathbf{p}}=(\mathbf{p},q)\in\mathcal{H}_N$ and $\vec{\bm{\eta}}=(\bm{\eta},\zeta)\in\mathcal{H}_D$, we define
\begin{equation}
\langle \vec{\mathbf{p}}, \vec{\bm{\eta}}\rangle := \langle\mathbf{p},\bm{\eta}\rangle_{\tau} + \langle q, \zeta\rangle_{\Gamma}.
\end{equation}
The importance of the spaces \eqref{H(curl) trace space}-\eqref{H(div) trace space} is highlighted by this next result.
\begin{lemma}[{See \cite[Lem. 3.2]{claeys2017first}}]
The compound traces $\mathcal{T}_D$ and $\mathcal{T}_N$ have continuous right inverses, i.e. lifting maps $\mathcal{R}_D:\mathcal{H}_D\rightarrow \mathbf{X}(\Delta,\Omega)$ and $\mathcal{R}_N\rightarrow \mathbf{X}(\Delta,\Omega)$, respectively.\label{lem: compound lifting}
\end{lemma}
\subsection{Boundary potentials}\label{sec: Boundary potentials}
\par By exploiting the radiating fundamental solution
\begin{equation*}
G_{\nu}(\mathbf{x}):=\exp\left(i\nu\abs{\mathbf{x}}\right)/4\pi\abs{\mathbf{x}}
\end{equation*}
for the scalar Helmholtz operator $\Delta-\nu^2\id$, it is shown in \cite[Sec. 4.2]{claeys2017first} that a distributional solution $\mathbf{U}\in\mathbf{L}^2(\mathbb{R}^3)$ such that $\mathbf{U}\vert_{\Omega_s}\in\mathbf{X}(\Delta,\Omega_s)$ and $\mathbf{U}\vert_{\Omega'}\in\mathbf{X}_{\text{loc}}(\Delta,\Omega')$ of the homogeneous (scaled) Hodge--Helmholtz/Laplace equation \eqref{eq: strong from coupled sys b} with constant coefficients $\eta>0$, $\kappa\geq 0$, stated in the whole of $\mathbb{R}^3$ with radiation conditions at infinity as considered in Section \ref{sec: Introduction}, affords a representation formula.
\newline\newline
\begin{greyFrameRepFormula}
\begin{equation}
\mathbf{U} = \mathcal{SL}_{\kappa}\cdot [\mathcal{T}_{N}(\mathbf{U})] + \mathcal{DL}_{\kappa}\cdot [\mathcal{T}_{D}(\mathbf{U})].\label{representation formula}
\end{equation}
\end{greyFrameRepFormula}
\newline
\par Letting $\tilde{\kappa}=\kappa/\sqrt{n}$, the Hodge-Helmholtz single layer potential is explicitly given by
\newline\newline
\begin{greyTracesDef}
\begin{equation}\label{Hodge-Helmholtz single layer potential}
\mathcal{SL}_{\kappa}\bra{\colvec{2}{\mathbf{p}}{q}}=-\bm{\Psi}_{\kappa}(\mathbf{p}) - \nabla\tilde{\psi}_k\bra{\text{div}_{\Gamma}(\mathbf{p})} + \nabla\psi_{\tilde{\kappa}}(q),
\end{equation}
\end{greyTracesDef}
\newline\newline
where the Helmholtz scalar single-layer, vector single-layer and the regular potentials are written individually for $\mathbf{p}\in\mathbf{H}^{-1/2}(\text{div}_{\Gamma},\Gamma)$ and $q\in H^{-1/2}(\Gamma)$ as
\newline\newline
\begin{greyTracesDef}
\begin{subequations}
\begin{align}
\psi_{\tilde{\kappa}}(q)(\mathbf{x})&:=\int_{\Gamma}q(\mathbf{y})G_{\tilde{\kappa}}(\mathbf{x}-\mathbf{y})\dif\sigma(\mathbf{y}), &\mathbf{x}\in\mathbb{R}^3\backslash\Gamma, \label{scalar single layer}\\
\bm{\Psi}_{\kappa}(\mathbf{p})(\mathbf{x})&:=\int_{\gamma}\mathbf{p}(\mathbf{y})G_{\kappa}(\mathbf{x}-\mathbf{y})\dif\sigma(\mathbf{y}), &\mathbf{x}\in\mathbb{R}^3\backslash\Gamma,\label{vector single layer}\\
\tilde{\psi}_{\kappa}(q)(\mathbf{x})&:=\int_{\Gamma}q(\mathbf{y})\frac{G_{\kappa}-G_{\tilde{\kappa}}}{\kappa^2}(\mathbf{x}-\mathbf{y})\dif\sigma(\mathbf{y}), &\mathbf{x}\in\mathbb{R}^3\backslash\Gamma,\label{regular potential}
\end{align}
\end{subequations}
\end{greyTracesDef}
\newline\newline
respectively. The expression \eqref{Hodge-Helmholtz single layer potential} is derived with \eqref{scalar single layer}-\eqref{regular potential} understood as duality pairings. However, if the essential supremum of $\mathbf{p}$, $q$ and $\text{div}_{\Gamma}(\mathbf{p})$ is bounded, then they can safely be computed as improper integrals \cite[Rmk. 4.2]{claeys2017first}.
\begin{lemma}[{See \cite[Lem. 4.1]{hiptmair2003coupling}}]\label{lem: single layer into H1}
The single layer potentials \eqref{scalar single layer} and \eqref{vector single layer} are continuous mappings $\psi_{\nu}:H^{-1/2}(\Gamma)\rightarrow H^1_{\emph{loc}}(\mathbb{R}^3)$ and $\bm{\Psi}_{\nu}:\mathbf{H}_T^{-1/2}(\Gamma)\rightarrow \mathbf{H}^1_{\emph{loc}}(\mathbb{R}^3)$.
\end{lemma}
The classical potentials solve the equations
\begin{subequations}
\begin{align}
-\,\text{div}\,\nabla \psi_{\tilde{\kappa}}(q)&= \tilde{\kappa}^2\psi_{\tilde{\kappa}}(q),\label{scalar helmholtz for scalar single layer}\\
-\,\Delta \bm{\Psi}_{\kappa}(\mathbf{p}) &= \kappa^2\bm{\Psi}_{\kappa}(\mathbf{p})\label{vector helmholtz for vector single layer},\\
-\text{div}\,\nabla\tilde{\psi}_{\kappa}(q) &= \psi_{\kappa}(q) + \frac{1}{\eta}\psi_{\tilde{\kappa}}(q),
\end{align}
\end{subequations}
and satisfy the identity \cite[Lem. 2.3]{maccamy1984solution}
\begin{align}\label{scalar and vector single layer potential identity}
\text{div}\,\bm{\Psi}_{\nu}(\mathbf{p}) &= \psi_{\nu}\left(\text{div}_{\Gamma}\mathbf{p}\right), & &\forall\,\mathbf{p}\in \mathbf{H}^{-1/2}(\text{div}_\Gamma,\Gamma).
\end{align}
These observations are used along with Lemma \ref{lem: single layer into H1} in the proof the following lemma.
\begin{lemma}[{See \cite[Sec. 5]{claeys2017first}}]\label{lem: single layers into delta}
The potentials $\nabla\psi_{\tilde{\kappa}}$, $\bm{\Psi}_{\kappa}$ and $\nabla\tilde{\psi}_{\kappa}$ are continuous mappings from $H^{-1/2}(\Gamma)$ and $\mathbf{H}^{-1/2}(\text{\emph{div}}_{\Gamma},\Gamma)$ into $\mathbf{X}(\Delta,\Omega_s)$ or $\mathbf{X}_{\text{\emph{loc}}}(\Delta,\Omega')$.
\end{lemma}
\begin{corollary}
The Hodge--Laplace/Helmholtz single layer potential is a continuous map from $\mathcal{H}_N$ into $\mathbf{X}(\Delta,\Omega_s)$ or $\mathbf{X}_{\text{\emph{loc}}}(\Delta,\Omega')$.
\end{corollary}
Ultimately, we will resort to a Fredholm alternative argument to prove well-posedness of the coupled system. It is therefore evident that the compactness properties of the boundary integral operators introduced in the next Lemma will be extensively used both explicitly and implicitly ---notably through exploiting the results found in \cite[Sec. 6]{claeys2017first}.
\newline\newline
\begin{whiteFrameResult}
\begin{lemma}[{see \cite[Lem. 3.9.8]{sauter2010boundary} and \cite[Lem. 7]{buffa2003galerkin}}]\label{lem: compactness of boundary integrals}
For any $\nu\geq 0$, the following operators are compact:
\begin{align*}
\gamma^{\pm}\bra{\psi_{\nu}-\psi_{0}}:&\,H^{-1/2}(\Gamma)\rightarrow H^{1/2}(\Gamma)\\
\gamma^{\pm}_n\bra{\nabla\psi_{\nu}-\nabla\psi_{0}}:&\,H^{-1/2}(\Gamma)\rightarrow H^{-1/2}(\Gamma)\\
\gamma^{\pm}_{t}\bra{\bm{\Psi}_{\nu}-\bm{\Psi}_{0}}:&\,\mathbf{H}^{-1/2}(\text{\emph{div}}_{\Gamma},\Gamma)\rightarrow \mathbf{H}^{-1/2}(\text{\emph{curl}}_{\Gamma},\Gamma)\\
\gamma^{\pm}_n\nabla\tilde{\psi}_{\nu}:&\,H^{-1/2}(\Gamma)\rightarrow H^{-1/2}(\Gamma)
\end{align*}
\end{lemma}
\end{whiteFrameResult}
\newline
\begin{proof}
Compactness of the second boundary integral operator listed in the statement of Lemma \ref{lem: compactness of boundary integrals} immediately entails compactness of
$$\nu^2\gamma^{\pm}_n\nabla\tilde{\psi}_{\nu}=\gamma^{\pm}_n\bra{\nabla\psi_{\nu}-\nabla\psi_{\tilde{\nu}}}=\gamma^{\pm}_n\bra{\nabla\psi_{\nu}-\nabla\psi_{0}} - \bra{\gamma^{\pm}_n\bra{\nabla\psi_{\tilde{\nu}}-\nabla\psi_{0}}}$$
by linearity. While it seems that blow-up occurs in $\tilde{\psi}_{\nu}$ as $\nu\rightarrow 0$, $\nabla\tilde{\psi}_{\nu}$ happens to be an entire function of $\nu$ that vanishes at $\nu = 0$ \cite[Sec. 4.1]{claeys2017first}.
\end{proof}
\par The Hodge-Helmholtz double layer potential is given for $\bm{\eta}\in\mathbf{H}^{-1/2}(\text{curl}_{\Gamma},\Gamma)$ and $\xi\in H^{1/2}(\Gamma)$ by
\newline\newline
\begin{greyTracesDef}
\begin{equation}\label{double layer potential}
\mathcal{DL}_{\kappa}\bra{\colvec{2}{\bm{\eta}}{\xi}} := \mathbf{curl}\,\bm{\Psi}_{\kappa}(\bm{\eta}\times\mathbf{n}) + \Upsilon_{\kappa}(\xi).
\end{equation}
\end{greyTracesDef}
\newline
\par We recognize in \eqref{double layer potential} the (electric) Maxwell double layer potential (c.f. \cite[Sec. 4]{hiptmair2003coupling}, \cite[Eq. 28]{buffa2003galerkin}) and the normal vector single-layer potential
\newline\newline
\begin{greyTracesDef}
\begin{align}
\Upsilon_{\kappa}(\xi)&:=\int_{\Gamma}\xi(\mathbf{y})\mathbf{G}_{\kappa}(\mathbf{x}-\mathbf{y})\mathbf{n}(\mathbf{y})\dif\sigma(\mathbf{y}), & \mathbf{x}\in\mathbb{R}^3\backslash\Gamma,
\end{align}
\end{greyTracesDef}
\newline
in which appears the matrix-valued fundamental solution
\begin{equation*}
\mathbf{G}_{\kappa} := G_{\kappa}\id +\frac{1}{\kappa^2}\nabla^2\bra{G_{\kappa}-G_{\tilde{\kappa}}}
\end{equation*}
satisfying $-\Delta_{\eta}\mathbf{G}_{\kappa} - \kappa^2\mathbf{G}_{\kappa}=\delta_0 \id$ exploited in \cite{claeys2017first} and detailed in \cite[App. A]{hazard1996solution}.
\begin{lemma}[{See \cite[Sec. 5.4]{claeys2017first}}]\label{lem: normal vector single layer into H1}
The normal vector single layer potential $\Upsilon_{\kappa}$ is a continuous mapping $\Upsilon_{\kappa}:H^{1/2}(\Gamma)\rightarrow \mathbf{H}^1_{\emph{loc}}(\mathbb{R}^3)$.
\end{lemma}
This surface potential solves the equation
\begin{equation}\label{eq: Hodge-Helmholtz Upsilon}
-\Delta_{\eta}\Upsilon_{\kappa}(\xi)=\kappa^2\Upsilon_{\kappa}(\xi)
\end{equation}
and satisfies the identity \cite[Sec.5.4]{claeys2017first}
\begin{equation}\label{eq: curl potentials identity}
\mathbf{curl}\,\Upsilon_{\kappa}(\xi)=\mathbf{curl}\bm{\Psi}_{\kappa}(\xi\mathbf{n})
\end{equation}
These results can be used in proving the following lemma.
\begin{lemma}[{See \cite[Sec. 5]{claeys2017first}}]\label{lem: double layers into delta}
The potentials $\mathbf{curl}\bm{\Psi}_{\kappa}(\cdot\times\mathbf{n})$ and $ \Upsilon_{\kappa}$ are continuous mappings from $\mathbf{H}^{-1/2}(\emph{curl}_{\Gamma},\Gamma)$ and $H^{1/2}(\Gamma)$ respectively, into $\mathbf{X}(\Delta,\Omega_s)$ and $\mathbf{X}_{\text{\emph{loc}}}(\Delta,\Omega')$.
\end{lemma}
\begin{corollary}
The Hodge-Helmholtz double layer potential is a continuous map from $\mathcal{H}_D$ into $\mathbf{X}(\Delta,\Omega_s)$ or $\mathbf{X}_{\text{\emph{loc}}}(\Delta,\Omega')$.
\end{corollary}
\subsection{Integral operators}\label{sec: Integral operators}
The well-known Cald\'eron identities are obtained from \eqref{representation formula} upon taking the classical compounded traces on both sides and utilizing the jump relations
\newline\newline
\begin{greyFrameJumpRelations}
\begin{subequations}
\begin{align}
[\mathcal{T}_D]\cdot\mathcal{DL}_{\kappa}(\vec{\bm{\eta}}) &= \vec{\bm{\eta}}, & [\mathcal{T}_N]\cdot\mathcal{DL}_{\kappa}(\vec{\bm{\eta}}) &= 0,\quad\quad \vec{\bm{\eta}}\in\mathcal{H}_D, \label{double layer jump}\\
[\mathcal{T}_D]\cdot\mathcal{SL}_{\kappa}(\vec{\mathbf{p}}) &= 0, & [\mathcal{T}_N]\cdot\mathcal{SL}_{\kappa}(\vec{\mathbf{p}}) &= \vec{\mathbf{p}},\quad\quad \vec{\mathbf{p}}\in\mathcal{H}_N, \label{single layer jump}
\end{align}
\end{subequations}
\end{greyFrameJumpRelations}
\newline\newline
given in \cite[Thm. 5.1]{claeys2017first}. The operator forms of the interior and exterior Cald\'eron projectors defined on $\mathcal{H}_D\times\mathcal{H}_N$, which we denote $\mathbb{P}^-_{\kappa}$ and $\mathbb{P}^+_{\kappa}$ respectively, enter the Cald\'eron identites:
\begin{subequations}
\begin{gather}
\underbrace{\begin{pmatrix}
\{\mathcal{T}_D\}\cdot\mathcal{DL}_k+\frac{1}{2}\id & \{\mathcal{T}_D\}\cdot\mathcal{SL}_k\\
\{\mathcal{T}_N\}\cdot\mathcal{DL}_k & \{\mathcal{T}_N\}\cdot\mathcal{SL}_k+\frac{1}{2}\id
\end{pmatrix}}_{\highlight{=:\mathbb{P}^-_{\kappa}}}
\begin{pmatrix}
\mathcal{T}_D^-\mathbf{U}\\
\mathcal{T}_N^-\mathbf{U}
\end{pmatrix}
= \begin{pmatrix}
\mathcal{T}_D^-\mathbf{U}\\
\mathcal{T}_N^-\mathbf{U}
\end{pmatrix},\label{interior calderon}\\
\underbrace{\begin{pmatrix}
-\{\mathcal{T}_D\}\cdot\mathcal{DL}_k+\frac{1}{2}\id & -\{\mathcal{T}_D\}\cdot\mathcal{SL}_k\\
-\{\mathcal{T}_N\}\cdot\mathcal{DL}_k & -\{\mathcal{T}_N\}\cdot\mathcal{SL}_k+\frac{1}{2}\id
\end{pmatrix}}_{\highlight{=:\mathbb{P}^+_{\kappa}}}
\begin{pmatrix}
\mathcal{T}_D^+\mathbf{U}^{\text{ext}}\\
\mathcal{T}_N^+\mathbf{U}^{\text{ext}}
\end{pmatrix}
= \begin{pmatrix}
\mathcal{T}_D^+\mathbf{U}^{\text{ext}}\\
\mathcal{T}_N^+\mathbf{U}^{\text{ext}}
\end{pmatrix},\label{exterior calderon}
\end{gather}
Note that $\mathbb{P}^-_{\kappa} + \mathbb{P}^+_{\kappa}=\id$ and that the range of $\mathbb{P}^+_{\kappa}$ coincides with the kernel of $\mathbb{P}^-_{\kappa}$ and vice-versa \cite[Sec. 5]{buffa2003galerkin}. As a consequence of the jump relations \eqref{double layer jump}-\eqref{single layer jump}, the representation formula \eqref{representation formula} and the Lemma \ref{lem: compound lifting}, we obtain the next proposition.
\end{subequations}
\begin{lemma}[{See \cite[Lem. 6.18]{steinbach2007numerical}, \cite[Thm. 8]{buffa2003galerkin} and \cite[Prop. 5.2]{claeys2017first}}]\label{prop: valid cauchy data}
The pair of ``magnetic" and ``electric" traces $\bra{\vec{\bm{\eta}}\,\,\,\vec{\mathbf{p}}}^{\top}\in\mathcal{H}_D\times\mathcal{H}_N$ is valid interior or exterior Cauchy data, if and only if it lies in the kernel of $\mathbb{P}^+_{\kappa}$ or $\mathbb{P}^-_{\kappa}$ respectively, i.e.
\begin{align*}
\ker&\bra{\mathbb{P}^+_{\kappa}} =\{\bra{\vec{\bm{\eta}}\,\,\,\vec{\mathbf{p}}}^{\top}:=\bra{\mathcal{T}_D^-\bra{\mathbf{U}},\mathcal{T}_N^-\bra{\mathbf{U}} }^{\top}\in\mathcal{H}_D\times\mathcal{H}_N\,\big\vert \,\mathbf{U}\in \mathbf{X}\bra{\Delta,\Omega_s},\\
&\Delta_{\eta}\mathbf{U} + \kappa^2\mathbf{U} = 0 \text{ \emph{in} } \Omega_s \},\\
\ker&\bra{\mathbb{P}^-_{\kappa}} =\{\bra{\vec{\bm{\eta}}\,\,\,\vec{\mathbf{p}}}^{\top}:=\bra{\mathcal{T}_D^-\bra{\mathbf{U}},\mathcal{T}_N^-\bra{\mathbf{U}} }^{\top}\in\mathcal{H}_D\times\mathcal{H}_N\,\big\vert \mathbf{U}\in \mathbf{X}_{\text{\emph{loc}}}\bra{\Delta,\Omega'},\\ &\Delta_{\eta}\mathbf{U} + \kappa^2\mathbf{U} = 0 \text{ \emph{in} } \Omega',\mathbf{U}\text{ \emph{satisfying radiation conditions at infinity as in \cite{hazard1996solution}}}\}.
\end{align*}
\end{lemma}
\par Inspecting equations \eqref{interior calderon}-\eqref{exterior calderon} reveals that the Cald\'eron projectors share a common structure. They can be written as
\newline\newline
\begin{greyTracesDef}
\begin{align*}
&\mathbb{P}_{\kappa}^- = \frac{1}{2}\id + \mathbb{A}_\kappa &\text{and}& &\mathbb{P}_{\kappa}^+ = \frac{1}{2}\id - \mathbb{A}_\kappa,
\end{align*}
\end{greyTracesDef}
\newline\newline
where the Cald\'eron operator is given by
\newline\newline
\begin{greyTracesDef}
\begin{align*}\label{eq: Ak blocks}
\mathbb{A}_\kappa =\begin{pmatrix}
\mathbb{A}_{\kappa}^{DD} & \mathbb{A}_{\kappa}^{ND}\\
\mathbb{A}_{\kappa}^{DN} & \mathbb{A}_{\kappa}^{NN}
\end{pmatrix}:= \begin{pmatrix}
\{\mathcal{T}_D\}\cdot\mathcal{DL}_\kappa & \{\mathcal{T}_D\}\cdot\mathcal{SL}_\kappa\\
\{\mathcal{T}_N\}\cdot\mathcal{DL}_\kappa & \{\mathcal{T}_N\}\cdot\mathcal{SL}_\kappa
\end{pmatrix}: \mathcal{H_D\times\mathcal{H}_N\rightarrow \mathcal{H}_D\times\mathcal{H}_N}.
\end{align*}
\end{greyTracesDef}
\newline
\par An analog of the operator matrix $\mathbb{A}_k$ was found convenient in the study of the boundary integral equations of electromagnetic scattering problems \cite[Sec. 6]{buffa2003galerkin}. It is known from \cite{claeys2017first} that the off-diagonal blocks of $\mathbb{A}_k$ independently satisfy generalized G{\aa}rding inequalities making them of Fredholm type with index 0. Injectivity holds when $\kappa^2$ lies outside a discrete set of ``forbidden resonant frequencies'' accumulating at infinity \cite[Sec. 3]{claeys2017first}. In the static case $\kappa=0$, the dimensions of $\ker\left(\{\mathcal{T}_N\}\cdot\mathcal{SL}_0\right)$ and $\ker\left(\{\mathcal{T}_D\}\cdot\mathcal{DL}_0\right)$ are finite and agree with the zeroth and first Betti number of $\Gamma$, respectively. \cite[Sec. 7]{claeys2017first}
In the case of the classical electric wave equation, the boundary integral operators involved in the Cald\'eron projectors enjoy a hidden symmetry:
\newline\newline
\begin{whiteFrameResult}
\begin{lemma}[{See \cite[Lem. 5.4]{hiptmair2003coupling} and \cite[Lem. 6]{buffa2003galerkin}}]\label{lem: adjoint maxwell potential}
There exists a compact linear operator $\mathbf{C}_k:\mathbf{H}^{-1/2}(\emph{div}_\Gamma,\Gamma)\rightarrow \mathbf{H}^{-1/2}(\emph{div}_\Gamma,\Gamma)$ such that
\begin{equation*}
\ip{\{\gamma_R\}\bm{\Psi}_k(\mathbf{p})}{\bm{\eta}}_{\tau} = \ip{\mathbf{p}}{\{\gamma_t\}\bm{\Psi}_{\kappa}\mathbf{curl}(\bm{\eta}\times\mathbf{n})}_{\tau} +\ip{\mathbf{C}_k\mathbf{p}}{\bm{\eta}}_{\tau}
\end{equation*}
for all $\mathbf{p}\in\mathbf{H}^{-1/2}(\emph{div}_\Gamma,\Gamma)$ and $\bm{\eta}\in\mathbf{H}^{-1/2}(\mathbf{curl}_\Gamma,\Gamma)$.
\end{lemma}
\end{whiteFrameResult}
\newline
\par We will extend Lemma \ref{lem: adjoint maxwell potential} to the integral operators defined for the scaled Hodge-Helmholtz/Laplace equation to better characterize the structure of \eqref{eq: Ak blocks}. The symmetry we are about to reveal in the diagonal blocks of the Cald\'eron projectors will be crucial in the derivation of the main T-coercivity estimate of this work. It will be exploited for complete cancellation, up to compact terms, of the operators lying on the off-diagonal of the block operator matrix associated to the coupled variational system introduced in Section \ref{sec: coupled problem}. The following lemmas are required.
\begin{greenFrameResultBeforeAfter}
\begin{lemma}\label{lem: grad phi term is anti hermitian}
There exists a compact linear operator $C_k:H^{-1/2}(\Gamma)\rightarrow H^{-1/2}(\Gamma)$ such that
\begin{equation*}
\langle\{\gamma_n\}\nabla\psi_{\tilde{\kappa}}(q), \xi\rangle_{\Gamma} =-\ip{q}{\{\eta\,\gamma_D\}\Upsilon_{\kappa}(\xi)}_{\Gamma} + \ip{C_kq}{\xi},
\end{equation*}
for all $q\in H^{-1/2}(\Gamma)$, $\xi\in H^{1/2}(\Gamma)$.
\end{lemma}
\end{greenFrameResultBeforeAfter}
\begin{proof}
This proof utilizes a strategy found in \cite[Lem. 5.4]{hiptmair2003coupling} and \cite[Thm. 3.9]{Buffa2002Boundary}. Let $\rho > 0 $ be such that $B_{\rho}$ is an open ball containing $\overline{\Omega}_s$. We will indicate with a hat (e.g. $\widehat{\gamma}$) the traces taken over the boundary $\partial B_{\rho}$ of that ball. Due to the lemmas \ref{lem: single layers into delta} and \ref{lem: double layers into delta}, we can use the extension of formula \eqref{IBP div} to compare the following terms.
On the one hand, using the scalar Helmholtz equation \eqref{scalar helmholtz for scalar single layer} and recalling that $\tilde{\kappa}=\kappa/\sqrt{\eta}$, we have
\begin{align}\label{eq: interior eq 1}
\langle\eta\,\hlo{ \gamma_D^-}\nabla\psi_{\tilde{\kappa}}(q), &\hlo{\gamma_n^-}\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} \nonumber\\
&= \int_{\Omega_s}\eta\,\text{div}\left(\nabla\psi_{\tilde{\kappa}}(q)\right)\text{div}\Upsilon_k(\xi) +\eta\nabla\text{div}\left(\nabla\psi_{\tilde{\kappa}}(q)\right)\cdot\Upsilon_{\kappa}(\xi)\dif\mathbf{x}\nonumber\\
&= \hly{-\int_{\Omega_s}\kappa^2\psi_{\tilde{\kappa}}(q)\text{div}\Upsilon_k(\xi)\dif\mathbf{x} -\int_{\Omega_s} \kappa^2\nabla\psi_{\tilde{\kappa}}(q)\cdot\Upsilon_{\kappa}(\xi)\dif\mathbf{x},}
\end{align}
and similarly,
\begin{multline*}
\langle\eta\,\gamma_D^+\nabla\psi_{\tilde{\kappa}}(q), \gamma_n^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} = \int_{\Omega'\cap B_{\rho}}\kappa^2\psi_{\tilde{\kappa}}(q)\,\text{div}\Upsilon_k(\xi) + \nabla\psi_{\tilde{\kappa}}(q)\cdot\Upsilon_{\kappa}(\xi)\dif\mathbf{x} \\
\hlb{+\langle\eta\,\widehat{\gamma}_D^+\nabla\psi_{\kappa}(q), \widehat{\gamma}_n^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}}}.
\end{multline*}
On the other hand, using \eqref{scalar helmholtz for scalar single layer} together with the scaled Hodge-Helmholtz equation \eqref{eq: Hodge-Helmholtz Upsilon}, we also have
\begin{align}\label{eq: interior eq 2}
\langle\hlo{\gamma_n^-}\nabla&\psi_{\tilde{\kappa}}(q), \eta\,\hlo{\gamma_D^-}\Upsilon_{\kappa}(\xi)\rangle_{\Gamma}\nonumber\\
&= \int_{\Omega_s} \eta\,\text{div}\left(\nabla\psi_{\tilde{\kappa}}(q)\right)\text{div}\Upsilon_{\kappa}(\xi)\dif\mathbf{x}
+ \int_{\Omega_s}\eta\,\nabla\psi_{\tilde{\kappa}}(q)\cdot\nabla\text{div}\Upsilon_{\kappa}(\xi)\dif{\mathbf{x}}\nonumber\\
&=\hly{-\int_{\Omega_s}\kappa^2\psi_{\tilde{\kappa}}(q)\,\text{div}\Upsilon_{\kappa}(\xi)\dif\mathbf{x}} + \hlm{\int_{\Omega_s}\nabla\psi_{\tilde{\kappa}}(q)\cdot\mathbf{curl}\,\mathbf{curl}\,\Upsilon_{\kappa}(\xi)\dif{\mathbf{x}}}\nonumber\\
&\qquad\qquad\hly{-\int_{\Omega_s}\kappa^2\nabla\psi_{\tilde{\kappa}}(q)\cdot\Upsilon_{\kappa}(\xi)\dif{\mathbf{x}}.}
\end{align}
Equations \cref{eq: interior eq 1} and \cref{eq: interior eq 2} together yield
\begin{multline*}
\langle\hlo{\gamma_n^-}\nabla\psi_{\tilde{\kappa}}(q), \eta\,\hlo{\gamma_D^-}\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} = \langle\eta\,\hlo{\gamma_D^-}\nabla\psi_{\kappa}(q), \hlo{\gamma_n^-}\Upsilon_{\kappa}(\xi)\rangle_{\Gamma}\\ +\hlm{\int_{\Omega_s}\nabla\psi_{\tilde{\kappa}}(q)\cdot\mathbf{curl}\,\mathbf{curl}\,\Upsilon_{\kappa}(\xi)\dif{\mathbf{x}}}.
\end{multline*}
Similarly, the terms involving the exterior traces satisfy
\begin{multline*}
\langle\gamma_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\gamma_D^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma}= \langle\eta\,\gamma_D^+\nabla\psi_{\kappa}(q), \gamma_n^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} - \hlb{\langle\eta\,\widehat{\gamma}_D^+\nabla\psi_{\kappa}(q), \widehat{\gamma}_n^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}}}\\
-\hlm{\int_{\Omega'\cap B_{\rho}}\nabla\psi_{\tilde{\kappa}}(q)\cdot\mathbf{curl}\,\mathbf{curl}\,\Upsilon_{\kappa}(\xi)\dif{\mathbf{x}}} + \hlb{\langle\widehat{\gamma}_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\widehat{\gamma}_D^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}}.}
\end{multline*}
From the first row of the jump properties \cite[Sec. 5]{claeys2017first}
\begin{subequations}
\begin{align}
&[\gamma_D]\nabla\psi_{\tilde{\kappa}}(q) = 0, & &[\gamma_n]\Upsilon_{\kappa}(\xi) = 0,\\
&[\gamma_D]\Upsilon_{\kappa}(\xi) = \xi/\eta, & &[\gamma_n]\nabla\psi_{\tilde{\kappa}}(q)=q \label{non-vanishing jumps psi upsilon},
\end{align}
\end{subequations}
we obtain, by gathering the above results, integrating by parts again and using the fact that $\mathbf{curl}\circ\nabla\equiv 0$,
\begin{align}\label{switching is compact perturbation psi upsilon}
\langle\gamma_n^-\nabla\psi_{\tilde{\kappa}}(q)&, \eta\,\gamma_D^-\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} = \langle\eta\,\gamma_D^+\nabla\psi_{\kappa}(q), \gamma_n^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} + \int_{\Omega_s}\kappa^2\nabla\psi_{\tilde{\kappa}}(q)\cdot\bm{\Psi}_{\kappa}(\xi\mathbf{n})\dif\mathbf{x}\nonumber\\
&= \langle\gamma_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\gamma_D^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} + \hlm{\int_{B_{\rho}}\nabla\psi_{\tilde{\kappa}}(q)\cdot\mathbf{curl}\,\mathbf{curl}\,\Upsilon_{\kappa}(\xi)\dif{\mathbf{x}}}\nonumber\\
&\qquad\hlb{+ \langle\eta\,\widehat{\gamma}_D^+\nabla\psi_{\kappa}(q), \widehat{\gamma}_n^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}} -\langle\widehat{\gamma}_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\widehat{\gamma}_D^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}}.}\nonumber\\
&= \langle\gamma_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\gamma_D^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} + \hlm{\langle\gamma_t\nabla\psi_{\tilde{\kappa}}(q), \gamma_R\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}}}\nonumber\\
&\qquad\hlb{+ \langle\eta\,\widehat{\gamma}_D^+\nabla\psi_{\kappa}(q), \widehat{\gamma}_n^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}} -\langle\widehat{\gamma}_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\widehat{\gamma}_D^+\Upsilon_{\kappa}(\xi)\rangle_{\partial B_{\rho}}.}
\end{align}
Fortunately, when restricted to domains away from $\Gamma$, the potentials are $C^{\infty}$-smoothing. Hence, their evaluation on $\partial B_{\rho}$, the highlighted terms in \eqref{switching is compact perturbation psi upsilon}, induce compact operators. This shows that for some compact operator $C_k:H^{-1/2}(\Gamma)\rightarrow H^{-1/2}(\Gamma)$,
\begin{equation}\label{identity with compact upsilon psi}
\langle\gamma_n^-\nabla\psi_{\tilde{\kappa}}(q), \eta\,\gamma_D^-\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} = \langle\gamma_n^+\nabla\psi_{\tilde{\kappa}}(q), \eta\,\gamma_D^+\Upsilon_{\kappa}(\xi)\rangle_{\Gamma} + \ip{C_k q}{\xi}_{\Gamma}.
\end{equation}
The jump identities \eqref{non-vanishing jumps psi upsilon} for the potentials yield formulas of the form $\{\gamma_*\}K = \gamma_*^{\pm}K\pm (1/2)\id$, where $*=n,\,D$ and $K=\nabla\psi_{\tilde{\kappa}},\,\Upsilon_{\kappa}$ accordingly. Substituting each one-sided trace involved in the two leftmost duality pairings of \eqref{identity with compact upsilon psi} for the integral operators using these equations completes the proof.
\end{proof}
\begin{greenFrameResultBeforeAfter}
\begin{lemma}\label{lem: symmetry of two last terms single layer}
For all $\mathbf{p}\in\mathbf{H}^{-1/2}(\emph{div}_\Gamma,\Gamma)$ and $\xi\in H^{1/2}(\Gamma)$, we have
\begin{equation*}
\ip{\mathbf{p}}{\gamma^{\pm}_t\Upsilon_{\kappa}(\xi)}_{\tau} = \langle\gamma^{\pm}_n\bm{\Psi}_{\kappa}(\mathbf{p}),\xi \rangle_{\Gamma} + \langle\gamma^{\pm}_n\nabla\tilde{\psi}_{\kappa}(\text{\emph{div}}_{\Gamma}(\mathbf{p})), \xi\rangle_{\Gamma}.
\end{equation*}
\end{lemma}
\end{greenFrameResultBeforeAfter}
\begin{proof}
In the following calculations, the boundary integrals are to be understood as duality pairings. Since $\mathbf{p}\in\mathbf{L}^2_t(\Gamma)$ is a tangent vector field lying in the image of $\gamma_t$, the tangential trace operator can safely be dropped in expanding these integrals using the definitions of Section \ref{sec: Boundary potentials}. On the one hand, this leads to
\begin{multline}\label{eq: tangential trace Gamma}
\ip{\mathbf{p}}{\gamma^{\pm}_t\Upsilon_{\kappa}(\xi)}_{\tau} = \int_{\Gamma}\int_{\Gamma}\xi(\mathbf{y})\mathbf{p}(\mathbf{x})\cdot\bra{\mathbf{G}_{\kappa}(\mathbf{x}-\mathbf{y})\mathbf{n}(\mathbf{y})}\dif\sigma(\mathbf{y})\dif\sigma(\mathbf{x})\\
= \int_{\Gamma}\int_{\Gamma}\xi(\mathbf{y})G_{\kappa}(\mathbf{x}-\mathbf{y})\mathbf{p}(\mathbf{x})\cdot \mathbf{n}(\mathbf{y})\dif\sigma(\mathbf{y})\dif\sigma(\mathbf{x}) \\
+\int_{\Gamma}\int_{\Gamma}\xi(\mathbf{y})\mathbf{p}(\mathbf{x})\cdot \bra{\nabla^2\tilde{G}_{\kappa}(\mathbf{x}-\mathbf{y})\mathbf{n}(\mathbf{y})}\dif\sigma(\mathbf{y})\dif\sigma(\mathbf{x}),
\end{multline}
where $\tilde{G}_{\kappa}:=\bra{G_{\kappa}-G_{\tilde{\kappa}}}/\kappa^2$.
On the other hand, the same observation implies that $\langle \mathbf{p},\nabla_\Gamma\gamma\mathbf{V})\rangle_{\tau}=\langle \mathbf{p},\gamma\nabla\mathbf{V})\rangle_{\tau}$ for any $\mathbf{V}\in\mathbf{H}_{\text{loc}}^1(\mathbb{R}^3)$, and thus that
\begin{align*}
\langle\gamma^{\pm}_n\nabla\tilde{\psi}_{\kappa}(\text{div}_{\Gamma}(\mathbf{p})), \xi\rangle_{\Gamma} &=\int_{\gamma}\int_{\gamma}\xi(\mathbf{y})\mathbf{n}(\mathbf{y})\cdot\nabla\tilde{G}_{\kappa}\bra{\mathbf{y}-\mathbf{x}}\text{div}_{\Gamma}\bra{\mathbf{p}(\mathbf{x})}\dif\sigma(\mathbf{y})\dif\sigma(\mathbf{x})\\
&=-\int_{\gamma}\int_{\gamma}\xi(\mathbf{y})\mathbf{p}(x)\nabla_{\mathbf{x}}\bra{\mathbf{n}(\mathbf{y})\cdot\nabla\tilde{G}_{\kappa}\bra{\mathbf{y}-\mathbf{x}}}\dif\sigma(\mathbf{y})\dif\sigma(\mathbf{x})\\
&=\int_{\gamma}\int_{\gamma}\xi(\mathbf{y})\mathbf{p}(x)\bra{\nabla^2\tilde{G}_{\kappa}(\mathbf{x}-\mathbf{y})\mathbf{n}(\mathbf{y})}\dif\sigma(\mathbf{y})\dif\sigma(\mathbf{x}),
\end{align*}
where we have remembered that the tangential divergence defined in Section \ref{sec: classical traces} was adjoint to the negative surface gradient. Recognizing the Helmholtz vector single-layer potential in the first expression on the right hand side of \eqref{eq: tangential trace Gamma} concludes the proof.
\end{proof}
\begin{greenFrameSymmetry}
\begin{proposition}\label{prop: adjointness}
There exists a compact operator $\mathcal{C}_k:\mathcal{H}_N\rightarrow \mathcal{H}_N$ such that
\begin{equation*}
\ip{\mathbb{A}^{NN}_\kappa(\vec{\mathbf{p}})}{\vec{\bm{\eta}}} = -\ip{\vec{\mathbf{p}}}{\mathbb{A}^{DD}_{\kappa}(\vec{\bm{\eta}})} + \ip{\mathcal{C}_k\vec{\mathbf{p}}}{\vec{\bm{\eta}}}
\end{equation*}
for all $\vec{\bm{\eta}}:=\bra{\bm{\eta},\,\, \xi}^{\top}\in\mathcal{H}_D$ and $\vec{\mathbf{p}}:=(\mathbf{p},\,\, q)^{\top}\in\mathcal{H}_N$.
\end{proposition}
\end{greenFrameSymmetry}
\begin{proof}
Recall that $\mathbb{A}^{NN}_{\kappa} = \{\mathcal{T}_N\}\cdot\mathcal{S}\mathcal{L}_{\kappa}$ and $\mathbb{A}^{DD}_{\kappa} = \{\mathcal{T}_D\}\cdot\mathcal{D}\mathcal{L}_{\kappa}$. Since $\mathbf{curl}\circ\nabla =0$,
$\langle\{\gamma_R\}\nabla\psi_{\tilde{k}}(q),\bm{\eta}\rangle_{\tau} = 0$ and $\langle\{\gamma_R\}\nabla\tilde{\psi}_{k}(\text{div}_{\Gamma}(\mathbf{p})),\bm{\eta}\rangle_{\tau} = 0$; therefore,
\begin{multline}\label{eq: average single layer}
\ip{\{\mathcal{T}_{N}\}\cdot\mathcal{SL}_k(\vec{\mathbf{p}})}{\vec{\bm{\eta}}}
= \langle-\{\gamma_R\}\bm{\Psi}_{\kappa}(\mathbf{p}),\bm{\eta}\rangle_{\tau} + \langle\{\gamma_n\}\nabla\psi_{\tilde{\kappa}}(q), \xi\rangle_{\Gamma}\\
- \langle\{\gamma_n\}\bm{\Psi}_{\kappa}(\mathbf{p}),\xi \rangle_{\Gamma} -\langle\{\gamma_n\}\nabla\tilde{\psi}_{\kappa}(\text{div}_{\Gamma}(\mathbf{p})), \xi\rangle_{\Gamma}.
\end{multline}
Since $\text{div}\circ\mathbf{curl}=0$, we also have $\{\gamma_D\}\,\mathbf{curl}\bm{\Psi_{\kappa}} =0$. Hence, we need to compare \eqref{eq: average single layer} with
\begin{equation*}
\ip{\vec{\mathbf{p}}}{\{\mathcal{T}_{D}\}\cdot\mathcal{DL}_k(\vec{\bm{\eta}})}=\ip{\mathbf{p}}{\{\gamma_t\}\mathbf{curl}\bm{\Psi}_k(\bm{\eta}\times\mathbf{n})}_{\tau} + \ip{q}{\{\eta\,\gamma_D\}\Upsilon_{\kappa}(\xi)}_{\Gamma} + \ip{\mathbf{p}}{\{\gamma_t\}\Upsilon_{\kappa}(\xi)}_{\tau}.
\end{equation*}
The desired result follows by combining Lemma \ref{lem: adjoint maxwell potential}, Lemma \ref{lem: grad phi term is anti hermitian} and Lemma \ref{lem: symmetry of two last terms single layer}.
\end{proof}
As consequence of Proposition \ref{prop: adjointness}, we have
\newline\newline
\begin{greenRectangle}
\begin{equation}\label{eq: symmetry in calderon projectors}
\bra{\mathbb{P}^+_{\kappa}}_{11}^* \hat{=}\bra{\mathbb{P}^-_{\kappa}}_{22},
\end{equation}
\end{greenRectangle}
\newline\newline
where $\hat{=}$ is used to indicate equality up to compact terms.
\section{Coupled problem}\label{sec: coupled problem}
\par In this section, we derive a variational formulation for the system \eqref{eq: strong from coupled sys a}-\eqref{eq: strong from coupled sys d} which couples a mixed variational formulation defined in the interior domain to a boundary integral equation of the first kind that arises in the exterior domain.
As proposed in \cite{arnold2006finite}, we introduce a new variable $P= -\text{div}\bra{\epsilon(\mathbf{x})\mathbf{U}}$ into equation \eqref{eq: strong from coupled sys a}. Applying Green's formulae \eqref{curl curl integral identity} in $\Omega_s$, we obtain
\begin{align}
\begin{split}
\int_{\Omega_s}\mu^{-1}\,\mathbf{curl}\,\mathbf{U}\cdot\mathbf{curl}\,\mathbf{V}\dif\mathbf{x} + \int_{\Omega_s}\epsilon\,\nabla P\cdot\mathbf{V}\dif\mathbf{x} \qquad {} & \\ -\omega^2\int_{\Omega_s}\epsilon\,\mathbf{U}\cdot\mathbf{V}\dif\mathbf{x}
+\langle \gamma_{R,\mu}^{-}\mathbf{U},\gamma^-_t\mathbf{V}\rangle_{\tau} &= \bra{\mathbf{J},\mathbf{V}}_{\Omega_s},\\
\int_{\Omega_s}P\,Q\dif\mathbf{x} - \int_{\Omega_s}\epsilon\,\mathbf{U}\cdot\nabla Q \dif\mathbf{x} +\langle \gamma^{-}_{n,\epsilon}\mathbf{U},\gamma^{-} Q\rangle_{\Gamma}&= 0
\label{eq: green mixed}
\end{split}
\end{align}
for all $\mathbf{V}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}$, $Q\in H^1(\Omega_s)$. The volume integrals in these equations enter the interior symmetric bi-linear form
\begin{multline}\label{eq: sym bilinear form}
\mathfrak{B}_{\kappa}\bra{\colvec{2}{\mathbf{U}}{P},\colvec{2}{\mathbf{V}}{Q}} := \int_{\Omega_s}\mu^{-1}\,\mathbf{curl}\,\mathbf{U}\cdot\mathbf{curl}\,\mathbf{V}\dif\mathbf{x} + \int_{\Omega_s}\epsilon\,\nabla P\cdot\mathbf{V}\dif\mathbf{x}\\
+ \int_{\Omega_s}P\,Q\dif\mathbf{x}
- \int_{\Omega_s}\epsilon\,\mathbf{U}\cdot\nabla Q \dif\mathbf{x} -\omega^2\int_{\Omega_s}\epsilon\,\mathbf{U}\cdot\mathbf{V}\dif\mathbf{x}
\end{multline}
related to the one supplied for the Hodge-Laplace operator in \cite[Sec. 3.2]{arnold2010finite}. We aim to couple \eqref{eq: sym bilinear form} with the BIEs replacing the PDEs in $\Omega'$. We may utilize the transmission conditions \eqref{eq: strong from coupled sys c}-\eqref{eq: strong from coupled sys d} to amend \eqref{eq: green mixed} to the variational equation
\begin{equation*}
\mathfrak{B}_{\kappa}\bra{\colvec{2}{\mathbf{U}}{P},\colvec{2}{\mathbf{V}}{Q}} + \hlly{\Big\langle \mathcal{T}^+_{N} (\mathbf{U}^{\text{ext}}),\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q} \Big\rangle} = \mathscr{G}\bra{\colvec{2}{\mathbf{V}}{Q}}, \label{eq: bilinear var. 1}
\end{equation*}
which sports a functional $\mathscr{G}\bra{(\mathbf{V}\, Q)^{\top}} := (\mathbf{J},\mathbf{V})_{\Omega_s} - \langle (\mathbf{g}_R\,g_n)^{\top},(\gamma^{-}_t\mathbf{V}\,\gamma^{-} Q)^{\top}\rangle$
bounded over the test space. The exterior Calder\'on projector can be used to express the so-called Dirichlet-to-Neumann operator in different ways.
Introducing the jump conditions into the {\color{cyan}first exterior Calder\'on identity} given on the first line of \eqref{exterior calderon} along with a new unknown $\vec{\mathbf{p}}=\mathcal{T}^+_{N} (\mathbf{U}^{\text{ext}})$ yields a variational system
\begin{equation}
\begin{gathered}
\mathfrak{B}_{\kappa}\bra{\colvec{2}{\mathbf{U}}{P},\colvec{2}{\mathbf{V}}{Q}} + \hlly{\Big\langle \vec{\mathbf{p}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q} \Big\rangle} = \mathscr{G}\bra{\colvec{2}{\mathbf{V}}{Q}},\\
\hllc{\Big\langle \bra{\{\mathcal{T_{D}}\}\cdot\mathcal{DL}_{\kappa}+\frac{1}{2}\id}\mathcal{T}_{D,\epsilon}^-(\mathbf{U}), \vec{\mathbf{a}}\Big\rangle + \Big\langle \{\mathcal{T}_{D}\}\cdot\mathcal{SL}_{\kappa}\bra{\vec{\mathbf{p}}},\vec{\mathbf{a}}\Big\rangle} = \mathscr{R}\bra{\vec{\mathbf{a}}},
\end{gathered}
\label{eq:Johnson-Nedelec}
\end{equation}
for all $(\mathbf{V}\,Q)^{\top}\in \mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$ and $\vec{\mathbf{a}}\in\mathcal{H}_N$,
resembling the original Johnson-Ned\'elec coupling \cite{aurada2013classical}. The new functional appearing on the right hand side of \eqref{eq:Johnson-Nedelec} is defined as $\mathscr{R}\bra{\vec{\mathbf{a}}}: = \langle \bra{\{\mathcal{T}_{D}\}\cdot\mathcal{DL}_{\kappa}+\frac{1}{2}\id}(\bm{\zeta}_t,\,\zeta_D)^{\top}, \vec{\mathbf{a}}\rangle$.
Following the exposition of Costabel in \cite{costabel1987symmetric}, we also retain the {\color{violet}second exterior Calder\'on identity} ---in which we again introduce the jump conditions to eliminate the dependence on the exterior solution--- and insert the resulting equation in \eqref{eq:Johnson-Nedelec} to obtain the symmetrically coupled problem: find $\vec{\mathbf{U}}:=(\mathbf{U},\,\,P)^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$ and $\vec{\mathbf{p}}\in \mathcal{H}_{N}$ such that
\begin{align}\label{Calderon coupled problem no Ak}
\begin{split}
\mathfrak{B}_{\kappa}\bra{\vec{\mathbf{U}},\vec{\mathbf{V}}} + \hllv{\Big\langle \bra{-\{\mathcal{T}_{N}\}\cdot\mathcal{SL}_{\kappa}+\frac{1}{2}\id}\vec{\mathbf{p}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q} \Big\rangle} \qquad {}\qquad {}\qquad {}&\\
\hllv{+\,\Big\langle-\{\mathcal{T}_{N}\}\cdot\mathcal{DL}_{\kappa}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q}\Big\rangle} &= \mathscr{F}\bra{\vec{\mathbf{V}}}\\
\hllc{\Big\langle \bra{\{\mathcal{T_{D}}\}\cdot\mathcal{DL}_{\kappa}+\frac{1}{2}\id}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}}, \vec{\mathbf{a}}\Big\rangle + \Big\langle \{\mathcal{T}_{D}\}\cdot\mathcal{SL}_{\kappa}\bra{\vec{\mathbf{p}}},\vec{\mathbf{a}}\Big\rangle} &= \mathscr{R}\bra{\vec{\mathbf{a}}},
\end{split}
\end{align}
for all $\vec{\mathbf{V}}:=(\mathbf{V},\,\,Q)^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$, $\vec{\mathbf{a}}\in \mathcal{H}_N$. Yet again, the right hand side of our system of equations has been modified to bear a new bounded linear functional $\mathscr{F}(\vec{\mathbf{V}}):= \mathscr{G}(V) + \langle -\{\mathcal{T}_{N}\}\cdot\mathcal{DL}_{\kappa}(\bm{\zeta}_t\,\zeta_D)^{\top},(\gamma^{-}_t\mathbf{V},\,\gamma^{-} Q)^{\top}\rangle$.
In terms of the Calder\'on projector, problem \eqref{Calderon coupled problem no Ak} can be rewritten more succinctly as
\begin{greenFrameCoupledProblem}
Find $\vec{\mathbf{U}}:=(\mathbf{U},\,\,P)^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$ and $\vec{\mathbf{p}}\in \mathcal{H}_{N}$ such that
\begin{align} \label{Calderon coupled problem}
\begin{split}
\mathfrak{B}_{\kappa}\bra{\vec{\mathbf{U}},\vec{\mathbf{V}}} + \Big\langle \bra{-\mathbb{A}_{\kappa}^{NN}+\frac{1}{2}\id}\vec{\mathbf{p}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q} \Big\rangle \qquad {}\qquad {}\qquad {}&\\
+\,\Big\langle-\mathbb{A}_{\kappa}^{DN}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q}\Big\rangle &= \mathscr{F}\bra{\vec{\mathbf{V}}}\\
\Big\langle \bra{\mathbb{A}^{DD}_{\kappa}+\frac{1}{2}\id}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}}, \vec{\mathbf{a}}\Big\rangle + \Big\langle \mathbb{A}^{DD}_{\kappa}\bra{\vec{\mathbf{p}}},\vec{\mathbf{a}}\Big\rangle &= \mathscr{R}\bra{\vec{\mathbf{a}}},
\end{split}
\end{align}
for all $\vec{\mathbf{V}}:=(\mathbf{V},\,\,Q)^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$, $\vec{\mathbf{a}}\in \mathcal{H}_N$.
\end{greenFrameCoupledProblem}
\begin{remark} Part of the justification for using mixed formulations for problems involving the Hodge--Helmholtz/Laplace operator is the need to move away from trial spaces contained in $\mathbf{H}(\mathbf{curl},\Omega_s)\cap\mathbf{H}(\text{div},\Omega_s)$, because the latter doesn't allow for viable discretizations using finite elements \cite{arnold2010finite}. While from \eqref{eq:Johnson-Nedelec} the issue seems to reappear after using the Cald\'eron identities, the benefits of the introduced new unknown $P\in H^1(\Omega_s)$ conveniently carries over to the coupled system \eqref{Calderon coupled problem} upon substituting $-\gamma^-\bra{P}$ in place of $\gamma_{D,\epsilon}(\mathbf{U})$ in $\mathcal{T}^-_{D,\epsilon}(\mathbf{U})$.
\end{remark}
In the following proposition, we call \emph{forbidden resonant frequencies} the interior ``Dirichlet" eigenvalues of the scaled Hodge-Laplace operator with constant coefficient $\eta = \mu_0\epsilon_0^2$. That is, $\kappa^2$ is a forbidden frequency if there exists $0\neq\mathbf{U}\in \mathbf{X}(\Delta,\Omega)$ with $\Delta_{\eta}\mathbf{U}=\kappa^2\mathbf{U}$ and $\mathcal{T}^{-}_D\mathbf{U}=0$. We refer the reader to \cite{claeys2017first}, where the spectrum of the scaled Hodge-Laplace operator is completely characterized. See for e.g. \cite{schulz2020b}, \cite{sauter2010boundary}, \cite{christiansen2004discrete}, \cite{demkowicz1994asymptotic} and \cite{colton2013integral} for an overview of the issue of spurious resonances in electromagnetic and acoustic scattering models based on integral equations.
\begin{greenFrameResultBeforeAfter}
\begin{proposition}\label{prop: variational system solves transmission system}
Suppose that $\kappa^2\in\mathbb{C}$ avoids forbidden resonant frequencies. By retaining an interior solution $U\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}$ and producing $\mathbf{U}^{\text{ext}}\in\mathbf{X}_{\text{\emph{loc}}}(\Delta,\Omega')$ using the representation formula \eqref{representation formula} for the Cauchy data $(\vec{\mathbf{p}},\mathcal{T}_{D,\epsilon}^-U - (\bm{\zeta}_t,\,\,\zeta_D)^{\top})$ with $\gamma^-_{D,\epsilon}(\mathbf{U}) = -\gamma^-\bra{P}$, a solution to \eqref{Calderon coupled problem} solves the transmission system \eqref{eq: strong from coupled sys a}-\eqref{eq: strong from coupled sys d} in the sense of distribution.
\end{proposition}
\end{greenFrameResultBeforeAfter}
\begin{proof}
\par The proof follows the approach in \cite[Lem. 6.1]{hiptmair2003coupling}. Since $\mathscr{D}(\Omega_s)^3\times C^{\infty}_0(\Omega_s)$ is a subset of the volume test space, any solution to the problem \eqref{Calderon coupled problem} solves \eqref{eq: strong from coupled sys a} in $\Omega_s$ in the sense of distribution. It follows that \eqref{eq: green mixed} holds for all admissible $\vec{\mathbf{V}}$, which reduces \eqref{Calderon coupled problem} to the variational system
\begin{align*}
-\Big\langle \vec{\mathbf{q}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q}\Big\rangle + \Big\langle \bra{-\mathbb{A}^{NN}_{\kappa}+\frac{1}{2}\id}\vec{\mathbf{p}},\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q} \Big\rangle
-\, \Big\langle \mathbb{A}^{DN}_{\kappa}(\vec{\bm{\xi}}),\colvec{2}{\gamma^{-}_t\mathbf{V}}{\gamma^{-} Q}\Big\rangle &= 0\\
\Big\langle \bra{\mathbb{A}^{DD}_{\kappa}+\frac{1}{2}\id}\vec{\bm{\xi}}, \vec{\bm{\eta}}\Big\rangle + \Big\langle \{\mathbb{A}^{ND}_{\kappa}\bra{\vec{\mathbf{p}}},\vec{\bm{\eta}}\Big\rangle &= 0
\end{align*}
where $\vec{\mathbf{q}} := \mathcal{T}_{N,\mu}^-(\mathbf{U}) - (\mathbf{g}_R,\,\,g_n)^{\top}$ and $\vec{\bm{\xi}}:=\mathcal{T}^-_{D,\epsilon}(\mathbf{U})-(\bm{\zeta}_t,\,\,\zeta_D)^{\top}$.
We recognize in the equivalent operator equation
\begin{equation}
\underbrace{\begin{pmatrix}
\mathbb{A}^{NN}_{\kappa}+\frac{1}{2}\id & \mathbb{A}^{DN}_{\kappa}\\
\mathbb{A}^{ND}_{\kappa} & \mathbb{A}^{DD}_{\kappa}+\frac{1}{2}\id
\end{pmatrix}}_{\mathbb{P}^-_{\kappa}}
\begin{pmatrix}
\vec{\mathbf{p}}\\
\vec{\bm{\xi}}
\end{pmatrix}
= \begin{pmatrix}
\vec{\mathbf{p}}-\vec{\mathbf{q}}\\
0
\end{pmatrix}
\label{op eq calderon}
\end{equation}
the interior Cald\'eron projector \eqref{interior calderon} whose image, by Lemma \ref{prop: valid cauchy data}, is the space of valid Cauchy data for the homogeneous (scaled) Hodge--Laplace/Helmholtz interior equation with constant coefficient $\eta$. In particular, $\vec{\mathbf{p}}-\vec{\mathbf{q}}=\mathcal{T}^-_N\bra{\tilde{\mathbf{U}}}$ for some vector-field $\tilde{\mathbf{U}}\in \mathbf{X}\bra{\Delta,\Omega_s}$ satisfying
\begin{align}\label{eq: hodge-laplace homogeneous dirichlet BVP}
\begin{split}
\Delta_{\eta} \tilde{\mathbf{U}} - \kappa^2 \tilde{\mathbf{U}}&= 0, \qquad\qquad\qquad\text{in }\Omega_s\\
\mathcal{T}^-_D\bra{\tilde{\mathbf{U}}} &= 0, \qquad\qquad\qquad \text{on }\Gamma.
\end{split}
\end{align}
If $\kappa^2\neq 0$, we rely on the hypothesis that $\kappa^2$ doesn't belong to the set of forbidden resonant frequencies to guarantee injectivity of the above boundary value problem \cite[Prop. 3.7]{claeys2017first}. Otherwise, a trivial de Rham cohomology implies that zero is not a Dirichlet eigenvalue. We conclude that $\tilde{\mathbf{U}}=0$ is the unique trivial solution to \eqref{eq: hodge-laplace homogeneous dirichlet BVP}. Therefore, for the right hand side of \eqref{op eq calderon} to exhibit valid Neumann data, it must be that $\vec{\mathbf{p}}=\vec{\mathbf{q}}$.
By Lemma \ref{prop: valid cauchy data} again, the null space of the interior Cald\'eron projector $\mathbb{P}^-_{\kappa}$ coincides with valid Cauchy data for the exterior boundary value problem \eqref{eq: strong from coupled sys b} complemented with the radiation conditions at infinity introduced in Section \ref{sec: Introduction}.
In particular $(\vec{\mathbf{p}},\,\,\vec{\bm{\xi}})^{\top}$ is valid Cauchy data for that exterior Hodge-Helmholtz/Laplace problem and $\mathbf{U}^{\text{ext}}=\mathcal{SL}_{\kappa}\bra{\vec{\mathbf{p}}} + \mathcal{DL}_{\kappa}\bra{\vec{\bm{\xi}}}$ solves \eqref{eq: strong from coupled sys b} and \eqref{eq: strong from coupled sys d} by construction. The fact that $\vec{\mathbf{p}}=\mathcal{T}^+_{N}\bra{\mathbf{U}^{\text{ext}}}$ solves \eqref{eq: strong from coupled sys c} is confirmed by the earlier observation that $\vec{\mathbf{p}}=\vec{\mathbf{q}}$.
\end{proof}
\begin{corollary}\label{cor: existence and uniqueness}
Suppose that $\kappa^2\in\mathbb{C}$ avoids forbidden resonant frequencies. A solution pair $\bra{\vec{\mathbf{U}},\,\vec{\mathbf{p}}}$ to the coupled problem \eqref{Calderon coupled problem} is unique.
\end{corollary}
\begin{remark}
We show in \cite{schulz2020b} that when $\kappa^2$ happens to be a resonant frequency, the interior solution $\mathbf{U}$ remains unique. This is no longer true for $\vec{\mathbf{p}}$ however, which is in general only unique up to Neumann traces of interior Dirichlet eigenfunctions of $\Delta_{\eta}$ associated to the eigenvalue $\kappa^2$. Fortunately, this kernel vanishes under the exterior representation formula obtained from \eqref{representation formula}.
\end{remark}
\begin{remark}
In principle, the CFIE-type stabilization strategy proposed in \cite{hiptmair2005stable} for the (scalar) Helmholtz transmission problem could also be attempted here to get rid of the spurious resonances haunting the coupled problem \eqref{Calderon coupled problem}, but such developments lie outside the scope of this work.
\end{remark}
\section{Space decompositions}\label{sec: space decomposition}
Using the classical Hodge decomposition, a general inf-sup condition for Hodge--Laplace problems posed on closed Hilbert complexes was derived in \cite{arnold2010finite}. However, as orthogonality won't be required, we rather opt for the enhanced regularity of the regular decomposition suggested in \cite{buffa2003galerkin} and \cite{claeys2017first}.
\begin{lemma}[{See \cite[Lem. 3.5]{amrouche1998vector}}]
There exists a continuous linear lifting $\mathsf{L}: \mathbf{H}(\text{\emph{div}}, \Omega_s)\cap\ker (\text{\emph{div}})\rightarrow \mathbf{H}^1(\Omega_s)$ such that $\text{\emph{div}}(\mathsf{L}\mathbf{U})=0$ and $\mathbf{curl}\bra{\mathsf{L}\mathbf{U}}=\mathbf{U}$.
\end{lemma}
\begin{lemma}\label{lem: proj volume}
The operator $\mathsf{Z}:\mathbf{H}\bra{\mathbf{curl},\Omega_s}\rightarrow\mathbf{H}^1(\Omega_s)$ defined by $$\mathsf{Z}\bra{\mathbf{U}}:=\mathsf{L}\bra{\mathbf{curl}\bra{\mathbf{U}}}$$ is a continuous projection with $\ker\bra{\mathsf{Z}}=\ker\bra{\mathbf{curl}}\cap\mathbf{H}\bra{\mathbf{curl},\Omega_s}$ and satisfying $\mathbf{curl}\bra{\mathsf{Z}(\mathbf{U})}=\mathbf{curl}\bra{\mathbf{U}}$.
\end{lemma}
The following corollary follows immediately from Rellich theorem.
\begin{corollary}\label{cor: volume proj is compact}
The projection operator of Lemma \ref{lem: proj volume} is compact as a mapping $\mathsf{Z}:\mathbf{H}\bra{\mathbf{curl},\Omega_s}\rightarrow \mathbf{L}^2(\Omega_s)$.
\end{corollary}
The subspaces $\mathbf{X}(\mathbf{curl},\Omega_s):=\mathsf{Z}\bra{\mathbf{H}\bra{\mathbf{curl},\Omega_s}}$ and $\mathbf{N}\bra{\mathbf{curl},\Omega_s}:=\ker\bra{\mathbf{curl}}\cap\mathbf{H}\bra{\mathbf{curl},\Omega_s}$ provide, by vitrue of the continuity of $\mathsf{Z}$, a stable direct regular decomposition
\begin{equation*}
\mathbf{H}\bra{\mathbf{curl},\Omega_s} = \mathbf{X}(\mathbf{curl},\Omega_s) \oplus \mathbf{N}\bra{\mathbf{curl},\Omega_s}.
\end{equation*}
A decomposition with similar properties can be designed for $\mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma}$. We define $\mathsf{J}: H^{-1/2}(\Gamma)\rightarrow \mathbf{H}^1(\Omega_s)$ by $\mathsf{J}(g) =\mathsf{L}\bra{\nabla U}$, where $U\in H_*^1(\Omega_s):=\{U\in H^1(\Omega_s) : \int_{\Omega_s} U \dif\mathbf{x}=0\}$ solves the Neumann problem
\begin{align}
\Delta U &= 0, &\text{in }\Omega_s,\label{eq: laplace neuman lifting}\\
\gamma_n^-\bra{\nabla U} &= g, &\text{on }\Gamma.\nonumber
\end{align}
This map is well-defined, since \eqref{eq: laplace neuman lifting} ensures that $\nabla U\in\text{Dom}(\mathsf{L})$.
\begin{lemma}\label{lem: trace projection}
The operator $\mathsf{Z}^{\Gamma}:\mathbf{H}^{-1/2}\bra{\text{\emph{div}}_{\Gamma},\Gamma}\rightarrow\mathbf{H}^{1/2}_R(\Gamma)$ defined by
\begin{equation}
\mathsf{Z}^{\Gamma} := \gamma_{\tau} \circ \mathsf{J}\circ \text{\emph{div}}_{\Gamma}
\end{equation}
is a continuous projection with $\ker(\mathsf{Z}^{\Gamma})=\ker\bra{\text{\emph{div}}_{\Gamma}}\cap\mathbf{H}^{-1/2}\bra{\text{\emph{div}}_{\Gamma},\Gamma}$ and satisfying
$
\text{\emph{div}}_{\Gamma}\bra{\mathsf{Z}^{\Gamma}(\mathbf{p})}=\text{\emph{div}}_{\Gamma}\bra{\mathbf{p}}.
$
\end{lemma}
As before, the extra regularity of the range, in this case provided by Lemma \ref{lem: Rellich for boundary}, leads to a valuable corollary.
\begin{corollary}\label{cor: trace proj is compact}
The projection operator of Lemma \ref{lem: trace projection} is compact as a mapping $\mathsf{Z}^{\Gamma}:\mathbf{H}^{-1/2}\bra{\text{\emph{div}}_{\Gamma},\Gamma}\rightarrow\mathbf{H}^{-1/2}_R(\Gamma)$.
\end{corollary}
The subspaces $\mathbf{X}\bra{\text{div}_{\Gamma},\Gamma}:=\mathsf{Z}^{\Gamma}\bra{\mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma}}$ and $\mathbf{N}\bra{\text{div}_{\Gamma},\Gamma}:=\ker\bra{\text{div}_{\Gamma}}\cap\mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma}$ provide a stable direct regular decomposition
\begin{equation*}
\mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma} = \mathbf{X}\bra{\text{div}_{\Gamma},\Gamma}\oplus \mathbf{N}\bra{\text{div}_{\Gamma},\Gamma}.
\end{equation*}
In the following, we may simplify notation by using $\mathbf{U}^{\perp}:=\mathsf{Z}\mathbf{U}$, $\mathbf{p}^{\perp}:=\mathsf{Z}^{\Gamma}\mathbf{p}$, $\mathbf{U}^0:=\bra{\id-\mathsf{Z}}\mathbf{U}$ and
$\mathbf{p}^0:=\bra{\id-\mathsf{Z}^{\Gamma}}\mathbf{p}$.
A very useful property of this pair of decompositions is stated in the next proposition.
\begin{lemma}[{See \cite[Lem. 8.1]{hiptmair2003coupling} and \cite[Lem. 8.2]{hiptmair2003coupling}}]\label{lem: compact sym parts}
The operators
\begin{equation*}
\bra{\gamma_t^{-}}^*\circ\bra{\{\gamma_R\}\bm{\Psi}_{\kappa}+\frac{1}{2}\id}:\mathbf{N}\bra{\text{\emph{div}}_{\Gamma},\Gamma}\rightarrow \mathbf{N}\bra{\mathbf{curl},\Omega_s}',
\end{equation*}
and
\begin{equation*}
\bra{\gamma_t^{-}}^*\circ\bra{\{\gamma_R\}\bm{\Psi}_{\kappa}+\frac{1}{2}\id}:\mathbf{X}\bra{\text{\emph{div}}_{\Gamma},\Gamma}\rightarrow \mathbf{X}\bra{\mathbf{curl},\Omega_s}'
\end{equation*}
are compact.
\iffalse
\begin{equation*}
\bra{\gamma_t^{-}}^*\circ\bra{\{\gamma_R\}\bm{\Psi}_{\kappa}+\frac{1}{2}\id}:\mathbf{N}\bra{\text{\emph{div}}_{\Gamma},\Gamma}\rightarrow \mathbf{N}\bra{\mathbf{curl},\Omega_s}'
\end{equation*}
and
\begin{equation*}\bra{\gamma_t^{-}}^*\circ\bra{\{\gamma_R\}\bm{\Psi}_{\kappa}+\frac{1}{2}\id}:\mathbf{X}\bra{\text{\emph{div}}_{\Gamma},\Gamma}\rightarrow \mathbf{X}\bra{\mathbf{curl},\Omega_s}'
\end{equation*}
\fi
\end{lemma}
Another benefit of this pair of regular decompositions will become explicit in the poof of Lemma \ref{lem: coercivity TNDL} found in the next section (see equation \eqref{eq: tau trace using commutative diagram}).
It follows from Lemma \ref{lem: properties of surface div and curl} that $\text{div}_{\Gamma}:\mathbf{X}\bra{\text{div}_{\Gamma},\Gamma}\rightarrow H^{-1/2}_*(\Gamma)$ is a continuous bijection. The bounded inverse theorem guarantees the existence of a continuous inverse $\bra{\text{div}_{\Gamma}}^{\dag}:H^{-1/2}_*(\Gamma)\rightarrow \mathbf{X}\bra{\text{div}_{\Gamma},\Gamma}$ such that
\begin{align*}
\bra{\text{div}_{\Gamma}}^{\dag}\circ\text{div}_{\Gamma} &=\id\Big\vert_{\mathbf{X}\bra{\text{div}_{\Gamma},\Gamma}}, & &\text{div}_{\Gamma}\circ\bra{\text{div}_{\Gamma}}^{\dag}=\id\Big\vert_{H^{-1/2}_*(\Gamma)}.
\end{align*}
We denote $Q_*:H^1(\Omega_s)\rightarrow H^1_*(\Omega_s)$ the projection onto mean zero functions.
\section{Well-posedness of the coupled variational problem}\label{sec: compactness and coercivity}
We use the direct decompositions introduced in Section \ref{sec: space decomposition} to prove that the bilinear form associated to the coupled system \eqref{prop: variational system solves transmission system} of Section \ref{sec: coupled problem} satisfies a generalized G{\aa}rding inequality.
\par The coupled variational problem \eqref{Calderon coupled problem} translates into the operator equation
\begin{equation}\label{eq: coupled system operator equation}
\mathbb{G}_{\kappa}
\begin{pmatrix}
\vec{\mathbf{U}}\\
\vec{\mathbf{p}}
\end{pmatrix} =
\colvec{2}{\mathscr{F}}{\mathscr{R}}\in \bra{\mathbf{H}\bra{\mathbf{curl},\Omega_s}\times H^1(\Omega)}'\times \bra{\mathcal{H}_N}'.
\quad
\end{equation}
Letting $\mathsf{B}_{\kappa}:\mathbf{H}\bra{\mathbf{curl},\Omega_s}\times H^1\bra{\Omega_s}\rightarrow\bra{\mathbf{H}\bra{\mathbf{curl},\Omega_s}\times H^1\bra{\Omega_s}}'$ be the operator
$$\langle\mathsf{B}_{\kappa}\bra{\vec{\mathbf{U}}}\vec{\mathbf{V}}\rangle:=\mathfrak{B}_{\kappa}\bra{\vec{\mathbf{U}},\vec{\mathbf{V}}}$$
associated with the Hodge-Helmholtz/Laplace volume contribution to the system, the operator $\mathbb{G}_{\kappa}:\bra{\mathbf{H}\bra{\mathbf{curl},\Omega_s}\times H^1(\Omega)}\times \mathcal{H}_N\rightarrow\bra{\mathbf{H}\bra{\mathbf{curl},\Omega_s}\times H^1(\Omega)}'\times \bra{\mathcal{H}_N}'$ can be represented by the block operator matrix
\newline\newline
\begin{greyTracesDef}
\begin{equation}\label{eq: coupled equation operator matrix}
\mathbb{G}_{\kappa}=\begin{pmatrix}
\begin{array}{c|c}
{\color{violet}\mathsf{B}_{\kappa}}{\color{purple}-\colvec{2}{\bra{\gamma_t^-}^*}{\bra{\gamma^-}^*}\cdot\mathbb{A}^{DN}_{\kappa}\cdot\colvec{2}{\gamma_t^-}{-\gamma^-}} & {\color{olive}\colvec{2}{\bra{\gamma_t^-}^*}{\bra{\gamma^-}^*}\cdot\bra{\mathbb{P}^+_{\kappa}}_{22}}\\
\hline
{\color{teal}\bra{\mathbb{P}_{\kappa}^-}_{11}\cdot\colvec{2}{\gamma_t^-}{-\gamma^-}} & {\color{magenta}\mathbb{A}_{\kappa}^{ND}}
\end{array}
\end{pmatrix},
\end{equation}
\end{greyTracesDef}
\newline\newline
shown here in ``variational arrangement".
The symmetry revealed in \eqref{eq: symmetry in calderon projectors} makes explicit much of the structure of the above operator. We have introduced colors to better highlight the contribution of each individual block in the following sections.
Our goal is to design an isomorphism $\mathbb{X}$ of the test space and resort to compact perturbations of $\mathbb{G}_{\kappa}\circ\mathbb{X}^{-1}$ to achieve an operator block structure with diagonal blocks that are elliptic over the splittings of Section \ref{sec: space decomposition} and off-diagonal blocks that fit a skew-symmetric pattern. Stability of the coupled system can then be obtained from the next theorem.
\begin{theorem}[{See \cite[Thm. 4]{buffa2003galerkin}}]\label{thm: coercivity implies fredholm}
If a bilinear form $a:V\times V\rightarrow \mathbb{C}$ on a reflexive Banach space $V$ is T-coercive:
\begin{equation}\label{eq: t-coercivity def}
\abs{a\bra{u,\mathbb{X}\overline{u}} + c\bra{u,\overline{u}}}\geq C\norm{u}^2_V\quad\forall u\in V,
\end{equation}
with $C>0$, $c:V\times V\rightarrow\mathbb{C}$ compact and $\mathbb{X}:V\rightarrow V$ an isomorphism of $V$, then the operator $A:V\rightarrow V'$ defined by $A:u\mapsto a(u,\cdot)$ is Fredholm with index 0.
\end{theorem}
The authors of \cite{Buffa2002Boundary} refer to \eqref{eq: t-coercivity def} as ``Generalized G{\aa}rding inequality", because
\begin{equation*}
\abs{a\bra{u,\mathbb{X}\overline{u}}}\geq C\norm{u}^2_V -\abs{c\bra{u,\overline{u}}},\qquad \forall\,u\in V,
\end{equation*}
generalizes the classical G{\aa}rding inequality which reads
\begin{equation*}
b(u,u) \geq C_1\norm{u}^2_{H^\ell(\Omega)}- C_2\norm{u}_{L^2\bra{\Omega}},\qquad \forall\,u\in H^\ell_0(\Omega),
\end{equation*}
for some $C_2\geq 0$, $C_1>0$, where $b$ is a bilinear form associated to a uniformly elliptic operator of even order $2\ell$. Assuming that \eqref{eq: t-coercivity def} holds with $\mathbb{X}=\id$, a simple proof of the stability estimate
\begin{equation*}
\norm{u}_V \leq C\norm{f}_{V'},
\end{equation*}
obtained for the unique solution of the operator equation $Au=f$ when $A$ is injective is given in \cite[Thm. 3.15]{steinbach2007numerical}. A proof of the general case can be deduced from \cite{hildebrandt1964constructive}.
T-coercivity theory is a reformulation of the Banach-Ne{\u c}as-Babu{\u s}ka theory. The former relies on the construction of explicit inf-sup operators at the discrete and continuous levels, whereas the later develops on an abstract inf-sup condition \cite{ciarlet2012t}.
In deriving the following results, it will be convenient to denote $\vec{\overline{\mathbf{U}}} := \bra{\overline{\mathbf{U}},\,\,\overline{P}}^{\top}$ and $\vec{\overline{\mathbf{p}}} := \bra{\overline{\mathbf{p}},\,\,\overline{q}}^{\top}$.
\subsection{Space isomorphisms}
In this section, we take up the challenge of finding a suitable isomorphism $\mathbb{X}$. We build it separately for the function spaces in $\Omega_s$ and on the boundary $\Gamma$. Crucial hints are offered by the construction of the sign-flip isomorphism of \cite{buffa2003galerkin}.
We start with devising an isomorphism $\Xi$ of the volume function spaces and show that the upper-left diagonal block of $\mathbb{G}_{\kappa}$ satisfy a generalized Gu{\aa}rding inequality.
Under the assumption that $\Omega_s$ has trivial de Rham cohomology, there exists a bijective ``scalar potential lifting" $\mathsf{S}:\mathbf{N}(\mathbf{curl},\Omega_s)\rightarrow H^1_*\bra{\Omega_s}$ satisfying $\nabla \mathsf{S}\bra{\mathbf{U}}=\mathbf{U}$. The Poincar\'e-Friedrichs inequality guarantees that this map is continuous.
Notice that since it also follows from the Poincar\'e-Friedrichs inequality that $\nabla:H_*^1(\Omega_s)\rightarrow\mathbf{N}(\mathbf{curl},\Omega_s)$ is injective, $\mathsf{S}\circ\nabla:H^1(\Omega_s)\rightarrow H_*^1(\Omega_s)$ is a bounded projection onto the space of Lebesgue measurable functions having zero mean. It's nullspace consists of the constant functions in $\Omega_s$.
\begin{greenFrameFunctionIso}
\begin{proposition}\label{prop: isomorphism volume space}
For any $\theta>0$ and $\beta>0$, the bounded linear operator $\Xi:\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\rightarrow\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$ defined by
\begin{equation*}
\Xi\bra{\vec{\mathbf{U}}}:=\colvec{2}{\mathbf{U}^{\perp}-\mathbf{U}^0+\beta\,\nabla P}{-\theta\bra{J\bra{\mathbf{U}^0}+\,\beta\,\mathbf{mean}\bra{P}}}.
\end{equation*}
has a continuous inverse. In other words, $\Xi$ is an isomorphism of Banach spaces.
\end{proposition}
\end{greenFrameFunctionIso}
\begin{proof}
By showing that $\Xi$ is a bijection, the theorem follows as a consequence of the bounded inverse theorem.
Let $\bra{\mathbf{V}\,\,Q}^{\top}\in \mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$. Since $\nabla Q\in\mathbf{N}\bra{\mathbf{curl},\Omega_s}$, we immediately have $\mathsf{Z}\bra{\mathbf{V}^{\perp}-\theta^{-1}\nabla Q}=\mathbf{V}^{\perp}$ and $\bra{\id-\mathsf{Z}}\bra{\mathbf{V}^{\perp}-\theta^{-1}\nabla Q}= -\theta^{-1}\nabla Q$. Hence, relying on the resulting observation that $\nabla \mathsf{S}\bra{\bra{\mathbf{V}^{\perp}-\theta^{-1}\nabla Q}^0} = -\theta^{-1}\nabla Q$ and exploiting that $\mathbf{mean}\bra{H^1_*(\Omega_s)}=\{0\}$, we have
\begin{align}\label{eq: inverse of Xi candidate}
\Xi\bra{\colvec{2}{\mathbf{V}^{\perp}-\theta^{-1}\nabla Q}{\beta^{-1}\bra{\mathsf{S}\bra{\mathbf{V}^0}-\theta^{-1}Q}}}
&=\colvec{2}{\mathbf{V}}{\mathsf{S}\bra{\nabla Q}+\mathbf{mean}\bra{Q}}.
\end{align}
Since $H^1(\Omega_s)$ decomposes into the stable direct sum of $H^1_*(\Omega_s)$ and the space of constant functions in $\Omega_s$, \eqref{eq: inverse of Xi candidate} shows that $\Xi$ is surjective.
Now, suppose that $\Xi\bra{\vec{\mathbf{V}}}=\Xi\bra{\vec{\mathbf{U}}}$. Then, we have
\begin{equation*}
\mathbf{U}^0-\mathbf{V}^0 = \nabla \mathsf{S}\bra{\mathbf{U}^0-\mathbf{V}^0} = \beta\,\nabla\bra{\mathbf{mean}\bra{Q-P}} = 0.
\end{equation*}
Since the considerations of Section \ref{sec: space decomposition} readily yield that $\mathbf{V}^\perp = \mathbf{U}^\perp$, we conclude that $\mathbf{V}=\mathbf{U}$. In turn, it follows that $\nabla P= \nabla Q$ and $\mathbf{mean}(P)=\mathbf{mean}(Q)$. Therefore, $\Xi$ is injective.
\end{proof}
We now turn to the design of an isomorphism for the Neumann trace space $\mathcal{H}_N$ and prove that the lower-right block $\mathbb{A}_{\kappa}^{ND}$ of $\mathbb{G}_{\kappa}$ satisfies a generalized Gu{\aa}rding inequality.
\begin{greenFrameTraceIso}
\begin{proposition}\label{prop: isomorphism trace space}
For any $\tau > 0$ and $\lambda>0$, the bounded linear operator $\Xi^{\Gamma}:\mathcal{H}_N\rightarrow\mathcal{H}_N$ defined by
\begin{equation*}
\Xi^{\Gamma}(\vec{\mathbf{p}}) :=\colvec{2}{\mathbf{p}^{\perp}-\mathbf{p}^0 -\lambda\bra{\text{\emph{div}}_{\Gamma}}^{\dag}\mathsf{Q}_*q}{-\tau\bra{\text{\emph{div}}_{\Gamma}\bra{\mathbf{p}}+\lambda\,\mathbf{mean}\bra{q}}}
\end{equation*}
has a continuous inverse. In other words, $\Xi^{\Gamma}$ is an isomorphism of Banach spaces.
\end{proposition}
\end{greenFrameTraceIso}
\begin{proof}
We proceed as in proposition \ref{prop: isomorphism volume space}. Since $\bra{\text{div}_{\Gamma}}^{\dag}\mathsf{Q}_*q\in\mathbf{X}(\text{div}_{\Gamma},\Gamma)$, we have $\mathsf{Z}^{\Gamma}\bra{\Xi_1^{\Gamma}(\vec{\mathbf{p}})}=\mathbf{p}^{\perp}-\bra{\text{div}_{\Gamma}}^{\dag}\mathsf{Q}_*q$. Using that $\mathbf{mean}\circ\text{div}_{\Gamma}=0$ and $\bra{\text{div}_{\Gamma}}^{\dag}\text{div}_{\Gamma}\mathbf{p}=\mathbf{p}^{\perp}$, we evaluate
\begin{align*}
\Xi^{\Gamma}\bra{\colvec{2}{-\mathbf{p}^0-\tau^{-1}\bra{\text{div}_{\Gamma}}^{\dag}\mathsf{Q}_*q}{\lambda^{-1}\bra{-\text{div}_{\Gamma}(\mathbf{p})-\tau^{-1}q}}} &=\colvec{2}{\mathbf{p}^0+\mathbf{p}^{\perp}}{\mathsf{Q}_*q+\mathbf{mean}\bra{q}}.
\end{align*}
This shows that $\Xi^{\Gamma}$ is surjective.
Suppose that $X^{\Gamma}(\vec{\mathbf{p}})=X^{\Gamma}(\vec{\mathbf{a}})$. It is immediate that $\mathbf{p}^0=\mathbf{a}^0$. On the one hand, we obtain from $X_1^{\Gamma}(\vec{\mathbf{p}})=X_1^{\Gamma}(\vec{\mathbf{a}})$ that
\begin{equation}\label{eq: injectivity Xi gamma proof obs 1}
\mathbf{p}^{\perp}-\mathbf{a}^{\perp} = \lambda\bra{\text{div}_{\Gamma}}^{\dag}\bra{\mathsf{Q}_*q -\mathsf{Q}_*b}.
\end{equation}
On the other hand, $X_2^{\Gamma}(\vec{\mathbf{p}})=X_2^{\Gamma}(\vec{\mathbf{a}})$ implies that
\begin{equation}\label{eq: injectivity Xi gamma proof obs 2}
\text{div}_{\Gamma}\bra{\mathbf{p}-\mathbf{a}} = \lambda\,\mathbf{mean}\bra{q-b}.
\end{equation}
Relying on the fact that $\text{div}_{\Gamma} = \text{div}_{\Gamma}\circ\mathsf{Z}^{\Gamma}$ again, combining \eqref{eq: injectivity Xi gamma proof obs 1} and \eqref{eq: injectivity Xi gamma proof obs 2} yields
\begin{equation*}
\mathsf{Q}_*q + \mathbf{mean}(q) = \mathsf{Q}_*b + \mathbf{mean}(b).
\end{equation*}
Evidently, \eqref{eq: injectivity Xi gamma proof obs 1} then also guarantees that $\mathbf{p}^{\perp}=\mathbf{a}^{\perp}$. We can finally conclude that $X^{\Gamma}$ is injective and thus the result follows from the bounded inverse theorem.
\end{proof}
In the following, we will write $\Xi^{\Gamma}_1$ and $\Xi^{\Gamma}_2$ for the components of the isomorphism of the trace space.
\subsection{Main result}
The main result of this work, stated in Theorem \ref{thm: main theorem}, states that the operator $\mathbb{G}_{\kappa}$ associated with the coupled system \eqref{Calderon coupled problem} is well-posed when $\kappa^2$ lies outside the discrete set of forbidden frequencies described in \cite{claeys2017first}. It relies on two main propositions, whose proofs are postponed until the end of section \ref{sec: compactness and coercivity}.
The first claims that the diagonal of $\mathbb{G}_{\kappa}$ (as a sum of block operators) is T-coercive.
\begin{greenFrameResultBeforeAfter}
\begin{proposition}\label{prop: coercivity volume and TNDL}
For any frequency $\omega\geq0$, there exist a compact operator $\mathsf{K}:\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\times\mathcal{H}_N\rightarrow\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\times\mathcal{H}_N$, a positive constant $C>0$ and parameters $\theta>0$ and $\tau>0$, possibly depending on $\Omega_s$, $\epsilon$, $\mu$, $\kappa$ and $\omega$, such that
\begin{multline*}
\mathfrak{Re}\,\Bigg\langle \diag\bra{\mathbb{G}_{\kappa}}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\colvec{2}{\Xi\,\vec{\overline{\mathbf{U}}}}{\Xi^{\Gamma}\vec{\overline{\mathbf{p}}}}\Big\rangle + \Big\langle\mathsf{K}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\colvec{2}{\vec{\overline{\mathbf{U}}}}{\vec{\overline{\mathbf{p}}}}\Bigg\rangle\\
\geq C \bra{\norm{\mathbf{U}}^2_{\mathbf{H}\bra{\mathbf{curl}, \Omega_s}} + \norm{P}^2_{H^1\bra{\Omega_s}}+\norm{\vec{\mathbf{p}}}^2_{\mathcal{H}_N}}
\end{multline*}
for all $\vec{\mathbf{U}}:=\bra{\mathbf{U}\,\,P}^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$ and $\vec{\mathbf{p}}\in \mathcal{H}_N$.
\end{proposition}
\end{greenFrameResultBeforeAfter}
The proof of this proposition will rely on several steps: Lemma \ref{lem: coercivity B}, Lemma \ref{lem: coercivity TNDL} and Lemma \ref{lem: coercivity of TDSL}.
The second proposition states that the off-diagonal blocks are compact operators. The proof of that fact relies on definitions and results that belong to the next technical section. It will materialize as the last piece of the puzzle that completes the proof of the T-coercivity of $\mathbb{G}_{\kappa}$.
\begin{greenFrameResultBeforeAfter}
\begin{proposition}\label{prop: compactness off-diagonal blocks}
For any frequency $\omega\geq0$, there exists, for a suitable choice of $\tau$, $\beta$, $\theta$ and $\lambda$, a continuous compact endomorphism $\mathsf{K}$ of the space $\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\times\mathcal{H}_N$ such that
\begin{equation}\label{eq: K kappa compact coercivity}
\mathfrak{Re}\,\Bigg\langle \bra{\mathbb{G}_{\kappa}-\diag\bra{\mathbb{G}_{\kappa}}}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\colvec{2}{\Xi\,\vec{\overline{\mathbf{U}}}}{\Xi^{\Gamma}\vec{\overline{\mathbf{p}}}}\Bigg\rangle
= \Big\langle \mathsf{K}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\colvec{2}{\vec{\overline{\mathbf{U}}}}{\vec{\overline{\mathbf{p}}}}\Big\rangle.
\end{equation}
\end{proposition}
\end{greenFrameResultBeforeAfter}
The main result seamlessly follows from the two previous propositions.
\begin{greenFrameTCoercivity}
\begin{theorem}\label{thm: main theorem}
For any $\omega\geq 0$, there exists an isomorphism $\mathbb{X}_{\kappa}$ of the trial space $\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\times\mathcal{H}_N$, and compact operator $\mathbb{K}:\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\times\mathcal{H}_N\rightarrow\bra{\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)}'\times\mathcal{H}_N'$ such that
\begin{equation*}
\mathfrak{Re}\,\Bigg\langle \bra{\mathbb{G}_{\kappa}+\mathbb{K}}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\mathbb{X}\colvec{2}{\vec{\overline{\mathbf{U}}}}{\vec{\overline{\mathbf{p}}}}\Bigg\rangle\geq C\bra{\norm{\mathbf{U}}^2_{\mathbf{H}\bra{\mathbf{curl},\Omega_s}} + \norm{P}_{H^1(\Omega_s)}^2+\norm{\vec{\mathbf{p}}}^2_{\mathcal{H}_N}}
\end{equation*}
for some positive constant $C>0$.
\end{theorem}
\end{greenFrameTCoercivity}
\begin{proof}
The proof will amount to the recognition that the choices of parameters in the previous Proposition \ref{prop: coercivity volume and TNDL} and Proposition \ref{prop: compactness off-diagonal blocks} are compatible.
\end{proof}
The following corollary is immediate upon applying Theorem \ref{thm: coercivity implies fredholm}.
\begin{corollary}
The system operator $\mathbb{G}_k:\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\times\mathcal{H}_N\rightarrow\bra{\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)}'\times\mathcal{H}_N'$ associated with the variational problem \eqref{Calderon coupled problem} is Fredholm of index 0.
\end{corollary}
Injectivity, guaranteed when $\kappa^2$ avoids resonant frequencies by corollary \ref{cor: existence and uniqueness}, yields well-posedness.
\subsection{Compactness and coercivity}
Equipped with the isomorphism $\Xi$, let us now study coercivity of the bilinear form $\mathfrak{B}_{\kappa}$ defined in \eqref{eq: sym bilinear form} and associated to the Hodge--Helmholtz/Laplace operator.
\begin{lemma}\label{lem: coercivity B}
For any frequency $\omega\geq0$ and parameter $\beta>0$, there exist a positive constant $C>0$ and a parameter $\theta>0$, possibly depending on $\Omega_s$, $\mu$, $\epsilon$ and $\omega$, and a compact bounded sesqui-linear form $\mathfrak{K}$ defined over $\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$, such that
\begin{equation*}
\mathfrak{Re}\bra{{\color{violet}\mathfrak{B}_{\kappa}}\bra{\vec{\mathbf{U}},\Xi\bra{\vec{\overline{\mathbf{U}}}}} - \mathfrak{K}\bra{\vec{\mathbf{U}},\vec{\mathbf{U}}}} \geq C \bra{\norm{\mathbf{U}}^2_{\mathbf{H}\bra{\mathbf{curl}, \Omega_s}} + \norm{P}^2_{H^1\bra{\Omega_s}}}
\end{equation*}
for all $\vec{\mathbf{U}}:=\bra{\mathbf{U},\,\,P}^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$.
\end{lemma}
\begin{proof}
As $\mathbf{curl}\bra{\mathbf{U}^0}=0$, $\mathbf{curl}\bra{\nabla P} =0$, and $\nabla \circ \mathbf{mean} = 0$, we evaluate
\begin{multline*}
\mathfrak{B}_{\kappa}\bra{\colvec{2}{\mathbf{U}}{P},\colvec{2}{\overline{\mathbf{U}}^{\perp}-\overline{\mathbf{U}}^0+\beta\overline{\nabla P}}{-\theta\bra{\mathsf{S}\bra{\overline{\mathbf{U}}^0}+\beta\,\mathbf{mean}\bra{\overline{P}}}}} \\=\bra{\mu^{-1}\mathbf{curl}\bra{\mathbf{U}^{\perp}},\mathbf{curl}\bra{\mathbf{U}^{\perp}}}_{\Omega_s}
+\bra{\epsilon\nabla P, \mathbf{U}^{\perp}}_{\Omega_s}
-\bra{\epsilon\nabla P, \mathbf{U}^{0}}_{\Omega_s} + \beta\bra{\epsilon\nabla P, \nabla P}_{\Omega_s} \\ +\theta\bra{\epsilon\mathbf{U}^{\perp},\mathbf{U}^0}_{\Omega_s}+\theta\bra{\epsilon\mathbf{U}^{0},\mathbf{U}^0}_{\Omega_s}-\omega^2\bra{\epsilon\mathbf{U}^{\perp},\mathbf{U}^{\perp}-\mathbf{U}^0+\beta\nabla P}_{\Omega_s}\\
-\omega^2\bra{\epsilon\mathbf{U}^0,\mathbf{U}^{\perp}}+\omega^2\bra{\epsilon\mathbf{U}^0,\mathbf{U}^{0}}-\beta\omega^2\bra{\epsilon\mathbf{U}^0,\nabla P}\\
-\bra{P,\theta \mathsf{S}\bra{\mathbf{U}^0}}_{\Omega_s}-\bra{P,\theta \beta\,\mathbf{mean}(P)}_{\Omega_s}.
\end{multline*}
Upon application of the Cauchy-Schwartz inequality, the bounded sesqui-linear form
\begin{multline*}
\mathfrak{K}\bra{\vec{\mathbf{U}},\vec{\mathbf{U}}} := \bra{\epsilon\nabla P, \mathbf{U}^{\perp}}_{\Omega_s}-\bra{P,\theta \mathsf{S}\bra{\mathbf{U}^0}}_{\Omega_s} +\theta\bra{\epsilon\mathbf{U}^{\perp},\mathbf{U}^0}_{\Omega_s}-\omega^2\bra{\epsilon\mathbf{U}^0,\mathbf{U}^{\perp}}_{\Omega_s}\\-\omega^2\bra{\epsilon\mathbf{U}^{\perp},\mathbf{U}^{\perp}-\mathbf{U}^0+\beta\nabla P}_{\Omega_s}-\bra{P,\theta \beta\,\mathbf{mean}(P)}_{\Omega_s}
\end{multline*}
is shown to be compact by Proposition \ref{cor: volume proj is compact} and the Rellich theorem. Using Young's inequality twice with $\delta>0$, we estimate
\begin{multline*}\label{eq: volume bilinear bounded below}
\mathfrak{Re}\bra{\mathfrak{B}_{\kappa}\bra{\vec{\mathbf{U}},\Xi\bra{\vec{\mathbf{U}}}} -\mathfrak{K}\bra{\vec{\mathbf{U}},\vec{\mathbf{U}}}} \geq\mu^{-1}_{\text{max}}\,\norm{\mathbf{curl}\,\mathbf{U}^{\perp}}^2_{\Omega_s}\\
+ \bra{\epsilon_{\text{min}}\bra{\theta+\omega^2}-\delta\,\epsilon_{\text{max}}\bra{1+\beta\omega^2}}\norm{\mathbf{U}^0}^2_{\Omega_s}\\
+\mathfrak{Re}\bra{\epsilon_{\text{min}}\,\beta -\frac{1}{\delta} \epsilon_{\text{max}}\,\bra{1 +\beta\omega^2}}\norm{\nabla P}^2_{\Omega_s}.
\end{multline*}
The operator $\mathbf{curl}:\mathsf{Z}\bra{\mathbf{H}\bra{\mathbf{curl},\Omega}}\rightarrow \mathbf{L}^2\bra{\Omega_s}$ is a continuous injection, hence since its image is closed in $\mathbf{L}^2\bra{\Omega_s}$, it is also bounded below. Hence, for any $\beta>0$, choose $\delta>0$ large enough, then $\theta>0$ accordingly large, and the desired inequality follows.
\end{proof}
\par The complex inner products
\begin{align*}
\bra{a,b}_{-1/2} &:=\int_{\Gamma}\int_{\Gamma}G_0\bra{\mathbf{x}-\mathbf{y}}a(\mathbf{x})\,\overline{b(\mathbf{y})}\dif\sigma(\mathbf{x})\dif\sigma(\mathbf{y}),\\
\bra{\mathbf{a},\mathbf{b}}_{-1/2} &:=\int_{\Gamma}\int_{\Gamma}G_0\bra{\mathbf{x}-\mathbf{y}}\mathbf{a}(\mathbf{x})\cdot\overline{\mathbf{b}(\mathbf{y})}\dif\sigma(\mathbf{x})\dif\sigma(\mathbf{y}),
\end{align*}
defined over $H^{-1/2}(\Gamma)$ and $\mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma}$ respectively, are positive definite Hermitian forms and they induce equivalent norms on the trace spaces. Combined with the stability of the decomposition introduced in Section \ref{sec: space decomposition}, this observation also allows us to conclude that
\begin{equation*}
\mathbf{a}\mapsto\norm{\text{div}_{\Gamma}\bra{\mathbf{a}}}_{-1/2} + \norm{(\id - P^{\Gamma})\,\mathbf{a}}_{-1/2}
\end{equation*}
also defines an equivalent norm in $\mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma}$.
\par Let us denote the two components of the isomorphism $\Xi$ by
\begin{align*}
&\Xi_1(\vec{\mathbf{U}}):=\mathbf{U}^{\perp}-\mathbf{U}^0+\nabla P,&\text{and}& &\Xi_2(\vec{\mathbf{U}}):=-\theta\bra{\mathsf{S}\bra{\mathbf{U}^0}+\,\mathbf{mean}\bra{P}}.
\end{align*}
We now derive an estimate similar to the one found in Lemma \ref{lem: coercivity B} that completes the proof of the coercivity of the upper-left diagonal block of $\mathbb{G}_{\kappa}$.
\begin{lemma}\label{lem: coercivity TNDL}
For any frequency $\omega\geq0$ and parameter $\beta>0$, there exist a positive constant $C>0$ and a parameter $\theta>0$, possibly depending on $\Omega_s$, $\mu$, $\epsilon$ and $\kappa$, and a compact linear operator $\mathcal{K}:\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)\rightarrow \mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$ such that
\begin{multline*}
\mathfrak{Re}\Bigg(\Big\langle {\color{purple}-\mathbb{A}^{DN}_{\kappa}}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},\colvec{2}{\gamma^{-}_t\Xi_1\vec{\overline{\mathbf{U}}}}{\gamma^{-} \Xi_2\vec{\overline{\mathbf{U}}}}\Big\rangle\\ +\Big\langle\mathcal{K}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},\colvec{2}{\gamma^{-}_t\Xi_1\vec{\overline{\mathbf{U}}}}{\gamma^{-} \Xi_2\vec{\overline{\mathbf{U}}}}\Big\rangle \Bigg)
\geq C\norm{\colvec{2}{\gamma_t^-\mathbf{U}}{\gamma^-(P)}}^2_{\mathcal{H}_D(\Omega_s)}
\end{multline*}
for all $\vec{\mathbf{U}}:=\bra{\mathbf{U}\,\,P}^{\top}\in\mathbf{H}\bra{\mathbf{curl}, \Omega_s}\times H^1(\Omega_s)$.
\end{lemma}
\begin{proof}
We indicate with a hat equality up to a compact perturbation (e.g. $\hat{=}$). The jump conditions \eqref{double layer jump} yield $\{\mathcal{T}_{N}\}\cdot\mathcal{DL}_{\kappa}=\mathcal{T}_{N}\cdot\mathcal{DL}_{\kappa}$. We deduce from \cite[Sec. 6.4]{claeys2017first} that,
\begin{multline}\label{eq: TNDL up to compact}
\Big\langle -\mathcal{T}_{N}\cdot\mathcal{DL}_{\kappa}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},\colvec{2}{\gamma^{-}_t\Xi_1\vec{\mathbf{U}}}{\gamma^{-} \Xi_2\vec{\mathbf{U}}}\Big\rangle \\
\hat{=} \bra{\text{div}_{\Gamma}\bra{\mathbf{n}\times\gamma_t^-\mathbf{U})},\text{div}_{\Gamma}\bra{\mathbf{n}\times\gamma^{-}_t\Xi_1\vec{\mathbf{U}})}}_{-1/2}
- \kappa^2\bra{\mathbf{n}\times\gamma_t^-\mathbf{U},\mathbf{n}\times\gamma^{-}_t\Xi_1\vec{\mathbf{U}}}_{-1/2}\\
+\bra{\mathbf{n}\times\gamma_t^-\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^{-} \Xi_2\vec{\mathbf{U}}}}_{-1/2}
-\bra{\mathbf{n}\times\gamma^{-}_t\Xi_1\vec{\mathbf{U}},\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}_{-1/2}\\
=\bra{\text{div}_{\Gamma}\bra{\gamma_{\tau}^-\mathbf{U})},\text{div}_{\Gamma}\bra{\gamma^{-}_{\tau}\Xi_1\vec{\mathbf{U}})}}_{-1/2}
- \kappa^2\bra{\gamma_{\tau}^-\mathbf{U},\gamma^{-}_{\tau}\Xi_1\vec{\mathbf{U}}}_{-1/2}\\
-\bra{\gamma_{\tau}^-\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^{-} \Xi_2\vec{\mathbf{U}}}}_{-1/2}
+\bra{\gamma^{-}_{\tau}\Xi_1\vec{\mathbf{U}},\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}_{-1/2}
\end{multline}
We consider each component of the isomorphim $\Xi$ in turn. Since Lemma \ref{lem: proj volume} guarantees that $\mathsf{Z}\bra{\mathbf{U}}\in\mathbf{H}^1(\Omega_s)$, $\gamma_{\tau}\circ \mathsf{Z}$ is a continuous mapping $\mathbf{H}\bra{\mathbf{curl},\Omega_s}\rightarrow\mathbf{H}_R^{1/2}\bra{\Omega_s}$ found compact by Lemma \ref{lem: Rellich for boundary}. Therefore,
\begin{align}
\gamma^-_{\tau}\Xi_1\bra{\vec{\mathbf{U}}} &= \gamma^-_\tau\mathbf{U}^{\perp}-\gamma^-_\tau\mathbf{U}^0+\beta\gamma^-_{\tau}\nabla P\nonumber\\
&\hat{=}\, \mathsf{Z}^{\Gamma}\bra{\gamma^-_{\tau}\mathbf{U}} - \bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U}+\beta\,\mathbf{curl}_{\Gamma}\bra{\gamma^-{P}}.\label{eq: tau trace using commutative diagram}
\end{align}
Let's introduce expression \eqref{eq: tau trace using commutative diagram} in the various terms of \eqref{eq: TNDL up to compact} involving $\Xi_1(\vec{\mathbf{U}})$. Lemma \ref{lem: trace projection} yields
\begin{multline*}
\bra{\text{div}_{\Gamma}\bra{\gamma_{\tau}^-\mathbf{U})},\text{div}_{\Gamma}\bra{\gamma^{-}_{\tau}\Xi_1\vec{\mathbf{U}})}}_{-1/2}
\hat{=}
\bra{\text{div}_{\Gamma}\bra{\gamma_{\tau}\mathbf{U}},\text{div}_{\Gamma}\bra{\mathsf{Z}^{\Gamma}\bra{\gamma^-_{\tau}\mathbf{U}}}}_{-1/2}\\
-\bra{\text{div}_{\Gamma}\bra{\gamma_{\tau}\mathbf{U}},\text{div}_{\Gamma}\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U}}}_{-1/2}\\
+\beta\bra{\text{div}_{\Gamma}\bra{\gamma_{\tau}\mathbf{U}},\text{div}_{\Gamma}\bra{\mathbf{curl}_{\Gamma}\bra{\gamma^-{P}}}}_{-1/2}\\
= \bra{\text{div}_{\Gamma}\bra{\gamma^-_{\tau}\mathbf{U}},\text{div}_{\Gamma}\bra{\gamma^-_{\tau}\mathbf{U}}}_{-1/2}.
\end{multline*}
Similarly,
\begin{multline*}
- \kappa^2\bra{\gamma_{\tau}^-\mathbf{U},\gamma^{-}_{\tau}\Xi_1\vec{\mathbf{U}}}_{-1/2}
\hat{=} \, \,\kappa^2\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma_{\tau}^-\mathbf{U},\bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U}}_{-1/2}\\
-\beta\kappa^2\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma_{\tau}^-\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^-{P}}}_{-1/2}
\end{multline*}
and
\begin{multline*}
\bra{\gamma^{-}_{\tau}\Xi_1\vec{\mathbf{U}},\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}_{-1/2}
\hat{=}
-\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}_{-1/2} \\
+\beta\bra{\mathbf{curl}_{\Gamma}\bra{\gamma^-{P}},\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}_{-1/2}.
\end{multline*}
We now want to evaluate the terms involving $\Xi_2(\vec{\mathbf{U}})$. We introduce
\begin{equation*}
\mathbf{curl}_{\Gamma}\bra{\gamma^{-} \Xi_2\vec{\mathbf{U}}}=-\theta\gamma^-_{\tau}\nabla\bra{\mathsf{S}(\mathbf{U}^0)+\mathbf{mean}(P)}=-\theta\bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U},
\end{equation*}
in \eqref{eq: TNDL up to compact} to obtain
\begin{equation*}
-\bra{\gamma_{\tau}^-\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^{-} \Xi_2\vec{\mathbf{U}}}}_{-1/2} =\theta\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U},\bra{\id - \mathsf{Z}^{\Gamma}}\gamma_{\tau}\mathbf{U}}_{-1/2}
\end{equation*}
Using Young's inequality twice with $\delta>0$,
\begin{multline*}
\mathfrak{Re}\bra{\Big\langle -\{\mathcal{T}_{N}\}\cdot\mathcal{DL}_{\kappa}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},\colvec{2}{\gamma^{-}_t\Xi_1\vec{\mathbf{U}}}{\gamma^{-} \Xi_2\vec{\mathbf{U}}}\Big\rangle}\\
\hat{=} \,\norm{\text{div}_{\Gamma}\bra{\gamma^-_{\tau}\mathbf{U}}}^2_{-1/2} + \bra{\mathfrak{Re}\bra{\kappa^2}+\theta}\norm{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma_{\tau}^-\mathbf{U}}^2_{-1/2}\\
+ \beta\norm{\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}^2
-\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma^-_{\tau}\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}_{-1/2}\\ -\beta\,\mathfrak{Re}\bra{\kappa^2}\bra{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma_{\tau}^-\mathbf{U},\mathbf{curl}_{\Gamma}\bra{\gamma^-{P}}}_{-1/2}
\end{multline*}
\begin{multline*}
\geq \,\,\norm{\text{div}_{\Gamma}\bra{\gamma^-_{\tau}\mathbf{U}}}^2_{-1/2}+ \bra{\beta - \frac{1}{\delta}\bra{1+\beta\,\mathfrak{Re}\bra{\kappa^2}}}\norm{\mathbf{curl}_{\Gamma}\bra{\gamma^-\bra{P}}}^2\\
+\bra{\mathfrak{Re}\bra{\kappa^2}+\theta-\delta\,\bra{1 +\beta\,\mathfrak{Re}\bra{\kappa^2}}}\norm{\bra{\id-\mathsf{Z}^{\Gamma}}\gamma_{\tau}^-\mathbf{U}}^2_{-1/2}. \\
\end{multline*}
The operator $\mathbf{curl}_{\Gamma}:H^1_*(\Omega_s)\rightarrow \mathbf{H}^{-1/2}\bra{\text{div}_{\Gamma},\Gamma}$ is a continuous injection \cite[Lem. 6.4]{claeys2017first}. It is thus bounded below. Since the mean operator has finite rank, it is compact. Therefore, for any $\beta>0$, choose $\delta>0$ large enough, then $\theta>0$ accordingly large, and the desired inequality follows by equivalence of norms.
\end{proof}
\begin{lemma}\label{lem: coercivity of TDSL}
For any frequency $\omega\geq0$, there exist a compact linear operator $\mathcal{K}:\mathcal{H}_N\rightarrow \mathcal{H}_D$, a positive constants $C>0$ and parameters $\tau>0$ and $\lambda>0$, possibly depending on $\Omega_s$, $\mu$, $\epsilon$ and $\kappa$, such that
\begin{equation*}
\mathfrak{Re}\bra{\Big\langle{\color{magenta}\mathbb{A}^{ND}_{\kappa}}\bra{\vec{\mathbf{p}}}, \Xi^{\Gamma}\vec{\overline{\mathbf{p}}}
\Big\rangle +\Big\langle\mathcal{K}\,\vec{\mathbf{p}},\vec{\overline{\mathbf{p}}}\Big\rangle} \geq C\norm{\vec{\mathbf{p}}}^2_{\mathcal{H}_N}
\end{equation*}
for all $\vec{\mathbf{p}}\in\mathcal{H}_N$. In particular, for $\mathfrak{Re}\bra{k^2}\neq 0$, the inequality holds with $\tau = 1/\kappa^2$.
\end{lemma}
\begin{proof}
We indicate with a hat equality up to a compact perturbation (e.g. $\hat{=}$). The jump conditions \eqref{single layer jump} yield $\{\mathcal{T}_D\}\cdot\mathcal{SL}\bra{\vec{\mathbf{p}}}=\mathcal{T}_D\cdot\mathcal{SL}\bra{\vec{\mathbf{p}}}$. We deduce from \cite[Sec. 6.3]{claeys2017first} and the compact embedding of $\mathbf{X}\bra{\text{div}_{\Gamma},\Gamma}$ into $\mathbf{H}_R^{-1/2}(\Gamma)$ that
\begin{multline*}
\Big\langle\mathcal{T}_D\cdot\mathcal{SL}\bra{\vec{\mathbf{p}}},\Xi^{\Gamma}\vec{\mathbf{p}}\Big\rangle \hat{=}-\bra{\mathbf{p}^0,\Xi^{\Gamma}_1(\mathbf{p})}_{-1/2} -\bra{q,\text{div}_{\Gamma}\bra{\Xi^{\Gamma}_1(\mathbf{p})}}_{-1/2} \\ -\bra{\text{div}_{\Gamma}(\mathbf{p}),\Xi^{\Gamma}_2\vec{\mathbf{p}}}_{-1/2} -\kappa^2\bra{q,\Xi^{\Gamma}_2\bra{\vec{\mathbf{p}}}}_{-1/2}\\
\hat{=}\bra{\mathbf{p}^0,\mathbf{p}^0}_{-1/2} -\bra{q,\text{div}_{\Gamma}(\mathbf{p}^{\perp})}_{-1/2} +\lambda\bra{q,\mathsf{Q}_*q}_{-1/2} \\
+\tau\bra{\text{div}_{\Gamma}(\mathbf{p}),\text{div}_{\Gamma}(\mathbf{p})}_{-1/2} + \tau\kappa^2\bra{q,\text{div}_{\Gamma}(\mathbf{p}^{\perp})}_{-1/2}.
\end{multline*}
When $\mathfrak{Re}\bra{\kappa^2} > 0$, setting $\tau=1/\kappa^2$ immediately yields the existence of a compact linear operator $\mathcal{K}:\mathcal{H}_N\rightarrow \mathcal{H}_D$ such that
\begin{equation*}
\Big\langle\mathcal{T}_D\cdot\mathcal{SL}\bra{\vec{\mathbf{p}}},\Xi^{\Gamma}\vec{\mathbf{p}}\Big\rangle
+ \Big\langle \mathcal{K}\vec{\mathbf{p}},\Xi^{\Gamma}\vec{\mathbf{p}}\Big\rangle \geq C\bra{ \norm{\text{div}_{\Gamma}\bra{\mathbf{p}}}^2_{-1/2} + \norm{\mathbf{p}^0}^2_{-1/2} + \norm{\mathsf{Q}_*q}^2_{-1/2}}.
\end{equation*}
When $\kappa^2 = 0$, the same inequality is obtained for any $\lambda>0$ by using Young's inequality as in the proof of Lemma \ref{lem: coercivity TNDL} and choosing $\tau$ large enough. The claimed inequality follows by equivalence of norms.
\end{proof}
Equipped with the previous three lemmas, we are now ready to prove Lemma \ref{prop: coercivity volume and TNDL}.
\begin{proof}[Proof of Proposition \ref{prop: coercivity volume and TNDL}]
For any parameters $\beta>0$ and $\lambda>0$, the choices of $\delta$ and $\theta$ in the proofs of Lemma \ref{lem: coercivity B} and Lemma \ref{lem: coercivity TNDL} are not mutually exclusive. The choice of $\tau$ in Lemma \ref{lem: coercivity of TDSL} is independent of that choice of $\theta$.
\end{proof}
Finally, The off-diagonal blocks remain to be considered. We will show that, up to compact perturbations, a suitable choice of parameters in the isomorphisms $\Xi$ and $\Xi^{\Gamma}$ of the test space leads to a skew-symmetric pattern in $\mathbb{G}_{\kappa}$. In other words, up to compact terms, the volume and boundary parts of the system decouples over the space decompositions introduced in Section \ref{sec: space decomposition}.
\begin{proof}[Proof of Proposition \ref{prop: compactness off-diagonal blocks}]
We indicate with a hat equality up to a compact perturbation (e.g. $\hat{=}$). The isomorphisms $\Xi$ and $\Xi^{\Gamma}$ were designed so that favorable cancellations occur in evaluating the left hand side of \eqref{eq: K kappa compact coercivity}.
From the jump properties \eqref{single layer jump}, we have $\{\mathcal{T}_N\}\mathcal{SL}_{\kappa} = \mathcal{T}_N^-\mathcal{SL}_{\kappa} -(1/2)\id$. Therefore, as in \eqref{eq: average single layer}, we evaluate
\begin{multline}\label{eq: off-diagonal first SL}
\Big\langle{\color{olive}\bra{\mathbb{P}_{\kappa}^+}_{22}\vec{\mathbf{p}}},\colvec{2}{\gamma^-_t\Xi_1\vec{\overline{\mathbf{U}}}}{\gamma^-\Xi_2\vec{\overline{\mathbf{U}}}} \Big\rangle
=\Big\langle\bra{-\{\mathcal{T}_N\}\cdot\mathcal{SL}_{\kappa}+\frac{1}{2}\id}\vec{\mathbf{p}},\colvec{2}{\gamma^-_t\Xi_1\vec{\overline{\mathbf{U}}}}{\gamma^-\Xi_2\vec{\overline{\mathbf{U}}}}\Big\rangle\\
= \Big\langle-\mathcal{T}_N^-\cdot\mathcal{SL}_{\kappa}\bra{\vec{\mathbf{p}}},\colvec{2}{\gamma^-_t\Xi_1\vec{\overline{\mathbf{U}}}}{\gamma^-\Xi_2\vec{\overline{\mathbf{U}}}}\Big\rangle + \Big\langle \vec{\mathbf{p}},\colvec{2}{\gamma^-_t\Xi_1\vec{\overline{\mathbf{U}}}}{\gamma^-\Xi_2\vec{\overline{\mathbf{U}}}}\Big\rangle\\
= \langle \gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}},\gamma_t^-\Xi_1\vec{\overline{\mathbf{U}}}\rangle_{\tau} - \langle\gamma_n^-\nabla\psi_{\tilde{\kappa}}\bra{q},\gamma^-\Xi_2\vec{\overline{\mathbf{U}}} \rangle_{\Gamma} + \langle \gamma_n^-\bm{\Psi}_{\kappa}\bra{\mathbf{p}}, \gamma^-\Xi_2\vec{\overline{\mathbf{U}}}\rangle_{\Gamma}\\
+\langle\gamma^-_n\nabla\tilde{\psi}_{\kappa}\bra{\text{div}_{\Gamma}\mathbf{p}},\gamma^-\Xi_2\vec{\overline{\mathbf{U}}}\rangle_{\Gamma} + \langle \mathbf{p},\gamma_t^-\Xi_1\vec{\overline{\mathbf{U}}}\rangle_{\tau} + \langle q, \gamma^-\Xi_2\vec{\overline{\mathbf{U}}}\rangle_{\Gamma}\\
\hat{=}\,
{\color{red}\langle \gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}^0},\gamma_t\overline{\mathbf{U}}^{\perp}\rangle_{\tau}}
{\color{blue} - \langle \gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}^0},\gamma_t\overline{\mathbf{U}^0}\rangle_{\tau}}
{\color{blue} + \beta\,\langle \gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}^0},\gamma_t\nabla\overline{P}\rangle_{\tau}}\\
+{\color{blue}\langle\gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}^{\perp}},\gamma_t\overline{\mathbf{U}}^{\perp}\rangle_{\tau}}
{\color{red} - \langle \gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}^{\perp}},\gamma_t\overline{\mathbf{U}^0}\rangle_{\tau}}
+ \beta\,\langle \gamma^-_R\bm{\Psi}_{\kappa}\bra{\mathbf{p}^{\perp}},\gamma_t\nabla\overline{P}\rangle_{\tau}\\
+\theta\,\langle\gamma^-_n\nabla\psi_{\tilde{\kappa}}\bra{q}, \gamma^-\mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}
-\theta\,\langle \gamma^-_n\bm{\Psi}_{\kappa}\bra{\mathbf{p}},\gamma^- \mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}\\
-\langle \gamma^-_n\nabla\tilde{\psi}_{\kappa}\bra{\text{div}_{\Gamma}\mathbf{p}},\theta\,\gamma^- \mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}
{\color{red} + \langle\mathbf{p}^0,\gamma_t^-\overline{\mathbf{U}}^{\perp} \rangle_{\tau}}
{\color{blue} + \langle\mathbf{p}^\perp,\gamma_t^-\overline{\mathbf{U}}^{\perp} \rangle_{\tau}}
{\color{blue} - \langle\mathbf{p}^0,\gamma_t^-\overline{\mathbf{U}}^{0} \rangle_{\tau}}\\
{\color{red} - \langle\mathbf{p}^{\perp},\gamma_t^-\overline{\mathbf{U}}^{0} \rangle_{\tau}}
{\color{blue} + \beta \,\langle \mathbf{p}^0,\gamma_t^-\nabla\overline{P}\rangle_{\tau}}
+ \beta \,\langle \mathbf{p}^{\perp},\gamma_t^-\nabla\overline{P}\rangle_{\tau}
- \theta\,\langle q,\gamma^- \mathsf{S}\bra{\mathbf{U}^0}\rangle_{\Gamma},
\end{multline}
where we have used that the finite rank of the mean operator implies compactness.
Similarly, using Proposition \ref{prop: adjointness}, we find
\begin{multline}\label{eq: off-diagonal second SL}
\Big\langle{\color{teal}\bra{\mathbb{P}_{\kappa}^-}_{11}}\colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}}, \Xi^{\Gamma}\vec{\overline{\mathbf{p}}}\Big\rangle = \Big\langle \colvec{2}{\gamma_t^-\mathbf{U}}{-\gamma^-\bra{P}},{\color{olive}\bra{\mathbb{P}_{\kappa}^+}_{22}}\Xi^{\Gamma}\vec{\overline{\mathbf{p}}}\Big\rangle\\
\hat{=}\, {\color{red}\langle\gamma^-_R\bm{\Psi}_{\kappa}\bra{\overline{\mathbf{p}}^{\perp}},\gamma_t\mathbf{U}^0\rangle_{\tau}}
{\color{blue}-\langle\gamma^-_R\bm{\Psi}_{\kappa}\bra{\overline{\mathbf{p}}^{0}},\gamma_t\mathbf{U}^{\perp}\rangle_{\tau}} - \lambda\,\langle\gamma_R^-\bm{\Psi}_{\kappa}\bra{\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}},\gamma_t^-\mathbf{U}^0 \rangle_{\tau}\\
+{\color{blue}\langle\gamma^-_R\bm{\Psi}_{\kappa}\bra{\overline{\mathbf{p}}^{0}},\gamma_t\mathbf{U}^{\perp}\rangle_{\tau}}
{\color{red}-\langle\gamma^-_R\bm{\Psi}_{\kappa}\bra{\overline{\mathbf{p}}^{0}},\gamma_t\mathbf{U}^{\perp}\rangle_{\tau}}
{\color{blue}-\lambda\,\langle\gamma_R^-\bm{\Psi}_{\kappa}\bra{\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}},\gamma_t^-\mathbf{U}^{\perp} \rangle_{\tau}}\\
-\tau\,\langle \gamma_n^-\nabla\psi_{\tilde{\kappa}}\bra{\text{div}_{\Gamma}\overline{\mathbf{p}}^{\perp}},\gamma^-P\rangle
-\langle\gamma^-_n\bm{\Psi}_{\kappa}\bra{\overline{\mathbf{p}}^{\perp}},\gamma^-P\rangle_{\Gamma}
+\langle\gamma^-_n\bm{\Psi}_{\kappa}\bra{\overline{\mathbf{p}}^{0}},\gamma^-P\rangle_{\Gamma}\\
+\lambda\,\langle\gamma^-_n\bm{\Psi}_{\kappa}\bra{\bra{\text{div}}^{\dag}Q_*\overline{q}},\gamma^-P\rangle_{\Gamma}
-\langle\gamma_n^-\nabla\tilde{\psi}_{\kappa}\bra{\text{div}_{\Gamma}\overline{\mathbf{p}}^{\perp}},\gamma^-P\rangle_{\Gamma}\\
+ \lambda\,\langle\gamma_n^-\nabla\tilde{\psi}_{\kappa}\bra{Q_*\overline{q}},\gamma^-P\rangle_{\Gamma}
{\color{red}+\langle\gamma^-_t\mathbf{U}^0,\overline{\mathbf{p}}^{\perp}\rangle_{\tau}}
{\color{blue}+\langle\gamma^-_t\mathbf{U}^{\perp},\overline{\mathbf{p}}^{\perp}\rangle_{\tau}}
{\color{red}-\langle\gamma^-_t\mathbf{U}^{\perp},\overline{\mathbf{p}}^{0}\rangle_{\tau}}
{\color{blue}-\langle\gamma^-_t\mathbf{U}^0,\overline{\mathbf{p}}^{0}\rangle_{\tau}}\\
{\color{blue}- \lambda\,\langle\gamma^-_t\mathbf{U}^{\perp},\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}\rangle_{\Gamma}}
- \lambda\,\langle\gamma^-_t\mathbf{U}^{0},\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}\rangle
+\tau\,\langle \gamma^- P,\text{div}_{\Gamma}\bra{\overline{\mathbf{p}}^{\perp}}\rangle_{\Gamma}.
\end{multline}
Many terms in these equations can be combined and asserted compact by Lemma \ref{lem: compact sym parts}. They are indicated in {\color{blue}blue}. When summing the real parts of \eqref{eq: off-diagonal first SL} and \eqref{eq: off-diagonal second SL}, the terms in {\color{red}red} cancel. Relying on Lemma \ref{lem: compactness of boundary integrals}, some terms amount to compact perturbations so that we may replace $\kappa$ and $\tilde{\kappa}$ by $0$ in those instances.
We have arrived at the following identity:
\begin{align}\label{eq: off-diagonal final identity}
\begin{split}
\mathfrak{Re}&\bra{\Big\langle \bra{\mathbb{G}_{\kappa}-\diag\bra{\mathbb{G}_{\kappa}}}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\colvec{2}{\Xi\,\vec{\overline{\mathbf{U}}}}{\Xi^{\Gamma}\vec{\overline{\mathbf{p}}}}\Big\rangle}\\
&\hat{=}\,\,\mathfrak{Re}\Bigg(
\beta\,\langle \gamma^-_R\bm{\Psi}_{0}\bra{\mathbf{p}^{\perp}},\gamma_t\nabla\overline{P}\rangle_{\tau}
+\theta\,\langle\gamma^-_n\nabla\psi_{0}\bra{q}, \gamma^-\mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}\\
&\quad{\color[rgb]{0,0.5,0.15}-\theta\,\langle \gamma^-_n\bm{\Psi}_0\bra{\mathbf{p}},\gamma^- \mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}}
{\color[rgb]{0.8,0.33,0}+ \beta \,\langle \mathbf{p}^{\perp},\gamma_t^-\nabla\overline{P}\rangle_{\tau}
- \theta\,\langle q,\gamma^- \mathsf{S}\bra{\mathbf{U}^0}\rangle_{\Gamma}}\\
&\quad- \lambda\,\langle\gamma_R^-\bm{\Psi}_{0}\bra{\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}},\gamma_t^-\mathbf{U}^0 \rangle_{\tau}
-\tau\,\langle \gamma_n^-\nabla\psi_{0}\bra{\text{div}_{\Gamma}\overline{\mathbf{p}}^{\perp}},\gamma^-P\rangle_{\Gamma}\\
&\quad{\color[rgb]{0,0.5,0.15}-\langle\gamma^-_n\bm{\Psi}_0\bra{\overline{\mathbf{p}}^{\perp}},\gamma^-P\rangle_{\Gamma}
+\langle\gamma^-_n\bm{\Psi}_0\bra{\overline{\mathbf{p}}^{0}},\gamma^-P\rangle_{\Gamma}
+\lambda\,\langle\gamma^-_n\bm{\Psi}_0\bra{\bra{\text{div}}^{\dag}Q_*\overline{q}},\gamma^-P\rangle_{\Gamma}}\\
&\quad{\color[rgb]{0.8,0.33,0}- \lambda\,\langle\gamma^-_t\mathbf{U}^{0},\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}\rangle_{\tau}
+\tau\,\langle \gamma^- P,\text{div}_{\Gamma}\bra{\overline{\mathbf{p}}^{\perp}}\rangle_{\Gamma}}\Bigg).
\end{split}
\end{align}
We claim that the terms colored in {\color[rgb]{0,0.5,0.15}green} are compact. Indeed, the integral identities of Section \ref{sec: classical traces} together with equality \eqref{scalar and vector single layer potential identity} yield
\begin{align*}
\langle \gamma^-_n\bm{\Psi}_0\bra{\mathbf{p}},\gamma^- \mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}
&\leq \bra{\norm{\psi_0\bra{\text{div}_{\Gamma}\mathbf{p}}}_{L^2(\Omega_s)}+\norm{\bm{\Psi}_0\bra{\mathbf{p}}}_{\mathbf{L}^2(\Omega_s)}}\norm{\overline{\mathbf{U}}^0}_{\mathbf{L}^2(\Omega_s)},\\
\langle \gamma_n^-\bm{\Psi}_0\bra{\overline{\mathbf{p}}},\gamma^-P\rangle_{\Gamma}
&\leq \bra{\norm{\psi_0\bra{\text{div}_{\Gamma}\overline{\mathbf{p}}}}_{L^2(\Omega_s)}+\norm{\bm{\Psi}_0\bra{\overline{\mathbf{p}}}}_{\mathbf{L}^2(\Omega_s)}}\norm{P}_{H^1(\Omega_s)}\\
\langle\gamma^-_n\bm{\Psi}_0\bra{\bra{\text{div}}^{\dag}Q_*\overline{q}},\gamma^-P\rangle_{\Gamma}
&\leq \bra{\norm{\psi_0\bra{Q_*q}}_{L^2(\Omega_s)}+\norm{\bm{\Psi}_0\bra{\text{div}_{\Gamma}\overline{\mathbf{p}}}}_{\mathbf{L}^2(\Omega_s)}}\norm{P}_{H^1(\Omega_s)}
\end{align*}
Since Lemma \ref{lem: single layer into H1} states that $\psi_{0}:H^{-1/2}(\Gamma)\rightarrow H^1(\Omega_s)$ and $\bm{\Psi}_{0}:\mathbf{H}^{-1/2}(\Gamma)\rightarrow \mathbf{H}^1(\Omega_s)$ are continuous, compactness is guaranteed by Rellich Theorem.
To go further, we need to settle for a choice of parameters in the volume and boundary isomorphisms. Choose $\tau$ to satisfy the requirements of Lemma \ref{lem: coercivity of TDSL}, then set $\beta = \tau$. We are still free to let $\theta$ satisfy both Lemma \ref{lem: coercivity B} and Lemma \ref{lem: coercivity TNDL}, and then choose $\lambda = \theta$.
Under this choice of parameters, the terms in {\color[rgb]{0.8,0.33,0}orange} vanish, because we have $
\langle\mathbf{p}^{\perp}\gamma_t^-\nabla\overline{P}\rangle_{\tau} = \langle\mathbf{p}^{\perp},\nabla_{\Gamma}\gamma^-\overline{P}\rangle_{\tau}=-\langle\text{div}_{\Gamma}\bra{\mathbf{p}^{\perp}},\gamma^-\overline{P}\rangle_{\Gamma}
$, and similarly
\begin{equation*}
\langle \gamma^-_t\mathbf{U}^0,\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}\rangle_{\tau} =\langle \gamma^-_t\nabla \mathsf{S}\bra{\mathbf{U}^0},\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}\rangle_{\tau}=-\langle \gamma^-\mathsf{S}\bra{\mathbf{U}^0},Q_*\overline{q}\rangle_{\Gamma}.
\end{equation*}
Finally, relying on \eqref{scalar helmholtz for scalar single layer}, \eqref{vector helmholtz for vector single layer} and \eqref{scalar and vector single layer potential identity} once more, we observe that
\begin{multline*}
\langle\gamma_R^-\bm{\Psi}_0\bra{\mathbf{p}^{\perp}},\gamma_t^-\nabla \overline{P}\rangle_{\tau} =\bra{\mathbf{curl}\,\mathbf{curl}\,\bm{\Psi}_0\bra{\mathbf{p}^{\perp}},\nabla P}_{\Omega_s}\\
=\bra{\nabla\psi_0\bra{\text{div}_{\Gamma}\mathbf{p}^{\perp}},\nabla P}_{\Omega_s}
= \langle\gamma^-_n\nabla\psi_0\bra{\text{div}_{\Gamma}\mathbf{p}^{\perp}},\gamma^-\overline{P}\rangle_{\Gamma}.
\end{multline*}
A similar derivation shows that
\begin{equation*}
\langle\gamma^-_n\nabla\psi_{0}\bra{q}, \gamma^-\mathsf{S}\bra{\overline{\mathbf{U}}^0}\rangle_{\Gamma}\,\hat{=}\, \langle\gamma_R^-\bm{\Psi}_{0}\bra{\bra{\text{div}_{\Gamma}}^{\dag}Q_*\overline{q}},\gamma_t^-\mathbf{U}^0 \rangle_{\tau}.
\end{equation*}
We conclude that for such a choice of parameters,
\begin{equation*}
\mathfrak{Re}\bra{\Big\langle \bra{\mathbb{G}_{\kappa}-\diag\bra{\mathbb{G}_{\kappa}}}\colvec{2}{\vec{\mathbf{U}}}{\vec{\mathbf{p}}},\colvec{2}{\Xi\,\vec{\overline{\mathbf{U}}}}{\Xi^{\Gamma}\vec{\overline{\mathbf{p}}}}\Big\rangle}
\,\hat{=}\, 0,
\end{equation*}
which concludes the proof of this proposition.
\end{proof}
\section{Conclusion}
Section \ref{sec: coupled problem} offers a system of equations coupling the mixed formulation of the variational form of the Hodge-Helmholtz and Hodge-Laplace equation with \emph{first-kind} boundary integral equations. Well-posedness of the coupled problem was obtained using a T-coercivity argument demonstrating that the operator associated to the coupled variational problem was Fredholm of index 0. When $\kappa^2\in\mathbb{C}$ avoids resonant frequencies, the operator's injectivity was guaranteed, and thus stability of the problem was obtained along with the existence and uniqueness of the solution. For such $\kappa^2$, Proposition \ref{prop: variational system solves transmission system} showed how solution to the coupled variational problem are in one-to-one correspondence with solutions of the transmission system.
The symmetrically coupled system \eqref{Calderon coupled problem} offers a variational formulation of the transmission problem \eqref{eq: transmission problem} in well-known energy spaces suited for discretization by finite and boundary elements. It is therefore a promising starting point for Galerkin discretization.
\bibliographystyle{siamplain}
\bibliography{bibliography.bib}
\end{document} | 63,066 |
Continue reading “Bookbed goes to: Bookish Destinations in Hong Kong”
Category Archives: Places
Welcome Aboard MV Logos Hope, the World’s Largest Floating Book Paradise
byContinue reading “Welcome Aboard MV Logos Hope, the World’s Largest Floating Book Paradise”.Continue reading “Bookbed goes to: Mt. Cloud Bookshop”Continue reading “Bookbed goes to: Café Talk”
Love inContinue reading “Love in the Time of Secondhand Bookstores”
Bookbed goes to: Kumazawa Bookstore
by KB Meniado I was in Tokyo last month and one of the places I loved visiting the most was this bookstore! I think you’ll know why. The Rurouni Kenshin films were in cinemas back then so it was imperative these existed: I am not the biggest Samurai X fan but I AM DEFINITELY A MOON CRYSTAL POWERContinue reading “Bookbed goes to: Kumazawa Bookstore” | 303,140 |
TITLE: How find the function if such $f(P(x)+Q(f(y))+Q(y)P(f(x)))=Q(y)+P(f(x))+P(x)Q(f(y))$
QUESTION [3 upvotes]: Let $f:\Bbb{R}\to \Bbb{R}$, and $P(x)$, $Q(x)$ be given real coefficient polynomials, where the degree is odd and for every $x, y\in\Bbb{R}$:
$$f(P(x)+Q(f(y))+Q(y)P(f(x)))=Q(y)+P(f(x))+P(x)Q(f(y))$$
Find $f(x)$.
I tried many times, I found the final result is only $f(x)=x$. Is it correct? If yes, how to prove it?
REPLY [2 votes]: Let $P = a_m x^m + \cdots$, $Q = b_n x^n + \cdots$ and $f = c_p x^p + \cdots$, with $a_m, b_n, c_p \ne 0$ and $m, n$ odd. We shall look only at the leading terms, i.e. the terms of maximum degree in both sides of the equality (both sides being understood as elements of $\Bbb R [x,y]$).
In the left-hand side, you have
$$c_p \Big( (a_m x^m + \cdots) + \color{blue} {\big( b_n (f(y))^n + \cdots \big)} + (b_n y^n + \cdots) \color{green} {\big( a_m (f(x))^m + \cdots \big)} \Big) ^p + \cdots = \\
c_p \Big( (a_m x^m + \cdots) + \color{blue} {\big( b_n (c_p y^p + \cdots)^n + \cdots \big)} + (b_n y^n + \cdots) \color{green} {\big( a_m (c_p x^p + \cdots)^m + \cdots \big)} \Big) ^p + \cdots \\
= c_p (a_m x^m + b_n c_p ^n y^{pn} + b_n a_m c_p ^m y^n x^{pm} + \cdots)^p + \cdots .$$
Let us now look at the right-hand side:
$$(b_n y^n + \cdots) + \color{blue} {\big( a_m (f(x))^m + \cdots \big)} + (a_m x^m + \cdots) \color{green} {\big( b_n (f(y))^n + \cdots \big)} = \\
(b_n y^n + \cdots) + \color{blue} {\big( a_m (c_p x^p + \cdots)^m + \cdots \big)} + (a_m x^m + \cdots) \color{green} {\big( b_n (c_p y^p + \cdots)^n + \cdots \big)} = \\
b_n y^n + a_m c_p ^m x^{pm} + a_m b_n c_p ^n x^m y^{pn} + \cdots .$$
Let us agree to call "total degree" of a monomial $Ax^M y^N$ the number $M+N$. Let us agree to call "the total degree of a polynomial" the largest of the total degrees of its monomials.
Let us assume $m \le n$. Let us also assume $p > 1$ and try to obtain a contradiction.
In the left-hand side, the total degree of
$$a_m x^m + b_n c_p ^n y^{pn} + b_n a_m c_p ^m y^n x^{pm} + \cdots$$
is $n + pm$, therefore the total degree of
$$(a_m x^m + b_n c_p ^n y^{pn} + b_n a_m c_p ^m y^n x^{pm} + \cdots)^p + \cdots$$
is $pn + p^2 m$.
In the right-hand side, the total degree of
$$b_n y^n + a_m c_p ^m x^{pm} + a_m b_n c_p ^n x^m y^{pn} + \cdots$$
is $m + pn$.
Since both sides are equal, their total degrees as polynomials in $\Bbb R [x,y]$ must be equal, therefore $pn + p^2 m = m + pn$, whence follows $p^2 m = m$. There are two possibilities:
$m = 0$ - this is forbidden by the condition that $m$ be odd;
$m \ne 0$, whence it follows that $p^2 = 1$, so $p=1$, which is again impossible because we have assumed $p>1$.
It follows that our assumption was wrong, therefore $p \le 1$.
We cannot have $f=0$, because (replacing $f$ by $0$ in the original statement) we would get $Q=0$, which is not possible since the degree of $Q$ is odd (therefore it cannot be $-\infty$, unless you define $-\infty$ to be both odd and even).
We also cannot have $\deg f = 0$, because $\deg f$ is required to be odd.
It remains that $f(x) = cx + d$ with $c \ne 0$. Let us plug this back into the original statement and see what we get:
$$c P(x) + c Q(cy + d) + c Q(y) P(cx + d) + d = Q(y) + P(cx + d) + P(x) Q(cy + d) .$$
The leading term in the left-hand side is $c b_n y^n a_m c^m x^m$, coming from $c Q(y) P(cx + d)$. The leading term in the right-hand side is $a_m x^m b_n c^n y^n$, coming from $P(x) Q(cy + d)$. Equating them, we get $c^{m+1} a_m b_n = c^n a_m b_n$ and, since $c, a_m, b_n \ne 0$, it follows that $c^{n-m-1} = 1$. Since $n,m$ are odd, it follows that $n-m-1$ is odd, and the only real root of $c^{\color{blue} {\text{odd}}} = 1$ is $c=1$. Therefore, $f(x) = x + d$ and the original equality becomes
$$\tag{*} P(x) + Q(y + d) + Q(y) P(x + d) + d = Q(y) + P(x + d) + P(x) Q(y + d) .$$
Let us finally show that $d=0$. Begin by noticing that
$$(x+d)^i = \sum _{a=0} ^i \binom i a x^{i-a} d^a = x^i + d \sum _{a=1} ^i \binom i a x^{i-a} d^{a-1} ,$$
whence it follows that there exist polynomials $F_d$ and $G_d$ (with coefficients possibly depending on $d$) such that $P(x+d) = P(x) + d F_d (x)$ and $Q(x+d) = Q(x) + d G_d (x)$. This means that equality (*) can be rewritten as
$$P(x) + Q(y) + d G_d (y) + Q(y) P(x) + d Q(y) F_d (x) + d = Q(y) + P(x) + d F_d (x) + P(x) Q(y) + d P(x) G_d (y) ,$$
which, after simplifications, becomes
$$d \big( G_d (y) - Q(y) F_d (x) + 1 - F_d (x) - P(x) G_d (y) \big) = 0 .$$
There are two possibilities: the first one is $d=0$. The second one is
$$G_d (y) - Q(y) F_d (x) + 1 - F_d (x) - P(x) G_d (y) = 0 .$$
Let us show that this is in fact not possible. View the polynomial
$$\tag{**} G_d (y) - Q(y) F_d (x) + 1 - F_d (x) - P(x) G_d (y)$$
as a polynomial in $y$ with coefficients in $\Bbb R[x]$. As such, the leading term in $y$ is $-b_n y^n F_d (x)$, coming from the term $- Q(y) F_d (x)$. Since the polynomial is equal to $0$ and $b_n \ne 0$, it follows that $F_d = 0$, which means that equality (**) gets rewritten as
$$\tag{***} G_d (y) + 1 - P(x) G_d (y) = 0 .$$
Viewing this polynomial as a polynomial in $x$ with coefficients in $\Bbb R[y]$, its leading term is $-a_m x^m G_d (y)$, coming from the term $- P(x) G_d (y)$. Again, equality to $0$ and the fact that $a_m \ne 0$ imply that $G_d = 0$, whence (***) becomes $1 = 0$, clearly false. Therefore, $d=0$ is the only possibility.
We have proved, therefore, that $f(x) = x$ is the only solution. The only technique systematically used was looking at the leading terms of the various polynomials encountered. | 49,876 |
Many of nature’s means of doing business on land has aquatic parallels. Seagrasses produce seeds and pollen the same way as terrestrial grasses do. However, there are certainly some differences. The pollen grains of seagrasses dwarf their land-based counterparts at almost 50 times larger, and the physics of the water in which pollination occurs certainly follows rules vastly different than those air currents.
Scientists have been postulating for some time about how marine plants get the job done in such a different environment, evident in this 1976 abstract for a submission in Letters to Nature. Many aquatic plants may reproduce by variations of self-cloning where a sexual partner is not needed to give rise to new individuals. Sea grasses often dabble in a little of both worlds, as both have their evolutionary advantages. Cloning, often referred to more specifically as vegetative propagation in plants, allows underwater flora to quickly take up real estate without expending as many resources. However, sexual reproduction confers the benefits of genetic variation and adaptability into their populations.
Initially, it was suggested water bore the primary burden of transporting pollen grains from male to female seagrass plants. However, in 2009, a team of researchers from National Autonomous University of Mexico led by Brigitta van Tussenbroek observed small invertebrate crustaceans visiting turtlegrass (Thalassia testudinum) flowers in a manner that reminded them of terrestrial bees.
To explore whether or not these small creatures had the capacity to truly serve as pollinators, they moved their exploration in the controlled setting of a laboratory. They put turtlegrass and pollinators in tanks sans current, and monitored the movement of pollen grains as well as successful instances of pollination. Both successfully occurred in tanks where crustaceans had been added, but not in tanks where the animals were absent. The research team coined a new term, zoobenthophilous pollination, to describe the marine process they observed.
Seagrasses are vital to human wellbeing in ways similar to oyster reefs, coral reefs, and mangroves, and provide a wealth of services to the environments they thrive in. They are an important food source for manatees, dugongs, turtles, and other creatures. Their root systems stabilizes sea floors. They may serve as nursery grounds for minuscule juvenile fish needing protection from predators. But seagrasses have seen vast decline worldwide, in what could be called a global crisis. Now that we are learning seagrass survival may also have dependence on pollinators, we can only hope we do a better job of conserving these little sea bees than we have their terrestrial counterparts. | 96,154 |
Abstract
Granulocytes obtained by continuous flow filtration leukopheresis (CFFL) were transfused to 21 patients on 131 occasions. An average of 28.2 × 109 granulocytes were administered per transfusion. These cells were more than 90 per cent viable by dye exclusion, ingested latex particles normally and had almost normal bactericidal activity. Migration to skin windows was demonstrated on four of six attempts, but 51Cr-labeling studies failed to show localization in infected areas on six occasions. Post-transfusion granulocyte count increments averaged 225/μl and were transient. Significant transfusion reactions occurred during 35 transfusions to 13 patients. Reactions occurred in some patients without demonstrable alloimmunization and after six infusions of HL-A identical or compatible cells. Definite clinical improvement was noted in three recipients. Stabilization of infection with patient survival occurred nine times, and progression of infection with death eight times. Granulocytes obtained by CFFL are viable and functional. Their transfusion is not without risk and must still be considered an investigative procedure of suggestive but as yet unproved clinical efficacy.
All Science Journal Classification (ASJC) codes
- Medicine(all) | 199,073 |
Antwerp, by Nicholas Royle
Review by Terry Gates-Grimwood
To quote the late, lamented Phil Lynott, there's a killer on the loose.
What? Another one, and he’s killing women? Oh, and of course there’s the killer’s calling card, a cinematic one in this case.
So, who is it? Independent film maker Johnny Vos who is obsessed with the paintings of Paul Delvaux and is making a film of the artist’s life using real life prostitutes as extras, or the mysterious diamond dealer who owns a brothel, itself named after a cult horror film, which he has filled with web cams so on-line subscribers can watch the working girls twenty-four hours a day, seven days a week? And will cinema journalist Frank Warner find his own girlfriend who has gone missing in the city? And so on and so forth.
In the hands of a lesser author, or worse, a churner-out of supermarket glossy crime best-sellers, Antwerp might be as shallow as I’ve made it sound. But in the hands of Nicholas Royle you have a dark and disturbing narrative that is always surprising and always a cut above the average crime blockbuster. Yes, the story is unoriginal at its basic level, but crime is big at the moment and it must be difficult to think up something new. This novel stands out, however, because of the writing, the feel, the atmosphere and its structure.
Be warned, Antwerp is never literary muzak.
The writing is crisp and seemingly effortless. The descriptions are vivid, the characterisation convincing. And, refreshingly, all the players are flawed; even Frank, who is, ostensibly, the hero of the piece, has darkness in his past and some less than endearing traits. A sense of unease is created by this honesty. Who exactly can you trust? Who is telling the truth? In some ways you begin to wonder if the author is lying. Even the constant references to cinema, particularly the world of independent film-making where the personalities of the film makers themselves, uncluttered by the need to please the men in grey suits, play a more defining role in the finished product, give the impression that nothing is as it seems.
One of the things that makes a good thriller a great thriller is its use of setting, lifting it from the role of blank canvas to that of a non-organic character, a player in the game, an accessory to the act. In Antwerp, Royle paints a picture of a country divided, a place of no unified national identity, a fractured, hostile place which breeds mistrust and distorts the psyches of those who live there. And, on a more intimate level, there are the empty buildings, landscapes within the greater landscape, so vividly described, filled with menace and atmosphere you can almost smell the decay.
Most interesting of all is the way in which the characters are presented. Frank, his girlfriend and Voss are shown in third person, the diamond dealer is first person. Now here is a disturbing character. You are inside his head yet somehow separated from him. He lets you look through his eyes, lets you know what he is thinking and feeling, but at the same time keeps you out, leaves you with the suspicion that there’s something he’s not telling you. The killer however speaks to you in the second person, a rarely used point of view, yet it seems to draw you close to the character’s soul. The effect of these varying points of view is stunning in the fast-paced, cliff-hanger passages where the focus switches swiftly from character to character.
The ending is astonishing, an odd shock that seems absolutely right once you’ve thought about what Royle has done. Disturbingly, loose ends are not tied and the unease lasts right up to the last page.
To me, it was a shame that such a tired old monster as the serial killer had to be wheeled out yet again. My experience of Royle’s short fiction is one of inventiveness and originality and someone who can create menace and tension without resorting to any of the usual suspects. However, the author handles the monster deftly and manages to create a nightmare that is original, fresh and unnerving.
Antwerp by Nicholas Royle. Pb, 288pp, £7.99. Published by Serpent’s Tail.
This review originally appeared on Whispers of Wickedness, and is reproduced here with permission. | 92,685 |
\begin{document}
\title[$p$-adic monodromy of Barsotti-Tate groups]
{$p$-adic monodromy of the universal deformation of a HW-cyclic
Barsotti-Tate group}
\author{TIAN Yichao}
\address{LAGA, Institut Galil\'ee, Universit\'e Paris 13,
93430 Villetaneuse, France}
\email{[email protected]}
\begin{abstract}Let $k$ be an algebraically closed field of characteristic $p>0$, and $G$ be a Barsotti-Tate over $k$. We denote by $\bS$ the ``algebraic'' local moduli in characteristic $p$ of $G$, by $\bG$ the universal deformation of $G$ over $\bS$, and by $\bU\subset\bS$ the ordinary locus of $\bG$. The \'etale part of $\bG$ over $\bU$ gives rise to a monodromy representation $\rho_{\bG}$ of the fundamental group of $\bU$ on the Tate module of $\bG$. Motivated by a famous theorem of Igusa, we prove in this article that $\rho_{\bG}$ is surjective if $G$ is connected and HW-cyclic. This latter condition is equivalent to saying that Oort's $a$-number of $G$ equals $1$, and it is satisfied by all connected one-dimensional Barsotti-Tate groups over $k$.
\end{abstract}
\maketitle
\section{Introduction}
\subsection{} A classical theorem of Igusa says that the monodromy representation associated with a versal family
of ordinary elliptic curves in characteristic $p>0$ is surjective \cite{Ig,Ka}. This important result has deep consequences in the theory of $p$-adic modular forms, and inpired various generalizations.
Faltings and Chai \cite{Ch,FC} extended it to the universal family over the moduli space of higher dimensional principally polarized ordinary abelian varieties in characteristic $p$, and Ekedahl \cite{Ek} generalized it to the jacobian of the universal $n$-pointed curve in characteristic $p$, equipped with a symplectic level structure. We refer to Deligne-Ribet \cite{DR} and Hida \cite{Hi} for other generalizations to some moduli spaces of PEL-type and their arithmetic applications.
Though it has been formulated in a global setting, the proof of Igusa's theorem is purely local,
and it has got also local generalizations.
Gross \cite{Go} generalized it to one-dimensional formal $\cO$-modules over a complete discrete valuation ring of characteristic $p$, where $\cO$ is the integral closure of $\Z_p$ in a finite extension of $\Q_p$. We refer to Chai \cite{Ch} and Achter-Norman \cite{AN} for more results on local monodromy of Barsotti-Tate groups.
Motivated by these results, it has been longly expected/conjectured that the monodromy of a \emph{versal}
family of ordinary Barsotti-Tate groups in characteristic $p>0$ is maximal.
The aim of this paper is to prove the surjectivity of the monodromy representation
associated with the universal deformation in characteristic $p$ of a certain class of Barsotti-Tate groups.
\subsection{} To describe our main result, we introduce first the notion of HW-cyclic Barsotti-Tate groups. Let $k$ be an algebraically closed field of characteristic $p>0$, and $G$ be a Barsotti-Tate group over $k$. We denote by $G^\vee$ the Serre dual of $G$, and by $\Lie(G^\vee)$ its Lie algebra.
The Frobenius homomorphism of $G$ (or dually the Verschiebung of $G^\vee$) induces a semi-linear endomorphism $\varphi_G$ on $\Lie(G^\vee)$,
called the Hasse-Witt map of $G$ \eqref{BT-HW}.
We say that $G$ is \emph{HW-cyclic}, if $c=\dim(G^\vee)\geq 1$ and there is a $v\in \Lie(G^\vee)$ such that $v,\varphi_G(v),\cdots, \varphi^{c-1}_G(v)$ form a basis of $\Lie(G^\vee)$ over $k$ (\ref{defn-cyclic}).
We prove in \ref{prop-HW-a} that $G$ is HW-cyclic and non-ordinary if and only if the $a$-number of $G$,
defined previously by Oort, equals $1$. We can construct HW-cyclic Barsotti-Tate groups as follows.
Let $r,s$ be relatively prime integers such that $0\leq s\leq r$ and $r\neq 0$, $\lambda=s/r$, $G^\lambda$ be the Barsotti-Tate group over $k$ whose (contravariant) Dieudonn\'e module is generated by an element $e$ over the non-commutative Dieudonn\'e ring with the relation $(F^{r-s}-V^s)\cdot e=0$ \eqref{HW-exem}. It is easy to see that $G^\lambda$ is HW-cyclic for any $0<\lambda<1$.
Any connected Barsotti-Tate group over $k$ of dimension $1$ and height $h$ is isomorphic to $G^{1/h}$ \cite[Chap.IV \S8]{De}.
Let $G$ be a Barsotti-Tate group of dimension $d$ and height $c+d$ over $k$; assume $c\geq 1$. We denote by $\bS$ the ``algebraic'' local moduli of $G$ in characteristic $p$, and by $\bG$ be the universal deformation of $G$ over $\bS$
(cf. \ref{defn-moduli}). The scheme $\bS$ is affine of ring $R\simeq k[[(t_{i,j})_{1\leq i \leq c,1\leq j\leq d}]]$, and the Barsotti-Tate group $\bG$ is obtained by algebraizing the formal universal deformation of $G$ over $\Spf(R)$ (\ref{cor-alg-univ}). Let $\bU$ be the ordinary locus of $\bG$ (\ie the open subscheme of $\bS$ parametrizing the ordinary fibers of $\bG$), and $\etab$ a geometric point over the generic point of $\bU$. For any integer $n\geq 1$, we denote by $\bG(n)$ the kernel of the multiplication by $p^n$ on $\bG$, and by
\[
\rT_p(\bG,\etab)=\varprojlim_n\bG(n)(\etab)
\] the Tate module of $\bG$ at $\etab$. This is a free $\Z_p$-module of rank $c$. We consider the monodromy representation attached to the \'etale part of $\bG$ over $\bU$
\begin{equation}\label{mono-rep-univ}
\rho_{\bG}:\pi_1(\bU,\etab)\ra \Aut_{\Z_p}(\rT_p(\bG,\etab))\simeq \GL_{c}(\Z_p).\end{equation}
The aim of this paper is to prove the following~:
\begin{thm}\label{thm-main} If $G$ is connected and HW-cyclic, then
the monodromy representation $\rho_\bG$ is surjective.
\end{thm}
Igusa's theorem mentioned above corresponds to Theorem \ref{thm-main} for $G=G^{1/2}$ (cf. \ref{thm-Igusa}). My interest in the $p$-adic monodromy problem started with the second part of my PhD thesis \cite{tian},
where I guessed \ref{thm-main} for $G=G^{\lam}$ with $0< \lam <1$ and proved it for $G^{1/3}$.
After I posted the manuscript on ArXiv \cite{tian2},
Strauch proved the one-dimensional case of \ref{thm-main} by using Drinfeld's level structures \cite[Theorem 2.1]{str}. Later on, Lau \cite{lau} proved \ref{thm-main} without the assumption that $G$ is HW-cyclic.
By using the Newton stratification of the universal deformation space of $G$ due to Oort, Lau reduced
the higher dimensional case to the one-dimensional case treated by Strauch.
In fact, Strauch and Lau considered more generally the monodromy representation
over each $p$-rank stratum of the universal deformation space. Recently, Chai and Oort \cite{CO} proved the maximality of the $p$-adic monodromy over each ``central leaf'' in the moduli space of abelian varieties which is not contained in the supersingular locus.
In this paper, we provide first a different proof of the one-dimensional case of \ref{thm-main}.
Our approach is purely characteristic $p$, while Strauch used Drinfeld's level structure in characteristic $0$.
Then by following Lau's strategy, we give a new (and easier) argument to reduce the general case of \ref{thm-main}
to the one-dimensional case for HW-cyclic groups. The essential part of our argument is a versality criterion by Hasse-Witt maps of deformations of a connected one-dimensional Barsotti-Tate group (Prop. \ref{prop-HW-versal}). This criterion can be considered as a generalization of another theorem of Igusa
which claims that the Hasse invariant of a versal family of elliptic curves in characteristic $p$ has simple zeros. Compared with Strauch's approch, our characteristic $p$ approach has the advantage of giving also results
on the monodromy of Barsotti-Tate groups over a discrete valuation ring of characteristic $p$.
\subsection{} Let $A=k[[\pi]]$ be the ring of formal power series over $k$ in the variable $\pi$, $K$ its fraction field, and $\tv$ the valuation on $K$ normalized by $\tv(\pi)=1$. We fix an algebraic closure $\Kb$ of $K$, and let $\Ks$ be the separable closure of $K$ contained in $\Kb$, $I$ be the Galois group of $\Ks$ over $K$, $I_p\subset I$ be the wild inertia subgroup, and $I_t=I/I_p$ the tame inertia group. For every integer $n\geq 1$, there is a canonical surjective character $\theta_{p^n-1}:I_t\ra \F^{\times}_{p^n}$ \eqref{galois-char}, where $\F_{p^n}$ is the finite subfield of $k$ with $p^n$ elements.
We put $S=\Spec(A)$. Let $G$ be a Barsotti-Tate group over $S$, $G^\vee$ be its Serre dual, and $\Lie(G^\vee)$ the Lie algebrasof $G^\vee$. Recall that the Frobenius homomorphism of $G$ induces a semi-linear endomorphism $\varphi_G$ of $\Lie(G^\vee)$, called the Hasse-Witt map of $G$. We define $h(G)$ to be the valuation of the determinant of a matrix of $\varphi_G$, and call it the \emph{Hasse invariant} of $G$ \eqref{defn-hw-index}. We see easily that $h(G)=0$ if and only if $G$ is ordinary over $S$, and $h(G)<\infty$ if and only if $G$ is generically ordinary. If $G$ is connected of height $2$ and dimension $1$, then $h(G)=1$ is equivalent to that $G$ is versal \eqref{thm-Igusa}.
\begin{prop} Let $S=\Spec(A)$ be as above, $G$ be a connected HW-cyclic Barsotti-Tate group with Hasse invariant $h(G)=1$, and $G(1)$ the kernel of the multiplication by $p$ on $G$. Then the action of $I$ on $G(1)(\Kb)$ is tame; moverover, $G(1)(\Kb)$ is an $\F_{p^c}$-vector space of dimension $1$ on which the induced action of $I_t$ is given by the surjective character $\theta_{p^c-1}:I_t\ra \F_{p^c}^\times$.
\end{prop}
This proposition is an analogue in characteristic $p$ of Serre's result \cite[Prop. 9]{Se} on the tameness of the monodromy associated with one-dimensional formal groups over a trait of mixed characteristic. We refer to \ref{prop-mono-trait} for the proof of this proposition and more results on the $p$-adic monodromy of HW-cyclic Barsotti-Tate groups over a trait in characteristic $p$.
\subsection{} This paper is organized as follows. In Section 2, we review some well known facts on ordinary Barsotti-Tate groups. Section 3 contains some preliminaries on the Dieudonn\'e theory and the deformation theory of Barsotti-Tate groups. In Section 4, after establishing some basic properties of HW-cyclic groups, we give the fundamental relation between the versality of a Barsotti-Tate group and the coefficients of its Hasse-Witt matrix (Prop. \ref{prop-HW-versal}). Section 5 is devoted to the study of the monodromy of a HW-cyclic Barsotti-Tate group over a complete trait of characteristic $p$. Section 6 is totally elementary, and contains a criterion \eqref{lemma-gp-1} for the surjectivity of a homomorphism from a profinite group to $\GL_n(\Z_p)$. In Section 7, we prove the one-dimensional case of Theorem \ref{thm-main}. Finally in Section 8, we follow Lau's strategy and complete the proof of \ref{thm-main} by reducing the general case to the one-dimensional case treated in Section 7.
\subsection{Acknowledgement} This paper is an expanded version of the second part of my Ph.D. thesis at University Paris 13. I would like to express my great gratitude to my thesis advisor Prof. A. Abbes for his encouragement during this work, and also for his various helpful comments on earlier versions of this paper. I also thank heartily E. Lau, F. Oort and M. Strauch for interesting discussions and valuable suggestions.
\subsection{Notations}\label{Notations} Let $S$ be a scheme of characteristic $p>0$. A {\em BT-group} over $S$
stands for a Barsotti-Tate group over $S$. Let $G$ be a commutative
finite group scheme (\resp a BT-group) over $S$. We denote by
$G^\vee$ its Cartier dual (\resp its Serre dual), by $\omega_G$ the
sheaf of invariant differentials of $G$ over $S$, and by $\Lie(G)$
the sheaf of Lie algebras of $G$. If $S=\Spec(A)$ is affine and there is no risk of confusions, we also use $\omega_G$ and $\Lie(G)$ to denote the correponding $A$-modules of global sections.
We put $G^{(p)}$ the pull-back of $G$ by the absolute Frobenius
of $S$, $F_G\colon G\ra G^{(p)}$ the Frobenius homomorphism and
$V_G\colon G^{(p)}\ra G$ the Verschiebung homomorphism. If $G$ is a
BT-group and $n$ an integer $\geq 1$, we denote by $G(n)$ the
kernel of { the} multiplication by $p^n$ on $G$; we have
$G^\vee(n)=(G^\vee)(n)$ by definition. For an $\cO_S$-module $M$, we denote by $M^{(p)}=\cO_S\otimes_{F_S}M$ the scalar extension of $M$ by the absolute Frobenius of $\cO_S$. If $\varphi: M\ra N$ be a semi-linear homomorphism of $\cO_S$-modules, we denote by $\widetilde{\varphi}:M^{(p)}\ra N$ the linearization of $\varphi$, \ie we have $\widetilde{\varphi}(\lambda\otimes x)=\lambda\cdot \varphi(x)$, where $\lambda$ (\resp $x$) is a local section of $\cO_S$ (\resp of $M$).
Starting from Section 5, $k$ will denote an algebraically closed field of characteristic $p>0$.
\section{Review of ordinary Barsotti-Tate groups}
In this section, $S$ denotes a scheme of \car $p>0$.
\subsection{}\label{Lie-alg}
Let $G$ be a commutative group scheme, locally free of finite type
over $S$. We have a canonical isomorphism of coherent
$\cO_S$-modules \cite[2.1]{Il}
\begin{equation}\label{dual-Groth}\Lie(G^\vee)\simeq
\cHom_{S_{\mathrm{fppf}}}(G,\G_a) ,
\end{equation}
where $\cHom_{S_{\mathrm{fppf}}}$ is the sheaf of homomorphisms in
the category of abelian $\fppf$-sheaves over $S$, and $\G_a$ is the
additive group scheme. Since $\G_a^{(p)}\simeq \G_a$, the Frobenius homomorphism of $\G_a$ induces
an endomorphism
\begin{equation}\label{hw-finite}
\varphi_G:\Lie(G^\vee)\ra \Lie(G^\vee),
\end{equation} semi-linear with respect
to the absolute Frobenius map $F_S:\cO_S\ra \cO_S$; we call it the
\emph{Hasse-Witt} map of $G$. By the functoriality of Frobenius, $\varphi_G$ is also the canonical map induced by the Frobenius of $G$, or dually by the Verschiebung of $G^\vee$.
\subsection{}\label{sect-GV} By a \emph{commutative $p$-Lie algebra} over $S$, we mean a pair
$(L,\varphi)$, where $L$ is an $\cO_S$-module locally free of finite type, and $\varphi:L\ra L$ is a semi-linear endomorphism with
respect to the absolute Frobenius $F_S:\cO_S\ra \cO_S$. When there
is no risk of confusions, we omit $\varphi$ from the notation. We
denote by $\pLie_S$ the category of commutative $p$-Lie algebras
over $S$.
Let $(L,\varphi)$ be an object of $\pLie_S$. We denote by
\[
\cU(L)=\mathrm{Sym}(L)=\oplus_{n\geq 0}\ \mathrm{Sym}^{n} (L),
\] the symmetric algebra of $L$ over $\cO_S$. Let $\cI_p(L)$ be
the ideal sheaf of $\cU(L)$ defined, for an open subset $V\subset
S$, by
\[
\Gamma(V,\cI_p(L))=\{x^{\otimes p}-\varphi(x)\ ;\ x\in \Gamma(V,\cU(L))\},
\]
where $x^{\otimes p}=x\otimes x\otimes\cdots\otimes x\in \Gamma(V,\mathrm{Sym}^{p} (L))$.
We put $\cU_p(L)=\cU(L)/\cI_p(L)$, and call it the
\emph{$p$-enveloping algebra of $(L,\varphi)$}. We endow $\cU_p(L)$ with the
structure of a Hopf-algebra with the comultiplication given by
$\Delta(x)=1\otimes x +x\otimes 1 $ and the coinverse given by
$i(x)=-x$.
Let $G$ be a commutative group scheme, locally free of finite type
over $S$. We say that $G$ is \emph{of coheight one} if the
Verschiebung $V_G: G^{(p)}\ra G$ is the zero homomorphism. We denote by $\GV_S$
the category of such objects. For an object $G$ of
$\GV_S$, the Frobenius $F_{G^\vee}$ of $G^\vee$ is zero, so the Lie algebra $\Lie(G^\vee)$ is
locally free of finite type over $\cO_S$ (\cite{DG} $\mathrm{VII_A}$
Th\'eo. 7.4(iii)). The Hasse-Witt map of $G$ \eqref{hw-finite}
endows $\Lie(G^\vee)$ with a commutative $p$-Lie algebra structure
over $S$.
\begin{prop}[\cite{DG} $\mathrm{VII_{A}}$, Th\'eo. 7.2 et
7.4]\label{GV-pLie} The functor $\GV_S\ra \pLie_S$ defined by
$G\mapsto \Lie(G^\vee)$ is an anti-equivalence of categories; a
quasi-inverse is given by
$(L,\varphi)\mapsto \Spec (\cU_p(L))$.
\end{prop}
\subsection{}\label{desc-ex}
Assume $S=\Spec(A)$ affine. Let $(L,\varphi)$ be an object of $
\pLie_S$ such that $L$ is free of rank $n$ over $\cO_S$,
$(e_1,\cdots, e_n)$ be a basis of $L$ over $\cO_S$,
$(h_{ij})_{1\leq i, j\leq n}$ be the matrix of $\varphi$ under the basis $(e_1,\cdots,e_n)$, \ie
$\varphi(e_j)=\sum_{i=1}^n
h_{ij}e_i$ for $1\leq j\leq n$. Then the group scheme associated to $(L,\varphi)$ is
explicitly given by
\[
\Spec(\cU_p(L))=\Spec \biggl (A[X_1,\cdots,X_n]/(X_j^p-\sum_{i=1}^nh_{ij}X_i)_{1\leq j\leq n}\biggr),
\]
with the comultiplication $\Delta(X_j)=1\otimes X_j+X_j\otimes 1$.
By the Jacobian criterion of \'etaleness [EGA $\mathrm{IV_0}$
22.6.7], the finite group scheme $\Spec(\cU_p(L))$ is \'etale over
$S$ if and only if the matrix $(h_{ij})_{1\leq i, j\leq n}$ is
invertible. This condition is equivalent to that the linearization
of $\varphi$ is an isomorphism.
\begin{cor}\label{cor-etale-GV} An object $G$ of
$\GV_S$ is \'etale over $S$, if and only if the linearization of
its Hasse-Witt map \eqref{hw-finite} is an isomorphism.
\end{cor}
\begin{proof}
The problem being local over $S$, we may assume $S$ affine and
$L=\Lie(G^\vee)$ free over $\cO_S$. By Theorem \ref{GV-pLie}, $G$ is
isomorphic to $\Spec(\cU_p(L))$, and we conclude by the last remark
of \ref{desc-ex}.
\end{proof}
\subsection{} Let $G$ be a BT-group over $S$ of height $c+d$ and dimension $d$, $G^\vee$ be its Serre dual. The Lie algebra $\Lie(G^\vee)$ is an $\cO_S$-module locally free of rank $c$, and canonically identified with $\Lie(G^\vee(1))$(\cite{BBM} 3.3.2).
We define the {\em Hasse-Witt map} of $G$
\begin{equation}\label{BT-HW}
\HW_G:\Lie(G^\vee)\ra\Lie(G^\vee)
\end{equation}
to be that of $G(1)$ \eqref{hw-finite}.
\subsection{}Let $k$ be a field of characteristic $p>0$, $G$ be a BT-group over $k$. Recall that we have a canonical exact
sequence of BT-groups over $k$
\begin{equation}\label{decomp-BT}0\ra G^{\circ}\ra G\ra G^{\et}\ra 0\end{equation}
with $G^{\circ}$ connected and $G^\et$ \'etale (\cite{De} Chap.II, \S 7). This induces an exact sequence of Lie algebras
\begin{equation}\label{decomp-Lie}
0\ra \Lie(G^{\et\vee})\ra \Lie(G^\vee)\ra \Lie(G^{\circ\vee})\ra 0,
\end{equation}
compatible with Hasse-Witt maps.
\begin{prop}\label{prop-etale-HW} Let $k$ be a field of characteristic $p>0$, $G$ be a BT-group over $k$. Then
$\Lie(G^{\et\vee})$ is the unique maximal $k$-subspace $V$ of $ \Lie(G^\vee)$ with the following properties:
\emph{(a)} $V$ is stable under $\varphi_G$;
\emph{(b)} the restriction of $\varphi_G$ to $V$ is injective.
\end{prop}
\begin{proof} It is clear that $\Lie(G^{\et\vee})$ satisfies property (a). We note that the Verschiebung of $G^\et(1)$ vanishes; so $G^\et(1)$ is in the category $\GV_{\Spec(k)}$. Since $k$ is a field, \ref{cor-etale-GV} implies that the restriction of $\varphi_G$ to $\Lie(G^{\et\vee})$, which coincides with $\varphi_{G^\et}$, is injective. This proves that $\Lie(G^{\et\vee})$ verifies (b). Conversely, let $V$ be an arbitrary $k$-subspace of $\Lie(G^\vee)$ with properties (a) and (b). We have to show that $V\subset \Lie(G^{\et\vee})$. Let $\sigma$ be the Frobenius endomorphism of $k$. If $M$ is a $k$-vector space, for each integer $n\geq 1$, we put $M^{(p^n)}=k\otimes_{\sigma^n}M$, \ie we have $1\otimes ax=\sigma^n(a)\otimes x$ in $k\otimes_{\sigma^n}M$. Since $\varphi_G|_V:V\ra V$ is injective by assumption, the linearization $\widetilde{\varphi^n_{G}}|_{V^{(p^n)}}:V^{(p^n)}\ra V$ of $\varphi^n_G|_V$ is injective (hence bijective) for any $n\geq 1$. We have $V=\widetilde{\varphi_G^n}(V^{(p^n)})$. Since $G^\circ$ is connected, there is an integer $n\geq 1$ such that the $n$-th iterated Frobenius $F^n_{G^\circ(1)}:G^\circ(1)\ra G^\circ(1)^{(p^n)}$ vanishes. Hence by definition, the linearized $n$-iterated Hasse-Witt map $\widetilde{\varphi^n_{G^\circ}}:\Lie(G^{\circ\vee})^{(p^n)}\ra \Lie(G^{\circ\vee})$ is zero. By the compatibility of Hasse-Witt maps, we have $\widetilde{\varphi_G^n}(\Lie(G^\vee)^{(p^n)})\subset \Lie(G^{\et\vee})$; in particular, we have $V=\widetilde{\varphi^n_G}(V^{(p^n)})\subset \Lie(G^{\et\vee})$. This completes the proof.
\end{proof}
\begin{cor}\label{cor-nilp-HW} Let $k$ be a field of characteristic $p>0$, $G$ be a BT-group over $k$. Then $G$ is connected if and only if $\varphi_G$ is nilpotent.
\end{cor}
\begin{proof}
In the proof of the proposition, we have seen that the Hasse-Witt map of the connected part of $G$ is nilpotent. So the ``only if'' part is verified. Conversely, if $\varphi_G$ is nilpotent, $\Lie(G^{\et\vee})$ is zero by the proposition. Therefore $G$ is connected.
\end{proof}
\begin{defn} Let $S$ be a scheme of characteristic $p>0$, $G$ be a BT-group over $S$. We say that $G$ is \emph{ordinary}
if there exists an exact sequence of BT-groups over $S$
\begin{equation}\label{decom-ord}
0\ra G^{\mult}\ra G\ra G^{\et}\ra 0,
\end{equation}
such that $G^{\mult}$ is multiplicative and $G^{\et}$ is \'etale.
\end{defn}
We note that when it exists, the exact sequence
\eqref{decom-ord}
is unique up to a unique isomorphism, because there is no
non-trivial homomorphisms between a multiplicative BT-group and an
\'etale one in characteristic $p>0$. The property of being ordinary is clearly stable under arbitrary
base change and Serre duality. If $S$ is the spectrum of a field of characteristic $p>0$, $G$ is
ordinary if and only if its connected part $G^{\circ}$ is of
multiplicative type.
\begin{prop}\label{prop-ord} Let $G$ be a BT-group over $S$. The following
conditions are equivalent:
\emph{(a)} $G$ is ordinary over $S$.
\emph{(b)} For every $x\in S$, the fiber $G_x=G\otimes_S \kappa(x)$ is ordinary over $\kappa(x)$.
\emph{(c)} The finite group scheme $\Ker V_G$ is \'etale over $S$.
\emph{(c')} The finite group scheme $\Ker F_G$ is of multiplicative type
over $S$.
\emph{(d)} The linearization of the Hasse-Witt map $\HW_G$
is an isomorphism.
\end{prop}
First, we prove the following lemmas.
\begin{lemma}\label{lemma-et-mul} Let $T$ be a scheme, $H$ be a commutative group
scheme locally free of finite type over $T$. Then $H$ is \'etale (\resp of
multiplicative type) over $T$ if and only if, for every
$x\in T$, the fiber $H\otimes_T\kappa(x)$ is \'etale (\resp of multiplicative type) over
$\kappa(x)$.
\end{lemma}
\begin{proof}We will consider only the \'etale case; the multiplicative
case follows by duality. Since $H$ is $T$-flat, it is \'etale over
$T$ if and only if it is unramified over $T$. By [EGA IV 17.4.2],
this condition is equivalent to that $H\otimes_T\kappa(x)$ is unramified over $\kappa(x)$ for every point $x\in T$. Hence the conclusion follows.
\end{proof}
\begin{lemma}\label{lemma-KerV} Let $G$ be a BT-group over $S$. Then
$\Ker V_G$ is an object of the category $\GV_S$, \ie it is locally
free of finite type over $S$, and its Verschiebung is zero. Moreover, we have a canonical isomorphism $(\Ker
V_G)^\vee\simeq \Ker F_{G^\vee}$, which induces an isomorphism of
Lie algebras $\Lie\bigl((\Ker V_G)^\vee\bigr)\simeq \Lie (\Ker
F_{G^\vee})= \Lie (G^\vee)$, and the Hasse-Witt map
\eqref{hw-finite} of $\Ker V_G$ is identified with $\HW_G$
\eqref{BT-HW}.
\end{lemma}
\begin{proof} The group scheme $\Ker V_G$ is locally free of finite
type over $S$ (\cite{Il} 1.3(b)), and we have a commutative diagram
\[\xymatrix{(\Ker V_G)^{(p)}\ar[rr]^{V_{\Ker V_G}}\ar@{^(->}[d] &&\Ker V_G\ar@{^(->}[d]\\
(G^{(p)})^{(p)}\ar[rr]^{V_{G^{(p)}}}&&G^{(p)}}\] By the
functoriality of Verschiebung, we have $ V_{G^{(p)}}=( V_{G})^{(p)}$
and $\Ker V_{G^{(p)}}=(\Ker V_G)^{(p)}$. Hence the composition of
the left vertical arrow with $V_{G^{(p)}}$ vanishes, and the
Verschiebung of $\Ker V_{G}$ is zero.
By Cartier duality, we have $(\Ker V_G)^\vee=\Coker
(F_{G^\vee(1)})$. Moreover, the exact sequence
\[\cdots \ra G^\vee(1)\xra{F_{G^\vee(1)}}\bigl(G^\vee(1)\bigr)^{(p)}\xra{V_{G^\vee(1)}}G^\vee(1)\ra \cdots,\]
induces a canonical isomorphism
\begin{equation}\label{isom-Ker-Coker}\Coker
(F_{G^\vee(1)})\xra{\sim} \im (V_{G^\vee(1)})=\Ker
F_{G^\vee(1)}=\Ker F_{G^\vee}.\end{equation} Hence, we deduce that
\begin{equation}\label{isom-Ker-Coker2}(\Ker
V_G)^\vee\simeq\Coker(F_{G^\vee(1)})\xra{\sim} \Ker
F_{G^\vee}\hookrightarrow G^\vee(1).\end{equation} Since the natural
injection $\Ker F_{G^\vee}\ra G^\vee(1)$ induces an isomorphism of
Lie algebras, we get \begin{equation}\label{isom-Lie}\Lie\bigl((\Ker
V_G)^\vee\bigr) \simeq \Lie (\Ker F_{G^\vee})=\Lie
(G^\vee(1))=\Lie(G^\vee).\end{equation} It remains to prove the
compatibility of the Hasse-Witt maps with \eqref{isom-Lie}. We note
that the dual of the morphism \eqref{isom-Ker-Coker2}
is the canonical map $F:G(1)\ra \Ker V_G=\im
(F_{G(1)})$ induced by $F_{G(1)}$. Hence by \eqref{dual-Groth}, the
isomorphism \eqref{isom-Lie} is identified with the functorial map
\[\cHom_{S_{\fppf}}(\Ker V_G,\G_a)\ra \cHom_{S_{\fppf}}(G(1),\G_a)\]
induced by $F$, and its compatibility with the Hasse-Witt maps
follows easily from the definition \eqref{hw-finite}.
\end{proof}
\begin{proof}[Proof of \ref{prop-ord}] (a)$\Rightarrow$(b). Indeed,
the ordinarity of $G$ is stable by base change.
(b)$\Rightarrow$(c). By Lemma \ref{lemma-et-mul}, it suffices to verify that for every point $x\in S$,
the fiber $(\Ker V_G)\otimes_S{\kappa(x)}\simeq \Ker V_{G_x}$ is \'etale over
$\kappa(x)$. Since $G_x$ is assumed to be ordinary, its connected part
$(G_x)^{\circ}$ is multiplicative. Hence, the Verschiebung of
$(G_x)^{\circ}$ is an isomorphism, and $\Ker V_{G_x}$ is canonically isomorphic to
$ \Ker V_{G_x^\et}\subset (G_x^\et)^{(p)}\simeq (G_x^{(p)})^\et$, so our assertion follows.
$(c)\Leftrightarrow(d)$. It follows
immediately from Lemma \ref{lemma-KerV} and Corollary
\ref{cor-etale-GV}.
(c)$\Leftrightarrow$(c').
By \ref{lemma-et-mul}, we may assume that $S$ is the spectrum of a field.
So the category of commutative
finite group schemes over $S$ is abelian. We will just prove
(c)$\Rightarrow$(c'); the converse can be proved by duality.
We have a fundamental short exact
sequence of finite group schemes
\begin{equation}\label{exseq-fund-BT}0\ra \Ker F_G\ra G(1)\xra{F} \Ker
V_G\ra0,
\end{equation} where $F$ is induced by $F_{G(1)}$,
That induces a commutative diagram
\[\xymatrix{0\ar[r]&\bigl(\Ker F_G\bigr)^{(p)}\ar[d]^{V'}\ar[r]&\bigl(G(1)\bigr)^{(p)}\ar[r]^{F^{(p)}}\ar[d]^{V_{G(1)}}
&\bigl(\Ker V_G\bigr)^{(p)}\ar[r]\ar[d]^{V''}&0\\
0\ar[r]&\Ker F_G\ar[r]& G(1)\ar[r]^{F}&\Ker V_G\ar[r]&0}\]
where vertical arrows are the Verschiebung homomorphisms. We have seen
that $V''=0$ (\ref{lemma-KerV}).
Therefore, by the snake lemma, we have a long exact sequence
\begin{equation}\label{long-ext-seq}0\ra \Ker V'\ra \Ker V_{G(1)}\xra{\alpha}
\bigl(\Ker V_G\bigr)^{(p)}\ra \Coker V'\ra \Coker V_{G(1)}\xra{\beta}\Ker V_G\ra
0,
\end{equation}
where the map $\alpha$ is the Frobenius of $\Ker V_G$ and $\beta$
is the composed isomorphism
$$\Coker (V_{G(1)})\simeq G(1)/\Ker F_{G(1)}\xra{\sim} \im (F_{G(1)})\simeq
\Ker V_G.$$ Then condition (c) is equivalent to that $\alpha$ is an
isomorphism; it implies that $\Ker V'=\Coker V'=0$, \ie the
Verschiebung of $\Ker F_G$ is an isomorphism, and hence (c').
(c)$\Rightarrow$(a). For every integer $n>0$, we denote
by $F^n_G$ the composed homomorphism
\[G\xra{F_G}G^{(p)}\xra{F_{G^{(p)}}}\cdots\xra{F_{G^{(p^{n-1})}}}G^{(p^n)},\] and by $V^n_G$
the composed homomorphism
\[G^{(p^n)}\xra{V_{G^{(p^{n-1})}}}G^{(p^{n-1})}\xra{V_{G^{(p^{n-2})}}}\cdots\xra{V_{G}}G;\]
$F_G^n$ and $V_G^n$ are isogenies of BT-groups. From the relation
$V^n_G\circ F^n_G=p^n$, we deduce an exact sequence
\beq\label{exseq-FV}0\ra \Ker F^n_G\ra G(n)\xra{F^n} \Ker V^n_G\ra
0,\eeq where $F^n$ is induced by $F_{G}^n$. For $1\leq j<n$, we
have a commutative diagram
\beq\label{diag-V_G} \xymatrix{G^{(p^n)}\ar[rr]^{V^{n-j}_{G^{(p^j)}}}\ar[rd]_{V_G^n}&&G^{(p^j)}\ar[ld]^{V_{G}^j}\\
&G.}\eeq One notices by the functoriality of Verschiebung that
$\Ker V^{n-j}_{G^{(p^j)}}=(\Ker V^{n-j}_G) ^{(p^j)}$. Since all maps
in \eqref{diag-V_G} are isogenies, we have an exact sequence
\beq\label{exseq-V}0\ra (\Ker V^{n-j}_G)^{(p^j)}\xra{i'_{n-j,n}}
\Ker V^n_G\xra{p_{n,j}} \Ker V^j_G\ra 0. \eeq Therefore, condition
(c) implies by induction that $\Ker V^n_G$ is an \'etale group
scheme over $S$. Hence the $j$-th iteration of the Frobenius $\Ker
V^{n-j}_G\ra (\Ker V^{n-j}_G)^{(p^j)}$ is an isomorphism,
and $\Ker V^{n-j}_G$ is identified with a closed subgroup scheme of $\Ker
V^{n}_G $ by the composed map
\[
i_{n-j,n}:\;\Ker V^{n-j}_G\xra{\sim} (\Ker
V^{n-j}_{G})^{(p^j)}\xra{i'_{n-j,n}}\Ker V^n_G.
\]
We claim that the kernel of the multiplication by $p^{n-j}$ on $\Ker
V^n_G$ is $\Ker V^{n-j}_G$. Indeed, from the relation $p^{n-j}\cdot
\Id_{G^{(p^n)}}=F^{n-j}_{G^{(p^j)}}\circ V^{n-j}_{G^{(p^j)}}$, we
deduce a commutative diagram (without dotted arrows)
\begin{equation}\label{diag-1}
\xymatrix{\Ker
V^n_G\ar[rr]\ar[dd]_-(0.7){p^{n-j}}\ar@{-->}[rd]^{p_{n,j}}&&G^{(p^n)}
\ar[dd]_-(0.7){p^{n-j}}\ar[rd]^{V^{n-j}_{G^{(p^j)}}}\\
&\Ker
V^j_{G}\ar@{-->}[rr]\ar@{-->}[ld]^{i_{j,n}}&&G^{(p^j)}\ar[ld]^{F^{n-j}_{G^{(p^{j})}}}\\
\Ker V_G^n\ar[rr]&&G^{(p^n)}.}
\end{equation}
It follows from \eqref{exseq-V} that the subgroup $\Ker V^n_G$ of $
G^{(p^n)}$ is sent by $V^{n-j}_{G^{(p^j)}}$ onto $\Ker V^j_G$.
Therefore diagram \eqref{diag-1} remains commutative when completed
by the dotted arrows, hence our claim.
It follows from the claim that $(\Ker V_G^n)_{n\geq 1}$ constitutes
an \'etale BT-group over
$S$, denoted by $G^\et$. By duality, we have an exact sequence
\begin{equation}\label{exseq-F}0\ra \Ker
F^j_G\ra \Ker F^n_G\ra (\Ker F^{n-j}_G)^{(p^j)}\ra 0.\end{equation}
Condition (c') implies by induction that $\Ker F^n_G$ is of
multiplicative type. Hence the $j$-th iteration of Verschiebung
$(\Ker F^{n-j}_G)^{(p^j)}\ra \Ker F^{n-j}_G$ is an isomorphism. We
deduce from \eqref{exseq-F} that $(\Ker F^n_G)_{n \geq 1}$ form a
multiplicative BT-group over $S$ that we denote by $G^{\mult}$.
Then the exact sequences \eqref{exseq-FV} give a decomposition of
$G$ of the form \eqref{decom-ord}.
\end{proof}
\begin{cor}\label{cor-ord-loc} Let $G$ be a BT-group over $S$, and $S^{\ord}$ be the
locus in $S$ of the points $x\in S$ such that
$G_x=G\otimes_S\kappa(x)$ is ordinary over $\kappa(x)$. Then
$S^{\ord}$ is open in $S$, and the canonical inclusion $S^{\ord}\ra
S$ is affine.
\end{cor}
The open subscheme $S^{\ord}$ of $S$ is called the \emph{ordinary
locus }of $G$.
\section{Preliminaries on Dieudonn\'e Theory and Deformation Theory}
\subsection{}\label{pre-Dieud} We will use freely the conventions of \ref{Notations}. Let $S$ be a scheme of characteristic $p>0$, $G$ be a Barsotti-Tate group over $S$, and $\M(G)$ be the coherent $\cO_S$-module obtained by evaluating the (contravariant) Dieudonn\'e crystal of $G$ at the trivial divided power immersion $S\hra S$. Recall that $\M(G)$ is an $\cO_S$-module locally free of finite type satisfying the following properties:
(i) Let $F_M:\M(G)^{(p)}\ra \M(G)$ and $V_M:\M(G)\ra \M(G)^{(p)}$ be the $\cO_S$-linear maps induced respectively by the Frobenius and the Verschiebung of $G$. We have the following exact sequence:
\[\cdots\ra \M(G)^{(p)}\xra{F_M}\M(G)\xra{V_M}\M(G)^{(p)}\ra\cdots.\]
(ii) There is a connection $\nabla: \M(G)\ra \M(G)\otimes_{\cO_S}\Omega^1_{S/\F_p}$ for which $F_M$ and $V_M$ are horizontal morphisms.
(iii) We have two canonical filtrations by $\cO_S$-modules on $\M(G)$:
\begin{equation}\label{filt-Hodge}
0\ra \omega_G\ra \M(G)\ra \Lie(G^\vee)\ra 0,
\end{equation}
called the \emph{Hodge filtration} on $\M(G)$, and
\begin{equation}\label{filt-cong}
0\ra \Lie(G^\vee)^{(p)}\xra{\phi_G} \M(G)\ra\omega^{(p)}_{G}\ra 0,
\end{equation}
called the \emph{conjugate filtration} on $\M(G)$. Moreover, we have the following commutative diagram (cf. \cite[2.3.2 and 2.3.4]{Kz})
\begin{equation}\label{diag-Dieud}
\xymatrix{&0\ar[d]&&0\ar[d]&&0\ar[d]&\\
&\omega^{(p)}_G\ar[d]&&\omega_G\ar[d]\ar[rr]^{\psi_G}&&\omega_G^{(p)}\ar[d]&\\
\ar[r]&\M(G)^{(p)}\ar[rr]^{F_M}\ar[d]&&\M(G)\ar[d]\ar[rr]^{V_M}\ar@^{->>}[urr]&&\M(G)^{(p)}\ar[d]\ar[r]&,\\
&\Lie(G^\vee)^{(p)}\ar[d]\ar@^{(->}[urr]^{\phi_G}\ar[rr]^{\widetilde{\varphi_G}}&&\Lie(G^\vee)\ar[d]&&\Lie(G^\vee)^{(p)}\ar[d]&\\
&0&&0&&0&}
\end{equation}
where the columns are the Hodge filtrations and the anti-diagonal is the conjugate filtration. By functoriality, we see easily that $\widetilde{\varphi_G}$ above is nothing but the linearization of the Hasse-Witt map $\varphi_G$ \eqref{BT-HW}, and the morphism $\psi_G^*:\Lie(G)^{(p)}\ra \Lie(G)$, which is obtained by applying the functor $\cHom_{\cO_S}(\_,\cO_S)$ to $\psi_G$, is identified with the linearization $\widetilde{\varphi_{G^\vee}}$ of $\varphi_{G^\vee}$.
The formation of these structures on $\M(G)$ commutes with arbitrary base changes of $S$.
In the sequel, we will use $(\M(G),F_M,\nabla)$ to emphasize these structures on $\M(G)$.
\subsection{}\label{versal} In the reminder of this section, $k$ will denote an algebraically closed field of characteristic $p>0$. Let $S$ be a scheme formally smooth over $k$ such that $\Omega^1_{S/\F_p}=\Omega^1_{S/k}$ is an $\cO_S$-module locally free of finite type, \emph{e.g.} $S=\Spec(A)$ with $A$ a formally smooth $k$-algebra with a finite $p$-basis over $k$. Let $G$ be a BT-group over $S$. We put $\KS$ to be the composed morphism
\begin{equation}\label{morph-KS}
\KS: \omega_{G}\ra \M(G)\xra{\nabla}\M(G)\otimes_{\cO_S}\Omega^1_{S/k}\xra{pr}\Lie(G^\vee)\otimes_{\cO_{S}}\Omega^1_{S/k}
\end{equation}
which is $\cO_{S}$-linear. We put $\cT_{S/k}=\cHom_{\cO_S}(\Omega^1_{S/k},\cO_S)$, and define the \emph{Kodaira-Spencer map} of $G$
\begin{equation}\label{Kod-map}
\Kod: \cT_{S/k}\ra \cHom_{\cO_S}(\omega_G,\Lie(G^\vee))
\end{equation}
to be the morphism induced by $\KS$. We say that $G$ is \emph{versal} if $\Kod$ is surjective.
\subsection{}\label{versal-formal} Let $r$ be an integer $\geq 1$, $R=k[[t_1,\cdots,t_r]]$, $\m$ be the maximal ideal of $R$. We put $\cS=\Spf(R)$, $S=\Spec(R)$, and for each integer $n\geq 0$, $S_n=\Spec (R/\m^{n+1})$. By a { BT-group}
$\cG$ over the formal scheme $\cS$, we mean a sequence
of BT-groups $(G_n)_{n\geq 0}$ over $(S_n)_{n\geq 0}$ equipped with
isomorphisms $G_{n+1}\times_{S_{n+1}}S_n\simeq G_n$.
According to (\cite{dJ} 2.4.4), \emph{the functor $G\mapsto
(G\times_S S_n)_{n\geq0}$ induces an equivalence of categories
between the category of BT-groups over $S$ and the category of
BT-groups over $\cS$.} For a BT-group $\cG$ over $\cS$, the
corresponding BT-group $G$ over $S$ is called the
\emph{algebraization }of $\cG$. We say that $\cG$ is \emph{versal} over $\cS$, if its algebraization $G$ is versal over $S$. Since $S$ is local, by Nakayama's Lemma, $\cG$ or $G$ is versal if and only if the reduction of $\Kod$ modulo the maximal ideal
\begin{equation}\label{Kod-0}\Kod_0:\cT_{S/k}\otimes_{\cO_S} k\lra \Hom_{k}(\omega_{G_0},\Lie(G^\vee_0))\end{equation}
is surjective.
\subsection{} We recall briefly the deformation theory of a BT-group.
Let $\AL_k$ be the category of local artinian $k$-algebras with
residue field $k$. We notice that all
morphisms of $\AL_k$ are local. A morphism $A'\ra A$ in $\AL_k$ is
called a \emph{small extension}, if it is surjective and its kernel
$I$ satisfies $I\cdot \m_{A'}=0$, where $\m_{A'}$ is the maximal ideal of $A'$.
Let $G_0$ be a BT-group over $k$, and $A$ an object of $\AL_k$. A
deformation of $G_0$ over $A$ is a pair $(G,\phi)$, where $G$ is a
BT-group over $\Spec (A)$ and $\phi$ is an isomorphism
$\phi:G\otimes_Ak\xra{\sim} G_0$. When there is no risk of
confusions, we will denote a deformation $(G,\phi)$ simply by $G$. Two deformations $(G,\phi)$
and $(G',\phi')$ over $A$ are isomorphic if there exists an
isomorphism of BT-groups $\psi:G\xra{\sim} G'$ over $A$ such that
$\phi=\phi'\circ(\psi\otimes_A k)$. Let's denote by $\D$ the functor
which associates with each object $A$ of $\AL_k$ the set of
isomorphic classes of deformations of $G_0$ over $A$. If $f:A\ra
B$ is a morphism of $\AL_k$, then the map $\D(f):\D(A)\ra \D(B)$ is
given by extension of scalars. We call $\D$ the \emph{deformation
functor} of $G_0$ over $\AL_k$.
\begin{prop}[\cite{Il} 4.8]\label{prop-deform} Let $G_0$ be a BT-group over $k$ of dimension $d$ and height
$c+d$, $\D$ be the deformation functor of $G_0$ over
$\AL_k$.
\emph{(i)} Let $A'\ra A$ be a small extension in $\AL_k$ with ideal $I$, $x=(G,\phi)$ be an element in $\D(A)$, $\D_x(A')$ be the subset of $\D(A')$ with image $x$ in $\D(A)$. Then the set $\D_x(A')$ is a nonempty homogenous space under the group $\Hom_k(\omega_{G_0},\Lie(G_0^\vee))\otimes_k I$.
\emph{(ii)} The functor $\D$ is pro-representable by a formally
smooth formal scheme $\cS$ over $k$ of relative dimension $cd$,
\ie $\cS=\Spf(R)$ with $R\simeq k[[(t_{ij})_{1\leq i \leq c, 1\leq
j\leq d}]]$, and there exists a unique deformation $(\cG,\psi)$ of $G_0$ over $\cS$ such that,
for any object $A$ of $\AL_k$ and any deformation $(G,\phi)$ of
$G_0$ over $A$, there is a unique homomorphism of local $k$-algebras
$\varphi:R\ra A$ with $(G,\phi)=\D(\varphi)(\cG,\psi)$.
\emph{(iii)} Let $\cT_{\cS/k}(0)=\cT_{\cS/k}\otimes_{\cO_{\cS}} k$ be the
tangent space of $\cS$ at its unique closed point,
\begin{equation*}\Kod_0: \cT_{\cS/k}(0)\lra \Hom_k(\omega_{G_0},\Lie(G_0^\vee))
\end{equation*} be the Kodaira-Spencer map of $\cG$
evaluated at the closed point of $\cS$. Then $\Kod_0$ is bijective,
and it can be described
as follows. For an element $f\in \cT_{\cS/k}(0)$, \ie a homomorphism of
local $k$-algebras
$f:R\ra k[\epsilon]/\epsilon^2$, $\Kod_0(f)$ is the difference of
deformations \[[\cG\otimes_R(k[\epsilon]/\epsilon^2)]-[G_0\otimes_{
k} (k[\epsilon]/\epsilon^2)],\] which is a well-defined element in $
\Hom_k(\omega_{G_0},\Lie(G^\vee_0))$ by \emph{(i)}.
\end{prop}
\begin{rem}\label{rem-basis} Let $(e_j)_{1\leq j \leq d}$ be a basis of $\omega_{G_0}$,
$(f_i)_{1\leq i\leq c}$ be a basis of $\Lie(G_0^\vee)$. In view of
\ref{prop-deform}(iii), we can choose a system of parameters
$(t_{ij})_{1\leq i \leq c, 1\leq j\leq d}$ of $\cS$ such that
$$\Kod_0(\frac{\partial\empty}{\partial t_{ij}})=e_j^*\otimes f_i,$$
where $(e_j^*)_{1\leq j\leq d}$ is the dual basis of $(e_j)_{1\leq
j\leq d}$. Moreover, if $\m$ is the maximal ideal of $R$, the
parameters $t_{ij}$ are determined uniquely modulo $\m^2$.
\end{rem}
\begin{cor}[\textbf{Algebraization of the universal deformation}]\label{cor-alg-univ}
The assumptions being those of
$\eqref{prop-deform}$, we put moreover $\bS=\Spec(R)$ and $\bG$ the
algebraization of the universal formal deformation $\cG$. Then the BT-group $\bG$ is versal over $\bS$, and satisfies the following universal property: Let
$A$ be a noetherian complete local $k$-algebra with residue field
$k$, $G$ be a BT-group over $A$ endowed with an isomorphism
$G\otimes_Ak\simeq G_0$. Then there exists a unique continuous
homomorphism of local $k$-algebras $\varphi:R\ra A$ such that
$G\simeq \bG\otimes_RA$.
\end{cor}
\begin{proof} By the last remark of \ref{versal-formal}, $\bG$ is clearly versal. It remains to prove that it satisfies the universal property in the corollary. Let $G$ be a deformation of $G_0$ over a
noetherian complete local $k$-algebra $A$ with residue field $k$. We
denote by $\m_A$ the maximal ideal of $A$, and put
$A_n=A/\m_A^{n+1}$ for each integer $n\geq0$. Then by
\ref{prop-deform}(b), there exists a unique local homomorphism
$\varphi_n:R\ra A_n$ such that $G\otimes A_n\simeq \bG\otimes_RA_n$.
The $\varphi_n$'s form a projective system $(\varphi_n)_{n\geq0}$,
whose projective limit $\varphi:R\ra A$ answers the question.
\end{proof}
\begin{defn}\label{defn-moduli}
The notations are those of \eqref{cor-alg-univ}. We call $\bS$ the
\emph{local moduli in characteristic $p$} of $G_0$, and $\bG$ the
\emph{universal deformation of $G_0$ in characteristic $p$}.
\end{defn}
If there is no confusions, we will
omit ``in characteristic $p$'' for short.
\subsection{} Let $G$ be a BT-group over $k$, $G^\circ$ be its connected part, and $G^\et$ be its \'etale part. Let $r$ be the height of $G^\et$. Then we have $G^\et\simeq (\Q_p/\Z_p)^r$, since $k$ is algebraically closed. Let $\D_{G}$ (\resp $\D_{G^{\circ}}$) be the deformation functor of $G$ (\resp $G^\circ$) over $\AL_k$. If $A$ is an object in $\AL_k$ and $\cG$ is a deformation of $G$ (\resp $G^\circ$) over $A$, we denote by $[\cG]$ its isomorphic class in $\D_{G}(A)$ (\resp in $\D_{G^\circ(A)}$).
\begin{prop}\label{prop-def-surj} The assumptions are as above, let
$\Theta: \D_{G}\ra \D_{G^{\circ}}$ be the morphism of functors that maps a deformation of $G$
to its connected component.
\emph{(i)} The morphism $\Theta$ is formally smooth of relative dimension $r$.
\emph{(ii)} Let $A$ be an object of $\AL_k$, and $\cG^\circ$ be a deformation of $G^\circ$ over $A$. Then the subset $\Theta_{A}^{-1}([\cG^{\circ}])$ of $\D_{G}(A)$ is canonically identified with $\Ext^1_{A}(\Q_p/\Z_p,\cG^\circ)^r$, where $\Ext^1_A$ means the group of extensions in the category of abelian $\fppf$-sheaves on $\Spec(A)$.
\end{prop}
\begin{proof} (i) Since $\D_{G}$ and $\D_{G^\circ}$ are both
pro-representable by a noetherian local complete $k$-algebra and
formally smooth over $k$ (\ref{prop-deform}), by a formal
completion version of [EGA $\mathrm{IV} 17.11.1(d)$], we only need to check that the tangent map $$\Theta_{k[\epsilon]/\epsilon^2}:\D_{G}(k[\epsilon]/\epsilon^2)\ra \D_{G^{\circ}}(k[\epsilon]/\epsilon^2)$$
is surjective with kernel of dimension $r$ over $k$. By
\ref{prop-deform}(iii), $\D_{G}(k[\epsilon]/\epsilon^2)$
(\resp $\D_{G^\circ}(k[\epsilon]/\epsilon^2)$) is
isomorphic to $\Hom_{k}(\omega_{G},\Lie(G^\vee))$
(\resp
$\Hom_{k}(\omega_{G^{\circ}},\Lie(G^{\circ\vee}))$)
by the Kodaira-Spencer morphism. In view of the canonical isomorphism $\omega_{G}\simeq \omega_{G^{\circ}}$, $\Theta_{k[\epsilon]/\epsilon^2}$ corresponds to the map
\[\Theta'_{k[\epsilon]/\epsilon^2}: \Hom_{k}(\omega_{G},\Lie(G^\vee))\ra \Hom_{k}(\omega_{G},\Lie(G^{\circ\vee}))\]
induced by the canonical surjection $\Lie(G^{\vee})\ra \Lie(G^{\circ\vee})$. It is clear that $\Theta'_{k[\epsilon]/\epsilon^2}$ is surjective of kernel $\Hom_{k}(\omega_{G},\Lie(G^{\et\vee}))$, which has dimension $r$ over $k$.
(ii) Since $G^{\et}$ is isomorphic to $(\Q_p/\Z_p)^r$, every element in $\Ext^1_{A}(\Q_p/\Z_p,\cG^{\circ})^r$ defines clearly an element of $\D_{G}(A)$ with image $[\cG^\circ]$ in $\D_{G^{\circ}}(A)$. Conversely, for any $\cG\in \D_{G}(A)$ with connected component isomorphic to $\cG^{\circ}$, the isomorphism $G^{\et}\simeq (\Q_p/\Z_p)^r$ lifts uniquely to an isomorphism $\cG^\et\simeq (\Q_p/\Z_p)^r$ because $A$ is henselian. The canonical exact sequence $0\ra \cG^{\circ}\ra \cG\ra \cG^{\et}\ra 0$ shows that $\cG$ comes from an element of $\Ext^1_{A}(\Q_p/\Z_p,\cG^{\circ})^r$.
\end{proof}
\section{HW-cyclic Barsotti-Tate Groups}
\begin{defn}\label{defn-cyclic} Let $S$ be a scheme of characteristic
$p>0$, $G$ be a BT-group over $S$ such that $c=\dim(G^\vee)$ is constant.
We say that $G$ is \emph{HW-cyclic}, if $c\geq 1$ and there exists an element $v\in \Gamma(S,\Lie(G^\vee))$ such that
\[v, \HW_G(v), \cdots, \HW_G^{c-1}(v)\]
generate $\Lie(G^\vee)$ as an $\cO_S$-module, where $\varphi_G$ is the Hasse-Witt map \eqref{BT-HW} of $G$.
\end{defn}
\begin{rem}
It is clear that a BT-group $G$ over $S$ is HW-cyclic, if and only if $\Lie(G^\vee)$ is free over $\cO_S$ and there exists a basis of $\Lie(G^\vee)$ over $\cO_S$ under which
$\varphi_{{G}}$ is expressed by a matrix of the form
\begin{equation}\label{HW-matrix}
\begin{pmatrix}0 &0 &\cdots &0 &-a_1\\
1 &0 &\cdots &0 &-a_2\\
0 &1 &\cdots &0 &-a_3\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&-a_{c}\end{pmatrix},
\end{equation}
where $a_i\in \Gamma(S,\cO_S)$ for $1\leq i\leq c$.
\end{rem}
\begin{lemma}\label{lemma-cyclic} Let $R$ be a local ring of characteristic $p>0$, $k$ be its residue field.
\emph{(i)} A BT-group $G$ over $R$ is HW-cyclic if and only if so is $G\otimes k$.
\emph{(ii)} Let $0\ra G'\ra G\ra G''\ra 0$ be an exact sequence of
BT-groups over $R$. If $G$ is HW-cyclic, then so is $G'$. In
particular, if $R$ is henselian, the connected part of a HW-cyclic BT-group over $R$ is HW-cyclic.
\end{lemma}
\begin{proof}
(i) The property of being HW-cyclic is clearly stable under arbitrary
base changes, so the ``only if'' part is clear. Assume that
$G_0=G\otimes k$ is HW-cyclic. Let $\overline{v}$ be an element of
$\Lie(G^\vee_0)=\Lie(G^\vee)\otimes k$ such that
$(\overline{v},\varphi_{G_0}(\overline{v}),\cdots,
\varphi^{c-1}_{G_0}(\overline{v}))$ is a basis of
$\Lie(G^\vee_0)$. Let $v$ be any lift of $\overline{v}$ in
$\Lie(G^\vee)$. Then by Nakayama's lemma, $(v, \varphi_G(v),\cdots,
\varphi_G^{c-1}(v))$ is a basis of $\Lie(G^\vee)$.
(ii) By statement (i), we may assume $R=k$. The exact sequence of BT-groups induces an exact sequence of
Lie algebras
\begin{equation}\label{exseq-Lie-alg}
0\ra\Lie(G''^\vee)\ra\Lie(G^\vee)\ra \Lie(G'^\vee)\ra 0,
\end{equation}
and the Hasse-Witt map $\HW_{G'}$ is
induced by $\HW_G$ by functoriality. Assume that $G$ is HW-cyclic
and $G^\vee$ has dimension $c$. Let $u$ be an
element of $\Lie(G^\vee)$ such that $${u},
\HW_{G}({u}), \cdots, \HW_{G}^{c-1}({u})$$
form a basis of $\Lie(G^\vee)$ over $k$. We denote by $u'$ the
image of $u$ in $\Lie(G'^{\vee})$.
Let $r\leq c$ be the maximal integer such
that the vectors
$$
u',\;\,
\HW_{G'}(u'),\;\, \cdots,\;\,
\HW_{G'}^{r-1}(u')
$$ are linearly
independent over $k$. It is easy to see that they form a basis of the $k$-vector
space $\Lie(G'^{\vee})$. Hence $G'$ is HW-cyclic.
\end{proof}
\begin{lemma}\label{lemma-HW-V} Let $S=\Spec(R)$ be an affine scheme
of characteristic $p>0$, $G$ be a HW-cyclic BT-group over $R$ with
$c=\dim(G^\vee)$ constant, and
\[\begin{pmatrix}0 &0 &\cdots &0 &-a_1\\
1 &0 &\cdots &0 &-a_2\\
0 &1 &\cdots &0 &-a_3\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&-a_{c}\end{pmatrix}\in \rM_{c\times c}(R),\] be a matrix
of $\varphi_G$. Put $a_{c+1}=1$, and $P(X)=\sum_{i=0}^c
a_{i+1}X^{p^i}\in R[X]$.
\emph{(i)} Let $V_G:G^{(p)}\ra G$ be the Verschiebung homomorphism of $G$.
Then $\Ker V_G$ is isomorphic to the group scheme $\Spec(R[X]/P(X))$ with comultiplication given by $X\mapsto 1\otimes X+ X\otimes 1$.
\emph{(ii)} Let $x\in S$, and $G_x$ be the fibre of $G$ at $x$. Put
\begin{equation}\label{integer-i_0}
i_0(x)=\min_{0\leq i\leq c}\{i; a_{i+1}(x)\neq 0\},
\end{equation}
where $a_i(x)$ denotes the image of $a_i$ in the residue field of $x$. Then the \'etale part of $G_x$ has height $c-i_0(x)$, and the connected part of $G_x$ has height $d+i_0(x)$. In particular, $G_x$ is connected if and only if $a_i(x)=0$ for $1\leq i\leq c$.
\end{lemma}
\begin{proof}(i) By \ref{GV-pLie} and \ref{lemma-KerV}, $\Ker V_G$ is isomorphic to the group scheme
\begin{equation*}\Spec\biggl(R[X_1,\dots,X_c]/(X_1^p-X_2,\cdots, X_{c-1}^p-X_c,X_c^p+a_1X_1+\cdots+a_{c}X_c)\biggr)\end{equation*}
with comultiplication $\Delta(X_i)=1\otimes X_i+X_i\otimes 1$ for $1\leq i \leq c$. By sending $(X_1,X_2,\cdots, X_c)\mapsto (X,X^p,\cdots, X^{p^{c-1}})$, we see that the above group scheme is isomorphic to $\Spec(R[X]/P(X))$ with comultiplication $\Delta(X)=1\otimes X+X\otimes 1$.
(ii) By base change, we may assume that $S=x=\Spec(k)$ and hence $G=G_x$. Let $G(1)$ be the kernel of the multiplication by $p$ on $G$. Then we have an exact sequence
\[0\ra\Ker F_G\ra G(1)\ra \Ker V_G\ra 0. \]
Since $\Ker F_G$ is an infinitesimal group scheme over $k$, we have $G(1)(\kb)=(\Ker V_G)(\kb)$, where $\kb$ is an algebraic closure of $k$.
By the definition of $i_0(x)$, we have $P(X)=Q(X^{p^{i_0(x)}})$, where $Q(X)$ is an additive sepearable polynomial in $k[X]$ with $\deg(Q)=p^{c-i_0(x)}$. Hence the roots of $P(X)$ in $\kb$ form an $\F_p$-vector space of dimension $c-i_0(x)$. By (i), $(\Ker V_G)(\kb)$ can be identified with the additive group consisting of the roots of $P(X)$ in $\kb$. Therefore, the \'etale part of $G$ has height $c-i_0(x)$, and the connected part of $G$ has height $d+i_0(x)$.
\end{proof}
\subsection{}\label{sect-a-num} Let $k$ be a perfect field of
characteristic $p>0$, and $\alpha_p=\Spec(k[X]/X^p)$ be the finite group scheme over $k$ with comultiplication map $\Delta(X)=1\otimes X+X\otimes 1$. Let $G$ be a BT-group over $k$. Following Oort, we call
\[a(G)=\dim_{k} \Hom_{k_{\fppf}}(\alpha_p,G)\]
the $a$-number of $G$,
where $\Hom_{k_{\fppf}}$ means the homomorphisms in the category of abelian $\fppf$-sheaves over $k$. Since the Frobenius of $\alpha_p$ vanishes, any morphism of $\alpha_p$ in $G$ factorize through $\Ker(F_G)$. Therefore we have
\begin{align*}\Hom_{k_{\fppf}}(\alpha_p,G)&=\Hom_{k-gr}(\alpha_p,\Ker(F_G))\\
&=\Hom_{k-gr}(\Ker(F_G)^\vee,\alpha_p)\\
&=\Hom_{\pLie_k}(\Lie(\alpha_p),\Lie(\Ker(F_G))),
\end{align*}
where $\Hom_{k-gr}$ denotes the homomorphisms in the category of commutative group schemes over $k$, and the last equality uses Proposition \ref{GV-pLie}. Since we have a canonical isomorphism $\Lie(\Ker(F_G))\simeq \Lie(G)$ and $\Lie(\alpha_p)$ has dimension one over $k$ with $\varphi_{\alpha_p}=0$, we get
\begin{equation}\label{a-number}a(G)=\dim_k\{x\in \Lie(G)| \varphi_{G^\vee}(x)=0\}=\dim_{k}\Ker(\varphi_{G^\vee}).\end{equation}
Due to the perfectness of $k$, we have also $a(G)=\dim_k\Ker
(\widetilde{\varphi_{G^\vee}})$, where $ \widetilde{\varphi_{G^\vee}}$ is the linearization of $\varphi_{G^\vee}$.
By Proposition \ref{prop-ord}, we see that $a(G)=0$ if and only if $G$ is ordinary.
\begin{lemma}\label{lemma-a-number}Let $G$ be a BT-group over $k$, and $G^\vee$ its Serre dual. Then we have $a(G)=a(G^\vee)$.
\end{lemma}
\begin{proof} Let $\psi_G:\omega_G\ra \omega_G^{(p)}$ be the $k$-linear map induced by the Verschiebung of $G$. Then $\psi^*_G$, the morphism obtained by applying the functor $\Hom_k(\_,k)$ to $\psi_G$, is identified with $\widetilde{\varphi_{G^\vee}}$. By \eqref{a-number} and the exactitude of the functor $\Hom_k(\_,k)$, we have $a(G)=\dim_k\Ker(\psi^*_G)=\dim_k\Coker(\psi_G)$. Using the additivity of $\dim_k$, we get finally $a(G)=\dim_k\Ker(\psi_G)$. By considering the commutative diagram \eqref{diag-Dieud}, we have
\[a(G)=\dim_k \biggl(\omega_G\cap \phi_G(\Lie(G^\vee)^{(p)})\biggr).\]
On the other hand, it follows also from \eqref{diag-Dieud} that
\[a(G^\vee)=\dim_k\Ker(\widetilde{\varphi_G})=\dim_{k}\biggl(\phi_G(\Lie(G^\vee)^{(p)})\cap \omega_G\biggr).\]
The lemma now follows immediately.
\end{proof}
\begin{prop}\label{prop-HW-a} Let $k$ be a perfect field of characteristic $p>0$, $G$ a BT-group over $k$.
Consider the following conditions:
\emph{(i)} $G$ is HW-cyclic and non-ordinary;
\emph{(ii)} the connected part $G^\circ$ of $G$ is HW-cyclic and not of multiplicative type;
\emph{(iii)} $a(G^\vee)=a(G)=1$.
We have $\mathrm{(i)}\Rightarrow \mathrm{(ii)}\Leftrightarrow \mathrm{(iii)}$. If $k$ is algebraically closed, we have moreover $\mathrm{(ii)}\Rightarrow \mathrm{(i)}$.
\end{prop}
\begin{rem} In \cite[Lemma 2.2]{oort}, Oort proved the following assertion, which is a generalization of $\mathrm{(iii)}\Rightarrow\mathrm{(ii)}$: Let $k$ be an algebraically closed field of characteristic $p>0$, and $G$ be a connected BT-group with $a(G)=1$. Then there exists a basis of the Dieudonn\'e module $M$ of $G$ over $W(k)$, such that the action of Frobenius on $M$ is given by a display-matrix of ``normal form'' in the sense of \cite[2.1]{oort}.
\end{rem}
\begin{proof} $\mathrm{(i)}\Rightarrow \mathrm{(ii)}$ follows from \ref{lemma-cyclic}(ii).
$\mathrm{(ii)}\Rightarrow\mathrm{(iii)}$. First, we note that $a(G)=a(G^\circ)$, so we may assume $G$ connected. Since $G$ is not of multiplicative type, we have $c=\dim(G^\vee)\geq 1$. By Lemma \ref{lemma-HW-V}(ii), there exists a basis of $\Lie(G^\vee)$ over $k$ under which $\varphi_G$ is expressed by
\[\begin{pmatrix}0 &0 &\cdots &0 &0\\
1 &0 &\cdots &0 &0\\
0&1&\cdots &0&0\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&0\end{pmatrix}\in \rM_{c\times c}(k).\] According to \eqref{a-number}, $a(G^\vee)$ equals to $\dim_k\Ker(\varphi_G)$, \ie the $k$-dimension of the solutions of the equation system in $(x_1,\cdots, x_c)$
\[\begin{pmatrix}0 &0 &\cdots &0 &0\\
1 &0 &\cdots &0 &0\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&0\end{pmatrix}\begin{pmatrix}x_1^p\\
x_2^p\\
\vdots\\x_c^p\end{pmatrix}=0\]
The solutions $(x_1,\cdots,x_c)$ form clearly a vector space over $k$ of dimension $1$, \ie we have $a(G^\vee)=1$.
$\mathrm{(iii)}\Rightarrow\mathrm{(ii)}.$ Let $G^\et$ be the \'etale part of $G$. Since $k$ is perfect, the exact sequence \eqref{decomp-BT} splits \cite[Chap. II \S7]{De}; so we have $G\simeq G^\circ\times G^\et$. We put $M=\Lie(G^\vee)$, $M_1=\Lie(G^{\circ\vee})$ and $M_2=\Lie(G^{\et\vee})$ for short. By \ref{prop-etale-HW} and \ref{cor-nilp-HW}, we have a decomposition $M=M_1\oplus M_2$, such that $M_1,M_2$ are stable under $\varphi_G$, and the action of $\varphi_G$ is nilpotent on $M_1$ and bijective on $M_2$. We note that $a(G^{\circ\vee})=a(G^\circ)=a(G)=1$. By the last remark of \ref{sect-a-num}, $G^\circ$ is not of multiplicative type, hence $\dim_kM_1=\dim(G^{\circ \vee})\geq 1$. It remains to prove that $G^\circ$ is HW-cyclic.
Let $n$ be the minimal integer such that $\varphi^n_G(M_1)=0$. We have a strictly increasing filtration
\[0\subsetneq \Ker(\varphi_G)\subsetneq \cdots \subsetneq \Ker(\varphi_G^n)=M_1.\]
If $n=1$, then $M_1$ is one-dimensional, hence $G^\circ$ is clearly HW-cyclic. Assume $n\geq 2$.
For $2\leq m\leq n$, $\varphi_G^{m-1}$ induces an injective map
\[\overline{\varphi_G^{m-1}}:\Ker(\varphi_G^{m})/\Ker(\varphi_G^{m-1})\lra \Ker(\varphi_G).\]
Since $\dim_k\Ker (\varphi_G)=a(G^{\circ\vee})=1$, $\overline{\varphi_G^{m-1}}$ is
necessarily bijective. So we have $\dim_k \Ker(\varphi_G^{m})=m$ for $1\leq m \leq n$. Let $v$ be an element of $M_1$ but not in $\Ker(\varphi_G^{n-1})$. Then $v,\varphi_G(v),\cdots,\varphi_G^{n-1}(v)$ are linearly independant, hence they form a basis of $M_1$ over $k$. This proves that $G^\circ$ is HW-cyclic.
Assume $k$ algebraically closed. We prove that $\mathrm{(ii)}\Rightarrow \mathrm{(i)}$. Noting that $G$ is ordinary if and only if $G^\circ$ is of multiplicative type, we only need to check that $G$ is HW-cyclic. We conserve the notations above. Since $\varphi_G$ is bijective on $M_2$ and $k$ algebraically closed, there exists a basis $(e_1,\cdots, e_m)$ of $M_2$ such that $\varphi_G(e_i)=e_i$ for $1\leq i \leq m$. Let $v\in M_1$ but not in $\Ker(\varphi_G^{n-1})$ as above, and put $u=v+\lambda_1 e_1+\cdots \lambda_m e_m$, where $\lambda_i(1\leq i\leq m)$ are some elements in $k$ to be determined later. Then we have
\[\begin{pmatrix}\varphi_G^{n}(u)\\
\vdots\\ \varphi_G^{n+m-1}(u)\end{pmatrix}=
\begin{pmatrix}\lambda_1^{p^n}& \cdots &\lambda_m^{p^n}\\
\vdots &\ddots & \vdots\\
\lambda_1^{p^{n+m-1}}&\cdots & \lambda_m^{p^{n+m-1}}
\end{pmatrix}\begin{pmatrix}e_1\\ \vdots\\ e_m \end{pmatrix}.\]
Let $L(\lambda_1,\cdots,\lambda_m)\in k[\lambda_1,\cdots, \lambda_m]$
be the determinant polynomial of the matrix on the right side. An
elementary computation shows that the polynomial
$L(\lambda_1,\cdots,\lambda_m)$ is not null. We can choose
$\lambda_1,\cdots, \lambda_m\in k$ such that
$L(\lambda_1,\cdots,\lambda_m)\neq 0$ because $k$ is algebraically closed. So $\varphi_G^n(u),\cdots, \varphi_G^{n+m-1}(u)$ form a basis of $M_2$ over $k$. Since
$$\varphi^{i}_G(u)\equiv \varphi^{i}_G(v) \mod M_2 \quad\text{ for} \quad 0\leq i \leq n,$$
by the choice of $u$, we see that $\{u,\varphi_G(u), \cdots, \varphi_G^{n+m-1}(u)\}$ form a basis of $M=\Lie(G^\vee)$ over $k$.
\end{proof}
By combining \ref{lemma-a-number} and \ref{prop-HW-a}, we obtain the following
\begin{cor} Let $k$ be an algebraically closed field of characteristic $p>0$. Then a BT-group over $k$ is HW-cyclic if and only if so is its Serre dual.
\end{cor}
\subsection{Examples}\label{HW-exem} Let $k$ be a perfect field,
$W(k)$ be the ring of Witt vectors with coefficients in $k$, and $\sigma$ be the Frobenius automorphism of $W(k)$. Let $s,r$ be relatively prime integers such that $0\leq s\leq r$ and $r\neq 0$; put $\lambda=\frac{s}{r}$. We consider the Dieudonn\'e module $M^\lambda\simeq W(k)[F,V]/(F^{r-s}-V^s)$, where $W(k)[F,V]$ is the non-commutative ring with relations $FV=VF=p$, $Fa=\sigma(a)F$ and $V\sigma(a)=aV$ for all $a\in W(k)$.
We note that $M^\lambda$ is free of rank $r$ over $W(k)$ and
$M^\lambda/VM^\lambda\simeq k[F]/F^{r-s}$. By the contravariant Dieudonn\'e theory, $M^\lambda$ corresponds to a BT-group $G^\lambda$ over $k$ of height $r$ with $\Lie(G^{\lambda\vee})=M^\lambda/VM^\lambda$. We see easily that $G^\lambda$ is HW-cyclic, and we call it the \emph{elementary BT-group of slope $\lambda$.} We note that $G^0\simeq \Q_p/\Z_p$, $G^1\simeq \mu_{p^\infty}$, and $(G^{\lambda})^\vee\simeq G^{1-\lambda}$ for $0\leq \lambda\leq 1$.
Assume $k$ algebraically closed. Then by the Dieudonn\'e-Manin's classification of isocrystals \cite[Chap.IV \S4]{De}, any BT-group over $k$ is isogenous to a finite product of $G^\lambda$'s; moreover, any connected one-dimensional BT-group over $k$ of height $r$ is necessarily isomorphic to $G^{1/r}$ \cite[Chap.IV \S8]{De}, hence in particular HW-cyclic.
\begin{prop}\label{prop-HW-versal}
Let $k$ be an algebraically closed field of characteristic $p>0$, $R$
be a noetherian complete regular local $k$-algebra with residue field
$k$, and $S=\Spec(R)$. Let $G$ be a connected HW-cyclic BT-group over $R$ of dimension $d\geq 1$ and height $c+d$,
\[\h=\begin{pmatrix}0 &0 &\cdots &0 &-a_1\\
1 &0 &\cdots &0 &-a_2\\
0&1&\cdots&0&-a_3\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&-a_c\end{pmatrix}\in \rM_{c\times c}(R)\] be a matrix of $\HW_G$.
\emph{(i)} If $G$ is versal over $S$, then $\{a_1,\cdots, a_c\}$ is a subset of a regular system of parameters of $R$.
\emph{(ii)} Assume that $d=1$. The converse of \emph{(i)} is also true, \ie if $\{a_1,\cdots,a_c\}$ is a subset of a regular system of parameters of $R$ then $G$ is versal over $S$. Furthermore, $G$ is the universal deformation of its special
fiber if and only if $\{a_1,\cdots, a_c\}$ is a system
of regular parameters of $R$.
\end{prop}
\begin{proof} Let $(\M(G),F_M,\nabla)$ be the finite free $\cO_S$-module equipped with a semi-linear endomorphism $F_M$ and a connection $\nabla:\M(G)\ra \M(G)\otimes_{\cO_{S}}\Omega^1_{S/k}$, obtained by evaluating the Dieudonn\'e crystal of $G$ at the trivial immersion $S\hra S$ (cf. \ref{pre-Dieud}). Recall that
we have a commutative diagram
\begin{equation}\label{diag-F-phi}
\xymatrix{\M(G)^{(p)}\ar[rr]^{F_M}\ar[d]_{pr}&&\M(G)\ar[d]^{pr}\\
\Lie(G^\vee)^{(p)}\ar[rr]^{\widetilde{\varphi_G}}\ar@{^(->}[urr]^{\phi_G}&&\Lie(G^\vee),}\end{equation}
where $\phi_G$ is universally injective \eqref{diag-Dieud}.
Let
$\{v_1,\cdots, v_c\}$ be a basis of $\Lie(G^\vee)$ over $\cO_S$ under
which $\HW_G$
is expressed by $\h$, \ie we have $\varphi_G^{i-1}(v_1)=v_i$ for
$1\leq i\leq c$ and
$\varphi_G^{c}(v_1)=\varphi_G(v_c)=-\sum_{i=1}^ca_iv_i$. Let $f_1$ be
a lift of $v_1$ to $\Gamma(S,\M(G))$, and put
$f_{i+1}=\phi_G(v_i^{(p)})$ for $1\leq i \leq c-1$, where
$v_i^{(p)}=1\otimes v_i\in\Gamma(S,\Lie(G^\vee)^{(p)})$. The image
of $f_i$ in $\Gamma(S,\Lie(G^\vee))$ is thus $v_i$ for $1\leq i\leq c$ by \eqref{diag-F-phi}. We put
\begin{equation}\label{defn-e1}
e_1=\phi_G(v_c^{(p)})+a_1f_1+\cdots +a_cf_c\in \Gamma(S,\M(G)).
\end{equation}
The image of $e_1$ in $\Gamma(S,\Lie(G^\vee))$ is
$\varphi_G(v_c)+\sum_{i=1}^ca_iv_i=0$; so we have $e_1\in
\Gamma(S,\omega_G).$ By \ref{lemma-HW-V}(ii), we notice that $a_1,\cdots,
a_c$ belong to the maximal ideal $\m_R$ of $R$, as $G$ is
connected. Hence, we have $\overline{e_1}=\overline{\phi_G(v^{(p)}_c)}$, where for a $R$-module $M$ and $x\in M$, we denote by $\xb$ the
canonical image of $x$ in $M\otimes k$. Since $\phi_G$ commutes with
base change and is universally
injective, we get
$\overline{e_1}=\overline{\phi_G(v^{(p)}_c)}=\phi_{G\otimes k}(\overline{v^{(p)}_c})\neq 0$. Therefore, we can choose
$e_2,\cdots, e_d\in \Gamma(S,\omega_G)$ such that $(e_1,\cdots, e_d)$
becomes a basis of $\omega_G$ over $\cO_S$, so $(e_1,\cdots,e_d,f_1,\cdots, f_c)$ is a basis of $\M(G)$.
Since $F_M$ is horizontal for the connection $\nabla$ (cf. \ref{pre-Dieud}(ii)), we have \[\nabla(\phi_G(v^{(p)}_c))=\nabla(F_M(f_c^{(p)}))=0.\] In view of \eqref{defn-e1}, we get
\begin{align}\nabla(e_1)&=\sum_{i=1}^c f_i\otimes da_i+\sum_{i=1}^c
a_i\nabla(f_i)\nonumber\\
&\equiv \sum_{i=1}^c f_i\otimes da_i \quad (\mathrm{mod}\; \m_R).\label{nabla-e_1}
\end{align}
Let $\KS_0$ and $\Kod_0$ be respectively the reductions modulo $\m_R$ of \eqref{morph-KS} and \eqref{Kod-map}. Since $(\overline{v_i})_{1\leq i\leq c}$ is a base of $\Lie(G^\vee)\otimes k$, we can write
\[\KS_0(e_j)=\sum_{i=1}^c\overline{v_i}\otimes \theta_{i,j} \quad \quad \text{for $1\leq j\leq d$,}\]
where $\theta_{i,j}\in \Omega_{S/k}\otimes k$.
From \eqref{nabla-e_1}, we deduce that $\theta_{i,1}=da_i$.
By the definition of $\Kod_0$, we have
\begin{equation}\label{equ-Kod_0}
\Kod_0(\partial)=\sum_{j=1}^d\sum_{i=1}^c<\partial,\theta_{i,j}>\overline{e_j}^*\otimes \overline{v_i}
\end{equation}
where
${\partial}\in \cT_{S/k}\otimes k$, $<\bullet,\bullet>$ is the canonical pairing between
$\cT_{S/k}\otimes k$ and $\Omega^1_{S/k}\otimes k$, and $(\overline{e_i}^*)_{1\leq i\leq d}$ denotes the dual basis of $(\overline{e_i})_{1\leq i\leq d}$. Now assume that $G$ is versal over $S$, \ie $\Kod_0$ is surjective by definition \eqref{versal}. In particular, there are $\partial_1,\cdots, \partial_c\in \cT_{S/k}\otimes k$ such that $\Kod_0(\partial_i)=\overline{e_1}^*\otimes v_i$ for $1\leq i\leq c$, \ie we have
\begin{equation}\label{formula-versal-1}<\partial_i,da_j>=\begin{cases}1 & \text{if $i=j$}\\
0& \text{if $i\neq j$} \end{cases}\quad \text{for $1\leq i,j\leq c$,}\end{equation}
and \[ <\partial_i, \theta_{\ell,j}>=0 \quad \quad \text{for $1\leq i,j \leq c, 2\leq\ell\leq d $}.\]
From \eqref{formula-versal-1}, we see easily that $da_1,\cdots, da_c$ are linearly independent in $\Omega_{S/k}\otimes k\simeq \m_R/\m_R^2$; therefore, $(a_1,\cdots, a_c)$ is a part of a regular system of parameters of $R$. Statement (i) is proved.
For statement (ii), we assume $d=1$ and that $(a_1,\cdots,a_c)$ is a part of a regular system of parameters of $R$. Then the formula \eqref{equ-Kod_0} is simplified as
\[\Kod_0(\partial)=\sum_{i=1}^c<\partial,da_i>\overline{e_1}^*\otimes \overline{v_i}.\]
Since $da_1,\cdots, da_c$ are linearly independent in $\Omega_{S/k}^1\otimes k$, there exist $\partial_1,\cdots, \partial_c\in \cT_{S/k}\otimes k$ such that \eqref{formula-versal-1} holds, \ie $(\overline{e_1}^*\otimes \overline{v_i})_{1\leq i\leq c}$ are in the image of $\Kod_0$. But the elements $(\overline{e_1}^*\otimes\overline{v_i})_{1\leq i\leq c}$ form already a basis of $\cHom_{\cO_S}(\omega_G,\Lie(G^\vee))\otimes k$. So $\Kod_0$ is surjective, and hence $G$ is versal over $S$ by Nakayama's lemma. Let $G_0$ be the special fiber of $G$. It remains to prove that when $d=1$, $G$ is the universal
deformation of $G_0$ if and only if
$\dim(S)=c$ and $G$ is versal over $S$. Let $\bS$ be the local moduli in characteristic $p$ of $G_0$. By the universal property of $\bG$ \eqref{cor-alg-univ}, there exists a unique morphism
$f:S\ra \bS$ such that $G\simeq \bG\times_{\bS}S$. Since $S$ and $\bS$ are local complete regular schemes
over $k$ with residue field $k$ of the same dimension, $f$ is an
isomorphism if and only if the tangent map of $f$ at the closed point
of $S$, denoted by $T_f$, is an isomorphism. By the functoriality of Kodaira-Spencer maps \eqref{Kod-map}, we have a commutative diagram
\[\xymatrix{\cT_{S/k}\otimes_{\cO_S} k\ar[d]_{T_f}\ar[rr]^{\Kod_0^S}&&\Hom_k(\omega_{G_0},\Lie(G_0^\vee))\ar@{=}[d]\\
\cT_{\bS/k}\otimes_{\cO_{\bS}}k\ar[rr]^{\Kod_0^{\bS}} &&\Hom_k(\omega_{G_0},\Lie(G_0^\vee))},\]
where horizontal arrows are the Kodaira-Spencer maps evaluated at the closed points \eqref{Kod-0}. Since $\Kod_0^S$ and $\Kod_0^{\bS}$ are isomorphisms according to the first part of this propostion, we deduce that so is $T_f$. This completes the proof.
\end{proof}
\section{Monodromy of a HW-cyclic BT-group over a Complete Trait of Characteristic $p>0$}
\subsection{}\label{nota-dvr}
Let $k$ be an algebraically closed field of \car $p>0$, $A$ be a
complete discrete valuation ring of \car $p$, with residue field $k$
and fraction field $K$. We put $S=\Spec(A)$, and denote by $s$ its closed
point, by $\eta$ its generic point. Let $\Kb$ be an algebraic closure of $K$, $\Ks$
be the maximal separable extension of $K$ contained in $\Kb$,
$K^{\mathrm{t}}$ be the maximal tamely ramified extension of $K$
contained in $\Ks$. We put $I=\Gal(\Ks/K)$, $I_p=\Gal(\Ks/K^\tame)$
and $I_t=I/I_p=\Gal(K^\tame/K)$.
Let $\pi$ be a uniformizer of $A$; so we have $A\simeq k[[\pi]]$.
Let $\tv$ be the valuation on $K$ normalized by $\tv(\pi)=1$; we
denote also by $\tv$ the unique extension of $\tv$ to $\Kb$. For
every $\alpha\in \Q$, we denote by $ \m_{\alpha}$ (\resp by
$\m_{\alpha}^+$) the set of elements $x\in \Ks$ such that
$\tv(x)\geq \alpha$ (\resp $\tv(x)> \alpha$). We put
\begin{equation}\label{defn-V_alpha}V_\alpha=\m_\alpha/\m_\alpha^+,\end{equation}
which is a $k$-vector space of dimension 1 equipped with a continuous action of the Galois group
$I$.
\subsection{}\label{galois-char} First, we recall some properties of the inertia groups
$I_p$ and
$I_t$ \cite[Chap. IV]{CL}. The subgroup $I_p$, called the
\emph{wild inertia subgroup}, is the unique maximal pro-$p$-group contained
in $I$ and hence normal in $I$. The quotient $I_t=I/I_p$ is a
commutative profinite group, called the \emph{tame inertia group.}
We have a canonical isomorphism
\begin{equation}\label{theta1}
\theta: I_t\xra{\sim}
\varprojlim_{(d,p)=1}\mu_d,
\end{equation}
where the projective
system is taken over positive integers prime to $p$, $\mu_d$ is the
group of $d$-th roots of unity in $k$, and the transition maps
$\mu_{m}\ra \mu_d$ are given by $\zeta\mapsto \zeta^{m/d}$,
whenever $d$ divides $m$. We denote by $\theta_d:I_t\ra \mu_d$ the
projection induced by \eqref{theta1}. Let $q$ be a power of $p$,
$\F_q$ be the finite subfield of $k$ with $q$ elements. Then
$\mu_{q-1}=\F_q^\times$, and we can write $\theta_{q-1}:I_t\ra
\F_q^\times$. The character $\theta_d$ is characterized by the
following property.
\begin{prop}[\cite{Se} Prop.7]\label{prop-V-alpha}
Let $a,d$ be relatively prime positive integers with $d$ prime to $p$. Then the natural action of $I_p$ on the $k$-vector
space $V_{a/d}$ \eqref{defn-V_alpha} is trivial, and the induced
action of $I_t$ on $V_{a/d}$ is given by the character
$(\theta_d)^a:I_t\ra \mu_d$. In particular, if $q$ is a power of
$p$, the action of $I_t$ on $V_{1/(q-1)}$ is given by the character
$\theta_{q-1}:I_t\ra \F_q^\times$ and any $I$-equivariant
$\F_p$-subspace of $V_{1/(q-1)}$ is an $\F_{q}$-vector space.
\end{prop}
\subsection{}\label{defn-hw-index} Let $G$ be a BT-group over $S$. We define $h(G)$ to be the
valuation of the determinant of a matrix of $\HW_G$ if $\dim(G^\vee)\geq 1$, and $h(G)=0$ if $\dim(G^\vee)=0$. We call $h(G)$
the \emph{Hasse invariant} of $G$.
{(a)} $h(G)$ does not
depend on the choice of the matrix representing $\HW_G$. Indeed, let
$c$ be the rank of $\Lie(G^\vee)$ over $A$,
$\h\in \rM_{c\times c}(A)$ be a matrix of $\HW_G$. Any other matrix representing $\HW_G$
can be written in the form $U^{-1}\cdot \h\cdot U^{(p)}$, where
$U\in \GL_{c}(A)$, $U^{-1}$ is the inverse of $U$, and $U^{(p)}$
is the matrix obtained by applying the Frobenius map of $A$ to the
coefficients of $U$.
{(b)} By \ref{prop-ord}, the generic fiber $G_\eta$ is ordinary if
and only if $h(G)<\infty$; $G$ is ordinary over $T$ if and only
$h(G)=0$.
{(c)} Let $0\ra G'\ra G\ra G''\ra 0$ be a short exact sequence of
BT-groups over $T$, then we have $h(G)=h(G')+h(G'')$. Indeed, the
exact sequence of BT-groups induces a short exact sequence of Lie
algebras (cf. \cite{BBM} 3.3.2)
\[
0\ra\Lie(G''^\vee)\ra\Lie(G^\vee)\ra \Lie(G'^\vee)\ra 0,
\]
from which our assertion follows easily.
\begin{prop}\label{prop-Hasse}
Let $G$ be a BT-group over $S$. Then we have $h(G)=h(G^\vee)$.
\end{prop}
\begin{proof} The proof is very similar to that of Lemma \ref{lemma-a-number}. First, we have
\[h(G)=\leng\bigl(\Lie(G^\vee)/\widetilde{\varphi_G}(\Lie(G^\vee)^{(p)})\bigr),\]
where $\widetilde{\varphi_G}$ is the linearization of $\varphi_G$, and ``$\leng$'' means the length of a finite $A$-module (note that this formulae holds even if $\dim(G^\vee)=0$). By the commutative diagram \eqref{diag-Dieud}, we have
\[h(G)=\leng\M(G)/(\phi_G(\Lie(G^\vee)^{(p)})+ \omega_G).\]
On the other hand, by applying the functor $\Hom_A(\_,A)$ to the $A$-linear map $\widetilde{\varphi_{G^\vee}}:\Lie(G)^{(p)}\ra \Lie(G)$, we obtain a map $\psi_G:\omega_{G}\ra \omega_G^{(p)}$. If $U$ is a matrix of $\widetilde{\varphi_{G^\vee}}$, then the transpose of $U$, denoted by $U^t$, is a matrix of $\psi_G$. So we have
\[h(G^\vee)=\tv(\det(U))=\tv(\det(U^t))=\leng\bigl(\omega_G^{(p)}/\psi_G(\omega_G)\bigr).\]
By diagram \ref{diag-Dieud}, we get
\[h(G^\vee)=\leng \M(G)/(\phi_G(\Lie(G^\vee)^{(p)})+ \omega_G)=h(G).\]
\end{proof}
\subsection{} Let $G$ be a BT-group over $S$, $c=\dim(G^\vee)$.
We put
\begin{equation}\label{Tate-mod}\rT_p(G)=\myprojlim G(n)(\Kb)\end{equation}
the Tate
module of $G$, where $G(n)$ is the kernel of $p^n:G\ra G$.
It is a free $\Z_p$-module of rank $\leq c$, and
the equality holds if and only if
the generic fiber $G_\eta$ is ordinary. The Galois group $I$ acts continuously on
$\rT_p(G)$.
We are interested in the image of the monodromy representation
\begin{equation}\label{rep-trait}\rho:I=\Gal(\Ks/K)\ra
\Aut_{\Z_p}(\rT_p(G)).\end{equation}
We denote by \begin{equation}\label{rho-0}\rhob: I=\Gal(\Ks/K)\ra
\Aut_{\F_p}\bigl(G(1)(\Kb)\bigr)\end{equation} its reduction mod
$p$.
\begin{thm}[Reformulation of Igusa's theorem]\label{thm-Igusa} Let $G$ be
a connected BT-group over $S$ of height $2$ and dimension $1$. Then $G$ is versal \eqref{versal} if and only if
$h(G)=1$; moreover, if this condition is satisfied, the monodromy
representation $\rho:I\ra
\Aut_{\Z_p}(\rT_p(G))\simeq \Z_p^\times $
is surjective.
\end{thm}
\begin{proof}
Since $\Lie(G^\vee)$ is an $\cO_S$-module free of rank 1, the
condition that $h(G)=1$ is equivalent to that any matrix of
$\varphi_G$ is represented by a uniformizer of $A$. Hence the first part of this theorem follows from Proposition \ref{prop-HW-versal}(ii).
We follow \cite[Thm 4.3]{Ka} to prove the surjectivity of $\rho$ under the assumption that $h(G)=1$.
For each integer $n\geq 1$, let
$$\rho_n:I\ra \Aut_{\Z/p^n\Z}(G(n)(\Kb))\simeq
(\Z/p^n\Z)^\times$$ be the reduction mod $p^n$ of $\rho$, $K_n$ be
the subfield of $\Ks$ fixed by the kernel of $\rho_n$. Then $\rho_n$
induces an injective homomorphism $\Gal(K_n/K)\ra
(\Z/p^n\Z)^\times$. By taking projective limits, we are reduced to proving
the surjectivity of $\rho_n$ for every $n\geq1$. It suffices to
verify that $$|\im(\rho_n)|=[K_n:K]\geq p^{n-1}(p-1)$$ (then the
equality holds automatically).
We regard $G$ as a formal group over $S$. Then by \cite[3.6]{Ka},
there exists a parameter $X$ of the formal group $G$ normalized by
the condition that $[\xi](X)=\xi(X)$ for all $(p-1)$-th root of unity
$\xi\in \Z_p$. For such a parameter, we have
\[[p](X)=a_1X^{p}+\alpha X^{p^2}+\sum_{m\geq 2}c_mX^{p(1+m(p-1))}\in A[[X]],\]
where we have $\tv(a_1)=h(G)=1$ by \cite[3.6.1 and 3.6.5]{Ka}, and
$\tv(\alpha)=0$, as $G$ is of height $2$. For each integer $i\geq
0$, we put
\[V^{(p^i)}(X)=a_1^{p^i}X+\alpha^{p^i} X^p+\sum_{m\geq 2}c^{p^i}_mX^{1+m(p-1)}\in A[[X]];\]
then we have $[p^n](X)=V^{(p^{n-1})}\circ V^{(p^{n-2})}\circ\cdots
\circ V(X^{p^n})$. Hence each point of $G(n)(\Kb)$ is given by a
sequence $y_1,\cdots, y_n\in \Ks$ (or simply an element $y_n\in
\Ks$) satisfying the equations
\[\begin{cases}V(y_1)=a_1y_1+\alpha y_1^p+\cdots=0;\\
V^{(p)}(y_2)=a_1^py_2+\alpha^p y_2^p+\cdots=y_1;\\
\vdots\\
V^{(p^{n-1})}(y_n)=a_1^{p^{n-1}}y_{n}+\alpha^{p^{n-1}}y_{n}^{p}+\cdots=y_{n-1}.\end{cases}\]
Let $y_n\in \Ks$ be such that $y_1\neq0$. By considering the Newton
polygons of the equations above, we verify that
$$\tv(y_i)=\frac{1}{p^{i-1}(p-1)}\quad \quad \text{for }1\leq i\leq n.$$ In particular, the
ramification index $e(K_n/K)$ is at least $p^{n-1}(p-1)$. By the
definition of $K_n$, the Galois group $\Gal(\Ks/K_n)$ must fix
$y_n\in \Ks$, \ie $K_n$ is an extension of $K(y_n)$. Therefore, we
have $[K_n:K]\geq [K(y_n):K]\geq e(K(y_n)/K)\geq p^{n-1}(p-1)$.
\end{proof}
\begin{prop}\label{prop-mono-trait} Let $G$ be a HW-cyclic BT-group
over $S$ of height $c+d$ and dimension $d$ such that $G\otimes K$ is ordinary,
$$\h=\begin{pmatrix}0& 0 &\cdots&0 &-a_1\\
1&0&\cdots &0 &-a_2\\
0&1 &\cdots &0 &-a_3\\
\vdots&&\ddots&& \vdots\\
0&0&\cdots &1&-a_{c} \end{pmatrix}$$ be a matrix of $\HW_G$. Put
$q=p^{c}$, $a_{c+1}=1$, and $P(X)=\sum_{i=0}^c a_{i+1}X^{p^i}\in A[X]$.
\emph{(i)} Assume that $G$ is connected and the Hasse invariant $h(G)=1$. Then the representation $\rhob$ \eqref{rho-0} is
tame, $G(1)(\Kb)$
is endowed with the structure of an $\F_{q}$-vector space of
dimension $1$, and the induced action of $I_t$
is given by the character $\theta_{q-1}:I_t\ra
\F_{q}^\times$.
\emph{(ii)} Assume that $c>1$, $\tv(a_{i})\geq 2$ for $1\leq i\leq c-1$ and $\tv(a_{c})=1$.
Then the order of $\im (\rhob)$ is divisible by $p^{c-1}(p-1)$.
\emph{(iii)} Put $i_0=\min_{0\leq i\leq c}\{i;\tv(a_{i+1})=0\}$. Assume that there exists $\alpha\in k$ such that
$\tv(P(\alpha))=1$. Then we have $i_0\leq c-1$ and the order of
$\im(\rhob)$ is divisible by $p^{i_0}$.
\end{prop}
\begin{proof}
Since $G$ is generically ordinary, we have $a_1\neq 0$ by \ref{prop-ord}(d). Hence $P(X)\in K[X]$ is a separable polynomial. By \ref{lemma-HW-V},
$G(1)(\Kb)\simeq (\Ker V_G)(\Ks)$ is identified with the additive group consisting of the roots of $P(X)$ in $\Ks$.
(i) By definition of the Hasse invariant, we have $\tv(a_1)=h(G)=1$. By \ref{lemma-HW-V}(ii), the assumption that $G$ is connected
is equivalent to saying $\tv(a_i)\geq 1$ for $1\leq i\leq
c$. From the Newton polygon of $P(X)$, we deduce that all the
non-zero roots of $P(X)$ in $\Ks$ have the same valuation $1/(q-1)$.
We denote by
\[
\psi: G(1)(\Kb)\ra V_{1/(q-1)}
\] the map
which sends each root $x\in \Ks$ of $P(X)$
to the class of $x$ in $V_{1/(q-1)}=\m_{1/(q-1)}/\m^+_{1/(q-1)}$ \eqref{defn-V_alpha}. We
remark that $G(1)(\Kb)$ is an $\F_p$-vector space of dimension
$c$. Hence $G(1)(\Kb)$ is automatically of dimension $1$ over
$\F_q$ once we know it is an $\F_q$-vector space. By
\ref{prop-V-alpha}, it suffices to show that $\psi$ is an injective
$I$-equivariant homomorphism of groups. By
\ref{lemma-HW-V}(i), $\psi$ is obviously an $I$-equivariant
homomorphism of groups. Let $x_0$ be a root of $P(X)$, and put
$Q(y)=P(x_0y)$. Then the polynomial $Q(y)$ has the form
$Q(y)=x_0^{q}Q_1(y)$, where
$$Q_1(y)=y^{q}+b_{c}y^{p^{c-1}}+\cdots +b_2 y^{p}+b_1y$$
with $b_i=a_i/x_0^{(q-p^{i-1})}\in \Ks$. We have $\tv(b_i)> 0$ for
$2\leq i\leq c$ and $\tv(b_1)=0$. Let $\overline{b}_1$ be the
class of $b_1$ in the residue field $k=\m_0/\m_0^+$. Then the images
of the roots of $P(X)$ in $V_{1/(q-1)}$ are $x_0
\overline{b}_1^{1/(q-1)}\zeta,$ where $\zeta$ runs over the finite
field $\F_q$. Therefore, $\psi$ is injective.
(ii) By computing the slopes of the Newton polygon of $P(X)$, we see
that $P(X)$ has $p^{c-1}(p-1)$ roots of valuation
$1/(p^{c}-p^{c-1})$. Let $L$ be the sub-extension of $\Ks$
obtained by adding to $K$ all the roots of $P(x)$. Then the
ramification index $e(L/K)$ is divisible by
$p^{c-1}(p-1)$. Let $\widetilde{L}$ be the
sub-extension of $\Ks$ fixed by the kernel of $\rhob$ \eqref{rho-0}.
The Galois group $\Gal(\Ks/\widetilde{L})$ fixes the roots of $P(x)$
by definition. Hence we have $L\subset \widetilde{L}$, and $|\im
(\rhob)|=[\widetilde{L}:K]$ is divisible by $[L:K]$; in particular,
it is divisible by $p^{c-1}(p-1)$.
(iii) Note that the relation $ i_0\leq c-1$ is equivalent to saying that $G$ is not connected by \ref{lemma-HW-V}(ii). Assume conversely
$i_0=c$, \ie $G$ is connected. Then we would have
$${P}(X)\equiv X^{q} \mod (\pi A[X]).$$ But
$\tv(P(\alpha))=1$ implies that $\alpha^{p^{c}}\in \pi A$, \ie
$\alpha=0$; hence we would have $P(\alpha)=0$, which contradicts the
condition $\tv(P(\alpha))=1$.
We put $Q(X)=P(X+\alpha)=P(X)+P(\alpha)$. As $\tv(P(\alpha))=1$,
then $(0,1)$ and $(p^{i_0}, 0)$ are the first two break points of
the Newton polygon of $Q(X)$. Hence there exists $p^{i_0}$ roots of
$Q(X)$ of valuation $1/p^{i_0}$. Let $L$ be the subextension of $K$
in $\Ks$ generated by the roots of $P(X)$. The ramification index
$e(L/K)$ is divisible by $p^{i_0}$. As in the proof of (ii), if $\widetilde{L}$ is the
subextension of $\Ks$ fixed by the kernel of $\rhob$, then it is an
extension of $L$. Therefore, we have $|\im(\rhob)|=[\widetilde{L}:K]$ is divisible by
$[L:K]$, and in particular, divisible by $p^{i_0}$.
\end{proof}
\subsection{} Let $G$ be a BT-group over $S$ with connected part $G^\circ$, and
\'etale part $G^\et$ of height $r$.
We have a canonical exact
sequence of $I$-modules
\begin{equation}\label{exseq-Tate-mod-p}
0\ra G^\circ(1)(\Kb)\ra G(1)(\Kb)\ra G^\et(1)(\Kb)\ra 0
\end{equation} giving rise to a class $\cb\in
\Ext^1_{\F_p[I]}(G^\et(1)(\Kb), G^\circ(1)(\Kb))$, which vanishes if
and only if \eqref{exseq-Tate-mod-p} splits. Since $I$ acts
trivially on $G^\et(1)(\Kb)$, we have an isomorphism of $I$-modules
$G^\et(1)(\Kb)\simeq\F_p^r$. Recall that for any $\F_p[I]$-module
$M$, we have a canonical isomorphism (\cite{CL} Chap.VII, \S2)
\begin{equation*}\Ext^1_{\F_p[I]}(\F_p,M)\simeq
H^1(I,M).\end{equation*} Hence we deduce that
\begin{equation}\label{class-cb}\overline{C}\in
\Ext^1_{\F_p[I]}(G^\et(1)(\Kb), G^\circ(1)(\Kb))\simeq H^1(I,
G^\circ(1)(\Kb))^{r}.\end{equation}
\begin{prop}\label{prop-class-coh} Let $G$ be a HW-cyclic BT-group over $S$ such that
$h(G)=1$, $\overline{\rho}$ \eqref{rho-0} be the representation
of $I$ on $G(1)(\Kb)$. Then the cohomology class $\overline{C}$
does not vanish if and only if the order of the group
$\im(\rhob)$ is divisible by $p$.
\end{prop}
First, we prove the following result on cohomology of groups.
\begin{lemma}\label{lemma-comm-coh} Let $F$ be a field,
$\Gamma$ be a commutative group,
and $\chi:\Gamma\ra F^\times$ be a non-trivial character of $\Gamma$. We denote
by $F(\chi)$ an $F$-vector space of dimension $1$ endowed with an
action
of $\Gamma$ given by $\chi$. Then we have $H^1(\Gamma, F(\chi))=0$.
\end{lemma}
\begin{proof}
Let $C$ be a $1$-cocycle of $\Gamma$ with values in $F(\chi)$. We
prove that $C$ is a $1$-coboundary. For any $g,h\in \Gamma$, we have
\begin{align*}C(gh)=C(g)+{\chi}(g)C(h),\\
C(hg)=C(h)+{\chi}(h)C(g).\end{align*} Since $\Gamma$ is
commutative, it follows from the relation $C(gh)=C(hg)$ that
\begin{equation}\label{formula-1}({\chi}(g)-1)C(h)=({\chi}(h)-1)C(g).\end{equation} If
${\chi}(g)\neq 1$ and ${\chi}(h)\neq 1$, then
$$\frac{1}{{\chi}(g)-1}C(g)=\frac{1}{{\chi}(h)-1}C(h).$$
Therefore, there exists $x\in \F_q(\chib)$ such that
$C(g)=({\chi}(g)-1)x$ for all $g\in \Gamma$ with ${\chi}(g)\neq 1$.
If ${\chi}(g)=1$, we have also $C(g)=0=({\chi}(g)-1)x$ by
\eqref{formula-1}. This shows that $C$ is a 1-coboundary.
\end{proof}
\begin{proof}[Proof of $\ref{prop-class-coh}$]
By \ref{lemma-cyclic}(ii) and \ref{defn-hw-index}(c), the connected
part $G^\circ$ of $G$ is HW-cyclic with $h(G^\circ)=h(G)=1$.
Assume that $\rT_p(G^\circ)$ has rank $\ell$ over $\Z_p$, and
$\rT_p(G^\et)$ has rank $r$. Then by \ref{prop-mono-trait}(a),
$G^\circ(1)(\Kb)$ is an $\F_q$-vector space of dimension 1 with
$q=p^\ell$, and the action of $I$ on $G^\circ(1)(\Kb)$ factors through
the character $\chib:I\ra I_t\xra{\theta_{q-1}} \F_q^{\times}$. We
write $G^\circ(1)(\Kb)=\F_q(\chib)$ for short. If the cohomology class
$\overline{C}$ is zero, then the exact sequence
\eqref{exseq-Tate-mod-p} splits, \ie we have an isomorphism of
Galois modules $G(1)(\Kb)\simeq \F_q(\chi)\oplus \F_p^r$. It is
clear that the group $\im(\rhob)$ has order $q-1$.
Conversely, if the cohomology class $\cb$ is not zero, we will show
that there exists an element in $\im(\rhob)$ of order $p$. We choose
a basis adapted to the exact sequence \eqref{exseq-Tate-mod-p} such
that the action of $g\in I$ is given by
\begin{equation}\label{formula-action}\rhob(g)=\begin{pmatrix}\chib(g)&\cb(g)\\
0&\mathbf{1}_r\end{pmatrix},\end{equation}
where $\mathbf{1}_r$ is the unit matrix of
type $(r,r)$ with coefficients in $\F_p$, and the map
$g\mapsto\cb(g)$ gives rise to a 1-cocycle representing the
cohomology class $\cb$. Let $I_1$ be the kernel of $\chib:I\ra
\F_q^\times$, $\Gamma$ be the quotient $I/I_1$, so $\chib$ induces
an isomorphism $\chib: \Gamma\xra{\sim} \F_q^\times$. We have an
exact sequence
\[0\ra H^1(\Gamma,\F_q(\chib))^r\xra{\mathrm{Inf}}H^1(I,\F_q(\chib))^r\xra{\mathrm{Res}}H^1(I_1,\F_q(\chib))^r,\]
where ``Inf'' and ``Res'' are respectively the inflation and
restriction homomorphisms in group cohomology. Since $H^1(\Gamma,
\F_q(\chib))^r=0$ by \ref{lemma-comm-coh}, the restriction of the
cohomology class $\cb$ to $H^1(I_1,\F_q(\chib))^r$ is non-zero.
Hence there exists $h\in I_1$ such that $\cb(h)\neq 0$. As we have
$\chib(h)=1$, then
\[\rhob(h)^{p}=\begin{pmatrix}\mathbf{1}_\ell&p\cb(h)\\
0&\mathbf{1}_r\end{pmatrix}=\mathbf{1}_{\ell+r}.\] Thus the order of
$\rhob(h)$ is $p$.
\end{proof}
\begin{cor}\label{cor-non-zero-coh} Let $G$ be a HW-cyclic BT-group over $S$,
$$\h=\begin{pmatrix}0& 0 &\cdots&0 &-a_1\\
1&0&\cdots &0 &-a_2\\
0&1&\cdots &0 &-a_3\\
\vdots&&\ddots&& \vdots\\
0&0&\cdots &1&-a_{c} \end{pmatrix}$$ be
a matrix
of $\HW_G$, $P(X)=X^{p^{c}}+a_{c}X^{p^{c-1}}+\cdots+a_1X\in
A[X]$. If $h(G)=1$ and if there exists $\alpha\in k\subset A$ such
that $\tv(P(\alpha))=1$, then the cohomology class \eqref{class-cb}
is not zero, \ie the extension
of $I$-modules \eqref{exseq-Tate-mod-p} does not split.
\end{cor}
\begin{proof} Since $\tv(a_1)=h(G)=1$, the integer $i_0$ defined in
\ref{prop-mono-trait}(iii) is at least $1$. Then the corollary follows
from \ref{prop-mono-trait}(iii) and \ref{prop-class-coh}.
\end{proof}
\section{Lemmas in Group Theory}
In this section, we fix a prime number $p\geq 2$ and an integer $n\geq 1$.
\subsection{} Recall that the general linear group $\GL_n(\Z_p)$ admits a natural exhaustive
decreasing filtration by normal subgroups
\[\GL_n(\Z_p)\supset 1+p\rM_n(\Z_p)\supset \cdots\supset 1+p^m\rM_n(\Z_p)\supset\cdots,\]
where $\rM_n(\Z_p)$ denotes the ring of matrix of type $(n,n)$ with
coefficients in $\Z_p$. We endow $\GL_n(\Z_p)$ with the topology
for which $(1+p^m\rM_n(\Z_p))_{m\geq 1}$ form a fundamental system
of neighborhoods of $1$. Then $\GL_n(\Z_p)$ is a complete and
separated topological group.
\subsection{}Let $\fG$ be a profinite group, $\rho:\fG\ra
\GL_n(\Z_p)$ be a continuous homomorphism of topological groups. By
taking inverse images, we obtain a decreasing filtration $(F^m\fG,
m\in \Z_{\geq 0})$ on $\fG$ by open normal subgroups:
\[F^0\fG=\fG, \quad \text{and }\quad
F^m\fG=\rho^{-1}(1+p^m\rM_n(\Z_p))\;\, \text{for $m\geq 1$}.\]
Furthermore, the homomorphism $\rho$ induces a sequence of injective
homomorphisms of finite groups
\begin{align}\label{phi-gradue}&\rho_0\colon F^0\fG/F^1\fG\lra \GL_n(\F_p)\\
&\rho_m\colon F^m\fG/F^{m+1}\fG\ra \rM_n(\F_p),\quad \text{for
}m\geq 1.\end{align}
\begin{lemma}\label{lemma-gp-1} The homomorphism $\rho$ is surjective if and only if
the following conditions are satisfied:
\emph{(i)} The homomorphism $\rho_0$ is surjective.
\emph{(ii)} For every integer $m\geq 1$, the subgroup $\im(\rho_m)$
of $ \rM_n(\F_p)$ contains an element of the form
$$\begin{pmatrix}x& 0&\cdots &0\\0&0&\cdots&0\\
\vdots&\vdots &\ddots&\vdots\\
0&0&\cdots &0\end{pmatrix}$$ with $x\neq 0$; or
equivalently, there exists, for every $m\geq 1$, an element $g_m\in
\fG$ such that $\rho(g_m)$ is of the
form $$\begin{pmatrix}1+p^ma_{1,1} &p^{m+1}a_{1,2} &\cdots &p^{m+1}a_{1,n}\\
p^{m+1}a_{2,1} &1+p^{m+1}a_{2,2} &\cdots & p^{m+1}a_{2,n}\\
\vdots&\vdots&\ddots&\vdots\\
p^{m+1}a_{n,1}& p^{m+1}a_{n,2}&\cdots& 1+p^{m+1}a_{n,n}
\end{pmatrix},$$ where $a_{i,j}\in \Z_p$ for $1\leq i,j\leq n$ and $a_{1,1}$
is not divisible by $p$.
\end{lemma}
\begin{proof}We notice first that $\rho$ is surjective if and only if $\rho_m$ is surjective for every $m\geq 0$, because $\fG$ is complete and $\GL_n(\Z_p)$ is separated \cite[Chap. III \S2 $\mathrm{n}^\circ8$ Cor.2 au Th\'eo. 1]{Bou}. The surjectivity of $\rho_0$ is condition (i). Condition (ii) is clearly necessary. We prove that it implies the surjectivity of $\rho_m$ for all $m\geq 1$, under the assumption of (i). First, we remark that under condition (i), if $A$ lies in $\im(\rho_m)$, then for any $U\in \GL_n(\F_p)$ the conjuagate matrix $U\cdot A\cdot U^{-1}$ lies also in $\im(\rho_m)$. In fact, let $\widetilde{A}$ be a lift of $A$ in $\rM_n(\Z_p)$ and $\widetilde{U}\in \GL_n(\Z_p)$ a lift of $U$. By assumption, there exist $g,h\in \fG$ such that
\[\rho(g)\equiv 1+p^m\widetilde{A}\mod (1+p^{m+1}M_n(\Z_p))\quad\text{and}\quad \rho(h)\equiv \widetilde{U}\mod(1+p\rM_n(\Z_p)).\]
Therefore, we have $\rho(hgh^{-1})\equiv (1+p^m\widetilde{U}\cdot\widetilde{A}\cdot \widetilde{U}^{-1})\mod (1+p^{m+1}\rM_n(\Z_p))$. Hence $hgh^{-1}\in F^m\fG$ and $\rho_m(hgh^{-1})=U\cdot A \cdot U^{-1}$.
For $1\leq i,j\leq n$, let $E_{i,j}\in \rM_n(\F_p)$ be the matrix
whose $(i,j)$-th entry is $0$ and the other entries are $0$. The
matrices $E_{i,j}(1\leq i,j\leq n)$ form clearly a basis of $\rM_n(\F_p)$ over $\F_p$. To prove the surjectivity of $\rho_m$, we only need to verify that $E_{i,j}\in \im(\rho_m)$ for $1\leq i,j \leq n$, because $\im(\rho_m)$ is an $\F_p$-subspace of $\rM_n(\F_p)$. By assumption, we have $E_{1,1}\in \im(\rho_m)$. For $2\leq i \leq n$, we put $U_i=E_{1,i}-E_{i,1}+\sum_{j\neq 1,i}E_{j,j}$. Then we have $U_i\in \GL_n(\Z_p)$ and $U_i\cdot E_{1,1}\cdot U_i^{-1}=E_{i,i}\in \im(\rho_m)$. For $1\leq i< j\leq n$, we put $U_{i,j}=I+E_{i,j}$ where $I$ is the unit matrix. Then we have $U_{i,j}\cdot E_{i,i}\cdot U^{-1}_{i,j}=E_{i,i}+E_{i,j}\in \im(\rho_m)$, and hence $E_{i,j}\in \im(\rho_m)$. This completes the proof.
\end{proof}
\begin{rem} By using the arguments in \cite[Chap. IV 3.4 Lemma 3]{Se2}, we can prove the following stronger form of Lemma \ref{lemma-gp-1}: \emph{If $p=2$, condition $\mathrm{(i)}$ and $\mathrm{(ii)}$ for $m=1,2$ are sufficient to guarantee the surjectivity of $\rho$; if $p\geq 3$, then $\mathrm{(i)}$ and $\mathrm{(ii)}$ just for $m=1$ suffice already.}
\end{rem}
A subgroup $C$ of $\GL_n(\F_p)$ is called a \emph{non-split Cartan subgroup}, if the subset $C\cup \{0\}$ of the matrix algebra $\rM_n(\F_p)$ is a field isomorphic to $\F_{p^n}$; such a group is cyclic of order $p^n-1$.
\begin{lemma}\label{lemma-gp-2}
Assume that $n\geq 2$. We denote by $H$ the subgroup of $\GL_n(\F_p)$ consisting of all the elements of the form
$\begin{pmatrix}A&b\\
0&1\end{pmatrix},$ where $A\in \GL_{n-1}(\F_p)$ and $b=\begin{pmatrix}b_1\\ \vdots\\ b_{n-1}
\end{pmatrix}$ with $b_i\in \F_p(1\leq i\leq n-1)$.
Let $G$ be a subgroup of $\GL_n(\F_p)$. Then $G=\GL_n(\F_p)$ if and only if $G$ contains $H$ and a non-split Cartan subgroup of $\GL_n(\F_p)$.
\end{lemma}
\begin{proof} The ``only if'' part is clear. For the ``if'' part, let
$C$ be a non-split Cartan subgroup contained in $G$. For a finite
group $\Lambda$, we denote by $|\Lambda|$ its order. An easy
computation shows that $|\GL_n(\F_p)|=|H| \cdot |C|$. So we just
need to prove that $U\cap C=\{1\}$; since then we will have
$|\GL_n(\F_p)|=|G|$, hence $G=\GL_n(\F_p)$. Let $g\in H\cap C$, and
$P(T)\in \F_p[T]$ be its characteristic polynomial. We fix an
isomorphism $C\simeq \F_{p^n}^\times$, and let $\zeta\in
\F_{p^n}^\times$ be the element corresponding to $g$. We have
$P(T)=\prod_{\sigma\in \Gal(\F_{p^n}/\F_p)}(T-\sigma(\zeta))$ in
$\F_{p^n}[T]$. On the other hand, the fact that $g\in H$ implies that $(T-1)$ divise $P(T)$. Therefore, we get $\zeta=1$, \ie $g=1$.
\end{proof}
\begin{rem} E. Lau point out the following strengthened version of \ref{lemma-gp-2}: \emph{When $n\geq 3$, a subgroup $G\subset\GL_n(\F_p)$ coincides with $\GL_n(\F_p)$ if and only if $G$ contains a non-split Cartan subgroup and the subgroup $\begin{pmatrix}\GL_{n-1}(\F_p)&0\\
0&1\end{pmatrix}$}. This can be used to simplify the induction process in the proof of Theorem \ref{thm-one-dim} when $n\geq 3$.
\end{rem}
\section{Proof of Theorem \ref{thm-main} in the One-dimensional Case}
\subsection{} We start with a general remark on the monodromy of
BT-groups. Let $X$ be a scheme, $G$ be an ordinary BT-group over a scheme $X$,
$G^\et$ be its \'etale part \eqref{decom-ord}. If $\etab$ is a
geometric point of $X$, we denote by
\[\rT_p(G,\etab)=\varprojlim_nG(n)(\etab)=\varprojlim_n G^\et(n)(\etab)\]
the Tate module of $G$ at $\etab$, and by $\rho(G)$ the monodromy
representation of $\pi_1(X,\etab)$ on $\rT_p(G,\etab)$. Let $f:Y\ra
X$ be a morphism of schemes, $\xib$ be a geometric point of $Y$,
$G_Y=G\times_X Y$. Then by the functoriality, we have a commutative
diagram
\begin{equation}\label{funct-mono}\xymatrix{\pi_1(Y,\xi)\ar[r]^{\pi_1(f)}\ar[d]_{\rho(G_Y)}&\pi_1(X,f(\xib))\ar[d]^{\rho(G)}\\
\Aut_{\Z_p}(\rT_p(G_Y,\xib))\ar@{=}[r]&\Aut_{\Z_p}(\rT_p(G,f(\xib)))}\end{equation}
In particular, the monodromy of $G_Y$ is a subgroup of the monodromy
of $G$. In the sequel, diagram \eqref{funct-mono} will be
refereed as the \emph{functoriality of monodromy} for the BT-group
$G$ and the morphism $f$.
\subsection{} Let $k$ be an algebraically closed field of
characteristic $p>0$, $G$ be the unique connected BT-group over $k$
of dimension $1$ and height $n+1\geq 2$ \eqref{HW-exem}. We denote by
$\bS$ the algebraic local moduli of $G$ in characteristic $p$, by
$\bG$ the universal deformation of $G$ over $\bS$, and by $\bU$ the
ordinary locus of $\bG$ over $\bS$ \eqref{defn-moduli}. Recall that $\bS$ is affine of
ring $R\simeq k[[t_1,\cdots, t_n]]$ \eqref{cor-alg-univ}, and that $G$
and $\bG$ are HW-cyclic (cf. \ref{lemma-cyclic}(i) and \ref{HW-exem}). Let $\etab$ be a geometric point of $\bU$ over its generic point. We put
$$\rT_p(\bG,\etab)=\varprojlim_{m\in \Z_{\geq 1}}\bG(m)(\etab)$$ to be the Tate module of $\bG$ at the point $\etab$. This is a free $\Z_p$-module of rank $n$. We have the monodromy representation
\[\rho_n:\pi_1(\bU,\etab)\ra \Aut_{\Z_p}(\rT_p(\bG,\etab))\simeq \GL_n(\Z_p).\]
The following is the one-dimensional case of Theorem \ref{thm-main}.
\begin{thm}\label{thm-one-dim}
Under the above assumptions, the homomorphism $\rho_n$ is surjective for $n\geq 1$.
\end{thm}
\subsection{}\label{nota-one-dim} First, we assume $n\geq 2$. By Proposition \ref{prop-HW-versal}(ii), we may assume that
\begin{equation}\label{HW-one-dim}
\h=\begin{pmatrix}0& 0 &\cdots&0 &-t_1\\
1&0&\cdots &0 &-t_2\\
0&1&\cdots &0 &-t_3\\
\vdots&&\ddots&& \vdots\\
0&0&\cdots &1&-t_{n}\end{pmatrix}
\end{equation}is a matrix of the Hasse-Witt map $\varphi_{\bG}$.
Let $\fp$ be the prime ideal of $R$ generated by
$t_1,\cdots,t_{n-1}$, $K_0\simeq k((t_n))$ be the fraction field of
$R/\fp$, $R'=\widehat{R}_{\fp}$ be the completion of the localization
of $R$ at $\fp$, and $\cG_{R'}=\bG\otimes_{R}R'$. Since the natural
map $R\ra R'$ is injective, for any $a\in R$, we will denote also by
$a$ its image in $R'$. Since the Hasse-Witt map commutes with base
change, the image of $\h$ in $\rM_{n\times n}(R')$, denoted also by
$\h$, is a matrix of $\varphi_{\cG_{R'}}$. Applying \ref{lemma-HW-V}(ii) to the closed point of $\Spec(R')$, we see that the \'etale part of $\cG_{R'}$ has height $1$ and its connected part $\cG^\circ_{R'}$ has height $n$. We have an exact sequence of BT-groups over $R'$
\begin{equation}\label{seq-1}
0\ra \cG^{\circ}_{R'}\ra \cG_{R'}\ra \cG_{R'}^\et\ra 0.
\end{equation}
We fix an imbedding $i:K_0\ra \Kb_0$ of $K_0$
into an algebraically closed field. Put
$\cG^{*}_{\Kb_0}=\cG^{*}_{R'}\otimes \Kb_0$ for
$*=\emptyset,\et,\circ$. We have $\cG^\et_{\Kb_0}\simeq \Q_p/\Z_p$,
and $\cG^\circ_{\Kb_0}$ is the unique connected one-dimensional
BT-group over $\Kb_0$ of height $n$ (cf. \ref{HW-exem}). We put
$\Rb=\Kb_0[[x_1,\cdots, x_{n-1}]]$, and
\begin{equation}\label{defn-Sigma}
\Sigma=\{\text{ring homomorphisms }\sigma:R'\ra \Rb \text{ lifting }R'\ra K_0\xra{i} \Kb_0\}
\end{equation}
Let $\sigma\in \Sigma$. We deduce from \eqref{seq-1} by base change an exact sequence of BT-groups over $\Rb$
\begin{equation}\label{seq-BT-R'}
0\ra \cG^\circ_{\Rb,\sigma}\ra \cG_{\Rb,\sigma}\ra \cG^\et_{\Rb,\sigma}\ra 0,
\end{equation}
where we have put $\cG^*_{\Rb,\sigma}=\cG^*_{R'}\otimes_{\sigma}\Rb$ for $*=\circ, \emptyset, \et$.
Due to the henselian property of $\Rb$, the isomorphism $\cG^\et_{\Kb_0}\simeq \Q_p/\Z_p$ lifts uniquely to an isomorphism $\cG^\et_{\Rb,\sigma}\simeq \Q_p/\Z_p$ .
Assume that $\cG^\circ_{\Rb,\sigma}$ is generically ordinary over $\Sb=\Spec(\Rb)$. Let $\widetilde{U}_\sigma'\subset \Sb$ be its ordinary locus, and $\xb$ be a geometric point over the generic point of $\widetilde{U}_\sigma'$. The exact sequence \eqref{seq-BT-R'} induces an exact sequence of Tate modules
\begin{equation}\label{filt-Tate-R'}
0\ra \rT_p(\cG^\circ_{\Rb,\sigma},\xb)\ra \rT_p(\cG_{\Rb,\sigma},\xb)\ra \rT_p(\cG^{\et}_{\Rb,\sigma},\xb)\ra0
\end{equation}
compatible with the actions of $\pi_1(\widetilde{U}_\sigma',\xb)$. Since we have $\rT_p(\cG^\et_{\Rb,\sigma},\xb)\simeq\rT_p(\Q_p/\Z_p,\xb)= \Z_p$, this determines a cohomology
class
\begin{equation}\label{class-one}C_\sigma\in \Ext^1_{\Z_p[\pi_1(\widetilde{U}_\sigma',\xb)]}(\Z_p,\rT_p(\cG^\circ_{\Rb,\sigma},\xb))\simeq H^1(\pi_1(\widetilde{U}_\sigma',\xb),\rT_p(\cG^\circ_{\Rb,\sigma},\xb)).\end{equation}
We consider also the ``mod-$p$ version'' of \eqref{filt-Tate-R'}
\[0\ra \cG^\circ_{\Rb,\sigma}(1)(\xb)\ra \cG_{\Rb,\sigma}(1)(\xb)\ra \F_p\ra 0,\]
which determines a cohomology class
\begin{equation}\label{class-mod-p-one}\overline{C}_\sigma\in \Ext^1_{\F_p[\pi_1(\widetilde{U}_\sigma',\xb)]}(\F_p,\cG^\circ_{\Rb,\sigma}(1)(\xb))\simeq H^1(\pi_1(\widetilde{U}_\sigma',\xb),\cG^\circ_{\Rb,\sigma}(1)(\xb)).\end{equation}
It is clear that $\overline{C}_\sigma$ is the image of $C_\sigma$ by the canonical reduction map \[H^1(\pi_1(\widetilde{U}_\sigma',\xb),\rT_p(\cG^\circ_{\Rb,\sigma},\xb))\ra H^1(\pi_1(\widetilde{U}_\sigma',\xb),\cG^\circ_{\Rb,\sigma}(1)(\xb)).\]
\begin{lemma}\label{lemma-key} Under the above assumptions, there exist $\sigma_1,\sigma_2\in \Sigma$ satisfying the following properties:
\emph{(i)} We have $\cG^\circ_{\Rb,\sigma_1}=\cG^\circ_{\Rb,\sigma_2}$, and it is the universal deformation of $\cG^\circ_{\Kb_0}$.
\emph{(ii)} We have $C_{\sigma_1}=0$ and $\cb_{\sigma_2}\neq0$.
\end{lemma}
Before proving this lemma, we prove first Theorem \ref{thm-one-dim}.
\begin{proof}[Proof of \ref{thm-one-dim}] First, we notice that the monodromy of a BT-group is independent of the base point. So we can change $\etab$ to any other geometric point of $\bU$ when discussing the monodromy of $\bG$. We make an induction on the codimension $n=\dim(G^\vee)$. The case of $n=1$ is proved in Theorem \ref{thm-Igusa}. Assume that $n\geq 2$ and the theorem is proved for $n-1$. We denote by
$$\rhob_n:\pi_1(\bU,\etab)\ra \Aut_{\F_p}(\bG(1)(\etab))\simeq
\GL_n(\F_p)$$ the reduction of $\rho_n$ modulo by $p$. By Lemma
\ref{lemma-gp-1} and \ref{lemma-gp-2}, to prove the surjectivity of
$\rho_n$, we only need to verify the following conditions:
(a) $\im(\rhob_n)$ contains a non-split Cartan subgroup of $\GL_n(\F_p)$;
(b) $\im(\rho_n)$ contains the subgroup $H\subset\GL_n(\Z_p)$ consisting of all the elements of the form $\begin{pmatrix}B&b\\0&1\end{pmatrix}\in \GL_n(\Z_p)$,
with $B\in \GL_{n-1}(\Z_p)$ and $b=\rM_{n-1\times 1}(\Z_p)$;
For condition (a), let $A=k[[\pi]]$, $T=\Spec(A)$, $\xi$ be its generic point, $\xib$ be a geometric point over $\xi$, and $I=\Gal(\xib/\xi)$ be the absolute Galois group over $\xi$. We keep the notations of \ref{nota-one-dim}. Let $f^*:R\ra A$ be the homomorphism of $k$-algebras such that $f^*(t_1)=\pi$ and $f^*(t_i)=0$ for $2\leq i\leq n$. We denote by $f:T\ra \bS$ the corresponding morphism of schemes, and put $G_T=\bG\times_{\bS}T$. By the functoriality of Hasse-Witt maps,
\[\h_T=\begin{pmatrix}0&0&\cdots&0 &-\pi\\
1&0&\cdots&0&0\\
\vdots&&\ddots&&\vdots\\
0&0&\cdots &1&0\end{pmatrix}\]
is a matrix of $\varphi_{G_T}$. By definition \ref{defn-hw-index}, the Hasse invariant of $G_T$ is $h(G)=1$. Hence $G_T$ is generically ordinary; so $f(\xi)\in \bU$. Let
\[\rhob_T:I=\Gal(\xib/\xi)\ra \Aut_{\F_p}(G_T(1)(\xib))\] be the mod-$p$ monodromy representation attached to $G_T$. Proposition \ref{prop-mono-trait}(i) implies that $\im(\rhob_T)$ is a non-split Cartan subgroup of $\GL_n(\F_p)$. On the other hand, by the functoriality of monodromy, we get $\im(\rhob_T)\subset \im(\rhob_n)$. This verifies condition (a).
To check condition (b), we consider the constructions in \ref{nota-one-dim}. Let $S'=\Spec(R')$, $f:S'\ra \bS$ be the morphism of schemes corresponding to the natural ring homomorphism $R\ra R'$, $U'$ be the ordinary locus of $\cG_{R'}$, and $\xib$ be a geometric point of $U'$. From \eqref{seq-1}, we deduce an exact sequence of Tate modules
\begin{equation}\label{seq-Tate-1}0\ra \rT_p(\cG_{R'}^\circ,\xib)\ra \rT_p(\cG_{R'},\xib)\ra \rT_p(\cG_{R'}^\et,\xib)\ra0.\end{equation}
Let $\rho_{\cG'}:\pi_1(U',\xib)\ra
\Aut_{\Z_p}(\rT_p(\cG_{R'},\xib))\simeq \GL_n(\Z_p)$ be the monodromy
represention of $\cG_{R'}$. Under any basis of $\rT_p(\cG_{R'},\xib)$
adapted to \eqref{seq-Tate-1}, the action of $\pi_1(U',\xib)$ on $\rT_p(\cG_{R'},\xib)$ is given by
\[\rho_{\cG_{R'}}\colon g\in \pi_1(U',\xib)\mapsto\begin{pmatrix}\rho_{\cG^\circ_{R'}}(g)&*\\
0&\rho_{\cG^{\et}_{R'}}(g),\end{pmatrix}\]
where $g\mapsto\rho_{\cG^\circ_{R'}}(g)\in \GL_{n-1}(\Z_p)$ (\resp
$g\mapsto\rho_{\cG^\et_{R'}}(g)\in \Z_p^\times$) gives the action of
$\pi_1(U',\xib)$ on $\rT_p(\cG^{\circ}_{R'},\xib)$ (\resp on
$\rT_p(\cG^{\et}_{R'},\xib)$). Note that $f(U')\subset \bU$. So by the functoriality of monodromy, we get $\im(\rho_{\cG'})\subset \im(\rho_n)$. To complete the proof of Theorem \ref{thm-one-dim}, it suffices to check condition (b)
with $\rho_{n}$ replaced by $\rho_{\cG_{R'}}$ under the induction
hypothesis that \ref{thm-one-dim} is valide for $n-1$. Let $\sigma_1, \sigma_2:R'\ra \Rb$ be the homomorphisms given by \ref{lemma-key}. For $i=1,2$, we denote by $f_{i}:\Sb=\Spec(\Rb)\ra S'=\Spec(R')$ the morphism of schemes corresponding to $\sigma_i$, and put $\cG_i=\cG_{\Rb,\sigma_i}=\cG_{R'}\otimes_{\sigma_i}\Rb$ to simply the notations. By condition \ref{lemma-key}(i), we can denote by $\cG^\circ$ the common connected component of $\cG_{1}$ and $\cG_{2}$. Let $\Ub\subset \Sb$ be the ordinary locus of $\cG^\circ$. Then we have $f_i(\Ub)\subset U'$ for $i=1,2$. Let $\xb$ be a geometric point over the generic point of $\Ub$. We have an exact sequence of Tate modules
\begin{equation}\label{seq-proof-one}
0\ra \rT_p(\cG^\circ,\xb)\ra \rT_p(\cG_i,\xb)\ra \rT_p(\Q_p/\Z_p,\xb)\ra 0
\end{equation}
compatible with the actions of $\pi_1(\Ub,\xb)$. We denote by
$$\rho_{\cG_i}:\pi_1(\Ub,\xb)\ra \Aut_{\Z_p}(\rT_p(\cG_i,\xb))\simeq \GL_n(\Z_p)$$ the monodromy representation of $\cG_i$. In a basis adapted to \eqref{seq-proof-one}, the action of $\pi_1(\Ub,\xb)$ on $\rT_p(\cG_i,\xb)$ is given by
\[\rho_{\cG_i}: g\mapsto \begin{pmatrix}\rho_{\cG^\circ}(g)&C_{\sigma_i}(g)\\
0&1\end{pmatrix},\]
where $\rho_{\cG^\circ}:\pi_1(\Ub,\xb)\ra \GL_{n-1}(\Z_p)$ is the monodromy representation of $\cG^\circ$, and the cohomology class in $H^1(\pi_1(\Ub,\xb),\rT_p(\cG^\circ))$ given by $g\mapsto C_{\sigma_i}(g)$ is nothing but the class defined in \eqref{class-one}. By \ref{lemma-key}(i) and the induction hypothesis, $\rho_{\cG^\circ}$ is surjective. Since the cohomology class $C_{\sigma_1}=0$ by \ref{lemma-key}(ii), we may assume $C_{\sigma_1}(g)=0$ for all $g\in \pi_1(U',\xb)$. Therefore $\im(\rho_{\cG_1})$ contains all the matrix of the form $\begin{pmatrix}B&0\\
0&1\end{pmatrix}$ with $B\in \GL_{n-1}(\Z_p)$. By the functoriality of monodromy, $\im(\rho_{\cG_{R'}})$ contains $\im(\rho_{\cG_1})$. Hence we have
\begin{equation}\label{mono-im-1}
\begin{pmatrix}\GL_{n-1}(\Z_p)&0\\
0&1\end{pmatrix}\subset \im(\rho_{\cG_1})\subset\im(\rho_{\cG_{R'}}).
\end{equation} On the other hand, since the cohomology class $\cb_{\sigma_2}\neq 0$, there exists a $g\in \pi_1(\Ub,\xb)$ such that $b_2=\cb_{\sigma_2}(g)\neq0$. Hence the matrix $\rho_{\cG_2}(g)$ has the form
$\begin{pmatrix}B_2&b_2\\
0&1\end{pmatrix}$ such that $B_2\in \GL_{n-1}(\Z_p)$ and the image of $b_2\in\rM_{1\times n-1}(\Z_p)$ in $\rM_{1\times n-1}(\F_p)$ is non-zero. By the functoriality of monodromy, we have $\im(\rho_{\cG_2})\subset\im(\rho_{\cG_{R'}})$; in particular, we have $\begin{pmatrix}B_2&b_2\\
0&1\end{pmatrix}\in \im(\rho_{\cG_{R'}})$. In view of \eqref{mono-im-1}, we get
\[\begin{pmatrix}\GL_{n-1}(\Z_p)&0\\
0&1\end{pmatrix}
\begin{pmatrix}B_2&b_2\\
0&1\end{pmatrix}
\begin{pmatrix}\GL_{n-1}(\Z_p)&0\\
0&1\end{pmatrix}
\subset \im(\rho_{\cG_{R'}}).\]
But the subset of $\GL_n(\Z_p)$ on the left hand side is just the subgroup $H$ described in condition (b). Therefore, condition (b) is verified for $\rho_{\cG_{R'}}$, and the proof of \ref{thm-one-dim} is complete.
\end{proof}
The rest of this section is dedicated to the proof of Lemma \ref{lemma-key}.
\begin{lemma}\label{sublemma-1}
Let $k$ be an algebraically closed field of characteristic $p>0$, $A$ be a noetherian henselian local $k$-algebra with residue field $k$, $G$ be a BT-group over $A$, and $G^\et$ be its \'etale part. Put
\[\Lie(G^\vee)^{\varphi=1}=\{x\in \Lie(G^\vee)\;\text{such that}\;\, \varphi_G(x)=x\}.\]Then $\Lie(G^\vee)^{\varphi=1}$ is an $\F_p$-vector space of dimension equal to the rank of $\Lie(G^{\et\vee})$, and the $A$-submodule $\Lie(G^{\et\vee})$ of $\Lie(G^\vee)$ is generated by $\Lie(G^\vee)^{\varphi=1}$.
\end{lemma}
\begin{proof} Let $r$ be the rank of $\Lie(G^{\et\vee})$, $G^\circ$ be the connected part of $G$, and $s$ be the height of $\Lie(G^{\circ\vee})$. We have an exact sequence of $A$-modules
\[0\ra \Lie(G^{\et\vee})\ra \Lie(G^\vee)\ra \Lie(G^{\circ\vee})\ra 0,\]
compatible with Hasse-Witt maps. We choose a basis of $\Lie(G^\vee)$ adapted to this exact sequence, so that $\varphi_{G}$ is expressed by a matrix of the form $\begin{pmatrix}U&W\\
0&V\end{pmatrix}$ with $U\in \rM_{r\times r}(A)$, $V\in \rM_{s\times s}(A)$, and $W\in \rM_{r\times s}(A)$. An element of $\Lie(G^\vee)^{\varphi=1}$ is given by a vector $\begin{pmatrix}x\\y\end{pmatrix}$, where $x=\begin{pmatrix}x_1\\ \vdots \\ x_{r}\end{pmatrix}$ and $y=\begin{pmatrix}y_1\\ \vdots \\ y_{s}\end{pmatrix}$ with $x_i, y_j\in A$, satisfying
\begin{equation}\label{formula-sol}\begin{pmatrix}U&W\\
0&V\end{pmatrix}\cdot \begin{pmatrix}x^{(p)}\\ y^{(p)}\end{pmatrix}=\begin{pmatrix}x\\y\end{pmatrix}\quad \Leftrightarrow \quad\begin{cases}U\cdot x^{(p)}+W\cdot y^{(p)}=x\\
V\cdot y^{(p)}=y.\end{cases}
\end{equation}
where $x^{(p)}$ (\resp $y^{(p)}$) is the vector obtained by applying $a\mapsto a^p$ to each $x_i(1\leq i\leq r)$ (\resp $y_j(1\leq j\leq s)$). By \ref{cor-nilp-HW}, the Hasse-Witt map of the special fiber of $G^\circ$ is nilpotent. So there exists an integer $N\geq 1$ such that $\varphi_{G^\circ}^N(\Lie(G^{\circ\vee}))\subset\m_{A}\cdot \Lie(G^{\circ\vee})$, \ie we have $V\cdot V^{(p)} \cdots V^{(p^{N-1})}\equiv 0\quad(\mathrm{mod}\; \m_A)$. From the equation $V\cdot y^{(p)}=y$, we deduce that
\[y=V\cdot V^{(p)}\cdots V^{(p^{N-1})}\cdot y^{(p^N)}\equiv 0 \quad(\mathrm{mod}\; \m_A).\]
But this implies that $y^{(p^N)}\equiv 0\quad(\mathrm{mod}\; \m_A^{p^N})$. Hence we get $y=V\cdot y^{(p)}\equiv 0\quad(\mathrm{mod}\; \m_A^{p^N+1})$. Repeting this argument, we get finally $y\equiv 0\quad(\mathrm{mod}\; \m_A^\ell)$ for all integers $\ell\geq 1$, so $y=0$. This implies that $\Lie(G^\vee)^{\varphi=1}\subset \Lie(G^{\et\vee})$, and the equation \eqref{formula-sol} is simplified as $U\cdot x^{(p)}=x$. Since the linearization of $\varphi_{G^\et}$ is bijective by \ref{prop-ord}, we have $U\in\GL_r(A)$. Let $\overline{U}$ be the image of $U$ in $\GL_r(k)$, and $\Sol$ be the solutions of the equation
$\overline{U}\cdot x^{(p)}=x.$ As $k$ is algebraically closed, $\Sol$ is an $\F_p$-space of dimension $r$, and $\Lie(G^{\et\vee})\otimes k$ is generated by $\Sol$ (cf. \cite[Prop. 4.1]{Ka}). By the henselian property of $A$, every elements in $\Sol$ lifts uniquely to a solution of $U\cdot x^{(p)}=x$, \ie the reduction map $\Lie(G^\vee)^{\varphi=1}\xra{\sim} \Sol$ is bijective. By Nakayama's lemma, $\Lie(G^\vee)^{\varphi=1}$ generates the $A$-module $\Lie(G^{\et\vee})$.
\end{proof}
\subsection{} We keep the notations of \ref{nota-one-dim}. Let $\Comp$ be the category of neotherian complete local $\Kb_0$-algebras with residue field $\Kb_0$, $\D_{\cG_{\Kb_0}}$ (\resp $\D_{\cG^{\circ}_{\Kb_0}}$) be the functor which associates to every object $A$ of $\Comp$ the set of isomorphic classes of deformations of $\cG_{\Kb_0}$ (\resp $\cG^\circ_{\Kb_0}$) . If $A$ is an object in $\Comp$ and $G$ is a deformation of $\cG_{\Kb_0}$ (\resp $\cG^\circ_{\Kb_0}$) over $A$, we denote by $[G]$ its isomorphic class in $\D_{\cG_{\Kb_0}}(A)$ (\resp in $\D_{\cG^\circ_{\Kb_0}}$).
\begin{lemma}\label{lemma-one-dim-2} Let $\Sigma$ be the set defined in \eqref{defn-Sigma}.
\emph{(i)} The morphism of sets $\Phi:\Sigma\ra \D_{\cG_{\Kb_0}}(\Rb)$ given by $\sigma\mapsto [\cG_{\Rb,\sigma}]$ is bijective.
\emph{(ii)} Let $\sigma \in \Sigma$. Then there exists a basis of $\Lie(\cG^{\circ\vee}_{\Rb,\sigma})$ such that $\varphi_{\cG^{\circ}_{\Rb,\sigma}}$ is represented by a matrix of the form
\begin{equation}\label{congru-matrix}
\h^\circ_{\sigma}= \begin{pmatrix}0& 0 &\cdots&0 &a_1\\
1&0&\cdots &0 &a_2\\
\vdots&&\ddots&& \vdots\\
0&0&\cdots &1& a_{n-1}\end{pmatrix}
\end{equation}
with $a_i\equiv \alpha\cdot \sigma(t_i)\;(\mathrm{mod}\, \m^2_{\Rb})$ for $1\leq i \leq n-1$, where $\alpha\in \Rb^\times$ and $\m_{\Rb}$ is the maximal ideal of $\Rb$. In particular,
$\cG^\circ_{\Rb,\sigma}$ is the universal deformation of $\cG^\circ_{\Kb_0}$ if and only if $\{\sigma(t_1),\cdots,\sigma(t_{n-1})\}$ is a system of regular parameters of $\Rb$.
\end{lemma}
\begin{proof} (i) We begin with a remark on the Kodaira-Spencer map of $\cG_{R'}$. Let $\cT_{\bS/k}=\cHom_{\cO_{\bS}}(\Omega^1_{\bS/k},\cO_{\bS})$ be the tangent sheaf of $\bS$. Since $\bG$ is universal, the Kodaira-Spencer map \eqref{Kod-map}
\[\Kod: \cT_{\bS/k}\xra{\sim} \cHom_{\cO_{\bS}}(\omega_{\bG},\Lie(\bG^\vee)) \]
is an isomorphism. By functoriality, this induces an isomorphism of $R'$-modules
\begin{equation}\label{Kod-R'}
\Kod_{R'}:T_{R'/k}\xra{\sim} \Hom_{R'}(\omega_{\cG_{R'}},\Lie(\cG_{R'}^\vee)),
\end{equation}
where $T_{R'/k}=\Hom_{R'}(\Omega^1_{R'/k},R')=\Gamma(\bS,\cT_{\bS/k})\otimes_R R'$.
For each integer $\nu\geq 0$, we put $\Rb_{\nu}=\Rb/\m_{\Rb}^{\nu+1}$, $\Sigma_{\nu}$ to be the set of liftings of $R\ra K_0\ra \Kb_0$ to $R\ra \Rb_{\nu}$, and $\Phi_{\nu}:\Sigma_{\nu}\ra \D_{\cG_{\Kb_0}}(\Rb_\nu)$ to be the morphism of sets $\sigma_{\nu}\mapsto [\cG_{R'}\otimes_{\sigma_{\nu}} \Rb_{\nu}]$. We prove by induction on $\nu$ that $\Phi_\nu$ is bijective for all $\nu\geq 0$. This will complete the proof of (i). For $\nu=0$, the claim holds trivially. Assume that it holds for $\nu-1$ with $\nu\geq 1$. We have a commutative diagram
\[\xymatrix{\Sigma_{\nu}\ar[d]\ar[rr]^{\Phi_{\nu}}&&\D_{\cG_{\Kb_0}}(\Rb_{\nu})\ar[d]\\
\Sigma_{\nu-1}\ar[rr]^{\Phi_{\nu-1}}&&\D_{\cG_{\Kb_0}}(\Rb_{\nu-1}),}\]
where the vertical arrows are the canonical reductions, and the lower arrow is an isomorphism by induction hypothesis. Let $\tau$ be an arbitrary element of $\Sigma_{\nu-1}$. We denote by $\Sigma_{\nu,\tau}\subset \Sigma_{\nu}$ the preimage of $\tau$, and by $\D_{\Phi_{\nu-1}(\tau)}(\Rb_\nu)\subset \D_{\cG_{\Kb_0}}(\Rb_\nu)$ the preimage of $\Phi_{\nu-1}(\tau)$. It suffices to prove that $\Phi_{\nu}$ induces a bijection between $\Sigma_{\nu,\tau}$ and $\D_{\Phi_{\nu-1}(\tau)}(\Rb_\nu)$. Let $I_{\nu}=\m_{\Rb}^{\nu}/\m_{\Rb}^{\nu+1}$ be the ideal of the reduction map $\Rb_{\nu}\ra \Rb_{\nu-1}$. By [EGA $\mathrm{0_{IV}}$ 21.2.5 and 21.9.4], we have $\Omega^1_{R'/k}\simeq \widehat{\Omega}^1_{R'/k}$, and they are free over $A$ of rank $n$. By [EGA $\mathrm{0_{IV}}$ 20.1.3], $\Sigma_{\nu,\tau}$ is a (nonempty) homogenous space under the group
$$
\Hom_{K_0}(\Omega^1_{R'/k}\otimes_{R'}K_0, I_{\nu})=T_{R'/k}\otimes_{R'}I_{\nu}.
$$
On the other hand, according to \ref{prop-deform}(i), $\D_{\Phi_{\nu-1}(\tau)}(\Rb_{\nu})$ is a homogenous space under the group
$$\Hom_{\Kb_0}(\omega_{\cG_{\Kb_0}},\Lie(\cG^{\vee}_{\Kb_0}))\otimes_{\Kb_0}I_{\nu}=\Hom_{R'}(\omega_{\cG_{R'}},\Lie(\cG^\vee_{R'}))\otimes_{R'} I_{\nu}.$$
Moreover, it is easy to check that the morphism of sets $\Phi_{\nu}:\Sigma_{\nu,\tau}\ra \D_{\Phi_{\nu-1}(\tau)}(\Rb_{\nu})$ is compatible with the homomorphism of groups
\[\Kod_{R'}\otimes_{R'}\Id:T_{R'/k}\otimes_{R'}I_{\nu}\ra \Hom_{R'}(\omega_{\cG_{R'}},\Lie(\cG^\vee_{R'}))\otimes_{R'} I_{\nu},\]
where $\Kod_{R'}$ is the Kodaira-Spencer map \eqref{Kod-R'} associated to $\cG_{R'}$. The bijectivity of $\Phi_{\nu}$ now follows from the fact that $\Kod_{R'}$ is an isomorphism.
(ii) First, we determine the submodule $\Lie(\cG^{\et\vee}_{\Rb,\sigma})$ of $\Lie(\cG^\vee_{\Rb,\sigma})$. We choose a basis of $\Lie(\bG^\vee)$ over $\cO_{\bS}$ such that $\varphi_{\bG}$ is expressed by the matrix $\h$ \eqref{HW-one-dim}. As $\cG_{\Rb,\sigma}$ derives from $\bG$ by base change $R\ra R'\xra{\sigma}\Rb$, there exists a basis $(e_1,\cdots,e_n)$ of $\Lie(\cG_{\Rb,\sigma}^\vee)$ such that $\varphi_{\cG_{\Rb,\sigma}}$ is expressed by
\[\h^{\sigma}=\begin{pmatrix}
0&0 &\cdots& 0& -\sigma(t_1)\\
1&0&\cdots& 0& -\sigma(t_2)\\
\vdots& &\ddots &&\vdots\\
0 &0&\cdots &1&-\sigma(t_n)
\end{pmatrix}.
\] By Lemma \ref{sublemma-1}, $\Lie(\cG^{\et\vee}_{\Rb,\sigma})$ is generated by $\Lie(\cG^{\vee}_{\Rb,\sigma})^{\varphi=1}$. If $\sum_{i=1}^nx_ne_n\in \Lie(\cG_{\Rb,\sigma}^\vee)^{\varphi=1}$ with $x_i\in \Rb$ for $1\leq i\leq n$, then $(x_i)_{1\leq i\leq n}$ must satisfy the equation
$\h^\sigma\cdot
\begin{pmatrix}x^p_1\\
\vdots \\
x^p_n\end{pmatrix}=\begin{pmatrix}x_1\\ \vdots \\ x_n
\end{pmatrix};$
or equivalently,
\begin{equation}\label{equ-sol}
\begin{cases}x_1=-\sigma(t_1)x_n^p\\
x_2=-\sigma(t_2)x_n^p-\sigma(t_1)^px_n^{p^2}\\
\cdots\\
x_{n-1}=-\sigma(t_{n-1})x_n^p-\cdots-\sigma(t_1)^{p^{n-2}}x_n^{p^{n-1}}\\
\sigma(t_1)^{p^{n-1}}x_n^{p^n}+\sigma(t_2)^{p^{n-2}}x_n^{p^{n-1}}+\cdots +\sigma(t_n)x_n^{p}+x_n=0.
\end{cases}
\end{equation}
We note that $\sigma(t_i)\in \m_{\Rb}$ for $1\leq i\leq n-1$ and $\sigma(t_n)\in\Rb^\times$ with image $i(t_n)\in \Kb_0$, where $i:K_0\ra \Kb_0$ is the fixed immbedding. By Hensel's lemma, every solution in $\Kb_0$ of the equation $i(t_n)x_n^{p}+x_n=0$ lifts uniquely to a solution of \eqref{equ-sol}. As $\Lie(\cG^{\et\vee}_{\Rb,\sigma})$ has rank $1$, by Lemma \ref{sublemma-1}, these are all the solutions. Let $(\lambda_1,\cdots,\lambda_n)$ be a non-zero solution of \eqref{equ-sol}. We have
\begin{equation}\label{congru-lam}\lambda_n\in \Rb^\times\quad \text{and}\quad \lambda_i\equiv -\lambda_n^p\sigma(t_i)\quad(\mathrm{mod}\;\m_{\Rb}^2).\end{equation}
We put $v=\lambda_1e_1+\cdots+\lambda_ne_n$; so $v$ is a basis of $\Lie(\cG^{\et\vee}_{\Rb,\sigma})$ by \ref{sublemma-1}. For $1\leq i\leq n$, let $f_i$ be the image of $e_i$ in $\Lie(\cG^{\circ\vee}_{\Rb,\sigma})$. Then $f_1,\cdots,f_n$ clearly generate $\Lie(\cG^{\circ\vee}_{\Rb,\sigma})$. By the explicit description above of $\Lie(\cG^{\et\vee}_{\Rb,\sigma})$, we have $f_n=-\lambda^{-1}_n(\lambda_1f_1\cdots+\lambda_{n-1}f_{n-1})$. Hence $f_1,\cdots,f_{n-1}$ form a basis of $\Lie(\cG^{\circ\vee}_{\Rb,\sigma})$. By the functoriality of Hasse-Witt maps, we have $\varphi_{\cG^{\circ}_{\Rb}}(f_i)=f_{i+1}$ for $1\leq i\leq n-1$, or equivalently,
\[\varphi_{\cG^\circ_{\Rb,\sigma}}(f_1,\cdots,f_{n-1})=(f_1,\cdots,f_{n-1})\cdot
\begin{pmatrix}0&0&\cdots&0&-\lambda_n^{-1}\lambda_1\\
1&0&\cdots&0&-\lambda_n^{-1}\lambda_2\\
\vdots &&\ddots&&\vdots\\
0&0&\cdots&1&-\lambda_n^{-1}\lambda_{n-1}
\end{pmatrix}.\]
In view of \eqref{congru-lam}, we see that the above matrix has the form of \eqref{congru-matrix} by setting $\alpha=\lambda_n^{p-1}\in \Rb^\times$. The second part of statement (ii) follows immediately from Proposition \ref{prop-HW-versal}(ii) and the description above of $\varphi_{\cG^{\circ}_{\Rb,\sigma}}$.
\end{proof}
\begin{lemma}\label{lemma-lifting}
Let $F$ be a field with the discrete topology, $A$ be a noetherian local complete and formally smooth $F$-algebra, $C$ be an adic topological $F$-algebra, $J\subset C$ be an ideal of definition $($\ie $C=\varprojlim_{n}C/J^{n+1})$, $g:A\ra C/J$ be a continuous homomorphism of topological $F$-algebras. Let $t_1,\cdots,t_n$ be elements in $A$ such that $dt_1,\cdots, dt_{n}$ form a basis of $\widehat{\Omega}^1_{A/F}$ over $A$, and $a_1,\cdots, a_n\in C$ be such that the image of $a_i$ in $C/J$ is $g(t_i)$ for $1\leq i\leq n$. Then there exists a unique continuous homomorphism of topological $F$-algebras $h:A\ra C$ which lifts $g$ and satisfies $h(t_i)=a_i$ for $1\leq i\leq n$.
\end{lemma}
\begin{proof} For each integer $\nu\geq 0$, we put $C_{\nu}=C/J^{\nu+1}$. It suffices to prove that there exists, for every integer $\nu\geq 0$, a unique continuous homomorphism of topological $F$-algebras $h_\nu:A\ra C_\nu$ which lifts $g=h_0$ and verifies $h_\nu(t_i)\equiv a_i\quad (\mathrm{mod}\;J^{\nu+1})$. We proceed by induction on $\nu\geq 0$. For $\nu= 0$, the assertion is trivial. Suppose that $\nu\geq 1$ and the required homomorphism $h_{\nu-1}:A\ra C_{\nu-1}$ exists uniquely. Since $A$ is formally smooth over $F$, by [EGA $\mathrm{0_{IV}}$ 20.7.14.4 and 20.1.3], the set of continuous homomorphisms $A\ra C_{\nu}$ lifting $h_{\nu-1}$ is a homogeneous space under the group
$\mathrm{Hom.cont}_{A}(\widehat{\Omega}^1_{A/F},J^{\nu}/J^{\nu+1}),$
where $\mathrm{Hom.cont}_A$ denotes the group of continuous homomorphisms of topological modules over $A$. Since $C/J$ is a discrete topological ring, there exists an inteter $\ell\geq 0$, such that the continuous map $g:A\ra C/J$ factors through the canonical surjection $A\ra A/\m_A^{\ell}$, where $\m_A$ is the maximal ideal of $A$. Note that $J^{\nu}/J^{\nu+1}$ is a $C/J$-module; so we have
\[\mathrm{Hom.cont}_A(\widehat{\Omega}^1_{A/F},J^{\nu}/J^{\nu+1})=\Hom_{A/\m_A^{\ell}}(\widehat{\Omega}^1_{A/F}\otimes A/\m_{A}^\ell,J^{\nu}/J^{\nu+1}).\]
Now let $\widetilde{h}_{\nu}:A\ra C_{\nu}$ be an arbitrary continuous lifting of $h_{\nu-1}$; then any other liftings of $h_{\nu-1}$ to $C_{\nu}$ writes as $\widetilde{h}_{\nu}+\delta$ with $\delta \in \Hom_{A/\m_A^{\ell}}(\widehat{\Omega}^1_{A/F}\otimes A/\m_{A}^\ell,J^{\nu}/J^{\nu+1})$. By assumption, $dt_1,\cdots, dt_{n}$ being a basis of $\widehat{\Omega}^1_{A/F}$, there exists thus a unique $\delta_0$ such that
$\delta_0(t_i)\equiv a_i-\widetilde{h}_{\nu}(t_i) \quad (\mathrm{mod}\; J^{\nu+1}).$
Then $h_{\nu}=\widetilde{h}_{\nu}+\delta_0$ is the unique continuous homomorphism $A\ra C_{\nu}$ lifting $g$ and satisfying $h_\nu(t_i)\equiv a_i \quad (\mathrm{mod}\; J^{\nu+1}).$ This completes the induction.
\end{proof}
Now we can turn to the proof of \ref{lemma-key}.
\subsection{Proof of Lemma \ref{lemma-key}} First, suppose that we have found a $\sigma_2\in \Sigma$ such that $\cb_{\sigma_2}\neq 0$ and $\cG^\circ_{\Rb,\sigma_2}$ is the universal deformation of $\cG^\circ_{\Kb_0}$. Since $\Phi:\Sigma\xra{\sim} \D_{\cG_{\Kb_0}}(\Rb)$ is bijective by \ref{lemma-one-dim-2}(i), there exists a $\sigma_1\in \Sigma$ corresponding to the deformation $[\cG^\circ_{\Rb,\sigma_2}\oplus \Q_p/\Z_p]\in \D_{\cG_{\Kb_0}}(\Rb)$. It is clear that $\cG^{\circ}_{\Rb,\sigma_1}\simeq \cG^{\circ}_{\Rb,\sigma_2}$. Besides, the exact sequence \eqref{filt-Tate-R'} for $\sigma_1$ splits; so we have $C_{\sigma_1}=0$. It remains to prove the existence of $\sigma_2$. We note first that $\Kb_0$ can be canonically imbedded into $\Rb$, since it is perfect. Since $R'$ is formally smooth over $k$ and $(dt_1,\cdots,dt_n)$ is a basis of $\widehat{\Omega}^1_{R'/k}\simeq \Omega^1_{R'/k}$, Lemma \ref{lemma-lifting} implies that there is a $\sigma \in \Sigma$ such that $\sigma(t_i)\;(1\leq i\leq n-1)$ form a system of regular parameters of $\Rb$ and $\sigma(t_n)\in \Kb_0\subset \Rb$. We claim that $\sigma_2=\sigma$ answers the question. In fact, Lemma \ref{lemma-one-dim-2}(ii) implies that $\cG^\circ_{\Rb,\sigma}$ is the universal deformation of $\cG^\circ_{\Kb_0}$. It remains to verify that $\cb_{\sigma}\neq 0$.
Let $A=\Kb_0[[\pi]]$ be a complete discrete valuation ring of characteristic $p$ with residue field $\Kb_0$, $T=\Spec(A)$, $\xi$ be the generic point of $T$, $\xib$ be a geometric over $\xi$, and $I=\Gal(\xib/\xi)$ the Galois group. We define a homomorphism of $\Kb_0$-algebras $f^*:\Rb\ra A$ by putting $f^*(\sigma(t_1))=\pi$ and $f^*(\sigma(t_i))=0$ for $2\leq i\leq n-1$. This is possible, since $(\sigma(t_1),\cdots, \sigma(t_{n-1}))$ is a system of regular parameters of $\Rb$. Let $f:T\ra \Sb$ be the homomorphism of schemes corresponding to $f^*$, and $\cG_T=\cG_{\Rb,\sigma}\times_{\Sb}T$. By the functoriality of Hasse-Witt maps,
\[\h_{T}=\begin{pmatrix}0&0&\cdots&0 &-\pi\\
1&0&\cdots&0&0\\
0&1&\cdots&0&0\\
\vdots&&\ddots&&\vdots\\
0&0&\cdots&1&-f^*(\sigma(t_n))\end{pmatrix}\in \rM_{n\times n}(\Rb)\]
is a matrix of $\varphi_{\cG_{T}}.$ By definition \eqref{defn-hw-index}, the Hasse invariant of $\cG_T$ is $h(\cG_{T})=1$. In particular, $\cG_T$ is generically ordinary. Let $\widetilde{U}'_\sigma\subset \Sb$ be the ordinary locus of $\cG_{\Rb,\sigma}$. We have $f(\xi)\in \widetilde{U}'_\sigma$. By the functoriality of fundamental groups, $f$ induces a homomorphism of groups
\[\pi_1(f):I=\Gal(\xib/\xi)\ra \pi_1(\widetilde{U}'_\sigma,f(\xib))\simeq \pi_1(\widetilde{U}'_\sigma,\xb).\]
Let $\cG^\circ_T$ be the connected part of $\cG_T$, and $\cG_{T}^\et$ be the \'etale part of $\cG_{T}$. Then $\cG^\et_{T}\simeq\Q_p/\Z_p$. We have an exact sequence of $\F_p[I]$-modules
\[0\ra \cG^\circ_T(1)(\xib)\ra \cG_T(1)(\xib)\ra \cG^\et_T(1)(\xib)\ra 0,\]
which determines a cohomology class $\cb_T\in H^1(I,\cG_T^\circ(1)(\xib))$. We notice that $\cG_T(1)(\xib)$ is isomorphic to $\cG_{\Rb,\sigma}(1)(\xb)$ as an abelian group, and the action of $I$ on $\cG_T(1)(\xib)$ is induced by the action of $\pi_1(\widetilde{U}'_\sigma,\xb)$ on $\cG_{\Rb,\sigma}(1)(\xb)$. Therefore, $\cb_T$ is the image of $\cb_\sigma$ by the functorial map
\[H^1\bigl(\pi_1(\widetilde{U}'_\sigma,\xb),\cG^\circ_{\Rb,\sigma}(1)(\xb)\bigr)\ra H^1\bigl(I,\cG^\circ_T(1)(\xib)\bigr).\]
To verify that $\cb_\sigma\neq 0$, it suffices to check that $\cb_T\neq0$. We consider the polynomial $P(X)=X^{p^n}+f^*(\sigma(t_n))X^{p^{n-1}}+\pi X\in A[X]$. According to \ref{cor-non-zero-coh}, it suffices to find a $\alpha\in \Kb_0\subset A$ such that $P(\alpha)$ is a uniformizer of $A$. But by the choice of $\sigma$, we have $\sigma(t_n)\in \Kb_0$ and $\sigma(t_n)\neq 0$; so $f^*(\sigma(t_n))\neq 0$ lies in $\Kb_0$. Let $\alpha$ be a $p^{n-1}(p-1)$-th root of $-f^*(\sigma(t_n))$ in $\Kb_0$. Then we have $\alpha\in \Kb_0^\times$, and $P(\alpha)=\alpha\pi$ is a uniformizer of $A$. This completes the proof of \ref{lemma-key}.
\section{End of the Proof of Theorem \ref{thm-main}}
In this section, $k$ denotes an algebraically closed field of characteristic $p>0$.
\subsection{}\label{New-strata} First, we recall some preliminaries on Newton stratification due to F. Oort. Let $G$ be an arbitrary BT-group over $k$, $\bS$ be the local moduli of $G$ in characteristic $p$, and $\bG$ be the universal deformation of $G$ over $\bS$ \eqref{defn-moduli}. Put $d=\dim (G)$ and $c=\dim(G^\vee)$. We denote by $\cN(G)$ the Newton polygon of $G$ which has endpoints $(0,0)$ and $(c+d,d)$. Here we use the normalization of Newton polygons such that slope 0 corresponds to \'etale BT- groups and slope 1 corresponds to groups of multiplicative type.
Let $\Nt(c+d,d)$ be the set of Newton polygons with endpoints $(0,0)$ and $(c+d,d)$ and slopes in $(0,1)$. For $\alpha,\beta \in \Nt(c+d,d)$, we say that $\alpha\preceq \beta$ if no point of $\alpha$ lies below $\beta$; then ``$\preceq$'' is a partial order on $\Nt(c+d,d)$. For each $\beta\in \Nt(c+d,d)$, we denote by $V_\beta$ the subset of $\bS$ consisting of points $x$ with $\cN(\bG_x)\preceq\beta$, and by $V_\beta^\circ$ the subset of $\bS$ consisting of points $x$ with $\cN(\bG_x)=\beta$. By Grothendieck-Katz's specialization theorem of Newton polygons, $V_\beta$ is closed in $\bS$, and $V_\beta^\circ$ is open (maybe empty) in $V_\beta$. We put
$$
\diamondsuit(\beta)=\{(x,y)\in \Z\times \Z\;|\; 0\leq y<d, y<x<c+d, (x,y) \text{ lies on or above the polygon } \beta\},
$$ and $\dim(\beta)=\#(\diamondsuit(\beta))$.
\begin{thm}[\cite{oort2} Theorem 2.11]\label{thm-oort} Under the above assumptions, for each $\beta\in \Nt(c+d,d)$, the subset $V^\circ_\beta$ is non-empty if and only if $\cN(G)\preceq \beta$. In that case, $V_\beta$ is the closure of $V^\circ_\beta$ and
all irreducible components of $V_\beta$ have dimension $\dim(\beta)$.
\end{thm}
\subsection{} Let $G$ be a connected
and HW-cyclic BT-group over $k$ of dimension $d=\dim(G)\geq 2$. Let $\beta\in \Nt(c+d,d)$ be the Newton polygon given by the following slope sequence:
\[ \beta=(\underbrace{1/(c+1),\cdots,1/(c+1)}_{c+1},\underbrace{1,\cdots,1}_{d-1}).\]
We have $\cN(G)\preceq \beta$ since $G$ is supposed to be
connected. By Oort's Theorem \ref{thm-oort}, $V_\beta$ is a equal dimensional
closed subset of the local moduli $\bS$ of dimension $c(d-1)$. We endow $V_\beta$ with the structure of a reduced closed subscheme of $\bS$.
\begin{lemma} Under the above assumptions, let $R$ be the ring of $\bS$,
and
\[
\begin{pmatrix}0 &0 &\cdots &0 &-a_1\\
1 &0 &\cdots &0 &-a_2\\
0&1&\cdots&0&-a_3\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&-a_c\end{pmatrix}\in \rM_{c\times c}(R)
\]
be a matrix of the Hasse-Witt map $\HW_G$. Then the closed reduced subscheme $V_\beta$ of $\bS$ is defined by the prime ideal $(a_1,\cdots,a_c)$. In particular, $V_\beta$ is irreducible.
\end{lemma}
\begin{proof} Note first that $\{a_1,\cdots, a_c\}$ is a subset of a system of regular parameters of $R$ by
\ref{prop-HW-versal}(i). Let $I$ be the ideal of $R$ defining $V_\beta$. Let $x$ be an arbitrary point of $V_\beta$, we denote by $\fp_x$ the prime ideal of $R$ corresponding to $x$. Since the Newton polygon of the fibre $\bG_x$ lies above
$\beta$, $\bG_x$ is connected. By Lemma \ref{lemma-HW-V}, we have $a_i\in \fp_x$
for $1\leq i\leq c$. Since $V_\beta$ is reduced, we have $a_i\in
I$. Let $\mathfrak{P}=(a_1,\cdots, a_c)$, and $V(\mathfrak{P})$ the closed subscheme of
$\bS$ defined by $\mathfrak{P}$. Then $V(\mathfrak{P})$ is an integral scheme of
dimension $c(d-1)$ and $V_\beta\subset V(\mathfrak{P})$. Since Theorem
\ref{thm-oort} implies that $\dim V_\beta=c(d-1)$, we have necessarily $V_\beta=V(\mathfrak{P})$.
\end{proof}
We keep the assumptions above. Let $(t_{i,j})_{1\leq i\leq c,
1\leq j \leq d}$ be a regular system of
parameters of $R$ such
that $t_{i,d}=a_i$ for all $1\leq i \leq c$. Let $x$ be the generic
point of the Newton strata $V_\beta$, $k'=\kappa(x)$, and
$R'=\widehat{\cO}_{\bS,x}$. Since $R$ is noetherian and integral, the
canonical ring homomorphism $R\ra \cO_{\bS,x}\ra R'$ is injective.
The image in $R'$ of an element $a\in
R$ will be denoted also by $a$. By choosing a $k$-section $k'\ra R'$ of the canonical projection
$R'\ra k'$, we get a (non-canonical) isomorphism of $k$-algebras $R'\simeq
k'[[t_{1,d},\cdots, t_{c,d}]]$. Let $k''$ be an algebraic closure of
$k'$, and $R''=k''[[t_{1,d},\cdots,t_{c,d}]]$. Then we have a natural
injective homomorphism of $k$-algebras $R'\ra R''$ mapping $t_{i,d}$
to $t_{i,d}$ for $ 1\leq i\leq c$.
Let $S''=\Spec (R'')$, $\xb$ be its closed point. By the construction of $S''$, we have a morphism
of $k$-schemes
\begin{equation}f: S''\ra \bS \end{equation}
sending $\xb$ to $x$. We put $\cG=\bG\times_{\bS} S''$. By the choice of the Newton polygon $\beta$, the closed fibre $\cG_{\xb}$ has a BT-subgroup $\cH_{\xb}$ of multiplicative type of height $d-1$. Since $S''$ is henselian, $\cH_{\xb}$ lifts uniquely to a BT-subgroup $\cH$ of $\cG$. We put $\cG''=\cG/\cH$. It is a connected BT-group over $S''$ of dimension $1$ and height $c+1$. \\
\begin{lemma}\label{rem-Lau} Under the above assumptions, $\cG''$ is the universal deformation in equal characteristic of its special fiber.
\end{lemma}
This lemma is a particular case of \cite[Lemma 3.1]{lau}. Here, we use \ref{prop-HW-versal}(ii) to give a simpler proof.
\begin{proof} We have an exact sequence of BT-groups over $S''$
\[0\ra \cH\ra \cG\ra \cG''\ra 0,\]
which induces an exact sequence of Lie algebras $0\ra
\Lie(\cG''^\vee)\ra \Lie(\cG^\vee)\ra \Lie(\cH^\vee)\ra 0$ compatible
with Hasse-Witt maps. Since $\cH$ is of multiplicative type, we get
$\Lie(\cH^\vee)=0$ and an isomorphism of Lie algebras
$\Lie(\cG''^\vee)\simeq \Lie(\cG^\vee)$. By the choice of the regular system $(t_{i,j})_{1\leq i \leq c, 1\leq j \leq d}$, there is a basis $(v_1,\cdots, v_c)$ of $\Lie(\cG''^\vee)$ over $\cO_{S''}$ such that $\HW_{\cG''}$ is given by the matrix
\[\h=\begin{pmatrix}
0 &0 &\cdots &0 &-t_{1,d}\\
1 &0 &\cdots &0 &-t_{2,d}\\
0 &1 &\cdots &0 &-t_{3,d}\\
\vdots &&\ddots &&\vdots\\
0&0&\cdots&1&-t_{c,d}\end{pmatrix}.\]
Now the lemma results from Proposition \ref{prop-HW-versal}(ii).
\end{proof}
\subsection{Proof of Theorem \ref{thm-main}} The one-dimensional case is treated in \ref{thm-one-dim}. If
$\dim(G)\geq 2$, we apply the preceding discussion to obtain the
morphism $f\colon S''\ra \bS$ and the BT-groups $\cG=\bG\times_{\bS}S''$ and $\cG''$,
which is the quotient of $\cG$ by the maximal subgroup of $\cG$ of
multiplicative type. Let $U''$ be the common ordinary locus of $\cG$
and $\cG''$ over
$S''$, and $\xib$ be a geometric point of $U''$. Then $f$ maps $U''$ into the ordinary locus $\bU$ of $\bG$. We denote by
$$
\rho_\cG:\pi_1(U'',\xib)\ra \Aut_{\Z_p}(\rT_p(\cG,\xib))
$$ the monodromy representation associated to $\cG$, and the same notation for $\rho_{\cG''}$. By the functoriality of monodromy, we have $\im(\rho_{\cG})\subset\im(\rho_{\bG})$. On the other hand, the canonical map $\cG\ra \cG''$ induces an isomorphism of Tate modules
$\rT_p(\cG,\etab)\xra{\sim}\rT_p(\cG'',\etab)$ compatible with the action of $\pi_1(U'',\etab)$. Therefore, the group $\im(\rho_{\cG})$ is identified with $\im(\rho_{\cG''})$. Since $\cG''$ is one-dimensional, we conclude the proof by Lemma \ref{rem-Lau} and Theorem \ref{thm-one-dim}. | 169,301 |
TITLE: Difference in the definitions of density in topology?
QUESTION [0 upvotes]: I have been trying to understand the concept of density and have so far come across two definitions. That is, the set $A$ is dense in a space $B$ if
cl(A) = B,
however, I have also come across the following definition.
The set $A$ is dense in $B$ if
$A \subset B$ and $B \subseteq cl(A)$.
Just wondering if the two definitions are indeed the same.
Also, additionally, we have it given that
$B \subseteq cl(A)$.
Taking the closure of both sides yields
$cl(B) \subseteq cl(A)$.
I am also just wondering if this is also indeed true.
Thanks for the help.
REPLY [0 votes]: So these are slightly different notions linguistically:
the first is the more standard one and is for spaces $X$: $A$ (a subset of $X$) is called
dense in $X$ when $\operatorname{cl}(A) = X$.
The second notion is for subsets of a space $X$ : the set $A$ is called dense in $B$ whenever $A \subseteq B$ and $B \subseteq \operatorname{cl}(B)$.
But this second notion is not really different in content: it means that $\operatorname{cl}_B(A) = A$, where $\operatorname{cl}_B$ is the closure operator in $B$ with the subspace topology seen as a space in its own right.
This can be seen by applying the following general fact for subspaces and their closure operator, and how it relates to the closure in the ambient space:
Fact: If $B$ is a subset of $X$ in the subspace topology w.r.t. $X$, and $A \subseteq B$ then $\operatorname{cl}_B(A) = \operatorname{cl}(A) \cap B$; or in words: the closure of $A$ in the subspace $B$ is just the part of the full closure of $A$ (in $X$) that lies inside $B$.
This fact can be proved directly from the definition of the subspace topology and the characterisation of the closure of $A$ as all points $x$ such that all neighbourhoods of $x$ intersect $A$, or from the fact that closed sets of $B$ are closed sets of $X$ intersected with $B$, and the closure of $A$ is the smallest closed set that contains $A$.
Now suppose $A \subseteq B$ and $A$ is dense (in the first sense) in the subspace $B$, so $\operatorname{cl}_B(A) = B$ and by the fact we can rewrite this as
$$\operatorname{cl}_B(A) = \operatorname{cl}(A) \cap B = B$$ but simple set theory tells us that $$C \cap B = B \text{ iff } B \subseteq C$$ for all $B,C$ and so we actually get that
$$\operatorname{cl}_B(A) = B \text{ iff } B \subseteq \operatorname{cl}(A)$$
So the "notions" of dense mean the same thing: $A$ is dense in $B$ if we can "approximate all points of $B$ by points of $A$", or $B$ is a subset of $\operatorname{cl}(A)$ (where $\operatorname{cl}(A)$ can be thought of as "all points that can be approximated by points of $A$", intuitively (or more formally e.g. if you use filters or nets) | 194,421 |
<<
I was torn on whether to use the Canadian or American spelling on this one. This is about wedding cheques or wedding checks, depending on your country of origin! I’ll use a bit of both, to spice things up.
Normally, cheque writing is easy as the recipient is just one person or business. However, weddings throw?
We ran into this problem when we got married. We mostly got cash, however a few were too Mr & Mrs Dickinson and it caused a bit of an issue. It was mostly from older generations that this was an issue.
We just do cash now, or a bank transfer when the option is available, so much easier. Plus you haven't got to wait two weeks for them cash it after their honeymoon.
Bank transfers are the best. We got a few and now do them whenever possible! I also liked having less cash in the "giant wad" before we were able to get to the bank.
We were given several checks with "Mr. & Mrs. XXXXX." You're right: that delayed everything by weeks. Give me my money! Daddy needs his money….
Hahaha, come to poppa. Half the time people need it to cover the cost of the wedding!
i wonder if i could just paypal the money over to folks when they get married and then let them do with it what they want :) of course, we then have to have a slightly awkward conversation where we ask for their paypal address..
Paypal would be a great one as well!! (Depending on the fee tier that you're in, anyway.) Good thought! I've asked people for their emails before to send eTransfers, it's always been no big deal.
I have always wired the amount, usually to a list (with no specific gift targeted, they could use it to buy any gift on the list or get cash from the store) or a personal account. Never heard about presents missing after a wedding, but I imagine you wouldn't go to your guests saying "did you bring me something, I didn't get it!" and vice versa, if someone doesn't thank me for a gift, I wouldn't go ask for gratefulness.
Yes, it ends up being such a sticky situation!! Often people end up asking about gifts when they don't get thank-yous, because they're worried the couple never received the gift! Wiring the amount is also a really good solution.
I normally just give them cash in a card from me. I know that it is easily lost, but with our wedding we really appreciated the simplicity of getting a lump sum of cash to buy what we needed.
Ah, I found it overwhelming to have a huge wad of cash, myself. Plus I've heard way too many unfortunate stories of things going missing or getting stolen. Money is definitely the best gift, though.
Easy way to decide between "cheque" and "check": look at the geographical breakdown of your traffic. More than two thirds of my traffic comes from HRM QEII's colonies so, well, it's "cheque". Communication is all about the audience.
Can't decide = playing to the researched audiences and anticipated organic traffic ;-)
I think I'd use Pay Pal, though a lot of people might prefer having something physical to give. Either way, I'd love to get cash when/if I get married!
I use "cheque" myself, but I've seen "check" a lot, even in Canada.
Paypal would be okay, but I think the fees would become quite large, quite quickly. I see "check" a lot too, must be thanks to all of the American influence.
We received a lot of cheques for our wedding and the only saving grace for us was that a friend of mine works at a bank and allowed us to cash them without the formal name change yet, what a pain in the butt if she hadn't!!
Yay for being well connected!
I usually just put some cash in the card. As impersonal as cash is, a bank transfer seems even more so. Of course I've always attended smaller affairs that are unlikely to be targeted by theives.
Bank transfers can actually be more personal! Picking the password is lots of fun :-)
How do you feel about Visa or Amex gift cards in place of checks/cheques? I know they can be seens as somewhat impersonal, but they can be good for paying off misc small wedding expenses, taking on the honeymoon for misc little expenses, or even buying things left on the registry after the wedding.
Also if they're lost, they can usually be replaced (after a suitable waiting period).
My recent post Cat crafting
Visa and Amex gift cards are also awesome :-) There's always that "cash is so impersonal" stigma, but almost everyone I've ever met or read online REALLY wants cash (or cash equivalents) as wedding gifts.
PS – excellent incorporation of cheques and checks!
When I got married, someone from another wedding tried to take off with my gift box full of cards with cash. Thank god for my brother who kept checking the box ! Cheques are the way to go !
Yikes!! It's really upsetting how many times I've heard similar stories. | 350,523 |
Welcome to Nightingale-Conant
Details
In sales, what might have worked yesterday is just not good enough in today's competitive marketplace.
In Power Selling,. | 157,747 |
Accession 1580 Death 1612
It was in the long, peaceful and prosperous reign of the fifth king
Muhammad Quli Qutb Shah that the Qutb Shahi dynasty reached its zenith.
Muhammad Quli was crowned king when only 15 years of age. He is best remembered as the great planner and founder of the city of Hyderabad. According to popular legend, the king was enamoured of a dancer calledBhagmati, belonging to the small village of Chichelam where the famous Charminar now stands. He found Bhagnagar to perpetuate his love for her. Later, when the title of “Hyder Mahal” was bestowed upon her, the name of the city was accordingly changed by the king to Hyderabad.
Muhammad Quli Qutb Shah, like Ibrahim, patronised fairly long reign of 32 years. He was a humane and just king, besides being a great builder and a man of letters. Hayath Bakshi Begum was his only daughter who was married to his nephew and successor, Sultan Muhammad Qutb Shah. | 187,079 |
For Sale / To Donate: This Item is for Sale or Available for Donation
The HP Tech Support Number is the key to finding the help which you are looking for the issue created by your system. The HP Support Number is available at your service 24*7.
To know more, you can visit at
Submitted by Aaliaalzraa on August 28, 2017 - 6:15am
Searching for Translators?
¿Hablas Español?
Connect in the | 314,065 |
TITLE: Limit of ideals?
QUESTION [3 upvotes]: I've been playing around with an idea in my head for a while. Consider the ideals $n \mathbb{Z}$ of $\mathbb{Z}$. As $n \to \infty$, I want to say that $n \mathbb{Z} \to 0 \mathbb{Z}$, for a couple of reasons. Firstly, the density of $n \mathbb{Z}$ in $\mathbb{Z}$ goes to 0 as $n$ grows. Secondly, it's intuitive that working (mod 0) should be equivalent to working (mod $\infty$) in a number-theoretical sense.
I want to know whether this makes sense. If so, how would you best formalise this notion of a sequence of ideals converging to a "limiting" ideal?
REPLY [6 votes]: Note that this sequence of principal ideals, when ordered by inclusion $n{\bf Z}\subseteq d{\bf Z}$ for $d\mid n$, is not totally ordered (in the sense that $n{\bf Z}$ and $m{\bf Z}$ are incomprable when neither of $n,m$ are divisors of the other). However, we can speak of all maximal chains in this poset (lattice, actually) of ideals, and defining the 'limit' of a chain $A_1\supset A_2\supset A_2\supset\cdots$ to be the intersection $\bigcap_{n\ge0}A_n$; it is a fairly straightforward logic exercise to verify arbitrary intersections of ideals are ideals. In this case, the limit of every single maximal chain is $0{\bf Z}$ (another relatively straightforward exercise).
There are different notions of limit, however, that complicate this picture with some exotic new algebraic structures. If $n{\bf Z}\to0{\bf Z}$ as ideals, we might naively expect ${\bf Z}/n{\bf Z}\to{\bf Z}/0{\bf Z}\cong{\bf Z}$ in some or other sense as rings (informally put, 'taking quotients is continuous'). In an arbitrary category (of groups, rings, fields, modules, lattices, ...), an inverse system of objects and morphisms can define an inverse limit. For our purposes, we can consider a sequence of rings with onto homomorphisms
$$R_0\xleftarrow{\varphi_1} R_1\xleftarrow{\varphi_2} R_2\xleftarrow{\varphi_3}R_3\xleftarrow{\varphi_4}\cdots.$$
We want to view the $R_i$ terms, as $i\to1$ from $\infty$, as successively further collapsed versions (which is to say, quotients, or homomorphic images of) some limiting object. The categorical definition of the limit involves universal properties and commutative diagrams, but for our purposes it suffices to look at the explicit construction: Define $\Pi:=\prod_{n\ge0}R_n$ to be the infinite product of the $R_i$s, and define $\varprojlim R_i$ to be the subring of $\Pi$ comprised of those tuples that are 'coherent,' in the sense that for these tuples $(r_0,r_1,r_2,\cdots)$, for each coordinate $r_n$ ($n>0$), $\varphi_n(r_n)=r_{n-1}$. That is, each coordinate is the homomorphic image of the next coordinate in line.
It makes sense that this is called an inverse limit, since the morphisms are flowing from right to left, and so to take the limit we are swimming upstream so to speak. It is also called the projective limit, which also makes sense because these transition morphisms may be called projections.
Some fun inverse systems to consider:
$${\bf Z}/p^0{\bf Z}\leftarrow{\bf Z}/p^1{\bf Z}\leftarrow {\bf Z}/p^2{\bf Z}\leftarrow\cdots $$
$$F\leftarrow F[T]/(T)\leftarrow F[T]/(T)^2\cdots$$
The inverse limit of the first is ${\bf Z}_p$, the $p$-adic integers, which has no zero divisors at all (so it is not to be confused with the integers modulo $p$). The inverse limit of the second, with $F$ an arbitrary field (actually it could be any ring), is $F[[T]]$, the ring of formal power series with coefficients taken from $F$. The latter is actually the "function field" analogue of the former (which is a "number field" fact); we use the term "global" fields for those that are either function fields or number fields, and these inverse limits are completions of them (or technically, of their integers). The fraction fields of the rings ${\bf Z}_p$ and $F[[T]]$, which are ${\bf Q}_p$ and $F((T))$ respectively, are called local fields.
Note that $\varprojlim{\bf Z}/n{\bf Z}$, taken over all $n\ne0$ (so our inverse system is no longer totally ordered, and our original sequence definition would need upgrading), is the direct product $\prod_p{\bf Z}_p$ over all prime numbers $p$. This is, arguably, high-tiered language for Sun-Ze's theorem (aka CRT).
There is a notion dual to inverse limits, called direct limits (or colimits). If we have a sequence
$$A_0\to A_1\to A_2\to A_3\to\cdots, $$
we can, roughly speaking, consider each morphism a sort of embedding of each term into the next, in which case the direct limit is the 'final object' that they are all embedded in, intuitively. For our algebraic-number-theoretic example, we have (as additive groups, not rings)
$$\varinjlim\, {\bf Z}/p^k{\bf Z}={\bf Z}(p^\infty)\cong{\bf Z}[1/p]/{\bf Z}\cong{\bf Q}_p/{\bf Z}_p$$
These are called the Prufer $p$-groups, and they occur in the $p$-primary decomposition of ${\bf Q}/{\bf Z}$; an analogous picture manifests when we look towards function fields, which tells us these Prufer groups encode the number field analogue of partial fraction decomposition in function fields.
Funnily enough, the Prufer $p$-groups ${\bf Z}(p^\infty)$ and $p$-adic integers ${\bf Z}_p$ are each other's dual groups.
This was rather tangential, so perhaps only the first paragraph was relevant to you, OP. It was also written summarily, so it'd be perfectly expected that others might be interested in expanding on any of the above points. (Mostly, it's an advertisement for studying number theory, I guess.) | 194,343 |
- ULTIMO is a leading player on the Polish debt management market. Due to its strong financing position the Group concentrates on the acquisition of non-performing loan portfolios (NPL) particularly from banking sector.
- Many years of experience in managing liabilities have given ULTIMO a unique ability to shape collection procedures and processes to maximize returns from NPL portfolios.
- The ULTIMO Group has been present on the Polish debt management market since 2002. In August 2014, Norwegian B2 Holding, a leading European specialty finance business, signed an agreement to acquire the entire share capital of ULTIMO.
- ULTIMO is one of the founding members of the Association of Financial Enterprises (KPF) in Poland and is at the forefront of setting industry standards. Principles of Best Practice and ULTIMO’s own Code of Ethics define a consistent approach across business.
- Since 2010, the Group has a license from the Financial Supervision Commission to manage securitized receivables. | 103,847 |
The Italian Women’s Volleyball side Savino Del Bene Scandicci announced that it has registered Russian player Tatyana Kosheleva!
A 30-year-old Russian star Kosheleva has officially signed with Scandicci for the 2018/19 Samsung Volley Cup A1 playoffs, the Italian club published on its website. The transfer was only a matter of time after the statement of an outside hitter, yesterday.
One-time FIVB World Championship and FIVB Club World Championship winner, as well as two-time CEV European Championship and CEV Cup winner, arrived in Scandicci after the experience in the Brazilian powerhouse SESC RJ.
Kosheleva has never played in Italy before and will likely make her debut for Scandicci in Game 2 of the playoff quarterfinals against Èpiù Pomì Casalmaggiore, on April! | 173,488 |
Based on 66 guest reviews
Lambert Airport(30 miles), Mid-America Airport(5 miles)Airport Shuttle? Sorry, no airport shuttle available.
Below are the meeting, banquet, conference and event spaces at Ramada Fairview Heights.
Feel free to use the Ramada Fairview Heights meeting space capacities chart below to help in your event planning. Hotel Planner specializes in Fairview Heights event planning for sleeping rooms and meeting space for corporate events, weddings, parties, conventions, negotiated rates and trade shows. | 18,353 |
/cdn.vox-cdn.com/uploads/chorus_image/image/69799879/1336979629.0.jpg)
The Denver Broncos had to trim their roster from 80 to 53 players before the NFL deadline today at 2 PM MT. While the roster will remain fluid throughout the regular season because of injuries, additions, etc., we have our first look at what George Paton’s 2021 team will actually look like.
3 Specialists
Brandon McManus, Sam Martin, Jacob Bobenmoyer
- We’ve known these were the guys since Max Duffy was released in June.
Preseason scoring leaders:— Andersen Pickard (@AndersenPickard) August 30, 2021
1. Rhamondre Stevenson - 30
2. Brandon McManus - 24
3. Ka'imi Fairbairn - 22
4. Sam Ficken - 21
5. Tyler Bass - 20
6. Rodrigo Blankenship - 19
T7. Jake Verity, Chris Boswell, Josh Lambo - 18
A rookie running back amidst a group of kickers. Special. pic.twitter.com/aCQNOFNXiX
11 Defensive Backs
Kyle Fuller, Ronald Darby, Bryce Callahan, Patrick Surtain II, Kareem Jackson, Justin Simmons, Caden Sterns, P.J. Locke III, Jamar Johnson, Kary Vincent, and Michael Ojemudia+
- Earlier this week Sports Illustrated’s Albert Breer reported that NFL teams had shown interest in Kyle Fuller, Bryce Callahan, Nate Hairston, and Saivion Smith, which could make Hairston and Smith’s trip through waivers interesting.
- Duke Dawson and Essang Bassey will open the season on the Physically Unable to Perform (PUP) list. Neither has left the list throughout training camp after tearing their ACLs late in the 2020 season. Landing on PUP means they’ll be eligible to return to play after six weeks. They do not count against the 53-man roster limit.
After the cut deadline any player on IR can return in 3 weeks. This looks likely for Mike Boone and Michael Ojemudia.— Joe Rowles (@JoRo_NFL) August 31, 2021
Any player on PUP must miss 6 weeks. This could be the case for Duke Dawson and Essang Bassey. Both Broncos corners tore ACLs late in 2020.
5 Linebackers
Alexander Johnson, Josey Jewell, Justin Strnad, Baron Browning, and Jonas Griffith
- It wouldn’t surprise me if undrafted rookie Curtis Robinson or Barrington Wade make their way to the practice squad. Both could help on special teams down the road.
- The Broncos sent a 2022 sixth-round pick and 2023 seventh-round pick to the San Francisco 49ers for linebacker Jonas Griffith this morning.
5 Edge Rushers
Von Miller, Bradley Chubb, Malik Reed, Jonathon Cooper, and Andre Mintze
- If Miller, Chubb, and Reed stay healthy this is going to be one of the best edge rooms in the NFL. If Cooper can build on what he showed in camp, the group is going to be incredible.
160-ish pound #15 Tutu Atwell attempting to block 275-ish pound #55 Bradley Chubb pic.twitter.com/pP0TZudptT— Nate Tice (@Nate_Tice) August 31, 2021
6 Defensive Linemen
Shelby Harris, Dre’Mont Jones, Mike Purcell, McTelvin Agim, DeShawn Williams, and Jonathan Harris
- There’s been little question about this group throughout camp. While they won’t receive the fanfare, the Broncos look like they’ll have one of the best interior lines in the NFL this season.
- I’m surprised Shamar Stephen was cut. He outplayed Jonathan Harris by what I saw in the preseason and dumping him means the Broncos will eat $750,000 in a dead cap hit.
The #Broncos Dre'Mont Jones with the 16th highest True Sack Rate for interior D-line players.— T. Kothe (@tkothe_nfl) August 30, 2021
He's also the 3rd youngest player in the top 25.
Jones is going to do some fun stuff this year with Miller & Chubb demanding extra attention on both ends of the line.
5 Wide Receivers
Courtland Sutton, Jerry Jeudy, K.J. Hamler, Tim Patrick, Diontae Spencer
- Pat Shurmur used three receiver sets on 66% of the Broncos’ offensive plays in 2020, so I expect the Broncos to bring back Seth Williams, Kendall Hinton, or Tyrie Cleveland if they do not add outside help.
- Health is going to be a big question for this group. Sutton was limited to 31 snaps in 2020 because of a torn ACL, while Hamler missed time because of a hamstring injury and eventually landed on Injured Reserve because of a concussion. Patrick also played in just eight games in 2019 because of a hand injury.
Jerry Jeudy posted a 75% success rate vs. man coverage in #ReceptionPerception. That was 3rd best among rookies last year and falls at the 87th percentile in RP history. pic.twitter.com/0hpi5dGQca— Matt Harmon (@MattHarmon_BYB) August 25, 2021
8 Offensive Linemen
Garett Bolles, Dalton Risner, Lloyd Cushenberry, Graham Glasgow, Bobby Massie, Calvin Anderson, Quinn Meinerz, Netane Muti
- The Broncos look set to start Bolles, Risner, Cushenberry, Glasgow, and Massie along the offensive line.
- Anderson looks like the swing tackle.
- Hopefully it’s a question that never needs answering, but I’m curious if the coaching staff would slide Glasgow to center or roll with Meinerz at the pivot in the event of a Cushenberry injury.
- Cutting Cameron Fleming cost the Broncos $1 million in dead money.
I've enjoyed studying OT Calvin Anderson from the Broncos this offseason. Has his flaws and limitations but he is an aggressive setting player with good grip strength. Love how he is able to control this block with his left arm on Alton Robinson pic.twitter.com/W7sxTB1k3I— Zach Hicks (@ZachHicks2) August 26, 2021
4 Tight ends/fullbacks
Noah Fant, Albert Okwuegbunam, Eric Saubert, Andrew Beck
- We’ve known since at least last Thursday that Saubert and Beck would make the roster.
- Beck’s the “starting” fullback as well as the TE4.
Everybody just take a chill pill toady. It’s obviously much needed— Noah Fant (@nrfant) August 25, 2021
4 Running Backs
Melvin Gordon, Javonte “Pookie” Williams, Royce Freeman, and Mike Boone+
- The Denver Post’s Ryan O’Halloran has confirmed my suspicions that Mike Boone will open the season on Injured Reserve. The way IR works in today’s NFL means the Broncos have to carry him through the 53-man deadline today before they can place him on the reserve if they wish to activate him again in 2021. This means Boone counts against the 53-man roster today, but won’t suit up until the Baltimore Ravens game in week 4 at the earliest.
- Once Boone is healthy I expect the Broncos to try and move Freeman. One reason I expect Damarea Crockett to land on the practice squad.
Javonte Williams has all the tools to become a star with the Broncos. pic.twitter.com/ndNC7eBPtx— Joe Rowles (@JoRo_NFL) May 17, 2021
2 Quarterbacks
Teddy Bridgewater and Drew Lock
- As I write this it looks like Brett Rypien will be the Broncos QB3 off of the practice squad.
The parallels are remarkable— Tyler Forness (@TheRealForno) August 30, 2021
-Traded up for their 2nd round rookie RB
-Loaded defense
-Same OC
-Deep weapons
-Hard nosed, defensive coach
:no_upscale()/cdn.vox-cdn.com/uploads/chorus_asset/file/22818088/Roster1_0.png) | 14,566 |
Subsets and Splits